Finite Group Databasebeta

← #29718454

Subgroups — #29718454

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×#16840102C211yesmaximal, characteristic, complemented
2C2×#16839755C211yesmaximal, characteristic
3#21449450C211yesmaximal, characteristic
4#21448708C211yesmaximal, characteristic, complemented
5#21447213C211yesmaximal, characteristic, complemented
6#21447205C211yesmaximal, characteristic, complemented
7#21447063C211yesmaximal, characteristic, complemented
8#21447055C211yesmaximal, characteristic, complemented
9#21446740C211yesmaximal, characteristic, complemented
10#21446453C211yesmaximal, characteristic, complemented
11#21446449C211yesmaximal, characteristic, complemented
12#21446239C211yesmaximal, characteristic, complemented
13#21445995C211yesmaximal, characteristic, complemented
14#21445756C211yesmaximal, characteristic
15C61×#1983C211yesmaximal, characteristic, complemented
16C22×#11758224C2211yescharacteristic
17C22×C8×D61C2211yescharacteristic
18C2×#14519775C2211yescharacteristic
19C2×#14519753C2211yescharacteristic
20C2×#14519735C2211yescharacteristic
21#16840102—24no—
22C2×#14519711C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×#14519706C2211yescharacteristic
25#16839755—24no—
26C2×#14519702C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519701C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×#14519700C2211yescharacteristic
32C2×#14519698C2211yescharacteristic
33C2×#14519698C2211yescharacteristic
34C2×#14519689C2211yescharacteristic
35#16839677—21no—
36#16839677—21no—
37#16839668C2211yescharacteristic
38#16839665C2211yescharacteristic
39#16839660C2211yescharacteristic
40#16839658C2211yescharacteristic
41#16839653—21no—
42#16839651—21no—
43#16839634—21no—
44#16839634—21no—
45#16839629—21no—
46#16839619C2211yescharacteristic
47C4⋊C4×Dic61C2211yescharacteristic
48C4×#11758196—21no—
49C4×#11758198C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839517C2211yescharacteristic
53#16839516C2211yescharacteristic
54C2×#124×C61C2211yescharacteristic
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×#102×C61C2211yescharacteristic
59C2×#102×C61C2211yescharacteristic
60C4×C4⋊C4×C61C2211yescharacteristic
61C2×#11758224—22no—
62C2×#11758224—22no—
63C2×#11758224—22no—
64C2×#11758224—22no—
65C2×#11758224—41no—
66C2×#11758224—41no—
67C22×Dic122C2311yescharacteristic
68C22×C22⋊D61—21no—
69C23×Dic61—21no—
70C22×C4×D61C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73C2×C8×D61—22no—
74C2×C8×D61—41no—
75C2×C8×D61—41no—
76#14519775—22no—
77#14519775—41no—
78#14519753—22no—
79#14519753—22no—
80#14519735—22no—
81#14519735D414yes—
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×C22⋊Dic61—21no—
91C2×C22⋊Dic61—21no—
92C2×#11758198—21no—
93C2×#11758198—21no—
94C2×#11758198—21no—
95C2×#11758198D411yescharacteristic
96C2×#11758198D411yescharacteristic
97C2×C4×Dic61—21no—
98C2×C4×Dic61—21no—
99C2×C4×Dic61C2311yescharacteristic
100#14519711—22no—
101#14519708—22no—
102#14519706—41no—
103#14519702—22no—
104#14519702—41no—
105#14519701—22no—
106#14519701—41no—
107#14519700—22no—
108#14519700—22no—
109#14519698—22no—
110#14519698—22no—
111#14519689—22no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684D411yescharacteristic
119#14519684D411yescharacteristic
120C22×Q8×C61C2311yescharacteristic
121C22×D4×C61C2311yescharacteristic
122C22×C8×C61C2311yescharacteristic
123C22×C8×C61C2311yescharacteristic
124#124×C61—22no—
125#124×C61—41no—
126C2×C4⋊C4×C61C2311yescharacteristic
127C2×C4⋊C4×C61C2311yescharacteristic
128C2×C22⋊C4×C61—21no—
129C2×C22⋊C4×C61—21no—
130C2×C42×C61—21no—
131C2×C42×C61C2311yescharacteristic
132C4⋊C8×C61—41no—
133#103×C61—22no—
134#103×C61—41no—
135#102×C61—22no—
136#102×C61—41no—
137#95×C61—21no—
138C24×C61—21no—
139C23×D61C2×D411yescharacteristic
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61#4112yes—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2411yescharacteristic
158C2×C22⋊D61—42no—
159C2×C22⋊D61—42no—
160C2×C22⋊D61—42no—
161C22×Dic61—21no—
162C22×Dic61—21no—
163C22×Dic61—21no—
164C22×Dic61—21no—
165C22×Dic61—42no—
166C22×Dic61—42no—
167C22×Dic61—42no—
168C22×Dic61C2×D411yescharacteristic
169C22×Dic61C2×D411yescharacteristic
170C22×Dic61C2×D411yescharacteristic
171#11758224—28no—
172#11758224—44no—
173#11758224—44no—
174#11758224—44no—
175C2×C4×D61—21no—
176C2×C4×D61—21no—
177C2×C4×D61—21no—
178C2×C4×D61—21no—
179C2×C4×D61C2×D412yes—
180C2×C4×D61#4112yes—
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—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—21no—
196C4⋊C4×C61C2×D412yes—
197C22⋊C4×C61—42no—
198C22⋊C4×C61—42no—
199C42×C61—21no—
200C42×C61—21no—
201C42×C61—81no—
202C22⋊Dic61—42no—
203C22⋊Dic61—42no—
204#11758198—24no—
205#11758198—22no—
206#11758198—41no—
207#11758198—42no—
208#11758198—81no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61—21no—
213C4⋊Dic61—21no—
214C4⋊Dic61C2×D412yes—
215#11758196—42no—
216#11758196—42no—
217#11758196—42no—
218#11758196—42no—
219#11758196—42no—
220C4×Dic61—21no—
221C4×Dic61—21no—
222C4×Dic61—24no—
223C4×Dic61—81no—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418—21no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418—21no—
230C8×D61—44no—
231C23×C61—22no—
232C23×C61—41no—
233C23×C61—41no—
234C23×C61—41no—
235C23×C61—41no—
236C23×C61—41no—
237C23×C61—41no—
238C23×C61C22×D411yescharacteristic
239C22×D61—21no—
240C22×D61—21no—
241C22×D61—21no—
242C22×D61—21no—
243C22×D61—21no—
244C22×D61—21no—
245C22×D61#12112yes—
246Q8×C61—41no—
247Q8×C61—41no—
248Q8×C61—41no—
249Q8×C61—41no—
250D4×C61—22no—
251D4×C61—22no—
252D4×C61—41no—
253D4×C61—41no—
254C2×C4×C61—22no—
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—41no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61—41no—
265C2×C4×C61—41no—
266C2×C4×C61—41no—
267C2×C4×C61—41no—
268C2×C4×C61—41no—
269C2×C4×C61C22×D411yescharacteristic
270C2×C4×C61C2×#4111yescharacteristic
271C2×C4×C61C2×#4111yescharacteristic
272C2×C4×C61C2×#4111yescharacteristic
273C2×C4×C61H(5,𝔽2)11yescharacteristic
274C2×C4×C61H(5,𝔽2)11yescharacteristic
275C22⋊D61—48no—
276C2×Dic61—21no—
277C2×Dic61—21no—
278C2×Dic61—21no—
279C2×Dic61—21no—
280C2×Dic61—22no—
281C2×Dic61—21no—
282C2×Dic61—22no—
283C2×Dic61—21no—
284C2×Dic61—21no—
285C2×Dic61—48no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C2×Dic61—41no—
290C2×Dic61—41no—
291C2×Dic61—41no—
292C2×Dic61—41no—
293C2×Dic61—41no—
294C2×Dic61—41no—
295C2×Dic61—41no—
296C2×Dic61#12112yes—
297C2×Dic61#13512yes—
298C4×D61—22no—
299C4×D61—22no—
300C4×D61—22no—
301C4×D61—22no—
302Dic122—22no—
303Dic122—22no—
304Dic122—41no—
305Dic122—41no—
306C8×C61—41no—
307C8×C61—41no—
308#92418—41no—
309#92418—41no—
310#24485—611nomaximal, Sylow
311C22×C61—42no—
312C22×C61—42no—
313C22×C61—42no—
314C22×C61—81no—
315C22×C61—81no—
316C22×C61—81no—
317C22×C61—81no—
318C22×C61C2×#12111yescharacteristic
319C22×C61#54611yescharacteristic
320C22×C61#54611yescharacteristic
321C22×C61C2×#13511yescharacteristic
322C22×C61#57511yescharacteristic
323C22×C61#57711yescharacteristic
324C22×C61#57811yescharacteristic
325C2×D61—22no—
326C2×D61—22no—
327C2×D61—22no—
328C2×D61—41no—
329C2×D61—41no—
330C2×D61—41no—
331C2×D61—41no—
332C4×C61—42no—
333C4×C61—41no—
334C4×C61—41no—
335C4×C61—81no—
336C4×C61—81no—
337C4×C61#52511yescharacteristic
338C4×C61#53511yescharacteristic
339C4×C61#55111yescharacteristic
340C4×C61#55111yescharacteristic
341Dic61—22no—
342Dic61—22no—
343Dic61—42no—
344Dic61—42no—
345Dic61—41no—
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#1983D6111yescharacteristic, complemented
357#1981—611no—
358#1927—611no—
359#1926—611no—
360#1892—611no—
361#1850—611no—
362#1788—611no—
363C2×C61—82no—
364C2×C61#241111yescharacteristic
365C2×C61#303511yescharacteristic
366C2×C61#304611yescharacteristic
367C2×C61#330111yescharacteristic
368C2×C61#330211yescharacteristic
369C2×C61#330711yescharacteristic
370C2×C61#331311yescharacteristic
371D61—42no—
372C22×#41—611no—
373C23×C8—611no—
374C2×C4⋊Q8—611no—
375C2×#124C2×D6111yescharacteristic
376C2×C22⋊Q8—611no—
377C2×#117—611no—
378#485—1224no—
379C2×#107—611no—
380C2×#106—611no—
381C2×C4⋊C8—611no—
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×#103C2×D6111yescharacteristic
391C2×#103C2×D6111yescharacteristic
392C2×#102—611no—
393C2×#102—611no—
394C2×#102C2×D6111yescharacteristic
395C2×#102C2×D6111yescharacteristic
396C2×C22⋊C8—611no—
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—1221no—
411C4×C4⋊C4C2×D6111yescharacteristic
412C4×C22⋊C4—611no—
413#340—611no—
414#340—611no—
415#336—611no—
416#336—611no—
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×Q8C22×D6111yescharacteristic
426C22×D4—1221no—
427C22×D4C22×D6111yescharacteristic
428C23×C4—611no—
429C23×C4—1221no—
430C22×C8—611no—
431C22×C8—611no—
432C22×C8—1222no—
433C22×C8—2441no—
434C22×C8—2441no—
435C22×C8C22×D6111yescharacteristic
436C22×C8C22×D6111yescharacteristic
437C4⋊Q8—1222no—
438C4⋊Q8—2441no—
439#124—22no—
440#124—41no—
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—1221no—
449C2×C4⋊C4—1221no—
450C2×C4⋊C4—1221no—
451C2×C4⋊C4—1221no—
452C2×C4⋊C4—1221no—
453C2×C4⋊C4C22×D6111yescharacteristic
454C2×C4⋊C4C22×D6111yescharacteristic
455C2×C22⋊C4—21no—
456C2×C22⋊C4—21no—
457C2×C22⋊C4—611no—
458C2×C22⋊C4—611no—
459C2×C22⋊C4—1221no—
460C2×C22⋊C4—1221no—
461C2×C22⋊C4—1221no—
462C2×C22⋊C4—1221no—
463C2×C22⋊C4—1221no—
464C2×C42—21no—
465C2×C42—611no—
466C2×C42—1221no—
467C2×C42—1221no—
468C2×C42C22×D6111yescharacteristic
469#107—1222no—
470#106—1222no—
471C4⋊C8—41no—
472C4⋊C8—2441no—
473#103—22no—
474#103—41no—
475#103—1222no—
476#103—1222no—
477#103—1222no—
478#103—1222no—
479#103—2441no—
480#103—2441no—
481#102—22no—
482#102—41no—
483#102—1222no—
484#102—1222no—
485C22⋊C8—1222no—
486#95—21no—
487#95—611no—
488#95—611no—
489#95—1221no—
490#95—1221no—
491#95—1221no—
492#95—1221no—
493#95—1221no—
494#95—1221no—
495C24—21no—
496C24—611no—
497#41—1228no—
498#41—2444no—
499#41—2444no—
500#41—2444no—
501C2×Q8—21no—
502C2×Q8—21no—
503C2×Q8—21no—
504C2×Q8—21no—
505C2×Q8—21no—
506C2×Q8—21no—
507C2×Q8—612no—
508C2×Q8—1221no—
509C2×Q8—1221no—
510C2×Q8—1221no—
511C2×Q8—1221no—
512C2×Q8—1221no—
513C2×D4—21no—
514C2×D4—21no—
515C2×D4—21no—
516C2×D4—21no—
517C2×D4—21no—
518C2×D4—2442no—
519C2×D4—2442no—
520C2×D4—2442no—
521C2×D4#1175822212yes—
522C22×C4—21no—
523C22×C4—21no—
524C22×C4—21no—
525C22×C4—611no—
526C22×C4—611no—
527C22×C4—612no—
528C22×C4—612no—
529C22×C4—611no—
530C22×C4—1221no—
531C22×C4—1221no—
532C22×C4—1221no—
533C22×C4—1221no—
534C22×C4—1221no—
535C22×C4—1221no—
536C22×C4—1221no—
537C22×C4—1221no—
538C22×C4—2442no—
539C22×C4—2442no—
540C22×C4—2442no—
541C22×C4D4×D6111yescharacteristic
542C22×C4D4×D6111yescharacteristic
543C22×C4C23×D6111yescharacteristic
544C2×C8—21no—
545C2×C8—21no—
546C2×C8—21no—
547C2×C8—21no—
548C2×C8—21no—
549C2×C8—21no—
550C2×C8—1221no—
551C2×C8—1221no—
552C2×C8—1221no—
553C2×C8—1221no—
554C2×C8—1221no—
555C2×C8—1221no—
556C2×C8—2444no—
557C4⋊C4—21no—
558C4⋊C4—21no—
559C4⋊C4—21no—
560C4⋊C4—612no—
561C4⋊C4—1221no—
562C4⋊C4—1221no—
563C4⋊C4—1221no—
564C4⋊C4—1221no—
565C4⋊C4—1221no—
566C4⋊C4—2442no—
567C4⋊C4—2442no—
568C4⋊C4—2442no—
569C4⋊C4—2442no—
570C4⋊C4—2442no—
571C4⋊C4D4×D6112yes—
572C22⋊C4—42no—
573C22⋊C4—42no—
574C22⋊C4—1222no—
575C22⋊C4—1224no—
576C22⋊C4—4881no—
577C22⋊C4—2441no—
578C22⋊C4—2442no—
579C22⋊C4—2442no—
580C22⋊C4—2442no—
581C42—21no—
582C42—21no—
583C42—81no—
584C42—1224no—
585C42—1221no—
586C42—1221no—
587C42—4881no—
588C23—22no—
589C23—41no—
590C23—41no—
591C23—41no—
592C23—41no—
593C23—41no—
594C23—41no—
595C23—612no—
596C23—1221no—
597C23—1221no—
598C23—1221no—
599C23—1221no—
600C23—1221no—
601C23—1221no—
602C23C2×D4×D6111yescharacteristic
603Q8—41no—
604Q8—41no—
605Q8—41no—
606Q8—41no—
607Q8—1222no—
608Q8—1222no—
609Q8—2441no—
610Q8—2441no—
611D4—22no—
612D4—22no—
613D4—41no—
614D4—41no—
615D4—2448no—
616C2×C4—22no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—612no—
632C2×C4—612no—
633C2×C4—1221no—
634C2×C4—1221no—
635C2×C4—1221no—
636C2×C4—1221no—
637C2×C4—1222no—
638C2×C4—1222no—
639C2×C4—1222no—
640C2×C4—1222no—
641C2×C4—1222no—
642C2×C4—1221no—
643C2×C4—1221no—
644C2×C4—1221no—
645C2×C4—1222no—
646C2×C4—2448no—
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—2441no—
656C2×C4—2441no—
657C2×C4C2×D4×D6111yescharacteristic
658C2×C4#41×D6111yescharacteristic
659C2×C4#41×D6111yescharacteristic
660C2×C4C2×#1175822211yescharacteristic
661C2×C4#1451982211yescharacteristic
662C2×C4#1451982211yescharacteristic
663C8—41no—
664C8—41no—
665C8—2441no—
666C8—2441no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—81no—
671C22—81no—
672C22—81no—
673C22—81no—
674C22—1222no—
675C22—1222no—
676C22—1222no—
677C22—2441no—
678C22—2441no—
679C22—2441no—
680C22—2441no—
681C22#121×D6111yescharacteristic
682C22#1684040211yescharacteristic
683C22#1684040911yescharacteristic
684C22#135×D6111yescharacteristic
685C22#1684056111yescharacteristic
686C22#1684057211yescharacteristic
687C22#1684057611yescharacteristic
688C4—42no—
689C4—41no—
690C4—41no—
691C4—81no—
692C4—81no—
693C4—1222no—
694C4—1222no—
695C4—4881no—
696C4—4881no—
697C4—2442no—
698C4—2441no—
699C4—2442no—
700C4#1684027111yescharacteristic
701C4#1684032511yescharacteristic
702C4#1684044111yescharacteristic
703C4#1684044211yescharacteristic
704C2—82no—
705C2—2442no—
706C2#2144920111yescharacteristic
707C2#2145527311yescharacteristic
708C2#2145537011yescharacteristic
709C2#2145772811yescharacteristic
710C2#2145773211yescharacteristic
711C2#2145779611yescharacteristic
712C2#2145789211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1983C2
socleC23×C61C22×D4
prime radicalC22×#11758224C22
hypercenter#1983D61
hypocenterC61#24485
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1983

Complements

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

NH ≅ G/Ntype
1. C2×#16840102C2semidirect
4. #21448708C2semidirect
5. #21447213C2semidirect
6. #21447205C2semidirect
7. #21447063C2semidirect
8. #21447055C2semidirect
9. #21446740C2semidirect
10. #21446453C2semidirect
11. #21446449C2semidirect
12. #21446239C2semidirect
13. #21445995C2semidirect
15. C61×#1983C2semidirect
356. #1983D61semidirect
417. C61#24485semidirect

Maximal subgroups

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

subgroupindex
C2×#168401022
C2×#168397552
#214494502
#214487082
#214472132
#214472052
#214470632
#214470552
#214467402
#214464532
#214464492
#214462392
#214459952
#214457562
C61×#19832
#2448561