Finite Group Databasebeta

← #94698

Subgroups — #94698

Order 512.

209 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 209 conjugacy classes of subgroups — too many to draw. Only its 22 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#10388C211yesmaximal, characteristic
2#7339C211yesmaximal, characteristic
3#7318C211yesmaximal, characteristic
4#2608C411yescharacteristic, complemented
5#1789—21no—
6#1782—21no—
7#1782C2211yescharacteristic
8#1317—21no—
9#1306C411yescharacteristic, complemented
10#1284—21no—
11C22×#41D411yescharacteristic
12C2×#104—41no—
13C2×#104—41no—
14C2×#99—41no—
15C2×#99—41no—
16C2×#99—41no—
17#407—21no—
18#396—41no—
19#389—41no—
20#387—41no—
21#381—22no—
22#381C2×C411yescharacteristic
23#381D411yescharacteristic
24C4×C22⋊C4—21no—
25#323—41no—
26C2×#41—41no—
27C2×#41—41no—
28C2×#41—82no—
29C22×Q8Q1611yescharacteristic
30C22×D4—21no—
31C22×D4SD1611yescharacteristic
32C23×C4—21no—
33C23×C4C22⋊C411yescharacteristic
34C2×M16—41no—
35C2×C4⋊C4—41no—
36C2×C4⋊C4—41no—
37C2×C4⋊C4—41no—
38C2×C22⋊C4—21no—
39C2×C22⋊C4—21no—
40C2×C22⋊C4—41no—
41C2×C22⋊C4—42no—
42C2×C42—21no—
43C2×C42—21no—
44#104—82no—
45#104—82no—
46#99—82no—
47#99—82no—
48#99—81no—
49#99—81no—
50C22⋊C8—41no—
51C22⋊C8—41no—
52C22⋊C8—81no—
53#95—41no—
54#95—41no—
55#95—41no—
56#95—41no—
57#95—82no—
58C24—21no—
59C24#9911yescharacteristic
60#41—42no—
61#41—42no—
62#41—81no—
63#41—81no—
64#41—82no—
65#41—82no—
66C2×Q8—41no—
67C2×Q8—81no—
68C2×D4—41no—
69C2×D4—42no—
70C2×D4—42no—
71C2×D4—41no—
72C2×D4—41no—
73C2×D4—81no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—42no—
78C22×C4—41no—
79C22×C4—41no—
80C22×C4—41no—
81C22×C4—41no—
82C22×C4—81no—
83C22×C4—81no—
84C22×C4—81no—
85C22×C4—82no—
86C22×C4#10311yescharacteristic
87C22×C4#10411yescharacteristic
88M16—81no—
89M16—81no—
90C2×C8—81no—
91C2×C8—161no—
92C4⋊C4—82no—
93C4⋊C4—82no—
94C4⋊C4—161no—
95C22⋊C4—82no—
96C22⋊C4—81no—
97C22⋊C4—81no—
98C22⋊C4—82no—
99C22⋊C4—82no—
100C42—81no—
101C42—81no—
102C23—22no—
103C23—41no—
104C23—41no—
105C23—41no—
106C23—81no—
107C23—161no—
108C23#32811yescharacteristic
109C23AΣL(1,𝔽16)11yescharacteristic
110C23#35411yescharacteristic
111Q8—42no—
112Q8—81no—
113D4—42no—
114D4—81no—
115D4—82no—
116D4—82no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—42no—
126C2×C4—41no—
127C2×C4—81no—
128C2×C4—82no—
129C2×C4—82no—
130C2×C4—81no—
131C2×C4—81no—
132C2×C4—81no—
133C2×C4—81no—
134C2×C4—81no—
135C2×C4—81no—
136C2×C4—81no—
137C2×C4—81no—
138C2×C4—81no—
139C2×C4—161no—
140C2×C4—162no—
141C2×C4—164no—
142C8—161no—
143C8—321no—
144C22—21no—
145C22—21no—
146C22—21no—
147C22—41no—
148C22—41no—
149C22—41no—
150C22—42no—
151C22—82no—
152C22—161no—
153C22—161no—
154C22—161no—
155C22#135111yescharacteristic
156C4—41no—
157C4—41no—
158C4—82no—
159C4—81no—
160C4—161no—
161C4—161no—
162C4—161no—
163C4—322no—
164C2—21no—
165C2—41no—
166C2—41no—
167C2—41no—
168C2—82no—
169C2#742611yescharacteristic
subgroupG/H
center, socleC2#7426
commutator#381C2×C4
Frattini, cosocle#1782C22
prime radicalC22×#41D4
LCSG ▷ #381 ▷ C23×C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×C4 ◁ #1782 ◁ G
LECSG ▷ #1782 ▷ C23×C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #2608C4semidirect
9. #1306C4semidirect

Maximal subgroups

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

subgroupindex
#103882
#73392
#73182