Finite Group Databasebeta

← #9119

Subgroups — #9119

Order 256.

348 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 79 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
1C22×#97C211yesmaximal, characteristic, complemented
2#2850C211yesmaximal, characteristic
3#1749C211yesmaximal, characteristic, complemented
4C2×#329C212yesmaximal, complemented
5#1295C212yesmaximal, complemented
6C23×C8—21no—
7C2×#124C2211yescharacteristic
8C2×C22⋊Q8C411yescharacteristic
9C2×#121C411yescharacteristic
10C22×C42C2211yescharacteristic
11C2×C22⋊C8—21no—
12C2×#97—21no—
13C2×#97—24no—
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—24no—
32C22×C8—24no—
33C22×C8—24no—
34C22×C8—22no—
35C22×C8—22no—
36#124—22no—
37#124—22no—
38C22⋊Q8—42no—
39#121—42no—
40C2×C4⋊C4—21no—
41C2×C4⋊C4C2×C411yescharacteristic
42C2×C4⋊C4C2×C411yescharacteristic
43C2×C22⋊C4—21no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C22⋊C4—21no—
48C2×C22⋊C4—21no—
49C2×C42—22no—
50C2×C42—22no—
51C2×C42C2×C411yescharacteristic
52C2×C42D411yescharacteristic
53C2×C42D411yescharacteristic
54C2×C42C2311yescharacteristic
55C22⋊C8—24no—
56#97—24no—
57#97—22no—
58#97—22no—
59#97—22no—
60#97D414yescomplemented
61#95—21no—
62#95—21no—
63#95—21no—
64#95—21no—
65C24—21no—
66C24C22⋊C411yescharacteristic
67C2×Q8—22no—
68C2×Q8—41no—
69C2×Q8—41no—
70C2×D4—22no—
71C2×D4—41no—
72C2×D4—41no—
73C2×D4—42no—
74C2×D4—42no—
75C2×D4—42no—
76C22×C4—22no—
77C22×C4—22no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—22no—
81C22×C4—22no—
82C22×C4—22no—
83C22×C4—22no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—41no—
88C22×C4—41no—
89C22×C4C22⋊C411yescharacteristic
90C22×C4C22⋊C411yescharacteristic
91C22×C4C22⋊C411yescharacteristic
92C22×C4C22×C411yescharacteristic
93C22×C4C2×D411yescharacteristic
94C22×C4C2×D411yescharacteristic
95C2×C8—216no—
96C2×C8—24no—
97C2×C8—24no—
98C2×C8—24no—
99C2×C8—24no—
100C2×C8—42no—
101C2×C8—42no—
102C4⋊C4—41no—
103C4⋊C4—41no—
104C4⋊C4—42no—
105C22⋊C4—42no—
106C22⋊C4—42no—
107C22⋊C4—42no—
108C22⋊C4—42no—
109C22⋊C4—41no—
110C22⋊C4—42no—
111C22⋊C4—41no—
112C42—22no—
113C42—21no—
114C42—22no—
115C42—21no—
116C42C2×D412yes—
117C42#4112yes—
118C23—21no—
119C23—21no—
120C23—21no—
121C23—21no—
122C23—41no—
123C23—41no—
124C23—41no—
125C23—41no—
126C23—41no—
127C23—41no—
128C23—41no—
129C23#10012yes—
130C23#10412yes—
131C23#10412yes—
132C23C2×C22⋊C411yescharacteristic
133Q8—82no—
134D4—82no—
135D4—84no—
136C2×C4—24no—
137C2×C4—24no—
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—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—44no—
153C2×C4—44no—
154C2×C4—41no—
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#10412yes—
164C2×C4C4×D412yes—
165C2×C4#12011yescharacteristic
166C2×C4#12112yes—
167C2×C4#12311yescharacteristic
168C8—28no—
169C8—44no—
170C22—22no—
171C22—22no—
172C22—22no—
173C22—22no—
174C22—22no—
175C22—22no—
176C22—81no—
177C22—81no—
178C22—81no—
179C22—81no—
180C22—81no—
181C22—81no—
182C22—81no—
183C22#32914yescomplemented
184C22#38611yescharacteristic
185C22C2×#10011yescharacteristic
186C22#41311yescharacteristic
187C22C2×#10411yescharacteristic
188C22C2×#10411yescharacteristic
189C22C2×#10411yescharacteristic
190C22#42111yescharacteristic
191C4—22no—
192C4—21no—
193C4—22no—
194C4—21no—
195C4—22no—
196C4—41no—
197C4—41no—
198C4—81no—
199C4—81no—
200C4—81no—
201C2—24no—
202C2—161no—
203C2C2×#32911yescharacteristic
204C2#152411yescharacteristic
205C2#153011yescharacteristic
206C2#153211yescharacteristic
207C2#187111yescharacteristic
208C2#189611yescharacteristic
209C2#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. C22×#97C2semidirect
3. #1749C2semidirect
4. C2×#329C2semidirect
5. #1295C2semidirect
60. #97D4semidirect
183. C22#329semidirect

Maximal subgroups

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

subgroupindex
C22×#972
#28502
#17492
C2×#3292
#12952