Finite Group Databasebeta

← #9276

Subgroups — #9276

Order 256.

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

normal subgroups only

This group has 261 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#3189C211yesmaximal, characteristic
2#2368C211yesmaximal, characteristic
3#1996C211yesmaximal, characteristic
4#1520C211yesmaximal, characteristic
5C2×#328C211yesmaximal, characteristic
6#1294C211yesmaximal, characteristic
7#1282C211yesmaximal, characteristic
8C2×C22⋊Q8C2211yescharacteristic
9C2×C22⋊Q8C2211yescharacteristic
10#484—21no—
11#484C412yescomplemented
12#467—22no—
13#451—21no—
14#451C412yescomplemented
15C2×C22⋊C8C2211yescharacteristic
16C2×C22⋊C8C2211yescharacteristic
17#391—21no—
18#387—21no—
19#387—21no—
20#385—21no—
21#380C2211yescharacteristic
22C4×C4⋊C4—21no—
23C4×C22⋊C4—21no—
24C4×C22⋊C4C2211yescharacteristic
25C2×#95C2211yescharacteristic
26#342—21no—
27#342—21no—
28#328—22no—
29#328—21no—
30#328—21no—
31C22×Q8C2×C411yescharacteristic
32C22×Q8D411yescharacteristic
33C23×C4—21no—
34C23×C4C2311yescharacteristic
35C2×Q16—22no—
36C22×C8—21no—
37C22×C8—21no—
38C22⋊Q8—21no—
39C22⋊Q8—21no—
40C22⋊Q8—21no—
41C22⋊Q8—21no—
42C22⋊Q8—21no—
43C22⋊Q8—21no—
44C22⋊Q8C2×C412yes—
45C22⋊Q8D412yes—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4C2×C411yescharacteristic
52C2×C4⋊C4D411yescharacteristic
53C2×C22⋊C4—21no—
54C2×C22⋊C4—21no—
55C2×C22⋊C4—21no—
56C2×C22⋊C4D411yescharacteristic
57C2×C22⋊C4D411yescharacteristic
58C2×C42—21no—
59C2×C42—21no—
60C2×C42—21no—
61#107—22no—
62#103—22no—
63#103—22no—
64C22⋊C8—21no—
65C22⋊C8—21no—
66C22⋊C8—21no—
67C22⋊C8C2×C412yes—
68#95—21no—
69#95—21no—
70#95—21no—
71#95—21no—
72#95—21no—
73#95—21no—
74#95—21no—
75#95D411yescharacteristic
76#95D411yescharacteristic
77C24C2×D411yescharacteristic
78C2×Q8—21no—
79C2×Q8—21no—
80C2×Q8—21no—
81C2×Q8—81no—
82C2×Q8—81no—
83C2×Q8C22⋊C412yes—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—42no—
93C22×C4—41no—
94C22×C4—41no—
95C22×C4—41no—
96C22×C4—81no—
97C22×C4—81no—
98C22×C4C22×C411yescharacteristic
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D411yescharacteristic
101C22×C4C2×D411yescharacteristic
102C22×C4#4111yescharacteristic
103C22×C4#4111yescharacteristic
104Q16—82no—
105C2×C8—22no—
106C2×C8—41no—
107C2×C8—81no—
108C2×C8—81no—
109C4⋊C4—22no—
110C4⋊C4—22no—
111C4⋊C4—21no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C4⋊C4—42no—
115C4⋊C4—41no—
116C4⋊C4—41no—
117C4⋊C4—81no—
118C4⋊C4—81no—
119C4⋊C4C22⋊C412yes—
120C22⋊C4—22no—
121C22⋊C4—22no—
122C22⋊C4—21no—
123C22⋊C4—21no—
124C22⋊C4—21no—
125C22⋊C4—41no—
126C22⋊C4—41no—
127C22⋊C4—41no—
128C22⋊C4—41no—
129C22⋊C4Q1612yescomplemented
130C42—42no—
131C42—81no—
132C42—81no—
133C23—41no—
134C23—41no—
135C23C4×D411yescharacteristic
136C23#12011yescharacteristic
137C23#12111yescharacteristic
138C23C2×Q1611yescharacteristic
139C23#13511yescharacteristic
140C23#13711yescharacteristic
141C23#13711yescharacteristic
142Q8—42no—
143Q8—42no—
144Q8—81no—
145Q8—81no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—22no—
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—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—82no—
173C2×C4—82no—
174C2×C4—81no—
175C2×C4—81no—
176C2×C4—81no—
177C2×C4—81no—
178C2×C4—81no—
179C2×C4C2×C22⋊C411yescharacteristic
180C2×C4C4×D411yescharacteristic
181C2×C4#12111yescharacteristic
182C2×C4#12311yescharacteristic
183C8—42no—
184C8—81no—
185C22—21no—
186C22—21no—
187C22—21no—
188C22—21no—
189C22—21no—
190C22—21no—
191C22—41no—
192C22—41no—
193C22—41no—
194C22—41no—
195C22#38611yescharacteristic
196C22C4×Q1611yescharacteristic
197C22#44111yescharacteristic
198C22#45111yescharacteristic
199C22#45211yescharacteristic
200C22#46211yescharacteristic
201C22#46411yescharacteristic
202C4—21no—
203C4—21no—
204C4—21no—
205C4—21no—
206C4—42no—
207C4—42no—
208C4—42no—
209C4—42no—
210C4—81no—
211C4—81no—
212C4—81no—
213C4—81no—
214C4—161no—
215C2—21no—
216C2—21no—
217C2—21no—
218C2—21no—
219C2—21no—
220C2—21no—
221C2#187811yescharacteristic
222C2#190611yescharacteristic
223C2#191311yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC24C2×D4
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
11. #484C4semidirect
14. #451C4semidirect
129. C22⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
#31892
#23682
#19962
#15202
C2×#3282
#12942
#12822