Finite Group Databasebeta

← #23647

Subgroups — #23647

Order 256.

457 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 457 conjugacy classes of subgroups — too many to draw. Only its 151 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
1C2×#428C211yesmaximal, characteristic, complemented
2#2896C211yesmaximal, characteristic
3C2×#417C211yesmaximal, characteristic
4#2879C211yesmaximal, characteristic
5#2079C214yesmaximal, complemented
6#2078C214yesmaximal, complemented
7#1742C211yesmaximal, characteristic, complemented
8C2×#343C211yesmaximal, characteristic, complemented
9#1738C211yesmaximal, characteristic, complemented
10C22×#41C2211yescharacteristic
11C22×M16C2211yescharacteristic
12C22×M16C2211yescharacteristic
13C22×M16C2211yescharacteristic
14C22×M16C2211yescharacteristic
15C2×#117C2211yescharacteristic
16C22×C4⋊C4C2211yescharacteristic
17#428—22no—
18#428—22no—
19C2×#107C2212yescomplemented
20C2×#106C2212yescomplemented
21#419—21no—
22#418—21no—
23#417—22no—
24#417C2212yes—
25#417C2214yes—
26#417C2212yes—
27#416—22no—
28C2×#103—21no—
29C2×#102—21no—
30#408—22no—
31#408C2214yes—
32#407—22no—
33#343C2212yescomplemented
34#343C2214yescomplemented
35#343C2212yescomplemented
36#341C2214yescomplemented
37C2×#41—21no—
38C2×#41—21no—
39C2×#41—22no—
40C2×#41—22no—
41C2×#41—22no—
42C2×#41—22no—
43C2×#41D412yes—
44C2×#41C2312yes—
45C22×Q8D411yescharacteristic
46C22×D4—21no—
47C22×D4D411yescharacteristic
48C23×C4—21no—
49C23×C4—21no—
50C23×C4C2311yescharacteristic
51C2×M16—22no—
52C2×M16—22no—
53C2×M16—21no—
54C2×M16—21no—
55C2×M16—21no—
56C2×M16—22no—
57C2×M16—22no—
58C2×M16—22no—
59C2×M16—22no—
60C2×M16—22no—
61C2×M16—22no—
62C2×M16D412yes—
63C2×M16D412yes—
64C2×M16C2312yes—
65C2×M16C2312yes—
66C2×M16C2312yes—
67C22×C8—21no—
68C22×C8—21no—
69C22×C8—21no—
70C22×C8C2312yes—
71#117—21no—
72#117—21no—
73#117—21no—
74#117—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4C2312yes—
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C22⋊C4—21no—
84C2×C42—21no—
85#107—24no—
86#106—24no—
87#103—24no—
88#103—22no—
89#103—22no—
90#102—24no—
91#102—22no—
92#102—22no—
93C22⋊C8—28no—
94C24—21no—
95C24#4111yescharacteristic
96#41—24no—
97#41—24no—
98#41—22no—
99#41—22no—
100#41—22no—
101#41—22no—
102#41—42no—
103#41—42no—
104#41—42no—
105#41—42no—
106C2×Q8—21no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8C2×D412yes—
110C2×D4—22no—
111C2×D4—22no—
112C2×D4—21no—
113C2×D4—22no—
114C2×D4—22no—
115C2×D4—42no—
116C2×D4—42no—
117C2×D4C2×D412yes—
118C22×C4—21no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—22no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4—82no—
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4#4111yescharacteristic
134C22×C4#4112yes—
135C22×C4#4112yes—
136C22×C4#4112yes—
137C22×C4#4112yes—
138C22×C4C2411yescharacteristic
139M16—24no—
140M16—24no—
141M16—22no—
142M16—22no—
143M16—22no—
144M16—22no—
145M16—22no—
146M16—22no—
147M16—42no—
148M16—42no—
149M16—42no—
150C2×C8—22no—
151C2×C8—22no—
152C2×C8—22no—
153C2×C8—22no—
154C2×C8—22no—
155C2×C8—42no—
156C2×C8—42no—
157C2×C8—42no—
158C2×C8C2×D414yes—
159C4⋊C4—22no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—22no—
164C4⋊C4—21no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C22⋊C4—41no—
168C22⋊C4—41no—
169C42—41no—
170C42—41no—
171C23—22no—
172C23—41no—
173C23—41no—
174C23—42no—
175C23—81no—
176C23—81no—
177C23#12111yescharacteristic
178C23#12111yescharacteristic
179C23#12612yes—
180C23#12612yes—
181C23C2×#4111yescharacteristic
182Q8—22no—
183Q8—22no—
184Q8—42no—
185D4—22no—
186D4—22no—
187D4—42no—
188D4—42no—
189D4—44no—
190D4—42no—
191C2×C4—22no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—44no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4—82no—
204C2×C4#12111yescharacteristic
205C2×C4#12111yescharacteristic
206C2×C4#12312yes—
207C2×C4#12312yes—
208C2×C4#12312yes—
209C2×C4#12312yes—
210C2×C4#12312yes—
211C2×C4#12312yes—
212C2×C4#12412yes—
213C2×C4#12412yes—
214C2×C4#12612yes—
215C2×C4#12612yes—
216C2×C4C22×D411yescharacteristic
217C2×C4C22×D411yescharacteristic
218C2×C4C2×#4111yescharacteristic
219C2×C4C2×#4111yescharacteristic
220C2×C4C2×#4111yescharacteristic
221C2×C4C2×#4111yescharacteristic
222C8—24no—
223C8—42no—
224C8—42no—
225C8—42no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—21no—
230C22—21no—
231C22—21no—
232C22—42no—
233C22—42no—
234C22—42no—
235C22—42no—
236C22—42no—
237C22—84no—
238C22#39712yes—
239C22#39712yes—
240C22C2×#12111yescharacteristic
241C22C2×#12611yescharacteristic
242C22C2×#12611yescharacteristic
243C4—42no—
244C4—81no—
245C4—82no—
246C4—81no—
247C4#39712yes—
248C4#39712yes—
249C4C2×#12311yescharacteristic
250C4C2×#12311yescharacteristic
251C4C2×#12311yescharacteristic
252C4C2×#12411yescharacteristic
253C2—22no—
254C2—21no—
255C2—21no—
256C2—22no—
257C2—82no—
258C2C2×#39711yescharacteristic
259C2#306811yescharacteristic
260C2#306911yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#428C2semidirect
5. #2079C2semidirect
6. #2078C2semidirect
7. #1742C2semidirect
8. C2×#343C2semidirect
9. #1738C2semidirect
19. C2×#107C22semidirect
20. C2×#106C22semidirect
33. #343C22semidirect
34. #343C22semidirect
35. #343C22semidirect
36. #341C22semidirect

Maximal subgroups

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

subgroupindex
C2×#4282
#28962
C2×#4172
#28792
#20792
#20782
#17422
C2×#3432
#17382