Finite Group Databasebeta

← #29712521

Subgroups — #29712521

Order 15616.

1164 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
1#21455220C211yesmaximal, characteristic
2#21455197C211yesmaximal, characteristic, complemented
3#21454367C211yesmaximal, characteristic, complemented
4#21446988C212yesmaximal, complemented
5#21446848C212yesmaximal, complemented
6#21446367C211yesmaximal, characteristic, complemented
7#21446351C212yesmaximal, complemented
8#21446342C211yesmaximal, characteristic, complemented
9#21446313C212yesmaximal, complemented
10#21445923C211yesmaximal, characteristic, complemented
11C61×#3510C211yesmaximal, characteristic, complemented
12#16840598C2211yescharacteristic
13C2×#14519808C2212yes—
14#16840529C2211yescharacteristic
15#16840329C2212yes—
16#16839948—24no—
17#16839944—22no—
18#16839939—22no—
19#16839780—22no—
20#16839780C2212yes—
21#16839780C2212yes—
22C2×#14519705—22no—
23C2×#14519705C2212yes—
24#16839771C2212yescomplemented
25#16839746C2211yescharacteristic
26C2×#14519694C2212yescomplemented
27#16839736C2211yescharacteristic
28#16839735C2212yes—
29#16839532C2212yescomplemented
30#16839522C2212yescomplemented
31#16839518C2212yescomplemented
32#16839518C2212yescomplemented
33C61×#559—22no—
34C61×#556—22no—
35C61×#551—24no—
36C61×#542—22no—
37C61×#536C2211yescharacteristic
38C61×#535C2211yescharacteristic
39C61×#533C2211yescharacteristic, complemented
40C61×#529—22no—
41C2×#124×C61C2211yescharacteristic, complemented
42C61×#519C2211yescharacteristic, complemented
43C61×#518C2211yescharacteristic, complemented
44C4×#41×C61C2211yescharacteristic
45#14519825—21no—
46#14519824—22no—
47#14519823—22no—
48#14519822—21no—
49#14519822—22no—
50C2×#11758222—22no—
51C2×#11758222C2312yes—
52#41×D61—42no—
53#14519808—24no—
54#14519808—42no—
55C2×#11758202—22no—
56C2×#11758201—22no—
57#14519784—42no—
58C2×#11758194—22no—
59C2×#11758194C2312yes—
60C2×#11758194C2312yes—
61#14519773—22no—
62#14519764—22no—
63#14519748—42no—
64#14519746—24no—
65#14519736C2312yes—
66#14519705—24no—
67#14519705—22no—
68#14519705—24no—
69#14519705—22no—
70#14519701—42no—
71#14519696—21no—
72#14519694—21no—
73#14519694—42no—
74#14519692—22no—
75#14519691—24no—
76C2×#41×C61C2311yescharacteristic
77C22×Q8×C61C2311yescharacteristic
78C22×D4×C61C2311yescharacteristic
79C4⋊Q8×C61—21no—
80C4⋊Q8×C61D412yescomplemented
81#127×C61—21no—
82#126×C61—24no—
83#126×C61—24no—
84#125×C61—21no—
85#124×C61—24no—
86#124×C61—21no—
87#124×C61—22no—
88#124×C61—21no—
89#124×C61—21no—
90#124×C61D412yescomplemented
91#123×C61—24no—
92#123×C61—24no—
93#123×C61—22no—
94C22⋊Q8×C61—24no—
95C22⋊Q8×C61—22no—
96C22⋊Q8×C61—22no—
97#121×C61—22no—
98#120×C61D414yes—
99C4×Q8×C61—21no—
100C4×Q8×C61—22no—
101C4×Q8×C61—22no—
102C4×Q8×C61—22no—
103C4×D4×C61—24no—
104C4×D4×C61—22no—
105C4×D4×C61—21no—
106C4×D4×C61—22no—
107C4×D4×C61—22no—
108#117×C61—24no—
109#117×C61—22no—
110#117×C61D412yescomplemented
111#117×C61C2311yescharacteristic
112#117×C61C2311yescharacteristic
113#117×C61C2311yescharacteristic
114C2×C22⋊C4×C61—22no—
115C2×C22⋊C4×C61—22no—
116C2×C42×C61D412yescomplemented
117C2×C42×C61C2311yescharacteristic
118C24×C61C2×D412yes—
119#41×C61—22no—
120#41×C61—22no—
121C2×Q8×C61—22no—
122C2×Q8×C61—21no—
123C2×Q8×C61—22no—
124C2×Q8×C61—22no—
125C2×Q8×C61C2×D412yes—
126C2×Q8×C61C2×D411yescharacteristic
127C2×D4×C61—24no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—22no—
131C2×D4×C61—22no—
132C2×D4×C61C2×D412yes—
133C2×D4×C61C2×D411yescharacteristic
134C22×C4×C61—21no—
135C22×C4×C61—24no—
136C22×C4×C61—22no—
137C22×C4×C61—21no—
138C22×C4×C61C2×D412yes—
139C22×C4×C61C2411yescharacteristic
140C2×C22⋊D61—22no—
141C2×C22⋊D61—42no—
142C2×C22⋊D61—44no—
143#11758226—42no—
144Q8×D61—42no—
145#11758224—42no—
146#11758224—44no—
147D4×D61—42no—
148D4×D61—44no—
149#11758222—24no—
150#11758222—22no—
151#11758222—22no—
152#11758222—22no—
153#11758222—22no—
154#11758222—42no—
155#11758222—42no—
156C2×D244—22no—
157C2×D244#4112yes—
158C2×C4×D61—22no—
159C2×C4×D61—42no—
160C2×Dic122—22no—
161C2×Dic122#4112yes—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—24no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—24no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C22⋊C4×C61—24no—
172C22⋊C4×C61—24no—
173C22⋊C4×C61—24no—
174C22⋊C4×C61—22no—
175C22⋊C4×C61—24no—
176C22⋊C4×C61—21no—
177C22⋊C4×C61—21no—
178C22⋊C4×C61—24no—
179C22⋊C4×C61C2×D412yes—
180C42×C61—24no—
181C42×C61—22no—
182C42×C61—21no—
183C42×C61—21no—
184C42×C61C2×D411yescharacteristic
185C42×C61C2×D412yes—
186C42×C61C2×D411yescharacteristic
187C42×C61C2×D412yes—
188C22⋊Dic61—22no—
189C22⋊Dic61—44no—
190#11758202—44no—
191#11758201—44no—
192C4⋊Dic61—22no—
193C4×Dic61—22no—
194#11758194—22no—
195#11758194—22no—
196#11758194—22no—
197#11758194—22no—
198#11758194—42no—
199C2×#92418—22no—
200C2×#92418—22no—
201C2×#92418—42no—
202C23×C61—22no—
203C23×C61—24no—
204C23×C61—82no—
205C23×C61#12012yes—
206C23×C61C22×D411yescharacteristic
207C22×D61—22no—
208C22×D61—42no—
209Q8×C61—22no—
210Q8×C61—24no—
211Q8×C61—41no—
212Q8×C61—41no—
213Q8×C61—82no—
214D4×C61—24no—
215D4×C61—22no—
216D4×C61—41no—
217D4×C61—41no—
218D4×C61—42no—
219D4×C61—84no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—21no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—21no—
227C2×C4×C61—22no—
228C2×C4×C61—42no—
229C2×C4×C61—82no—
230C2×C4×C61—84no—
231C2×C4×C61—81no—
232C2×C4×C61#12012yes—
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C2×#4112yes—
236C2×C4×C61H(5,𝔽2)12yes—
237C22⋊D61—44no—
238C22⋊D61—42no—
239C22⋊D61—42no—
240C22⋊D61—164no—
241C2×Dic61—22no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244D244—22no—
245D244—22no—
246D244—42no—
247C4×D61—44no—
248C4×D61—42no—
249C4×D61—42no—
250Dic122—22no—
251Dic122—22no—
252Dic122—42no—
253#92418—42no—
254#92418—42no—
255#23991—611nomaximal, Sylow
256C22×C61—22no—
257C22×C61—41no—
258C22×C61—42no—
259C22×C61—44no—
260C22×C61—42no—
261C22×C61—84no—
262C22×C61C2×#12011yescharacteristic
263C22×C61#54612yes—
264C2×D61—42no—
265C2×D61—82no—
266C2×D61—162no—
267C4×C61—22no—
268C4×C61—42no—
269C4×C61—41no—
270C4×C61—42no—
271C4×C61—41no—
272C4×C61—42no—
273C4×C61—42no—
274C4×C61#52511yescharacteristic
275C4×C61#53511yescharacteristic
276C4×C61#54612yes—
277Dic61—42no—
278Dic61—82no—
279#3510D6111yescharacteristic, complemented
280#3027—611no—
281#3022—611no—
282#2901—611no—
283#2031—612no—
284#2004—612no—
285#1914—611no—
286#1911—612no—
287#1909—611no—
288#1905—612no—
289#1827—611no—
290C2×C61—22no—
291C2×C61—21no—
292C2×C61—42no—
293C2×C61—82no—
294C2×C61#240411yescharacteristic
295D61—82no—
296#585—611no—
297C2×#137—612no—
298#568—611no—
299#559—22no—
300#556—22no—
301#551—24no—
302#542—22no—
303#536C2×D6111yescharacteristic
304#535—612no—
305#535C2×D6111yescharacteristic
306#533C2×D6111yescharacteristic, complemented
307#529—22no—
308C2×#124C2×D6111yescharacteristic, complemented
309#519C2×D6111yescharacteristic, complemented
310#518C2×D6111yescharacteristic, complemented
311C4×#41C2×D6111yescharacteristic
312B(4,𝔽2)—1224no—
313#456—1222no—
314#455—1222no—
315#421—612no—
316#421—612no—
317#421—1222no—
318C2×#104—612no—
319C2×#104—1222no—
320#419—612no—
321#413—611no—
322C2×#100—612no—
323#410—611no—
324#410—612no—
325#344—612no—
326#339—612no—
327H(3,ℤ/4)—612no—
328H(3,ℤ/4)—612no—
329C61#2399111yescharacteristic, complemented, Sylow
3302−1+4—1222no—
3312−1+4—1221no—
332H(5,𝔽2)—1222no—
333H(5,𝔽2)—1222no—
334H(5,𝔽2)—1221no—
335C2×#41—612no—
336C2×#41—1222no—
337C2×#41—2442no—
338C2×#41C22×D6111yescharacteristic
339C22×Q8C22×D6111yescharacteristic
340C22×D4C22×D6111yescharacteristic
341#137—1224no—
342#137—2442no—
343C2×Q16—1222no—
344C2×SD16—1222no—
345#131—2442no—
346C2×M16—612no—
347C2×M16—612no—
348C2×M16—1222no—
349C4⋊Q8—21no—
350C4⋊Q8C22⋊D6112yescomplemented
351#127—21no—
352#127—1222no—
353#126—24no—
354#126—24no—
355#125—21no—
356#124—24no—
357#124—22no—
358#124—21no—
359#124—21no—
360#124—21no—
361#124—1222no—
362#124C22⋊D6112yescomplemented
363#123—24no—
364#123—24no—
365#123—22no—
366C22⋊Q8—24no—
367C22⋊Q8—22no—
368C22⋊Q8—22no—
369#121—22no—
370#121—2442no—
371#120—1224no—
372#120C22⋊D6114yes—
373C4×Q8—21no—
374C4×Q8—22no—
375C4×Q8—22no—
376C4×Q8—22no—
377C4×D4—24no—
378C4×D4—22no—
379C4×D4—21no—
380C4×D4—22no—
381C4×D4—22no—
382#117—24no—
383#117—22no—
384#117—612no—
385#117C22⋊D6112yescomplemented
386#117C22×D6111yescharacteristic
387#117C22×D6111yescharacteristic
388#117C22×D6111yescharacteristic
389C2×C22⋊C4—22no—
390C2×C22⋊C4—22no—
391C2×C42C22⋊D6112yescomplemented
392C2×C42C22×D6111yescharacteristic
393#104—1222no—
394#104—1222no—
395#104—1224no—
396#104—1224no—
397#103—2442no—
398#101—1221no—
399#100—1221no—
400#100—2442no—
401#99—1222no—
402#99—1224no—
403C24C2×C22⋊D6112yes—
404#41—22no—
405#41—22no—
406#41—1224no—
407#41—1222no—
408#41—1222no—
409#41—1222no—
410#41—1222no—
411#41—2442no—
412#41—2444no—
413#41—2442no—
414#41—2442no—
415#41—2442no—
416C2×Q8—22no—
417C2×Q8—21no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—612no—
421C2×Q8—1222no—
422C2×Q8—2442no—
423C2×Q8C2×C22⋊D6112yes—
424C2×Q8C2×C22⋊D6111yescharacteristic
425C2×D4—24no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—612no—
431C2×D4—1222no—
432C2×D4—1222no—
433C2×D4—2442no—
434C2×D4—2444no—
435C2×D4—2444no—
436C2×D4—2442no—
437C2×D4C2×C22⋊D6111yescharacteristic
438C2×D4C2×C22⋊D6112yes—
439C22×C4—21no—
440C22×C4—24no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—1222no—
444C22×C4—2442no—
445C22×C4C2×C22⋊D6112yes—
446C22×C4C23×D6111yescharacteristic
447Q16—2444no—
448SD16—2444no—
449M16—1222no—
450M16—1222no—
451M16—1222no—
452M16—1222no—
453M16—2442no—
454C2×C8—1222no—
455C2×C8—1222no—
456C2×C8—2442no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—1222no—
467C22⋊C4—24no—
468C22⋊C4—24no—
469C22⋊C4—24no—
470C22⋊C4—21no—
471C22⋊C4—21no—
472C22⋊C4—22no—
473C22⋊C4—24no—
474C22⋊C4—24no—
475C22⋊C4—1222no—
476C22⋊C4—2444no—
477C22⋊C4C2×C22⋊D6112yes—
478C42—24no—
479C42—21no—
480C42—21no—
481C42—22no—
482C42—1222no—
483C42C2×C22⋊D6111yescharacteristic
484C42C2×C22⋊D6112yes—
485C42C2×C22⋊D6111yescharacteristic
486C42C2×C22⋊D6112yes—
487C23—22no—
488C23—24no—
489C23—82no—
490C23—1222no—
491C23—2442no—
492C23#1451974612yes—
493C23C22×C22⋊D6111yescharacteristic
494Q8—22no—
495Q8—24no—
496Q8—41no—
497Q8—41no—
498Q8—82no—
499Q8—1222no—
500Q8—1222no—
501Q8—2442no—
502D4—24no—
503D4—22no—
504D4—41no—
505D4—41no—
506D4—42no—
507D4—84no—
508D4—1222no—
509D4—1222no—
510D4—9764no—
511D4—2442no—
512D4—2444no—
513D4—2442no—
514D4—2442no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—42no—
524C2×C4—82no—
525C2×C4—84no—
526C2×C4—81no—
527C2×C4—1222no—
528C2×C4—2444no—
529C2×C4—2442no—
530C2×C4—2442no—
531C2×C4—2442no—
532C2×C4—2442no—
533C2×C4#1451974612yes—
534C2×C4C22×C22⋊D6111yescharacteristic
535C2×C4C22×C22⋊D6111yescharacteristic
536C2×C4#41×D6112yes—
537C2×C4#1451982312yes—
538C8—2442no—
539C8—2442no—
540C22—22no—
541C22—41no—
542C22—42no—
543C22—44no—
544C22—42no—
545C22—84no—
546C22—9762no—
547C22—4882no—
548C22—2442no—
549C22C2×#1451974611yescharacteristic
550C22#1684040012yes—
551C4—22no—
552C4—42no—
553C4—41no—
554C4—42no—
555C4—41no—
556C4—42no—
557C4—42no—
558C4—4882no—
559C4—2442no—
560C4#1684026911yescharacteristic
561C4#1684032611yescharacteristic
562C4#1684040012yes—
563C2—21no—
564C2—22no—
565C2—42no—
566C2—82no—
567C2—4882no—
568C2#2144913411yescharacteristic
subgroupG/H
centerC2#21449134
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3510C2
socleC2×C61#2404
prime radical#21455220C2
hypercenter#3510D61
hypocenterC61#23991
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #3510

Complements

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

NH ≅ G/Ntype
2. #21455197C2semidirect
3. #21454367C2semidirect
4. #21446988C2semidirect
5. #21446848C2semidirect
6. #21446367C2semidirect
7. #21446351C2semidirect
8. #21446342C2semidirect
9. #21446313C2semidirect
10. #21445923C2semidirect
11. C61×#3510C2semidirect
24. #16839771C22semidirect
26. C2×#14519694C22semidirect
29. #16839532C22semidirect
30. #16839522C22semidirect
31. #16839518C22semidirect
32. #16839518C22semidirect
39. C61×#533C22semidirect
41. C2×#124×C61C22semidirect
42. C61×#519C22semidirect
43. C61×#518C22semidirect
80. C4⋊Q8×C61D4semidirect
90. #124×C61D4semidirect
110. #117×C61D4semidirect
116. C2×C42×C61D4semidirect
279. #3510D61semidirect
306. #533C2×D61semidirect
308. C2×#124C2×D61semidirect
309. #519C2×D61semidirect
310. #518C2×D61semidirect
329. C61#23991semidirect
350. C4⋊Q8C22⋊D61semidirect
362. #124C22⋊D61semidirect
385. #117C22⋊D61semidirect
391. C2×C42C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214552202
#214551972
#214543672
#214469882
#214468482
#214463672
#214463512
#214463422
#214463132
#214459232
C61×#35102
#2399161