Finite Group Databasebeta

← #150537

Subgroups — #150537

Order 512.

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

normal subgroups only

This group has 284 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#29146C211yesmaximal, characteristic
2C4×#355C211yesmaximal, characteristic
3#11531C211yesmaximal, characteristic
4#7429C214yesmaximal, complemented
5#2859C411yescharacteristic
6C4×#124C2211yescharacteristic
7C2×#355—21no—
8#2026C2212yes—
9#2025C412yes—
10#1959—21no—
11#1836—21no—
12#1805—21no—
13#1804—21no—
14#1767C411yescharacteristic
15C4×#101—21no—
16#1465—21no—
17#1394C2212yes—
18#1324—21no—
19#1319C2212yes—
20C2×#125C2×C412yes—
21C2×#124D411yescharacteristic
22C2×C4×Q8D411yescharacteristic
23C2×C4×D4D411yescharacteristic
24C2×C4⋊C8—21no—
25C2×#103—22no—
26C2×#102—22no—
27C2×#101—22no—
28#407—41no—
29C4×M16—41no—
30C2×#97—21no—
31#391C2311yescharacteristic
32#390C2×C411yescharacteristic
33#389—41no—
34#389—41no—
35#388—41no—
36#384—21no—
37#382—21no—
38#382D411yescharacteristic
39C4×C22⋊C4—41no—
40C43C2×C411yescharacteristic
41#355—24no—
42#343—41no—
43#340—22no—
44#340—22no—
45#336C2×C412yes—
46#324—22no—
47#323—41no—
48C22×Q8C2×D411yescharacteristic
49C22×D4C22⋊C411yescharacteristic
50C23×C4—21no—
51C2×M16—42no—
52C2×M16—81no—
53C22×C8—22no—
54#125—82no—
55#124—22no—
56#124—22no—
57C4×Q8—41no—
58C4×Q8—41no—
59C4×D4—41no—
60C4×D4—41no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—42no—
64C2×C4⋊C4—41no—
65C2×C4⋊C4—41no—
66C2×C4⋊C4C22⋊C411yescharacteristic
67C2×C4⋊C4C2×D411yescharacteristic
68C2×C22⋊C4—21no—
69C2×C22⋊C4—41no—
70C2×C22⋊C4—41no—
71C2×C42—21no—
72C2×C42—21no—
73C2×C42—21no—
74C2×C42C22⋊C411yescharacteristic
75C2×C42C22⋊C411yescharacteristic
76C2×C42C22×C411yescharacteristic
77C4⋊C8—41no—
78C4⋊C8—41no—
79#103—82no—
80#102—82no—
81#101—24no—
82#101—42no—
83C22⋊C8—42no—
84C22⋊C8—81no—
85#97—42no—
86#97—81no—
87C4×C8—81no—
88#95—41no—
89#95—41no—
90#95—41no—
91#95—41no—
92C24—21no—
93C2×Q8—21no—
94C2×Q8—41no—
95C2×Q8—41no—
96C2×Q8#13712yes—
97C2×D4—21no—
98C2×D4—21no—
99C2×D4—41no—
100C2×D4—41no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—41no—
107C22×C4—41no—
108C22×C4—42no—
109C22×C4—82no—
110C22×C4#9911yescharacteristic
111C22×C4#9911yescharacteristic
112C22×C4C2×C22⋊C411yescharacteristic
113M16—44no—
114M16—82no—
115M16—162no—
116C2×C8—42no—
117C2×C8—42no—
118C2×C8—42no—
119C2×C8—82no—
120C2×C8—82no—
121C4⋊C4—41no—
122C4⋊C4—41no—
123C4⋊C4—42no—
124C4⋊C4—42no—
125C4⋊C4—81no—
126C4⋊C4—84no—
127C4⋊C4—82no—
128C4⋊C4—82no—
129C22⋊C4—44no—
130C22⋊C4—81no—
131C22⋊C4—82no—
132C42—21no—
133C42—21no—
134C42—21no—
135C42—41no—
136C42—41no—
137C42—41no—
138C42—81no—
139C42—81no—
140C42#10412yes—
141C23—41no—
142C23—41no—
143C23—41no—
144C23—81no—
145C23—81no—
146C23C2×#9911yescharacteristic
147Q8—42no—
148Q8—81no—
149D4—42no—
150D4—81no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—82no—
168C2×C4—81no—
169C2×C4—81no—
170C2×C4—82no—
171C2×C4—82no—
172C2×C4—82no—
173C2×C4—161no—
174C2×C4#35411yescharacteristic
175C2×C4#35411yescharacteristic
176C2×C4#41911yescharacteristic
177C2×C4C2×#10411yescharacteristic
178C8—82no—
179C8—84no—
180C8—162no—
181C22—21no—
182C22—21no—
183C22—82no—
184C22—81no—
185C22—81no—
186C22—81no—
187C22—161no—
188C22#150811yescharacteristic
189C22#212211yescharacteristic
190C22C2×#35411yescharacteristic
191C4—21no—
192C4—21no—
193C4—21no—
194C4—21no—
195C4—42no—
196C4—42no—
197C4—41no—
198C4—81no—
199C4—81no—
200C4—81no—
201C4—162no—
202C2—21no—
203C2—21no—
204C2—161no—
205C2#1178711yescharacteristic
206C2#1359811yescharacteristic
207C2#1360311yescharacteristic
subgroupG/H
center, socleC22C2×#354
commutatorC2×C42C22×C4
Frattini, cosocle#391C23
prime radicalC22×D4C22⋊C4
LCSG ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#124 ◁ G
LECSG ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7429C2semidirect

Maximal subgroups

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

subgroupindex
#291462
C4×#3552
#115312
#74292