Finite Group Databasebeta

← #46849088

Subgroups — #46849088

Order 34688.

382 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774623C211yesmaximal, characteristic, complemented
2#31774567C211yesmaximal, characteristic
3#31774261C211yesmaximal, characteristic, complemented
4#31774260C211yesmaximal, characteristic, complemented
5#31774242C211yesmaximal, characteristic, complemented
6#31774199C211yesmaximal, characteristic, complemented
7#31774198C211yesmaximal, characteristic, complemented
8#31774175C211yesmaximal, characteristic, complemented
9#31774095C211yesmaximal, characteristic, complemented
10#31774095C211yesmaximal, characteristic, complemented
11#31774026C211yesmaximal, characteristic, complemented
12#31773998C211yesmaximal, characteristic, complemented
13#31773995C211yesmaximal, characteristic, complemented
14SD16×Dic271C211yesmaximal, characteristic, complemented
15#460×C271C211yesmaximal, characteristic, complemented
16C2×#16864653—21no—
17C2×#16864641C2211yescharacteristic
18#22576159—21no—
19#22576159C2211yescharacteristic
20#22576156C2211yescharacteristic
21#22576156C2211yescharacteristic
22#22576152—21no—
23#22576152C2211yescharacteristic
24#22576151—21no—
25#22576151C2211yescharacteristic
26#22576132—21no—
27#22576132—21no—
28#22576131—21no—
29Q8×Dic271C2211yescharacteristic
30Q8×Dic271C2211yescharacteristic
31C4×Dic542—21no—
32C4×Dic542C2211yescharacteristic
33C4×D1084C2211yescharacteristic
34D4×Dic271C2211yescharacteristic
35D4×Dic271C2211yescharacteristic
36#22576094C2211yescharacteristic
37#22576093C2211yescharacteristic
38#22576092C2211yescharacteristic
39Dic271⋊C8C2211yescharacteristic
40Dic271⋊C8C2211yescharacteristic
41#22576084C2211yescharacteristic
42#22576082C2211yescharacteristic
43#22576082C2211yescharacteristic
44#22576081C2211yescharacteristic
45#22576081C2211yescharacteristic
46#22576081C2211yescharacteristic
47#22576081C2211yescharacteristic
48#22576080C2211yescharacteristic
49#22576071C2211yescharacteristic
50#22576070C2211yescharacteristic
51C8×Dic271C2211yescharacteristic
52C2×SD16×C271C2211yescharacteristic
53C2×SD16×C271C2211yescharacteristic
54#127×C271C2211yescharacteristic
55C4×Q8×C271C2211yescharacteristic
56C4⋊C8×C271C2211yescharacteristic
57#102×C271C2211yescharacteristic
58#102×C271C2211yescharacteristic
59#16864653—44no—
60C2×Dic542—21no—
61C2×Dic542D411yescharacteristic
62C2×D1084C2311yescharacteristic
63C2×C22⋊D271—21no—
64C2×C22⋊D271—21no—
65C2×C22⋊D271—21no—
66C2×C22⋊D271—21no—
67C22×Dic271—21no—
68C22×Dic271—21no—
69C2×C4×D271—21no—
70#16864641—22no—
71C2×C271⋊C8C2311yescharacteristic
72C2×C271⋊C8C2311yescharacteristic
73C4⋊Dic271—21no—
74C4⋊Dic271—21no—
75C4⋊Dic271—21no—
76C4⋊Dic271D411yescharacteristic
77C4⋊Dic271C2311yescharacteristic
78C4⋊Dic271C2311yescharacteristic
79C4⋊Dic271C2311yescharacteristic
80Dic271⋊C4—21no—
81Dic271⋊C4—21no—
82Dic271⋊C4—21no—
83Dic271⋊C4—21no—
84Dic271⋊C4D411yescharacteristic
85Dic271⋊C4D411yescharacteristic
86C22⋊Dic271—21no—
87C22⋊Dic271—21no—
88(C2×D271)⋊C4—21no—
89C4×Dic271—21no—
90C4×Dic271—21no—
91C4×Dic271C2311yescharacteristic
92C4×Dic271C2311yescharacteristic
93C2×Q8×C271C2311yescharacteristic
94C2×D4×C271—21no—
95C2×D4×C271—21no—
96C2×D4×C271C2311yescharacteristic
97C2×D4×C271C2311yescharacteristic
98SD16×C271—22no—
99SD16×C271—22no—
100C2×C8×C271C2311yescharacteristic
101C2×C8×C271C2311yescharacteristic
102C4⋊C4×C271—21no—
103C4⋊C4×C271C2311yescharacteristic
104C42×C271—21no—
105C42×C271C2311yescharacteristic
106C22×D271—21no—
107Dic542—22no—
108Dic542—42no—
109D1084—21no—
110D1084—21no—
111C22⋊D271—42no—
112C22⋊D271—42no—
113C22⋊D271—81no—
114C22⋊D271—81no—
115C2×Dic271—21no—
116C2×Dic271—21no—
117C2×Dic271—21no—
118C2×Dic271—42no—
119C2×Dic271—42no—
120C2×Dic271C2×D411yescharacteristic
121C2×Dic271C2×D411yescharacteristic
122C2×Dic271C2×D411yescharacteristic
123C2×Dic271C2×D411yescharacteristic
124C4×D271—42no—
125C271⋊C8—21no—
126C271⋊C8—21no—
127C23×C271—21no—
128C23×C271—21no—
129Q8×C271—21no—
130Q8×C271#4112yescomplemented
131D4×C271—21no—
132D4×C271—21no—
133D4×C271—21no—
134D4×C271—21no—
135D4×C271—42no—
136D4×C271—81no—
137C2×C4×C271—21no—
138C2×C4×C271—21no—
139C2×C4×C271C2×D411yescharacteristic
140C2×C4×C271C2×D411yescharacteristic
141C2×C4×C271C2411yescharacteristic
142C8×C271—21no—
143C8×C271—21no—
144C22×C271—41no—
145C22×C271—41no—
146C22×C271—41no—
147C22×C271—41no—
148C22×C271—41no—
149C22×C271—41no—
150C22×C271C22×D411yescharacteristic
151C2×D271—41no—
152C2×D271—41no—
153C2×D271—41no—
154C4×C271—22no—
155C4×C271—21no—
156C4×C271—21no—
157C4×C271—41no—
158C4×C271C2×#4111yescharacteristic
159C4×C2712−1+411yescharacteristic
160Dic271—21no—
161Dic271—21no—
162Dic271—21no—
163Dic271—41no—
164Dic271—41no—
165Dic271—41no—
166Dic271AGL(1,ℤ/8)12yes—
167C2×C271—81no—
168C2×C271—81no—
169C2×C271#54811yescharacteristic
170C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
171C2×C271#57711yescharacteristic
172D271—81no—
173C271#336811yescharacteristic, complemented, Sylow
174#3368—2711nomaximal, Sylow
175#557—2711no—
176#550—2711no—
177#490—2711no—
178#490—2711no—
179#486—2711no—
180#478—2711no—
181#478—2711no—
182#474—2711no—
183#460—2711no—
184#460—2711no—
185#460D27111yescharacteristic, complemented
186#446—2711no—
187#440—2711no—
188#440—2711no—
189C4×SD16—2711no—
190C2×#41—5421no—
191C2×SD16—2711no—
192C2×SD16C2×D27111yescharacteristic
193C2×SD16C2×D27111yescharacteristic
194C4⋊Q8—2711no—
195C4⋊Q8—5421no—
196#127—2711no—
197#127—2711no—
198#127C2×D27111yescharacteristic
199#125—2711no—
200#125—2711no—
201#125—5421no—
202#125—5421no—
203#121—5421no—
204#121—5421no—
205#121—5421no—
206C4×Q8—2711no—
207C4×Q8—2711no—
208C4×Q8—2711no—
209C4×Q8—5421no—
210C4×Q8C2×D27111yescharacteristic
211C4×D4—2711no—
212C4×D4—2711no—
213C4×D4—2711no—
214#107—2711no—
215#107—2711no—
216#106—2711no—
217C4⋊C8—2711no—
218C4⋊C8—2711no—
219C4⋊C8C2×D27111yescharacteristic
220#103—2711no—
221#102—2711no—
222#102—2711no—
223#102—2711no—
224#102—2711no—
225#102—2711no—
226#102—2711no—
227#102—2711no—
228#102C2×D27111yescharacteristic
229#102C2×D27111yescharacteristic
230#97—2711no—
231#97—2711no—
232C4×C8—2711no—
233#41—10844no—
234C2×Q8—2711no—
235C2×Q8—5421no—
236C2×Q8C22×D27111yescharacteristic
237C2×D4—21no—
238C2×D4—21no—
239C2×D4—2711no—
240C2×D4—5421no—
241C2×D4—5421no—
242C2×D4—5421no—
243C2×D4—5421no—
244C2×D4C22×D27111yescharacteristic
245C2×D4C22×D27111yescharacteristic
246C22×C4—5421no—
247C22×C4—5421no—
248C22×C4—5421no—
249SD16—22no—
250SD16—22no—
251SD16—5422no—
252C2×C8—2711no—
253C2×C8—2711no—
254C2×C8C22×D27111yescharacteristic
255C2×C8C22×D27111yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—2711no—
258C4⋊C4—2711no—
259C4⋊C4—2711no—
260C4⋊C4—2711no—
261C4⋊C4—2711no—
262C4⋊C4—2711no—
263C4⋊C4—5421no—
264C4⋊C4—5421no—
265C4⋊C4—5421no—
266C4⋊C4—5421no—
267C4⋊C4—5421no—
268C4⋊C4—5421no—
269C4⋊C4—5421no—
270C4⋊C4C22×D27111yescharacteristic
271C22⋊C4—5421no—
272C22⋊C4—5421no—
273C22⋊C4—5421no—
274C42—21no—
275C42—2711no—
276C42—2711no—
277C42—5421no—
278C42—5421no—
279C42C22×D27111yescharacteristic
280C23—21no—
281C23—21no—
282C23—5421no—
283Q8—21no—
284Q8—5422no—
285Q8—10842no—
286Q8#1686465312yescomplemented
287D4—21no—
288D4—21no—
289D4—21no—
290D4—21no—
291D4—42no—
292D4—81no—
293D4—5421no—
294D4—5421no—
295D4—10842no—
296D4—10842no—
297D4—21681no—
298D4—21681no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—2711no—
302C2×C4—2711no—
303C2×C4—2711no—
304C2×C4—2711no—
305C2×C4—5421no—
306C2×C4—5421no—
307C2×C4—5421no—
308C2×C4—10842no—
309C2×C4—10842no—
310C2×C4—10842no—
311C2×C4D4×D27111yescharacteristic
312C2×C4D4×D27111yescharacteristic
313C2×C4C23×D27111yescharacteristic
314C8—21no—
315C8—21no—
316C8—5421no—
317C8—5421no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—10841no—
325C22—10841no—
326C22—10841no—
327C22C2×D4×D27111yescharacteristic
328C4—22no—
329C4—21no—
330C4—21no—
331C4—41no—
332C4—2712no—
333C4—5421no—
334C4—5421no—
335C4—5421no—
336C4—10841no—
337C4—10841no—
338C4—10841no—
339C4C2×#1686465311yescharacteristic
340C4#2257620711yescharacteristic
341C2—81no—
342C2—81no—
343C2—21681no—
344C2#3177455311yescharacteristic
345C2AGL(1,ℤ/8)×D27111yescharacteristic
346C2#3177470311yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#460×C271C2
socleC22×C271C22×D4
prime radical#31774567C2
hypercenter#460D271
hypocenterC271#3368
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. #31774623C2semidirect
3. #31774261C2semidirect
4. #31774260C2semidirect
5. #31774242C2semidirect
6. #31774199C2semidirect
7. #31774198C2semidirect
8. #31774175C2semidirect
9. #31774095C2semidirect
10. #31774095C2semidirect
11. #31774026C2semidirect
12. #31773998C2semidirect
13. #31773995C2semidirect
14. SD16×Dic271C2semidirect
15. #460×C271C2semidirect
130. Q8×C271#41semidirect
173. C271#3368semidirect
185. #460D271semidirect
286. Q8#16864653semidirect

Maximal subgroups

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

subgroupindex
#317746232
#317745672
#317742612
#317742602
#317742422
#317741992
#317741982
#317741752
#317740952
#317740262
#317739982
#317739952
SD16×Dic2712
#460×C2712
#3368271