Finite Group Databasebeta

← #27792

Subgroups — #27792

Order 256.

405 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 405 conjugacy classes of subgroups — too many to draw. Only its 135 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×#500C211yesmaximal, characteristic, complemented
2C2×#467C211yesmaximal, characteristic
3#2492C212yesmaximal
4#2036C212yesmaximal
5#2024C212yesmaximal, complemented
6#1969C211yesmaximal, characteristic, complemented
7#1969C211yesmaximal, characteristic, complemented
8#1928C212yesmaximal, complemented
9#1808C211yesmaximal, characteristic
10C4×#107C211yesmaximal, characteristic, complemented
11C8×C22⋊C4C211yesmaximal, characteristic
12C22×Q16C2211yescharacteristic
13C23×C8C2211yescharacteristic
14C2×C4⋊Q8C2212yes—
15C2×C22⋊Q8C2212yes—
16#500—24no—
17#467—24no—
18C2×#107C2211yescharacteristic
19C2×#107C2211yescharacteristic
20C2×#107C2212yes—
21C2×#107C2211yescharacteristic
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×C22⋊C8C2211yescharacteristic
26C2×C4×C8C2211yescharacteristic
27C2×C4×C8C2211yescharacteristic
28#400—22no—
29#393—22no—
30#391—22no—
31#391C2212yes—
32#391C2212yes—
33#388—22no—
34#380C2212yes—
35C4×C4⋊C4C2212yes—
36C4×C22⋊C4C2211yescharacteristic
37#340C2212yes—
38#340C2212yes—
39#336C2211yescharacteristic
40#336C2211yescharacteristic
41C22×Q8—22no—
42C22×Q8C2312yes—
43C23×C4C2311yescharacteristic
44C2×Q16—22no—
45C2×Q16—22no—
46C2×Q16—22no—
47C22×C8—22no—
48C22×C8—22no—
49C22×C8—22no—
50C22×C8C2311yescharacteristic
51C22×C8C2311yescharacteristic
52C22×C8C2311yescharacteristic
53C22×C8C2311yescharacteristic
54C4⋊Q8—24no—
55C4⋊Q8—42no—
56C22⋊Q8—24no—
57C22⋊Q8—42no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2312yes—
63C2×C22⋊C4—22no—
64C2×C22⋊C4—22no—
65C2×C22⋊C4D411yescharacteristic
66C2×C22⋊C4D411yescharacteristic
67C2×C42—22no—
68C2×C42C2311yescharacteristic
69C2×C42C2311yescharacteristic
70#107—22no—
71#107—22no—
72#107—24no—
73#107D414yes—
74#103—24no—
75#103—24no—
76#103—42no—
77C22⋊C8—22no—
78C4×C8—22no—
79C4×C8—22no—
80#95—22no—
81#95—22no—
82#95—22no—
83#95—22no—
84#95—22no—
85#95—22no—
86#95D411yescharacteristic
87#95D411yescharacteristic
88C24C2×D411yescharacteristic
89C2×Q8—22no—
90C2×Q8—22no—
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—22no—
95C2×Q8—44no—
96C2×Q8—44no—
97C2×Q8—82no—
98C22×C4—22no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D411yescharacteristic
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4C2411yescharacteristic
114Q16—44no—
115C2×C8—21no—
116C2×C8—21no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—44no—
124C2×C8C2×D412yes—
125C2×C8#4112yes—
126C2×C8#4112yes—
127C2×C8#4112yes—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—44no—
134C4⋊C4—44no—
135C4⋊C4C2×D414yes—
136C22⋊C4—22no—
137C22⋊C4—44no—
138C22⋊C4—82no—
139C22⋊C4Q1614yescomplemented
140C42—21no—
141C42—21no—
142C42—21no—
143C42—21no—
144C42—82no—
145C23—21no—
146C23—21no—
147C23—21no—
148C23—21no—
149C23—21no—
150C23—21no—
151C23C2×Q1612yes—
152C23C22×D411yescharacteristic
153Q8—42no—
154Q8—42no—
155Q8—42no—
156Q8—42no—
157Q8—88no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—24no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4C2×Q1612yes—
187C2×C4C2×Q1612yes—
188C2×C4#13512yes—
189C2×C4C22×D411yescharacteristic
190C2×C4C2×#4111yescharacteristic
191C2×C4C2×#4111yescharacteristic
192C2×C4C2×#4111yescharacteristic
193C2×C4H(5,𝔽2)11yescharacteristic
194C2×C42−1+411yescharacteristic
195C8—22no—
196C8—41no—
197C8—41no—
198C8#12414yes—
199C22—22no—
200C22—22no—
201C22—22no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22#53211yescharacteristic
207C22D4211yescharacteristic
208C22#54711yescharacteristic
209C22C22×Q1611yescharacteristic
210C22C2×#13511yescharacteristic
211C22#57611yescharacteristic
212C22#57811yescharacteristic
213C4—22no—
214C4—22no—
215C4—22no—
216C4—41no—
217C4—41no—
218C4—44no—
219C4—82no—
220C4—82no—
221C4—82no—
222C4C2×#12411yescharacteristic
223C4#54211yescharacteristic
224C4#54811yescharacteristic
225C4#55511yescharacteristic
226C2—42no—
227C2#249111yescharacteristic
228C2#317011yescharacteristic
229C2#317511yescharacteristic
230C2#328011yescharacteristic
231C2D4×Q1611yescharacteristic
232C2#333811yescharacteristic
233C2#334211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#500C2semidirect
5. #2024C2semidirect
6. #1969C2semidirect
7. #1969C2semidirect
8. #1928C2semidirect
10. C4×#107C2semidirect
139. C22⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#5002
C2×#4672
#24922
#20362
#20242
#19692
#19282
#18082
C4×#1072
C8×C22⋊C42