Finite Group Databasebeta

← #8855

Subgroups — #8855

Order 256.

284 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 75 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#2313C211yesmaximal, characteristic
2#1853C211yesmaximal, characteristic
3#1852C211yesmaximal, characteristic
4C2×#324C211yesmaximal, characteristic
5#1441C211yesmaximal, characteristic
6#1294C211yesmaximal, characteristic
7#1294C211yesmaximal, characteristic
8C22×C4⋊C4C2211yescharacteristic
9C22×C4⋊C4C2211yescharacteristic
10C2×#107—21no—
11C2×#106—21no—
12C2×C22⋊C8C2211yescharacteristic
13C2×C22⋊C8C2211yescharacteristic
14#387—21no—
15#387—21no—
16#387—21no—
17#387—21no—
18#381—21no—
19#380C2211yescharacteristic
20#380C2211yescharacteristic
21C4×C22⋊C4—21no—
22C2×#95—21no—
23C2×#95—21no—
24C2×#95C2211yescharacteristic
25#340—21no—
26#340—21no—
27#324C414yescomplemented
28C23×C4—21no—
29C23×C4—21no—
30C23×C4D411yescharacteristic
31C23×C4D411yescharacteristic
32C23×C4C2311yescharacteristic
33C22×C8—21no—
34C22×C8—21no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4—21no—
41C2×C4⋊C4—42no—
42C2×C4⋊C4C2×C412yes—
43C2×C4⋊C4D411yescharacteristic
44C2×C4⋊C4D411yescharacteristic
45C2×C4⋊C4D411yescharacteristic
46C2×C4⋊C4D411yescharacteristic
47C2×C4⋊C4Q812yes—
48C2×C22⋊C4—21no—
49C2×C22⋊C4—21no—
50C2×C42—21no—
51C2×C42—21no—
52#107—42no—
53#106—42no—
54C22⋊C8—21no—
55C22⋊C8—21no—
56C22⋊C8C2×C412yes—
57C22⋊C8C2×C412yes—
58#95—21no—
59#95—21no—
60#95—21no—
61#95—21no—
62#95—21no—
63#95—21no—
64#95—21no—
65#95—21no—
66#95—21no—
67#95—21no—
68#95—42no—
69#95—42no—
70C24C2×D411yescharacteristic
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—22no—
80C22×C4—22no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—41no—
86C22×C4—41no—
87C22×C4—41no—
88C22×C4—41no—
89C22×C4—41no—
90C22×C4—41no—
91C22×C4—41no—
92C22×C4—41no—
93C22×C4—41no—
94C22×C4—41no—
95C22×C4—41no—
96C22×C4—41no—
97C22×C4—41no—
98C22×C4—41no—
99C22×C4—41no—
100C22×C4—41no—
101C22×C4C4⋊C412yes—
102C22×C4C4⋊C412yes—
103C22×C4C22×C411yescharacteristic
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D411yescharacteristic
106C22×C4C2×Q811yescharacteristic
107C22×C4#4111yescharacteristic
108C22×C4#4111yescharacteristic
109C2×C8—22no—
110C2×C8—22no—
111C2×C8—41no—
112C2×C8—41no—
113C2×C8—41no—
114C2×C8—41no—
115C4⋊C4—22no—
116C4⋊C4—22no—
117C4⋊C4—22no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—41no—
122C4⋊C4—41no—
123C4⋊C4—41no—
124C4⋊C4—41no—
125C4⋊C4—41no—
126C4⋊C4—41no—
127C22⋊C4—41no—
128C22⋊C4—41no—
129C22⋊C4—41no—
130C22⋊C4—41no—
131C42—81no—
132C42—81no—
133C23—21no—
134C23—21no—
135C23—21no—
136C23—21no—
137C23C2×C4⋊C411yescharacteristic
138C23#12011yescharacteristic
139C23#12311yescharacteristic
140C23#13511yescharacteristic
141C23#13511yescharacteristic
142C23AGL(1,ℤ/8)11yescharacteristic
143C23#13711yescharacteristic
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—22no—
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—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—42no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—81no—
182C2×C4—81no—
183C2×C4—81no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4—81no—
187C2×C4—81no—
188C2×C4—81no—
189C2×C4—81no—
190C2×C4C4×D411yescharacteristic
191C2×C4C4×D411yescharacteristic
192C2×C4C22⋊Q811yescharacteristic
193C2×C4C22⋊Q811yescharacteristic
194C8—42no—
195C8—42no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22#38511yescharacteristic
209C22#42711yescharacteristic
210C22#42811yescharacteristic
211C22#44911yescharacteristic
212C22#45211yescharacteristic
213C22#45411yescharacteristic
214C22#45611yescharacteristic
215C22B(4,𝔽2)11yescharacteristic
216C22#45811yescharacteristic
217C22#48211yescharacteristic
218C22#48511yescharacteristic
219C4—21no—
220C4—21no—
221C4—21no—
222C4—21no—
223C4—42no—
224C4—42no—
225C4—82no—
226C4—81no—
227C4—82no—
228C4—81no—
229C4—81no—
230C4—81no—
231C2—21no—
232C2—21no—
233C2—21no—
234C2—21no—
235C2#159711yescharacteristic
236C2#160311yescharacteristic
237C2#160911yescharacteristic
238C2#161511yescharacteristic
239C2#185411yescharacteristic
240C2#185711yescharacteristic
241C2#186711yescharacteristic
subgroupG/H
center, socleC23#120
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
27. #324C4semidirect

Maximal subgroups

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

subgroupindex
#23132
#18532
#18522
C2×#3242
#14412
#12942