Finite Group Databasebeta

← #9123

Subgroups — #9123

Order 256.

300 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 71 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
1C2×#432C211yesmaximal, characteristic, complemented
2#2850C211yesmaximal, characteristic
3#1749C211yesmaximal, characteristic, complemented
4C2×#329C212yesmaximal, complemented
5#1295C212yesmaximal, complemented
6C2×#124C2211yescharacteristic
7C2×C22⋊Q8C411yescharacteristic
8C2×#121C411yescharacteristic
9C22×C42C2211yescharacteristic
10#432—24no—
11C2×C4⋊C8—21no—
12C2×C22⋊C8—21no—
13C2×C22⋊C8—21no—
14C2×#97C2212yes—
15C2×#97C2212yes—
16#391C411yescharacteristic
17#390C411yescharacteristic
18#388—21no—
19#388—21no—
20#386—21no—
21#386—21no—
22#382C2211yescharacteristic
23#336—21no—
24#329—24no—
25C22×Q8C2×C411yescharacteristic
26C22×D4—21no—
27C22×D4C2×C411yescharacteristic
28C23×C4C2×C411yescharacteristic
29C23×C4D411yescharacteristic
30C23×C4D411yescharacteristic
31C22×C8—22no—
32C22×C8—22no—
33#124—22no—
34#124—22no—
35C22⋊Q8—42no—
36#121—42no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4C2×C411yescharacteristic
39C2×C4⋊C4C2×C411yescharacteristic
40C2×C22⋊C4—21no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4—21no—
46C2×C42—22no—
47C2×C42—22no—
48C2×C42C2×C411yescharacteristic
49C2×C42D411yescharacteristic
50C2×C42D411yescharacteristic
51C2×C42C2311yescharacteristic
52C4⋊C8—24no—
53C22⋊C8—24no—
54C22⋊C8—24no—
55#97—22no—
56#97—22no—
57#97—22no—
58#97D414yescomplemented
59#95—21no—
60#95—21no—
61#95—21no—
62#95—21no—
63C24—21no—
64C24C22⋊C411yescharacteristic
65C2×Q8—22no—
66C2×Q8—41no—
67C2×Q8—41no—
68C2×D4—22no—
69C2×D4—41no—
70C2×D4—41no—
71C2×D4—42no—
72C2×D4—42no—
73C2×D4—42no—
74C22×C4—22no—
75C22×C4—22no—
76C22×C4—22no—
77C22×C4—21no—
78C22×C4—22no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—41no—
84C22×C4—41no—
85C22×C4—41no—
86C22×C4—41no—
87C22×C4C22⋊C411yescharacteristic
88C22×C4C22⋊C411yescharacteristic
89C22×C4C22⋊C411yescharacteristic
90C22×C4C22×C411yescharacteristic
91C22×C4C2×D411yescharacteristic
92C22×C4C2×D411yescharacteristic
93C2×C8—24no—
94C2×C8—24no—
95C2×C8—42no—
96C2×C8—42no—
97C2×C8—42no—
98C2×C8—42no—
99C4⋊C4—41no—
100C4⋊C4—41no—
101C4⋊C4—42no—
102C22⋊C4—42no—
103C22⋊C4—42no—
104C22⋊C4—42no—
105C22⋊C4—42no—
106C22⋊C4—41no—
107C22⋊C4—42no—
108C22⋊C4—41no—
109C42—22no—
110C42—21no—
111C42—22no—
112C42—21no—
113C42C2×D412yes—
114C42#4112yes—
115C23—21no—
116C23—21no—
117C23—21no—
118C23—21no—
119C23—21no—
120C23—21no—
121C23—41no—
122C23—41no—
123C23—41no—
124C23—41no—
125C23—41no—
126C23—41no—
127C23—41no—
128C23#10412yes—
129C23C2×C22⋊C411yescharacteristic
130Q8—82no—
131D4—82no—
132D4—84no—
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—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—44no—
148C2×C4—44no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4#10012yes—
161C2×C4C4×D412yes—
162C2×C4#12011yescharacteristic
163C2×C4#12112yes—
164C2×C4#12311yescharacteristic
165C8—44no—
166C8—44no—
167C22—22no—
168C22—22no—
169C22—22no—
170C22—41no—
171C22—41no—
172C22—41no—
173C22—41no—
174C22—81no—
175C22—81no—
176C22—81no—
177C22—81no—
178C22—81no—
179C22—81no—
180C22—81no—
181C22#38611yescharacteristic
182C22C2×#10011yescharacteristic
183C22#41311yescharacteristic
184C22C2×#10411yescharacteristic
185C22#42111yescharacteristic
186C22#42111yescharacteristic
187C22#42111yescharacteristic
188C4—22no—
189C4—21no—
190C4—21no—
191C4—22no—
192C4—22no—
193C4—41no—
194C4—41no—
195C4—81no—
196C4—81no—
197C4—81no—
198C2—42no—
199C2—161no—
200C2#152711yescharacteristic
201C2#153211yescharacteristic
202C2#153511yescharacteristic
203C2#153611yescharacteristic
204C2#187111yescharacteristic
205C2#189711yescharacteristic
206C2#190011yescharacteristic
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 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#432C2semidirect
3. #1749C2semidirect
4. C2×#329C2semidirect
5. #1295C2semidirect
58. #97D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4322
#28502
#17492
C2×#3292
#12952