Finite Group Databasebeta

← #94156

Subgroups — #94156

Order 512.

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

normal subgroups only

This group has 225 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
1C4×#356C211yesmaximal, characteristic
2#11478C211yesmaximal, characteristic
3#9641C211yesmaximal, characteristic
4C4×C4⋊Q8C2211yescharacteristic
5C2×#356C412yes—
6#1831C411yescharacteristic
7C4×#101—21no—
8#1744C411yescharacteristic
9C2×Q8⋊C8—21no—
10C2×Q8⋊C8—21no—
11#1461—21no—
12#1394—21no—
13#1325C412yes—
14#1319—21no—
15C2×C4⋊Q8C2×C411yescharacteristic
16C2×C4×Q8D411yescharacteristic
17C2×C4×Q8D411yescharacteristic
18#431—41no—
19C2×C4⋊C8—21no—
20C2×#101—22no—
21C4×M16—41no—
22C2×C4×C8—21no—
23#391D411yescharacteristic
24#391Q811yescharacteristic
25#389—41no—
26#384C2×C411yescharacteristic
27C4×C4⋊C4—41no—
28C43C2×C411yescharacteristic
29#356—22no—
30#356C814yescomplemented
31#343—41no—
32#336—41no—
33#336—41no—
34Q8⋊C8—42no—
35Q8⋊C8—42no—
36#324—22no—
37#324—41no—
38C22×Q8C22⋊C411yescharacteristic
39C22×Q8C4⋊C411yescharacteristic
40C2×M16—42no—
41C22×C8—41no—
42C22×C8—41no—
43C4⋊Q8—41no—
44C4⋊Q8C2×C812yes—
45C4⋊Q8M1612yes—
46C4×Q8—41no—
47C4×Q8—41no—
48C4×Q8—41no—
49C4×Q8—41no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—41no—
53C2×C4⋊C4—41no—
54C2×C4⋊C4C22⋊C411yescharacteristic
55C2×C4⋊C4C4⋊C411yescharacteristic
56C2×C42—21no—
57C2×C42—21no—
58C2×C42—21no—
59C2×C42—21no—
60C2×C42C4211yescharacteristic
61C2×C42C22⋊C411yescharacteristic
62C2×C42C4⋊C411yescharacteristic
63C4⋊C8—41no—
64C4⋊C8—41no—
65C4⋊C8—81no—
66#101—24no—
67#101—42no—
68C22⋊C8—42no—
69#97—81no—
70C4×C8—22no—
71C4×C8—41no—
72C4×C8—81no—
73C4×C8—81no—
74#95—41no—
75#95—41no—
76C2×Q8—21no—
77C2×Q8—21no—
78C2×Q8—41no—
79C2×Q8—41no—
80C2×Q8—41no—
81C2×Q8—41no—
82C2×Q8C22⋊C812yes—
83C2×Q8C4⋊C812yes—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4#9511yescharacteristic
91C22×C4#9911yescharacteristic
92C22×C4#9911yescharacteristic
93M16—44no—
94M16—82no—
95C2×C8—42no—
96C2×C8—82no—
97C2×C8—82no—
98C2×C8—82no—
99C2×C8—82no—
100C2×C8—81no—
101C4⋊C4—42no—
102C4⋊C4—42no—
103C4⋊C4—41no—
104C4⋊C4—41no—
105C4⋊C4—81no—
106C4⋊C4—81no—
107C4⋊C4—82no—
108C42—21no—
109C42—21no—
110C42—21no—
111C42—41no—
112C42—42no—
113C42—41no—
114C42—41no—
115C42—41no—
116C42—41no—
117C42—81no—
118C42C4×C811yescharacteristic
119C42#9711yescharacteristic
120C23#34211yescharacteristic
121Q8—42no—
122Q8—42no—
123Q8—81no—
124Q8—81no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—22no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—21no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—81no—
146C2×C4—81no—
147C2×C4#33611yescharacteristic
148C2×C4#34411yescharacteristic
149C2×C4#35311yescharacteristic
150C2×C4#35411yescharacteristic
151C8—82no—
152C8—84no—
153C8—161no—
154C8—162no—
155C22—21no—
156C22—21no—
157C22#128211yescharacteristic
158C22#139311yescharacteristic
159C22#139511yescharacteristic
160C4—21no—
161C4—21no—
162C4—21no—
163C4—41no—
164C4—41no—
165C4—42no—
166C4—42no—
167C4—41no—
168C4—81no—
169C4—81no—
170C4—81no—
171C4#141312yes—
172C2—21no—
173C2—21no—
174C2#720511yescharacteristic
175C2#750311yescharacteristic
176C2#750511yescharacteristic
subgroupG/H
centerC2×C4#354
commutatorC42C4×C8
Frattini, cosocleC4×C4⋊Q8C22
socleC22#1393
prime radicalC23#342
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C4×C4⋊Q8 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
30. #356C8semidirect

Maximal subgroups

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

subgroupindex
C4×#3562
#114782
#96412