Finite Group Databasebeta

← #10123

Subgroups — #10123

Order 256.

221 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 221 conjugacy classes of subgroups — too many to draw. Only its 49 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#3236C211yesmaximal, characteristic, complemented
2#3235C211yesmaximal, characteristic
3#3142C211yesmaximal, characteristic
4#1496C211yesmaximal, characteristic
5#1485C211yesmaximal, characteristic, complemented
6#1470C211yesmaximal, characteristic, complemented
7#1291C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9#532C2211yescharacteristic
10C2×#125C2211yescharacteristic
11#501—21no—
12#501—21no—
13#491—21no—
14#491—21no—
15#490—21no—
16#490—21no—
17#488—22no—
18#484—21no—
19#483—21no—
20#482—22no—
21#479—22no—
22#476—21no—
23#474—21no—
24#432C2211yescharacteristic
25#432C2211yescharacteristic
26#431C2211yescharacteristic
27#417—21no—
28#417—21no—
29C4×M16C2211yescharacteristic
30Q8⋊C8—22no—
31D4⋊C8—22no—
32#324—22no—
33C2×#41—21no—
34C2×M16—21no—
35C4⋊Q8D411yescharacteristic
36C4⋊Q8D411yescharacteristic
37C4⋊Q8D411yescharacteristic
38#127D411yescharacteristic
39#125—21no—
40#125—21no—
41#125—21no—
42#125—21no—
43#125—21no—
44#124—21no—
45C22⋊Q8—21no—
46C22⋊Q8—21no—
47#121—21no—
48#121—21no—
49C4×Q8—21no—
50C4×Q8—21no—
51C4×D4—21no—
52C4×D4—21no—
53#117—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4D411yescharacteristic
57C2×C4⋊C4D411yescharacteristic
58C2×C42C2311yescharacteristic
59#107—22no—
60#107—22no—
61#106—22no—
62#106—22no—
63C4⋊C8—21no—
64C4⋊C8—21no—
65C4⋊C8D412yes—
66#103—22no—
67#103—22no—
68#103—22no—
69#103—22no—
70#102—22no—
71#102—22no—
72#102—22no—
73#102—22no—
74C22⋊C8—21no—
75C22⋊C8—21no—
76C22⋊C8D412yes—
77#97—21no—
78#97—21no—
79#97—21no—
80C4×C8—21no—
81C4×C8D412yes—
82#41—44no—
83C2×Q8—21no—
84C2×Q8—21no—
85C2×Q8—21no—
86C2×D4—21no—
87C2×D4—21no—
88C2×D4—21no—
89C2×D4—41no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—41no—
93C22×C4C2×D411yescharacteristic
94C22×C4C2×D411yescharacteristic
95C22×C4#4111yescharacteristic
96M16—42no—
97C2×C8—22no—
98C2×C8—22no—
99C2×C8—22no—
100C2×C8—41no—
101C4⋊C4—21no—
102C4⋊C4—21no—
103C4⋊C4—21no—
104C4⋊C4—21no—
105C4⋊C4—21no—
106C4⋊C4—21no—
107C4⋊C4—21no—
108C4⋊C4—21no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—41no—
112C4⋊C4—41no—
113C4⋊C4—41no—
114C4⋊C4—41no—
115C4⋊C4—41no—
116C4⋊C4—41no—
117C22⋊C4—41no—
118C22⋊C4—41no—
119C22⋊C4—41no—
120C42—41no—
121C42C2×D411yescharacteristic
122C42C2×D411yescharacteristic
123C42C2×D411yescharacteristic
124C42C2×D411yescharacteristic
125C23—41no—
126C23#12111yescharacteristic
127Q8—42no—
128Q8—42no—
129Q8—42no—
130D4—42no—
131D4—42no—
132D4—42no—
133D4—82no—
134C2×C4—22no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—42no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—82no—
144C2×C4—81no—
145C2×C4#12011yescharacteristic
146C2×C4#12011yescharacteristic
147C2×C4#12011yescharacteristic
148C2×C4#12111yescharacteristic
149C2×C4#12111yescharacteristic
150C2×C4#12711yescharacteristic
151C8—42no—
152C8—42no—
153C8—42no—
154C8—81no—
155C22—21no—
156C22—21no—
157C22—21no—
158C22—81no—
159C22—81no—
160C22—81no—
161C22#39211yescharacteristic
162C4—21no—
163C4—21no—
164C4—21no—
165C4—21no—
166C4—21no—
167C4—81no—
168C4—81no—
169C4—81no—
170C4—81no—
171C4—81no—
172C4#45412yes—
173C2—41no—
174C2—161no—
175C2#200311yescharacteristic
176C2#200911yescharacteristic
177C2#201711yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radical#532C22
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. #3236C2semidirect
5. #1485C2semidirect
6. #1470C2semidirect
7. #1291C2semidirect

Maximal subgroups

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

subgroupindex
#32362
#32352
#31422
#14962
#14852
#14702
#12912