Finite Group Databasebeta

← #29654875

Subgroups — #29654875

Order 15616.

1190 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456757C211yesmaximal, characteristic, complemented
2#21456734C211yesmaximal, characteristic, complemented
3#21450765C211yesmaximal, characteristic, complemented
4#21444765C218yesmaximal, complemented
5#21444318C211yesmaximal, characteristic, complemented
6#21444314C211yesmaximal, characteristic, complemented
7#21444140C211yesmaximal, characteristic, complemented
8C61×#3506C211yesmaximal, characteristic, complemented
9C2×#14519774C2211yescharacteristic
10C2×#14519773—21no—
11C2×#14519751—22no—
12#16840138—21no—
13#16840136—21no—
14#16840110—21no—
15#16840110—21no—
16#16839962—22no—
17#16839957—22no—
18#16839912C2214yes—
19#16839895C2214yes—
20#16839815C2211yescharacteristic
21#16839815C2211yescharacteristic
22#16839671C2214yes—
23#16839663—21no—
24C4×C4⋊Dic61C2211yescharacteristic
25#16839499C2214yes—
26#16839496C2214yes—
27#16839486C2214yes—
28C61×#559—22no—
29C61×#556—21no—
30C61×#555—21no—
31C61×#553—22no—
32C61×#550—21no—
33C61×#547—21no—
34C61×#538C2214yes—
35C61×#535—21no—
36C61×#533C2211yescharacteristic
37C61×#532C2211yescharacteristic
38C61×#525—22no—
39C61×#520—21no—
40C2×C4×D4×C61C2211yescharacteristic
41C22×D244—21no—
42C22×D244—21no—
43C22×D244C2312yes—
44C22×C22⋊D61—24no—
45C2×#11758201—24no—
46C2×#11758199—24no—
47#14519774—21no—
48#14519774—21no—
49#14519774—21no—
50#14519774—21no—
51#14519773—44no—
52#14519751—48no—
53C2×C4⋊Dic61—21no—
54C2×#11758198—24no—
55C2×C4×Dic61—21no—
56C4⋊#92418—21no—
57C4⋊#92418—21no—
58#14519697—24no—
59#14519697—24no—
60C22⋊#92418C2312yes—
61C22⋊#92418C2312yes—
62C4×#92418—21no—
63C4×#92418—21no—
64#14519684C2312yes—
65C2×#41×C61—22no—
66C2×#41×C61—21no—
67C22×D4×C61D411yescharacteristic
68C23×C4×C61D412yes—
69C4⋊Q8×C61D411yescharacteristic, complemented
70C4⋊Q8×C61D411yescharacteristic, complemented
71C4⋊Q8×C61D411yescharacteristic, complemented
72#127×C61—22no—
73#127×C61D411yescharacteristic, complemented
74#126×C61—24no—
75#125×C61—21no—
76#124×C61—22no—
77#124×C61—24no—
78#124×C61—21no—
79#123×C61—24no—
80#123×C61—24no—
81C22⋊Q8×C61—22no—
82C22⋊Q8×C61—21no—
83C22⋊Q8×C61C2312yes—
84#121×C61—22no—
85#121×C61—22no—
86#121×C61—22no—
87#121×C61—24no—
88#121×C61—21no—
89#121×C61C2312yes—
90#120×C61—24no—
91C4×Q8×C61—22no—
92C4×Q8×C61—21no—
93C4×Q8×C61D412yescomplemented
94C4×D4×C61—22no—
95C4×D4×C61—24no—
96C4×D4×C61—24no—
97C4×D4×C61—21no—
98C4×D4×C61—21no—
99C4×D4×C61—21no—
100C4×D4×C61—21no—
101C4×D4×C61—21no—
102C4×D4×C61D412yescomplemented
103#117×C61—22no—
104#117×C61—22no—
105#117×C61—21no—
106C2×C4⋊C4×C61—22no—
107C2×C4⋊C4×C61D411yescharacteristic
108C2×C22⋊C4×C61C2312yes—
109C2×C42×C61C2311yescharacteristic
110C24×C61C2×D412yes—
111C23×D61—22no—
112#41×C61—42no—
113#41×C61—42no—
114#41×C61—84no—
115C2×Q8×C61—21no—
116C2×Q8×C61—21no—
117C2×Q8×C61C2×D412yes—
118C2×D4×C61—22no—
119C2×D4×C61—24no—
120C2×D4×C61—24no—
121C2×D4×C61—21no—
122C2×D4×C61—21no—
123C2×D4×C61—21no—
124C2×D4×C61—21no—
125C2×D4×C61—22no—
126C2×D4×C61—22no—
127C2×D4×C61C2×D412yes—
128C22×C4×C61—24no—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—44no—
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D412yes—
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2411yescharacteristic
137C2×C22⋊D61—48no—
138C2×C22⋊D61—48no—
139C2×C22⋊D61—84no—
140C22×Dic61—22no—
141C2×D244—22no—
142C2×D244—22no—
143C2×D244—44no—
144C2×D244—41no—
145C2×D244—41no—
146C2×D244—41no—
147C2×D244—41no—
148C2×D244—42no—
149C2×D244—42no—
150C2×D244—82no—
151C4⋊C4×C61—24no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61C2×D412yes—
161C4⋊C4×C61C2×D412yes—
162C22⋊C4×C61—24no—
163C22⋊C4×C61—24no—
164C22⋊C4×C61—24no—
165C22⋊C4×C61—22no—
166C22⋊C4×C61—22no—
167C22⋊C4×C61—22no—
168C22⋊C4×C61—22no—
169C22⋊C4×C61—22no—
170C42×C61—24no—
171C42×C61—22no—
172C42×C61C2×D411yescharacteristic
173C42×C61C2×D411yescharacteristic
174C42×C61C2×D411yescharacteristic
175C42×C61C2×D411yescharacteristic
176#11758201—84no—
177#11758199—84no—
178#11758198—48no—
179C4⋊Dic61—24no—
180C4×Dic61—42no—
181C2×#92418—22no—
182C2×#92418—22no—
183C23×C61—22no—
184C23×C61—22no—
185C23×C61—22no—
186C23×C61—22no—
187C23×C61—42no—
188C23×C61—42no—
189C23×C61C22×D411yescharacteristic
190C22×D61—44no—
191C22×D61—44no—
192C22×D61—42no—
193C22×D61—42no—
194C22×D61—42no—
195Q8×C61—42no—
196Q8×C61—42no—
197Q8×C61—42no—
198D4×C61—42no—
199D4×C61—42no—
200D4×C61—42no—
201D4×C61—42no—
202D4×C61—42no—
203D4×C61—42no—
204D4×C61—42no—
205D4×C61—84no—
206D4×C61—84no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—41no—
213C2×C4×C61—44no—
214C2×C4×C61—44no—
215C2×C4×C61—42no—
216C2×C4×C61—82no—
217C2×C4×C61—84no—
218C2×C4×C61#12011yescharacteristic
219C2×C4×C61#12011yescharacteristic
220C2×C4×C61#12011yescharacteristic
221C2×C4×C61#12011yescharacteristic
222C2×C4×C61#13512yes—
223C2×C4×C61AGL(1,ℤ/8)12yes—
224C2×C4×C61C22×D411yescharacteristic
225C2×C4×C61C22×D411yescharacteristic
226C22⋊D61—816no—
227C2×Dic61—24no—
228C2×Dic61—42no—
229C2×Dic61—42no—
230D244—44no—
231D244—44no—
232D244—82no—
233D244—82no—
234D244—82no—
235D244—82no—
236#92418—42no—
237#92418—42no—
238#18405—611nomaximal, Sylow
239C22×C61—21no—
240C22×C61—21no—
241C22×C61—44no—
242C22×C61—42no—
243C22×C61—42no—
244C22×C61—42no—
245C22×C61—42no—
246C22×C61—42no—
247C22×C61—42no—
248C22×C61—82no—
249C22×C61C2×#12011yescharacteristic
250C22×C61C2×#13511yescharacteristic
251C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
252C2×D61—84no—
253C2×D61—84no—
254C2×D61—82no—
255C2×D61—82no—
256C2×D61—82no—
257C4×C61—21no—
258C4×C61—21no—
259C4×C61—21no—
260C4×C61—42no—
261C4×C61—42no—
262C4×C61—42no—
263C4×C61—42no—
264C4×C61#44912yes—
265C4×C61#44912yes—
266C4×C61#45312yes—
267Dic61—44no—
268#3506D6111yescharacteristic, complemented
269#3208—611no—
270#3205—611no—
271#2541—611no—
272#1600—618no—
273#1503—611no—
274#1502—611no—
275#1460—611no—
276C2×C61—21no—
277C2×C61—21no—
278C2×C61—82no—
279C2×C61—82no—
280C2×C61C2×#44911yescharacteristic
281C2×C61C2×#45311yescharacteristic
282C2×C61#302711yescharacteristic
283D61—162no—
284#559—22no—
285#556—21no—
286#555—21no—
287#553—22no—
288#550—21no—
289#547—21no—
290#538C2×D6114yes—
291#535—21no—
292#533C2×D6111yescharacteristic
293#532C2×D6111yescharacteristic
294C2×#127—611no—
295C2×#127—1221no—
296#525—22no—
297C2×#121—1222no—
298#520—21no—
299C2×C4×D4C2×D6111yescharacteristic
300#493—1221no—
301#492—1221no—
302#486—1221no—
303#486—1221no—
304#460—1222no—
305#459—1222no—
306#450—614no—
307#447—614no—
308#431—611no—
309#431—611no—
310#392—614no—
311#390—1221no—
312C4×C4⋊C4—611no—
313#328—614no—
314#327—614no—
315#323—614no—
316C61#1840511yescharacteristic, complemented, Sylow
317C2×#41—22no—
318C2×#41—21no—
319C22×D4—612no—
320C22×D4—1224no—
321C22×D4—1221no—
322C22×D4—1221no—
323C22×D4C22⋊D6111yescharacteristic
324C23×C4C22⋊D6112yes—
325C2×SD16—1224no—
326C2×D8—1224no—
327C4⋊Q8C22⋊D6111yescharacteristic, complemented
328C4⋊Q8C22⋊D6111yescharacteristic, complemented
329C4⋊Q8C22⋊D6111yescharacteristic, complemented
330#127—22no—
331#127—1221no—
332#127—1221no—
333#127—1221no—
334#127—1221no—
335#127—2444no—
336#127C22⋊D6111yescharacteristic, complemented
337#126—24no—
338#125—21no—
339#124—22no—
340#124—24no—
341#124—21no—
342#123—24no—
343#123—24no—
344C22⋊Q8—22no—
345C22⋊Q8—21no—
346C22⋊Q8C22×D6112yes—
347#121—22no—
348#121—22no—
349#121—22no—
350#121—24no—
351#121—21no—
352#121—2448no—
353#121C22×D6112yes—
354#120—24no—
355C4×Q8—22no—
356C4×Q8—21no—
357C4×Q8C22⋊D6112yescomplemented
358C4×D4—22no—
359C4×D4—24no—
360C4×D4—24no—
361C4×D4—21no—
362C4×D4—21no—
363C4×D4—21no—
364C4×D4—21no—
365C4×D4—21no—
366C4×D4C22⋊D6112yescomplemented
367#117—22no—
368#117—22no—
369#117—21no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4C22⋊D6111yescharacteristic
373C2×C22⋊C4—1224no—
374C2×C22⋊C4C22×D6112yes—
375C2×C42—1221no—
376C2×C42C22×D6111yescharacteristic
377C4⋊C8—1221no—
378C4⋊C8—1221no—
379#102—1224no—
380#102—1224no—
381C22⋊C8—612no—
382C22⋊C8—612no—
383C4×C8—1221no—
384C4×C8—1221no—
385#95—612no—
386C24—1222no—
387C24C2×C22⋊D6112yes—
388#41—42no—
389#41—42no—
390#41—84no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8C2×C22⋊D6112yes—
394C2×D4—22no—
395C2×D4—24no—
396C2×D4—24no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—1222no—
404C2×D4—1222no—
405C2×D4—4882no—
406C2×D4—4884no—
407C2×D4—2444no—
408C2×D4—2448no—
409C2×D4—2448no—
410C2×D4—2441no—
411C2×D4—2441no—
412C2×D4—2441no—
413C2×D4—2441no—
414C2×D4—2442no—
415C2×D4—2442no—
416C2×D4C2×C22⋊D6112yes—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—44no—
422C22×C4—1222no—
423C22×C4C2×C22⋊D6111yescharacteristic
424C22×C4C2×C22⋊D6112yes—
425C22×C4C2×C22⋊D6111yescharacteristic
426C22×C4C23×D6111yescharacteristic
427SD16—4884no—
428D8—4884no—
429C2×C8—1222no—
430C2×C8—1222no—
431C4⋊C4—24no—
432C4⋊C4—22no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—22no—
440C4⋊C4—1224no—
441C4⋊C4C2×C22⋊D6112yes—
442C4⋊C4C2×C22⋊D6112yes—
443C22⋊C4—24no—
444C22⋊C4—24no—
445C22⋊C4—24no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—2448no—
452C42—24no—
453C42—22no—
454C42—2442no—
455C42C2×C22⋊D6111yescharacteristic
456C42C2×C22⋊D6111yescharacteristic
457C42C2×C22⋊D6111yescharacteristic
458C42C2×C22⋊D6111yescharacteristic
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—42no—
464C23—42no—
465C23—2444no—
466C23—2444no—
467C23—2442no—
468C23—2442no—
469C23—2442no—
470C23C22×C22⋊D6111yescharacteristic
471Q8—42no—
472Q8—42no—
473Q8—42no—
474D4—42no—
475D4—42no—
476D4—42no—
477D4—42no—
478D4—42no—
479D4—42no—
480D4—42no—
481D4—84no—
482D4—84no—
483D4—4882no—
484D4—48816no—
485D4—4882no—
486D4—4882no—
487D4—4882no—
488D4—2444no—
489D4—2444no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—41no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—82no—
500C2×C4—84no—
501C2×C4—1224no—
502C2×C4—2442no—
503C2×C4—2442no—
504C2×C4#1451974611yescharacteristic
505C2×C4#1451974611yescharacteristic
506C2×C4#1451974611yescharacteristic
507C2×C4#1451974611yescharacteristic
508C2×C4#1451979712yes—
509C2×C4#1451980212yes—
510C2×C4C22×C22⋊D6111yescharacteristic
511C2×C4C22×C22⋊D6111yescharacteristic
512C8—2442no—
513C8—2442no—
514C22—21no—
515C22—21no—
516C22—44no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—82no—
524C22—4884no—
525C22—4884no—
526C22—4882no—
527C22—4882no—
528C22—4882no—
529C22C2×#1451974611yescharacteristic
530C22C2×#1451979711yescharacteristic
531C22C2×#1451980211yescharacteristic
532C4—21no—
533C4—21no—
534C4—21no—
535C4—42no—
536C4—42no—
537C4—42no—
538C4—42no—
539C4—2444no—
540C4#1683990712yes—
541C4#1683990712yes—
542C4#1683993012yes—
543C2—21no—
544C2—21no—
545C2—82no—
546C2—82no—
547C2—9762no—
548C2C2×#1683990711yescharacteristic
549C2C2×#1683993011yescharacteristic
550C2#2145522111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3506C2
socleC22×C61C2×#120
hypercenter#3506D61
hypocenterC61#18405
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3506

Complements

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

NH ≅ G/Ntype
1. #21456757C2semidirect
2. #21456734C2semidirect
3. #21450765C2semidirect
4. #21444765C2semidirect
5. #21444318C2semidirect
6. #21444314C2semidirect
7. #21444140C2semidirect
8. C61×#3506C2semidirect
69. C4⋊Q8×C61D4semidirect
70. C4⋊Q8×C61D4semidirect
71. C4⋊Q8×C61D4semidirect
73. #127×C61D4semidirect
93. C4×Q8×C61D4semidirect
102. C4×D4×C61D4semidirect
268. #3506D61semidirect
316. C61#18405semidirect
327. C4⋊Q8C22⋊D61semidirect
328. C4⋊Q8C22⋊D61semidirect
329. C4⋊Q8C22⋊D61semidirect
336. #127C22⋊D61semidirect
357. C4×Q8C22⋊D61semidirect
366. C4×D4C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214567572
#214567342
#214507652
#214447652
#214443182
#214443142
#214441402
C61×#35062
#1840561