Finite Group Databasebeta

← #8461

Subgroups — #8461

Order 256.

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

normal subgroups only

This group has 341 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#3507C211yesmaximal, characteristic
2#1985C211yesmaximal, characteristic, complemented
3#1552C211yesmaximal, characteristic, complemented
4#1551C211yesmaximal, characteristic
5#1531C211yesmaximal, characteristic, complemented
6#1530C211yesmaximal, characteristic
7#1467C211yesmaximal, characteristic, complemented
8#559—21no—
9#557—21no—
10#556—21no—
11#556—22no—
12#554—21no—
13#552—22no—
14#539—22no—
15#536C412yes—
16#533C2211yescharacteristic
17#532C2211yescharacteristic
18#529—22no—
19#520C412yes—
20C2×C4×Q8C2211yescharacteristic
21#432C2211yescharacteristic
22#432C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×#97C2211yescharacteristic
25#336—22no—
26#333—21no—
27#332—21no—
28#332—21no—
29#331—21no—
30#330—22no—
31#329—22no—
32#324—22no—
33C2×#41C2×C412yes—
34C22×Q8C2×C411yescharacteristic
35C22×C8—21no—
36C22×C8—21no—
37C4⋊Q8—22no—
38C4⋊Q8D411yescharacteristic
39C4⋊Q8D411yescharacteristic
40C4⋊Q8D411yescharacteristic
41#127D411yescharacteristic
42#126—24no—
43#126—24no—
44#125—22no—
45#125—24no—
46#125D412yes—
47#124—22no—
48#124—22no—
49#124D412yes—
50#123—24no—
51#123—24no—
52C22⋊Q8—24no—
53C22⋊Q8—21no—
54C22⋊Q8—21no—
55C22⋊Q8—22no—
56C22⋊Q8—22no—
57#121—21no—
58#121—21no—
59C4×Q8—22no—
60C4×Q8—22no—
61C4×Q8—21no—
62C4×Q8—21no—
63C4×Q8—21no—
64C4×Q8—21no—
65C4×Q8—21no—
66C4×Q8—21no—
67C4×D4—22no—
68C4×D4—22no—
69C4×D4—24no—
70C4×D4—21no—
71C4×D4—21no—
72#117—24no—
73#117—22no—
74#117C2×C412yes—
75C2×C4⋊C4—22no—
76C2×C4⋊C4C2×C411yescharacteristic
77C2×C4⋊C4D412yes—
78C2×C42D412yes—
79C2×C42C2311yescharacteristic
80C4⋊C8—21no—
81C4⋊C8—21no—
82C4⋊C8—21no—
83C4⋊C8—21no—
84C22⋊C8—21no—
85C22⋊C8—21no—
86#97—21no—
87#97—21no—
88#97—21no—
89#97—21no—
90#41—44no—
91C2×Q8—21no—
92C2×Q8—21no—
93C2×Q8—21no—
94C2×Q8—21no—
95C2×Q8—42no—
96C2×Q8C22⋊C412yes—
97C2×D4—22no—
98C2×D4—21no—
99C2×D4—21no—
100C2×D4C22⋊C412yes—
101C22×C4—24no—
102C22×C4—22no—
103C22×C4C22⋊C412yes—
104C22×C4C22⋊C412yes—
105C22×C4C22×C411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D411yescharacteristic
108C2×C8—22no—
109C2×C8—41no—
110C2×C8—41no—
111C2×C8—42no—
112C2×C8—41no—
113C2×C8—41no—
114C2×C8—41no—
115C4⋊C4—24no—
116C4⋊C4—24no—
117C4⋊C4—24no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—22no—
127C4⋊C4—21no—
128C4⋊C4C22⋊C412yes—
129C4⋊C4C22⋊C412yes—
130C22⋊C4—24no—
131C22⋊C4—24no—
132C22⋊C4—24no—
133C22⋊C4—22no—
134C22⋊C4—22no—
135C22⋊C4—22no—
136C42—24no—
137C42—22no—
138C42—22no—
139C42—22no—
140C42C2×D411yescharacteristic
141C42C2×D411yescharacteristic
142C42C2×D411yescharacteristic
143C42C2×D411yescharacteristic
144C23—22no—
145C23C2×C22⋊C411yescharacteristic
146Q8—42no—
147Q8—42no—
148Q8—42no—
149Q8—41no—
150Q8—41no—
151Q8—81no—
152D4—42no—
153D4—44no—
154D4—41no—
155D4—41no—
156D4—81no—
157C2×C4—22no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—24no—
166C2×C4—41no—
167C2×C4—44no—
168C2×C4—42no—
169C2×C4—84no—
170C2×C4C2×C22⋊C411yescharacteristic
171C2×C4C2×C22⋊C411yescharacteristic
172C2×C4#12011yescharacteristic
173C2×C4#12011yescharacteristic
174C2×C4#12011yescharacteristic
175C2×C4#12011yescharacteristic
176C8—42no—
177C8—81no—
178C8—81no—
179C8—81no—
180C22—21no—
181C22—21no—
182C22—42no—
183C22—42no—
184C22—42no—
185C22#37911yescharacteristic
186C22#41312yes—
187C4—21no—
188C4—21no—
189C4—21no—
190C4—21no—
191C4—21no—
192C4—42no—
193C4—42no—
194C4—42no—
195C4—42no—
196C4—42no—
197C4#45411yescharacteristic
198C4#45611yescharacteristic
199C2—22no—
200C2—82no—
201C2#178511yescharacteristic
202C2#179311yescharacteristic
203C2#179411yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radical#532C22
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. #1985C2semidirect
3. #1552C2semidirect
5. #1531C2semidirect
7. #1467C2semidirect

Maximal subgroups

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

subgroupindex
#35072
#19852
#15522
#15512
#15312
#15302
#14672