Finite Group Databasebeta

← #10117

Subgroups — #10117

Order 256.

235 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 235 conjugacy classes of subgroups — too many to draw. Only its 53 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#3255C211yesmaximal, characteristic
2#3230C211yesmaximal, characteristic
3#3165C211yesmaximal, characteristic
4#1966C211yesmaximal, characteristic
5#1526C211yesmaximal, characteristic
6#1485C211yesmaximal, characteristic
7#1299C211yesmaximal, characteristic
8#533C2211yescharacteristic
9#533C2211yescharacteristic
10C2×C4⋊Q8C2211yescharacteristic
11#501—22no—
12#500—21no—
13#500—21no—
14#498—21no—
15#487—21no—
16#484—22no—
17#477—22no—
18#475—21no—
19#475—21no—
20#470—22no—
21#462—21no—
22#462—21no—
23#431C2211yescharacteristic
24#428—22no—
25#423C2211yescharacteristic
26C2×#101—22no—
27C4×M16C2211yescharacteristic
28C4×M16C2211yescharacteristic
29#330—22no—
30Q8⋊C8—22no—
31C22×Q8—21no—
32C2×Q16—22no—
33C2×M16—21no—
34C2×M16—21no—
35C2×M16D412yes—
36C4⋊Q8—21no—
37C4⋊Q8—21no—
38C4⋊Q8—21no—
39C4⋊Q8—21no—
40C4⋊Q8D411yescharacteristic
41C4⋊Q8D411yescharacteristic
42C4⋊Q8D411yescharacteristic
43C4⋊Q8D411yescharacteristic
44#125—21no—
45#125—21no—
46C22⋊Q8—21no—
47C22⋊Q8—21no—
48C22⋊Q8—21no—
49C22⋊Q8—21no—
50C4×Q8—21no—
51C4×Q8—21no—
52C4×Q8—21no—
53C4×Q8—21no—
54#117—21no—
55#117—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4D412yes—
58C2×C42C2311yescharacteristic
59#107—24no—
60#107—24no—
61#106—24no—
62#106—22no—
63C4⋊C8—21no—
64C4⋊C8—21no—
65C4⋊C8D412yes—
66#103—22no—
67#103—22no—
68#103—24no—
69#101—44no—
70C22⋊C8—21no—
71#97—21no—
72#97D412yes—
73C4×C8—21no—
74C4×C8—21no—
75C4×C8—21no—
76C2×Q8—21no—
77C2×Q8—21no—
78C2×Q8—21no—
79C2×Q8—21no—
80C2×Q8—42no—
81C2×Q8—42no—
82C2×Q8—41no—
83C2×Q8—41no—
84C22×C4—22no—
85C22×C4C2×D411yescharacteristic
86C22×C4C2×D411yescharacteristic
87C22×C4#4111yescharacteristic
88Q16—82no—
89M16—24no—
90M16—22no—
91M16—41no—
92M16—42no—
93C2×C8—22no—
94C2×C8—22no—
95C2×C8—22no—
96C2×C8—41no—
97C4⋊C4—21no—
98C4⋊C4—21no—
99C4⋊C4—22no—
100C4⋊C4—21no—
101C4⋊C4—22no—
102C4⋊C4—21no—
103C4⋊C4—21no—
104C4⋊C4—21no—
105C4⋊C4—21no—
106C4⋊C4—21no—
107C4⋊C4—41no—
108C4⋊C4—41no—
109C4⋊C4—41no—
110C4⋊C4—41no—
111C4⋊C4—41no—
112C4⋊C4—41no—
113C22⋊C4—41no—
114C22⋊C4—41no—
115C42—41no—
116C42—41no—
117C42C2×D411yescharacteristic
118C42C2×D411yescharacteristic
119C42C2×D411yescharacteristic
120C42C2×D411yescharacteristic
121C23#12111yescharacteristic
122Q8—41no—
123Q8—41no—
124Q8—41no—
125Q8—41no—
126Q8—42no—
127Q8—42no—
128Q8—82no—
129Q8—81no—
130Q8—81no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—44no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—42no—
141C2×C4#12011yescharacteristic
142C2×C4#12011yescharacteristic
143C2×C4#12011yescharacteristic
144C2×C4#12111yescharacteristic
145C2×C4#12111yescharacteristic
146C2×C4#12711yescharacteristic
147C8—24no—
148C8—42no—
149C8—42no—
150C8—42no—
151C22—21no—
152C22—21no—
153C22—21no—
154C22#39211yescharacteristic
155C4—21no—
156C4—21no—
157C4—21no—
158C4—81no—
159C4—82no—
160C4—81no—
161C4—81no—
162C4—81no—
163C4#45311yescharacteristic
164C4#45411yescharacteristic
165C4#45612yes—
166C4#47312yes—
167C2—41no—
168C2#200511yescharacteristic
169C2#200611yescharacteristic
170C2#202011yescharacteristic
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
#32552
#32302
#31652
#19662
#15262
#14852
#12992