Finite Group Databasebeta

← #29833334

Subgroups — #29833334

Order 15616.

738 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840196C211yesmaximal, characteristic
2C4×#14519796C211yesmaximal, characteristic, complemented
3C4×D8×D61C211yesmaximal, characteristic, complemented
4#21447693C211yesmaximal, characteristic, complemented
5#21447685C211yesmaximal, characteristic, complemented
6#21447683C211yesmaximal, characteristic, complemented
7#21447674C211yesmaximal, characteristic, complemented
8#21447670C211yesmaximal, characteristic, complemented
9D16×Dic61C211yesmaximal, characteristic, complemented
10#21447663C211yesmaximal, characteristic, complemented
11#21447606C211yesmaximal, characteristic, complemented
12#21447533C211yesmaximal, characteristic, complemented
13#21447528C211yesmaximal, characteristic, complemented
14#21447412C211yesmaximal, characteristic, complemented
15C61×#2177C211yesmaximal, characteristic, complemented
16C2×#14519796C2211yescharacteristic
17C2×D8×D61C2211yescharacteristic
18C4×#11758224—21no—
19C4×D4×D61—21no—
20#16840196C418yescomplemented
21C2×#14519722C2211yescharacteristic
22C2×#14519721C2211yescharacteristic
23C2×#14519718C2211yescharacteristic
24C2×#14519716C2211yescharacteristic
25C2×#14519716C2211yescharacteristic
26C4×Dic244C2211yescharacteristic, complemented
27#16839856C2211yescharacteristic, complemented
28#16839853—21no—
29C4×#11758200—21no—
30C4×D488C2211yescharacteristic
31D8×Dic61C2211yescharacteristic
32D8×Dic61C2211yescharacteristic, complemented
33#16839844C2211yescharacteristic
34C4×#11758199—21no—
35#16839842—21no—
36#16839799C2211yescharacteristic
37#107×D61C2211yescharacteristic
38#16839755—21no—
39#102×D61—21no—
40C4×C8×D61C2211yescharacteristic
41#16839589C2211yescharacteristic, complemented
42#16839588C2211yescharacteristic, complemented
43#16839587C2211yescharacteristic, complemented
44#16839567C2211yescharacteristic, complemented
45#16839565C2211yescharacteristic, complemented
46#16839564C2211yescharacteristic
47#16839563C2211yescharacteristic
48#16839563C2211yescharacteristic, complemented
49#16839562C2211yescharacteristic
50#16839538C2211yescharacteristic, complemented
51C16×Dic61C2211yescharacteristic, complemented
52C16×Dic61C2211yescharacteristic, complemented
53C2×D16×C61C2211yescharacteristic
54C4×D8×C61C2211yescharacteristic
55C4×D8×C61C2211yescharacteristic, complemented
56C61×#367C2211yescharacteristic, complemented
57C61×#357C2211yescharacteristic
58C61×#357C2211yescharacteristic, complemented
59C61×#346C2211yescharacteristic, complemented
60C2×#11758224—21no—
61C2×D4×D61—21no—
62C22×C4×D61—41no—
63#14519796—22no—
64#14519796C2×C414yescomplemented
65C2×Dic244C2311yescharacteristic
66C2×#11758200—21no—
67C2×D488C2311yescharacteristic
68C2×#11758199—21no—
69D8×D61—22no—
70D8×D61C2×C414yescomplemented
71C2×C8×D61D411yescharacteristic
72C2×C8×D61C2311yescharacteristic
73C4×Dic122—21no—
74#14519742—21no—
75C4×D244—21no—
76D4×Dic61—21no—
77D4×Dic61—21no—
78#14519739—21no—
79C4×C22⋊D61—41no—
80C4×C22⋊D61—41no—
81#14519737—41no—
82#14519737—41no—
83#14519735—21no—
84#14519733—41no—
85C4⋊C4×D61—21no—
86C22⋊C4×D61—41no—
87C2×C4×Dic61—41no—
88C42×D61D411yescharacteristic
89#14519722C2×C414yescomplemented
90#14519721C2×C414yescomplemented
91#14519718C2×C414yescomplemented
92#14519716—22no—
93#14519716C2×C414yescomplemented
94C2×#11758185C2311yescharacteristic
95C2×#11758185C2311yescharacteristic
96#14519711C2311yescharacteristic
97#14519711C2311yescharacteristic
98#14519710—21no—
99#14519708—21no—
100#14519702—21no—
101#14519700—21no—
102#14519699—21no—
103#14519698—21no—
104#14519698—21no—
105#14519697—21no—
106C4×#92418D411yescharacteristic
107C8×Dic61D411yescharacteristic
108C8×Dic61C2311yescharacteristic
109C2×D8×C61C2311yescharacteristic
110C2×D8×C61C2311yescharacteristic
111C4×D4×C61—21no—
112C4×D4×C61—21no—
113D16×C61C2×C414yescomplemented
114C2×C16×C61C2311yescharacteristic
115C2×C16×C61C2311yescharacteristic
116#107×C61C2311yescharacteristic
117#107×C61C2311yescharacteristic
118#102×C61—21no—
119#102×C61—21no—
120C4×C8×C61C2311yescharacteristic
121C23×D61—41no—
122C2×D4×C61—21no—
123C2×D4×C61—21no—
124C22×C4×C61—41no—
125C22×C4×C61—41no—
126C2×C22⋊D61—41no—
127C2×C22⋊D61—41no—
128C22×Dic61—41no—
129C22×Dic61—41no—
130C22×Dic61—41no—
131#11758224—24no—
132#11758224—42no—
133D4×D61—24no—
134D4×D61—42no—
135C2×D244—21no—
136C2×C4×D61—41no—
137C2×C4×D61—41no—
138C2×C4×D61—41no—
139C2×C4×D61—84no—
140C2×C4×D61D811yescharacteristic
141C2×C4×D61D811yescharacteristic
142C2×C4×D61C2×D411yescharacteristic
143C2×Dic122—21no—
144D8×C61—21no—
145D8×C61—21no—
146D8×C61C22×C412yes—
147D8×C61C22×C412yescomplemented
148C2×C8×C61C2×D411yescharacteristic
149C2×C8×C61C2411yescharacteristic
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C22⋊C4×C61—41no—
153C22⋊C4×C61—41no—
154C42×C61C2×D411yescharacteristic
155C22⋊Dic61—41no—
156C22⋊Dic61—41no—
157#11758200—24no—
158#11758199—24no—
159#11758198—41no—
160#11758198—41no—
161#11758198—41no—
162C4⋊Dic61—21no—
163C4⋊Dic61—21no—
164#11758196—41no—
165#11758196—41no—
166#11758196—41no—
167C4×Dic61—41no—
168C4×Dic61—41no—
169C4×Dic61—41no—
170C4×Dic61D811yescharacteristic
171C4×Dic61D811yescharacteristic
172C4×Dic61C2×D411yescharacteristic
173C2×#92418C2×D411yescharacteristic
174C2×#92418C2×D411yescharacteristic
175Dic244—21no—
176Dic244C22×C412yescomplemented
177D488—21no—
178D488C22×C412yes—
179C8×D61—22no—
180C8×D61C22×C412yes—
181C8×D61#4112yes—
182C16×C61—21no—
183C16×C61C22×C412yescomplemented
184#11758185—21no—
185#11758185C22×C412yescomplemented
186C23×C61—41no—
187C23×C61—41no—
188C22×D61—41no—
189C22×D61—44no—
190C22×D61—82no—
191C22×D61—82no—
192C22×D61C2×D811yescharacteristic
193D4×C61—22no—
194D4×C61—22no—
195D4×C61—41no—
196D4×C61—41no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61—82no—
200C2×C4×C61—82no—
201C2×C4×C61C2×D811yescharacteristic
202C2×C4×C61C2×D811yescharacteristic
203C2×C4×C61C22×D411yescharacteristic
204C22⋊D61—44no—
205C22⋊D61—44no—
206C2×Dic61—41no—
207C2×Dic61—44no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61—82no—
211C2×Dic61—82no—
212C2×Dic61C2×D811yescharacteristic
213C2×Dic61C2×D811yescharacteristic
214C2×Dic61C2×D811yescharacteristic
215D244—22no—
216D244—41no—
217C4×D61—22no—
218C4×D61—41no—
219C4×D61—82no—
220C4×D61—82no—
221C4×D61—82no—
222C4×D61C4×D412yes—
223C4×D61#13512yes—
224Dic122—22no—
225Dic122—41no—
226C8×C61—21no—
227C8×C61C23×C411yescharacteristic
228C8×C61C2×#4111yescharacteristic
229#92418—21no—
230#92418C4×D412yes—
231C4×#509—611nomaximal, Sylow
232C22×C61—42no—
233C22×C61—42no—
234C22×C61—81no—
235C22×C61—81no—
236C22×C61—81no—
237C22×C61—81no—
238C22×C61C22×D811yescharacteristic
239C2×D61—21no—
240C2×D61—21no—
241C2×D61—42no—
242C2×D61—88no—
243C2×D61—81no—
244C2×D61—81no—
245C2×D61C4×D812yes—
246C4×C61—21no—
247C4×C61—21no—
248C4×C61—81no—
249C4×C61—81no—
250C4×C61C2×C4×D411yescharacteristic
251C4×C61C2×#13511yescharacteristic
252Dic61—21no—
253Dic61—42no—
254Dic61—81no—
255Dic61—81no—
256Dic61C4×D812yes—
257Dic61#50912yescomplemented
258C2×#509—611no—
259C4×#135—611no—
260C2×C4×D8—611no—
261#2177—611no—
262#2177D6111yescharacteristic, complemented
263#2175—611no—
264#2175—611no—
265C4×SD32—611no—
266C4×SD32—611no—
267C4×D16—611no—
268C4×D16—611no—
269#2159—611no—
270#2140—611no—
271#2139—611no—
272C4×M32—611no—
273C2×C61—82no—
274C2×C61—82no—
275C2×C61C2×C4×D811yescharacteristic
276C2×C61C2×#50911yescharacteristic
277C2×C61#341411yescharacteristic
278D61—22no—
279D61—82no—
280C2×#135—611no—
281C22×D8—611no—
282C4×#41—1221no—
283C2×C4×D4—1221no—
284#509—618no—
285C2×SD32—611no—
286C2×SD32—611no—
287C2×D16—611no—
288C2×D16C2×D6111yescharacteristic
289C2×M32—611no—
290C2×M32—611no—
291C4×Q16—611no—
292C4×Q16—611no—
293C4×SD16—1221no—
294C4×SD16—1221no—
295C4×D8—611no—
296C4×D8—611no—
297C4×D8—611no—
298C4×D8—611no—
299C4×D8—1221no—
300C4×D8—1221no—
301C4×D8C2×D6111yescharacteristic
302C4×D8C2×D6111yescharacteristic, complemented
303#427—611no—
304C2×#107—611no—
305#416—1221no—
306C2×#102—1221no—
307C2×C4×C8—611no—
308#367—611no—
309#367C2×D6111yescharacteristic, complemented
310#366—611no—
311#366—611no—
312#358—611no—
313#358—611no—
314#357—611no—
315#357—611no—
316#357—611no—
317#357—611no—
318#357C2×D6111yescharacteristic
319#357C2×D6111yescharacteristic, complemented
320#346—611no—
321#346C2×D6111yescharacteristic, complemented
322C4×C16—611no—
323C4×C16—611no—
324C61C4×#50911yescharacteristic, complemented, Sylow
325C2×#41—1221no—
326C22×D4—1221no—
327C23×C4—2441no—
328#135—614no—
329#135—1222no—
330C2×Q16—611no—
331C2×SD16—1221no—
332C2×D8—614no—
333C2×D8—611no—
334C2×D8—1222no—
335C2×D8—1221no—
336C2×D8C22×D6111yescharacteristic
337C2×D8C22×D6111yescharacteristic
338C22×C8—611no—
339C22×C8—611no—
340C4×Q8—1221no—
341C4×Q8—1221no—
342C4×D4—21no—
343C4×D4—21no—
344C4×D4—1221no—
345C4×D4—1221no—
346C4×D4—1221no—
347C4×D4—1221no—
348C4×D4—2441no—
349C4×D4—2441no—
350C4×D4—2441no—
351C4×D4—2441no—
352#117—1221no—
353#117—2441no—
354C2×C4⋊C4—1221no—
355C2×C22⋊C4—2441no—
356C2×C42—611no—
357C2×C42—2441no—
358SD32—614no—
359SD32—614no—
360D16—614no—
361D16C4×D6114yescomplemented
362M32—614no—
363M32—1222no—
364C2×C16—611no—
365C2×C16—611no—
366C2×C16C22×D6111yescharacteristic
367C2×C16C22×D6111yescharacteristic
368#107—611no—
369#107—611no—
370#107—1221no—
371#107C22×D6111yescharacteristic
372#107C22×D6111yescharacteristic
373#106—1221no—
374#103—1221no—
375#103—1221no—
376#102—21no—
377#102—21no—
378#102—1221no—
379#102—1221no—
380#102—1221no—
381#102—1221no—
382C4×C8—611no—
383C4×C8—611no—
384C4×C8—611no—
385C4×C8C22×D6111yescharacteristic
386C24—2441no—
387#41—1224no—
388#41—2442no—
389C2×Q8—1221no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—1224no—
393C2×D4—1221no—
394C2×D4—2442no—
395C2×D4—2441no—
396C2×D4—2441no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—611no—
400C22×C4—611no—
401C22×C4—611no—
402C22×C4—4884no—
403C22×C4—2441no—
404C22×C4—2441no—
405C22×C4—2441no—
406C22×C4—2441no—
407C22×C4—2441no—
408C22×C4—2441no—
409Q16—612no—
410Q16—1221no—
411SD16—1224no—
412D8—21no—
413D8—21no—
414D8—612no—
415D8—1224no—
416D8—1221no—
417D8C2×C4×D6112yes—
418D8C2×C4×D6112yescomplemented
419C2×C8—611no—
420C2×C8—612no—
421C2×C8—612no—
422C2×C8—611no—
423C2×C8—1222no—
424C2×C8D4×D6111yescharacteristic
425C2×C8C23×D6111yescharacteristic
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—1221no—
429C4⋊C4—1221no—
430C4⋊C4—2441no—
431C4⋊C4—2441no—
432C4⋊C4—2441no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—2441no—
436C22⋊C4—2441no—
437C22⋊C4—2441no—
438C22⋊C4—2441no—
439C22⋊C4—2441no—
440C42—611no—
441C42—611no—
442C42—611no—
443C42—2441no—
444C42—2441no—
445C42—2441no—
446C42D4×D6111yescharacteristic
447C16—21no—
448C16—612no—
449C16—1221no—
450C16C2×C4×D6112yescomplemented
451C23—41no—
452C23—41no—
453C23—611no—
454C23—4882no—
455C23—4882no—
456C23—2444no—
457C23—2441no—
458Q8—1222no—
459Q8—2441no—
460D4—22no—
461D4—22no—
462D4—41no—
463D4—41no—
464D4—1222no—
465D4—2444no—
466D4—2444no—
467D4—2441no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—611no—
473C2×C4—611no—
474C2×C4—612no—
475C2×C4—612no—
476C2×C4—611no—
477C2×C4—1222no—
478C2×C4—4882no—
479C2×C4—4882no—
480C2×C4—4882no—
481C2×C4—4882no—
482C2×C4—4882no—
483C2×C4—2441no—
484C2×C4—2444no—
485C2×C4—2441no—
486C2×C4—2441no—
487C2×C4—2441no—
488C2×C4D8×D6111yescharacteristic
489C2×C4D8×D6111yescharacteristic
490C2×C4C2×D4×D6111yescharacteristic
491C8—21no—
492C8—612no—
493C8—1221no—
494C8C22×C4×D6111yescharacteristic
495C8#41×D6111yescharacteristic
496C22—42no—
497C22—42no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22—612no—
503C22—1221no—
504C22—1221no—
505C22—4888no—
506C22—4881no—
507C22—4881no—
508C22—2442no—
509C22C2×D8×D6111yescharacteristic
510C4—21no—
511C4—21no—
512C4—81no—
513C4—81no—
514C4—612no—
515C4—612no—
516C4—1221no—
517C4—4881no—
518C4—4881no—
519C4—2442no—
520C4C4×D4×D6111yescharacteristic
521C4#135×D6111yescharacteristic
522C2—82no—
523C2—82no—
524C2—1222no—
525C2—4882no—
526C2C4×D8×D6111yescharacteristic
527C2#509×D6111yescharacteristic
528C2#2145895111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutatorC8×C61C23×C4
FrattiniC2×C8C23×D61
FittingC61×#2177C2
socleC22×C61C22×D8
prime radicalC2×#16840196C2
cosocleC2×C8×C61C24
hypercenter#2177D61
hypocenterC61C4×#509
LCSG ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2177

Complements

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

NH ≅ G/Ntype
2. C4×#14519796C2semidirect
3. C4×D8×D61C2semidirect
4. #21447693C2semidirect
5. #21447685C2semidirect
6. #21447683C2semidirect
7. #21447674C2semidirect
8. #21447670C2semidirect
9. D16×Dic61C2semidirect
10. #21447663C2semidirect
11. #21447606C2semidirect
12. #21447533C2semidirect
13. #21447528C2semidirect
14. #21447412C2semidirect
15. C61×#2177C2semidirect
20. #16840196C4semidirect
26. C4×Dic244C22semidirect
27. #16839856C22semidirect
32. D8×Dic61C22semidirect
41. #16839589C22semidirect
42. #16839588C22semidirect
43. #16839587C22semidirect
44. #16839567C22semidirect
45. #16839565C22semidirect
48. #16839563C22semidirect
50. #16839538C22semidirect
51. C16×Dic61C22semidirect
52. C16×Dic61C22semidirect
55. C4×D8×C61C22semidirect
56. C61×#367C22semidirect
58. C61×#357C22semidirect
59. C61×#346C22semidirect
64. #14519796C2×C4semidirect
70. D8×D61C2×C4semidirect
89. #14519722C2×C4semidirect
90. #14519721C2×C4semidirect
91. #14519718C2×C4semidirect
93. #14519716C2×C4semidirect
113. D16×C61C2×C4semidirect
147. D8×C61C22×C4semidirect
176. Dic244C22×C4semidirect
183. C16×C61C22×C4semidirect
185. #11758185C22×C4semidirect
257. Dic61#509semidirect
262. #2177D61semidirect
302. C4×D8C2×D61semidirect
309. #367C2×D61semidirect
319. #357C2×D61semidirect
321. #346C2×D61semidirect
324. C61C4×#509semidirect
361. D16C4×D61semidirect
418. D8C2×C4×D61semidirect
450. C16C2×C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168401962
C4×#145197962
C4×D8×D612
#214476932
#214476852
#214476832
#214476742
#214476702
D16×Dic612
#214476632
#214476062
#214475332
#214475282
#214474122
C61×#21772
C4×#50961