Finite Group Databasebeta

← #9364

Subgroups — #9364

Order 256.

366 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 65 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#3007C211yesmaximal, characteristic
2#2341C211yesmaximal, characteristic
3#1996C211yesmaximal, characteristic, complemented
4#1789C211yesmaximal, characteristic
5#1508C211yesmaximal, characteristic, complemented
6#1305C211yesmaximal, characteristic, complemented
7#1297C211yesmaximal, characteristic, complemented
8C22×#41C2211yescharacteristic
9C2×#137C412yescomplemented
10C2×C22⋊Q8C2211yescharacteristic
11C2×C4×Q8—21no—
12#452—22no—
13#451—21no—
14#450—21no—
15C2×#104—22no—
16#417C412yescomplemented
17#407C2211yescharacteristic
18#407C2211yescharacteristic
19#390—21no—
20#390—22no—
21#380—21no—
22#380C2211yescharacteristic
23C4×C22⋊C4—21no—
24C4×C22⋊C4C2211yescharacteristic
25C4×C22⋊C4C2211yescharacteristic
26#342—22no—
27#328—21no—
28#328—21no—
29C2×#41—21no—
30C2×#41—21no—
31C2×#41—44no—
32C2×#41C2×C412yes—
33C2×#41D412yes—
34C22×Q8C2×C411yescharacteristic
35C22×Q8D411yescharacteristic
36C22×D4—21no—
37C22×D4D411yescharacteristic
38C23×C4—21no—
39C23×C4C2311yescharacteristic
40#137—44no—
41C2×Q16—22no—
42C2×SD16—22no—
43C2×M16—21no—
44C2×M16C2×C412yes—
45C22⋊Q8—21no—
46C22⋊Q8—21no—
47C22⋊Q8—21no—
48C22⋊Q8—21no—
49C4×Q8—44no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4C2×C411yescharacteristic
54C2×C22⋊C4—21no—
55C2×C22⋊C4—22no—
56C2×C22⋊C4—21no—
57C2×C22⋊C4—21no—
58C2×C22⋊C4—22no—
59C2×C22⋊C4D412yes—
60C2×C42—22no—
61C2×C42—21no—
62C2×C42—21no—
63#104—44no—
64#103—22no—
65#102—22no—
66C22⋊C8—21no—
67C22⋊C8—21no—
68C22⋊C8—21no—
69C22⋊C8—21no—
70#95—21no—
71#95—21no—
72#95D411yescharacteristic
73#95D411yescharacteristic
74C24—21no—
75C24C2×D411yescharacteristic
76#41—24no—
77#41—24no—
78#41—42no—
79#41—42no—
80#41—44no—
81#41—44no—
82C2×Q8—24no—
83C2×Q8—21no—
84C2×Q8—21no—
85C2×Q8—21no—
86C2×Q8—42no—
87C2×Q8C22⋊C412yes—
88C2×D4—22no—
89C2×D4—22no—
90C2×D4—21no—
91C2×D4—42no—
92C2×D4—44no—
93C2×D4C22⋊C412yes—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—42no—
103C22×C4—44no—
104C22×C4—41no—
105C22×C4C22×C411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D411yescharacteristic
109C22×C4#4111yescharacteristic
110C22×C4#4111yescharacteristic
111Q16—44no—
112SD16—44no—
113M16—42no—
114M16—42no—
115C2×C8—22no—
116C2×C8—41no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—44no—
120C4⋊C4—42no—
121C4⋊C4—41no—
122C4⋊C4—41no—
123C22⋊C4—22no—
124C22⋊C4—42no—
125C22⋊C4—44no—
126C22⋊C4—41no—
127C22⋊C4—41no—
128C22⋊C4—41no—
129C22⋊C4—42no—
130C22⋊C4—82no—
131C42—44no—
132C42—42no—
133C42—42no—
134C23—21no—
135C23—21no—
136C23—22no—
137C23—41no—
138C23—41no—
139C23—42no—
140C23—42no—
141C23—41no—
142C23—41no—
143C23C4×D411yescharacteristic
144C23#12011yescharacteristic
145C23#12111yescharacteristic
146Q8—24no—
147Q8—22no—
148Q8—42no—
149Q8—81no—
150Q8#9914yes—
151D4—22no—
152D4—22no—
153D4—44no—
154D4—44no—
155D4—42no—
156C2×C4—24no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—44no—
166C2×C4—44no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—42no—
177C2×C4—41no—
178C2×C4—81no—
179C2×C4—81no—
180C2×C4—81no—
181C2×C4—81no—
182C2×C4C2×C22⋊C411yescharacteristic
183C2×C4C4×D411yescharacteristic
184C2×C4#12111yescharacteristic
185C2×C4#12311yescharacteristic
186C8—42no—
187C8—81no—
188C22—21no—
189C22—21no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—21no—
195C22—21no—
196C22—42no—
197C22—42no—
198C22—42no—
199C22—81no—
200C22—81no—
201C22—82no—
202C22—82no—
203C22#38611yescharacteristic
204C22#45511yescharacteristic
205C22#45611yescharacteristic
206C4—21no—
207C4—24no—
208C4—21no—
209C4—21no—
210C4—42no—
211C4—82no—
212C4—81no—
213C4—82no—
214C4—81no—
215C4—81no—
216C4C2×#9912yes—
217C2—21no—
218C2—21no—
219C2—41no—
220C2—41no—
221C2—82no—
222C2#188211yescharacteristic
223C2#189811yescharacteristic
224C2#190611yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC22×#41C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1996C2semidirect
5. #1508C2semidirect
6. #1305C2semidirect
7. #1297C2semidirect
9. C2×#137C4semidirect
16. #417C4semidirect

Maximal subgroups

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

subgroupindex
#30072
#23412
#19962
#17892
#15082
#13052
#12972