Finite Group Databasebeta

← #8869

Subgroups — #8869

Order 256.

329 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 329 conjugacy classes of subgroups — too many to draw. Only its 77 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#2325C211yesmaximal, characteristic
2#1874C211yesmaximal, characteristic
3#1852C211yesmaximal, characteristic, complemented
4C2×#327C211yesmaximal, characteristic
5#1442C211yesmaximal, characteristic, complemented
6#1294C211yesmaximal, characteristic, complemented
7#1280C211yesmaximal, characteristic, complemented
8C2×#121C2211yescharacteristic
9C2×C4×D4—21no—
10C22×C4⋊C4C2211yescharacteristic
11C2×#107—21no—
12C2×#102—21no—
13C2×C22⋊C8C2211yescharacteristic
14C2×C22⋊C8C2211yescharacteristic
15#388—21no—
16#387—21no—
17#386—21no—
18#386—21no—
19#385—21no—
20#380C2211yescharacteristic
21C2×#95—21no—
22C2×#95—21no—
23C2×#95C2211yescharacteristic
24C2×#95C2211yescharacteristic
25#340—21no—
26#340—21no—
27#327C414yescomplemented
28C22×D4—21no—
29C22×D4D411yescharacteristic
30C23×C4—21no—
31C23×C4—21no—
32C23×C4—21no—
33C23×C4D411yescharacteristic
34C23×C4D411yescharacteristic
35C23×C4C2311yescharacteristic
36C22×C8—21no—
37C22×C8—21no—
38#121—21no—
39#121—21no—
40#121C2×C412yes—
41#121Q812yes—
42C4×D4—42no—
43C4×D4—42no—
44C2×C4⋊C4—21no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—42no—
48C2×C4⋊C4D411yescharacteristic
49C2×C4⋊C4D411yescharacteristic
50C2×C4⋊C4D411yescharacteristic
51C2×C22⋊C4—21no—
52C2×C22⋊C4—21no—
53C2×C22⋊C4—21no—
54C2×C22⋊C4—21no—
55C2×C22⋊C4—21no—
56C2×C42—21no—
57#107—42no—
58#102—42no—
59C22⋊C8—21no—
60C22⋊C8—21no—
61C22⋊C8—21no—
62C22⋊C8C2×C412yescomplemented
63#95—21no—
64#95—21no—
65#95—21no—
66#95—21no—
67#95—21no—
68#95—21no—
69#95—42no—
70#95—42no—
71#95C2×C412yescomplemented
72C24—21no—
73C24C2×D411yescharacteristic
74C2×D4—22no—
75C2×D4—22no—
76C2×D4—21no—
77C2×D4—41no—
78C2×D4—41no—
79C2×D4—41no—
80C2×D4—41no—
81C2×D4—41no—
82C2×D4—41no—
83C2×D4C4⋊C412yes—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—22no—
88C22×C4—22no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—41no—
95C22×C4—41no—
96C22×C4—42no—
97C22×C4—41no—
98C22×C4—41no—
99C22×C4—41no—
100C22×C4—41no—
101C22×C4—41no—
102C22×C4—41no—
103C22×C4—41no—
104C22×C4—41no—
105C22×C4—41no—
106C22×C4—41no—
107C22×C4—41no—
108C22×C4—41no—
109C22×C4—41no—
110C22×C4—41no—
111C22×C4—41no—
112C22×C4—41no—
113C22×C4—81no—
114C22×C4—81no—
115C22×C4D812yes—
116C22×C4C22×C411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2×Q811yescharacteristic
120C22×C4#4111yescharacteristic
121C22×C4#4111yescharacteristic
122C2×C8—22no—
123C2×C8—41no—
124C2×C8—41no—
125C2×C8—41no—
126C2×C8—41no—
127C2×C8—41no—
128C4⋊C4—21no—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—41no—
132C4⋊C4—41no—
133C4⋊C4—41no—
134C4⋊C4—41no—
135C4⋊C4—42no—
136C4⋊C4C4⋊C412yes—
137C22⋊C4—22no—
138C22⋊C4—22no—
139C22⋊C4—42no—
140C22⋊C4—41no—
141C22⋊C4—41no—
142C22⋊C4—81no—
143C22⋊C4—81no—
144C42—42no—
145C23—21no—
146C23—21no—
147C23—21no—
148C23—21no—
149C23—22no—
150C23—41no—
151C23—41no—
152C23—41no—
153C23—41no—
154C23—41no—
155C23—41no—
156C23C4×D411yescharacteristic
157C23#12011yescharacteristic
158C23C22⋊Q811yescharacteristic
159C23C2×D811yescharacteristic
160C23#13511yescharacteristic
161C23AGL(1,ℤ/8)11yescharacteristic
162C23#13711yescharacteristic
163D4—42no—
164D4—42no—
165D4—42no—
166D4—42no—
167D4—42no—
168D4—81no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—81no—
201C2×C4—84no—
202C2×C4—81no—
203C2×C4—81no—
204C2×C4—81no—
205C2×C4—81no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4C2×C4⋊C411yescharacteristic
212C2×C4C4×D411yescharacteristic
213C2×C4C22⋊Q811yescharacteristic
214C2×C4#12311yescharacteristic
215C8—42no—
216C8—81no—
217C22—21no—
218C22—21no—
219C22—21no—
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—42no—
230C22—42no—
231C22—42no—
232C22—81no—
233C22—81no—
234C22—81no—
235C22—81no—
236C22#38511yescharacteristic
237C22C4×D811yescharacteristic
238C22#44011yescharacteristic
239C22#44711yescharacteristic
240C22#45211yescharacteristic
241C22#45411yescharacteristic
242C22#45511yescharacteristic
243C22B(4,𝔽2)11yescharacteristic
244C22#45811yescharacteristic
245C22#47411yescharacteristic
246C22#47811yescharacteristic
247C4—21no—
248C4—21no—
249C4—21no—
250C4—21no—
251C4—42no—
252C4—42no—
253C4—82no—
254C4—81no—
255C4—82no—
256C4—81no—
257C4—81no—
258C2—21no—
259C2—21no—
260C2—21no—
261C2—21no—
262C2—82no—
263C2#159811yescharacteristic
264C2#160611yescharacteristic
265C2#160711yescharacteristic
266C2#161511yescharacteristic
267C2#186011yescharacteristic
268C2#186211yescharacteristic
269C2#186711yescharacteristic
subgroupG/H
center, socleC23#120
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC2×#121C22
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
3. #1852C2semidirect
5. #1442C2semidirect
6. #1294C2semidirect
7. #1280C2semidirect
27. #327C4semidirect
62. C22⋊C8C2×C4semidirect
71. #95C2×C4semidirect

Maximal subgroups

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

subgroupindex
#23252
#18742
#18522
C2×#3272
#14422
#12942
#12802