Finite Group Databasebeta

← #150548

Subgroups — #150548

Order 512.

325 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 325 conjugacy classes of subgroups — too many to draw. Only its 58 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#28512C211yesmaximal, characteristic
2C4×#356C211yesmaximal, characteristic
3#11532C211yesmaximal, characteristic
4#7437C214yesmaximal
5C2×#475—21no—
6C2×#475—21no—
7#2868C411yescharacteristic
8C4×C4⋊Q8C2211yescharacteristic
9C2×#356—21no—
10#2096C412yes—
11#2096C2212yes—
12#1956—21no—
13#1844—21no—
14C4×C4⋊C8C411yescharacteristic
15C4×#101—21no—
16#1467—21no—
17#1394C2212yes—
18#1325—21no—
19#1319C2212yes—
20C2×C4⋊Q8—21no—
21C2×C4⋊Q8—41no—
22C2×C4⋊Q8D411yescharacteristic
23C2×C4×Q8D411yescharacteristic
24C2×C4×Q8D411yescharacteristic
25#475—42no—
26#475—42no—
27#475—42no—
28#475—42no—
29#432—41no—
30C2×#106—22no—
31C2×#106—22no—
32C2×C4⋊C8—21no—
33C2×#103—22no—
34C2×#103—22no—
35C2×#101—22no—
36C4×M16—41no—
37C2×C4×C8—21no—
38#391C2×C411yescharacteristic
39#391C2311yescharacteristic
40#389—41no—
41#384—21no—
42#384—41no—
43#384C2×C412yes—
44#384D411yescharacteristic
45C4×C4⋊C4—41no—
46C43C2×C411yescharacteristic
47#356—24no—
48#343—41no—
49#336C2×C412yes—
50#324—22no—
51#324—41no—
52C22×Q8—41no—
53C22×Q8C22⋊C411yescharacteristic
54C22×Q8C2×D411yescharacteristic
55C2×M16—42no—
56C22×C8—22no—
57C4⋊Q8—22no—
58C4⋊Q8—22no—
59C4⋊Q8—41no—
60C4⋊Q8—41no—
61C4⋊Q8—41no—
62C4⋊Q8—41no—
63C4⋊Q8—84no—
64C4×Q8—41no—
65C4×Q8—41no—
66C4×Q8—41no—
67C4×Q8—41no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—41no—
72C2×C4⋊C4—41no—
73C2×C4⋊C4—42no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4C22⋊C411yescharacteristic
77C2×C4⋊C4C2×D411yescharacteristic
78C2×C42—21no—
79C2×C42—21no—
80C2×C42—21no—
81C2×C42—21no—
82C2×C42C22⋊C411yescharacteristic
83C2×C42C22⋊C411yescharacteristic
84C2×C42C22×C411yescharacteristic
85#106—44no—
86#106—44no—
87C4⋊C8—22no—
88C4⋊C8—22no—
89C4⋊C8—81no—
90#103—44no—
91#103—44no—
92#101—24no—
93#101—42no—
94C22⋊C8—42no—
95#97—81no—
96#97—81no—
97C4×C8—42no—
98C4×C8—81no—
99#95—41no—
100#95—41no—
101C2×Q8—21no—
102C2×Q8—21no—
103C2×Q8—21no—
104C2×Q8—41no—
105C2×Q8—41no—
106C2×Q8—41no—
107C2×Q8—41no—
108C2×Q8—84no—
109C2×Q8—81no—
110C2×Q8—81no—
111C2×Q8#13712yes—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—42no—
119C22×C4#9911yescharacteristic
120C22×C4#9911yescharacteristic
121C22×C4C2×C22⋊C411yescharacteristic
122M16—44no—
123M16—82no—
124C2×C8—24no—
125C2×C8—42no—
126C2×C8—42no—
127C2×C8—82no—
128C2×C8—82no—
129C4⋊C4—44no—
130C4⋊C4—41no—
131C4⋊C4—41no—
132C4⋊C4—42no—
133C4⋊C4—42no—
134C4⋊C4—81no—
135C4⋊C4—81no—
136C4⋊C4—82no—
137C4⋊C4—81no—
138C4⋊C4—81no—
139C4⋊C4—84no—
140C4⋊C4—84no—
141C42—24no—
142C42—21no—
143C42—21no—
144C42—21no—
145C42—41no—
146C42—42no—
147C42—41no—
148C42—41no—
149C42—41no—
150C42—41no—
151C42#10412yes—
152C23C2×#9911yescharacteristic
153Q8—42no—
154Q8—42no—
155Q8—81no—
156Q8—81no—
157Q8—162no—
158Q8—162no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—21no—
167C2×C4—22no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—81no—
179C2×C4—81no—
180C2×C4—84no—
181C2×C4—82no—
182C2×C4#35311yescharacteristic
183C2×C4#35311yescharacteristic
184C2×C4#41911yescharacteristic
185C2×C4C2×#10411yescharacteristic
186C8—44no—
187C8—84no—
188C8—162no—
189C22—21no—
190C22—21no—
191C22#150811yescharacteristic
192C22#212311yescharacteristic
193C22C2×#35311yescharacteristic
194C4—24no—
195C4—21no—
196C4—21no—
197C4—21no—
198C4—42no—
199C4—42no—
200C4—41no—
201C4—81no—
202C4—81no—
203C4—81no—
204C4—162no—
205C4#141012yes—
206C2—21no—
207C2—21no—
208C2#1178811yescharacteristic
209C2#1358011yescharacteristic
210C2C2×#141011yescharacteristic
subgroupG/H
center, socleC22C2×#353
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
#285122
C4×#3562
#115322
#74372