Finite Group Databasebeta

← #29733373

Subgroups — #29733373

Order 15616.

1126 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840452C211yesmaximal, characteristic
2C2×#16840130C211yesmaximal, characteristic, complemented
3C2×#16839969C211yesmaximal, characteristic, complemented
4#21446827C212yesmaximal, complemented
5#21446827C212yesmaximal, complemented
6#21446776C211yesmaximal, characteristic, complemented
7#21446618C211yesmaximal, characteristic
8#21446132C212yesmaximal, complemented
9#21446127C212yesmaximal, complemented
10#21445652C211yesmaximal, characteristic, complemented
11C61×#2542C211yesmaximal, characteristic
12C22×#11758202C2212yes—
13#16840452—24no—
14#16840452—24no—
15C2×#14519770—21no—
16C2×#14519769C2211yescharacteristic
17C2×#14519767—21no—
18C2×#14519758—21no—
19C2×#14519756C2211yescharacteristic
20C2×#14519756C2211yescharacteristic
21C2×C4×Dic122C2211yescharacteristic
22C2×#14519736C2211yescharacteristic
23C2×#14519733—21no—
24C22×C4⋊Dic61C2211yescharacteristic
25#16840130—24no—
26#16839969—24no—
27C2×C4⋊#92418C2211yescharacteristic
28C2×#14519701C2212yes—
29C2×#14519701C2212yes—
30C2×#14519701C2212yes—
31C2×#14519700C2212yes—
32C2×C22⋊#92418C2212yes—
33C2×C22⋊#92418C2212yes—
34C2×#14519688C2211yescharacteristic
35#16839524C2212yes—
36#16839524C2212yes—
37#16839524C2212yes—
38C2×C4⋊Q8×C61C2211yescharacteristic
39C2×C22⋊Q8×C61C2212yes—
40C2×C22⋊Q8×C61C2212yes—
41C2×C4×D4×C61C2211yescharacteristic
42C61×#399—21no—
43C61×#399—21no—
44C61×#390—21no—
45C61×#384—21no—
46C4×C4⋊C4×C61C2211yescharacteristic
47C22×Dic122C2311yescharacteristic
48C23×Dic61—21no—
49C2×#11758202—24no—
50C2×#11758202—42no—
51C2×#11758202—42no—
52C22×#92418C2312yes—
53C22×#92418C2312yes—
54#14519770—28no—
55#14519769—22no—
56#14519769D414yes—
57#14519767—28no—
58#14519758—28no—
59#14519756—22no—
60#14519756—22no—
61#14519756—22no—
62#14519756—22no—
63C4×Dic122—22no—
64C4×Dic122D414yes—
65#14519736—22no—
66#14519736—22no—
67#14519733—28no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—22no—
70C2×C4⋊Dic61—22no—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C4⋊#92418—22no—
85#14519701—24no—
86#14519701—24no—
87#14519701—24no—
88#14519700—24no—
89C22⋊#92418—42no—
90C22⋊#92418—42no—
91#14519688—22no—
92C22×Q8×C61C2312yes—
93C22×D4×C61D411yescharacteristic
94C23×C4×C61C2311yescharacteristic
95C23×C4×C61C2311yescharacteristic
96C4⋊Q8×C61—22no—
97C4⋊Q8×C61—22no—
98C22⋊Q8×C61—24no—
99C22⋊Q8×C61—24no—
100C22⋊Q8×C61—42no—
101C22⋊Q8×C61—42no—
102C4×D4×C61—22no—
103C4×D4×C61—22no—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61—22no—
107C2×C4⋊C4×C61—22no—
108C2×C4⋊C4×C61D411yescharacteristic
109C2×C4⋊C4×C61C2312yes—
110C2×C22⋊C4×C61—22no—
111C2×C22⋊C4×C61—22no—
112C2×C22⋊C4×C61D411yescharacteristic
113C2×C22⋊C4×C61D411yescharacteristic
114C2×C42×C61—21no—
115C2×C42×C61C2311yescharacteristic
116#95×C61—21no—
117C24×C61C2×D411yescharacteristic
118C24×C61C2×D411yescharacteristic
119C2×Q8×C61—22no—
120C2×Q8×C61—22no—
121C2×Q8×C61—22no—
122C2×Q8×C61—22no—
123C2×Q8×C61—22no—
124C2×Q8×C61—22no—
125C2×D4×C61—22no—
126C2×D4×C61—22no—
127C2×D4×C61—41no—
128C22×C4×C61—22no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61—22no—
137C22×C4×C61—22no—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2411yescharacteristic
143C22×Dic61—24no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—42no—
149C22×Dic61—42no—
150C2×Dic122—22no—
151C2×Dic122—22no—
152C2×Dic122—21no—
153C2×Dic122C2×D412yes—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—44no—
157C4⋊C4×C61—44no—
158C4⋊C4×C61—42no—
159C4⋊C4×C61—42no—
160C4⋊C4×C61#4114yes—
161C22⋊C4×C61—22no—
162C22⋊C4×C61—22no—
163C22⋊C4×C61—44no—
164C22⋊C4×C61—44no—
165C42×C61—42no—
166C42×C61C2×D412yes—
167C42×C61C2×D412yes—
168C22⋊Dic61—24no—
169C22⋊Dic61—24no—
170C22⋊Dic61—24no—
171#11758202—48no—
172C4⋊Dic61—24no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61C2×D412yes—
177C4⋊Dic61C2×D412yes—
178C4⋊Dic61C2×D412yes—
179#11758196—24no—
180#11758196—24no—
181#11758196—24no—
182#11758196—24no—
183#11758196—24no—
184C4×Dic61—24no—
185C4×Dic61—24no—
186C2×#92418—22no—
187C2×#92418—22no—
188C2×#92418—22no—
189C2×#92418—22no—
190C2×#92418—22no—
191C2×#92418—22no—
192C23×C61—21no—
193C23×C61—22no—
194C23×C61—21no—
195C23×C61—22no—
196C23×C61—21no—
197C23×C61—22no—
198C23×C61—21no—
199C23×C61—21no—
200C23×C61—22no—
201C23×C61#13712yes—
202C23×C61C22×D411yescharacteristic
203Q8×C61—42no—
204Q8×C61—42no—
205Q8×C61—42no—
206Q8×C61—42no—
207D4×C61—44no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—22no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—24no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61#12012yes—
228C2×C4×C61#12012yes—
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C2×#4112yes—
232C2×C4×C61H(5,𝔽2)12yes—
233C2×Dic61—22no—
234C2×Dic61—22no—
235C2×Dic61—22no—
236C2×Dic61—22no—
237C2×Dic61—48no—
238C2×Dic61—42no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—42no—
244Dic122—22no—
245Dic122—22no—
246Dic122—42no—
247#92418—42no—
248#92418—42no—
249#25584—611nomaximal, Sylow
250C22×C61—22no—
251C22×C61—24no—
252C22×C61—41no—
253C22×C61—41no—
254C22×C61—42no—
255C22×C61—41no—
256C22×C61—41no—
257C22×C61—41no—
258C22×C61—42no—
259C22×C61—42no—
260C22×C61C2×#12011yescharacteristic
261C22×C61#54612yes—
262C22×C61C2×#13711yescharacteristic
263C22×C61#57511yescharacteristic
264C22×C61#57812yes—
265C4×C61—22no—
266C4×C61—22no—
267C4×C61—41no—
268C4×C61—41no—
269C4×C61—44no—
270C4×C61—82no—
271C4×C61#53211yescharacteristic
272C4×C61#53511yescharacteristic
273C4×C61#55012yes—
274Dic61—42no—
275Dic61—42no—
276Dic61—42no—
277Dic61—42no—
278C2×#552—611no—
279C2×#491—611no—
280C2×#462—611no—
281#2542D6111yescharacteristic
282#2001—612no—
283#2001—612no—
284#1989—611no—
285#1955—611no—
286#1873—612no—
287#1872—612no—
288#1763—611no—
289C2×C61—42no—
290C2×C61—81no—
291C2×C61#243111yescharacteristic
292C2×C61#300711yescharacteristic
293C2×C61#301111yescharacteristic
294C2×C61#331212yes—
295C2×C61#331512yes—
296C22×Q16—612no—
297#552—1224no—
298#552—1224no—
299C2×C4⋊Q8C2×D6111yescharacteristic
300C2×#126—1221no—
301C2×#125—611no—
302C2×#125—1221no—
303C2×#123—1221no—
304C2×C22⋊Q8—611no—
305C2×C22⋊Q8—611no—
306C2×C22⋊Q8C2×D6112yes—
307C2×C22⋊Q8C2×D6112yes—
308C2×C4×Q8—611no—
309C2×C4×D4C2×D6111yescharacteristic
310C2×#117—611no—
311C2×#117—1221no—
312C22×C4⋊C4—611no—
313#491—1224no—
314#462—1224no—
315C2×C4⋊C8—611no—
316C2×#103—612no—
317C2×#103—612no—
318C2×#103—612no—
319C2×#103—612no—
320C2×C22⋊C8—612no—
321C2×C22⋊C8—612no—
322C2×#97—611no—
323#399—21no—
324#399—21no—
325#390—21no—
326#384—21no—
327C4×C4⋊C4C2×D6111yescharacteristic
328#340—612no—
329#340—612no—
330#340—612no—
331C61#2558411yescharacteristic, complemented, Sylow
332C22×Q8—611no—
333C22×Q8C22×D6112yes—
334C22×D4C22⋊D6111yescharacteristic
335C23×C4—1221no—
336C23×C4C22×D6111yescharacteristic
337C23×C4C22×D6111yescharacteristic
338C2×Q16—1224no—
339C2×Q16—2442no—
340C2×Q16—2442no—
341C22×C8—612no—
342C22×C8—612no—
343C4⋊Q8—22no—
344C4⋊Q8—22no—
345#126—1228no—
346#125—614no—
347#125—1228no—
348#125—1222no—
349#123—1228no—
350C22⋊Q8—24no—
351C22⋊Q8—24no—
352C22⋊Q8—42no—
353C22⋊Q8—42no—
354C22⋊Q8—1222no—
355C22⋊Q8—1222no—
356C22⋊Q8—1222no—
357C22⋊Q8—1222no—
358C4×Q8—614no—
359C4×Q8—1222no—
360C4×D4—22no—
361C4×D4—22no—
362#117—1228no—
363#117—1222no—
364#117—1222no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—611no—
370C2×C4⋊C4—611no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—1222no—
373C2×C4⋊C4—1222no—
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⋊C4C22⋊D6111yescharacteristic
381C2×C4⋊C4C22×D6112yes—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—1221no—
385C2×C22⋊C4—1221no—
386C2×C22⋊C4—1221no—
387C2×C22⋊C4C22⋊D6111yescharacteristic
388C2×C22⋊C4C22⋊D6111yescharacteristic
389C2×C42—21no—
390C2×C42—1221no—
391C2×C42—1221no—
392C2×C42C22×D6111yescharacteristic
393C4⋊C8—1222no—
394#103—1224no—
395#103—1224no—
396#103—1224no—
397#103—1224no—
398C22⋊C8—2442no—
399C22⋊C8—2442no—
400#97—1222no—
401#95—21no—
402C24C2×C22⋊D6111yescharacteristic
403C24C2×C22⋊D6111yescharacteristic
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—612no—
411C2×Q8—1221no—
412C2×Q8—1222no—
413C2×Q8—1222no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—41no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—1224no—
428C22×C4—1221no—
429C22×C4—1221no—
430C22×C4—1221no—
431C22×C4—1221no—
432C22×C4—2442no—
433C22×C4—2442no—
434C22×C4C2×C22⋊D6111yescharacteristic
435C22×C4C2×C22⋊D6111yescharacteristic
436C22×C4C2×C22⋊D6111yescharacteristic
437C22×C4C2×C22⋊D6111yescharacteristic
438C22×C4C23×D6111yescharacteristic
439Q16—2448no—
440C2×C8—1222no—
441C2×C8—1222no—
442C2×C8—1222no—
443C2×C8—1222no—
444C2×C8—1222no—
445C2×C8—1222no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—612no—
453C4⋊C4—612no—
454C4⋊C4—612no—
455C4⋊C4—1224no—
456C4⋊C4—1224no—
457C4⋊C4—1224no—
458C4⋊C4—1221no—
459C4⋊C4—1221no—
460C4⋊C4—1221no—
461C4⋊C4—1224no—
462C4⋊C4—1224no—
463C4⋊C4—1224no—
464C4⋊C4#1175822414yes—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—1224no—
470C22⋊C4—1224no—
471C22⋊C4—1224no—
472C42—42no—
473C42—1224no—
474C42—1224no—
475C42D4×D6112yes—
476C42D4×D6112yes—
477C23—21no—
478C23—21no—
479C23—22no—
480C23—22no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—22no—
485C23—22no—
486C23#1451980812yes—
487C23C22×C22⋊D6111yescharacteristic
488Q8—42no—
489Q8—42no—
490Q8—42no—
491Q8—42no—
492Q8—1222no—
493Q8—1222no—
494Q8—2442no—
495D4—44no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—1222no—
516C2×C4—1222no—
517C2×C4—1222no—
518C2×C4—1222no—
519C2×C4—2448no—
520C2×C4—2441no—
521C2×C4—2441no—
522C2×C4—2441no—
523C2×C4—2441no—
524C2×C4—2442no—
525C2×C4—2442no—
526C2×C4#1451974512yes—
527C2×C4#1451974512yes—
528C2×C4C2×D4×D6111yescharacteristic
529C2×C4C2×D4×D6111yescharacteristic
530C2×C4C2×#1175822412yes—
531C2×C4#1451982212yes—
532C8—2442no—
533C8—2442no—
534C22—24no—
535C22—22no—
536C22—41no—
537C22—41no—
538C22—42no—
539C22—41no—
540C22—42no—
541C22—41no—
542C22—41no—
543C22—42no—
544C22C2×#1451974511yescharacteristic
545C22#1684040612yes—
546C22C2×#1451980811yescharacteristic
547C22#1684056011yescharacteristic
548C22#1684057712yes—
549C4—22no—
550C4—22no—
551C4—41no—
552C4—41no—
553C4—44no—
554C4—82no—
555C4—2442no—
556C4—2442no—
557C4—2442no—
558C4—2442no—
559C4#1684031011yescharacteristic
560C4#1684032511yescharacteristic
561C4#1684043212yes—
562C2—42no—
563C2—81no—
564C2#2144943511yescharacteristic
565C2#2145507711yescharacteristic
566C2#2145511211yescharacteristic
567C2#2145787012yes—
568C2#2145791512yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2542C2
socleC23×C61C22×D4
prime radicalC22×D4×C61D4
hypercenter#2542D61
hypocenterC61#25584
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2542

Complements

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

NH ≅ G/Ntype
2. C2×#16840130C2semidirect
3. C2×#16839969C2semidirect
4. #21446827C2semidirect
5. #21446827C2semidirect
6. #21446776C2semidirect
8. #21446132C2semidirect
9. #21446127C2semidirect
10. #21445652C2semidirect
331. C61#25584semidirect

Maximal subgroups

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

subgroupindex
C2×#168404522
C2×#168401302
C2×#168399692
#214468272
#214467762
#214466182
#214461322
#214461272
#214456522
C61×#25422
#2558461