Finite Group Databasebeta

← #150538

Subgroups — #150538

Order 512.

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

normal subgroups only

This group has 275 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#29144C211yesmaximal, characteristic
2C4×#356C211yesmaximal, characteristic
3#11532C211yesmaximal, characteristic
4#7429C214yesmaximal
5#2859C411yescharacteristic
6C4×C4⋊Q8C2211yescharacteristic
7C2×#356—21no—
8#2026C412yes—
9#2026C2212yes—
10#1956—21no—
11#1834—21no—
12#1805—21no—
13#1805—21no—
14C4×C4⋊C8C411yescharacteristic
15C4×#101—21no—
16#1467—21no—
17#1394C2212yes—
18#1325—21no—
19#1319C2212yes—
20C2×C4⋊Q8D411yescharacteristic
21C2×#125C2×C412yes—
22C2×C4×Q8D411yescharacteristic
23C2×C4×Q8D411yescharacteristic
24#432—41no—
25C2×C4⋊C8—21no—
26C2×#103—22no—
27C2×#103—22no—
28C2×#101—22no—
29C4×M16—41no—
30C2×C4×C8—21no—
31#391C2×C411yescharacteristic
32#391C2311yescharacteristic
33#389—41no—
34#389—41no—
35#389—41no—
36#384—21no—
37#384D411yescharacteristic
38#382—21no—
39C4×C4⋊C4—41no—
40C43C2×C411yescharacteristic
41#356—24no—
42#343—41no—
43#340—22no—
44#340—22no—
45#336C2×C412yes—
46#324—22no—
47#324—41no—
48C22×Q8C22⋊C411yescharacteristic
49C22×Q8C2×D411yescharacteristic
50C2×M16—42no—
51C22×C8—22no—
52C4⋊Q8—22no—
53C4⋊Q8—22no—
54#125—82no—
55C4×Q8—41no—
56C4×Q8—41no—
57C4×Q8—41no—
58C4×Q8—41no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—41no—
63C2×C4⋊C4—41no—
64C2×C4⋊C4—42no—
65C2×C4⋊C4—41no—
66C2×C4⋊C4—41no—
67C2×C4⋊C4C22⋊C411yescharacteristic
68C2×C4⋊C4C2×D411yescharacteristic
69C2×C42—21no—
70C2×C42—21no—
71C2×C42—21no—
72C2×C42—21no—
73C2×C42C22⋊C411yescharacteristic
74C2×C42C22⋊C411yescharacteristic
75C2×C42C22×C411yescharacteristic
76C4⋊C8—41no—
77C4⋊C8—41no—
78C4⋊C8—81no—
79#103—82no—
80#103—82no—
81#101—24no—
82#101—42no—
83C22⋊C8—42no—
84#97—81no—
85#97—81no—
86C4×C8—42no—
87C4×C8—81no—
88#95—41no—
89#95—41no—
90#95—41no—
91#95—41no—
92C2×Q8—21no—
93C2×Q8—21no—
94C2×Q8—21no—
95C2×Q8—41no—
96C2×Q8—41no—
97C2×Q8—41no—
98C2×Q8—41no—
99C2×Q8#13712yes—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—42no—
107C22×C4#9911yescharacteristic
108C22×C4#9911yescharacteristic
109C22×C4C2×C22⋊C411yescharacteristic
110M16—44no—
111M16—82no—
112C2×C8—42no—
113C2×C8—42no—
114C2×C8—42no—
115C2×C8—82no—
116C2×C8—82no—
117C4⋊C4—44no—
118C4⋊C4—41no—
119C4⋊C4—41no—
120C4⋊C4—42no—
121C4⋊C4—42no—
122C4⋊C4—81no—
123C4⋊C4—81no—
124C4⋊C4—82no—
125C4⋊C4—82no—
126C4⋊C4—82no—
127C4⋊C4—84no—
128C42—21no—
129C42—21no—
130C42—21no—
131C42—41no—
132C42—42no—
133C42—42no—
134C42—41no—
135C42—41no—
136C42—41no—
137C42—41no—
138C42#10412yes—
139C23C2×#9911yescharacteristic
140Q8—42no—
141Q8—42no—
142Q8—81no—
143Q8—81no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—22no—
149C2×C4—21no—
150C2×C4—22no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—81no—
164C2×C4—81no—
165C2×C4—82no—
166C2×C4—82no—
167C2×C4—82no—
168C2×C4#35411yescharacteristic
169C2×C4#35411yescharacteristic
170C2×C4#41911yescharacteristic
171C2×C4C2×#10411yescharacteristic
172C8—82no—
173C8—84no—
174C8—162no—
175C22—21no—
176C22—21no—
177C22#150811yescharacteristic
178C22#212211yescharacteristic
179C22C2×#35411yescharacteristic
180C4—24no—
181C4—21no—
182C4—21no—
183C4—21no—
184C4—21no—
185C4—42no—
186C4—41no—
187C4—81no—
188C4—81no—
189C4—81no—
190C4—81no—
191C4—162no—
192C2—21no—
193C2—21no—
194C2#1178711yescharacteristic
195C2#1359911yescharacteristic
196C2#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
#291442
C4×#3562
#115322
#74292