Finite Group Databasebeta

← #150694

Subgroups — #150694

Order 512.

268 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 56 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#23820C211yesmaximal, characteristic
2#12802C211yesmaximal, characteristic
3#11532C211yesmaximal, characteristic
4#7429C212yesmaximal
5#7413C212yesmaximal
6C4×C4⋊Q8C2211yescharacteristic
7C4×#125C411yescharacteristic
8C2×#356—21no—
9#2099—21no—
10#2074C411yescharacteristic
11#2074C2211yescharacteristic
12#2043—21no—
13#2026C411yescharacteristic
14#2026C2211yescharacteristic
15#1801—21no—
16C4×#103—21no—
17C4×#103—21no—
18#1744C411yescharacteristic
19#1467—21no—
20#1394C2212yes—
21#1325—21no—
22#1319C2212yes—
23C2×C4⋊Q8D411yescharacteristic
24C2×#125C2×C411yescharacteristic
25C2×C4×Q8D411yescharacteristic
26C2×C4×Q8D411yescharacteristic
27#432—41no—
28#423—41no—
29C2×#103—21no—
30C2×#103—21no—
31C2×#103—21no—
32C2×#103—21no—
33C2×#101—22no—
34C2×C4×C8—21no—
35#391C2×C411yescharacteristic
36#391C2311yescharacteristic
37#389—21no—
38#389—41no—
39#387—41no—
40#384D411yescharacteristic
41#382C2×C411yescharacteristic
42C4×C4⋊C4—21no—
43C4×C4⋊C4—41no—
44C4×C4⋊C4—41no—
45C43C2×C411yescharacteristic
46#356—42no—
47#343—41no—
48#340—21no—
49#340—21no—
50#340—21no—
51#340—21no—
52#336—21no—
53#336C2×C411yescharacteristic
54#336C2×C411yescharacteristic
55#324—22no—
56#324—41no—
57C22×Q8C22⋊C411yescharacteristic
58C22×Q8C2×D411yescharacteristic
59C2×M16—42no—
60C22×C8—21no—
61C22×C8—21no—
62C4⋊Q8—21no—
63C4⋊Q8—21no—
64C4⋊Q8—41no—
65#125—42no—
66C4×Q8—41no—
67C4×Q8—41no—
68C4×Q8—41no—
69C4×Q8—41no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4C22⋊C411yescharacteristic
79C2×C4⋊C4C2×D411yescharacteristic
80C2×C42—21no—
81C2×C42—21no—
82C2×C42—21no—
83C2×C42—21no—
84C2×C42—41no—
85C2×C42C22⋊C411yescharacteristic
86C2×C42C22⋊C411yescharacteristic
87C2×C42C22×C411yescharacteristic
88C4⋊C8—81no—
89C4⋊C8—81no—
90C4⋊C8—81no—
91#103—42no—
92#103—42no—
93#103—42no—
94#103—41no—
95#103—41no—
96#101—42no—
97#101—42no—
98C22⋊C8—42no—
99#97—81no—
100C4×C8—41no—
101C4×C8—41no—
102#95—41no—
103#95—41no—
104#95—41no—
105#95—41no—
106#95—41no—
107C2×Q8—21no—
108C2×Q8—21no—
109C2×Q8—21no—
110C2×Q8—41no—
111C2×Q8—41no—
112C2×Q8—41no—
113C2×Q8—41no—
114C2×Q8AGL(1,ℤ/8)11yescharacteristic
115C2×Q8#13711yescharacteristic
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—41no—
123C22×C4—41no—
124C22×C4#9911yescharacteristic
125C22×C4#9911yescharacteristic
126C22×C4C2×C22⋊C411yescharacteristic
127M16—82no—
128M16—82no—
129C2×C8—41no—
130C2×C8—41no—
131C2×C8—41no—
132C2×C8—41no—
133C2×C8—41no—
134C2×C8—41no—
135C2×C8—82no—
136C2×C8—82no—
137C4⋊C4—22no—
138C4⋊C4—41no—
139C4⋊C4—41no—
140C4⋊C4—41no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4—41no—
146C4⋊C4—41no—
147C4⋊C4—81no—
148C4⋊C4—81no—
149C4⋊C4—82no—
150C4⋊C4—82no—
151C42—21no—
152C42—21no—
153C42—21no—
154C42—21no—
155C42—41no—
156C42—42no—
157C42—42no—
158C42—41no—
159C42—41no—
160C42—41no—
161C42—41no—
162C42—82no—
163C23C2×#9911yescharacteristic
164Q8—41no—
165Q8—41no—
166Q8—42no—
167Q8—81no—
168Q8—81no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—81no—
191C2×C4—81no—
192C2×C4—81no—
193C2×C4—81no—
194C2×C4—81no—
195C2×C4—81no—
196C2×C4—81no—
197C2×C4#35411yescharacteristic
198C2×C4#35411yescharacteristic
199C2×C4#41711yescharacteristic
200C2×C4#42111yescharacteristic
201C8—81no—
202C8—81no—
203C8—162no—
204C8—162no—
205C22—21no—
206C22—21no—
207C22#151311yescharacteristic
208C22#212011yescharacteristic
209C22C2×#35411yescharacteristic
210C4—21no—
211C4—21no—
212C4—21no—
213C4—42no—
214C4—41no—
215C4—41no—
216C4—41no—
217C4—81no—
218C4—82no—
219C4—81no—
220C4—81no—
221C4—81no—
222C4—161no—
223C4#141312yes—
224C2—21no—
225C2—21no—
226C2#1182311yescharacteristic
227C2C2×#141311yescharacteristic
228C2#1360211yescharacteristic
subgroupG/H
center, socleC22C2×#354
commutatorC2×C42C22×C4
Frattini, cosocle#391C23
prime radicalC23C2×#99
LCSG ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#238202
#128022
#115322
#74292
#74132