Finite Group Databasebeta

← #29833401

Subgroups — #29833401

Order 15616.

674 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840193C211yesmaximal, characteristic
2#21454694C211yesmaximal, characteristic, complemented
3#21454689C211yesmaximal, characteristic, complemented
4#21447693C211yesmaximal, characteristic, complemented
5#21447692C211yesmaximal, characteristic, complemented
6#21447688C211yesmaximal, characteristic, complemented
7#21447687C211yesmaximal, characteristic, complemented
8C4×#14519722C211yesmaximal, characteristic, complemented
9SD32×Dic61C211yesmaximal, characteristic, complemented
10#21447670C211yesmaximal, characteristic, complemented
11#367×D61C211yesmaximal, characteristic, complemented
12#21447530C211yesmaximal, characteristic, complemented
13#21447528C211yesmaximal, characteristic, complemented
14#21447414C211yesmaximal, characteristic, complemented
15C4×SD32×C61C211yesmaximal, characteristic, complemented
16C2×#14519800C2211yescharacteristic
17C2×#14519796C2211yescharacteristic
18#16840244—21no—
19#16840241—21no—
20#16840193C418yescomplemented
21C2×#14519723C2211yescharacteristic
22C2×#14519722C2211yescharacteristic
23C2×#14519718C2211yescharacteristic
24C2×#14519716C2211yescharacteristic
25C2×C16×D61C2211yescharacteristic
26#16839865—21no—
27#16839862—21no—
28#16839860—21no—
29#16839859—21no—
30C4×Dic244C2211yescharacteristic
31Q16×Dic61C2211yescharacteristic
32#16839856C2211yescharacteristic
33C4×D488C2211yescharacteristic
34D8×Dic61C2211yescharacteristic
35#16839844C2211yescharacteristic
36#107×D61C2211yescharacteristic
37#107×D61C2211yescharacteristic
38#16839762—21no—
39#16839760—21no—
40#16839716C2211yescharacteristic
41#16839590C2211yescharacteristic
42#16839589C2211yescharacteristic
43#16839589C2211yescharacteristic
44#16839567C2211yescharacteristic
45#16839566C2211yescharacteristic
46#16839565C2211yescharacteristic
47#16839564C2211yescharacteristic
48#16839563C2211yescharacteristic
49#16839562C2211yescharacteristic
50#16839538C2211yescharacteristic
51#16839537C2211yescharacteristic
52C16×Dic61C2211yescharacteristic
53C2×SD32×C61C2211yescharacteristic
54C4×Q16×C61C2211yescharacteristic
55C4×D8×C61C2211yescharacteristic
56C61×#367C2211yescharacteristic
57C61×#358C2211yescharacteristic
58C61×#357C2211yescharacteristic
59C4×C16×C61C2211yescharacteristic
60C2×#11758226—21no—
61C2×#11758224—21no—
62#14519800—22no—
63#14519800C2×C414yescomplemented
64#14519796—22no—
65#14519796C2×C414yescomplemented
66C2×Dic244C2311yescharacteristic
67C2×#11758200—21no—
68C2×#11758201—21no—
69C2×D488C2311yescharacteristic
70C2×#11758189D411yescharacteristic
71C2×C8×D61C2311yescharacteristic
72Q8×Dic61—21no—
73C4×Dic122—21no—
74#14519742—21no—
75C4×D244—21no—
76C4×D244—41no—
77D4×Dic61—21no—
78#14519739—21no—
79#14519739—41no—
80C4×C22⋊D61—41no—
81#14519737—41no—
82#14519734—41no—
83#14519734D411yescharacteristic
84#14519733—41no—
85C2×#11758196—41no—
86C4⋊C4×D61—21no—
87C4⋊C4×D61—21no—
88C4⋊C4×D61—41no—
89#14519723C2×C414yescomplemented
90#14519722C2×C414yescomplemented
91#14519718C2×C414yescomplemented
92#14519716—22no—
93C2×#11758185C2311yescharacteristic
94C2×#11758185C2311yescharacteristic
95C16×D61C2×C414yescomplemented
96#14519711C2311yescharacteristic
97#14519711C2311yescharacteristic
98#14519710—21no—
99#14519710—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
121C2×Q8×C61—21no—
122C2×D4×C61—21no—
123C22×C4×C61—41no—
124C2×C22⋊D61—41no—
125C22×Dic61—41no—
126C22×Dic61—41no—
127#11758226—24no—
128#11758226—42no—
129#11758224—24no—
130#11758224—42no—
131C2×D244—21no—
132C2×D244—41no—
133C2×C4×D61—41no—
134C2×C4×D61—41no—
135C2×C4×D61—41no—
136C2×C4×D61—41no—
137C2×C4×D61—41no—
138C2×C4×D61C2×D411yescharacteristic
139C2×Dic122—21no—
140Q16×C61—21no—
141Q16×C61C22×C412yes—
142D8×C61—21no—
143D8×C61C22×C412yes—
144C2×C8×C61C2×D411yescharacteristic
145C2×C8×C61C2411yescharacteristic
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—41no—
149C22⋊C4×C61—41no—
150C42×C61—41no—
151C42×C61C2×D411yescharacteristic
152C22⋊Dic61—41no—
153#11758201—24no—
154#11758200—24no—
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#11758196—41no—
165#11758196—41no—
166#11758196D811yescharacteristic
167#11758196D811yescharacteristic
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×D61C2×D811yescharacteristic
188Q8×C61—22no—
189Q8×C61—41no—
190D4×C61—22no—
191D4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61—161no—
196C2×C4×C61C2×D811yescharacteristic
197C2×C4×C61C2×D811yescharacteristic
198C2×C4×C61C22×D411yescharacteristic
199C22⋊D61—44no—
200C2×Dic61—41no—
201C2×Dic61—44no—
202C2×Dic61—41no—
203C2×Dic61—41no—
204C2×Dic61—161no—
205C2×Dic61C2×D811yescharacteristic
206C2×Dic61C2×D811yescharacteristic
207C2×Dic61C2×D811yescharacteristic
208D244—22no—
209D244—44no—
210D244—41no—
211C4×D61—44no—
212C4×D61—161no—
213C4×D61—161no—
214C4×D61—161no—
215C4×D61—161no—
216C4×D61C4×D412yes—
217C4×D61#13512yes—
218Dic122—22no—
219Dic122—41no—
220C8×C61—21no—
221C8×C61C23×C411yescharacteristic
222C8×C61C2×#4111yescharacteristic
223#92418—21no—
224#92418C4×D412yes—
225#33460—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×D61C4×D812yes—
236C4×C61—21no—
237C4×C61—21no—
238C4×C61—42no—
239C4×C61—81no—
240C4×C61—81no—
241C4×C61C2×C4×D411yescharacteristic
242C4×C61C2×#13511yescharacteristic
243Dic61—21no—
244Dic61—21no—
245Dic61—42no—
246Dic61—81no—
247Dic61—81no—
248Dic61C4×D812yes—
249C2×#508—611no—
250#2951—611no—
251#2951—611no—
252#2177—611no—
253#2177—611no—
254#2176—611no—
255#2176—611no—
256C4×SD32—611no—
257C4×SD32—611no—
258C4×SD32—611no—
259C4×SD32D6111yescharacteristic, complemented
260C2×#367—611no—
261#2139—611no—
262#2139—611no—
263#2108—611no—
264C2×C61—82no—
265C2×C61C2×C4×D811yescharacteristic
266C2×C61#341611yescharacteristic
267C2×C61#341611yescharacteristic
268D61—22no—
269D61—82no—
270C2×#135—611no—
271C2×#135—611no—
272#520—1221no—
273#520—1221no—
274#508—618no—
275C2×Q32—611no—
276C2×SD32—611no—
277C2×SD32C2×D6111yescharacteristic
278C2×D16—611no—
279C2×M32—611no—
280C22×C16—611no—
281#440—1221no—
282#440—1221no—
283#440—1221no—
284#440—1221no—
285C4×Q16—611no—
286C4×Q16—611no—
287C4×Q16—611no—
288C4×Q16C2×D6111yescharacteristic
289C4×D8—611no—
290C4×D8—611no—
291C4×D8—611no—
292C4×D8C2×D6111yescharacteristic
293C2×#107—611no—
294C2×#107—611no—
295#417—1221no—
296#417—1221no—
297#405—611no—
298#367—611no—
299#367—611no—
300#367—611no—
301#367C2×D6111yescharacteristic
302#358—611no—
303#358—611no—
304#358—611no—
305#358C2×D6111yescharacteristic
306#357—611no—
307#357—611no—
308#357—611no—
309#357C2×D6111yescharacteristic
310#346—611no—
311#346—611no—
312C4×C16—611no—
313C4×C16C2×D6111yescharacteristic
314C61#3346011yescharacteristic, complemented, Sylow
315C2×#41—1221no—
316C2×#41—1221no—
317#135—614no—
318#135—614no—
319#135—1222no—
320#135—1222no—
321C2×Q16—611no—
322C2×Q16C22×D6111yescharacteristic
323C2×SD16—1221no—
324C2×SD16—1221no—
325C2×D8—611no—
326C2×D8C22×D6111yescharacteristic
327C2×M16—611no—
328C22×C8—611no—
329C4×Q8—21no—
330C4×Q8—1221no—
331C4×Q8—1221no—
332C4×Q8—1221no—
333C4×D4—21no—
334C4×D4—1221no—
335C4×D4—1221no—
336C4×D4—1221no—
337C4×D4—2441no—
338C4×D4—2441no—
339C4×D4—2441no—
340C4×D4—2441no—
341#117—611no—
342#117—2441no—
343#117—2441no—
344C2×C4⋊C4—1221no—
345C2×C4⋊C4—1221no—
346C2×C4⋊C4—2441no—
347C2×C4⋊C4—2441no—
348Q32—614no—
349SD32—614no—
350SD32C4×D6114yescomplemented
351D16—614no—
352M32—1222no—
353C2×C16—611no—
354C2×C16—614no—
355C2×C16—611no—
356C2×C16C22×D6111yescharacteristic
357C2×C16C22×D6111yescharacteristic
358#107—611no—
359#107—611no—
360#107—1221no—
361#107—1221no—
362#107C22×D6111yescharacteristic
363#107C22×D6111yescharacteristic
364#103—21no—
365#103—1221no—
366#103—1221no—
367#103—1221no—
368#102—21no—
369#102—1221no—
370#102—1221no—
371#102—1221no—
372#97—611no—
373#97—611no—
374C4×C8—611no—
375C4×C8C22×D6111yescharacteristic
376#41—1224no—
377#41—1224no—
378#41—2442no—
379#41—2442no—
380C2×Q8—21no—
381C2×Q8—1221no—
382C2×D4—21no—
383C2×D4—1221no—
384C2×D4—2441no—
385C2×D4—2441no—
386C22×C4—41no—
387C22×C4—611no—
388C22×C4—2441no—
389C22×C4—2441no—
390C22×C4—2441no—
391C22×C4—2441no—
392C22×C4—2441no—
393C22×C4—2441no—
394C22×C4—2441no—
395Q16—21no—
396Q16—612no—
397Q16—1221no—
398Q16C2×C4×D6112yes—
399SD16—1224no—
400SD16—1224no—
401D8—21no—
402D8—612no—
403D8—1221no—
404D8C2×C4×D6112yes—
405M16—1222no—
406C2×C8—611no—
407C2×C8—612no—
408C2×C8—612no—
409C2×C8—611no—
410C2×C8D4×D6111yescharacteristic
411C2×C8C23×D6111yescharacteristic
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—41no—
415C4⋊C4—611no—
416C4⋊C4—611no—
417C4⋊C4—1221no—
418C4⋊C4—1221no—
419C4⋊C4—2441no—
420C4⋊C4—2441no—
421C4⋊C4—2441no—
422C4⋊C4—2441no—
423C4⋊C4—2441no—
424C22⋊C4—41no—
425C22⋊C4—611no—
426C22⋊C4—611no—
427C22⋊C4—2441no—
428C22⋊C4—2441no—
429C22⋊C4—2441no—
430C42—41no—
431C42—611no—
432C42—2441no—
433C42—2441no—
434C42—2441no—
435C42D4×D6111yescharacteristic
436C16—21no—
437C16—612no—
438C16—1221no—
439C16C2×C4×D6112yes—
440C23—41no—
441C23—611no—
442C23—2441no—
443Q8—22no—
444Q8—41no—
445Q8—1222no—
446Q8—2441no—
447D4—22no—
448D4—41no—
449D4—1222no—
450D4—2444no—
451D4—2444no—
452D4—2441no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—161no—
457C2×C4—611no—
458C2×C4—612no—
459C2×C4—612no—
460C2×C4—611no—
461C2×C4—611no—
462C2×C4—9761no—
463C2×C4—9761no—
464C2×C4—9761no—
465C2×C4—9761no—
466C2×C4—9761no—
467C2×C4—2444no—
468C2×C4—2444no—
469C2×C4—2441no—
470C2×C4—2441no—
471C2×C4—2441no—
472C2×C4D8×D6111yescharacteristic
473C2×C4D8×D6111yescharacteristic
474C2×C4C2×D4×D6111yescharacteristic
475C8—21no—
476C8—612no—
477C8—1221no—
478C8C22×C4×D6111yescharacteristic
479C8#41×D6111yescharacteristic
480C22—42no—
481C22—81no—
482C22—81no—
483C22—612no—
484C22—1221no—
485C22—1221no—
486C22—4881no—
487C22—4881no—
488C22—2442no—
489C22C2×D8×D6111yescharacteristic
490C4—21no—
491C4—21no—
492C4—42no—
493C4—81no—
494C4—81no—
495C4—612no—
496C4—1221no—
497C4—1221no—
498C4—4881no—
499C4—4881no—
500C4—2442no—
501C4C4×D4×D6111yescharacteristic
502C4#135×D6111yescharacteristic
503C2—82no—
504C2—1222no—
505C2—4882no—
506C2C4×D8×D6111yescharacteristic
507C2#2145896111yescharacteristic
508C2#2145896211yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutatorC8×C61C23×C4
FrattiniC2×C8C23×D61
FittingC4×SD32×C61C2
socleC22×C61C22×D8
prime radicalC2×#16840193C2
cosocleC2×C8×C61C24
hypercenterC4×SD32D61
hypocenterC61#33460
LCSG ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ C4×SD32

Complements

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

NH ≅ G/Ntype
2. #21454694C2semidirect
3. #21454689C2semidirect
4. #21447693C2semidirect
5. #21447692C2semidirect
6. #21447688C2semidirect
7. #21447687C2semidirect
8. C4×#14519722C2semidirect
9. SD32×Dic61C2semidirect
10. #21447670C2semidirect
11. #367×D61C2semidirect
12. #21447530C2semidirect
13. #21447528C2semidirect
14. #21447414C2semidirect
15. C4×SD32×C61C2semidirect
20. #16840193C4semidirect
63. #14519800C2×C4semidirect
65. #14519796C2×C4semidirect
89. #14519723C2×C4semidirect
90. #14519722C2×C4semidirect
91. #14519718C2×C4semidirect
95. C16×D61C2×C4semidirect
113. SD32×C61C2×C4semidirect
259. C4×SD32D61semidirect
314. C61#33460semidirect
350. SD32C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168401932
#214546942
#214546892
#214476932
#214476922
#214476882
#214476872
C4×#145197222
SD32×Dic612
#214476702
#367×D612
#214475302
#214475282
#214474142
C4×SD32×C612
#3346061