Finite Group Databasebeta

← #9177

Subgroups — #9177

Order 256.

247 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 247 conjugacy classes of subgroups — too many to draw. Only its 67 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#3077C211yesmaximal, characteristic
2#2357C211yesmaximal, characteristic
3#1518C211yesmaximal, characteristic, complemented
4#1503C211yesmaximal, characteristic, complemented
5#1478C211yesmaximal, characteristic
6#1303C211yesmaximal, characteristic, complemented
7#1286C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9C2×C4×D4C2211yescharacteristic
10#478—22no—
11#476C412yescomplemented
12#474C412yescomplemented
13#431C2211yescharacteristic
14#431C2211yescharacteristic
15#428—21no—
16#423C2211yescharacteristic
17#418—21no—
18#390—21no—
19#388—22no—
20#382C2211yescharacteristic
21#381—21no—
22C4×C4⋊C4—21no—
23C4×C4⋊C4C2211yescharacteristic
24C4×C22⋊C4—21no—
25#328—21no—
26#328—21no—
27#328—22no—
28D4⋊C8—22no—
29C22×D4D411yescharacteristic
30C23×C4—21no—
31C2×M16—21no—
32C4⋊Q8C2×C411yescharacteristic
33C4⋊Q8C2×C411yescharacteristic
34#125—21no—
35C22⋊Q8—21no—
36C22⋊Q8—21no—
37C4×Q8—21no—
38C4×Q8—21no—
39C4×D4—21no—
40C4×D4—21no—
41C4×D4C2×C412yes—
42C4×D4D412yes—
43#117—21no—
44C2×C4⋊C4—21no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4D411yescharacteristic
47C2×C22⋊C4—21no—
48C2×C22⋊C4—22no—
49C2×C22⋊C4—22no—
50C2×C42—21no—
51C2×C42—22no—
52C2×C42C2311yescharacteristic
53#107—22no—
54#106—22no—
55C4⋊C8—21no—
56C4⋊C8—21no—
57C4⋊C8—21no—
58C4⋊C8C2×C412yes—
59#102—22no—
60#102—22no—
61C22⋊C8—21no—
62C22⋊C8—21no—
63C4×C8—21no—
64C4×C8—21no—
65#95—22no—
66#95—21no—
67#95D411yescharacteristic
68#95D411yescharacteristic
69#95D412yes—
70C24—21no—
71C2×Q8—21no—
72C2×Q8—21no—
73C2×D4—21no—
74C2×D4—21no—
75C2×D4—21no—
76C2×D4—21no—
77C2×D4—21no—
78C2×D4C22⋊C412yes—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—22no—
82C22×C4—22no—
83C22×C4—22no—
84C22×C4—21no—
85C22×C4—82no—
86C22×C4C2×D411yescharacteristic
87C22×C4C2×D411yescharacteristic
88C22×C4C2×D411yescharacteristic
89M16—41no—
90M16—41no—
91C2×C8—22no—
92C2×C8—41no—
93C2×C8—41no—
94C4⋊C4—21no—
95C4⋊C4—21no—
96C4⋊C4—21no—
97C4⋊C4—21no—
98C4⋊C4—21no—
99C4⋊C4—42no—
100C4⋊C4—41no—
101C4⋊C4—41no—
102C4⋊C4—41no—
103C4⋊C4—41no—
104C4⋊C4C22⋊C412yes—
105C22⋊C4—22no—
106C22⋊C4—22no—
107C22⋊C4—44no—
108C22⋊C4—41no—
109C22⋊C4—82no—
110C42—41no—
111C42—41no—
112C42—41no—
113C42—82no—
114C42C22×C411yescharacteristic
115C42C2×D411yescharacteristic
116C42#4111yescharacteristic
117C42#4111yescharacteristic
118C23—22no—
119C23—42no—
120C23—42no—
121C23—41no—
122C23—41no—
123C23#12011yescharacteristic
124Q8—81no—
125Q8—81no—
126D4—22no—
127D4—22no—
128D4—41no—
129D4—41no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—42no—
136C2×C4—41no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—42no—
145C2×C4—82no—
146C2×C4—82no—
147C2×C4C2×C22⋊C411yescharacteristic
148C2×C4C4×D411yescharacteristic
149C2×C4C4×D411yescharacteristic
150C2×C4#12111yescharacteristic
151C2×C4#12111yescharacteristic
152C2×C4#12311yescharacteristic
153C2×C4#13511yescharacteristic
154C2×C4#13511yescharacteristic
155C2×C4#13711yescharacteristic
156C2×C4#13711yescharacteristic
157C8—42no—
158C8—81no—
159C8—81no—
160C22—21no—
161C22—21no—
162C22—42no—
163C22—42no—
164C22—42no—
165C22—82no—
166C22—82no—
167C22#38611yescharacteristic
168C22#45211yescharacteristic
169C22#45211yescharacteristic
170C4—21no—
171C4—21no—
172C4—21no—
173C4—21no—
174C4—42no—
175C4—42no—
176C4—81no—
177C4—82no—
178C4—81no—
179C4—81no—
180C4#41611yescharacteristic
181C4#41911yescharacteristic
182C4#48511yescharacteristic
183C4#48511yescharacteristic
184C2—21no—
185C2—21no—
186C2—82no—
187C2#187311yescharacteristic
188C2#190411yescharacteristic
189C2#190611yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radicalC22×D4D4
LCSG ▷ 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
3. #1518C2semidirect
4. #1503C2semidirect
6. #1303C2semidirect
7. #1286C2semidirect
11. #476C4semidirect
12. #474C4semidirect

Maximal subgroups

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

subgroupindex
#30772
#23572
#15182
#15032
#14782
#13032
#12862