Finite Group Databasebeta

← #10173

Subgroups — #10173

Order 256.

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

normal subgroups only

This group has 238 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
2C2×#498C211yesmaximal, characteristic
3#3139C211yesmaximal, characteristic
4#1997C211yesmaximal, characteristic
5#1546C211yesmaximal, characteristic
6#1493C211yesmaximal, characteristic
7#1278C211yesmaximal, characteristic
8#533C2211yescharacteristic
9C2×C4⋊Q8C2211yescharacteristic
10C2×C4⋊Q8C2211yescharacteristic
11#500—21no—
12#498—22no—
13#498—22no—
14#491—22no—
15#487—21no—
16#487—21no—
17#487—21no—
18#484—22no—
19#477—22no—
20#431C2211yescharacteristic
21C2×#106—22no—
22#423C2211yescharacteristic
23#419—22no—
24C4×M16C2211yescharacteristic
25C2×C4×C8C2211yescharacteristic
26#343—22no—
27#332—22no—
28Q8⋊C8—22no—
29C22×Q8—21no—
30C22×Q8—21no—
31C2×M16—21no—
32C2×M16D412yes—
33C22×C8—21no—
34C4⋊Q8—21no—
35C4⋊Q8—21no—
36C4⋊Q8—21no—
37C4⋊Q8—21no—
38C4⋊Q8—21no—
39C4⋊Q8—21no—
40C4⋊Q8—21no—
41C4⋊Q8D412yes—
42C4⋊Q8D411yescharacteristic
43C4⋊Q8D411yescharacteristic
44#125—21no—
45C22⋊Q8—21no—
46C22⋊Q8—21no—
47C4×Q8—21no—
48C4×Q8—21no—
49#117—21no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4D412yes—
54C2×C42C2311yescharacteristic
55#107—24no—
56#106—28no—
57C4⋊C8—21no—
58C4⋊C8—21no—
59C4⋊C8D412yes—
60#103—24no—
61#103—24no—
62#103—24no—
63C22⋊C8—21no—
64#97—21no—
65C4×C8—21no—
66C4×C8—21no—
67C4×C8—21no—
68C4×C8D412yes—
69C2×Q8—21no—
70C2×Q8—22no—
71C2×Q8—21no—
72C2×Q8—41no—
73C2×Q8—41no—
74C2×Q8—41no—
75C2×Q8—42no—
76C2×Q8—42no—
77C2×Q8—82no—
78C22×C4—22no—
79C22×C4—41no—
80C22×C4C2×D411yescharacteristic
81C22×C4C2×D411yescharacteristic
82C22×C4#4111yescharacteristic
83M16—22no—
84M16—22no—
85M16—42no—
86C2×C8—24no—
87C2×C8—22no—
88C2×C8—22no—
89C2×C8—22no—
90C2×C8—41no—
91C4⋊C4—21no—
92C4⋊C4—21no—
93C4⋊C4—22no—
94C4⋊C4—22no—
95C4⋊C4—22no—
96C4⋊C4—21no—
97C4⋊C4—22no—
98C4⋊C4—21no—
99C4⋊C4—41no—
100C4⋊C4—41no—
101C4⋊C4—41no—
102C4⋊C4—41no—
103C4⋊C4—41no—
104C4⋊C4—41no—
105C22⋊C4—41no—
106C42—41no—
107C42C2×D411yescharacteristic
108C42C2×D411yescharacteristic
109C42C2×D411yescharacteristic
110C42C2×D411yescharacteristic
111C23#12111yescharacteristic
112Q8—44no—
113Q8—42no—
114Q8—42no—
115Q8—82no—
116Q8—82no—
117Q8—81no—
118Q8—81no—
119C2×C4—22no—
120C2×C4—22no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—42no—
124C2×C4—42no—
125C2×C4—82no—
126C2×C4—82no—
127C2×C4#12011yescharacteristic
128C2×C4#12011yescharacteristic
129C2×C4#12011yescharacteristic
130C2×C4#12111yescharacteristic
131C2×C4#12111yescharacteristic
132C2×C4#12711yescharacteristic
133C2×C4#13512yes—
134C2×C4#13712yes—
135C8—24no—
136C8—42no—
137C8—42no—
138C8—81no—
139C22—21no—
140C22—21no—
141C22#39211yescharacteristic
142C22#46412yes—
143C4—21no—
144C4—21no—
145C4—81no—
146C4—82no—
147C4—82no—
148C4—81no—
149C4#45212yes—
150C4#45612yes—
151C4#47112yes—
152C4#49511yescharacteristic
153C4#49711yescharacteristic
154C2—22no—
155C2#200511yescharacteristic
156C2#201511yescharacteristic
157C2#201911yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radicalC23#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#32302
C2×#4982
#31392
#19972
#15462
#14932
#12782