Finite Group Databasebeta

← #29661567

Subgroups — #29661567

Order 15616.

888 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456834C211yesmaximal, characteristic
2#21456195C211yesmaximal, characteristic, complemented
3C2×#16840128C211yesmaximal, characteristic, complemented
4#21445121C214yesmaximal, complemented
5#21445117C214yesmaximal, complemented
6#21444434C211yesmaximal, characteristic, complemented
7#21444421C211yesmaximal, characteristic, complemented
8#21444407C211yesmaximal, characteristic, complemented
9C61×#3563C211yesmaximal, characteristic, complemented
10C2×#14519774C2211yescharacteristic
11#16840152C2212yes—
12#16840128—22no—
13#16840128C2212yes—
14#16840128C2214yes—
15#16840128C2214yes—
16#16840110—21no—
17#16840110—21no—
18#16839962—21no—
19#16839957—21no—
20#16839912—21no—
21#16839895—21no—
22#16839819C2211yescharacteristic
23#16839766—22no—
24C2×#14519697—21no—
25C2×#14519697—21no—
26C4×#11758194C2211yescharacteristic
27C2×#14519688C2211yescharacteristic
28#16839504C2212yescomplemented
29#16839504C2214yescomplemented
30#16839504C2214yescomplemented
31#16839502C2212yescomplemented
32C61×#564C2212yescomplemented
33C61×#563C2212yescomplemented
34C61×#557—21no—
35C61×#554—21no—
36C61×#552—21no—
37C61×#552—22no—
38C61×#552—21no—
39C61×#544—21no—
40C61×#544—21no—
41C61×#541—21no—
42C61×#541—22no—
43C61×#539—21no—
44C61×#533C2211yescharacteristic
45C61×#529C2211yescharacteristic
46C2×#125×C61C2211yescharacteristic
47C22×D244—21no—
48C22×D244—21no—
49C22×D244—21no—
50C2×#11758201—22no—
51C2×#11758199—22no—
52C2×#11758194—21no—
53C22×#92418—21no—
54#14519774—21no—
55#14519774—21no—
56#14519774—21no—
57#14519774C2312yes—
58C4⋊#92418—21no—
59#14519697—22no—
60#14519697—22no—
61#14519697—24no—
62#14519697—24no—
63#14519697—24no—
64#14519697—24no—
65C22⋊#92418—21no—
66#14519688—21no—
67#14519688C2312yes—
68#14519688C2312yes—
69#14519688C2312yes—
70C4×#92418—21no—
71C4⋊Q8×C61—22no—
72C4⋊Q8×C61—21no—
73C4⋊Q8×C61—22no—
74C4⋊Q8×C61D411yescharacteristic
75C4⋊Q8×C61D411yescharacteristic
76#126×C61—22no—
77#126×C61—22no—
78#126×C61—22no—
79#126×C61—22no—
80#126×C61—21no—
81#125×C61—22no—
82#125×C61—22no—
83#125×C61—21no—
84#125×C61—22no—
85#125×C61—22no—
86#125×C61—22no—
87#125×C61—22no—
88#125×C61—22no—
89#125×C61—22no—
90#125×C61—21no—
91#125×C61—21no—
92#125×C61D412yes—
93#125×C61C2312yes—
94#125×C61C2312yes—
95#125×C61C2311yescharacteristic
96#124×C61C2311yescharacteristic
97#123×C61—22no—
98#123×C61—22no—
99#123×C61—21no—
100C22⋊Q8×C61—22no—
101C22⋊Q8×C61—21no—
102C22⋊Q8×C61—21no—
103C22⋊Q8×C61—22no—
104C22⋊Q8×C61—21no—
105C22⋊Q8×C61—21no—
106C22⋊Q8×C61—22no—
107C22⋊Q8×C61—22no—
108C22⋊Q8×C61D411yescharacteristic
109#121×C61D411yescharacteristic
110C4×Q8×C61—21no—
111C4×Q8×C61—22no—
112C4×Q8×C61—21no—
113C4×Q8×C61—21no—
114C4×Q8×C61—21no—
115C4×Q8×C61D411yescharacteristic
116C4×D4×C61—22no—
117C4×D4×C61—21no—
118C4×D4×C61D411yescharacteristic
119#117×C61—22no—
120#117×C61—21no—
121#117×C61—22no—
122#117×C61—21no—
123#117×C61D412yes—
124C2×C4⋊C4×C61—21no—
125C2×C4⋊C4×C61—21no—
126C2×C4⋊C4×C61—21no—
127C2×C4⋊C4×C61D411yescharacteristic
128C2×C4⋊C4×C61D411yescharacteristic
129C2×C42×C61C2311yescharacteristic
130C23×D61—41no—
131C2×Q8×C61—22no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61C2×D411yescharacteristic
135C2×D4×C61—21no—
136C2×D4×C61C2×D411yescharacteristic
137C22×C4×C61—21no—
138C22×C4×C61—22no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C2×D244—22no—
145C2×D244—22no—
146C2×D244—22no—
147C2×D244—44no—
148C2×D244—44no—
149C2×D244—41no—
150C2×D244—41no—
151C2×D244—41no—
152C2×D244—82no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61C2×D412yes—
176C4⋊C4×C61C2×D412yes—
177C4⋊C4×C61C2×D412yes—
178C4⋊C4×C61C2×D412yes—
179C4⋊C4×C61C2×D411yescharacteristic
180C4⋊C4×C61C2×D411yescharacteristic
181C22⋊C4×C61—22no—
182C22⋊C4×C61—22no—
183C22⋊C4×C61—22no—
184C22⋊C4×C61—21no—
185C22⋊C4×C61—22no—
186C22⋊C4×C61—21no—
187C22⋊C4×C61—21no—
188C22⋊C4×C61—21no—
189C22⋊C4×C61—21no—
190C42×C61—22no—
191C42×C61—22no—
192C42×C61—22no—
193C42×C61—22no—
194C42×C61—21no—
195C42×C61C2×D411yescharacteristic
196C42×C61C2×D411yescharacteristic
197C42×C61C2×D411yescharacteristic
198C42×C61C2411yescharacteristic
199#11758201—82no—
200#11758199—82no—
201#11758194—42no—
202C2×#92418—22no—
203C2×#92418—22no—
204C2×#92418—22no—
205C2×#92418—42no—
206C23×C61—21no—
207C23×C61#12011yescharacteristic
208C22×D61—42no—
209C22×D61—82no—
210C22×D61—82no—
211C22×D61—82no—
212Q8×C61—41no—
213Q8×C61—42no—
214Q8×C61—81no—
215Q8×C61—82no—
216D4×C61—41no—
217D4×C61—42no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—41no—
228C2×C4×C61—42no—
229C2×C4×C61—42no—
230C2×C4×C61—81no—
231C2×C4×C61—82no—
232C2×C4×C61#12011yescharacteristic
233C2×C4×C61#12011yescharacteristic
234C2×C4×C61#12011yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61C22×D411yescharacteristic
238D244—42no—
239D244—42no—
240D244—44no—
241D244—44no—
242D244—82no—
243D244—82no—
244D244—82no—
245#92418—42no—
246#92418—42no—
247#92418—42no—
248#19028—611nomaximal, Sylow
249C22×C61—21no—
250C22×C61—21no—
251C22×C61—41no—
252C22×C61—41no—
253C22×C61—41no—
254C22×C61#45412yes—
255C22×C61C2×#12011yescharacteristic
256C2×D61—82no—
257C2×D61—82no—
258C2×D61—82no—
259C2×D61—164no—
260C4×C61—21no—
261C4×C61—21no—
262C4×C61—21no—
263C4×C61—21no—
264C4×C61—21no—
265C4×C61—42no—
266C4×C61—41no—
267C4×C61—41no—
268C4×C61—42no—
269C4×C61—42no—
270C4×C61—42no—
271C4×C61—41no—
272C4×C61#45412yes—
273#3563D6111yescharacteristic, complemented
274#3217—611no—
275#3138—611no—
276C2×#490—611no—
277#1658—614no—
278#1657—614no—
279#1531—611no—
280#1528—611no—
281#1524—611no—
282C2×C61—22no—
283C2×C61—81no—
284C2×C61C2×#45411yescharacteristic
285C2×C61C2×#45411yescharacteristic
286C2×C61#302111yescharacteristic
287D61—162no—
288#564C2×D6112yescomplemented
289#563C2×D6112yescomplemented
290#557—21no—
291#554—21no—
292#552—21no—
293#552—22no—
294#552—21no—
295#544—21no—
296#544—21no—
297#541—21no—
298#541—22no—
299#539—21no—
300#533C2×D6111yescharacteristic
301C2×#127—611no—
302#529C2×D6111yescharacteristic
303C2×#125C2×D6111yescharacteristic
304#496—612no—
305#490—614no—
306#490—614no—
307#490—612no—
308#490—1222no—
309#486—1221no—
310#486—1221no—
311#460—1221no—
312#459—1221no—
313#450—1221no—
314#447—1221no—
315#432—611no—
316#418—1222no—
317C2×#102—1221no—
318C2×#102—1221no—
319C4×M16—611no—
320C2×#97—611no—
321#330—614no—
322#330—614no—
323#330—612no—
324#329—612no—
325C61#1902811yescharacteristic, complemented, Sylow
326C22×D4—1221no—
327C22×D4—1221no—
328C22×D4—1221no—
329C2×SD16—1222no—
330C2×D8—1222no—
331C2×M16—1221no—
332C22×C8—1221no—
333C4⋊Q8—22no—
334C4⋊Q8—21no—
335C4⋊Q8—22no—
336C4⋊Q8C22⋊D6111yescharacteristic
337C4⋊Q8C22⋊D6111yescharacteristic
338#127—612no—
339#127—1221no—
340#127—1221no—
341#127—1221no—
342#126—22no—
343#126—22no—
344#126—22no—
345#126—22no—
346#126—21no—
347#125—22no—
348#125—22no—
349#125—21no—
350#125—22no—
351#125—22no—
352#125—22no—
353#125—22no—
354#125—22no—
355#125—22no—
356#125—21no—
357#125—21no—
358#125C22⋊D6112yes—
359#125C22×D6112yes—
360#125C22×D6111yescharacteristic
361#125C22×D6112yes—
362#124C22×D6111yescharacteristic
363#123—22no—
364#123—22no—
365#123—21no—
366C22⋊Q8—22no—
367C22⋊Q8—21no—
368C22⋊Q8—21no—
369C22⋊Q8—22no—
370C22⋊Q8—21no—
371C22⋊Q8—21no—
372C22⋊Q8—22no—
373C22⋊Q8—22no—
374C22⋊Q8C22⋊D6111yescharacteristic
375#121C22⋊D6111yescharacteristic
376C4×Q8—21no—
377C4×Q8—22no—
378C4×Q8—21no—
379C4×Q8—21no—
380C4×Q8—21no—
381C4×Q8C22⋊D6111yescharacteristic
382C4×D4—22no—
383C4×D4—21no—
384C4×D4C22⋊D6111yescharacteristic
385#117—22no—
386#117—21no—
387#117—22no—
388#117—21no—
389#117C22⋊D6112yes—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4C22⋊D6111yescharacteristic
394C2×C4⋊C4C22⋊D6111yescharacteristic
395C2×C42C22×D6111yescharacteristic
396C4⋊C8—1221no—
397#102—1224no—
398#102—1224no—
399#102—1224no—
400#102—1224no—
401#102—1222no—
402#102—1222no—
403C22⋊C8—1221no—
404#97—612no—
405#97—612no—
406#97—612no—
407#97—1221no—
408C4×C8—1221no—
409C24—2441no—
410C2×Q8—22no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8C2×C22⋊D6111yescharacteristic
414C2×D4—21no—
415C2×D4—1222no—
416C2×D4—1222no—
417C2×D4—1222no—
418C2×D4—4882no—
419C2×D4—2444no—
420C2×D4—2444no—
421C2×D4—2441no—
422C2×D4—2441no—
423C2×D4—2441no—
424C2×D4C2×C22⋊D6111yescharacteristic
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4C2×C22⋊D6111yescharacteristic
430C22×C4C2×C22⋊D6111yescharacteristic
431C22×C4C2×C22⋊D6111yescharacteristic
432SD16—4882no—
433D8—4882no—
434M16—2442no—
435C2×C8—1222no—
436C2×C8—1222no—
437C2×C8—1222no—
438C2×C8—2442no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4C2×C22⋊D6112yes—
462C4⋊C4C2×C22⋊D6112yes—
463C4⋊C4C2×C22⋊D6112yes—
464C4⋊C4C2×C22⋊D6112yes—
465C4⋊C4C2×C22⋊D6111yescharacteristic
466C4⋊C4C2×C22⋊D6111yescharacteristic
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—21no—
471C22⋊C4—22no—
472C22⋊C4—21no—
473C22⋊C4—21no—
474C22⋊C4—21no—
475C22⋊C4—21no—
476C42—22no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—21no—
481C42C2×C22⋊D6111yescharacteristic
482C42C2×C22⋊D6111yescharacteristic
483C42C2×C22⋊D6111yescharacteristic
484C42C23×D6111yescharacteristic
485C23—21no—
486C23—4882no—
487C23—4882no—
488C23—4882no—
489C23—2442no—
490C23#1451974611yescharacteristic
491Q8—41no—
492Q8—42no—
493Q8—81no—
494Q8—82no—
495D4—41no—
496D4—42no—
497D4—4882no—
498D4—4882no—
499D4—4882no—
500D4—2442no—
501D4—2442no—
502D4—2444no—
503D4—2444no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—81no—
517C2×C4—82no—
518C2×C4#1451974611yescharacteristic
519C2×C4#1451974611yescharacteristic
520C2×C4#1451974611yescharacteristic
521C2×C4C22×C22⋊D6111yescharacteristic
522C2×C4C22×C22⋊D6111yescharacteristic
523C2×C4C22×C22⋊D6111yescharacteristic
524C8—2442no—
525C8—2442no—
526C8—2442no—
527C22—21no—
528C22—21no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22—9764no—
533C22—4882no—
534C22—4882no—
535C22—4882no—
536C22#1683993512yes—
537C22C2×#1451974611yescharacteristic
538C4—21no—
539C4—21no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—42no—
544C4—41no—
545C4—41no—
546C4—42no—
547C4—42no—
548C4—42no—
549C4—41no—
550C4#1683993512yes—
551C2—22no—
552C2—81no—
553C2—9762no—
554C2C2×#1683993511yescharacteristic
555C2C2×#1683993511yescharacteristic
556C2#2145519111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3563C2
socleC22×C61C2×#120
prime radical#21456834C2
hypercenter#3563D61
hypocenterC61#19028
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3563

Complements

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

NH ≅ G/Ntype
2. #21456195C2semidirect
3. C2×#16840128C2semidirect
4. #21445121C2semidirect
5. #21445117C2semidirect
6. #21444434C2semidirect
7. #21444421C2semidirect
8. #21444407C2semidirect
9. C61×#3563C2semidirect
28. #16839504C22semidirect
29. #16839504C22semidirect
30. #16839504C22semidirect
31. #16839502C22semidirect
32. C61×#564C22semidirect
33. C61×#563C22semidirect
273. #3563D61semidirect
288. #564C2×D61semidirect
289. #563C2×D61semidirect
325. C61#19028semidirect

Maximal subgroups

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

subgroupindex
#214568342
#214561952
C2×#168401282
#214451212
#214451172
#214444342
#214444212
#214444072
C61×#35632
#1902861