Finite Group Databasebeta

← #7633

Subgroups — #7633

Order 256.

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

normal subgroups only

This group has 320 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
1C4×#121C211yesmaximal, characteristic
2#1749C211yesmaximal, characteristic, complemented
3#1728C211yesmaximal, characteristic, complemented
4#1271C212yesmaximal, complemented
5#1270C212yesmaximal, complemented
6C22×M16—21no—
7C2×#121C411yescharacteristic
8C2×C4×D4—21no—
9C2×C4×D4C411yescharacteristic
10C22×C42C2211yescharacteristic
11C2×C22⋊C8—21no—
12C2×#97—21no—
13#390C2211yescharacteristic
14#389C2211yescharacteristic
15#388—21no—
16#386—21no—
17#380C411yescharacteristic
18C4×C4⋊C4C411yescharacteristic
19C4×C22⋊C4—21no—
20#336—21no—
21#336C2212yes—
22#336C2212yes—
23C22×D4—21no—
24C22×D4C2×C411yescharacteristic
25C23×C4—21no—
26C23×C4C2×C411yescharacteristic
27C23×C4D411yescharacteristic
28C23×C4D411yescharacteristic
29C2×M16—24no—
30C2×M16—42no—
31C2×M16—42no—
32C22×C8—22no—
33C22×C8—22no—
34#121—22no—
35#121—22no—
36C4×D4—42no—
37C4×D4—44no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4C2×C411yescharacteristic
40C2×C4⋊C4C2×C411yescharacteristic
41C2×C4⋊C4C2×C411yescharacteristic
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C42—21no—
48C2×C42—42no—
49C2×C42C2×C411yescharacteristic
50C2×C42D411yescharacteristic
51C2×C42D411yescharacteristic
52C2×C42C2311yescharacteristic
53C22⋊C8—24no—
54#97—42no—
55#95—21no—
56#95—21no—
57#95—21no—
58C24—21no—
59C24C22⋊C411yescharacteristic
60C2×D4—24no—
61C2×D4—21no—
62C2×D4—41no—
63C2×D4—41no—
64C2×D4—42no—
65C2×D4—42no—
66C2×D4M1612yescomplemented
67C22×C4—22no—
68C22×C4—22no—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—22no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—22no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—41no—
79C22×C4—41no—
80C22×C4—41no—
81C22×C4—42no—
82C22×C4—42no—
83C22×C4—42no—
84C22×C4C22⋊C411yescharacteristic
85C22×C4C22⋊C411yescharacteristic
86C22×C4C22⋊C411yescharacteristic
87C22×C4C22×C411yescharacteristic
88C22×C4C2×D411yescharacteristic
89C22×C4C2×D411yescharacteristic
90M16—48no—
91C2×C8—24no—
92C2×C8—24no—
93C2×C8—42no—
94C2×C8—42no—
95C2×C8—42no—
96C2×C8—42no—
97C4⋊C4—21no—
98C4⋊C4—41no—
99C4⋊C4—41no—
100C4⋊C4—42no—
101C4⋊C4M1612yescomplemented
102C22⋊C4—24no—
103C22⋊C4—42no—
104C22⋊C4—42no—
105C22⋊C4—41no—
106C22⋊C4—42no—
107C22⋊C4—41no—
108C42—22no—
109C42—22no—
110C42—41no—
111C42—41no—
112C42—42no—
113C23—21no—
114C23—21no—
115C23—21no—
116C23—21no—
117C23—21no—
118C23—21no—
119C23—22no—
120C23—41no—
121C23—41no—
122C23—41no—
123C23—41no—
124C23—41no—
125C23—41no—
126C23#10412yes—
127C23C2×C22⋊C411yescharacteristic
128D4—42no—
129D4—42no—
130D4—48no—
131C2×C4—24no—
132C2×C4—24no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—22no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—44no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—84no—
159C2×C4#9912yes—
160C2×C4#10412yes—
161C2×C4C2×M1611yescharacteristic
162C2×C4C2×M1611yescharacteristic
163C2×C4AGL(1,ℤ/8)11yescharacteristic
164C2×C4AGL(1,ℤ/8)12yes—
165C2×C4#13711yescharacteristic
166C8—44no—
167C8—44no—
168C22—22no—
169C22—22no—
170C22—22no—
171C22—41no—
172C22—41no—
173C22—41no—
174C22—41no—
175C22—42no—
176C22—42no—
177C22—42no—
178C22—81no—
179C22—81no—
180C22—81no—
181C22—81no—
182C22#40711yescharacteristic
183C22C2×#9911yescharacteristic
184C22#41311yescharacteristic
185C22#41711yescharacteristic
186C22#41811yescharacteristic
187C22C2×#10411yescharacteristic
188C22C2×#10411yescharacteristic
189C4—21no—
190C4—21no—
191C4—21no—
192C4—22no—
193C4—21no—
194C4—22no—
195C4—22no—
196C4—41no—
197C4—42no—
198C4—81no—
199C4—81no—
200C2—42no—
201C2—82no—
202C2#146511yescharacteristic
203C2#148911yescharacteristic
204C2#149011yescharacteristic
205C2#150411yescharacteristic
206C2#150711yescharacteristic
207C2#152411yescharacteristic
208C2#155711yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC2×#121C4
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. #1749C2semidirect
3. #1728C2semidirect
4. #1271C2semidirect
5. #1270C2semidirect
66. C2×D4M16semidirect
101. C4⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C4×#1212
#17492
#17282
#12712
#12702