Finite Group Databasebeta

← #29657173

Subgroups — #29657173

Order 15616.

1152 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#21456757C211yesmaximal, characteristic, complemented
2C2×#16840138C211yesmaximal, characteristic, complemented
3#21455299C211yesmaximal, characteristic, complemented
4#21444884C214yesmaximal, complemented
5#21444878C214yesmaximal, complemented
6#21444487C211yesmaximal, characteristic, complemented
7#21444274C211yesmaximal, characteristic, complemented
8#21444201C211yesmaximal, characteristic, complemented
9C61×#3508C211yesmaximal, characteristic, complemented
10C2×#14519802—21no—
11C22×#11758199—22no—
12C2×#14519774C2211yescharacteristic
13#16840138—22no—
14#16840138C2214yes—
15#16840136—21no—
16#16840110—21no—
17#16839962—21no—
18#16839962C2212yescomplemented
19#16839962C2214yescomplemented
20#16839957—21no—
21#16839957C2212yescomplemented
22#16839912—22no—
23#16839815C2211yescharacteristic
24#16839786C2211yescharacteristic
25#16839766—21no—
26C2×C4×#92418C2211yescharacteristic
27#16839506C2214yescomplemented
28Q8⋊#92418C2212yescomplemented
29Q8⋊#92418C2214yescomplemented
30D4⋊#92418C2212yescomplemented
31C61×#559—21no—
32C61×#559C2212yescomplemented
33C61×#550—21no—
34C61×#550C2212yescomplemented
35C61×#546—22no—
36C61×#546—22no—
37C61×#540—22no—
38C61×#535—21no—
39C61×#533C2211yescharacteristic
40C61×#532—21no—
41C61×#532—21no—
42C2×#127×C61C2211yescharacteristic
43C61×#518—21no—
44C4×#41×C61C2211yescharacteristic
45C22×D244—21no—
46C22×D244—21no—
47C22×D244—21no—
48#14519802—44no—
49C2×#11758201—22no—
50C2×#11758201—24no—
51C2×#11758199—22no—
52C2×#11758199—28no—
53C2×#11758199—48no—
54C2×#11758194—21no—
55C22×#92418—21no—
56#14519774—21no—
57#14519774—21no—
58#14519774—21no—
59#14519774C2312yes—
60C4⋊#92418—21no—
61C4⋊#92418C2312yes—
62C4⋊#92418C2312yes—
63#14519697—22no—
64#14519697—22no—
65#14519697—24no—
66C22⋊#92418—21no—
67C4×#92418—21no—
68C4×#92418—21no—
69C4×#92418C2312yes—
70C2×#41×C61—22no—
71C2×#41×C61—22no—
72C2×#41×C61D411yescharacteristic
73C22×D4×C61—21no—
74C22×D4×C61—21no—
75C22×D4×C61D412yes—
76C4⋊Q8×C61D411yescharacteristic
77C4⋊Q8×C61D411yescharacteristic
78#127×C61—22no—
79#127×C61—21no—
80#127×C61—21no—
81#127×C61D412yes—
82#127×C61C2312yes—
83#125×C61—21no—
84#124×C61—22no—
85#124×C61—22no—
86#124×C61—22no—
87#123×C61—24no—
88#123×C61—24no—
89C22⋊Q8×C61—21no—
90C22⋊Q8×C61D412yes—
91#121×C61—22no—
92#121×C61—22no—
93#121×C61—22no—
94#121×C61—24no—
95#121×C61—24no—
96#121×C61—22no—
97#120×C61—24no—
98C4×Q8×C61—21no—
99C4×Q8×C61D411yescharacteristic
100C4×Q8×C61C2312yes—
101C4×Q8×C61C2311yescharacteristic
102C4×D4×C61—22no—
103C4×D4×C61—22no—
104C4×D4×C61—22no—
105C4×D4×C61—21no—
106C4×D4×C61—21no—
107C4×D4×C61—22no—
108C4×D4×C61—22no—
109C4×D4×C61—22no—
110C4×D4×C61D411yescharacteristic
111C4×D4×C61C2311yescharacteristic
112#117×C61—21no—
113#117×C61—21no—
114#117×C61D411yescharacteristic
115C2×C22⋊C4×C61—24no—
116C2×C42×C61—21no—
117C2×C42×C61C2311yescharacteristic
118C24×C61—22no—
119C23×D61—41no—
120#41×C61—22no—
121#41×C61—22no—
122#41×C61—84no—
123#41×C61—84no—
124C2×Q8×C61—21no—
125C2×Q8×C61C2×D411yescharacteristic
126C2×Q8×C61C2×D412yes—
127C2×D4×C61—21no—
128C2×D4×C61—24no—
129C2×D4×C61—24no—
130C2×D4×C61—22no—
131C2×D4×C61—22no—
132C2×D4×C61—22no—
133C2×D4×C61—22no—
134C2×D4×C61—22no—
135C2×D4×C61—44no—
136C2×D4×C61—42no—
137C2×D4×C61—42no—
138C2×D4×C61—82no—
139C2×D4×C61C2×D411yescharacteristic
140C2×D4×C61C2×D414yes—
141C22×C4×C61—24no—
142C22×C4×C61—24no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C2×D244—22no—
150C2×D244—22no—
151C2×D244—22no—
152C2×D244—44no—
153C2×D244—42no—
154C2×D244—42no—
155C2×D244—41no—
156C2×D244—41no—
157C2×D244—41no—
158C2×D244—82no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61C2×D411yescharacteristic
167C4⋊C4×C61C2×D411yescharacteristic
168C4⋊C4×C61C2×D412yes—
169C22⋊C4×C61—24no—
170C22⋊C4×C61—24no—
171C22⋊C4×C61—24no—
172C22⋊C4×C61—24no—
173C22⋊C4×C61—21no—
174C22⋊C4×C61—22no—
175C22⋊C4×C61—21no—
176C22⋊C4×C61—21no—
177C42×C61—21no—
178C42×C61—21no—
179C42×C61—21no—
180C42×C61—22no—
181C42×C61C2×D411yescharacteristic
182C42×C61C2×D411yescharacteristic
183C42×C61C2×D411yescharacteristic
184C42×C61C2411yescharacteristic
185#11758201—44no—
186#11758201—84no—
187#11758199—416no—
188#11758199—44no—
189#11758194—42no—
190C2×#92418—24no—
191C2×#92418—22no—
192C2×#92418—22no—
193C2×#92418—22no—
194C23×C61—24no—
195C23×C61—21no—
196C23×C61—44no—
197C23×C61—82no—
198C23×C61—82no—
199C23×C61#12011yescharacteristic
200C22×D61—42no—
201C22×D61—82no—
202C22×D61—82no—
203C22×D61—82no—
204Q8×C61—21no—
205Q8×C61—21no—
206Q8×C61—24no—
207Q8×C61—42no—
208D4×C61—21no—
209D4×C61—21no—
210D4×C61—44no—
211D4×C61—42no—
212D4×C61—42no—
213D4×C61—42no—
214D4×C61—44no—
215D4×C61—44no—
216D4×C61—44no—
217D4×C61—82no—
218D4×C61—84no—
219D4×C61—84no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61—42no—
230C2×C4×C61—84no—
231C2×C4×C61—84no—
232C2×C4×C61#12011yescharacteristic
233C2×C4×C61#12011yescharacteristic
234C2×C4×C61#12011yescharacteristic
235C2×C4×C61#13512yes—
236C2×C4×C61AGL(1,ℤ/8)12yes—
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240D244—44no—
241D244—42no—
242D244—42no—
243D244—44no—
244D244—82no—
245D244—82no—
246D244—82no—
247#92418—24no—
248#92418—42no—
249#92418—42no—
250#18624—611nomaximal, Sylow
251C22×C61—21no—
252C22×C61—21no—
253C22×C61—41no—
254C22×C61—44no—
255C22×C61—41no—
256C22×C61—44no—
257C22×C61—41no—
258C22×C61—44no—
259C22×C61—164no—
260C22×C61#44912yes—
261C22×C61C2×#12011yescharacteristic
262C2×D61—82no—
263C2×D61—82no—
264C2×D61—82no—
265C2×D61—164no—
266C4×C61—21no—
267C4×C61—21no—
268C4×C61—41no—
269C4×C61—41no—
270C4×C61—42no—
271C4×C61—41no—
272C4×C61—42no—
273C4×C61#44912yes—
274C4×C61#45312yes—
275C4×C61C2×#13511yescharacteristic
276C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
277C4×C61#57612yes—
278#3508D6111yescharacteristic, complemented
279#3208—611no—
280C2×#493—611no—
281#3038—611no—
282#1621—614no—
283#1620—614no—
284#1545—611no—
285#1493—611no—
286#1476—611no—
287C2×C61—22no—
288C2×C61—81no—
289C2×C61—84no—
290C2×C61C2×#44911yescharacteristic
291C2×C61#300811yescharacteristic
292C2×C61C2×#45311yescharacteristic
293D61—162no—
294C2×AGL(1,ℤ/8)—1221no—
295C22×D8—1222no—
296#559—21no—
297#559C2×D6112yescomplemented
298#550—21no—
299#550C2×D6112yescomplemented
300#546—22no—
301#546—22no—
302#540—22no—
303#535—21no—
304#533C2×D6111yescharacteristic
305#532—21no—
306#532—21no—
307C2×#127—611no—
308C2×#127C2×D6111yescharacteristic
309#518—21no—
310C4×#41C2×D6111yescharacteristic
311#493—614no—
312#493—1222no—
313#492—1221no—
314#486—1221no—
315#460—612no—
316#460—614no—
317#460—1221no—
318#459—612no—
319#459—1221no—
320#450—1222no—
321#431—611no—
322#423—611no—
323#418—1221no—
324C2×C4×C8—611no—
325#331—614no—
326Q8⋊C8—612no—
327Q8⋊C8—614no—
328D4⋊C8—612no—
329C61#1862411yescharacteristic, complemented, Sylow
330C2×#41—22no—
331C2×#41—22no—
332C2×#41C22⋊D6111yescharacteristic
333C22×D4—21no—
334C22×D4—21no—
335C22×D4—1221no—
336C22×D4—1221no—
337C22×D4—1221no—
338C22×D4C22⋊D6112yes—
339AGL(1,ℤ/8)—2444no—
340C2×SD16—1224no—
341C2×SD16—1222no—
342C2×D8—1228no—
343C2×D8—1222no—
344C2×D8—2448no—
345C2×M16—1221no—
346C22×C8—1221no—
347C4⋊Q8C22⋊D6111yescharacteristic
348C4⋊Q8C22⋊D6111yescharacteristic
349#127—22no—
350#127—21no—
351#127—21no—
352#127—612no—
353#127—1221no—
354#127—1221no—
355#127—1221no—
356#127C22⋊D6112yes—
357#127C22×D6112yes—
358#125—21no—
359#124—22no—
360#124—22no—
361#124—22no—
362#123—24no—
363#123—24no—
364C22⋊Q8—21no—
365C22⋊Q8C22⋊D6112yes—
366#121—22no—
367#121—22no—
368#121—22no—
369#121—24no—
370#121—24no—
371#121—22no—
372#120—24no—
373C4×Q8—21no—
374C4×Q8C22⋊D6111yescharacteristic
375C4×Q8C22×D6111yescharacteristic
376C4×Q8C22×D6112yes—
377C4×D4—22no—
378C4×D4—22no—
379C4×D4—22no—
380C4×D4—21no—
381C4×D4—21no—
382C4×D4—22no—
383C4×D4—22no—
384C4×D4—22no—
385C4×D4C22⋊D6111yescharacteristic
386C4×D4C22×D6111yescharacteristic
387#117—21no—
388#117—21no—
389#117C22⋊D6111yescharacteristic
390C2×C22⋊C4—24no—
391C2×C42—21no—
392C2×C42C22×D6111yescharacteristic
393C4⋊C8—612no—
394C4⋊C8—612no—
395C4⋊C8—1221no—
396#102—1222no—
397#102—1222no—
398#102—1224no—
399C22⋊C8—1221no—
400C4×C8—612no—
401C4×C8—1221no—
402C4×C8—1221no—
403C24—22no—
404C24—2441no—
405#41—22no—
406#41—22no—
407#41—84no—
408#41—84no—
409C2×Q8—21no—
410C2×Q8C2×C22⋊D6111yescharacteristic
411C2×Q8C2×C22⋊D6112yes—
412C2×D4—21no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—44no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—82no—
424C2×D4—1222no—
425C2×D4—1222no—
426C2×D4—1222no—
427C2×D4—4882no—
428C2×D4—2444no—
429C2×D4—2441no—
430C2×D4—2441no—
431C2×D4—2441no—
432C2×D4—2442no—
433C2×D4—2442no—
434C2×D4C2×C22⋊D6111yescharacteristic
435C2×D4C2×C22⋊D6114yes—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4C2×C22⋊D6111yescharacteristic
442C22×C4C2×C22⋊D6111yescharacteristic
443C22×C4C2×C22⋊D6111yescharacteristic
444SD16—4884no—
445SD16—2444no—
446D8—24416no—
447D8—2444no—
448M16—2442no—
449C2×C8—1224no—
450C2×C8—1222no—
451C2×C8—1222no—
452C2×C8—1222no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4C2×C22⋊D6111yescharacteristic
461C4⋊C4C2×C22⋊D6111yescharacteristic
462C4⋊C4C2×C22⋊D6112yes—
463C22⋊C4—24no—
464C22⋊C4—24no—
465C22⋊C4—24no—
466C22⋊C4—24no—
467C22⋊C4—21no—
468C22⋊C4—22no—
469C22⋊C4—21no—
470C22⋊C4—21no—
471C42—21no—
472C42—22no—
473C42—21no—
474C42—21no—
475C42C2×C22⋊D6111yescharacteristic
476C42C2×C22⋊D6111yescharacteristic
477C42C2×C22⋊D6111yescharacteristic
478C42C23×D6111yescharacteristic
479C23—24no—
480C23—21no—
481C23—44no—
482C23—82no—
483C23—82no—
484C23—4882no—
485C23—4882no—
486C23—4882no—
487C23—2442no—
488C23#1451974611yescharacteristic
489Q8—21no—
490Q8—21no—
491Q8—24no—
492Q8—42no—
493D4—21no—
494D4—21no—
495D4—44no—
496D4—42no—
497D4—42no—
498D4—42no—
499D4—44no—
500D4—44no—
501D4—44no—
502D4—82no—
503D4—84no—
504D4—84no—
505D4—4882no—
506D4—4882no—
507D4—4882no—
508D4—2444no—
509D4—2442no—
510D4—2442no—
511D4—2444no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—84no—
523C2×C4—84no—
524C2×C4#1451974611yescharacteristic
525C2×C4#1451974611yescharacteristic
526C2×C4#1451974611yescharacteristic
527C2×C4#1451979712yes—
528C2×C4#1451980212yes—
529C2×C4C22×C22⋊D6111yescharacteristic
530C2×C4C22×C22⋊D6111yescharacteristic
531C2×C4C22×C22⋊D6111yescharacteristic
532C8—1224no—
533C8—2442no—
534C8—2442no—
535C22—21no—
536C22—21no—
537C22—41no—
538C22—44no—
539C22—41no—
540C22—44no—
541C22—41no—
542C22—44no—
543C22—164no—
544C22—9764no—
545C22—4882no—
546C22—4882no—
547C22—4882no—
548C22#1683990712yes—
549C22C2×#1451974611yescharacteristic
550C4—21no—
551C4—21no—
552C4—41no—
553C4—42no—
554C4—41no—
555C4—41no—
556C4—42no—
557C4#1683990712yes—
558C4#1683993012yes—
559C4C2×#1451979711yescharacteristic
560C4C2×#1451980211yescharacteristic
561C4#1684056512yes—
562C2—22no—
563C2—84no—
564C2—81no—
565C2—9762no—
566C2C2×#1683990711yescharacteristic
567C2#2145508711yescharacteristic
568C2C2×#1683993011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3508C2
socleC22×C61C2×#120
hypercenter#3508D61
hypocenterC61#18624
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3508

Complements

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

NH ≅ G/Ntype
1. #21456757C2semidirect
2. C2×#16840138C2semidirect
3. #21455299C2semidirect
4. #21444884C2semidirect
5. #21444878C2semidirect
6. #21444487C2semidirect
7. #21444274C2semidirect
8. #21444201C2semidirect
9. C61×#3508C2semidirect
18. #16839962C22semidirect
19. #16839962C22semidirect
21. #16839957C22semidirect
27. #16839506C22semidirect
28. Q8⋊#92418C22semidirect
29. Q8⋊#92418C22semidirect
30. D4⋊#92418C22semidirect
32. C61×#559C22semidirect
34. C61×#550C22semidirect
278. #3508D61semidirect
297. #559C2×D61semidirect
299. #550C2×D61semidirect
329. C61#18624semidirect

Maximal subgroups

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

subgroupindex
#214567572
C2×#168401382
#214552992
#214448842
#214448782
#214444872
#214442742
#214442012
C61×#35082
#1862461