Finite Group Databasebeta

← #46848217

Subgroups — #46848217

Order 34688.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774591C211yesmaximal, characteristic, complemented
2#31774544C211yesmaximal, characteristic, complemented
3#31774250C211yesmaximal, characteristic, complemented
4#31774234C211yesmaximal, characteristic, complemented
5#31774196C211yesmaximal, characteristic, complemented
6#31774131C211yesmaximal, characteristic, complemented
7#31774127C211yesmaximal, characteristic, complemented
8#31774111C211yesmaximal, characteristic, complemented
9#31774044C211yesmaximal, characteristic, complemented
10#31774032C211yesmaximal, characteristic, complemented
11#31773979C211yesmaximal, characteristic, complemented
12(C22⋊D271)⋊C8C211yesmaximal, characteristic, complemented
13#31773891C211yesmaximal, characteristic, complemented
14#102×D271C211yesmaximal, characteristic, complemented
15#482×C271C211yesmaximal, characteristic, complemented
16C2×#16864654C2211yescharacteristic
17C2×D4×D271C2211yescharacteristic
18C2×#16864643C2211yescharacteristic, complemented
19C2×D2168C2211yescharacteristic
20C2×#16864639C2211yescharacteristic
21C22×C271⋊C8C2211yescharacteristic, complemented
22C2×C8×D271C2211yescharacteristic
23#22576154—21no—
24#22576151C2211yescharacteristic, complemented
25#22576147C2211yescharacteristic
26#22576142—21no—
27#22576142—21no—
28#22576140C2211yescharacteristic
29#22576135C2211yescharacteristic
30#22576134C2211yescharacteristic, complemented
31#22576133—21no—
32#22576131—21no—
33#22576128—21no—
34C4×D1084—21no—
35#22576122C2211yescharacteristic
36#22576122C2211yescharacteristic, complemented
37C4×C22⋊D271C2211yescharacteristic
38C4⋊C4×D271C2211yescharacteristic
39C22⋊C4×D271—21no—
40#22576093C2211yescharacteristic, complemented
41Dic271⋊C8C2211yescharacteristic, complemented
42#22576085C2211yescharacteristic, complemented
43#22576083C2211yescharacteristic, complemented
44#22576083C2211yescharacteristic, complemented
45#22576082C2211yescharacteristic
46#22576081C2211yescharacteristic
47#22576080C2211yescharacteristic
48#22576080C2211yescharacteristic
49#22576080C2211yescharacteristic, complemented
50(C2×D271)⋊C8C2211yescharacteristic
51C8×Dic271C2211yescharacteristic, complemented
52#121×C271C2211yescharacteristic
53#117×C271C2211yescharacteristic, complemented
54#107×C271C2211yescharacteristic, complemented
55#106×C271C2211yescharacteristic, complemented
56#102×C271C2211yescharacteristic
57#102×C271C2211yescharacteristic
58C22⋊C8×C271C2211yescharacteristic, complemented
59C23×D271—21no—
60#16864654—22no—
61#16864654—41no—
62C2×Dic542C2311yescharacteristic
63C2×D1084—21no—
64C2×D1084C2311yescharacteristic
65C2×D1084C2311yescharacteristic
66C2×C22⋊D271—21no—
67C2×C22⋊D271—21no—
68C2×C22⋊D271—21no—
69C2×C22⋊D271D411yescharacteristic
70D4×D271—22no—
71D4×D271—41no—
72C2×C4×D271—21no—
73C2×C4×D271—21no—
74C2×C4×D271—21no—
75C2×C4×D271—21no—
76C2×C4×D271C2311yescharacteristic
77#16864643—41no—
78D2168—22no—
79#16864639—41no—
80C2×C271⋊C8—41no—
81C2×C271⋊C8C2311yescharacteristic
82C2×C271⋊C8C2311yescharacteristic
83C8×D271—41no—
84C4⋊Dic271—21no—
85C4⋊Dic271—21no—
86C4⋊Dic271C2311yescharacteristic
87Dic271⋊C4—21no—
88Dic271⋊C4—21no—
89Dic271⋊C4D411yescharacteristic
90C22⋊Dic271—21no—
91C22⋊Dic271—21no—
92C22⋊Dic271D411yescharacteristic
93(C2×D271)⋊C4—21no—
94(C2×D271)⋊C4—21no—
95(C2×D271)⋊C4—21no—
96(C2×D271)⋊C4—21no—
97(C2×D271)⋊C4—21no—
98(C2×D271)⋊C4D411yescharacteristic
99C4×Dic271C2311yescharacteristic
100C2×D4×C271—21no—
101C2×D4×C271C2311yescharacteristic
102C22×C4×C271C2311yescharacteristic
103C2×C8×C271C2311yescharacteristic
104C2×C8×C271C2311yescharacteristic
105C4⋊C4×C271C2311yescharacteristic
106C4⋊C4×C271C2311yescharacteristic
107C4⋊C4×C271C2311yescharacteristic
108C22⋊C4×C271—21no—
109C22⋊C4×C271—21no—
110C42×C271—21no—
111C22×D271—21no—
112C22×D271—21no—
113C22×D271—42no—
114C22×D271—42no—
115C22×D271—81no—
116C22×D271C2×D411yescharacteristic
117Dic542—21no—
118Dic542—21no—
119D1084—21no—
120D1084—21no—
121D1084—21no—
122D1084—42no—
123D1084#4112yes—
124C22⋊D271—22no—
125C22⋊D271—42no—
126C22⋊D271—42no—
127C22⋊D271—81no—
128C2×Dic271—21no—
129C2×Dic271—21no—
130C2×Dic271C2×D411yescharacteristic
131C2×Dic271C2×D411yescharacteristic
132C2×Dic271C2×D411yescharacteristic
133C4×D271—21no—
134C4×D271—21no—
135C4×D271—42no—
136C4×D271—81no—
137C4×D271—81no—
138C4×D271—81no—
139C271⋊C8—21no—
140C271⋊C8—21no—
141C23×C271—21no—
142C23×C271C2×D411yescharacteristic
143D4×C271—21no—
144D4×C271—21no—
145D4×C271—81no—
146C2×C4×C271—21no—
147C2×C4×C271—21no—
148C2×C4×C271—21no—
149C2×C4×C271—21no—
150C2×C4×C271—21no—
151C2×C4×C271C2×D411yescharacteristic
152C2×C4×C271C2411yescharacteristic
153C8×C271—21no—
154C8×C271—21no—
155C22×C271—21no—
156C22×C271—21no—
157C22×C271—21no—
158C22×C271—41no—
159C22×C271—41no—
160C22×C271—41no—
161C22×C271C22×D411yescharacteristic
162C2×D271—21no—
163C2×D271—21no—
164C2×D271—22no—
165C2×D271—41no—
166C2×D271—41no—
167C2×D271—41no—
168C2×D271—41no—
169C2×D271—41no—
170C2×D271—84no—
171C2×D271#13512yes—
172C4×C271—21no—
173C4×C271—41no—
174C4×C271—41no—
175C4×C271—41no—
176C4×C271C2×#4111yescharacteristic
177C4×C271H(5,𝔽2)11yescharacteristic
178Dic271—21no—
179Dic271—21no—
180Dic271—21no—
181Dic271—41no—
182Dic271—41no—
183C2×C271—41no—
184C2×C271—81no—
185C2×C271#54611yescharacteristic
186C2×C271C2×#13511yescharacteristic
187C2×C271#57611yescharacteristic
188D271—22no—
189D271—42no—
190D271—81no—
191C271#329711yescharacteristic, complemented, Sylow
192#3297—2711nomaximal, Sylow
193#553—2711no—
194#546—2711no—
195#488—2711no—
196#484—2711no—
197#482D27111yescharacteristic, complemented
198#478—2711no—
199#466—2711no—
200#465—2711no—
201#463—2711no—
202#449—2711no—
203#447—2711no—
204C4×D8—2711no—
205C8×D4—2711no—
206#416—2711no—
207C2×#102—2711no—
208C2×#41—2711no—
209C22×D4—2711no—
210C2×SD16—2711no—
211C2×D8—2711no—
212C2×D8—2711no—
213C22×C8—2711no—
214C22×C8—2711no—
215#126—5421no—
216#125—2711no—
217#124—2711no—
218#123—5421no—
219#123—5421no—
220C22⋊Q8—2711no—
221#121—2711no—
222#121—2711no—
223#121—5421no—
224#121—5421no—
225#121C2×D27111yescharacteristic
226#120—5421no—
227C4×D4—2711no—
228C4×D4—2711no—
229C4×D4—2711no—
230C4×D4—5421no—
231#117C2×D27111yescharacteristic, complemented
232C2×C4⋊C4—2711no—
233C2×C22⋊C4—5421no—
234#107—2711no—
235#107C2×D27111yescharacteristic, complemented
236#106C2×D27111yescharacteristic, complemented
237C4⋊C8—2711no—
238#103—2711no—
239#103—2711no—
240#103—2711no—
241#102—2711no—
242#102—2711no—
243#102—2711no—
244#102—2711no—
245#102—2711no—
246#102C2×D27111yescharacteristic
247#102C2×D27111yescharacteristic
248C22⋊C8—2711no—
249C22⋊C8C2×D27111yescharacteristic, complemented
250C4×C8—2711no—
251C24—5421no—
252#41—5422no—
253#41—10841no—
254C2×Q8—2711no—
255C2×D4—21no—
256C2×D4—2711no—
257C2×D4—2711no—
258C2×D4—2711no—
259C2×D4—5421no—
260C2×D4—5421no—
261C2×D4—5422no—
262C2×D4—5421no—
263C2×D4—5421no—
264C2×D4—10841no—
265C2×D4C22×D27111yescharacteristic
266C22×C4—2711no—
267C22×C4—5421no—
268C22×C4—5421no—
269C22×C4—5421no—
270C22×C4—5421no—
271C22×C4C22×D27111yescharacteristic
272SD16—10841no—
273D8—5422no—
274D8—10841no—
275C2×C8—2711no—
276C2×C8—2711no—
277C2×C8—10841no—
278C2×C8—10841no—
279C2×C8C22×D27111yescharacteristic
280C2×C8C22×D27111yescharacteristic
281C4⋊C4—2711no—
282C4⋊C4—2711no—
283C4⋊C4—5421no—
284C4⋊C4—5421no—
285C4⋊C4—5421no—
286C4⋊C4—5421no—
287C4⋊C4C22×D27111yescharacteristic
288C4⋊C4C22×D27111yescharacteristic
289C4⋊C4C22×D27111yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—2711no—
293C22⋊C4—2711no—
294C22⋊C4—5421no—
295C22⋊C4—5421no—
296C22⋊C4—5421no—
297C22⋊C4—5421no—
298C22⋊C4—5421no—
299C22⋊C4—5421no—
300C22⋊C4—5421no—
301C42—21no—
302C42—2711no—
303C23—21no—
304C23—2711no—
305C23—5421no—
306C23—5421no—
307C23—10842no—
308C23—10842no—
309C23—21681no—
310C23D4×D27111yescharacteristic
311Q8—5421no—
312Q8—5421no—
313D4—21no—
314D4—21no—
315D4—81no—
316D4—2712no—
317D4—5422no—
318D4—5421no—
319D4—5421no—
320D4—5421no—
321D4—10842no—
322D4—10842no—
323D4—10842no—
324D4—21681no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—2711no—
331C2×C4—2711no—
332C2×C4—2711no—
333C2×C4—5421no—
334C2×C4—5421no—
335C2×C4—5421no—
336C2×C4—5421no—
337C2×C4—10842no—
338C2×C4—21681no—
339C2×C4—21681no—
340C2×C4—21681no—
341C2×C4D4×D27111yescharacteristic
342C2×C4C23×D27111yescharacteristic
343C8—21no—
344C8—21no—
345C8—5421no—
346C8—5421no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—2712no—
354C22—5421no—
355C22—5421no—
356C22—5422no—
357C22—10841no—
358C22—10841no—
359C22—10841no—
360C22—10841no—
361C22—10841no—
362C22—21684no—
363C22C2×D4×D27111yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—5421no—
369C4—5421no—
370C4—5421no—
371C4—10841no—
372C4—10841no—
373C4#41×D27111yescharacteristic
374C4#2257620611yescharacteristic
375C2—41no—
376C2—81no—
377C2—5422no—
378C2—10842no—
379C2—21681no—
380C2#3177453911yescharacteristic
381C2#135×D27111yescharacteristic
382C2#3177470111yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#482×C271C2
socleC22×C271C22×D4
hypercenter#482D271
hypocenterC271#3297
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
1. #31774591C2semidirect
2. #31774544C2semidirect
3. #31774250C2semidirect
4. #31774234C2semidirect
5. #31774196C2semidirect
6. #31774131C2semidirect
7. #31774127C2semidirect
8. #31774111C2semidirect
9. #31774044C2semidirect
10. #31774032C2semidirect
11. #31773979C2semidirect
12. (C22⋊D271)⋊C8C2semidirect
13. #31773891C2semidirect
14. #102×D271C2semidirect
15. #482×C271C2semidirect
18. C2×#16864643C22semidirect
21. C22×C271⋊C8C22semidirect
24. #22576151C22semidirect
30. #22576134C22semidirect
36. #22576122C22semidirect
40. #22576093C22semidirect
41. Dic271⋊C8C22semidirect
42. #22576085C22semidirect
43. #22576083C22semidirect
44. #22576083C22semidirect
49. #22576080C22semidirect
51. C8×Dic271C22semidirect
53. #117×C271C22semidirect
54. #107×C271C22semidirect
55. #106×C271C22semidirect
58. C22⋊C8×C271C22semidirect
191. C271#3297semidirect
197. #482D271semidirect
231. #117C2×D271semidirect
235. #107C2×D271semidirect
236. #106C2×D271semidirect
249. C22⋊C8C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317745912
#317745442
#317742502
#317742342
#317741962
#317741312
#317741272
#317741112
#317740442
#317740322
#317739792
(C22⋊D271)⋊C82
#317738912
#102×D2712
#482×C2712
#3297271