Finite Group Databasebeta

← #46822699

Subgroups — #46822699

Order 34432.

494 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 139 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#31771259C211yesmaximal, characteristic, complemented
2D4×D1076C211yesmaximal, characteristic, complemented
3#31770952C211yesmaximal, characteristic, complemented
4#31770919C212yesmaximal, complemented
5#31770886C211yesmaximal, characteristic, complemented
6#31770841C212yesmaximal, complemented
7#31770801C211yesmaximal, characteristic, complemented
8#31770741C212yesmaximal, complemented
9C4×D2152C211yesmaximal, characteristic, complemented
10C2×#22575375C212yesmaximal, complemented
11C8×D4×C269C211yesmaximal, characteristic, complemented
12C2×#16864401—21no—
13C22×D1076C2212yes—
14C2×D4×D269—21no—
15C2×D2152C2211yescharacteristic
16C2×D2152C2212yescomplemented
17#22575453C2211yescharacteristic, complemented
18#22575450C2211yescharacteristic
19#22575437—22no—
20#22575433—21no—
21#22575428—21no—
22#22575427C2212yes—
23#22575427C2212yescomplemented
24#22575423—22no—
25C4×D1076C2211yescharacteristic
26C4×D1076C2211yescharacteristic, complemented
27C2×C4⋊Dic269C2212yescomplemented
28#22575387C2212yescomplemented
29#22575387C2211yescharacteristic, complemented
30#22575375C2212yes—
31#22575375C2211yescharacteristic
32#22575375C2212yescomplemented
33#22575375C2212yescomplemented
34#22575375C2211yescharacteristic, complemented
35#22575375C2212yescomplemented
36C22×C8×C269C2212yescomplemented
37C4×D4×C269C2211yescharacteristic
38C4⋊C8×C269C2211yescharacteristic, complemented
39C22⋊C8×C269C2212yescomplemented
40C4×C8×C269C2211yescharacteristic, complemented
41C23×D269—22no—
42#16864401—44no—
43C2×Dic538—21no—
44C2×D1076—24no—
45C2×D1076—22no—
46C2×D1076—21no—
47C2×D1076—21no—
48C2×D1076—42no—
49C2×D1076C2312yes—
50C2×D1076C2311yescharacteristic
51C2×D1076C2311yescharacteristic
52C2×C22⋊D269—22no—
53C2×C22⋊D269—22no—
54D4×D269—44no—
55C22×Dic269—22no—
56C2×C4×D269—21no—
57C2×C4×D269—21no—
58D2152—22no—
59D2152—42no—
60C4⋊Dic269—22no—
61C4⋊Dic269—21no—
62C4⋊Dic269C2311yescharacteristic
63C4⋊Dic269C2312yes—
64C4⋊Dic269C2311yescharacteristic
65#16864379—21no—
66#16864379—21no—
67#16864379—22no—
68#16864379—22no—
69C2×D4×C269D411yescharacteristic
70C22×C4×C269C2312yes—
71C2×C8×C269—42no—
72C2×C8×C269C2311yescharacteristic
73C2×C8×C269C2312yes—
74C2×C8×C269C2311yescharacteristic
75C4⋊C4×C269D411yescharacteristic
76C22⋊C4×C269D412yes—
77C42×C269C2311yescharacteristic
78C22×D269—21no—
79C22×D269—21no—
80C22×D269—22no—
81C22×D269—44no—
82C22×D269—44no—
83C22×D269—82no—
84Dic538—42no—
85D1076—21no—
86D1076—22no—
87D1076—21no—
88D1076—21no—
89D1076—22no—
90D1076—44no—
91D1076—42no—
92D1076—81no—
93D1076#4112yescomplemented
94C22⋊D269—44no—
95C22⋊D269—44no—
96C2×Dic269—21no—
97C2×Dic269—21no—
98C2×Dic269—22no—
99C2×Dic269—44no—
100C4×D269—42no—
101C4×D269—42no—
102C23×C269C2×D412yes—
103D4×C269D814yescomplemented
104C2×C4×C269—22no—
105C2×C4×C269—22no—
106C2×C4×C269C2×D411yescharacteristic
107C2×C4×C269C2×D412yes—
108C2×C4×C269C2×D411yescharacteristic
109C2×C4×C269C2411yescharacteristic
110C8×C269—21no—
111C8×C269—21no—
112C8×C269—22no—
113C22×C269—22no—
114C22×C269—22no—
115C22×C269C2×D814yes—
116C22×C269C22×D411yescharacteristic
117C2×D269—22no—
118C2×D269—41no—
119C2×D269—41no—
120C2×D269—42no—
121C2×D269—41no—
122C2×D269—41no—
123C2×D269—42no—
124C2×D269—42no—
125C2×D269—41no—
126C2×D269—88no—
127C4×C269—22no—
128C4×C269—21no—
129C4×C269C2×D812yes—
130C4×C269C2×#4111yescharacteristic
131C4×C269H(5,𝔽2)11yescharacteristic
132Dic269—41no—
133Dic269—41no—
134Dic269—42no—
135C2×C269—24no—
136C2×C269#54611yescharacteristic
137C2×C269C22×D811yescharacteristic
138C2×C269#57711yescharacteristic
139D269—42no—
140D269—81no—
141D269—82no—
142C269#329411yescharacteristic, complemented, Sylow
143#3294—2691nomaximal, Sylow
144#547—2691no—
145D42—2691no—
146#486—2691no—
147#480—2692no—
148#474—2691no—
149#466—2692no—
150#459—2691no—
151#447—2692no—
152C4×D8—2691no—
153C8×D4D26911yescharacteristic, complemented
154C2×#102—2692no—
155C2×#41—5381no—
156C22×D4—2692no—
157C22×D4—5381no—
158C2×D8—2691no—
159C2×D8—2692no—
160C22×C8C2×D26912yescomplemented
161C4⋊Q8—2691no—
162#127—2691no—
163#123—5382no—
164C22⋊Q8—5381no—
165#121—2692no—
166#121—2692no—
167#121—5381no—
168#120—5382no—
169C4×D4—2691no—
170C4×D4—2691no—
171C4×D4C2×D26911yescharacteristic
172C2×C4⋊C4—2692no—
173#107—2691no—
174#107—2692no—
175C4⋊C8C2×D26911yescharacteristic, complemented
176#102—2691no—
177#102—2692no—
178#102—2692no—
179#102—2692no—
180#102—2691no—
181#102—2692no—
182C22⋊C8C2×D26912yescomplemented
183C4×C8C2×D26911yescharacteristic, complemented
184C24—5382no—
185#41—10764no—
186C2×Q8—5381no—
187C2×D4—2691no—
188C2×D4—2692no—
189C2×D4—2691no—
190C2×D4—5384no—
191C2×D4—5382no—
192C2×D4—5382no—
193C2×D4—5382no—
194C2×D4—5381no—
195C2×D4—5381no—
196C2×D4—10762no—
197C2×D4—10764no—
198C2×D4D107611yescharacteristic
199C22×C4—5382no—
200C22×C4—5381no—
201C22×C4—5381no—
202C22×C4C22×D26912yes—
203D8—5382no—
204D8—10762no—
205C2×C8—42no—
206C2×C8C22×D26911yescharacteristic
207C2×C8C22×D26911yescharacteristic
208C2×C8C22×D26912yes—
209C4⋊C4—2691no—
210C4⋊C4—2692no—
211C4⋊C4—2691no—
212C4⋊C4—5382no—
213C4⋊C4—5381no—
214C4⋊C4D107611yescharacteristic
215C22⋊C4—5381no—
216C22⋊C4—5381no—
217C22⋊C4—5382no—
218C22⋊C4—5382no—
219C22⋊C4D107612yes—
220C42C22×D26911yescharacteristic
221C23—5381no—
222C23—5381no—
223C23—5382no—
224C23—10764no—
225C23—10764no—
226C23—21522no—
227C23C2×D107612yes—
228Q8—10762no—
229D4—2692no—
230D4—5381no—
231D4—5382no—
232D4—5381no—
233D4—5381no—
234D4—5382no—
235D4—10764no—
236D4—10764no—
237D4—10764no—
238D4—10762no—
239D4—21521no—
240D4D215214yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—5381no—
244C2×C4—5381no—
245C2×C4—5382no—
246C2×C4—10764no—
247C2×C4—10762no—
248C2×C4—10762no—
249C2×C4C2×D107611yescharacteristic
250C2×C4C2×D107612yes—
251C2×C4C2×D107611yescharacteristic
252C2×C4C23×D26911yescharacteristic
253C8—21no—
254C8—21no—
255C8—22no—
256C22—22no—
257C22—22no—
258C22—5382no—
259C22—10761no—
260C22—10761no—
261C22—10762no—
262C22—10761no—
263C22—10762no—
264C22—10761no—
265C22—10761no—
266C22—10762no—
267C22—21528no—
268C22C2×D215214yes—
269C22C22×D107611yescharacteristic
270C4—21no—
271C4—22no—
272C4—10761no—
273C4—10761no—
274C4—10762no—
275C4C2×D215212yes—
276C4#41×D26911yescharacteristic
277C4#2257549811yescharacteristic
278C2—24no—
279C2—10762no—
280C2—21522no—
281C2—21521no—
282C2#3177125011yescharacteristic
283C2C22×D215211yescharacteristic
284C2#3177141711yescharacteristic
subgroupG/H
centerC22C22×D1076
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingC8×D4×C269C2
socleC22×C269C22×D4
hypercenterC8×D4D269
hypocenterC269#3294
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #31771259C2semidirect
2. D4×D1076C2semidirect
3. #31770952C2semidirect
4. #31770919C2semidirect
5. #31770886C2semidirect
6. #31770841C2semidirect
7. #31770801C2semidirect
8. #31770741C2semidirect
9. C4×D2152C2semidirect
10. C2×#22575375C2semidirect
11. C8×D4×C269C2semidirect
16. C2×D2152C22semidirect
17. #22575453C22semidirect
23. #22575427C22semidirect
26. C4×D1076C22semidirect
27. C2×C4⋊Dic269C22semidirect
28. #22575387C22semidirect
29. #22575387C22semidirect
32. #22575375C22semidirect
33. #22575375C22semidirect
34. #22575375C22semidirect
35. #22575375C22semidirect
36. C22×C8×C269C22semidirect
38. C4⋊C8×C269C22semidirect
39. C22⋊C8×C269C22semidirect
40. C4×C8×C269C22semidirect
93. D1076#41semidirect
103. D4×C269D8semidirect
142. C269#3294semidirect
153. C8×D4D269semidirect
160. C22×C8C2×D269semidirect
175. C4⋊C8C2×D269semidirect
182. C22⋊C8C2×D269semidirect
183. C4×C8C2×D269semidirect
240. D4D2152semidirect

Maximal subgroups

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

subgroupindex
#317712592
D4×D10762
#317709522
#317709192
#317708862
#317708412
#317708012
#317707412
C4×D21522
C2×#225753752
C8×D4×C2692
#3294269