Finite Group Databasebeta

← #46848199

Subgroups — #46848199

Order 34688.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774587C211yesmaximal, characteristic, complemented
2#31774251C211yesmaximal, characteristic, complemented
3#31774225C211yesmaximal, characteristic, complemented
4#31774218C211yesmaximal, characteristic, complemented
5#31774197C211yesmaximal, characteristic, complemented
6#31774136C211yesmaximal, characteristic, complemented
7#31774124C211yesmaximal, characteristic, complemented
8#31774116C211yesmaximal, characteristic, complemented
9#31774059C211yesmaximal, characteristic, complemented
10#31774048C211yesmaximal, characteristic, complemented
11C4×#16864642C211yesmaximal, characteristic, complemented
12D4×C271⋊C8C211yesmaximal, characteristic, complemented
13#31773893C211yesmaximal, characteristic, complemented
14C2×#22576081C211yesmaximal, characteristic, complemented
15#546×C271C211yesmaximal, characteristic
16C2×#16864644C2211yescharacteristic, complemented
17C2×#16864642C2211yescharacteristic
18C2×#16864642C2211yescharacteristic
19C22×C271⋊C8C2211yescharacteristic
20C22×C271⋊C8C2211yescharacteristic, complemented
21#22576153—21no—
22#22576152C2211yescharacteristic, complemented
23#22576150—21no—
24#22576141—21no—
25#22576139C2211yescharacteristic
26#22576139C2211yescharacteristic, complemented
27C4×Dic542C2211yescharacteristic, complemented
28#22576119C2211yescharacteristic, complemented
29#22576116—21no—
30C2×C4⋊Dic271C2211yescharacteristic
31#22576093C2211yescharacteristic, complemented
32#22576091C2211yescharacteristic, complemented
33#22576091C2211yescharacteristic, complemented
34C4⋊(C271⋊C8)C2211yescharacteristic, complemented
35#22576084C2211yescharacteristic, complemented
36#22576084C2211yescharacteristic, complemented
37#22576084C2211yescharacteristic, complemented
38#22576083C2211yescharacteristic, complemented
39#22576081C2211yescharacteristic
40#22576081C2211yescharacteristic
41#22576081C2211yescharacteristic
42#22576081C2211yescharacteristic
43#22576081C2211yescharacteristic
44#22576081C2211yescharacteristic
45C22⋊(C271⋊C8)C2211yescharacteristic
46C22⋊(C271⋊C8)C2211yescharacteristic, complemented
47C4×C271⋊C8C2211yescharacteristic, complemented
48C2×#41×C271C2211yescharacteristic
49C22×D4×C271C2211yescharacteristic
50#124×C271C2211yescharacteristic
51#123×C271—21no—
52C22⋊Q8×C271C2211yescharacteristic
53#121×C271—21no—
54#121×C271C2211yescharacteristic
55#120×C271—21no—
56C4×D4×C271C2211yescharacteristic
57C4×D4×C271C2211yescharacteristic
58C2×C22⋊C4×C271—21no—
59C2×Dic542C2311yescharacteristic
60C22×Dic271—21no—
61#16864644—41no—
62#16864642—22no—
63#16864642—41no—
64C2×C271⋊C8—41no—
65C2×C271⋊C8—41no—
66C2×C271⋊C8C2311yescharacteristic
67C2×C271⋊C8C2311yescharacteristic
68C2×C271⋊C8C2311yescharacteristic
69C2×C271⋊C8C2311yescharacteristic
70C4⋊Dic271—21no—
71C4⋊Dic271C2311yescharacteristic
72C4⋊Dic271C2311yescharacteristic
73C4⋊Dic271C2311yescharacteristic
74Dic271⋊C4—21no—
75Dic271⋊C4—21no—
76Dic271⋊C4—21no—
77Dic271⋊C4—21no—
78Dic271⋊C4—21no—
79C22⋊Dic271—21no—
80C22⋊Dic271—21no—
81C22⋊Dic271—21no—
82C4×Dic271—21no—
83C4×Dic271—21no—
84C24×C271—21no—
85#41×C271—22no—
86#41×C271—41no—
87C2×Q8×C271C2311yescharacteristic
88C2×D4×C271—22no—
89C2×D4×C271—21no—
90C2×D4×C271—21no—
91C2×D4×C271—21no—
92C2×D4×C271—41no—
93C2×D4×C271D411yescharacteristic
94C2×D4×C271C2311yescharacteristic
95C2×D4×C271C2311yescharacteristic
96C22×C4×C271—21no—
97C22×C4×C271—21no—
98C22×C4×C271C2311yescharacteristic
99C22×C4×C271C2311yescharacteristic
100C4⋊C4×C271—21no—
101C4⋊C4×C271D411yescharacteristic
102C4⋊C4×C271C2311yescharacteristic
103C22⋊C4×C271—21no—
104C22⋊C4×C271—21no—
105C22⋊C4×C271—21no—
106C22⋊C4×C271—21no—
107C22⋊C4×C271—21no—
108C22⋊C4×C271D411yescharacteristic
109C22⋊C4×C271D411yescharacteristic
110C42×C271C2311yescharacteristic
111Dic542—21no—
112Dic542—21no—
113C2×Dic271—21no—
114C2×Dic271—21no—
115C2×Dic271—21no—
116C2×Dic271—21no—
117C2×Dic271—42no—
118C271⋊C8—21no—
119C271⋊C8—21no—
120C271⋊C8—21no—
121C271⋊C8—21no—
122C23×C271—21no—
123C23×C271—21no—
124C23×C271—42no—
125C23×C271—42no—
126C23×C271—81no—
127C23×C271C2×D411yescharacteristic
128C23×C271C2×D411yescharacteristic
129Q8×C271—21no—
130Q8×C271—21no—
131D4×C271—22no—
132D4×C271—21no—
133D4×C271—21no—
134D4×C271—21no—
135D4×C271—42no—
136D4×C271—42no—
137D4×C271—81no—
138D4×C271#4112yes—
139C2×C4×C271—21no—
140C2×C4×C271—21no—
141C2×C4×C271—21no—
142C2×C4×C271—21no—
143C2×C4×C271—21no—
144C2×C4×C271—21no—
145C2×C4×C271—42no—
146C2×C4×C271—81no—
147C2×C4×C271C2×D411yescharacteristic
148C2×C4×C271C2×D411yescharacteristic
149C2×C4×C271C2×D411yescharacteristic
150C2×C4×C271C2×D411yescharacteristic
151C2×C4×C271C2411yescharacteristic
152C22×C271—21no—
153C22×C271—21no—
154C22×C271—21no—
155C22×C271—22no—
156C22×C271—21no—
157C22×C271—21no—
158C22×C271—41no—
159C22×C271—41no—
160C22×C271—41no—
161C22×C271—41no—
162C22×C271—41no—
163C22×C271—84no—
164C22×C271#13512yes—
165C22×C271C22×D411yescharacteristic
166C4×C271—21no—
167C4×C271—21no—
168C4×C271—21no—
169C4×C271—21no—
170C4×C271—41no—
171C4×C271—41no—
172C4×C271C2×#4111yescharacteristic
173C4×C271H(5,𝔽2)11yescharacteristic
174Dic271—41no—
175Dic271—41no—
176Dic271—41no—
177Dic271—41no—
178C2×C271—22no—
179C2×C271—41no—
180C2×C271—42no—
181C2×C271—81no—
182C2×C271#54611yescharacteristic
183C2×C271C2×#13511yescharacteristic
184C2×C271#57711yescharacteristic
185C271#329611yescharacteristic, complemented, Sylow
186#3296—2711nomaximal, Sylow
187#552—2711no—
188#546D27111yescharacteristic
189#488—2711no—
190#483—2711no—
191#482—2711no—
192#478—2711no—
193#467—2711no—
194#465—2711no—
195#464—2711no—
196#452—2711no—
197#450—2711no—
198C4×SD16—2711no—
199C8×D4—2711no—
200#416—2711no—
201C2×#102—2711no—
202C2×#41C2×D27111yescharacteristic
203C22×D4C2×D27111yescharacteristic
204C2×Q16—2711no—
205C2×SD16—2711no—
206C2×SD16—2711no—
207C22×C8—2711no—
208C22×C8—2711no—
209#126—5421no—
210#125—2711no—
211#125—5421no—
212#124C2×D27111yescharacteristic
213#123—21no—
214#123—5421no—
215C22⋊Q8—2711no—
216C22⋊Q8—2711no—
217C22⋊Q8C2×D27111yescharacteristic
218#121—21no—
219#121C2×D27111yescharacteristic
220#120—21no—
221C4×Q8—2711no—
222C4×D4C2×D27111yescharacteristic
223C4×D4C2×D27111yescharacteristic
224#117—2711no—
225#117—5421no—
226C2×C4⋊C4—2711no—
227C2×C22⋊C4—21no—
228#107—2711no—
229#106—2711no—
230#106—2711no—
231C4⋊C8—2711no—
232#103—2711no—
233#103—2711no—
234#103—2711no—
235#103—2711no—
236#102—2711no—
237#102—2711no—
238#102—2711no—
239#102—2711no—
240#102—2711no—
241#102—2711no—
242C22⋊C8—2711no—
243C22⋊C8—2711no—
244C4×C8—2711no—
245C24—21no—
246#41—22no—
247#41—41no—
248C2×Q8—2711no—
249C2×Q8C22×D27111yescharacteristic
250C2×D4—22no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—41no—
255C2×D4C22⋊D27111yescharacteristic
256C2×D4C22×D27111yescharacteristic
257C2×D4C22×D27111yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—5421no—
261C22×C4C22×D27111yescharacteristic
262C22×C4C22×D27111yescharacteristic
263Q16—10841no—
264SD16—5422no—
265SD16—10841no—
266C2×C8—2711no—
267C2×C8—2711no—
268C2×C8—2711no—
269C2×C8—2711no—
270C2×C8—10841no—
271C2×C8—10841no—
272C4⋊C4—21no—
273C4⋊C4—2711no—
274C4⋊C4—2711no—
275C4⋊C4—2711no—
276C4⋊C4—5421no—
277C4⋊C4—5421no—
278C4⋊C4—5421no—
279C4⋊C4—5421no—
280C4⋊C4—5421no—
281C4⋊C4—5421no—
282C4⋊C4C22⋊D27111yescharacteristic
283C4⋊C4C22×D27111yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—5421no—
290C22⋊C4—5421no—
291C22⋊C4—5421no—
292C22⋊C4C22⋊D27111yescharacteristic
293C22⋊C4C22⋊D27111yescharacteristic
294C42—5421no—
295C42—5421no—
296C42C22×D27111yescharacteristic
297C23—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23C2×C22⋊D27111yescharacteristic
303C23C2×C22⋊D27111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—5421no—
307Q8—5421no—
308D4—22no—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—42no—
313D4—42no—
314D4—81no—
315D4#1686465312yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—81no—
324C2×C4—5421no—
325C2×C4—5421no—
326C2×C4—5421no—
327C2×C4—5421no—
328C2×C4—10842no—
329C2×C4C2×C22⋊D27111yescharacteristic
330C2×C4C2×C22⋊D27111yescharacteristic
331C2×C4C2×C22⋊D27111yescharacteristic
332C2×C4C2×C22⋊D27111yescharacteristic
333C2×C4C23×D27111yescharacteristic
334C8—5421no—
335C8—5421no—
336C8—5421no—
337C8—5421no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—22no—
342C22—21no—
343C22—21no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—84no—
350C22#2257618012yes—
351C22C22×C22⋊D27111yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—10841no—
359C4—10841no—
360C4—10841no—
361C4—10841no—
362C4C2×#1686465311yescharacteristic
363C4#2257620511yescharacteristic
364C2—22no—
365C2—41no—
366C2—42no—
367C2—81no—
368C2#3177454111yescharacteristic
369C2C2×#2257618011yescharacteristic
370C2#3177470811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting, prime radical#546×C271C2
socleC22×C271C22×D4
hypercenter#546D271
hypocenterC271#3296
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. #31774587C2semidirect
2. #31774251C2semidirect
3. #31774225C2semidirect
4. #31774218C2semidirect
5. #31774197C2semidirect
6. #31774136C2semidirect
7. #31774124C2semidirect
8. #31774116C2semidirect
9. #31774059C2semidirect
10. #31774048C2semidirect
11. C4×#16864642C2semidirect
12. D4×C271⋊C8C2semidirect
13. #31773893C2semidirect
14. C2×#22576081C2semidirect
16. C2×#16864644C22semidirect
20. C22×C271⋊C8C22semidirect
22. #22576152C22semidirect
26. #22576139C22semidirect
27. C4×Dic542C22semidirect
28. #22576119C22semidirect
31. #22576093C22semidirect
32. #22576091C22semidirect
33. #22576091C22semidirect
34. C4⋊(C271⋊C8)C22semidirect
35. #22576084C22semidirect
36. #22576084C22semidirect
37. #22576084C22semidirect
38. #22576083C22semidirect
46. C22⋊(C271⋊C8)C22semidirect
47. C4×C271⋊C8C22semidirect
185. C271#3296semidirect

Maximal subgroups

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

subgroupindex
#317745872
#317742512
#317742252
#317742182
#317741972
#317741362
#317741242
#317741162
#317740592
#317740482
C4×#168646422
D4×C271⋊C82
#317738932
C2×#225760812
#546×C2712
#3296271