Finite Group Databasebeta

← #29667830

Subgroups — #29667830

Order 15616.

820 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#21456394C211yesmaximal, characteristic, complemented
2#21456394C211yesmaximal, characteristic, complemented
3#21456379C211yesmaximal, characteristic, complemented
4#21456379C211yesmaximal, characteristic, complemented
5#21456247C211yesmaximal, characteristic, complemented
6#21445498C211yesmaximal, characteristic, complemented
7#21445487C211yesmaximal, characteristic, complemented
8#21445478C211yesmaximal, characteristic, complemented
9#21445474C211yesmaximal, characteristic, complemented
10#21445461C211yesmaximal, characteristic, complemented
11#21445460C211yesmaximal, characteristic, complemented
12#21445449C211yesmaximal, characteristic, complemented
13#21445443C211yesmaximal, characteristic, complemented
14C2×#16839484C211yesmaximal, characteristic, complemented
15C61×#3131C211yesmaximal, characteristic, complemented
16C2×#14519799—22no—
17#16840312C2211yescharacteristic
18#16840312C2211yescharacteristic
19#16840165C2211yescharacteristic
20#16840165C2211yescharacteristic
21#16840158C2211yescharacteristic
22#16840158C2211yescharacteristic
23#16840146—22no—
24#16840142C2211yescharacteristic
25#16840139C2211yescharacteristic
26#16840138C2211yescharacteristic
27#16840138C2211yescharacteristic
28#16840136C2211yescharacteristic
29#16840136C2211yescharacteristic
30#16840135C2211yescharacteristic
31#16840135C2211yescharacteristic
32#16840112C2211yescharacteristic
33#16840112C2211yescharacteristic
34#16840110C2211yescharacteristic
35#16840110C2211yescharacteristic
36#16840110C2211yescharacteristic
37#16840110C2211yescharacteristic
38#16840108C2211yescharacteristic
39#16840108C2211yescharacteristic
40#16839996—22no—
41#16839996—22no—
42#16839993—22no—
43#16839993—22no—
44C4×#11758201—22no—
45C4×#11758201—22no—
46C4×#11758199—22no—
47C4×#11758199—22no—
48C2×C4×#92418C2211yescharacteristic
49C2×C4×#92418C2211yescharacteristic
50#16839484C2211yescharacteristic
51#16839484C2211yescharacteristic
52#16839484C2211yescharacteristic
53#16839484C2211yescharacteristic
54C61×#533C2211yescharacteristic
55C61×#532C2211yescharacteristic
56C61×#488—22no—
57C61×#487C2211yescharacteristic
58C61×#487C2211yescharacteristic
59C61×#486C2211yescharacteristic
60C61×#486C2211yescharacteristic
61C61×#416—22no—
62C2×C4×C8×C61C2211yescharacteristic
63C2×#11758222—21no—
64C2×#11758222—21no—
65#14519799—48no—
66C2×Dic244—22no—
67C2×#11758190—22no—
68C2×#11758190—22no—
69C2×#11758201—22no—
70C2×#11758201—22no—
71C2×D488—22no—
72C2×#11758199—22no—
73C2×#11758199—22no—
74C22×#92418D412yes—
75C22×#92418D412yes—
76#14519777C2311yescharacteristic
77#14519777C2311yescharacteristic
78#14519774C2311yescharacteristic
79#14519774C2311yescharacteristic
80#14519765—21no—
81#14519765—21no—
82#14519751—21no—
83#14519751—21no—
84#14519751—21no—
85#14519751—21no—
86C4×D244—21no—
87C4×D244—21no—
88C4×D244—21no—
89C4×D244—21no—
90#14519710—22no—
91#14519710—22no—
92#14519708—22no—
93#14519708—22no—
94#14519701—22no—
95#14519701—22no—
96#14519698—22no—
97#14519698—22no—
98#14519697—22no—
99#14519697—22no—
100#14519697—22no—
101#14519697—22no—
102C4×#92418D412yes—
103C4×#92418D412yes—
104C4×#92418C2311yescharacteristic
105C4×#92418C2311yescharacteristic
106C4×#92418C2311yescharacteristic
107C4×#92418C2311yescharacteristic
108C2×#41×C61—21no—
109C22×C8×C61D412yes—
110C4⋊Q8×C61C2311yescharacteristic
111C4⋊Q8×C61C2311yescharacteristic
112C4⋊Q8×C61C2311yescharacteristic
113#127×C61C2311yescharacteristic
114#125×C61—21no—
115#124×C61—21no—
116C22⋊Q8×C61—21no—
117C22⋊Q8×C61—21no—
118#121×C61—21no—
119#121×C61—21no—
120C4×Q8×C61—21no—
121C4×Q8×C61—21no—
122C4×D4×C61—21no—
123C4×D4×C61—21no—
124#117×C61—21no—
125C2×C42×C61C2311yescharacteristic
126#103×C61—22no—
127#103×C61—22no—
128#102×C61—22no—
129#102×C61—22no—
130C4×C8×C61D412yes—
131C4×C8×C61C2311yescharacteristic
132C4×C8×C61C2311yescharacteristic
133#41×C61—44no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61—41no—
141C22×C4×C61—41no—
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C2×C22⋊D61—41no—
146C2×C22⋊D61—41no—
147#11758222—44no—
148#11758222—44no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—21no—
155C2×C4×D61—41no—
156C2×C4×D61—41no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×C8×C61—24no—
160C2×C8×C61C2×D412yes—
161C2×C8×C61C2×D412yes—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—41no—
169C4⋊C4×C61—41no—
170C22⋊C4×C61—41no—
171C22⋊C4×C61—41no—
172C22⋊C4×C61—41no—
173C42×C61—41no—
174C42×C61C2×D411yescharacteristic
175C42×C61C2×D411yescharacteristic
176C42×C61C2×D411yescharacteristic
177C42×C61C2411yescharacteristic
178#11758201—44no—
179#11758201—44no—
180#11758199—44no—
181#11758199—44no—
182#11758198—41no—
183#11758198—41no—
184#11758198—41no—
185#11758198—41no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C2×#92418—24no—
191C2×#92418—42no—
192C2×#92418C2×D412yes—
193C2×#92418C2×D412yes—
194C2×#92418C2×D412yes—
195C2×#92418C2×D412yes—
196Dic244—44no—
197D488—44no—
198#11758190—44no—
199#11758190—44no—
200C23×C61—41no—
201C23×C61#12711yescharacteristic
202C22×D61—41no—
203C22×D61—41no—
204Q8×C61—42no—
205Q8×C61—42no—
206Q8×C61—42no—
207D4×C61—42no—
208D4×C61—42no—
209D4×C61—42no—
210D4×C61—82no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—82no—
217C2×C4×C61#12711yescharacteristic
218C2×C4×C61#12711yescharacteristic
219C2×C4×C61#12711yescharacteristic
220C2×C4×C61AGL(1,ℤ/8)14yes—
221C2×C4×C61C22×D411yescharacteristic
222C2×C4×C61C22×D411yescharacteristic
223C2×C4×C61C22×D411yescharacteristic
224C22⋊D61—82no—
225C22⋊D61—82no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228D244—42no—
229D244—42no—
230D244—42no—
231D244—42no—
232D244—42no—
233D244—42no—
234C4×D61—82no—
235C4×D61—82no—
236Dic122—42no—
237Dic122—42no—
238C8×C61—22no—
239C8×C61—22no—
240#92418—22no—
241#92418—22no—
242#92418—22no—
243#92418#13514yescomplemented
244#19619—611nomaximal, Sylow
245C22×C61—21no—
246C22×C61—21no—
247C22×C61—81no—
248C22×C61—81no—
249C22×C61—81no—
250C22×C61#49612yes—
251C22×C61C2×#12711yescharacteristic
252C2×D61—81no—
253C2×D61—81no—
254C2×D61—81no—
255C2×D61—81no—
256C2×D61—81no—
257C2×D61—81no—
258C4×C61—21no—
259C4×C61—21no—
260C4×C61—81no—
261C4×C61—81no—
262C4×C61—81no—
263C4×C61#49612yes—
264C4×C61C2×#13512yes—
265C4×C61C2×AGL(1,ℤ/8)12yes—
266C4×C61#57512yes—
267Dic61—81no—
268Dic61—81no—
269#3169—611no—
270#3169—611no—
271#3167—611no—
272#3167—611no—
273#3147—611no—
274#3131D6111yescharacteristic, complemented
275#1723—611no—
276#1721—611no—
277#1720—611no—
278#1719—611no—
279#1717—611no—
280#1717—611no—
281#1715—611no—
282#1714—611no—
283C2×#322—611no—
284C2×C61—22no—
285C2×C61—161no—
286C2×C61#314711yescharacteristic
287C2×C61C2×#49611yescharacteristic
288C2×C61#315011yescharacteristic
289D61—161no—
290D61—161no—
291C2×#135—1222no—
292#533C2×D6111yescharacteristic
293#532—611no—
294#532—611no—
295#532C2×D6111yescharacteristic
296#500—611no—
297#500—611no—
298#498—611no—
299#498—611no—
300#495—1222no—
301#494—611no—
302#493—611no—
303#493—611no—
304#493—611no—
305#492—611no—
306#492—611no—
307#492—611no—
308#492—611no—
309#488—22no—
310#487—611no—
311#487—611no—
312#487C2×D6111yescharacteristic
313#487C2×D6111yescharacteristic
314#486—611no—
315#486—611no—
316#486—611no—
317#486—611no—
318#486—611no—
319#486—611no—
320#486C2×D6111yescharacteristic
321#486C2×D6111yescharacteristic
322#466—1222no—
323#466—1222no—
324#465—1222no—
325#465—1222no—
326C4×SD16—1222no—
327C4×SD16—1222no—
328C4×D8—1222no—
329C4×D8—1222no—
330#416—22no—
331C2×C4×C8—611no—
332C2×C4×C8—611no—
333C2×C4×C8C2×D6111yescharacteristic
334#322—611no—
335#322—611no—
336#322—611no—
337#322—611no—
338C61#1961911yescharacteristic, complemented, Sylow
339C2×#41—21no—
340C2×#41—1221no—
341C2×#41—1221no—
342#135—2448no—
343C2×Q16—1222no—
344C2×SD16—1222no—
345C2×SD16—1222no—
346C2×SD16—1222no—
347C2×SD16—1222no—
348C2×D8—1222no—
349C2×D8—1222no—
350C2×D8—1222no—
351C22×C8—612no—
352C22×C8—612no—
353C22×C8C22⋊D6112yes—
354C4⋊Q8—611no—
355C4⋊Q8—611no—
356C4⋊Q8C22×D6111yescharacteristic
357C4⋊Q8C22×D6111yescharacteristic
358C4⋊Q8C22×D6111yescharacteristic
359#127—611no—
360#127—611no—
361#127C22×D6111yescharacteristic
362#125—21no—
363#124—21no—
364#124—1221no—
365#124—1221no—
366C22⋊Q8—21no—
367C22⋊Q8—21no—
368#121—21no—
369#121—21no—
370#121—1221no—
371#121—1221no—
372#121—1221no—
373#121—1221no—
374C4×Q8—21no—
375C4×Q8—21no—
376C4×D4—21no—
377C4×D4—21no—
378C4×D4—1221no—
379C4×D4—1221no—
380C4×D4—1221no—
381C4×D4—1221no—
382#117—21no—
383C2×C42C22×D6111yescharacteristic
384#107—1222no—
385#107—1222no—
386#106—1222no—
387#106—1222no—
388#103—22no—
389#103—22no—
390#103—1222no—
391#103—1222no—
392#102—22no—
393#102—22no—
394#102—1222no—
395#102—1222no—
396#102—1222no—
397#102—1222no—
398#102—1222no—
399#102—1222no—
400C4×C8—612no—
401C4×C8—612no—
402C4×C8—611no—
403C4×C8—611no—
404C4×C8—611no—
405C4×C8—611no—
406C4×C8C22⋊D6112yes—
407C4×C8C22×D6111yescharacteristic
408C4×C8C22×D6111yescharacteristic
409#41—44no—
410#41—2444no—
411#41—2444no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—1221no—
416C2×Q8—1221no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—41no—
421C2×D4—1221no—
422C2×D4—1221no—
423C2×D4—1221no—
424C2×D4—1221no—
425C2×D4—1221no—
426C2×D4—1221no—
427C2×D4—2441no—
428C2×D4—2441no—
429C22×C4—41no—
430C22×C4—2441no—
431C22×C4—2441no—
432C22×C4D4×D6111yescharacteristic
433C22×C4D4×D6111yescharacteristic
434C22×C4C2×C22⋊D6111yescharacteristic
435Q16—2444no—
436SD16—2444no—
437SD16—2444no—
438SD16—2444no—
439SD16—2444no—
440D8—2444no—
441D8—2444no—
442D8—2444no—
443C2×C8—24no—
444C2×C8—612no—
445C2×C8—612no—
446C2×C8—612no—
447C2×C8—612no—
448C2×C8—1224no—
449C2×C8—2442no—
450C2×C8C2×C22⋊D6112yes—
451C2×C8C2×C22⋊D6112yes—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—1221no—
461C4⋊C4—1221no—
462C4⋊C4—1221no—
463C4⋊C4—1221no—
464C22⋊C4—41no—
465C22⋊C4—41no—
466C22⋊C4—41no—
467C22⋊C4—2441no—
468C22⋊C4—2441no—
469C22⋊C4—2441no—
470C22⋊C4—2441no—
471C42—41no—
472C42D4×D6111yescharacteristic
473C42D4×D6111yescharacteristic
474C42C2×C22⋊D6111yescharacteristic
475C42C23×D6111yescharacteristic
476C23—41no—
477C23—2441no—
478C23—2441no—
479C23#1451977311yescharacteristic
480Q8—42no—
481Q8—42no—
482Q8—42no—
483Q8—2442no—
484Q8—2442no—
485D4—42no—
486D4—42no—
487D4—42no—
488D4—82no—
489D4—4882no—
490D4—4882no—
491D4—2442no—
492D4—2442no—
493D4—2442no—
494D4—2442no—
495D4—2442no—
496D4—2442no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—82no—
503C2×C4—4882no—
504C2×C4—4882no—
505C2×C4—2441no—
506C2×C4—2441no—
507C2×C4#1451977311yescharacteristic
508C2×C4#1451977311yescharacteristic
509C2×C4#1451977311yescharacteristic
510C2×C4#1451980214yes—
511C2×C4C2×D4×D6111yescharacteristic
512C2×C4C2×D4×D6111yescharacteristic
513C2×C4C22×C22⋊D6111yescharacteristic
514C8—22no—
515C8—22no—
516C8—614no—
517C8—1222no—
518C8—1222no—
519C8—1222no—
520C22—21no—
521C22—21no—
522C22—81no—
523C22—81no—
524C22—81no—
525C22—4881no—
526C22—4881no—
527C22—4881no—
528C22—4881no—
529C22—4881no—
530C22—4881no—
531C22#1684015212yes—
532C22C2×#1451977311yescharacteristic
533C4—21no—
534C4—21no—
535C4—81no—
536C4—81no—
537C4—81no—
538C4—4881no—
539C4—4881no—
540C4#1684015212yes—
541C4#135×D6112yes—
542C4C2×#1451980212yes—
543C4#1684055912yes—
544C2—22no—
545C2—161no—
546C2—9761no—
547C2—9761no—
548C2#2145624311yescharacteristic
549C2C2×#1684015211yescharacteristic
550C2#2145626211yescharacteristic
subgroupG/H
centerC2×C4#14519773
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3131C2
socleC22×C61C2×#127
hypercenter#3131D61
hypocenterC61#19619
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3131

Complements

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

NH ≅ G/Ntype
1. #21456394C2semidirect
2. #21456394C2semidirect
3. #21456379C2semidirect
4. #21456379C2semidirect
5. #21456247C2semidirect
6. #21445498C2semidirect
7. #21445487C2semidirect
8. #21445478C2semidirect
9. #21445474C2semidirect
10. #21445461C2semidirect
11. #21445460C2semidirect
12. #21445449C2semidirect
13. #21445443C2semidirect
14. C2×#16839484C2semidirect
15. C61×#3131C2semidirect
243. #92418#135semidirect
274. #3131D61semidirect
338. C61#19619semidirect

Maximal subgroups

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

subgroupindex
#214563942
#214563792
#214562472
#214454982
#214454872
#214454782
#214454742
#214454612
#214454602
#214454492
#214454432
C2×#168394842
C61×#31312
#1961961