Finite Group Databasebeta

← #134760

Subgroups — #134760

Order 512.

348 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 90 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
1C8×C4⋊Q8C211yesmaximal, characteristic
2#12802C212yesmaximal
3#8617C212yesmaximal
4#7238C212yesmaximal
5C2×C8×Q8C411yescharacteristic
6C2×C8×Q8C2211yescharacteristic
7C4×C4⋊Q8C2211yescharacteristic
8C2×#356—22no—
9#1984C411yescharacteristic
10#1984C2211yescharacteristic
11#1939—22no—
12#1842C411yescharacteristic
13#1801C2214yes—
14C8×C4⋊C4—22no—
15C42×C8C411yescharacteristic
16#1567—22no—
17#1394—22no—
18#1292—22no—
19C2×C4⋊Q8D411yescharacteristic
20C2×C4×Q8C2×C411yescharacteristic
21C2×C4×Q8C2311yescharacteristic
22C8×Q8—41no—
23C8×Q8—41no—
24C8×Q8—41no—
25C8×Q8—41no—
26#423—24no—
27C2×C4⋊C8—22no—
28C2×C4⋊C8—21no—
29C2×C4⋊C8C2×C411yescharacteristic
30C2×C4⋊C8C2×C411yescharacteristic
31C2×C4⋊C8D412yes—
32C2×#101—24no—
33C2×C4×C8—21no—
34C2×C4×C8—21no—
35C2×C4×C8—21no—
36C2×C4×C8C2×C411yescharacteristic
37C2×C4×C8C2×C411yescharacteristic
38C2×C4×C8D412yes—
39#391D411yescharacteristic
40#391Q811yescharacteristic
41#389—22no—
42#384Q811yescharacteristic
43C4×C4⋊C4—22no—
44C43C2×C411yescharacteristic
45#356—44no—
46#343—24no—
47#336—22no—
48#336—22no—
49#335—42no—
50#335—42no—
51C22×Q8C4⋊C411yescharacteristic
52C22×Q8C2×D411yescharacteristic
53C2×M16—44no—
54C22×C8—21no—
55C22×C8—21no—
56C22×C8C22⋊C411yescharacteristic
57C22×C8C22⋊C412yes—
58C22×C8C22⋊C411yescharacteristic
59C4⋊Q8—22no—
60C4⋊Q8D812yes—
61C4⋊Q8Q1612yes—
62C4×Q8—41no—
63C4×Q8—41no—
64C4×Q8—41no—
65C4×Q8—41no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4C4⋊C411yescharacteristic
70C2×C4⋊C4C2×Q811yescharacteristic
71C2×C4⋊C4#4112yes—
72C2×C42—21no—
73C2×C42—21no—
74C2×C42—21no—
75C2×C42C4⋊C411yescharacteristic
76C2×C42C4⋊C411yescharacteristic
77C2×C42C22×C411yescharacteristic
78C2×C42C2×D412yes—
79C4⋊C8—41no—
80C4⋊C8—42no—
81C4⋊C8—41no—
82C4⋊C8—44no—
83C4⋊C8—44no—
84C4⋊C8—44no—
85C4⋊C8—42no—
86#101—44no—
87#101—44no—
88C4×C8—24no—
89C4×C8—22no—
90C4×C8—22no—
91C4×C8—21no—
92C4×C8—21no—
93C4×C8—41no—
94C4×C8—42no—
95C4×C8—41no—
96C4×C8—41no—
97#95—22no—
98#95—22no—
99C2×Q8—21no—
100C2×Q8—41no—
101C2×Q8—41no—
102C2×Q8—41no—
103C2×Q8—41no—
104C2×Q8#10712yes—
105C2×Q8C2×D811yescharacteristic
106C2×Q8C2×Q1611yescharacteristic
107C2×Q8#13512yes—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4C2×C22⋊C411yescharacteristic
113C22×C4C2×C4⋊C411yescharacteristic
114C22×C4#11711yescharacteristic
115C22×C4#12112yes—
116C22×C4C22⋊Q812yes—
117M16—84no—
118M16—84no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—21no—
122C2×C8—22no—
123C2×C8—41no—
124C2×C8—41no—
125C2×C8—41no—
126C2×C8—42no—
127C2×C8—42no—
128C2×C8—41no—
129C2×C8—84no—
130C2×C8#9911yescharacteristic
131C2×C8#9911yescharacteristic
132C4⋊C4—24no—
133C4⋊C4—24no—
134C4⋊C4—41no—
135C4⋊C4—42no—
136C4⋊C4—41no—
137C4⋊C4—44no—
138C4⋊C4—42no—
139C42—22no—
140C42—22no—
141C42—22no—
142C42—21no—
143C42—21no—
144C42—22no—
145C42—21no—
146C42—41no—
147C42—42no—
148C42—41no—
149C42—41no—
150C42#10712yes—
151C23#38011yescharacteristic
152Q8—41no—
153Q8—41no—
154Q8—42no—
155Q8—42no—
156Q8—42no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—21no—
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—41no—
178C2×C4C2×#9911yescharacteristic
179C2×C4#41011yescharacteristic
180C2×C4C2×#10711yescharacteristic
181C2×C4#42711yescharacteristic
182C2×C4#46612yes—
183C2×C4#46712yes—
184C8—21no—
185C8—42no—
186C8—41no—
187C8—42no—
188C8—81no—
189C8—164no—
190C8#35612yescomplemented
191C22—21no—
192C22—21no—
193C22#179911yescharacteristic
194C22#180811yescharacteristic
195C22#181211yescharacteristic
196C4—24no—
197C4—21no—
198C4—22no—
199C4—21no—
200C4—21no—
201C4—22no—
202C4—42no—
203C4—41no—
204C4—42no—
205C4—42no—
206C4—81no—
207C4C2×#35611yescharacteristic
208C4#213211yescharacteristic
209C2—21no—
210C2—21no—
211C2#860911yescharacteristic
212C2#1280211yescharacteristic
213C2#1280911yescharacteristic
subgroupG/H
center, socleC22#1799
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4×Q8C23
prime radicalC23#380
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
190. C8#356semidirect

Maximal subgroups

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

subgroupindex
C8×C4⋊Q82
#128022
#86172
#72382