Finite Group Databasebeta

← #99078

Subgroups — #99078

Order 512.

338 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 106 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#8785C211yesmaximal, characteristic
3#8785C211yesmaximal, characteristic
4#7652C212yesmaximal
5#7040C212yesmaximal
6C2×C8×Q8—21no—
7C4×C4⋊Q8C2211yescharacteristic
8#1984—21no—
9#1939—21no—
10#1939C412yes—
11#1842C2211yescharacteristic
12#1831C2212yes—
13#1831C2212yes—
14C8×C4⋊C4—21no—
15C8×C4⋊C4C412yes—
16C42×C8C2211yescharacteristic
17#1277—21no—
18#1277—21no—
19#1277—21no—
20#1277—21no—
21#1271—24no—
22C2×C4⋊Q8Q811yescharacteristic
23C2×C4×Q8—21no—
24C8×Q8—44no—
25C2×C4⋊C8—22no—
26C2×C4⋊C8—21no—
27C2×C4⋊C8—21no—
28C2×C4⋊C8—22no—
29C2×C4⋊C8—22no—
30C2×C4⋊C8C2×C412yes—
31C2×C4×C8—22no—
32C2×C4×C8C2×C411yescharacteristic
33C2×C4×C8C2×C411yescharacteristic
34C2×C4×C8D411yescharacteristic
35C2×C4×C8D411yescharacteristic
36C2×C4×C8D411yescharacteristic
37C2×C4×C8D411yescharacteristic
38#391—21no—
39#389—21no—
40#389D412yes—
41#384Q811yescharacteristic
42C4×C4⋊C4—21no—
43C4×C4⋊C4C2×C412yes—
44C43C2311yescharacteristic
45#336—22no—
46#336—22no—
47#336—22no—
48#336—22no—
49#336—22no—
50#336—22no—
51C22×Q8—21no—
52C22×C8—22no—
53C22×C8—42no—
54C22×C8—42no—
55C22×C8C22⋊C411yescharacteristic
56C22×C8C22⋊C411yescharacteristic
57C22×C8C22⋊C411yescharacteristic
58C22×C8C22⋊C411yescharacteristic
59C4⋊Q8—24no—
60C4×Q8—42no—
61C4×Q8—42no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4C4⋊C412yes—
67C2×C4⋊C4C4⋊C412yes—
68C2×C42—22no—
69C2×C42C22×C411yescharacteristic
70C2×C42C2×D411yescharacteristic
71C2×C42C2×D411yescharacteristic
72C2×C42C2×D411yescharacteristic
73C2×C42C2×Q811yescharacteristic
74C2×C42#4111yescharacteristic
75C2×C42#4111yescharacteristic
76C4⋊C8—42no—
77C4⋊C8—44no—
78C4⋊C8—42no—
79C4⋊C8—42no—
80C4⋊C8—42no—
81C4⋊C8—42no—
82C4⋊C8—42no—
83C4⋊C8—42no—
84C4×C8—22no—
85C4×C8—22no—
86C4×C8—22no—
87C4×C8—22no—
88C4×C8—22no—
89C4×C8—41no—
90C4×C8—44no—
91#95—22no—
92#95—22no—
93C2×Q8—24no—
94C2×Q8—42no—
95C2×Q8—42no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4C2×C22⋊C411yescharacteristic
99C22×C4C2×C4⋊C411yescharacteristic
100C22×C4#11711yescharacteristic
101C22×C4#12111yescharacteristic
102C22×C4#12111yescharacteristic
103C22×C4C22⋊Q811yescharacteristic
104C22×C4C22⋊Q811yescharacteristic
105C2×C8—21no—
106C2×C8—21no—
107C2×C8—21no—
108C2×C8—21no—
109C2×C8—21no—
110C2×C8—21no—
111C2×C8—21no—
112C2×C8—21no—
113C2×C8—21no—
114C2×C8—21no—
115C2×C8—21no—
116C2×C8—42no—
117C2×C8—42no—
118C2×C8—42no—
119C2×C8—82no—
120C2×C8—82no—
121C2×C8—82no—
122C2×C8—82no—
123C2×C8—82no—
124C2×C8—82no—
125C2×C8#10112yes—
126C4⋊C4—24no—
127C4⋊C4—22no—
128C4⋊C4—42no—
129C4⋊C4—42no—
130C4⋊C4—44no—
131C4⋊C4—42no—
132C4⋊C4#10814yes—
133C42—21no—
134C42—21no—
135C42—22no—
136C42—41no—
137C42—41no—
138C42—41no—
139C42—42no—
140C42—42no—
141C42#13512yes—
142C42#13512yes—
143C42AGL(1,ℤ/8)12yes—
144C42#13712yes—
145C23#38011yescharacteristic
146Q8—48no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—24no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4C2×#10111yescharacteristic
167C2×C4#41311yescharacteristic
168C2×C4#41611yescharacteristic
169C2×C4#41611yescharacteristic
170C2×C4#41811yescharacteristic
171C2×C4#41911yescharacteristic
172C2×C4C2×#10812yes—
173C2×C4#46312yes—
174C2×C4#46412yes—
175C2×C4#47812yes—
176C2×C4#47912yes—
177C8—22no—
178C8—41no—
179C8—41no—
180C8—41no—
181C8—82no—
182C8—162no—
183C8—162no—
184C22#180111yescharacteristic
185C22#180411yescharacteristic
186C22#180511yescharacteristic
187C22#181011yescharacteristic
188C22#181211yescharacteristic
189C22#181411yescharacteristic
190C22#181411yescharacteristic
191C4—22no—
192C4—22no—
193C4—22no—
194C4—22no—
195C4—22no—
196C4—41no—
197C4—44no—
198C4—82no—
199C4#154411yescharacteristic
200C4#155311yescharacteristic
201C4#155411yescharacteristic
202C4#155611yescharacteristic
203C2#859311yescharacteristic
204C2#862311yescharacteristic
205C2#863111yescharacteristic
206C2#866011yescharacteristic
207C2#866111yescharacteristic
208C2#867811yescharacteristic
209C2#867911yescharacteristic
subgroupG/H
center, socle, prime radicalC23#380
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
C8×C4⋊Q82
#87852
#76522
#70402