Finite Group Databasebeta

← #47986015

Subgroups — #47986015

Order 35456.

792 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22579838C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×D277C211yesmaximal, characteristic
3C2×C22⋊C4×D277C211yesmaximal, characteristic
4#31799896C211yesmaximal, characteristic, complemented
5#31799895C211yesmaximal, characteristic, complemented
6#31799890C211yesmaximal, characteristic, complemented
7#31799876C211yesmaximal, characteristic, complemented
8#31799873C211yesmaximal, characteristic, complemented
9#31799864C211yesmaximal, characteristic, complemented
10#31799861C211yesmaximal, characteristic, complemented
11#31799857C211yesmaximal, characteristic
12C4×#17984540C211yesmaximal, characteristic, complemented
13C4×#17984540C211yesmaximal, characteristic, complemented
14#31799849C211yesmaximal, characteristic, complemented
15#388×C277C211yesmaximal, characteristic, complemented
16C24×D277C2211yescharacteristic
17C22×C4×D277C2211yescharacteristic
18C22×C4×D277C2211yescharacteristic
19C22×C4×D277C2211yescharacteristic
20C22×C4×D277C2211yescharacteristic
21#22579838—24no—
22C2×C4⋊Dic277C2211yescharacteristic, complemented
23C2×#17984542C2211yescharacteristic, complemented
24C2×#17984542C2211yescharacteristic, complemented
25C2×C22⋊Dic277C2211yescharacteristic
26C2×C22⋊Dic277C2211yescharacteristic
27C2×C22⋊Dic277C2211yescharacteristic
28C2×#17984540C2211yescharacteristic
29C2×#17984540C2211yescharacteristic
30C2×#17984540C2211yescharacteristic
31C2×#17984540C2211yescharacteristic
32C2×#17984540C2211yescharacteristic
33C2×#17984540C2211yescharacteristic
34C2×#17984540C2211yescharacteristic
35C2×#17984540C2211yescharacteristic
36C22⋊C4×D277—24no—
37C22⋊C4×D277C418yescomplemented
38C2×C4×Dic277C2211yescharacteristic, complemented
39C2×C4×Dic277C2211yescharacteristic, complemented
40C2×C4×Dic277C2211yescharacteristic, complemented
41#22579787C2211yescharacteristic, complemented
42#22579787C2211yescharacteristic, complemented
43#22579787C2211yescharacteristic, complemented
44#22579787C2211yescharacteristic, complemented
45#22579787C2211yescharacteristic, complemented
46#22579787C2211yescharacteristic, complemented
47C2×C4⋊C4×C277C2211yescharacteristic, complemented
48C2×C22⋊C4×C277C2211yescharacteristic
49C2×C22⋊C4×C277C2211yescharacteristic
50C2×C22⋊C4×C277C2211yescharacteristic
51C2×C42×C277C2211yescharacteristic, complemented
52#95×C277C2211yescharacteristic, complemented
53#95×C277C2211yescharacteristic, complemented
54C23×D277—22no—
55C23×D277—22no—
56C23×D277—22no—
57C23×D277—22no—
58C23×D277—41no—
59C23×D277—41no—
60C23×D277C2×C414yes—
61C23×D277C2311yescharacteristic
62C23×D277C2311yescharacteristic
63C22×Dic277C2311yescharacteristic
64C22×Dic277C2311yescharacteristic
65C22×Dic277C2311yescharacteristic
66C22×Dic277C2311yescharacteristic
67C22×Dic277C2311yescharacteristic
68C22×Dic277C2311yescharacteristic
69C2×C4×D277—22no—
70C2×C4×D277—22no—
71C2×C4×D277—22no—
72C2×C4×D277—22no—
73C2×C4×D277—22no—
74C2×C4×D277—22no—
75C2×C4×D277—41no—
76C2×C4×D277—41no—
77C2×C4×D277—41no—
78C2×C4×D277—41no—
79C2×C4×D277C2×C414yescomplemented
80C2×C4×D277C2×C414yescomplemented
81C4⋊Dic277—22no—
82#17984542—41no—
83#17984542—41no—
84C22⋊Dic277—22no—
85C22⋊Dic277—41no—
86C22⋊Dic277C2×C414yescomplemented
87#17984540—22no—
88#17984540—22no—
89#17984540—22no—
90#17984540—22no—
91#17984540—22no—
92#17984540C2×C414yes—
93#17984540C2×C414yes—
94#17984540D414yes—
95C4×Dic277—22no—
96C4×Dic277—22no—
97C4×Dic277—41no—
98C24×C277C2311yescharacteristic
99C22×C4×C277C2311yescharacteristic
100C22×C4×C277C2311yescharacteristic
101C22×C4×C277C2311yescharacteristic
102C22×C4×C277C2311yescharacteristic
103C22×C4×C277C2311yescharacteristic
104C22×C4×C277C2311yescharacteristic
105C4⋊C4×C277—22no—
106C22⋊C4×C277—22no—
107C22⋊C4×C277—41no—
108C22⋊C4×C277C2×C414yescomplemented
109C42×C277—41no—
110C22×D277—21no—
111C22×D277—21no—
112C22×D277—21no—
113C22×D277—21no—
114C22×D277—21no—
115C22×D277—21no—
116C22×D277—21no—
117C22×D277—21no—
118C22×D277—28no—
119C22×D277—28no—
120C22×D277—44no—
121C22×D277—44no—
122C22×D277—44no—
123C22×D277—44no—
124C22×D277—44no—
125C22×D277C22×C412yes—
126C22×D277C22×C412yes—
127C22×D277C2×D412yes—
128C22×D277#4112yes—
129C22×D277#4112yes—
130C22×D277#4112yes—
131C2×Dic277—21no—
132C2×Dic277—21no—
133C2×Dic277—21no—
134C2×Dic277—21no—
135C2×Dic277—21no—
136C2×Dic277—21no—
137C2×Dic277—21no—
138C2×Dic277—21no—
139C2×Dic277—21no—
140C2×Dic277—21no—
141C2×Dic277—21no—
142C2×Dic277—21no—
143C2×Dic277—21no—
144C2×Dic277—21no—
145C2×Dic277—21no—
146C2×Dic277C22×C412yes—
147C2×Dic277C22×C412yes—
148C2×Dic277C2×D412yes—
149C4×D277—28no—
150C4×D277—44no—
151C4×D277—44no—
152C4×D277—44no—
153C23×C277—21no—
154C23×C277—21no—
155C23×C277—21no—
156C23×C277—21no—
157C23×C277—21no—
158C23×C277—21no—
159C23×C277C22×C412yes—
160C23×C277C2411yescharacteristic
161C2×C4×C277—21no—
162C2×C4×C277—21no—
163C2×C4×C277—21no—
164C2×C4×C277—21no—
165C2×C4×C277—21no—
166C2×C4×C277—21no—
167C2×C4×C277—21no—
168C2×C4×C277—21no—
169C2×C4×C277—21no—
170C2×C4×C277—21no—
171C2×C4×C277—21no—
172C2×C4×C277—21no—
173C2×C4×C277—21no—
174C2×C4×C277—21no—
175C2×C4×C277—21no—
176C2×C4×C277C22×C412yes—
177C2×C4×C277C22×C412yes—
178C2×C4×C277C2×D412yes—
179C22×C277—22no—
180C22×C277—22no—
181C22×C277—22no—
182C22×C277—41no—
183C22×C277—41no—
184C22×C277—41no—
185C22×C277—41no—
186C22×C277C23×C411yescharacteristic
187C22×C277C22×D411yescharacteristic
188C22×C277C2×#4111yescharacteristic
189C22×C277C2×#4111yescharacteristic
190C22×C277C2×#4111yescharacteristic
191C22×C277H(5,𝔽2)11yescharacteristic
192C22×C277H(5,𝔽2)11yescharacteristic
193C2×D277—22no—
194C2×D277—22no—
195C2×D277—22no—
196C2×D277—22no—
197C2×D277—22no—
198C2×D277—22no—
199C2×D277—22no—
200C2×D277—22no—
201C2×D277—22no—
202C2×D277—22no—
203C2×D277—22no—
204C2×D277—22no—
205C2×D277—416no—
206C2×D277#11714yescomplemented
207C2×D277C4×D414yes—
208C4×C277—22no—
209C4×C277—22no—
210C4×C277—22no—
211C4×C277—41no—
212C4×C277—41no—
213C4×C277—41no—
214Dic277—22no—
215Dic277—22no—
216Dic277—22no—
217Dic277—41no—
218Dic277—41no—
219Dic277—41no—
220C2×C277—42no—
221C2×C277C2×#11711yescharacteristic
222C2×C277C2×C4×D411yescharacteristic
223C2×C277#51811yescharacteristic
224C2×C277#54611yescharacteristic
225C2×C277#54611yescharacteristic
226C2×C277#55111yescharacteristic
227C2×C277#55111yescharacteristic
228D277—24no—
229D277—24no—
230C277#234211yescharacteristic, complemented, Sylow
231#2342—2771nomaximal, Sylow
232C2×#117—2771no—
233C22×C22⋊C4—2771no—
234C22×C22⋊C4—2771no—
235#388—2771no—
236#388—2771no—
237#388—2771no—
238#388D27711yescharacteristic, complemented
239#385—2771no—
240#385—2771no—
241#381—2771no—
242#380—2771no—
243#379—2771no—
244C4×C22⋊C4—2771no—
245C4×C22⋊C4—2771no—
246C4×C22⋊C4—2771no—
247C25—2771no—
248C23×C4—2771no—
249C23×C4—2771no—
250C23×C4—2771no—
251C23×C4—2771no—
252#117—5544no—
253C2×C4⋊C4—2771no—
254C2×C4⋊C4—2771no—
255C2×C4⋊C4—2771no—
256C2×C4⋊C4C2×D27711yescharacteristic, complemented
257C2×C22⋊C4—2771no—
258C2×C22⋊C4—2771no—
259C2×C22⋊C4—2771no—
260C2×C22⋊C4—2771no—
261C2×C22⋊C4—2778no—
262C2×C22⋊C4—2771no—
263C2×C22⋊C4—2771no—
264C2×C22⋊C4—2771no—
265C2×C22⋊C4—2771no—
266C2×C22⋊C4—2771no—
267C2×C22⋊C4—2771no—
268C2×C22⋊C4—2771no—
269C2×C22⋊C4—5544no—
270C2×C22⋊C4C2×D27711yescharacteristic
271C2×C22⋊C4C2×D27711yescharacteristic
272C2×C22⋊C4C2×D27711yescharacteristic
273C2×C42—2771no—
274C2×C42—2771no—
275C2×C42—2771no—
276C2×C42C2×D27711yescharacteristic, complemented
277#95—2771no—
278#95—2771no—
279#95—2771no—
280#95—2771no—
281#95—2771no—
282#95—2771no—
283#95C2×D27711yescharacteristic, complemented
284#95C2×D27711yescharacteristic, complemented
285C24—2771no—
286C24—2771no—
287C24—2774no—
288C24—5542no—
289C24—5542no—
290C24—5542no—
291C24—5542no—
292C24—11081no—
293C24—11081no—
294C24C22×D27711yescharacteristic
295C22×C4—2771no—
296C22×C4—2771no—
297C22×C4—2771no—
298C22×C4—2774no—
299C22×C4—2774no—
300C22×C4—2771no—
301C22×C4—2771no—
302C22×C4—2771no—
303C22×C4—5542no—
304C22×C4—5542no—
305C22×C4—5542no—
306C22×C4—5542no—
307C22×C4—5542no—
308C22×C4—5542no—
309C22×C4—11081no—
310C22×C4—11081no—
311C22×C4—11081no—
312C22×C4—11081no—
313C22×C4C22×D27711yescharacteristic
314C22×C4C22×D27711yescharacteristic
315C22×C4C22×D27711yescharacteristic
316C22×C4C22×D27711yescharacteristic
317C22×C4C22×D27711yescharacteristic
318C22×C4C22×D27711yescharacteristic
319C4⋊C4—22no—
320C4⋊C4—5542no—
321C4⋊C4—11081no—
322C4⋊C4—11081no—
323C22⋊C4—22no—
324C22⋊C4—41no—
325C22⋊C4—2774no—
326C22⋊C4—2774no—
327C22⋊C4—2774no—
328C22⋊C4—2774no—
329C22⋊C4—5542no—
330C22⋊C4—5542no—
331C22⋊C4—5542no—
332C22⋊C4—5542no—
333C22⋊C4—5542no—
334C22⋊C4—5542no—
335C22⋊C4—11081no—
336C22⋊C4C4×D27714yescomplemented
337C42—41no—
338C42—5542no—
339C42—5542no—
340C42—11081no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—2772no—
348C23—2772no—
349C23—2772no—
350C23—2772no—
351C23—2772no—
352C23—2772no—
353C23—5548no—
354C23—5548no—
355C23—5541no—
356C23—5541no—
357C23—5541no—
358C23—5541no—
359C23—5541no—
360C23—5541no—
361C23—5541no—
362C23—5541no—
363C23—11084no—
364C23—11084no—
365C23—11084no—
366C23—11084no—
367C23—11084no—
368C23C2×C4×D27712yes—
369C23C23×D27711yescharacteristic
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—2772no—
386C2×C4—2772no—
387C2×C4—2772no—
388C2×C4—5548no—
389C2×C4—5541no—
390C2×C4—5541no—
391C2×C4—5541no—
392C2×C4—5541no—
393C2×C4—5541no—
394C2×C4—5541no—
395C2×C4—5541no—
396C2×C4—5541no—
397C2×C4—5541no—
398C2×C4—5541no—
399C2×C4—5541no—
400C2×C4—5541no—
401C2×C4—5541no—
402C2×C4—5541no—
403C2×C4—5541no—
404C2×C4—11084no—
405C2×C4—11084no—
406C2×C4—11084no—
407C2×C4C2×C4×D27712yes—
408C2×C4C2×C4×D27712yes—
409C2×C4D4×D27712yes—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—2774no—
418C22—2774no—
419C22—5542no—
420C22—5542no—
421C22—5542no—
422C22—5542no—
423C22—5542no—
424C22—5542no—
425C22—5542no—
426C22—5542no—
427C22—5542no—
428C22—5542no—
429C22—5542no—
430C22—5542no—
431C22—110816no—
432C22C22×C4×D27711yescharacteristic
433C22C2×D4×D27711yescharacteristic
434C22#41×D27711yescharacteristic
435C22#41×D27711yescharacteristic
436C22#41×D27711yescharacteristic
437C22#2257992511yescharacteristic
438C22#2257992611yescharacteristic
439C4—22no—
440C4—22no—
441C4—22no—
442C4—41no—
443C4—41no—
444C4—41no—
445C4—5542no—
446C4—5542no—
447C4—5542no—
448C4—11081no—
449C4—11081no—
450C4—11081no—
451C2—42no—
452C2—5544no—
453C2—5544no—
454C2#117×D27711yescharacteristic
455C2C4×D4×D27711yescharacteristic
456C2#3180047411yescharacteristic
457C2#3180064111yescharacteristic
458C2#3180064511yescharacteristic
459C2#3180067711yescharacteristic
460C2#3180068011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D277
commutatorC22×C277C23×C4
Fitting#388×C277C2
socle, cosocleC23×C277C24
prime radicalC24×D277C22
hypercenter#388D277
hypocenterC277#2342
LCSG ▷ C22×C277 ▷ C277
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#22579838C2semidirect
4. #31799896C2semidirect
5. #31799895C2semidirect
6. #31799890C2semidirect
7. #31799876C2semidirect
8. #31799873C2semidirect
9. #31799864C2semidirect
10. #31799861C2semidirect
12. C4×#17984540C2semidirect
13. C4×#17984540C2semidirect
14. #31799849C2semidirect
15. #388×C277C2semidirect
22. C2×C4⋊Dic277C22semidirect
23. C2×#17984542C22semidirect
24. C2×#17984542C22semidirect
37. C22⋊C4×D277C4semidirect
38. C2×C4×Dic277C22semidirect
39. C2×C4×Dic277C22semidirect
40. C2×C4×Dic277C22semidirect
41. #22579787C22semidirect
42. #22579787C22semidirect
43. #22579787C22semidirect
44. #22579787C22semidirect
45. #22579787C22semidirect
46. #22579787C22semidirect
47. C2×C4⋊C4×C277C22semidirect
51. C2×C42×C277C22semidirect
52. #95×C277C22semidirect
53. #95×C277C22semidirect
79. C2×C4×D277C2×C4semidirect
80. C2×C4×D277C2×C4semidirect
86. C22⋊Dic277C2×C4semidirect
108. C22⋊C4×C277C2×C4semidirect
206. C2×D277#117semidirect
230. C277#2342semidirect
238. #388D277semidirect
256. C2×C4⋊C4C2×D277semidirect
276. C2×C42C2×D277semidirect
283. #95C2×D277semidirect
284. #95C2×D277semidirect
336. C22⋊C4C4×D277semidirect

Maximal subgroups

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

subgroupindex
C2×#225798382
C2×C22⋊C4×D2772
#317998962
#317998952
#317998902
#317998762
#317998732
#317998642
#317998612
#317998572
C4×#179845402
#317998492
#388×C2772
#2342277