Finite Group Databasebeta

← #46820679

Subgroups — #46820679

Order 34432.

440 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 143 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#31771158C211yesmaximal, characteristic, complemented
2C2×D4×Dic269C211yesmaximal, characteristic
3#31770907C212yesmaximal, complemented
4#31770903C212yesmaximal, complemented
5#31770893C212yesmaximal, complemented
6#31770883C212yesmaximal, complemented
7#31770648C211yesmaximal, characteristic, complemented
8#31770647C211yesmaximal, characteristic, complemented
9#31770628C211yesmaximal, characteristic, complemented
10#31770610C211yesmaximal, characteristic
11#418×C269C211yesmaximal, characteristic
12C23×Dic269—21no—
13C2×#16864386C2211yescharacteristic
14#22575452C2211yescharacteristic, complemented
15#22575452C2211yescharacteristic, complemented
16#22575444C2211yescharacteristic, complemented
17#22575444C2211yescharacteristic, complemented
18#22575430—21no—
19#22575430—21no—
20C4×Dic538—21no—
21C4×Dic538—21no—
22D4×Dic269—21no—
23D4×Dic269—21no—
24D4×Dic269C2212yes—
25D4×Dic269C2212yes—
26#22575412C2211yescharacteristic
27C2×C4⋊Dic269C2211yescharacteristic
28C2×C22⋊Dic269—21no—
29C2×C4×Dic269C2211yescharacteristic
30#22575387C2212yescomplemented
31#22575386C2212yescomplemented
32#22575385C2212yescomplemented
33#22575384C2212yescomplemented
34#22575382C2212yescomplemented
35#22575382C2212yescomplemented
36#22575374C2212yes—
37#22575374C2212yes—
38C22×D4×C269C2211yescharacteristic
39C2×M16×C269C2211yescharacteristic
40#117×C269C2211yescharacteristic
41#102×C269C2212yes—
42#102×C269C2212yes—
43C2×Dic538—21no—
44C2×Dic538—21no—
45C22×Dic269—22no—
46C22×Dic269—22no—
47C22×Dic269—21no—
48C22×Dic269—44no—
49C22×Dic269D411yescharacteristic
50C22×Dic269D411yescharacteristic
51#16864386—22no—
52C2×#14524130C2312yes—
53C4⋊Dic269—21no—
54C4⋊Dic269—21no—
55C4⋊Dic269—21no—
56C4⋊Dic269C2311yescharacteristic
57C4⋊Dic269C2311yescharacteristic
58C4⋊Dic269C2312yes—
59#16864381—21no—
60#16864381—21no—
61#16864381—21no—
62#16864381—21no—
63C22⋊Dic269—22no—
64C22⋊Dic269—22no—
65C22⋊Dic269—21no—
66C4×Dic269—21no—
67C4×Dic269D411yescharacteristic
68C4×Dic269D411yescharacteristic
69C4×Dic269C2311yescharacteristic
70C4×Dic269C2311yescharacteristic
71C24×C269—21no—
72C2×D4×C269—21no—
73C2×D4×C269—22no—
74C2×D4×C269Q814yescomplemented
75C2×D4×C269C2312yes—
76C22×C4×C269C2311yescharacteristic
77M16×C269—22no—
78C2×C8×C269C2312yes—
79C4⋊C4×C269C2311yescharacteristic
80C4⋊C4×C269C2311yescharacteristic
81C22⋊C4×C269—21no—
82C42×C269—21no—
83Dic538—42no—
84Dic538—42no—
85C2×Dic269—22no—
86C2×Dic269—21no—
87C2×Dic269—21no—
88C2×Dic269—22no—
89C2×Dic269—41no—
90C2×Dic269—44no—
91C2×Dic269—42no—
92C2×Dic269—44no—
93C2×Dic269C2×D411yescharacteristic
94C2×Dic269C2×D411yescharacteristic
95C2×Dic269C2×D411yescharacteristic
96C2×Dic269C2×D411yescharacteristic
97#14524130—22no—
98C23×C269—24no—
99C23×C269—22no—
100C23×C269—42no—
101C23×C269—42no—
102C23×C269C2×D411yescharacteristic
103D4×C269—24no—
104D4×C269—22no—
105D4×C269C2×Q814yes—
106C2×C4×C269—21no—
107C2×C4×C269—21no—
108C2×C4×C269C2×D411yescharacteristic
109C2×C4×C269C2×Q812yes—
110C2×C4×C269#4112yes—
111C2×C4×C269C2411yescharacteristic
112C8×C269—22no—
113C22×C269—21no—
114C22×C269—21no—
115C22×C269—24no—
116C22×C269—48no—
117C22×C269—42no—
118C22×C269—42no—
119C22×C269C22⋊Q812yes—
120C22×C269C22×D411yescharacteristic
121C4×C269—41no—
122C4×C269—41no—
123C4×C269C22⋊Q812yes—
124C4×C269C22×Q811yescharacteristic
125C4×C269C2×#4111yescharacteristic
126Dic269—21no—
127Dic269—21no—
128Dic269—21no—
129Dic269—42no—
130Dic269—41no—
131Dic269—41no—
132Dic269AGL(1,ℤ/8)12yes—
133C2×C269—22no—
134C2×C269—44no—
135C2×C269C2×C22⋊Q811yescharacteristic
136C2×C269C2×AGL(1,ℤ/8)11yescharacteristic
137C2×C269#57511yescharacteristic
138C269#307711yescharacteristic, complemented, Sylow
139#3077—2691nomaximal, Sylow
140#533—2691no—
141C2×C4×D4—2691no—
142#478—2692no—
143#478—2692no—
144#476—2692no—
145#474—2692no—
146#428—2691no—
147#428—2691no—
148#423—2691no—
149#418—2691no—
150#418D26911yescharacteristic
151C22×D4C2×D26911yescharacteristic
152C23×C4—5381no—
153C2×M16—2691no—
154C2×M16C2×D26911yescharacteristic
155C4⋊Q8—2691no—
156C4⋊Q8—2691no—
157#125—2691no—
158#125—2691no—
159C22⋊Q8—5381no—
160C22⋊Q8—5381no—
161C4×Q8—5381no—
162C4×Q8—5381no—
163C4×D4—2692no—
164C4×D4—2692no—
165C4×D4—5381no—
166C4×D4—5381no—
167#117—2691no—
168#117C2×D26911yescharacteristic
169C2×C4⋊C4—2691no—
170C2×C22⋊C4—5381no—
171C2×C42—2691no—
172#107—2692no—
173#107—2692no—
174#106—2692no—
175#106—2692no—
176C4⋊C8—2692no—
177C4⋊C8—2692no—
178#102—2692no—
179#102—2692no—
180#102C2×D26912yes—
181#102C2×D26912yes—
182C24—21no—
183C2×Q8—5381no—
184C2×Q8—5381no—
185C2×D4—21no—
186C2×D4—22no—
187C2×D4Dic53814yescomplemented
188C2×D4C22×D26912yes—
189C22×C4—2691no—
190C22×C4—2691no—
191C22×C4—5382no—
192C22×C4—5382no—
193C22×C4—5381no—
194C22×C4—10764no—
195C22×C4C22×D26911yescharacteristic
196M16—22no—
197M16—5382no—
198C2×C8—2692no—
199C2×C8C22×D26912yes—
200C4⋊C4—2691no—
201C4⋊C4—2691no—
202C4⋊C4—2692no—
203C4⋊C4—5381no—
204C4⋊C4—5381no—
205C4⋊C4—5381no—
206C4⋊C4—5381no—
207C4⋊C4—5381no—
208C4⋊C4—5381no—
209C4⋊C4—5381no—
210C4⋊C4C22×D26911yescharacteristic
211C4⋊C4C22×D26911yescharacteristic
212C22⋊C4—21no—
213C22⋊C4—5382no—
214C22⋊C4—5382no—
215C22⋊C4—5381no—
216C42—21no—
217C42—2691no—
218C42—2691no—
219C42—2691no—
220C42—2691no—
221C42—5381no—
222C23—24no—
223C23—22no—
224C23—42no—
225C23—42no—
226C23D4×D26911yescharacteristic
227Q8—10762no—
228Q8—10762no—
229D4—24no—
230D4—22no—
231D4C2×Dic53814yes—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—2691no—
235C2×C4—2691no—
236C2×C4—2691no—
237C2×C4—2691no—
238C2×C4—5382no—
239C2×C4—5381no—
240C2×C4—5381no—
241C2×C4—5382no—
242C2×C4—10764no—
243C2×C4—10761no—
244C2×C4—10764no—
245C2×C4—10762no—
246C2×C4D4×D26911yescharacteristic
247C2×C4C2×Dic53812yes—
248C2×C4#1686440112yes—
249C2×C4C23×D26911yescharacteristic
250C8—22no—
251C8—5382no—
252C22—21no—
253C22—21no—
254C22—24no—
255C22—48no—
256C22—42no—
257C22—42no—
258C22#2257543012yes—
259C22C2×D4×D26911yescharacteristic
260C4—41no—
261C4—41no—
262C4—2692no—
263C4—5381no—
264C4—5381no—
265C4—5381no—
266C4—10761no—
267C4—10762no—
268C4—10761no—
269C4#2257543012yes—
270C4C22×Dic53811yescharacteristic
271C4C2×#1686440111yescharacteristic
272C2—22no—
273C2—44no—
274C2C2×#2257543011yescharacteristic
275C2AGL(1,ℤ/8)×D26911yescharacteristic
276C2#3177140111yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#418×C269C2
socleC22×C269C22×D4
prime radicalC22×D4×C269C22
hypercenter#418D269
hypocenterC269#3077
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #31771158C2semidirect
3. #31770907C2semidirect
4. #31770903C2semidirect
5. #31770893C2semidirect
6. #31770883C2semidirect
7. #31770648C2semidirect
8. #31770647C2semidirect
9. #31770628C2semidirect
14. #22575452C22semidirect
15. #22575452C22semidirect
16. #22575444C22semidirect
17. #22575444C22semidirect
30. #22575387C22semidirect
31. #22575386C22semidirect
32. #22575385C22semidirect
33. #22575384C22semidirect
34. #22575382C22semidirect
35. #22575382C22semidirect
74. C2×D4×C269Q8semidirect
138. C269#3077semidirect
187. C2×D4Dic538semidirect

Maximal subgroups

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

subgroupindex
#317711582
C2×D4×Dic2692
#317709072
#317709032
#317708932
#317708832
#317706482
#317706472
#317706282
#317706102
#418×C2692
#3077269