Finite Group Databasebeta

← #29737240

Subgroups — #29737240

Order 15616.

958 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840070C211yesmaximal, characteristic, complemented
2C2×#16839759C211yesmaximal, characteristic
3#21449532C211yesmaximal, characteristic, complemented
4#21448659C211yesmaximal, characteristic
5#21447346C211yesmaximal, characteristic, complemented
6#21447206C211yesmaximal, characteristic, complemented
7#21447192C211yesmaximal, characteristic, complemented
8#21446966C211yesmaximal, characteristic, complemented
9#21446076C211yesmaximal, characteristic, complemented
10#21446076C211yesmaximal, characteristic, complemented
11#21446020C211yesmaximal, characteristic, complemented
12#21445848C211yesmaximal, characteristic, complemented
13#21445825C211yesmaximal, characteristic, complemented
14#21445760C211yesmaximal, characteristic
15C61×#1850C211yesmaximal, characteristic, complemented
16C22×#11758222C2211yescharacteristic
17C2×#14519769C2211yescharacteristic
18C2×C4×Dic122C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20C2×#14519736C2211yescharacteristic
21#16840070—24no—
22C2×#14519711C2211yescharacteristic
23C2×#14519711C2211yescharacteristic
24C2×#14519711C2211yescharacteristic
25C2×#14519709C2211yescharacteristic
26#16839759—24no—
27C2×#14519702C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519702C2211yescharacteristic
31C2×#14519699C2211yescharacteristic
32C2×#14519699C2211yescharacteristic
33C2×#14519699C2211yescharacteristic
34C2×#14519699C2211yescharacteristic
35#16839698—21no—
36#16839685—21no—
37#16839664C2211yescharacteristic
38#16839664C2211yescharacteristic
39#16839662—21no—
40#16839662—21no—
41#16839660C2211yescharacteristic
42#16839660C2211yescharacteristic
43#16839659—21no—
44#16839645—21no—
45#16839632C2211yescharacteristic
46#16839622C2211yescharacteristic
47#16839622C2211yescharacteristic
48#16839621—21no—
49#16839609—21no—
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
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—41no—
65C2×#11758222—41no—
66C2×#11758222Q814yes—
67C22×Dic122C2311yescharacteristic
68C22×D244C2311yescharacteristic
69C22×C22⋊D61—21no—
70C22×C4×D61—21no—
71#14519769—22no—
72#14519769—41no—
73C4×Dic122—22no—
74C4×Dic122—22no—
75C4×Dic122—41no—
76C4×Dic122—41no—
77#14519736—22no—
78#14519736—41no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61C2311yescharacteristic
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×#11758196—21no—
91C2×#11758196—21no—
92C2×#11758196—21no—
93C2×C22⋊Dic61—21no—
94C2×C22⋊Dic61—21no—
95C2×C22⋊Dic61—21no—
96C2×#11758198—21no—
97C2×#11758198—21no—
98C2×#11758198—21no—
99C2×#11758198—21no—
100C2×C4×Dic61—21no—
101C2×C4×Dic61—21no—
102C2×C4×Dic61—21no—
103#14519711—22no—
104#14519711—41no—
105#14519711—41no—
106#14519709—22no—
107#14519702—22no—
108#14519702—22no—
109#14519702—22no—
110#14519702D414yes—
111#14519699—22no—
112#14519699—22no—
113#14519699—41no—
114#14519699—41no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119C23×C4×C61C2311yescharacteristic
120C22×C8×C61—22no—
121C22×C8×C61—22no—
122C22×C8×C61—41no—
123C22×C8×C61C2311yescharacteristic
124C22×C8×C61C2311yescharacteristic
125C22×C8×C61C2311yescharacteristic
126C22×C8×C61C2311yescharacteristic
127C2×C22⋊C4×C61D411yescharacteristic
128C2×C22⋊C4×C61D411yescharacteristic
129C2×C42×C61C2311yescharacteristic
130C2×C42×C61C2311yescharacteristic
131C4⋊C8×C61—22no—
132C4⋊C8×C61—41no—
133C22⋊C8×C61—22no—
134#95×C61D411yescharacteristic
135#95×C61D411yescharacteristic
136C24×C61C2×D411yescharacteristic
137C23×D61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×Q812yes—
148C22×C4×C61#4112yes—
149C22×C4×C61C2411yescharacteristic
150C2×C22⋊D61—24no—
151C2×C22⋊D61—42no—
152C2×C22⋊D61—42no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160#11758222—28no—
161#11758222—44no—
162#11758222—44no—
163#11758222—44no—
164C2×D244—21no—
165C2×D244—21no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244—21no—
169C2×D244C2×Q812yes—
170C2×C4×D61—24no—
171C2×C4×D61—42no—
172C2×C4×D61—42no—
173C2×Dic122—21no—
174C2×Dic122—21no—
175C2×Dic122—21no—
176C2×Dic122C2×D412yes—
177C2×Dic122C2×Q812yes—
178C2×Dic122#4112yes—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61—21no—
186C2×C8×C61—21no—
187C2×C8×C61—21no—
188C2×C8×C61—21no—
189C2×C8×C61—21no—
190C2×C8×C61—44no—
191C2×C8×C61C2×D412yes—
192C22⋊C4×C61—22no—
193C22⋊C4×C61—41no—
194C42×C61—21no—
195C42×C61—21no—
196C42×C61—21no—
197C42×C61—21no—
198C22⋊Dic61—42no—
199C22⋊Dic61—81no—
200C22⋊Dic61—81no—
201#11758198—42no—
202#11758198—42no—
203#11758198—81no—
204#11758198—81no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61—21no—
213C4⋊Dic61—21no—
214C4⋊Dic61—21no—
215C4⋊Dic61—21no—
216C4⋊Dic61—42no—
217C4⋊Dic61—81no—
218C4⋊Dic61C2×D412yes—
219#11758196—42no—
220#11758196—42no—
221#11758196—42no—
222#11758196—42no—
223#11758196—42no—
224#11758196—81no—
225C4×Dic61—42no—
226C4×Dic61—42no—
227C4×Dic61—42no—
228C23×C61—21no—
229C23×C61—21no—
230C23×C61—21no—
231C23×C61—21no—
232C23×C61—21no—
233C23×C61—21no—
234C23×C61C22⋊Q812yes—
235C23×C61C22×D411yescharacteristic
236C22×D61—22no—
237C22×D61—41no—
238C22×D61—41no—
239C22×D61—41no—
240C22×D61—41no—
241C22×D61—41no—
242C22×D61—41no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—22no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—22no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
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×C61C22⋊Q812yes—
261C2×C4×C61#13512yes—
262C2×C4×C61C22×D411yescharacteristic
263C2×C4×C61C22×Q811yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×C4×C61H(5,𝔽2)11yescharacteristic
267C2×C4×C61H(5,𝔽2)11yescharacteristic
268C22⋊D61—48no—
269C2×Dic61—22no—
270C2×Dic61—22no—
271C2×Dic61—22no—
272C2×Dic61—22no—
273C2×Dic61—41no—
274C2×Dic61—41no—
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—
290D244—22no—
291D244—22no—
292D244—41no—
293D244—41no—
294C4×D61—48no—
295Dic122—22no—
296Dic122—22no—
297Dic122—22no—
298Dic122C22⋊Q814yes—
299C8×C61—22no—
300C8×C61—41no—
301C8×C61—41no—
302C8×C61—41no—
303#25961—611nomaximal, Sylow
304C22×C61—22no—
305C22×C61—22no—
306C22×C61—22no—
307C22×C61—41no—
308C22×C61—41no—
309C22×C61—41no—
310C22×C61—41no—
311C22×C61C2×C22⋊Q811yescharacteristic
312C22×C61D4211yescharacteristic
313C22×C61#54611yescharacteristic
314C22×C61C2×#13511yescharacteristic
315C22×C61#57511yescharacteristic
316C22×C61#57611yescharacteristic
317C22×C61#57811yescharacteristic
318C2×D61—42no—
319C2×D61—42no—
320C2×D61—42no—
321C2×D61—81no—
322C2×D61—81no—
323C2×D61—81no—
324C2×D61—81no—
325C4×C61—22no—
326C4×C61—22no—
327C4×C61—41no—
328C4×C61—41no—
329C4×C61—41no—
330C4×C61C2×C22⋊Q811yescharacteristic
331C4×C61#54311yescharacteristic
332C4×C61#54611yescharacteristic
333C4×C61#55111yescharacteristic
334Dic61—24no—
335Dic61—42no—
336Dic61—81no—
337Dic61—81no—
338Dic61—81no—
339Dic61—81no—
340Dic61—81no—
341C2×#479—611no—
342C2×#416—611no—
343#2441—611no—
344#2341—611no—
345#2093—611no—
346#2067—611no—
347#2065—611no—
348#2025—611no—
349#1863—611no—
350#1863—611no—
351#1854—611no—
352#1850D6111yescharacteristic, complemented
353#1808—611no—
354#1803—611no—
355#1788—611no—
356C2×C61—42no—
357C2×C61#243711yescharacteristic
358C2×C61#307611yescharacteristic
359C2×C61#308311yescharacteristic
360C2×C61#326911yescharacteristic
361C2×C61#328711yescharacteristic
362C2×C61#330011yescharacteristic
363C2×C61#330911yescharacteristic
364D61—82no—
365C22×#41—611no—
366C23×C8C2×D6111yescharacteristic
367C2×#125—611no—
368C2×C4×Q8—611no—
369C2×C4×Q8—611no—
370C2×#117—611no—
371#479—1224no—
372C2×#107—611no—
373C2×#107—611no—
374C2×#107—611no—
375C2×#106—611no—
376C2×C4⋊C8C2×D6111yescharacteristic
377C2×C4⋊C8C2×D6111yescharacteristic
378#416—1224no—
379C2×#103—611no—
380C2×#103—611no—
381C2×#103—611no—
382C2×#103—611no—
383C2×#102—611no—
384C2×#102—611no—
385C2×#102—611no—
386C2×#102—611no—
387C2×C22⋊C8C2×D6111yescharacteristic
388#399—1221no—
389#396—1221no—
390#390—611no—
391#390—611no—
392#390—1221no—
393#390—1221no—
394#389—611no—
395#389—611no—
396#389—1221no—
397#387—1221no—
398#384—611no—
399#380—611no—
400#380—611no—
401#380—1221no—
402C4×C22⋊C4—1221no—
403C4×C22⋊C4C2×D6111yescharacteristic
404#340—611no—
405#340—611no—
406#340—611no—
407#340—611no—
408#336C2×D6111yescharacteristic
409#336C2×D6111yescharacteristic
410C61#2596111yescharacteristic, complemented, Sylow
411C2×#41—614no—
412C2×#41—1222no—
413C2×#41—1222no—
414C2×#41—1222no—
415C2×#41—2441no—
416C2×#41—2441no—
417C22×Q8—611no—
418C22×D4—611no—
419C22×D4—1221no—
420C23×C4—1221no—
421C23×C4C22×D6111yescharacteristic
422C22×C8—22no—
423C22×C8—22no—
424C22×C8—41no—
425C22×C8C22×D6111yescharacteristic
426C22×C8C22×D6111yescharacteristic
427C22×C8C22×D6111yescharacteristic
428C22×C8C22×D6111yescharacteristic
429#125—1222no—
430#125—2441no—
431C4×Q8—1222no—
432C4×Q8—1222no—
433C4×Q8—2441no—
434C4×Q8—2441no—
435#117—1222no—
436#117—2441no—
437C2×C4⋊C4—611no—
438C2×C4⋊C4—611no—
439C2×C4⋊C4—611no—
440C2×C4⋊C4—611no—
441C2×C4⋊C4—611no—
442C2×C4⋊C4—611no—
443C2×C4⋊C4—1221no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4—1221no—
446C2×C4⋊C4—1221no—
447C2×C4⋊C4—1221no—
448C2×C4⋊C4—1221no—
449C2×C4⋊C4—1221no—
450C2×C4⋊C4—1221no—
451C2×C22⋊C4—1221no—
452C2×C22⋊C4—1221no—
453C2×C22⋊C4—1221no—
454C2×C22⋊C4—1221no—
455C2×C22⋊C4—1221no—
456C2×C22⋊C4—1221no—
457C2×C22⋊C4—1221no—
458C2×C22⋊C4D24411yescharacteristic
459C2×C22⋊C4D24411yescharacteristic
460C2×C42—1221no—
461C2×C42—1221no—
462C2×C42—1221no—
463C2×C42C22×D6111yescharacteristic
464C2×C42C22×D6111yescharacteristic
465#107—1222no—
466#107—2441no—
467#107—2441no—
468#106—1222no—
469C4⋊C8—22no—
470C4⋊C8—41no—
471#103—614no—
472#103—1222no—
473#103—1222no—
474#103—1222no—
475#102—1222no—
476#102—1222no—
477#102—2441no—
478#102—2441no—
479C22⋊C8—22no—
480#95—1221no—
481#95—1221no—
482#95—1221no—
483#95—1221no—
484#95D24411yescharacteristic
485#95D24411yescharacteristic
486C24—1221no—
487C24C2×D24411yescharacteristic
488#41—1228no—
489#41—2444no—
490#41—2444no—
491#41—2444no—
492C2×Q8—612no—
493C2×Q8—612no—
494C2×Q8—612no—
495C2×Q8—1221no—
496C2×Q8—1221no—
497C2×Q8—1221no—
498C2×D4—612no—
499C2×D4—1224no—
500C2×D4—1221no—
501C2×D4—1221no—
502C2×D4—1221no—
503C2×D4—1221no—
504C2×D4—1221no—
505C2×D4—2442no—
506C2×D4—2442no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—1221no—
512C22×C4—1221no—
513C22×C4—1224no—
514C22×C4—1221no—
515C22×C4—1221no—
516C22×C4—1221no—
517C22×C4—1221no—
518C22×C4—1221no—
519C22×C4—2442no—
520C22×C4—2442no—
521C22×C4C2×D24411yescharacteristic
522C22×C4C2×D24411yescharacteristic
523C22×C4C2×D24411yescharacteristic
524C22×C4C2×D24411yescharacteristic
525C22×C4C2×D24411yescharacteristic
526C22×C4#1175822412yes—
527C22×C4Q8×D6112yes—
528C22×C4C23×D6111yescharacteristic
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8—44no—
541C2×C8D4×D6112yes—
542C4⋊C4—612no—
543C4⋊C4—1221no—
544C4⋊C4—1221no—
545C4⋊C4—1221no—
546C4⋊C4—1221no—
547C4⋊C4—1221no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—1221no—
551C4⋊C4—1221no—
552C4⋊C4—1221no—
553C4⋊C4—1221no—
554C4⋊C4—4881no—
555C4⋊C4—4881no—
556C4⋊C4—2442no—
557C4⋊C4—2442no—
558C4⋊C4—2442no—
559C4⋊C4—2442no—
560C4⋊C4—2442no—
561C4⋊C4—2442no—
562C22⋊C4—22no—
563C22⋊C4—41no—
564C22⋊C4—4881no—
565C22⋊C4—4881no—
566C22⋊C4—4881no—
567C22⋊C4—4881no—
568C22⋊C4—2442no—
569C22⋊C4—2442no—
570C22⋊C4—2442no—
571C42—21no—
572C42—21no—
573C42—21no—
574C42—21no—
575C42—2442no—
576C42—2442no—
577C42—2442no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—1222no—
585C23—2441no—
586C23—2441no—
587C23—2441no—
588C23—2441no—
589C23—2441no—
590C23—2441no—
591C23#1451975712yes—
592C23C22×D24411yescharacteristic
593Q8—614no—
594Q8—1222no—
595Q8—1222no—
596Q8—1222no—
597D4—1222no—
598D4—1222no—
599D4—2448no—
600D4—2441no—
601D4—2441no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—1222no—
620C2×C4—1222no—
621C2×C4—1222no—
622C2×C4—1222no—
623C2×C4—2448no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4—2441no—
634C2×C4—2441no—
635C2×C4—2441no—
636C2×C4—2441no—
637C2×C4—2441no—
638C2×C4—2441no—
639C2×C4—2441no—
640C2×C4—2441no—
641C2×C4#1451975712yes—
642C2×C4#1451979912yes—
643C2×C4C2×D4×D6111yescharacteristic
644C2×C4C2×Q8×D6111yescharacteristic
645C2×C4#41×D6111yescharacteristic
646C2×C4C2×#1175822411yescharacteristic
647C2×C4#1451982211yescharacteristic
648C2×C4#1451982311yescharacteristic
649C8—22no—
650C8—41no—
651C8—41no—
652C8—41no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—41no—
657C22—41no—
658C22—41no—
659C22—41no—
660C22—4881no—
661C22—4881no—
662C22—4881no—
663C22—4881no—
664C22—2442no—
665C22—2442no—
666C22—2442no—
667C22C2×#1451975711yescharacteristic
668C22D4×D24411yescharacteristic
669C22#1684040711yescharacteristic
670C22C2×#1451979911yescharacteristic
671C22#1684056211yescharacteristic
672C22#1684056711yescharacteristic
673C22#1684057911yescharacteristic
674C4—22no—
675C4—22no—
676C4—41no—
677C4—41no—
678C4—41no—
679C4—1224no—
680C4—4881no—
681C4—4881no—
682C4—4881no—
683C4—4881no—
684C4—4881no—
685C4—2442no—
686C4C22⋊Q8×D6111yescharacteristic
687C4#1684038711yescharacteristic
688C4#1684039911yescharacteristic
689C4#1684043511yescharacteristic
690C2—42no—
691C2—4882no—
692C2#2144948311yescharacteristic
693C2#2145560911yescharacteristic
694C2#2145567211yescharacteristic
695C2#2145737911yescharacteristic
696C2#2145757911yescharacteristic
697C2#2145770811yescharacteristic
698C2#2145782711yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1850C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
hypercenter#1850D61
hypocenterC61#25961
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×#16840070C2semidirect
3. #21449532C2semidirect
5. #21447346C2semidirect
6. #21447206C2semidirect
7. #21447192C2semidirect
8. #21446966C2semidirect
9. #21446076C2semidirect
10. #21446076C2semidirect
11. #21446020C2semidirect
12. #21445848C2semidirect
13. #21445825C2semidirect
15. C61×#1850C2semidirect
352. #1850D61semidirect
410. C61#25961semidirect

Maximal subgroups

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

subgroupindex
C2×#168400702
C2×#168397592
#214495322
#214486592
#214473462
#214472062
#214471922
#214469662
#214460762
#214460202
#214458482
#214458252
#214457602
C61×#18502
#2596161