Finite Group Databasebeta

← #46846290

Subgroups — #46846290

Order 34688.

458 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 129 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774409C211yesmaximal, characteristic
2C2×C4⋊C4×D271C211yesmaximal, characteristic, complemented
3#31774234C212yesmaximal, complemented
4#31774229C212yesmaximal, complemented
5#31774217C212yesmaximal, complemented
6#31774212C212yesmaximal, complemented
7#31773940C211yesmaximal, characteristic, complemented
8#31773899C211yesmaximal, characteristic, complemented
9#31773896C211yesmaximal, characteristic, complemented
10#31773860C211yesmaximal, characteristic, complemented
11#428×C271C211yesmaximal, characteristic, complemented
12C2×#16864654C2211yescharacteristic
13C22×C4×D271—21no—
14C22×C4×D271C2211yescharacteristic
15C2×#16864638C2211yescharacteristic
16#22576144—21no—
17#22576140C2211yescharacteristic
18#22576140C2211yescharacteristic
19#22576135C2211yescharacteristic
20#22576135C2211yescharacteristic
21#22576130—21no—
22C4×D1084—21no—
23C4×D1084—21no—
24C2×C4⋊Dic271C2211yescharacteristic
25C2×Dic271⋊C4—21no—
26C4⋊C4×D271—21no—
27C4⋊C4×D271—21no—
28C4⋊C4×D271C2212yes—
29C4⋊C4×D271C2212yes—
30#22576093C2212yescomplemented
31#22576091C2212yescomplemented
32#22576085C2212yescomplemented
33#22576083C2212yescomplemented
34#22576082C2212yescomplemented
35#22576080C2212yescomplemented
36(C2×D271)⋊C8C2212yes—
37(C2×D271)⋊C8C2212yes—
38C2×M16×C271C2211yescharacteristic
39#117×C271C2211yescharacteristic
40C2×C4⋊C4×C271C2211yescharacteristic
41#107×C271C2212yescomplemented
42#106×C271C2212yescomplemented
43C23×D271D411yescharacteristic
44#16864654—22no—
45#16864654—22no—
46C2×Dic542C2311yescharacteristic
47C2×D1084—21no—
48C2×D1084—21no—
49C2×D1084C2311yescharacteristic
50C2×C22⋊D271—21no—
51C22×Dic271—21no—
52C22×Dic271D411yescharacteristic
53C2×C4×D271—22no—
54C2×C4×D271—22no—
55C2×C4×D271—21no—
56C2×C4×D271—22no—
57C2×C4×D271—22no—
58C2×C4×D271—44no—
59C2×C4×D271D411yescharacteristic
60C2×C4×D271D411yescharacteristic
61C2×C4×D271C2311yescharacteristic
62C2×C4×D271C2311yescharacteristic
63#16864638—22no—
64C2×C271⋊C8C2312yes—
65C4⋊Dic271—21no—
66C4⋊Dic271—21no—
67C4⋊Dic271—21no—
68C4⋊Dic271C2312yes—
69Dic271⋊C4—22no—
70Dic271⋊C4—22no—
71(C2×D271)⋊C4—21no—
72(C2×D271)⋊C4—21no—
73(C2×D271)⋊C4—21no—
74(C2×D271)⋊C4—21no—
75C22×C4×C271—21no—
76C22×C4×C271C2311yescharacteristic
77M16×C271—22no—
78C2×C8×C271C2312yes—
79C4⋊C4×C271—21no—
80C4⋊C4×C271C2311yescharacteristic
81C4⋊C4×C271C2311yescharacteristic
82C4⋊C4×C271C2312yes—
83C22⋊C4×C271—21no—
84C42×C271—21no—
85C22×D271—22no—
86C22×D271—22no—
87C22×D271—22no—
88C22×D271—21no—
89C22×D271C2×D411yescharacteristic
90C22×D271C2×D411yescharacteristic
91Dic542—21no—
92Dic542—21no—
93D1084—21no—
94D1084—21no—
95D1084—42no—
96D1084—42no—
97C22⋊D271—42no—
98C2×Dic271—22no—
99C2×Dic271—21no—
100C2×Dic271—22no—
101C2×Dic271—42no—
102C2×Dic271C2×D411yescharacteristic
103C2×Dic271C2×D411yescharacteristic
104C4×D271—22no—
105C4×D271—22no—
106C4×D271—21no—
107C4×D271—21no—
108C4×D271—21no—
109C4×D271—21no—
110C4×D271—44no—
111C4×D271—44no—
112C4×D271—42no—
113C271⋊C8—22no—
114C23×C271C2×D411yescharacteristic
115C2×C4×C271—21no—
116C2×C4×C271—21no—
117C2×C4×C271—22no—
118C2×C4×C271—42no—
119C2×C4×C271C2×D411yescharacteristic
120C2×C4×C271#4112yes—
121C2×C4×C271#4112yes—
122C2×C4×C271C2411yescharacteristic
123C8×C271—22no—
124C22×C271—21no—
125C22×C271—21no—
126C22×C271#12312yes—
127C22×C271C22×D411yescharacteristic
128C2×D271—28no—
129C2×D271—21no—
130C2×D271—21no—
131C2×D271—21no—
132C2×D271—21no—
133C2×D271—41no—
134C2×D271—41no—
135C2×D271—41no—
136C2×D271AGL(1,ℤ/8)12yes—
137C2×D271#13712yes—
138C4×C271—42no—
139C4×C271—41no—
140C4×C271—41no—
141C4×C271#12312yes—
142C4×C271C2×#4111yescharacteristic
143C4×C271C2×#4111yescharacteristic
144Dic271—21no—
145Dic271—21no—
146Dic271—41no—
147Dic271—42no—
148C2×C271—22no—
149C2×C271C2×#12311yescharacteristic
150C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
151C2×C271C2×#13711yescharacteristic
152D271—22no—
153D271—22no—
154D271—81no—
155C271#309211yescharacteristic, complemented, Sylow
156#3092—2711nomaximal, Sylow
157#525—2711no—
158C22×C4⋊C4—2711no—
159#484—2712no—
160#483—2712no—
161#481—2712no—
162#480—2712no—
163#428—2711no—
164#428D27111yescharacteristic, complemented
165#417—2711no—
166#417—2711no—
167#407—2711no—
168C2×#41—2711no—
169C23×C4—2711no—
170C23×C4—5421no—
171C2×M16—2711no—
172C2×M16C2×D27111yescharacteristic
173#123—5421no—
174C22⋊Q8—2711no—
175C22⋊Q8—2711no—
176#121—2711no—
177#121—2711no—
178#120—5421no—
179C4×D4—5421no—
180C4×D4—5421no—
181#117C2×D27111yescharacteristic
182C2×C4⋊C4—2711no—
183C2×C4⋊C4—2712no—
184C2×C4⋊C4—2712no—
185C2×C4⋊C4—5421no—
186C2×C4⋊C4—5421no—
187C2×C4⋊C4—5421no—
188C2×C4⋊C4C2×D27111yescharacteristic
189#107—2712no—
190#107C2×D27112yescomplemented
191#106—2712no—
192#106C2×D27112yescomplemented
193#103—2712no—
194#103—2712no—
195#102—2712no—
196#102—2712no—
197C22⋊C8—2712no—
198C22⋊C8—2712no—
199C24—2711no—
200#41—5422no—
201#41—5422no—
202C2×Q8—2711no—
203C2×D4—2711no—
204C2×D4—5421no—
205C2×D4—5421no—
206C2×D4—5421no—
207C22×C4—21no—
208C22×C4—2711no—
209C22×C4—2711no—
210C22×C4—2711no—
211C22×C4—2711no—
212C22×C4—2711no—
213C22×C4—5422no—
214C22×C4—5422no—
215C22×C4—5422no—
216C22×C4—5422no—
217C22×C4—5421no—
218C22×C4—5421no—
219C22×C4—10844no—
220C22×C4C22×D27111yescharacteristic
221M16—22no—
222M16—5422no—
223C2×C8—2712no—
224C2×C8C22×D27112yes—
225C4⋊C4—21no—
226C4⋊C4—2712no—
227C4⋊C4—5422no—
228C4⋊C4—5422no—
229C4⋊C4—5421no—
230C4⋊C4—5421no—
231C4⋊C4—5421no—
232C4⋊C4C22×D27112yes—
233C4⋊C4C22×D27111yescharacteristic
234C4⋊C4C22×D27111yescharacteristic
235C22⋊C4—21no—
236C22⋊C4—5421no—
237C22⋊C4—5421no—
238C22⋊C4—5421no—
239C22⋊C4—5421no—
240C42—21no—
241C23—2711no—
242C23—2711no—
243C23—5422no—
244C23—5422no—
245C23—5422no—
246C23—5421no—
247C23D4×D27111yescharacteristic
248Q8—5421no—
249Q8—5421no—
250D4—5421no—
251D4—5421no—
252D4—10842no—
253D4—10842no—
254D4—10842no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—42no—
259C2×C4—2711no—
260C2×C4—2711no—
261C2×C4—5422no—
262C2×C4—5422no—
263C2×C4—5422no—
264C2×C4—5421no—
265C2×C4—5421no—
266C2×C4—5421no—
267C2×C4—5421no—
268C2×C4—5421no—
269C2×C4—5422no—
270C2×C4—10844no—
271C2×C4—10844no—
272C2×C4—10842no—
273C2×C4—10842no—
274C2×C4D4×D27111yescharacteristic
275C2×C4#1686465412yes—
276C2×C4#1686465512yes—
277C2×C4C23×D27111yescharacteristic
278C8—22no—
279C8—5422no—
280C22—21no—
281C22—21no—
282C22—2712no—
283C22—2712no—
284C22—5428no—
285C22—5421no—
286C22—5421no—
287C22—5421no—
288C22—5421no—
289C22—10841no—
290C22—10841no—
291C22—10841no—
292C22#2257614512yes—
293C22C2×D4×D27111yescharacteristic
294C4—42no—
295C4—41no—
296C4—41no—
297C4—5421no—
298C4—5421no—
299C4—10841no—
300C4—10842no—
301C4#2257614512yes—
302C4C2×#1686465411yescharacteristic
303C4C2×#1686465511yescharacteristic
304C2—22no—
305C2—5422no—
306C2—5422no—
307C2—21681no—
308C2C2×#2257614511yescharacteristic
309C2AGL(1,ℤ/8)×D27111yescharacteristic
310C2#137×D27111yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#428×C271C2
socleC22×C271C22×D4
prime radical#31774409C2
hypercenter#428D271
hypocenterC271#3092
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D271C2semidirect
3. #31774234C2semidirect
4. #31774229C2semidirect
5. #31774217C2semidirect
6. #31774212C2semidirect
7. #31773940C2semidirect
8. #31773899C2semidirect
9. #31773896C2semidirect
10. #31773860C2semidirect
11. #428×C271C2semidirect
30. #22576093C22semidirect
31. #22576091C22semidirect
32. #22576085C22semidirect
33. #22576083C22semidirect
34. #22576082C22semidirect
35. #22576080C22semidirect
41. #107×C271C22semidirect
42. #106×C271C22semidirect
155. C271#3092semidirect
164. #428D271semidirect
190. #107C2×D271semidirect
192. #106C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317744092
C2×C4⋊C4×D2712
#317742342
#317742292
#317742172
#317742122
#317739402
#317738992
#317738962
#317738602
#428×C2712
#3092271