Finite Group Databasebeta

← #10149

Subgroups — #10149

Order 256.

264 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 61 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#3230C211yesmaximal, characteristic, complemented
2#3138C211yesmaximal, characteristic
3C2×#486C211yesmaximal, characteristic
4#1997C211yesmaximal, characteristic, complemented
5#1545C211yesmaximal, characteristic
6#1493C211yesmaximal, characteristic, complemented
7#1278C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9C2×C4⋊Q8C2211yescharacteristic
10C2×#127C2211yescharacteristic
11#500—21no—
12#490—22no—
13#487—21no—
14#486—21no—
15#486—21no—
16#486—22no—
17#486—22no—
18#484—22no—
19#477—22no—
20#431C2211yescharacteristic
21#423C2211yescharacteristic
22#418—22no—
23C2×#102—22no—
24C4×M16C2211yescharacteristic
25C2×C4×C8C2211yescharacteristic
26#343—22no—
27#331—22no—
28Q8⋊C8—22no—
29C22×Q8—21no—
30C22×D4—21no—
31C22×D4—21no—
32C22×D4—21no—
33C2×M16—21no—
34C2×M16D412yes—
35C22×C8—21no—
36C4⋊Q8—21no—
37C4⋊Q8—21no—
38C4⋊Q8—21no—
39C4⋊Q8—21no—
40C4⋊Q8D411yescharacteristic
41C4⋊Q8D411yescharacteristic
42#127—21no—
43#127—21no—
44#127—21no—
45#127D412yes—
46#125—21no—
47C22⋊Q8—21no—
48C22⋊Q8—21no—
49C4×Q8—21no—
50C4×Q8—21no—
51#117—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4D412yes—
54C2×C42C2311yescharacteristic
55#107—24no—
56C4⋊C8—21no—
57C4⋊C8—21no—
58C4⋊C8D412yes—
59#103—24no—
60#102—28no—
61#102—24no—
62#102—24no—
63C22⋊C8—21no—
64#97—21no—
65C4×C8—21no—
66C4×C8—21no—
67C4×C8—21no—
68C4×C8D412yes—
69C24—41no—
70C2×Q8—21no—
71C2×Q8—21no—
72C2×Q8—41no—
73C2×Q8—41no—
74C2×Q8—82no—
75C2×D4—22no—
76C2×D4—22no—
77C2×D4—22no—
78C2×D4—44no—
79C2×D4—41no—
80C2×D4—41no—
81C2×D4—41no—
82C2×D4—42no—
83C2×D4—42no—
84C2×D4—82no—
85C22×C4—22no—
86C22×C4C2×D411yescharacteristic
87C22×C4C2×D411yescharacteristic
88C22×C4#4111yescharacteristic
89M16—22no—
90M16—22no—
91M16—42no—
92C2×C8—22no—
93C2×C8—22no—
94C2×C8—22no—
95C2×C8—42no—
96C2×C8—41no—
97C4⋊C4—22no—
98C4⋊C4—21no—
99C4⋊C4—21no—
100C4⋊C4—22no—
101C4⋊C4—21no—
102C4⋊C4—21no—
103C4⋊C4—41no—
104C4⋊C4—41no—
105C4⋊C4—41no—
106C4⋊C4—41no—
107C22⋊C4—41no—
108C42—41no—
109C42C2×D411yescharacteristic
110C42C2×D411yescharacteristic
111C42C2×D411yescharacteristic
112C42C2×D411yescharacteristic
113C23—42no—
114C23—82no—
115C23—82no—
116C23—82no—
117C23#12111yescharacteristic
118Q8—42no—
119Q8—42no—
120Q8—82no—
121Q8—82no—
122D4—44no—
123D4—44no—
124D4—82no—
125D4—82no—
126D4—82no—
127D4—81no—
128D4—81no—
129C2×C4—22no—
130C2×C4—22no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—42no—
134C2×C4—82no—
135C2×C4#12011yescharacteristic
136C2×C4#12011yescharacteristic
137C2×C4#12011yescharacteristic
138C2×C4#12111yescharacteristic
139C2×C4#12111yescharacteristic
140C2×C4#12711yescharacteristic
141C2×C4#13512yes—
142C2×C4#13712yes—
143C8—42no—
144C8—42no—
145C8—42no—
146C8—81no—
147C22—21no—
148C22—21no—
149C22—82no—
150C22—82no—
151C22—82no—
152C22—164no—
153C22#39211yescharacteristic
154C22#46412yes—
155C4—21no—
156C4—21no—
157C4—81no—
158C4—82no—
159C4—81no—
160C4#45212yes—
161C4#45412yes—
162C4#47212yes—
163C4#49511yescharacteristic
164C4#49711yescharacteristic
165C2—22no—
166C2—162no—
167C2#200311yescharacteristic
168C2#201511yescharacteristic
169C2#201811yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radicalC2×#127C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3230C2semidirect
4. #1997C2semidirect
6. #1493C2semidirect
7. #1278C2semidirect

Maximal subgroups

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

subgroupindex
#32302
#31382
C2×#4862
#19972
#15452
#14932
#12782