Finite Group Databasebeta

← #46821257

Subgroups — #46821257

Order 34432.

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

normal subgroups only

This group has 484 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#31771158C211yesmaximal, characteristic, complemented
2C2×#22575449C211yesmaximal, characteristic
3#31770968C214yesmaximal, complemented
4#31770950C214yesmaximal, complemented
5#31770611C211yesmaximal, characteristic, complemented
6#31770610C211yesmaximal, characteristic, complemented
7#31770609C211yesmaximal, characteristic, complemented
8M16×Dic269C211yesmaximal, characteristic, complemented
9#418×C269C211yesmaximal, characteristic, complemented
10C22×D1076C2211yescharacteristic
11C22×C22⋊D269—22no—
12C2×#16864386C2211yescharacteristic
13#22575452C2212yescomplemented
14#22575449—22no—
15#22575449C2212yes—
16#22575449C2212yes—
17#22575444C2212yescomplemented
18#22575430—22no—
19C4×Dic538—22no—
20#22575412C2211yescharacteristic
21C2×C4×Dic269C2211yescharacteristic
22#22575375C2214yescomplemented
23#22575374C2214yescomplemented
24#22575373C2214yescomplemented
25#22575363C2212yescomplemented
26C8×Dic269C2212yescomplemented
27C22×D4×C269C2211yescharacteristic
28C2×M16×C269C2211yescharacteristic
29#117×C269C2211yescharacteristic
30#102×C269C2214yescomplemented
31C23×D269—21no—
32C2×Dic538—22no—
33C2×D1076—21no—
34C2×D1076—22no—
35C2×D1076—22no—
36C2×D1076C2312yes—
37C2×C22⋊D269—24no—
38C2×C22⋊D269—24no—
39C2×C22⋊D269—48no—
40C22×Dic269D412yes—
41#16864386—22no—
42C2×#14524130C2312yes—
43C4⋊Dic269—22no—
44C4⋊Dic269C2312yes—
45#16864381—22no—
46#16864381—22no—
47C22⋊Dic269—21no—
48C4×Dic269—21no—
49C4×Dic269D412yes—
50C4×Dic269C2311yescharacteristic
51C4×Dic269C2311yescharacteristic
52C24×C269—21no—
53C2×D4×C269—21no—
54C2×D4×C269—22no—
55C2×D4×C269—22no—
56C2×D4×C269C2312yes—
57C22×C4×C269C2311yescharacteristic
58M16×C269—22no—
59C2×C8×C269C2312yes—
60C4⋊C4×C269C2312yes—
61C22⋊C4×C269—21no—
62C42×C269—21no—
63C22×D269—22no—
64C22×D269—44no—
65C22×D269—42no—
66Dic538—44no—
67D1076—22no—
68D1076—22no—
69D1076—42no—
70C22⋊D269—48no—
71C22⋊D269—48no—
72C2×Dic269—24no—
73C2×Dic269—22no—
74C2×Dic269C2×D412yes—
75C2×Dic269C2×D412yes—
76#14524130—22no—
77C23×C269—22no—
78C23×C269—44no—
79C23×C269—42no—
80C23×C269C2×D411yescharacteristic
81D4×C269—22no—
82D4×C269—22no—
83D4×C269—42no—
84C2×C4×C269—22no—
85C2×C4×C269C2×D411yescharacteristic
86C2×C4×C269#4112yes—
87C2×C4×C269#4112yes—
88C2×C4×C269C2411yescharacteristic
89C8×C269—22no—
90C22×C269—22no—
91C22×C269—42no—
92C22×C269—44no—
93C22×C269—84no—
94C22×C269#12412yes—
95C22×C269C22×D411yescharacteristic
96C2×D269—42no—
97C2×D269—44no—
98C2×D269—84no—
99C4×C269—42no—
100C4×C269#12412yes—
101C4×C269C2×#4111yescharacteristic
102C4×C269C2×#4111yescharacteristic
103Dic269—22no—
104Dic269—42no—
105Dic269AGL(1,ℤ/8)14yes—
106C2×C269—22no—
107C2×C269—82no—
108C2×C269C2×#12411yescharacteristic
109C2×C269C2×AGL(1,ℤ/8)12yes—
110D269—82no—
111C269#313811yescharacteristic, complemented, Sylow
112#3138—2691nomaximal, Sylow
113#533—2691no—
114C2×#127—2691no—
115#490—2694no—
116#486—2694no—
117#418—2691no—
118#418—2691no—
119#418—2691no—
120#418D26911yescharacteristic, complemented
121C4×M16—2691no—
122C22×D4—2691no—
123C22×D4—5382no—
124C22×D4C2×D26911yescharacteristic
125C2×M16—2691no—
126C2×M16C2×D26911yescharacteristic
127C4⋊Q8—2692no—
128#127—2692no—
129#127—2692no—
130#127—5382no—
131#125—2692no—
132C22⋊Q8—5382no—
133C4×Q8—5382no—
134#117—2691no—
135#117C2×D26911yescharacteristic
136C2×C42—2691no—
137#102—2694no—
138#102—2694no—
139#102—2694no—
140#102C2×D26914yescomplemented
141#97—2692no—
142C4×C8—2692no—
143C24—21no—
144C24—5381no—
145C2×Q8—5382no—
146C2×D4—21no—
147C2×D4—22no—
148C2×D4—22no—
149C2×D4—2692no—
150C2×D4—5384no—
151C2×D4—5384no—
152C2×D4—5381no—
153C2×D4—5382no—
154C2×D4—5382no—
155C2×D4—10768no—
156C2×D4C22×D26912yes—
157C22×C4—2692no—
158C22×C4C22×D26911yescharacteristic
159M16—22no—
160M16—5382no—
161C2×C8—2692no—
162C2×C8C22×D26912yes—
163C4⋊C4—2692no—
164C4⋊C4—5382no—
165C4⋊C4—5382no—
166C4⋊C4—5382no—
167C4⋊C4C22×D26912yes—
168C22⋊C4—21no—
169C22⋊C4—5381no—
170C42—21no—
171C42—2692no—
172C42—2691no—
173C42—2691no—
174C42—5381no—
175C23—22no—
176C23—44no—
177C23—42no—
178C23—5382no—
179C23—10764no—
180C23—10762no—
181C23D4×D26911yescharacteristic
182Q8—10764no—
183D4—22no—
184D4—22no—
185D4—42no—
186D4—5382no—
187D4—5382no—
188D4—10768no—
189D4—10768no—
190D4—10762no—
191C2×C4—22no—
192C2×C4—2692no—
193C2×C4—2692no—
194C2×C4—5384no—
195C2×C4—5382no—
196C2×C4D4×D26911yescharacteristic
197C2×C4#1686440112yes—
198C2×C4#1686440212yes—
199C2×C4C23×D26911yescharacteristic
200C8—22no—
201C8—5382no—
202C22—22no—
203C22—42no—
204C22—44no—
205C22—84no—
206C22—10762no—
207C22—10764no—
208C22—21524no—
209C22#2257543912yes—
210C22C2×D4×D26911yescharacteristic
211C4—42no—
212C4—2694no—
213C4—5382no—
214C4—10762no—
215C4#2257543912yes—
216C4C2×#1686440111yescharacteristic
217C4C2×#1686440211yescharacteristic
218C2—22no—
219C2—82no—
220C2—21522no—
221C2C2×#2257543911yescharacteristic
222C2AGL(1,ℤ/8)×D26912yes—
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#418×C269C2
socleC22×C269C22×D4
prime radicalC2×#22575449C2
hypercenter#418D269
hypocenterC269#3138
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. #31770968C2semidirect
4. #31770950C2semidirect
5. #31770611C2semidirect
6. #31770610C2semidirect
7. #31770609C2semidirect
8. M16×Dic269C2semidirect
9. #418×C269C2semidirect
13. #22575452C22semidirect
17. #22575444C22semidirect
22. #22575375C22semidirect
23. #22575374C22semidirect
24. #22575373C22semidirect
25. #22575363C22semidirect
26. C8×Dic269C22semidirect
30. #102×C269C22semidirect
111. C269#3138semidirect
120. #418D269semidirect
140. #102C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317711582
C2×#225754492
#317709682
#317709502
#317706112
#317706102
#317706092
M16×Dic2692
#418×C2692
#3138269