Finite Group Databasebeta

← #29833379

Subgroups — #29833379

Order 15616.

682 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD32×D61C211yesmaximal, characteristic
2#21454682C211yesmaximal, characteristic, complemented
3#21454666C211yesmaximal, characteristic
4#21447685C211yesmaximal, characteristic, complemented
5#21447684C211yesmaximal, characteristic, complemented
6#21447682C211yesmaximal, characteristic, complemented
7#21447674C211yesmaximal, characteristic, complemented
8C4×#14519720C211yesmaximal, characteristic, complemented
9C4×#14519721C211yesmaximal, characteristic, complemented
10#21447668C211yesmaximal, characteristic, complemented
11#21447600C211yesmaximal, characteristic, complemented
12#21447537C211yesmaximal, characteristic, complemented
13#21447533C211yesmaximal, characteristic
14#21447417C211yesmaximal, characteristic, complemented
15C61×#2175C211yesmaximal, characteristic, complemented
16C2×Q16×D61C2211yescharacteristic
17C2×D8×D61C2211yescharacteristic
18#16840238—21no—
19#16840234—21no—
20C2×#14519722C2211yescharacteristic
21C2×#14519721C2211yescharacteristic
22C2×#14519720C2211yescharacteristic
23SD32×D61C418yescomplemented
24C2×#14519716C2211yescharacteristic
25C2×C16×D61C2211yescharacteristic
26#16839872—21no—
27#16839871—21no—
28#16839867—21no—
29#16839866—21no—
30C4×Dic244C2211yescharacteristic, complemented
31Q16×Dic61C2211yescharacteristic, complemented
32#16839856C2211yescharacteristic, complemented
33C4×D488C2211yescharacteristic
34D8×Dic61C2211yescharacteristic
35#16839844C2211yescharacteristic
36#16839799C2211yescharacteristic
37#16839799C2211yescharacteristic
38#16839769—21no—
39#16839765—21no—
40#16839716C2211yescharacteristic
41#16839589C2211yescharacteristic, complemented
42#16839589C2211yescharacteristic, complemented
43#16839588C2211yescharacteristic, complemented
44#16839567C2211yescharacteristic, complemented
45#16839566C2211yescharacteristic, complemented
46#16839565C2211yescharacteristic, complemented
47#16839564C2211yescharacteristic
48#16839563C2211yescharacteristic
49#16839562C2211yescharacteristic
50#16839537C2211yescharacteristic, complemented
51C4×#11758185C2211yescharacteristic, complemented
52C16×Dic61C2211yescharacteristic, complemented
53C2×SD32×C61C2211yescharacteristic
54C4×Q16×C61C2211yescharacteristic, complemented
55C4×D8×C61C2211yescharacteristic
56C61×#366C2211yescharacteristic, complemented
57C61×#358C2211yescharacteristic, complemented
58C61×#357C2211yescharacteristic
59C61×#346C2211yescharacteristic, complemented
60C2×Q8×D61—21no—
61C2×D4×D61—21no—
62C2×Dic244C2311yescharacteristic
63C2×#11758202—21no—
64Q16×D61—22no—
65Q16×D61C2×C414yescomplemented
66C2×D488C2311yescharacteristic
67C2×#11758199—21no—
68D8×D61—22no—
69D8×D61C2×C414yes—
70C2×#11758189D411yescharacteristic
71C2×C8×D61C2311yescharacteristic
72Q8×Dic61—21no—
73C4×Dic122—21no—
74C4×Dic122—41no—
75#14519742—21no—
76#14519742—41no—
77C4×D244—21no—
78D4×Dic61—21no—
79#14519739—21no—
80C4×C22⋊D61—41no—
81#14519737—41no—
82#14519735—21no—
83#14519735—21no—
84#14519735—41no—
85#14519734—41no—
86#14519734D411yescharacteristic
87C2×#11758198—41no—
88C22⋊C4×D61—41no—
89#14519722C2×C414yescomplemented
90#14519721C2×C414yescomplemented
91#14519720C2×C414yescomplemented
92#14519716—22no—
93C2×#11758185C2311yescharacteristic
94C2×#11758185C2311yescharacteristic
95C16×D61C2×C414yescomplemented
96#14519711C2311yescharacteristic
97#14519711C2311yescharacteristic
98#14519708—21no—
99#14519708—21no—
100#14519702—21no—
101#14519701—21no—
102#14519700—21no—
103#14519699—21no—
104#14519698—21no—
105#14519697—21no—
106#14519688D411yescharacteristic
107#14519687D411yescharacteristic
108C8×Dic61C2311yescharacteristic
109C2×Q16×C61C2311yescharacteristic
110C2×D8×C61C2311yescharacteristic
111C4×Q8×C61—21no—
112C4×D4×C61—21no—
113SD32×C61C2×C414yescomplemented
114C2×C16×C61C2311yescharacteristic
115C2×C16×C61C2311yescharacteristic
116#107×C61C2311yescharacteristic
117#107×C61C2311yescharacteristic
118#103×C61—21no—
119#102×C61—21no—
120C4×C8×C61C2311yescharacteristic
121C23×D61—41no—
122C2×Q8×C61—21no—
123C2×D4×C61—21no—
124C22×C4×C61—41no—
125C2×C22⋊D61—41no—
126C22×Dic61—41no—
127Q8×D61—24no—
128Q8×D61—42no—
129D4×D61—24no—
130D4×D61—42no—
131C2×D244—21no—
132C2×C4×D61—41no—
133C2×C4×D61—41no—
134C2×C4×D61—41no—
135C2×C4×D61C2×D411yescharacteristic
136C2×Dic122—21no—
137C2×Dic122—41no—
138Q16×C61—21no—
139Q16×C61C22×C412yes—
140D8×C61—21no—
141D8×C61C22×C412yes—
142C2×C8×C61C2×D411yescharacteristic
143C2×C8×C61C2411yescharacteristic
144C4⋊C4×C61—21no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—41no—
147C22⋊C4×C61—41no—
148C42×C61—41no—
149C42×C61C2×D411yescharacteristic
150C22⋊Dic61—41no—
151#11758202—24no—
152#11758199—24no—
153#11758198—41no—
154#11758198—41no—
155#11758198—41no—
156#11758198—41no—
157#11758198D811yescharacteristic
158#11758198D811yescharacteristic
159C4⋊Dic61—21no—
160C4⋊Dic61—21no—
161C4⋊Dic61—41no—
162#11758196—41no—
163#11758196—41no—
164#11758196D811yescharacteristic
165#11758196D811yescharacteristic
166C4×Dic61—41no—
167C4×Dic61—41no—
168C4×Dic61—41no—
169C4×Dic61—41no—
170C4×Dic61—41no—
171C4×Dic61C2×D411yescharacteristic
172C2×#92418C2×D411yescharacteristic
173C2×#92418C2×D411yescharacteristic
174Dic244—21no—
175Dic244C22×C412yes—
176D488—21no—
177D488C22×C412yes—
178#11758189—22no—
179C8×D61C22×C412yes—
180C8×D61#4112yes—
181C16×C61—21no—
182C16×C61C22×C412yes—
183#11758185—21no—
184#11758185C22×C412yes—
185C23×C61—41no—
186C22×D61—41no—
187C22×D61—44no—
188C22×D61—161no—
189C22×D61—161no—
190C22×D61C2×D811yescharacteristic
191Q8×C61—22no—
192Q8×C61—41no—
193D4×C61—22no—
194D4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61—41no—
198C2×C4×C61—161no—
199C2×C4×C61C2×D811yescharacteristic
200C2×C4×C61C2×D811yescharacteristic
201C2×C4×C61C22×D411yescharacteristic
202C22⋊D61—44no—
203C2×Dic61—41no—
204C2×Dic61—41no—
205C2×Dic61—41no—
206C2×Dic61—161no—
207C2×Dic61C2×D811yescharacteristic
208C2×Dic61C2×D811yescharacteristic
209C2×Dic61C2×D811yescharacteristic
210D244—22no—
211D244—41no—
212C4×D61—44no—
213C4×D61—161no—
214C4×D61—161no—
215C4×D61C4×D412yes—
216C4×D61#13512yes—
217Dic122—22no—
218Dic122—41no—
219Dic122—44no—
220C8×C61—21no—
221C8×C61C23×C411yescharacteristic
222C8×C61C2×#4111yescharacteristic
223#92418—21no—
224#92418C4×D412yes—
225#33457—611nomaximal, Sylow
226C22×C61—42no—
227C22×C61—81no—
228C22×C61—81no—
229C22×C61C22×D811yescharacteristic
230C2×D61—21no—
231C2×D61—21no—
232C2×D61—42no—
233C2×D61—81no—
234C2×D61—81no—
235C2×D61—164no—
236C2×D61C4×D812yes—
237C4×C61—21no—
238C4×C61—21no—
239C4×C61—42no—
240C4×C61—81no—
241C4×C61—81no—
242C4×C61C2×C4×D411yescharacteristic
243C4×C61C2×#13511yescharacteristic
244Dic61—21no—
245Dic61—21no—
246Dic61—42no—
247Dic61—81no—
248Dic61—81no—
249Dic61C4×D812yes—
250C22×SD32—611no—
251#2950—611no—
252#2948—611no—
253#2175—611no—
254#2175—611no—
255#2175—611no—
256#2175D6111yescharacteristic, complemented
257C4×SD32—611no—
258C4×SD32—611no—
259C4×SD32—611no—
260C4×SD32—611no—
261#2158—611no—
262#2141—611no—
263#2140—611no—
264#2108—611no—
265C2×C61—82no—
266C2×C61C2×C4×D811yescharacteristic
267C2×C61#341611yescharacteristic
268C2×C61#341611yescharacteristic
269D61—22no—
270D61—82no—
271C22×Q16—611no—
272C22×D8—611no—
273#519—1221no—
274#518—1221no—
275C2×SD32—611no—
276C2×SD32—611no—
277C2×SD32—618no—
278C2×SD32—611no—
279C2×SD32C2×D6111yescharacteristic
280C2×M32—611no—
281C22×C16—611no—
282#442—1221no—
283#442—1221no—
284#441—1221no—
285#441—1221no—
286C4×Q16—611no—
287C4×Q16—611no—
288C4×Q16—611no—
289C4×Q16C2×D6111yescharacteristic, complemented
290C4×D8—611no—
291C4×D8—611no—
292C4×D8—611no—
293C4×D8C2×D6111yescharacteristic
294#427—611no—
295#427—611no—
296#419—1221no—
297#418—1221no—
298#405—611no—
299#367—611no—
300#367—611no—
301#366—611no—
302#366C2×D6111yescharacteristic, complemented
303#358—611no—
304#358—611no—
305#358—611no—
306#358C2×D6111yescharacteristic, complemented
307#357—611no—
308#357—611no—
309#357—611no—
310#357C2×D6111yescharacteristic
311#346—611no—
312#346C2×D6111yescharacteristic, complemented
313C4×C16—611no—
314C4×C16—611no—
315C61#3345711yescharacteristic, complemented, Sylow
316C22×Q8—1221no—
317C22×D4—1221no—
318C2×Q16—614no—
319C2×Q16—611no—
320C2×Q16—1222no—
321C2×Q16—1221no—
322C2×Q16C22×D6111yescharacteristic
323C2×D8—614no—
324C2×D8—611no—
325C2×D8—1222no—
326C2×D8—1221no—
327C2×D8C22×D6111yescharacteristic
328C2×M16—611no—
329C22×C8—611no—
330C4×Q8—21no—
331C4×Q8—1221no—
332C4×Q8—1221no—
333C4×Q8—1221no—
334C4×Q8—2441no—
335C4×Q8—2441no—
336C4×D4—21no—
337C4×D4—1221no—
338C4×D4—1221no—
339C4×D4—1221no—
340C4×D4—2441no—
341C4×D4—2441no—
342#117—611no—
343#117—1221no—
344#117—1221no—
345#117—2441no—
346#117—2441no—
347C2×C22⋊C4—2441no—
348C2×C22⋊C4—2441no—
349SD32—614no—
350SD32—614no—
351SD32—614no—
352SD32C4×D6114yescomplemented
353M32—1222no—
354C2×C16—611no—
355C2×C16—614no—
356C2×C16—611no—
357C2×C16C22×D6111yescharacteristic
358C2×C16C22×D6111yescharacteristic
359#107—611no—
360#107—611no—
361#107C22×D6111yescharacteristic
362#107C22×D6111yescharacteristic
363#106—1221no—
364#106—1221no—
365#103—21no—
366#103—1221no—
367#103—1221no—
368#103—1221no—
369#102—21no—
370#102—1221no—
371#102—1221no—
372#102—1221no—
373#97—611no—
374#97—611no—
375C4×C8—611no—
376C4×C8C22×D6111yescharacteristic
377C24—2441no—
378C2×Q8—21no—
379C2×Q8—1224no—
380C2×Q8—1221no—
381C2×Q8—2441no—
382C2×Q8—2442no—
383C2×D4—21no—
384C2×D4—1224no—
385C2×D4—1221no—
386C2×D4—2442no—
387C2×D4—2441no—
388C22×C4—41no—
389C22×C4—611no—
390C22×C4—2441no—
391C22×C4—2441no—
392C22×C4—2441no—
393C22×C4—2441no—
394Q16—21no—
395Q16—612no—
396Q16—1224no—
397Q16—1221no—
398Q16C2×C4×D6112yes—
399D8—21no—
400D8—612no—
401D8—1224no—
402D8—1221no—
403D8C2×C4×D6112yes—
404M16—1222no—
405C2×C8—611no—
406C2×C8—612no—
407C2×C8—612no—
408C2×C8—611no—
409C2×C8D4×D6111yescharacteristic
410C2×C8C23×D6111yescharacteristic
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—41no—
414C4⋊C4—611no—
415C4⋊C4—611no—
416C4⋊C4—1221no—
417C4⋊C4—1221no—
418C4⋊C4—2441no—
419C4⋊C4—2441no—
420C4⋊C4—2441no—
421C22⋊C4—41no—
422C22⋊C4—611no—
423C22⋊C4—611no—
424C22⋊C4—2441no—
425C22⋊C4—2441no—
426C22⋊C4—2441no—
427C22⋊C4—2441no—
428C22⋊C4—2441no—
429C42—41no—
430C42—611no—
431C42—2441no—
432C42—2441no—
433C42—2441no—
434C42—2441no—
435C42—2441no—
436C42D4×D6111yescharacteristic
437C16—21no—
438C16—612no—
439C16—1221no—
440C16C2×C4×D6112yes—
441C23—41no—
442C23—611no—
443C23—9761no—
444C23—9761no—
445C23—2444no—
446C23—2441no—
447Q8—22no—
448Q8—41no—
449Q8—1222no—
450Q8—2444no—
451Q8—2441no—
452D4—22no—
453D4—41no—
454D4—1222no—
455D4—2444no—
456D4—2441no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—161no—
461C2×C4—611no—
462C2×C4—612no—
463C2×C4—612no—
464C2×C4—611no—
465C2×C4—611no—
466C2×C4—9761no—
467C2×C4—9761no—
468C2×C4—9761no—
469C2×C4—2444no—
470C2×C4—2441no—
471C2×C4—2441no—
472C2×C4—2441no—
473C2×C4D8×D6111yescharacteristic
474C2×C4D8×D6111yescharacteristic
475C2×C4C2×D4×D6111yescharacteristic
476C8—21no—
477C8—612no—
478C8—1221no—
479C8C22×C4×D6111yescharacteristic
480C8#41×D6111yescharacteristic
481C22—42no—
482C22—81no—
483C22—81no—
484C22—612no—
485C22—1221no—
486C22—1221no—
487C22—9764no—
488C22—4881no—
489C22—4881no—
490C22—2442no—
491C22C2×D8×D6111yescharacteristic
492C4—21no—
493C4—21no—
494C4—42no—
495C4—81no—
496C4—81no—
497C4—612no—
498C4—1221no—
499C4—1221no—
500C4—4881no—
501C4—4881no—
502C4—2442no—
503C4C4×D4×D6111yescharacteristic
504C4#135×D6111yescharacteristic
505C2—82no—
506C2—1222no—
507C2—4882no—
508C2C4×D8×D6111yescharacteristic
509C2#2145896411yescharacteristic
510C2#2145896511yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutatorC8×C61C23×C4
FrattiniC2×C8C23×D61
FittingC61×#2175C2
socleC22×C61C22×D8
prime radicalC2×D8×D61C22
cosocleC2×C8×C61C24
hypercenter#2175D61
hypocenterC61#33457
LCSG ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2175

Complements

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

NH ≅ G/Ntype
2. #21454682C2semidirect
4. #21447685C2semidirect
5. #21447684C2semidirect
6. #21447682C2semidirect
7. #21447674C2semidirect
8. C4×#14519720C2semidirect
9. C4×#14519721C2semidirect
10. #21447668C2semidirect
11. #21447600C2semidirect
12. #21447537C2semidirect
14. #21447417C2semidirect
15. C61×#2175C2semidirect
23. SD32×D61C4semidirect
30. C4×Dic244C22semidirect
31. Q16×Dic61C22semidirect
32. #16839856C22semidirect
41. #16839589C22semidirect
42. #16839589C22semidirect
43. #16839588C22semidirect
44. #16839567C22semidirect
45. #16839566C22semidirect
46. #16839565C22semidirect
50. #16839537C22semidirect
51. C4×#11758185C22semidirect
52. C16×Dic61C22semidirect
54. C4×Q16×C61C22semidirect
56. C61×#366C22semidirect
57. C61×#358C22semidirect
59. C61×#346C22semidirect
65. Q16×D61C2×C4semidirect
89. #14519722C2×C4semidirect
90. #14519721C2×C4semidirect
91. #14519720C2×C4semidirect
95. C16×D61C2×C4semidirect
113. SD32×C61C2×C4semidirect
256. #2175D61semidirect
289. C4×Q16C2×D61semidirect
302. #366C2×D61semidirect
306. #358C2×D61semidirect
312. #346C2×D61semidirect
315. C61#33457semidirect
352. SD32C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×SD32×D612
#214546822
#214546662
#214476852
#214476842
#214476822
#214476742
C4×#145197202
C4×#145197212
#214476682
#214476002
#214475372
#214475332
#214474172
C61×#21752
#3345761