Finite Group Databasebeta

← #29718455

Subgroups — #29718455

Order 15616.

954 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840104C211yesmaximal, characteristic, complemented
2C2×#16839759C211yesmaximal, characteristic
3#21449447C211yesmaximal, characteristic
4#21448710C211yesmaximal, characteristic, complemented
5#21447213C211yesmaximal, characteristic, complemented
6#21447206C211yesmaximal, characteristic, complemented
7#21447066C211yesmaximal, characteristic, complemented
8#21447061C211yesmaximal, characteristic, complemented
9#21446749C211yesmaximal, characteristic, complemented
10#21446741C211yesmaximal, characteristic, complemented
11#21446454C211yesmaximal, characteristic, complemented
12#21446447C211yesmaximal, characteristic, complemented
13#21446242C211yesmaximal, characteristic, complemented
14#21445760C211yesmaximal, characteristic
15C61×#1850C211yesmaximal, characteristic, complemented
16C22×#11758222C2211yescharacteristic
17C2×#14519777C2211yescharacteristic
18C2×#14519765C2211yescharacteristic
19C2×#14519756C2211yescharacteristic
20C2×#14519736C2211yescharacteristic
21#16840104—24no—
22C2×#14519711C2211yescharacteristic
23C2×#14519709C2211yescharacteristic
24#16839759—24no—
25C2×#14519702C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519702C2211yescharacteristic
31C2×#14519702C2211yescharacteristic
32C2×#14519702C2211yescharacteristic
33C2×#14519699C2211yescharacteristic
34C2×#14519699C2211yescharacteristic
35C2×#14519699C2211yescharacteristic
36C2×#14519699C2211yescharacteristic
37#16839678—21no—
38#16839678—21no—
39#16839668C2211yescharacteristic
40#16839664C2211yescharacteristic
41#16839660C2211yescharacteristic
42#16839660C2211yescharacteristic
43#16839656—21no—
44#16839656—21no—
45#16839636—21no—
46#16839636—21no—
47#16839629—21no—
48#16839622C2211yescharacteristic
49C4×C4⋊Dic61C2211yescharacteristic
50C4×C4⋊Dic61C2211yescharacteristic
51#16839613—21no—
52#16839525C2211yescharacteristic
53#16839525C2211yescharacteristic
54C23×C8×C61C2211yescharacteristic
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×C4⋊C8×C61C2211yescharacteristic
57C2×C22⋊C8×C61C2211yescharacteristic
58C4×C22⋊C4×C61C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C2×#11758222—22no—
62C2×#11758222—22no—
63C2×#11758222—22no—
64C2×#11758222—22no—
65C2×#11758222—41no—
66C2×#11758222—41no—
67C22×Dic122C2311yescharacteristic
68C22×Dic122C2311yescharacteristic
69C22×D244C2311yescharacteristic
70C22×C22⋊D61—21no—
71C22×C4×D61—21no—
72#14519777—22no—
73#14519777—41no—
74#14519765—22no—
75#14519765—41no—
76#14519756—22no—
77#14519756—22no—
78#14519736—22no—
79#14519736D414yes—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61—21no—
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×C22⋊Dic61—21no—
91C2×C22⋊Dic61—21no—
92C2×C22⋊Dic61—21no—
93C2×#11758198—21no—
94C2×#11758198—21no—
95C2×#11758198—21no—
96C2×#11758198—21no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61—21no—
99C2×C4×Dic61—21no—
100#14519711—22no—
101#14519709—22no—
102#14519702—22no—
103#14519702—22no—
104#14519702—22no—
105#14519702—22no—
106#14519702—22no—
107#14519702—41no—
108#14519702—41no—
109#14519702—41no—
110#14519699—22no—
111#14519699—22no—
112#14519699—22no—
113#14519699—41no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119#14519684—21no—
120#14519684—21no—
121C23×C4×C61C2311yescharacteristic
122C22×C8×C61—22no—
123C22×C8×C61—41no—
124C22×C8×C61—41no—
125C22×C8×C61C2311yescharacteristic
126C22×C8×C61C2311yescharacteristic
127C22×C8×C61C2311yescharacteristic
128C22×C8×C61C2311yescharacteristic
129C2×C22⋊C4×C61D411yescharacteristic
130C2×C22⋊C4×C61D411yescharacteristic
131C2×C42×C61C2311yescharacteristic
132C2×C42×C61C2311yescharacteristic
133C4⋊C8×C61—41no—
134C4⋊C8×C61—41no—
135C22⋊C8×C61—22no—
136#95×C61D411yescharacteristic
137#95×C61D411yescharacteristic
138C24×C61C2×D411yescharacteristic
139C23×D61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D412yes—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61#4112yes—
151C22×C4×C61C2411yescharacteristic
152C2×C22⋊D61—42no—
153C2×C22⋊D61—42no—
154C2×C22⋊D61—42no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—21no—
162#11758222—28no—
163#11758222—44no—
164#11758222—44no—
165#11758222—44no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244—21no—
169C2×D244—21no—
170C2×D244—21no—
171C2×D244#4112yes—
172C2×C4×D61—42no—
173C2×C4×D61—42no—
174C2×C4×D61—42no—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122—21no—
179C2×Dic122—21no—
180C2×Dic122—21no—
181C2×Dic122—21no—
182C2×Dic122—21no—
183C2×Dic122—21no—
184C2×Dic122—21no—
185C2×Dic122—21no—
186C2×Dic122#4112yes—
187C2×C8×C61—21no—
188C2×C8×C61—21no—
189C2×C8×C61—21no—
190C2×C8×C61—21no—
191C2×C8×C61—21no—
192C2×C8×C61—21no—
193C2×C8×C61—21no—
194C2×C8×C61—21no—
195C2×C8×C61—21no—
196C2×C8×C61—21no—
197C2×C8×C61—21no—
198C2×C8×C61—21no—
199C2×C8×C61—44no—
200C22⋊C4×C61—22no—
201C22⋊C4×C61—41no—
202C42×C61—21no—
203C42×C61—21no—
204C42×C61—21no—
205C42×C61—21no—
206C22⋊Dic61—24no—
207C22⋊Dic61—42no—
208C22⋊Dic61—81no—
209#11758198—42no—
210#11758198—42no—
211#11758198—42no—
212#11758198—42no—
213C4⋊Dic61—21no—
214C4⋊Dic61—21no—
215C4⋊Dic61—21no—
216C4⋊Dic61—21no—
217C4⋊Dic61—21no—
218C4⋊Dic61—21no—
219C4⋊Dic61—21no—
220C4⋊Dic61—21no—
221C4⋊Dic61—42no—
222C4⋊Dic61—42no—
223C4⋊Dic61C2×D412yes—
224C4⋊Dic61C2×D412yes—
225#11758196—42no—
226#11758196—42no—
227#11758196—42no—
228C4×Dic61—24no—
229C4×Dic61—81no—
230C4×Dic61—81no—
231C23×C61—21no—
232C23×C61—21no—
233C23×C61—21no—
234C23×C61—21no—
235C23×C61—21no—
236C23×C61—21no—
237C23×C61#12112yes—
238C23×C61C22×D411yescharacteristic
239C22×D61—22no—
240C22×D61—41no—
241C22×D61—41no—
242C22×D61—41no—
243C22×D61—41no—
244C22×D61—41no—
245C22×D61—41no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—22no—
251C2×C4×C61—22no—
252C2×C4×C61—22no—
253C2×C4×C61—22no—
254C2×C4×C61—21no—
255C2×C4×C61—21no—
256C2×C4×C61—21no—
257C2×C4×C61—21no—
258C2×C4×C61—21no—
259C2×C4×C61—21no—
260C2×C4×C61—21no—
261C2×C4×C61—21no—
262C2×C4×C61—21no—
263C2×C4×C61#12112yes—
264C2×C4×C61#13512yes—
265C2×C4×C61C22×D411yescharacteristic
266C2×C4×C61C2×#4111yescharacteristic
267C2×C4×C61C2×#4111yescharacteristic
268C2×C4×C61C2×#4111yescharacteristic
269C2×C4×C61H(5,𝔽2)11yescharacteristic
270C2×C4×C61H(5,𝔽2)11yescharacteristic
271C22⋊D61—48no—
272C2×Dic61—22no—
273C2×Dic61—22no—
274C2×Dic61—22no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C2×Dic61—41no—
290C2×Dic61—41no—
291C2×Dic61—41no—
292C2×Dic61—41no—
293D244—22no—
294D244—22no—
295D244—41no—
296D244—41no—
297C4×D61—48no—
298Dic122—22no—
299Dic122—22no—
300Dic122—41no—
301Dic122—41no—
302Dic122—41no—
303Dic122—41no—
304Dic122—41no—
305Dic122—41no—
306C8×C61—41no—
307C8×C61—41no—
308C8×C61—41no—
309C8×C61—41no—
310#24485—611nomaximal, Sylow
311C22×C61—22no—
312C22×C61—22no—
313C22×C61—22no—
314C22×C61—41no—
315C22×C61—41no—
316C22×C61—41no—
317C22×C61—41no—
318C22×C61C2×#12111yescharacteristic
319C22×C61#54611yescharacteristic
320C22×C61#54611yescharacteristic
321C22×C61C2×#13511yescharacteristic
322C22×C61#57511yescharacteristic
323C22×C61#57711yescharacteristic
324C22×C61#57811yescharacteristic
325C2×D61—42no—
326C2×D61—42no—
327C2×D61—42no—
328C2×D61—81no—
329C2×D61—81no—
330C2×D61—81no—
331C2×D61—81no—
332C4×C61—22no—
333C4×C61—22no—
334C4×C61—41no—
335C4×C61—41no—
336C4×C61—41no—
337C4×C61#52511yescharacteristic
338C4×C61#53511yescharacteristic
339C4×C61#55111yescharacteristic
340C4×C61#55111yescharacteristic
341Dic61—42no—
342Dic61—42no—
343Dic61—42no—
344Dic61—81no—
345Dic61—81no—
346Dic61—81no—
347Dic61—81no—
348C2×#485—611no—
349C2×#416—611no—
350#2432—611no—
351#2347—611no—
352#2068—611no—
353#2067—611no—
354#2043—611no—
355#2042—611no—
356#1983—611no—
357#1981—611no—
358#1927—611no—
359#1926—611no—
360#1892—611no—
361#1850D6111yescharacteristic, complemented
362#1788—611no—
363C2×C61—42no—
364C2×C61#241111yescharacteristic
365C2×C61#303511yescharacteristic
366C2×C61#304611yescharacteristic
367C2×C61#330111yescharacteristic
368C2×C61#330211yescharacteristic
369C2×C61#330711yescharacteristic
370C2×C61#331311yescharacteristic
371D61—82no—
372C22×#41—611no—
373C23×C8C2×D6111yescharacteristic
374C2×C4⋊Q8—611no—
375C2×#124—611no—
376C2×C22⋊Q8—611no—
377C2×#117—611no—
378#485—1224no—
379C2×#107—611no—
380C2×#106—611no—
381C2×C4⋊C8C2×D6111yescharacteristic
382C2×C4⋊C8C2×D6111yescharacteristic
383#416—1224no—
384C2×#103—611no—
385C2×#103—611no—
386C2×#103—611no—
387C2×#103—611no—
388C2×#103—611no—
389C2×#103—611no—
390C2×#103—611no—
391C2×#103—611no—
392C2×#102—611no—
393C2×#102—611no—
394C2×#102—611no—
395C2×#102—611no—
396C2×C22⋊C8C2×D6111yescharacteristic
397#394—1221no—
398#394—1221no—
399#391—611no—
400#390—611no—
401#389—611no—
402#389—611no—
403#388—1221no—
404#388—1221no—
405#385—1221no—
406#385—1221no—
407#383—1221no—
408#380—611no—
409C4×C4⋊C4—611no—
410C4×C4⋊C4—611no—
411C4×C4⋊C4—1221no—
412C4×C22⋊C4C2×D6111yescharacteristic
413#340—611no—
414#340—611no—
415#336C2×D6111yescharacteristic
416#336C2×D6111yescharacteristic
417C61#2448511yescharacteristic, complemented, Sylow
418C2×#41—1222no—
419C2×#41—1222no—
420C2×#41—1222no—
421C2×#41—1222no—
422C2×#41—2441no—
423C2×#41—2441no—
424C22×Q8—611no—
425C22×Q8—611no—
426C22×D4—611no—
427C22×D4—1221no—
428C23×C4—1221no—
429C23×C4C22×D6111yescharacteristic
430C22×C8—22no—
431C22×C8—41no—
432C22×C8—41no—
433C22×C8C22×D6111yescharacteristic
434C22×C8C22×D6111yescharacteristic
435C22×C8C22×D6111yescharacteristic
436C22×C8C22×D6111yescharacteristic
437C4⋊Q8—1222no—
438C4⋊Q8—2441no—
439#124—1222no—
440#124—2441no—
441C22⋊Q8—1222no—
442C22⋊Q8—1222no—
443#117—614no—
444#117—1222no—
445C2×C4⋊C4—611no—
446C2×C4⋊C4—611no—
447C2×C4⋊C4—611no—
448C2×C4⋊C4—611no—
449C2×C4⋊C4—611no—
450C2×C4⋊C4—1221no—
451C2×C4⋊C4—1221no—
452C2×C4⋊C4—1221no—
453C2×C4⋊C4—1221no—
454C2×C4⋊C4—1221no—
455C2×C22⋊C4—1221no—
456C2×C22⋊C4—1221no—
457C2×C22⋊C4—1221no—
458C2×C22⋊C4—1221no—
459C2×C22⋊C4—1221no—
460C2×C22⋊C4—1221no—
461C2×C22⋊C4—1221no—
462C2×C22⋊C4D24411yescharacteristic
463C2×C22⋊C4D24411yescharacteristic
464C2×C42—1221no—
465C2×C42—1221no—
466C2×C42—1221no—
467C2×C42C22×D6111yescharacteristic
468C2×C42C22×D6111yescharacteristic
469#107—1222no—
470#106—1222no—
471C4⋊C8—41no—
472C4⋊C8—41no—
473#103—1222no—
474#103—1222no—
475#103—1222no—
476#103—1222no—
477#103—1222no—
478#103—2441no—
479#103—2441no—
480#103—2441no—
481#102—1222no—
482#102—1222no—
483#102—1222no—
484#102—2441no—
485C22⋊C8—22no—
486#95—1221no—
487#95—1221no—
488#95—1221no—
489#95—1221no—
490#95—1221no—
491#95—1221no—
492#95—1221no—
493#95D24411yescharacteristic
494#95D24411yescharacteristic
495C24—1221no—
496C24C2×D24411yescharacteristic
497#41—1228no—
498#41—2444no—
499#41—2444no—
500#41—2444no—
501C2×Q8—612no—
502C2×Q8—1221no—
503C2×Q8—1221no—
504C2×Q8—1221no—
505C2×Q8—1221no—
506C2×Q8—1221no—
507C2×Q8—1221no—
508C2×Q8—1221no—
509C2×Q8—1221no—
510C2×Q8—1221no—
511C2×Q8—1221no—
512C2×Q8—1221no—
513C2×D4—612no—
514C2×D4—1221no—
515C2×D4—1221no—
516C2×D4—1221no—
517C2×D4—1221no—
518C2×D4—1221no—
519C2×D4—2442no—
520C2×D4—2442no—
521C2×D4—2442no—
522C22×C4—21no—
523C22×C4—21no—
524C22×C4—21no—
525C22×C4—21no—
526C22×C4—1221no—
527C22×C4—1221no—
528C22×C4—1221no—
529C22×C4—1221no—
530C22×C4—1221no—
531C22×C4—1221no—
532C22×C4—1221no—
533C22×C4—2442no—
534C22×C4—2442no—
535C22×C4—2442no—
536C22×C4C2×D24411yescharacteristic
537C22×C4C2×D24411yescharacteristic
538C22×C4C2×D24411yescharacteristic
539C22×C4C2×D24411yescharacteristic
540C22×C4C2×D24411yescharacteristic
541C22×C4D4×D6112yes—
542C22×C4#1175822612yes—
543C22×C4C23×D6111yescharacteristic
544C2×C8—21no—
545C2×C8—21no—
546C2×C8—21no—
547C2×C8—21no—
548C2×C8—21no—
549C2×C8—21no—
550C2×C8—21no—
551C2×C8—21no—
552C2×C8—21no—
553C2×C8—21no—
554C2×C8—21no—
555C2×C8—21no—
556C2×C8—44no—
557C4⋊C4—612no—
558C4⋊C4—612no—
559C4⋊C4—1221no—
560C4⋊C4—1221no—
561C4⋊C4—1221no—
562C4⋊C4—1221no—
563C4⋊C4—1221no—
564C4⋊C4—1221no—
565C4⋊C4—1221no—
566C4⋊C4—1221no—
567C4⋊C4—2442no—
568C4⋊C4—2442no—
569C4⋊C4—2442no—
570C4⋊C4—2442no—
571C4⋊C4—2442no—
572C22⋊C4—22no—
573C22⋊C4—41no—
574C22⋊C4—1224no—
575C22⋊C4—4881no—
576C22⋊C4—2442no—
577C22⋊C4—2442no—
578C22⋊C4—2442no—
579C22⋊C4—2442no—
580C22⋊C4—2442no—
581C42—21no—
582C42—21no—
583C42—21no—
584C42—21no—
585C42—1224no—
586C42—4881no—
587C42—4881no—
588C23—21no—
589C23—21no—
590C23—21no—
591C23—21no—
592C23—21no—
593C23—21no—
594C23—1222no—
595C23—2441no—
596C23—2441no—
597C23—2441no—
598C23—2441no—
599C23—2441no—
600C23—2441no—
601C23#1451975212yes—
602C23C22×D24411yescharacteristic
603Q8—1222no—
604Q8—1222no—
605Q8—2441no—
606Q8—2441no—
607Q8—2441no—
608Q8—2441no—
609Q8—2441no—
610Q8—2441no—
611D4—1222no—
612D4—1222no—
613D4—2448no—
614D4—2441no—
615D4—2441no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—1222no—
634C2×C4—1222no—
635C2×C4—1222no—
636C2×C4—2448no—
637C2×C4—2441no—
638C2×C4—2441no—
639C2×C4—2441no—
640C2×C4—2441no—
641C2×C4—2441no—
642C2×C4—2441no—
643C2×C4—2441no—
644C2×C4—2441no—
645C2×C4—2441no—
646C2×C4—2441no—
647C2×C4—2441no—
648C2×C4—2441no—
649C2×C4—2441no—
650C2×C4—2441no—
651C2×C4—2441no—
652C2×C4—2441no—
653C2×C4—2441no—
654C2×C4—2441no—
655C2×C4#1451975212yes—
656C2×C4#1451979912yes—
657C2×C4C2×D4×D6111yescharacteristic
658C2×C4#41×D6111yescharacteristic
659C2×C4#41×D6111yescharacteristic
660C2×C4C2×#1175822611yescharacteristic
661C2×C4#1451982211yescharacteristic
662C2×C4#1451982211yescharacteristic
663C8—41no—
664C8—41no—
665C8—41no—
666C8—41no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22—41no—
674C22—4881no—
675C22—4881no—
676C22—4881no—
677C22—4881no—
678C22—2442no—
679C22—2442no—
680C22—2442no—
681C22C2×#1451975211yescharacteristic
682C22#1684040711yescharacteristic
683C22#1684040711yescharacteristic
684C22C2×#1451979911yescharacteristic
685C22#1684056211yescharacteristic
686C22#1684057411yescharacteristic
687C22#1684057911yescharacteristic
688C4—22no—
689C4—22no—
690C4—41no—
691C4—41no—
692C4—41no—
693C4—4881no—
694C4—4881no—
695C4—4881no—
696C4—4881no—
697C4—2442no—
698C4—2442no—
699C4—2442no—
700C4#1684027011yescharacteristic
701C4#1684032711yescharacteristic
702C4#1684043911yescharacteristic
703C4#1684043911yescharacteristic
704C2—42no—
705C2—4882no—
706C2#2144920211yescharacteristic
707C2#2145527611yescharacteristic
708C2#2145537111yescharacteristic
709C2#2145772311yescharacteristic
710C2#2145773711yescharacteristic
711C2#2145779711yescharacteristic
712C2#2145788711yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1850C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
hypercenter#1850D61
hypocenterC61#24485
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1850

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#16840104C2semidirect
4. #21448710C2semidirect
5. #21447213C2semidirect
6. #21447206C2semidirect
7. #21447066C2semidirect
8. #21447061C2semidirect
9. #21446749C2semidirect
10. #21446741C2semidirect
11. #21446454C2semidirect
12. #21446447C2semidirect
13. #21446242C2semidirect
15. C61×#1850C2semidirect
361. #1850D61semidirect
417. C61#24485semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168401042
C2×#168397592
#214494472
#214487102
#214472132
#214472062
#214470662
#214470612
#214467492
#214467412
#214464542
#214464472
#214462422
#214457602
C61×#18502
#2448561