Finite Group Databasebeta

← #7612

Subgroups — #7612

Order 256.

328 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 69 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#2297C211yesmaximal, characteristic
2#1797C211yesmaximal, characteristic, complemented
3#1749C211yesmaximal, characteristic, complemented
4#1272C212yesmaximal, complemented
5#1270C212yesmaximal, complemented
6C22×#41C411yescharacteristic
7C2×#117C411yescharacteristic
8C22×C42C2211yescharacteristic
9C2×C4⋊C8—21no—
10C2×C22⋊C8—21no—
11C2×C22⋊C8—21no—
12C2×#97—21no—
13#391C411yescharacteristic
14#391C2211yescharacteristic
15#390C411yescharacteristic
16#390C2211yescharacteristic
17#386—21no—
18#386—21no—
19C4×C22⋊C4—21no—
20C4×C22⋊C4—21no—
21#336C2212yes—
22#336C2212yes—
23C2×#41—22no—
24C2×#41—22no—
25C2×#41—42no—
26C2×#41—42no—
27C22×Q8C2×C411yescharacteristic
28C22×D4—21no—
29C22×D4C2×C411yescharacteristic
30C23×C4—21no—
31C23×C4C2×C411yescharacteristic
32C23×C4D411yescharacteristic
33C23×C4D411yescharacteristic
34C22×C8—22no—
35C22×C8—22no—
36#117—42no—
37C2×C4⋊C4C2×C411yescharacteristic
38C2×C4⋊C4C2×C411yescharacteristic
39C2×C22⋊C4—21no—
40C2×C22⋊C4—21no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C42—21no—
45C2×C42—42no—
46C2×C42C2×C411yescharacteristic
47C2×C42D411yescharacteristic
48C2×C42D411yescharacteristic
49C2×C42C2311yescharacteristic
50C4⋊C8—42no—
51C22⋊C8—24no—
52C22⋊C8—42no—
53#97—42no—
54#95—21no—
55#95—21no—
56#95—21no—
57#95—21no—
58C24—21no—
59C24C22⋊C411yescharacteristic
60#41—48no—
61#41—48no—
62C2×Q8—21no—
63C2×Q8—41no—
64C2×Q8—41no—
65C2×Q8M1612yes—
66C2×D4—24no—
67C2×D4—21no—
68C2×D4—41no—
69C2×D4—41no—
70C2×D4—42no—
71C2×D4—42no—
72C2×D4M1612yes—
73C22×C4—22no—
74C22×C4—22no—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—24no—
78C22×C4—21no—
79C22×C4—22no—
80C22×C4—22no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—41no—
86C22×C4—41no—
87C22×C4—41no—
88C22×C4—42no—
89C22×C4—42no—
90C22×C4C22⋊C411yescharacteristic
91C22×C4C22⋊C411yescharacteristic
92C22×C4C22⋊C411yescharacteristic
93C22×C4C22×C411yescharacteristic
94C22×C4C2×D411yescharacteristic
95C22×C4C2×D411yescharacteristic
96C2×C8—24no—
97C2×C8—42no—
98C2×C8—42no—
99C2×C8—42no—
100C2×C8—42no—
101C2×C8—42no—
102C4⋊C4—41no—
103C4⋊C4—41no—
104C22⋊C4—42no—
105C22⋊C4—42no—
106C22⋊C4—42no—
107C22⋊C4—42no—
108C22⋊C4—41no—
109C22⋊C4—41no—
110C42—22no—
111C42—22no—
112C42—41no—
113C42—41no—
114C42—42no—
115C23—21no—
116C23—21no—
117C23—21no—
118C23—21no—
119C23—21no—
120C23—21no—
121C23—22no—
122C23—41no—
123C23—41no—
124C23—41no—
125C23—41no—
126C23—41no—
127C23—41no—
128C23#10012yes—
129C23C2×C22⋊C411yescharacteristic
130Q8—42no—
131Q8—42no—
132D4—42no—
133D4—42no—
134D4—48no—
135C2×C4—24no—
136C2×C4—24no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—22no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—44no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—48no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4#9912yes—
164C2×C4#10412yes—
165C2×C4C2×M1611yescharacteristic
166C2×C4C2×M1611yescharacteristic
167C2×C4#13512yes—
168C2×C4AGL(1,ℤ/8)11yescharacteristic
169C2×C4#13711yescharacteristic
170C8—44no—
171C8—82no—
172C22—22no—
173C22—22no—
174C22—22no—
175C22—41no—
176C22—41no—
177C22—41no—
178C22—41no—
179C22—42no—
180C22—42no—
181C22—42no—
182C22—81no—
183C22—81no—
184C22—81no—
185C22—81no—
186C22#40711yescharacteristic
187C22C2×#9911yescharacteristic
188C22C2×#10011yescharacteristic
189C22#41611yescharacteristic
190C22#41711yescharacteristic
191C22C2×#10411yescharacteristic
192C22#42111yescharacteristic
193C4—22no—
194C4—21no—
195C4—21no—
196C4—22no—
197C4—22no—
198C4—21no—
199C4—21no—
200C4—42no—
201C4—41no—
202C4—81no—
203C4—81no—
204C2—42no—
205C2—82no—
206C2#146211yescharacteristic
207C2#148811yescharacteristic
208C2#148911yescharacteristic
209C2#150111yescharacteristic
210C2#151311yescharacteristic
211C2#153011yescharacteristic
212C2#155111yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC22×#41C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #1797C2semidirect
3. #1749C2semidirect
4. #1272C2semidirect
5. #1270C2semidirect

Maximal subgroups

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

subgroupindex
#22972
#17972
#17492
#12722
#12702