Finite Group Databasebeta

← #46849151

Subgroups — #46849151

Order 34688.

384 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#31774629C211yesmaximal, characteristic, complemented
2D4×Dic542C211yesmaximal, characteristic, complemented
3#31774258C211yesmaximal, characteristic, complemented
4#31774256C211yesmaximal, characteristic, complemented
5#31774242C211yesmaximal, characteristic, complemented
6#31774188C211yesmaximal, characteristic, complemented
7#31774185C211yesmaximal, characteristic, complemented
8#31774180C211yesmaximal, characteristic, complemented
9#31774114C211yesmaximal, characteristic, complemented
10#31774090C211yesmaximal, characteristic
11#31774026C211yesmaximal, characteristic, complemented
12#31774009C211yesmaximal, characteristic, complemented
13#31774007C211yesmaximal, characteristic, complemented
14SD16×Dic271C211yesmaximal, characteristic, complemented
15#463×C271C211yesmaximal, characteristic, complemented
16C22×Dic542—21no—
17C2×#16864639C2211yescharacteristic
18#22576160C2211yescharacteristic
19#22576159C2211yescharacteristic
20#22576159C2211yescharacteristic
21#22576156C2211yescharacteristic
22#22576149C2211yescharacteristic
23#22576146—21no—
24#22576144—21no—
25#22576142—21no—
26#22576139—21no—
27#22576137—21no—
28#22576137—21no—
29Q8×Dic271C2211yescharacteristic
30C4×Dic542C2211yescharacteristic
31C4×D1084C2211yescharacteristic
32D4×Dic271C2211yescharacteristic
33D4×Dic271C2211yescharacteristic
34D4×Dic271C2211yescharacteristic
35(C22⋊D271)⋊C4—21no—
36#22576092C2211yescharacteristic
37#22576092C2211yescharacteristic
38#22576091C2211yescharacteristic
39Dic271⋊C8C2211yescharacteristic
40Dic271⋊C8C2211yescharacteristic
41#22576084C2211yescharacteristic
42#22576084C2211yescharacteristic
43#22576082C2211yescharacteristic
44#22576082C2211yescharacteristic
45#22576081C2211yescharacteristic
46#22576081C2211yescharacteristic
47#22576081C2211yescharacteristic
48#22576080C2211yescharacteristic
49#22576071C2211yescharacteristic
50#22576070C2211yescharacteristic
51C8×Dic271C2211yescharacteristic
52C2×SD16×C271C2211yescharacteristic
53C2×D8×C271C2211yescharacteristic
54#124×C271C2211yescharacteristic
55C4×D4×C271C2211yescharacteristic
56C4⋊C8×C271C2211yescharacteristic
57#103×C271C2211yescharacteristic
58#102×C271C2211yescharacteristic
59C2×Dic542—21no—
60C2×Dic542—21no—
61C2×Dic542—21no—
62C2×Dic542—44no—
63C2×Dic542D411yescharacteristic
64C2×D1084C2311yescharacteristic
65C2×C22⋊D271—21no—
66C2×C22⋊D271—21no—
67C22×Dic271—21no—
68C22×Dic271—21no—
69C22×Dic271—21no—
70C2×C4×D271—21no—
71#16864639—22no—
72C2×C271⋊C8C2311yescharacteristic
73C2×C271⋊C8C2311yescharacteristic
74C4⋊Dic271—21no—
75C4⋊Dic271—21no—
76C4⋊Dic271D411yescharacteristic
77C4⋊Dic271C2311yescharacteristic
78C4⋊Dic271C2311yescharacteristic
79C4⋊Dic271C2311yescharacteristic
80Dic271⋊C4—21no—
81Dic271⋊C4—21no—
82Dic271⋊C4D411yescharacteristic
83Dic271⋊C4D411yescharacteristic
84C22⋊Dic271—21no—
85C22⋊Dic271—21no—
86C22⋊Dic271—21no—
87(C2×D271)⋊C4—21no—
88(C2×D271)⋊C4—21no—
89(C2×D271)⋊C4—21no—
90C4×Dic271—21no—
91C4×Dic271C2311yescharacteristic
92C4×Dic271C2311yescharacteristic
93C2×Q8×C271C2311yescharacteristic
94C2×D4×C271C2311yescharacteristic
95C2×D4×C271C2311yescharacteristic
96C22×C4×C271—21no—
97SD16×C271—22no—
98D8×C271—22no—
99C2×C8×C271C2311yescharacteristic
100C2×C8×C271C2311yescharacteristic
101C4⋊C4×C271C2311yescharacteristic
102C22⋊C4×C271—21no—
103C22⋊C4×C271—21no—
104C22⋊C4×C271—21no—
105C42×C271C2311yescharacteristic
106C22×D271—21no—
107Dic542—22no—
108Dic542—42no—
109Dic542—42no—
110Dic542—42no—
111D1084—21no—
112D1084—21no—
113C22⋊D271—42no—
114C22⋊D271—81no—
115C2×Dic271—21no—
116C2×Dic271—21no—
117C2×Dic271—21no—
118C2×Dic271—42no—
119C2×Dic271—42no—
120C2×Dic271—81no—
121C2×Dic271C2×D411yescharacteristic
122C2×Dic271C2×D411yescharacteristic
123C2×Dic271C2×D411yescharacteristic
124C2×Dic271C2×D411yescharacteristic
125C4×D271—81no—
126C271⋊C8—21no—
127C271⋊C8—21no—
128C23×C271—21no—
129C23×C271—21no—
130Q8×C271—21no—
131Q8×C271—21no—
132D4×C271—21no—
133D4×C271—21no—
134D4×C271—21no—
135D4×C271#4112yes—
136C2×C4×C271—21no—
137C2×C4×C271—21no—
138C2×C4×C271—42no—
139C2×C4×C271C2×D411yescharacteristic
140C2×C4×C271C2×D411yescharacteristic
141C2×C4×C271C2411yescharacteristic
142C8×C271—21no—
143C8×C271—21no—
144C22×C271—22no—
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—21no—
155C4×C271—21no—
156C4×C271—41no—
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—42no—
168C2×C271—81no—
169C2×C271#54811yescharacteristic
170C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
171C2×C271#57711yescharacteristic
172D271—81no—
173C271#337311yescharacteristic, complemented, Sylow
174#3373—2711nomaximal, Sylow
175#559—2711no—
176D4×Q8—2711no—
177#489—2711no—
178#489—2711no—
179#486—2711no—
180#476—2711no—
181#476—2711no—
182#475—2711no—
183#463—2711no—
184#463D27111yescharacteristic, complemented
185#459—2711no—
186#446—2711no—
187#442—2711no—
188#442—2711no—
189C4×SD16—2711no—
190C22×Q8—5421no—
191C2×SD16C2×D27111yescharacteristic
192C2×D8—2711no—
193C2×D8C2×D27111yescharacteristic
194C4⋊Q8—2711no—
195C4⋊Q8—2711no—
196C4⋊Q8—2711no—
197#127—2711no—
198#124—2711no—
199#124—5421no—
200#124C2×D27111yescharacteristic
201#123—5421no—
202#123—5421no—
203C22⋊Q8—5421no—
204C22⋊Q8—5421no—
205C22⋊Q8—5421no—
206C4×Q8—2711no—
207C4×Q8—2711no—
208C4×D4—2711no—
209C4×D4—2711no—
210C4×D4—2711no—
211C4×D4—2711no—
212C4×D4—5421no—
213C4×D4C2×D27111yescharacteristic
214#106—2711no—
215#106—2711no—
216#106—2711no—
217C4⋊C8—2711no—
218C4⋊C8—2711no—
219C4⋊C8C2×D27111yescharacteristic
220#103—2711no—
221#103—2711no—
222#103C2×D27111yescharacteristic
223#102—2711no—
224#102—2711no—
225#102—2711no—
226#102—2711no—
227#102—2711no—
228#102—2711no—
229#102C2×D27111yescharacteristic
230#97—2711no—
231#97—2711no—
232C4×C8—2711no—
233C2×Q8—2711no—
234C2×Q8—5421no—
235C2×Q8—5421no—
236C2×Q8—5421no—
237C2×Q8—10844no—
238C2×Q8C22×D27111yescharacteristic
239C2×D4—2711no—
240C2×D4—5421no—
241C2×D4—5421no—
242C2×D4C22×D27111yescharacteristic
243C2×D4C22×D27111yescharacteristic
244C22×C4—21no—
245C22×C4—5421no—
246C22×C4—5421no—
247C22×C4—5421no—
248C22×C4—5421no—
249SD16—22no—
250D8—22no—
251D8—5422no—
252C2×C8—2711no—
253C2×C8—2711no—
254C2×C8C22×D27111yescharacteristic
255C2×C8C22×D27111yescharacteristic
256C4⋊C4—2711no—
257C4⋊C4—2711no—
258C4⋊C4—2711no—
259C4⋊C4—2711no—
260C4⋊C4—2711no—
261C4⋊C4—2711no—
262C4⋊C4—5421no—
263C4⋊C4—5421no—
264C4⋊C4—5421no—
265C4⋊C4—5421no—
266C4⋊C4C22×D27111yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—5421no—
271C22⋊C4—5421no—
272C22⋊C4—5421no—
273C22⋊C4—5421no—
274C22⋊C4—5421no—
275C22⋊C4—5421no—
276C42—2711no—
277C42—2711no—
278C42—5421no—
279C42C22×D27111yescharacteristic
280C23—21no—
281C23—21no—
282C23—5421no—
283Q8—21no—
284Q8—21no—
285Q8—5422no—
286Q8—10842no—
287Q8—10842no—
288Q8—10842no—
289D4—21no—
290D4—21no—
291D4—21no—
292D4—5421no—
293D4—5421no—
294D4—10842no—
295D4—21681no—
296D4#1686465312yes—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—42no—
300C2×C4—2711no—
301C2×C4—2711no—
302C2×C4—2711no—
303C2×C4—2711no—
304C2×C4—5421no—
305C2×C4—5421no—
306C2×C4—5421no—
307C2×C4—10842no—
308C2×C4—10842no—
309C2×C4—21681no—
310C2×C4—21681no—
311C2×C4D4×D27111yescharacteristic
312C2×C4D4×D27111yescharacteristic
313C2×C4C23×D27111yescharacteristic
314C8—21no—
315C8—21no—
316C8—5421no—
317C8—5421no—
318C22—22no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—10841no—
325C22—10841no—
326C22—10841no—
327C22C2×D4×D27111yescharacteristic
328C4—21no—
329C4—21no—
330C4—41no—
331C4—41no—
332C4—2712no—
333C4—5421no—
334C4—5421no—
335C4—5421no—
336C4—10841no—
337C4—10841no—
338C4—10841no—
339C4C2×#1686465311yescharacteristic
340C4#2257620711yescharacteristic
341C2—42no—
342C2—81no—
343C2—21681no—
344C2#3177455311yescharacteristic
345C2AGL(1,ℤ/8)×D27111yescharacteristic
346C2#3177470411yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#463×C271C2
socleC22×C271C22×D4
prime radical#31774090C2
hypercenter#463D271
hypocenterC271#3373
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. #31774629C2semidirect
2. D4×Dic542C2semidirect
3. #31774258C2semidirect
4. #31774256C2semidirect
5. #31774242C2semidirect
6. #31774188C2semidirect
7. #31774185C2semidirect
8. #31774180C2semidirect
9. #31774114C2semidirect
11. #31774026C2semidirect
12. #31774009C2semidirect
13. #31774007C2semidirect
14. SD16×Dic271C2semidirect
15. #463×C271C2semidirect
173. C271#3373semidirect
184. #463D271semidirect

Maximal subgroups

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

subgroupindex
#317746292
D4×Dic5422
#317742582
#317742562
#317742422
#317741882
#317741852
#317741802
#317741142
#317740902
#317740262
#317740092
#317740072
SD16×Dic2712
#463×C2712
#3373271