Finite Group Databasebeta

← #8458

Subgroups — #8458

Order 256.

383 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 383 conjugacy classes of subgroups — too many to draw. Only its 73 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#3506C211yesmaximal, characteristic
2#1955C211yesmaximal, characteristic, complemented
3#1552C211yesmaximal, characteristic, complemented
4#1551C211yesmaximal, characteristic, complemented
5#1531C211yesmaximal, characteristic, complemented
6#1530C211yesmaximal, characteristic, complemented
7#1466C211yesmaximal, characteristic, complemented
8#559—22no—
9#556—21no—
10#555—21no—
11#553—22no—
12#550—21no—
13#547—21no—
14#538—22no—
15#535C412yes—
16#533C2211yescharacteristic
17#532C2211yescharacteristic
18#525—22no—
19#520C412yes—
20C2×C4×D4C2211yescharacteristic
21#432C2211yescharacteristic
22#432C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×C22⋊C8—22no—
25C2×#97C2211yescharacteristic
26#333—21no—
27#332—21no—
28#332—21no—
29#331—21no—
30#330—22no—
31#329—22no—
32#323—22no—
33C2×#41—22no—
34C2×#41C2×C412yes—
35C22×D4C2×C411yescharacteristic
36C23×C4D412yes—
37C22×C8—21no—
38C22×C8—21no—
39C4⋊Q8D411yescharacteristic
40C4⋊Q8D411yescharacteristic
41C4⋊Q8D411yescharacteristic
42#127—22no—
43#127D411yescharacteristic
44#126—24no—
45#125D412yes—
46#124—24no—
47#124—22no—
48#124D412yes—
49#123—24no—
50#123—24no—
51C22⋊Q8—22no—
52C22⋊Q8—21no—
53C22⋊Q8—21no—
54#121—22no—
55#121—24no—
56#121—21no—
57#121—21no—
58#121—22no—
59#121—22no—
60#120—24no—
61C4×Q8—22no—
62C4×Q8—21no—
63C4×Q8—21no—
64C4×D4—24no—
65C4×D4—22no—
66C4×D4—24no—
67C4×D4—21no—
68C4×D4—21no—
69C4×D4—21no—
70C4×D4—21no—
71C4×D4—21no—
72C4×D4—21no—
73#117—22no—
74#117—22no—
75#117C2×C412yes—
76C2×C4⋊C4—22no—
77C2×C4⋊C4C2×C411yescharacteristic
78C2×C22⋊C4D412yes—
79C2×C42C2311yescharacteristic
80C4⋊C8—21no—
81C4⋊C8—21no—
82C4⋊C8—21no—
83C4⋊C8—21no—
84C22⋊C8—21no—
85C22⋊C8—21no—
86C22⋊C8—44no—
87#97—21no—
88#97—21no—
89#97—21no—
90#97—21no—
91C24C22⋊C412yes—
92#41—44no—
93#41—82no—
94#41—82no—
95C2×Q8—21no—
96C2×Q8—21no—
97C2×Q8C22⋊C412yes—
98C2×D4—22no—
99C2×D4—24no—
100C2×D4—24no—
101C2×D4—21no—
102C2×D4—21no—
103C2×D4—21no—
104C2×D4—21no—
105C2×D4—42no—
106C2×D4C22⋊C412yes—
107C22×C4—24no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4C22⋊C412yes—
114C22×C4C22×C411yescharacteristic
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D411yescharacteristic
117C2×C8—22no—
118C2×C8—41no—
119C2×C8—41no—
120C2×C8—42no—
121C2×C8—41no—
122C2×C8—41no—
123C2×C8—41no—
124C4⋊C4—24no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—22no—
130C4⋊C4—21no—
131C4⋊C4—22no—
132C4⋊C4—21no—
133C4⋊C4C22⋊C412yes—
134C4⋊C4C22⋊C412yes—
135C22⋊C4—24no—
136C22⋊C4—24no—
137C22⋊C4—24no—
138C22⋊C4—22no—
139C22⋊C4—22no—
140C22⋊C4—22no—
141C22⋊C4—22no—
142C22⋊C4—22no—
143C42—24no—
144C42—22no—
145C42C2×D411yescharacteristic
146C42C2×D411yescharacteristic
147C42C2×D411yescharacteristic
148C42C2×D411yescharacteristic
149C23—22no—
150C23—22no—
151C23—22no—
152C23—22no—
153C23—42no—
154C23—42no—
155C23C2×C22⋊C411yescharacteristic
156Q8—42no—
157Q8—41no—
158Q8—41no—
159Q8—81no—
160D4—42no—
161D4—42no—
162D4—42no—
163D4—44no—
164D4—41no—
165D4—41no—
166D4—81no—
167D4—84no—
168D4—84no—
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—41no—
177C2×C4—44no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—82no—
182C2×C4—84no—
183C2×C4C2×C22⋊C411yescharacteristic
184C2×C4C2×C22⋊C411yescharacteristic
185C2×C4#12011yescharacteristic
186C2×C4#12011yescharacteristic
187C2×C4#12011yescharacteristic
188C2×C4#12011yescharacteristic
189C8—42no—
190C8—81no—
191C8—81no—
192C8—81no—
193C22—21no—
194C22—21no—
195C22—44no—
196C22—42no—
197C22—42no—
198C22—42no—
199C22—42no—
200C22—42no—
201C22—42no—
202C22—82no—
203C22#37911yescharacteristic
204C22#41312yes—
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—21no—
210C4—42no—
211C4—42no—
212C4—42no—
213C4—42no—
214C4#45311yescharacteristic
215C4#45511yescharacteristic
216C2—22no—
217C2—82no—
218C2—82no—
219C2#178511yescharacteristic
220C2#179211yescharacteristic
221C2#179411yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radical#3506C2
LCSG ▷ C22×C4 ▷ 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
2. #1955C2semidirect
3. #1552C2semidirect
4. #1551C2semidirect
5. #1531C2semidirect
6. #1530C2semidirect
7. #1466C2semidirect

Maximal subgroups

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

subgroupindex
#35062
#19552
#15522
#15512
#15312
#15302
#14662