Finite Group Databasebeta

← #29738971

Subgroups — #29738971

Order 15616.

750 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450852C211yesmaximal, characteristic, complemented
2#21448764C211yesmaximal, characteristic
3#21447348C211yesmaximal, characteristic
4#21447344C211yesmaximal, characteristic
5#21447206C211yesmaximal, characteristic
6#21447202C211yesmaximal, characteristic
7#21446576C211yesmaximal, characteristic, complemented
8#21446574C211yesmaximal, characteristic, complemented
9#21446534C211yesmaximal, characteristic, complemented
10#21446393C211yesmaximal, characteristic, complemented
11#21446390C211yesmaximal, characteristic, complemented
12#21445977C211yesmaximal, characteristic, complemented
13#21445818C211yesmaximal, characteristic
14#21445633C211yesmaximal, characteristic
15C61×#1947C211yesmaximal, characteristic, complemented
16C2×#14519767—21no—
17C2×#14519739C2211yescharacteristic
18C2×#14519734—21no—
19C2×#14519711C2211yescharacteristic
20C2×#14519710C2211yescharacteristic
21C2×#14519710C2211yescharacteristic
22C2×#14519710C2211yescharacteristic
23C2×#14519706C2211yescharacteristic
24C2×#14519706C2211yescharacteristic
25C2×#14519699C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27C2×#14519697C2211yescharacteristic
28C2×#14519697C2211yescharacteristic
29C2×C8×Dic61C2211yescharacteristic
30#16839695—21no—
31#16839693—21no—
32#16839664C2211yescharacteristic
33#16839663C2211yescharacteristic
34#16839661C2211yescharacteristic
35#16839660C2211yescharacteristic
36#16839657—21no—
37#16839653—21no—
38#16839651—21no—
39#16839631C2211yescharacteristic
40#16839631C2211yescharacteristic
41#16839631C2211yescharacteristic
42#16839621—21no—
43C4⋊C4×Dic61C2211yescharacteristic
44C4⋊C4×Dic61C2211yescharacteristic
45C4×#11758196C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839517C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×#107×C61C2211yescharacteristic
53C2×#107×C61C2211yescharacteristic
54C2×C4⋊C8×C61C2211yescharacteristic
55C61×#384C2211yescharacteristic
56C4×C4⋊C4×C61C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C22×D244C2311yescharacteristic
60C22×C4×D61—21no—
61C22×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63#14519767—44no—
64#14519739—22no—
65#14519739Q814yes—
66#14519734—44no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758196—21no—
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196D411yescharacteristic
79C2×#11758196D411yescharacteristic
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61C2311yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89#14519711—41no—
90#14519710—22no—
91#14519710—22no—
92#14519710—41no—
93#14519706—22no—
94#14519706—22no—
95#14519699—22no—
96#14519699—41no—
97#14519697—22no—
98#14519697—41no—
99C8×Dic61—22no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684D411yescharacteristic
105#14519684D411yescharacteristic
106C22×C8×C61C2311yescharacteristic
107C22×C8×C61C2311yescharacteristic
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C42×C61—21no—
115C2×C42×C61C2311yescharacteristic
116#107×C61—41no—
117#107×C61—41no—
118C4⋊C8×C61—22no—
119#95×C61—21no—
120C23×D61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2411yescharacteristic
128C22×Dic61—21no—
129C22×Dic61—21no—
130C22×Dic61—21no—
131C22×Dic61C2×D411yescharacteristic
132C22×Dic61C2×D411yescharacteristic
133C22×Dic61C2×D411yescharacteristic
134C22×Dic61C2×D411yescharacteristic
135C2×D244—21no—
136C2×D244—21no—
137C2×D244—21no—
138C2×D244—21no—
139C2×D244—21no—
140C2×D244C2×Q812yes—
141C2×C4×D61—24no—
142C2×C4×D61—42no—
143C2×C4×D61—42no—
144C2×C8×C61—21no—
145C2×C8×C61—21no—
146C2×C8×C61—21no—
147C2×C8×C61—21no—
148C2×C8×C61—21no—
149C2×C8×C61#4112yes—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—42no—
158C4⋊C4×C61—81no—
159C4⋊C4×C61C2×Q812yes—
160C42×C61—21no—
161C42×C61—21no—
162C42×C61—42no—
163#11758198—24no—
164#11758198—42no—
165#11758198—42no—
166#11758198—81no—
167#11758198—81no—
168C4⋊Dic61—21no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—42no—
175#11758196—22no—
176#11758196—42no—
177#11758196—42no—
178#11758196—42no—
179#11758196—42no—
180#11758196—81no—
181#11758196—81no—
182#11758196—81no—
183#11758196D814yes—
184C4×Dic61—21no—
185C4×Dic61—21no—
186C4×Dic61—42no—
187C4×Dic61—42no—
188C4×Dic61C2×Q812yes—
189C4×Dic61#4112yes—
190C2×#92418—21no—
191C2×#92418—21no—
192C2×#92418—21no—
193C2×#92418—21no—
194C2×#92418—21no—
195C2×#92418#4112yes—
196C23×C61C22×D411yescharacteristic
197C22×D61—22no—
198C22×D61—41no—
199C22×D61—41no—
200C22×D61—41no—
201C22×D61—41no—
202C22×D61—41no—
203C22×D61—41no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—22no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61C2×D812yes—
222C2×C4×C61C22×Q811yescharacteristic
223C2×C4×C61C2×#4111yescharacteristic
224C2×C4×C61C2×#4111yescharacteristic
225C2×C4×C61C2×#4111yescharacteristic
226C2×C4×C61H(5,𝔽2)11yescharacteristic
227C2×C4×C612−1+411yescharacteristic
228C2×Dic61—21no—
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61C22⋊Q812yes—
245C2×Dic61C22⋊Q812yes—
246C2×Dic61C2×D812yes—
247C2×Dic61C2×D812yes—
248C2×Dic61#13712yes—
249D244—22no—
250D244—22no—
251D244—41no—
252D244—41no—
253C4×D61—48no—
254C8×C61—22no—
255C8×C61—41no—
256#92418—22no—
257#92418—41no—
258#26086—611nomaximal, Sylow
259C22×C61C2×C22⋊Q811yescharacteristic
260C22×C61#54711yescharacteristic
261C22×C61#55011yescharacteristic
262C22×C61C22×D811yescharacteristic
263C22×C61C2×#13711yescharacteristic
264C22×C61#57611yescharacteristic
265C22×C61#57811yescharacteristic
266C2×D61—42no—
267C2×D61—42no—
268C2×D61—42no—
269C2×D61—81no—
270C2×D61—81no—
271C2×D61—81no—
272C2×D61—81no—
273C4×C61—22no—
274C4×C61—41no—
275C4×C61—42no—
276C4×C61—81no—
277C4×C61—81no—
278C4×C61—81no—
279C4×C61#53311yescharacteristic
280C4×C61#54411yescharacteristic
281C4×C61#55211yescharacteristic
282C4×C61#55311yescharacteristic
283Dic61—22no—
284Dic61—22no—
285Dic61—41no—
286Dic61—81no—
287Dic61—81no—
288Dic61—81no—
289Dic61#47414yes—
290#2552—611no—
291#2354—611no—
292#2093—611no—
293#2093—611no—
294#2067—611no—
295#2067—611no—
296#1947—611no—
297#1947—611no—
298#1947D6111yescharacteristic, complemented
299#1939—611no—
300#1918—611no—
301#1918—611no—
302#1845—611no—
303#1802—611no—
304C4×#102—611no—
305C2×C61#244111yescharacteristic
306C2×C61C2×#47411yescharacteristic
307C2×C61#308311yescharacteristic
308C2×C61#333411yescharacteristic
309C2×C61#335711yescharacteristic
310C2×C61#339111yescharacteristic
311C2×C61#339811yescharacteristic
312D61—82no—
313C2×#125—1221no—
314C2×C4×D4—611no—
315C2×#117—1221no—
316C2×#107—611no—
317C2×#107—611no—
318C2×#107—611no—
319C2×#107—611no—
320C2×#107C2×D6111yescharacteristic
321C2×#107C2×D6111yescharacteristic
322C2×C4⋊C8—611no—
323C2×C4⋊C8—611no—
324C2×C4⋊C8C2×D6111yescharacteristic
325C2×#102—611no—
326C2×#102—611no—
327C2×#102—611no—
328C2×#102—611no—
329C2×C4×C8—611no—
330#398—1221no—
331#398—1221no—
332#390—611no—
333#390—611no—
334#389—611no—
335#389—611no—
336#389—1221no—
337#388—1221no—
338#388—1221no—
339#384—611no—
340#384—611no—
341#384—611no—
342#384C2×D6111yescharacteristic
343#380—1221no—
344C4×C4⋊C4—611no—
345C4×C4⋊C4—611no—
346C4×C4⋊C4—611no—
347C4×C4⋊C4C2×D6111yescharacteristic
348#340—611no—
349#340—611no—
350#340—611no—
351#340—611no—
352#340C2×D6111yescharacteristic
353#340C2×D6111yescharacteristic
354#336—611no—
355#336—611no—
356C61#2608611yescharacteristic, complemented, Sylow
357C22×D4—611no—
358C23×C4—1221no—
359C22×C8—611no—
360C22×C8—611no—
361C22×C8C22×D6111yescharacteristic
362C22×C8C22×D6111yescharacteristic
363#125—2444no—
364C4×D4—614no—
365C4×D4—1222no—
366#117—2444no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—611no—
370C2×C4⋊C4—611no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—611no—
373C2×C4⋊C4—611no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4C22×D6111yescharacteristic
383C2×C4⋊C4C22×D6111yescharacteristic
384C2×C4⋊C4C22×D6111yescharacteristic
385C2×C4⋊C4C22×D6111yescharacteristic
386C2×C22⋊C4—1221no—
387C2×C22⋊C4—1221no—
388C2×C22⋊C4—1221no—
389C2×C22⋊C4—1221no—
390C2×C22⋊C4—1221no—
391C2×C42—21no—
392C2×C42—611no—
393C2×C42—611no—
394C2×C42—1221no—
395C2×C42—1221no—
396C2×C42C22×D6111yescharacteristic
397#107—41no—
398#107—41no—
399#107—1222no—
400#107—1222no—
401#107—2441no—
402#107—2441no—
403C4⋊C8—22no—
404C4⋊C8—1222no—
405C4⋊C8—1222no—
406#102—1222no—
407#102—1222no—
408#102—2441no—
409#102—2441no—
410C4×C8—1222no—
411#95—21no—
412#95—611no—
413#95—611no—
414#95—1221no—
415#95—1221no—
416#95—1221no—
417#95—1221no—
418C24—1221no—
419C2×D4—612no—
420C2×D4—1221no—
421C2×D4—1221no—
422C2×D4—1221no—
423C2×D4—1221no—
424C2×D4—1221no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—611no—
430C22×C4—611no—
431C22×C4—611no—
432C22×C4—611no—
433C22×C4—1224no—
434C22×C4—1221no—
435C22×C4—1221no—
436C22×C4—1221no—
437C22×C4—2442no—
438C22×C4—2442no—
439C22×C4D4×D6111yescharacteristic
440C22×C4D4×D6111yescharacteristic
441C22×C4C23×D6111yescharacteristic
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—612no—
448C2×C8—1221no—
449C2×C8—1221no—
450C2×C8—1221no—
451C2×C8—1221no—
452C2×C8—1221no—
453C2×C8#1175822212yes—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—42no—
462C4⋊C4—81no—
463C4⋊C4—614no—
464C4⋊C4—1222no—
465C4⋊C4—1221no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4—4881no—
472C4⋊C4—4881no—
473C4⋊C4—4881no—
474C4⋊C4—2442no—
475C4⋊C4—2442no—
476C4⋊C4—2442no—
477C4⋊C4—2442no—
478C4⋊C4—2442no—
479C4⋊C4Q8×D6112yes—
480C22⋊C4—1224no—
481C22⋊C4—4881no—
482C22⋊C4—4881no—
483C22⋊C4—2442no—
484C22⋊C4—2442no—
485C42—21no—
486C42—21no—
487C42—42no—
488C42—612no—
489C42—612no—
490C42—1221no—
491C42—1221no—
492C42—2442no—
493C42—2442no—
494C23—1222no—
495C23—2441no—
496C23—2441no—
497C23—2441no—
498C23—2441no—
499C23—2441no—
500C23—2441no—
501C23C2×D4×D6111yescharacteristic
502D4—1222no—
503D4—1222no—
504D4—2441no—
505D4—2441no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—612no—
524C2×C4—612no—
525C2×C4—612no—
526C2×C4—612no—
527C2×C4—612no—
528C2×C4—1221no—
529C2×C4—1221no—
530C2×C4—1221no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—1221no—
534C2×C4—1221no—
535C2×C4—2448no—
536C2×C4—2441no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4—2441no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4D8×D6112yes—
546C2×C4C2×Q8×D6111yescharacteristic
547C2×C4#41×D6111yescharacteristic
548C2×C4#41×D6111yescharacteristic
549C2×C4C2×#1175822211yescharacteristic
550C2×C4#1451982211yescharacteristic
551C2×C4#1451982511yescharacteristic
552C8—22no—
553C8—41no—
554C8—1222no—
555C8—2441no—
556C22—4881no—
557C22—4881no—
558C22—4881no—
559C22—4881no—
560C22—2442no—
561C22—2442no—
562C22—2442no—
563C22C22⋊Q8×D6111yescharacteristic
564C22#1684041311yescharacteristic
565C22#1684043011yescharacteristic
566C22C2×D8×D6111yescharacteristic
567C22#137×D6111yescharacteristic
568C22#1684056611yescharacteristic
569C22#1684057511yescharacteristic
570C4—22no—
571C4—41no—
572C4—42no—
573C4—81no—
574C4—81no—
575C4—81no—
576C4—614no—
577C4—1222no—
578C4—1222no—
579C4—4881no—
580C4—4881no—
581C4—4881no—
582C4—2441no—
583C4#1684031611yescharacteristic
584C4#1684038911yescharacteristic
585C4#1684044411yescharacteristic
586C4#1684046111yescharacteristic
587C2—4882no—
588C2#2144952911yescharacteristic
589C2#474×D6111yescharacteristic
590C2#2145566711yescharacteristic
591C2#2145816411yescharacteristic
592C2#2145844111yescharacteristic
593C2#2145879011yescharacteristic
594C2#2145884511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1947C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1947D61
hypocenterC61#26086
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1947

Complements

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

NH ≅ G/Ntype
1. #21450852C2semidirect
7. #21446576C2semidirect
8. #21446574C2semidirect
9. #21446534C2semidirect
10. #21446393C2semidirect
11. #21446390C2semidirect
12. #21445977C2semidirect
15. C61×#1947C2semidirect
298. #1947D61semidirect
356. C61#26086semidirect

Maximal subgroups

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

subgroupindex
#214508522
#214487642
#214473482
#214473442
#214472062
#214472022
#214465762
#214465742
#214465342
#214463932
#214463902
#214459772
#214458182
#214456332
C61×#19472
#2608661