Finite Group Databasebeta

← #46813868

Subgroups — #46813868

Order 34432.

928 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 928 conjugacy classes of subgroups — too many to draw. Only its 209 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
1C2×#22575421C211yesmaximal, characteristic
2C2×C4×C22⋊D269C212yesmaximal, complemented
3C2×C22⋊C4×D269C212yesmaximal, complemented
4#31770508C211yesmaximal, characteristic, complemented
5#31770499C212yesmaximal, complemented
6#31770481C212yesmaximal, complemented
7#31770479C212yesmaximal, complemented
8#31770452C212yesmaximal, complemented
9#386×C269C211yesmaximal, characteristic, complemented
10C24×D269C2211yescharacteristic
11C22×C22⋊D269C2212yes—
12C22×C4×D269C2212yes—
13C22×C4×D269C2212yes—
14#22575421C418yescomplemented
15C4×C22⋊D269—28no—
16C2×C4⋊Dic269C2211yescharacteristic, complemented
17C2×#16864381C2212yescomplemented
18C2×C22⋊Dic269C2212yes—
19C2×C22⋊Dic269C2211yescharacteristic
20C2×C22⋊Dic269C2212yescomplemented
21C2×#16864379C2212yes—
22C2×#16864379C2212yes—
23C2×#16864379C2212yes—
24C22⋊C4×D269—28no—
25C2×C4×Dic269C2212yescomplemented
26C2×C4×Dic269C2211yescharacteristic, complemented
27#22575360C2212yescomplemented
28#22575360C2212yescomplemented
29C22×D4×C269C2211yescharacteristic
30C23×C4×C269C2211yescharacteristic, complemented
31C2×C22⋊C4×C269C2212yes—
32C2×C22⋊C4×C269C2211yescharacteristic, complemented
33#95×C269C2212yescomplemented
34C23×D269—24no—
35C23×D269—24no—
36C23×D269—41no—
37C23×D269—41no—
38C23×D269C2×C414yescomplemented
39C23×D269C2312yes—
40C2×C22⋊D269—24no—
41C2×C22⋊D269—24no—
42C2×C22⋊D269C2×C418yescomplemented
43C22×Dic269C2312yes—
44C22×Dic269C2312yes—
45C22×Dic269C2312yes—
46C2×C4×D269—24no—
47C2×C4×D269—24no—
48C2×C4×D269—24no—
49C2×C4×D269—24no—
50C2×C4×D269—42no—
51C2×C4×D269—42no—
52C4⋊Dic269—41no—
53#16864381—24no—
54C22⋊Dic269—24no—
55C22⋊Dic269—24no—
56C22⋊Dic269C2×C414yescomplemented
57#16864379—24no—
58#16864379C2×C418yescomplemented
59#16864379D418yescomplemented
60C4×Dic269—24no—
61C4×Dic269—41no—
62C24×C269C2311yescharacteristic
63C24×C269C2311yescharacteristic
64C2×D4×C269—41no—
65C2×D4×C269—41no—
66C2×D4×C269C2×C414yescomplemented
67C22×C4×C269—24no—
68C22×C4×C269—41no—
69C22×C4×C269C2312yes—
70C22×C4×C269C2311yescharacteristic
71C22×C4×C269C2312yes—
72C22⋊C4×C269—24no—
73C22⋊C4×C269—41no—
74C22×D269—22no—
75C22×D269—22no—
76C22×D269—22no—
77C22×D269—22no—
78C22×D269—216no—
79C22×D269—44no—
80C22×D269—44no—
81C22×D269—48no—
82C22×D269—44no—
83C22×D269C22×C414yes—
84C22×D269C2×D414yes—
85C22×D269#4114yes—
86C22⋊D269—216no—
87C2×Dic269—22no—
88C2×Dic269—22no—
89C2×Dic269—22no—
90C2×Dic269—22no—
91C2×Dic269—22no—
92C2×Dic269—22no—
93C2×Dic269—22no—
94C2×Dic269C22×C414yes—
95C2×Dic269C2×D414yes—
96C4×D269—48no—
97C4×D269—48no—
98C23×C269—22no—
99C23×C269—22no—
100C23×C269—22no—
101C23×C269—22no—
102C23×C269—21no—
103C23×C269—21no—
104C23×C269—21no—
105C23×C269—21no—
106C23×C269C22×C412yes—
107C23×C269C22×C412yes—
108C23×C269C2411yescharacteristic
109D4×C269—44no—
110C2×C4×C269—22no—
111C2×C4×C269—22no—
112C2×C4×C269—21no—
113C2×C4×C269—21no—
114C2×C4×C269—22no—
115C2×C4×C269—22no—
116C2×C4×C269—22no—
117C2×C4×C269—44no—
118C2×C4×C269C22×C412yes—
119C2×C4×C269C2×D414yes—
120C22×C269—24no—
121C22×C269—24no—
122C22×C269—22no—
123C22×C269—22no—
124C22×C269—41no—
125C22×C269—41no—
126C22×C269—42no—
127C22×C269—42no—
128C22×C269—41no—
129C22×C269—41no—
130C22×C269C23×C411yescharacteristic
131C22×C269C22×D412yes—
132C22×C269C2×#4112yes—
133C22×C269H(5,𝔽2)11yescharacteristic
134C22×C269H(5,𝔽2)11yescharacteristic
135C2×D269—24no—
136C2×D269—24no—
137C2×D269—24no—
138C2×D269—24no—
139C2×D269—24no—
140C2×D269—24no—
141C2×D269—416no—
142C2×D269C4×D418yescomplemented
143C4×C269—22no—
144C4×C269—24no—
145C4×C269—42no—
146Dic269—24no—
147Dic269—24no—
148Dic269—42no—
149C2×C269—42no—
150C2×C269—42no—
151C2×C269C2×C4×D412yes—
152C2×C269#51811yescharacteristic
153C2×C269D4211yescharacteristic
154C2×C269#54612yes—
155C2×C269#55111yescharacteristic
156D269—28no—
157C269#235811yescharacteristic, complemented, Sylow
158#2358—2691nomaximal, Sylow
159C2×#120—2691no—
160C2×C4×D4—2692no—
161C22×C22⋊C4—2692no—
162#390—2691no—
163#388—2692no—
164#386—2692no—
165#386D26911yescharacteristic, complemented
166#385—2692no—
167C4×C22⋊C4—2692no—
168C25—2691no—
169C22×D4—2692no—
170C22×D4C2×D26911yescharacteristic
171C23×C4—2692no—
172C23×C4—2692no—
173C23×C4C2×D26911yescharacteristic, complemented
174#120—2698no—
175C4×D4—5388no—
176C2×C4⋊C4—2692no—
177C2×C4⋊C4—2691no—
178C2×C22⋊C4—2691no—
179C2×C22⋊C4—2692no—
180C2×C22⋊C4—2692no—
181C2×C22⋊C4—2692no—
182C2×C22⋊C4—2692no—
183C2×C22⋊C4—2692no—
184C2×C22⋊C4—5388no—
185C2×C22⋊C4C2×D26912yes—
186C2×C22⋊C4C2×D26911yescharacteristic, complemented
187C2×C42—2692no—
188C2×C42—2691no—
189#95—2692no—
190#95—2692no—
191#95C2×D26912yescomplemented
192C24—2692no—
193C24—2694no—
194C24—5384no—
195C24—5384no—
196C24—10761no—
197C24—10761no—
198C24C22×D26911yescharacteristic
199C24C22×D26911yescharacteristic
200C2×D4—41no—
201C2×D4—41no—
202C2×D4—2698no—
203C2×D4—5384no—
204C2×D4—5384no—
205C2×D4C4×D26914yescomplemented
206C22×C4—24no—
207C22×C4—41no—
208C22×C4—2692no—
209C22×C4—2692no—
210C22×C4—2692no—
211C22×C4—5384no—
212C22×C4—5384no—
213C22×C4—5384no—
214C22×C4—5384no—
215C22×C4—10762no—
216C22×C4—10762no—
217C22×C4C22×D26912yes—
218C22×C4C22×D26911yescharacteristic
219C22×C4C22×D26912yes—
220C4⋊C4—5384no—
221C4⋊C4—10761no—
222C22⋊C4—24no—
223C22⋊C4—41no—
224C22⋊C4—2694no—
225C22⋊C4—2698no—
226C22⋊C4—2698no—
227C22⋊C4—5384no—
228C22⋊C4—5384no—
229C22⋊C4—5384no—
230C42—5384no—
231C42—10761no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—2694no—
241C23—2694no—
242C23—2694no—
243C23—53816no—
244C23—5382no—
245C23—5382no—
246C23—5382no—
247C23—5382no—
248C23—10764no—
249C23—10764no—
250C23—10768no—
251C23—10764no—
252C23C2×C4×D26912yes—
253C23C2×C4×D26912yes—
254C23C23×D26911yescharacteristic
255D4—44no—
256D4—53816no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—44no—
265C2×C4—2694no—
266C2×C4—2694no—
267C2×C4—5382no—
268C2×C4—5382no—
269C2×C4—5382no—
270C2×C4—5382no—
271C2×C4—5382no—
272C2×C4—5382no—
273C2×C4—5382no—
274C2×C4—10768no—
275C2×C4—10768no—
276C2×C4C2×C4×D26912yes—
277C2×C4D4×D26914yes—
278C22—24no—
279C22—24no—
280C22—22no—
281C22—22no—
282C22—41no—
283C22—41no—
284C22—42no—
285C22—42no—
286C22—41no—
287C22—41no—
288C22—2698no—
289C22—5384no—
290C22—5384no—
291C22—5384no—
292C22—5384no—
293C22—5384no—
294C22—5384no—
295C22—107616no—
296C22C22×C4×D26911yescharacteristic
297C22C2×D4×D26912yes—
298C22#41×D26912yes—
299C22#2257549811yescharacteristic
300C22#2257549811yescharacteristic
301C4—24no—
302C4—22no—
303C4—42no—
304C4—5384no—
305C4—5384no—
306C4—10762no—
307C2—42no—
308C2—42no—
309C2—5388no—
310C2C4×D4×D26912yes—
311C2#3177107811yescharacteristic
312C2#3177123711yescharacteristic
313C2#3177125212yes—
314C2#3177128311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D269
commutatorC22×C269C23×C4
Fitting#386×C269C2
socle, cosocleC23×C269C24
prime radicalC2×#22575421C2
hypercenter#386D269
hypocenterC269#2358
LCSG ▷ C22×C269 ▷ C269
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D269C2semidirect
3. C2×C22⋊C4×D269C2semidirect
4. #31770508C2semidirect
5. #31770499C2semidirect
6. #31770481C2semidirect
7. #31770479C2semidirect
8. #31770452C2semidirect
9. #386×C269C2semidirect
14. #22575421C4semidirect
16. C2×C4⋊Dic269C22semidirect
17. C2×#16864381C22semidirect
20. C2×C22⋊Dic269C22semidirect
25. C2×C4×Dic269C22semidirect
26. C2×C4×Dic269C22semidirect
27. #22575360C22semidirect
28. #22575360C22semidirect
30. C23×C4×C269C22semidirect
32. C2×C22⋊C4×C269C22semidirect
33. #95×C269C22semidirect
38. C23×D269C2×C4semidirect
42. C2×C22⋊D269C2×C4semidirect
56. C22⋊Dic269C2×C4semidirect
58. #16864379C2×C4semidirect
59. #16864379D4semidirect
66. C2×D4×C269C2×C4semidirect
142. C2×D269C4×D4semidirect
157. C269#2358semidirect
165. #386D269semidirect
173. C23×C4C2×D269semidirect
186. C2×C22⋊C4C2×D269semidirect
191. #95C2×D269semidirect
205. C2×D4C4×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754212
C2×C4×C22⋊D2692
C2×C22⋊C4×D2692
#317705082
#317704992
#317704812
#317704792
#317704522
#386×C2692
#2358269