Finite Group Databasebeta

← #10133

Subgroups — #10133

Order 256.

268 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 51 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#3215C211yesmaximal, characteristic
3C2×#497C211yesmaximal, characteristic
4#1556C211yesmaximal, characteristic, complemented
5#1531C211yesmaximal, characteristic, complemented
6#1469C211yesmaximal, characteristic
7#1302C211yesmaximal, characteristic, complemented
8C22×Q16—21no—
9C22×SD16—21no—
10#533C2211yescharacteristic
11C2×C4⋊Q8C2211yescharacteristic
12C2×#124C2211yescharacteristic
13#500—21no—
14#497—22no—
15#497—21no—
16#497—22no—
17#489—21no—
18#487—21no—
19#484—22no—
20#477—22no—
21#464—22no—
22#451—21no—
23#450—21no—
24#432C2211yescharacteristic
25#432C2211yescharacteristic
26#431C2211yescharacteristic
27C2×#97C2211yescharacteristic
28#332—22no—
29#330—22no—
30#323—22no—
31C22×Q8—21no—
32C22×Q8D411yescharacteristic
33C22×D4D411yescharacteristic
34C2×Q16—24no—
35C2×Q16—22no—
36C2×Q16—44no—
37C2×SD16—24no—
38C2×SD16—22no—
39C2×SD16—44no—
40C22×C8—21no—
41C4⋊Q8—21no—
42C4⋊Q8—21no—
43C4⋊Q8—21no—
44C4⋊Q8D412yes—
45C4⋊Q8D411yescharacteristic
46C4⋊Q8D411yescharacteristic
47#125—21no—
48#124—21no—
49#124—21no—
50#124—21no—
51#124—21no—
52C22⋊Q8—21no—
53C22⋊Q8—21no—
54C4×Q8—21no—
55C4×Q8—21no—
56#117—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C22⋊C4—21no—
60C2×C22⋊C4—21no—
61C2×C42C2311yescharacteristic
62#107—24no—
63C4⋊C8—21no—
64C4⋊C8—21no—
65C4⋊C8D412yes—
66#103—22no—
67#103—24no—
68#102—22no—
69C22⋊C8—21no—
70C22⋊C8—21no—
71C22⋊C8D412yes—
72#97—21no—
73#97—21no—
74#97—21no—
75#97D412yes—
76C4×C8—21no—
77C24—21no—
78C2×Q8—21no—
79C2×Q8—21no—
80C2×Q8—21no—
81C2×Q8—21no—
82C2×Q8—22no—
83C2×Q8—42no—
84C2×Q8—42no—
85C2×Q8—42no—
86C2×Q8—41no—
87C2×D4—21no—
88C2×D4—21no—
89C2×D4—42no—
90C22×C4—21no—
91C22×C4—41no—
92C22×C4C2×D411yescharacteristic
93C22×C4C2×D411yescharacteristic
94C22×C4#4111yescharacteristic
95Q16—48no—
96Q16—82no—
97SD16—48no—
98SD16—82no—
99C2×C8—24no—
100C2×C8—22no—
101C2×C8—22no—
102C2×C8—22no—
103C2×C8—41no—
104C4⋊C4—21no—
105C4⋊C4—22no—
106C4⋊C4—21no—
107C4⋊C4—22no—
108C4⋊C4—21no—
109C4⋊C4—21no—
110C4⋊C4—41no—
111C4⋊C4—41no—
112C4⋊C4—41no—
113C4⋊C4—41no—
114C22⋊C4—41no—
115C22⋊C4—41no—
116C22⋊C4—41no—
117C22⋊C4—41no—
118C22⋊C4—41no—
119C42—41no—
120C42C2×D411yescharacteristic
121C42C2×D411yescharacteristic
122C42C2×D411yescharacteristic
123C42C2×D411yescharacteristic
124C23—41no—
125C23—42no—
126C23—81no—
127C23—81no—
128C23#12111yescharacteristic
129Q8—42no—
130Q8—42no—
131Q8—42no—
132Q8—42no—
133Q8—42no—
134Q8—42no—
135Q8—82no—
136D4—42no—
137D4—42no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—42no—
145C2×C4—82no—
146C2×C4—81no—
147C2×C4#12011yescharacteristic
148C2×C4#12011yescharacteristic
149C2×C4#12011yescharacteristic
150C2×C4#12111yescharacteristic
151C2×C4#12111yescharacteristic
152C2×C4#12711yescharacteristic
153C8—24no—
154C8—42no—
155C8—42no—
156C8—81no—
157C22—21no—
158C22—21no—
159C22—81no—
160C22—81no—
161C22—81no—
162C22—162no—
163C22#39211yescharacteristic
164C22#47312yes—
165C4—21no—
166C4—21no—
167C4—21no—
168C4—21no—
169C4—21no—
170C4—82no—
171C4—81no—
172C4—81no—
173C4—81no—
174C4#45312yes—
175C2—22no—
176C2—161no—
177C2#200211yescharacteristic
178C2#200911yescharacteristic
179C2#202011yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radicalC22×D4D4
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. #1556C2semidirect
5. #1531C2semidirect
7. #1302C2semidirect

Maximal subgroups

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

subgroupindex
#32302
#32152
C2×#4972
#15562
#15312
#14692
#13022