Finite Group Databasebeta

← #27789

Subgroups — #27789

Order 256.

389 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 389 conjugacy classes of subgroups — too many to draw. Only its 127 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×#499C211yesmaximal, characteristic, complemented
2C2×#467C211yesmaximal, characteristic
3#2489C212yesmaximal
4#2035C212yesmaximal
5#2026C212yesmaximal, complemented
6#1969C211yesmaximal, characteristic, complemented
7#1969C211yesmaximal, characteristic, complemented
8#1927C212yesmaximal, complemented
9#1808C211yesmaximal, characteristic
10C4×#107C211yesmaximal, characteristic, complemented
11C8×C22⋊C4C211yesmaximal, characteristic
12C22×Q16C2211yescharacteristic
13C23×C8C2211yescharacteristic
14C2×#125C2212yes—
15C2×C22⋊Q8C2212yes—
16#499—24no—
17#467—24no—
18C2×#107C2211yescharacteristic
19C2×#107C2211yescharacteristic
20C2×#107C2211yescharacteristic
21C2×#106C2212yes—
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×C22⋊C8C2211yescharacteristic
26C2×C4×C8C2211yescharacteristic
27C2×C4×C8C2211yescharacteristic
28#400—22no—
29#397—22no—
30#391C2212yes—
31#391C2212yes—
32#388—22no—
33#382—22no—
34#380C2212yes—
35C4×C4⋊C4C2212yes—
36C4×C22⋊C4C2211yescharacteristic
37#340C2212yes—
38#340C2212yes—
39#336C2211yescharacteristic
40#336C2211yescharacteristic
41C22×Q8C2312yes—
42C23×C4C2311yescharacteristic
43C2×Q16—22no—
44C2×Q16—22no—
45C2×Q16—22no—
46C22×C8—22no—
47C22×C8—22no—
48C22×C8—22no—
49C22×C8C2311yescharacteristic
50C22×C8C2311yescharacteristic
51C22×C8C2311yescharacteristic
52C22×C8C2311yescharacteristic
53#125—24no—
54#125—42no—
55C22⋊Q8—24no—
56C22⋊Q8—42no—
57C2×C4⋊C4—22no—
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#107D414yes—
73#106—24no—
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—
95C22×C4—22no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D411yescharacteristic
110C22×C4C2411yescharacteristic
111Q16—44no—
112C2×C8—21no—
113C2×C8—21no—
114C2×C8—21no—
115C2×C8—21no—
116C2×C8—21no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—44no—
121C2×C8C2×D412yes—
122C2×C8#4112yes—
123C2×C8#4112yes—
124C2×C8#4112yes—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—44no—
131C4⋊C4—44no—
132C4⋊C4—44no—
133C4⋊C4C2×D414yes—
134C22⋊C4—22no—
135C22⋊C4—22no—
136C22⋊C4—44no—
137C22⋊C4—44no—
138C42—21no—
139C42—21no—
140C42—21no—
141C42—21no—
142C42—44no—
143C23—21no—
144C23—21no—
145C23—21no—
146C23—21no—
147C23—21no—
148C23—21no—
149C23#13512yes—
150C23C22×D411yescharacteristic
151Q8—42no—
152Q8—42no—
153Q8—42no—
154Q8—42no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—24no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—42no—
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#13512yes—
186C2×C4C22×D411yescharacteristic
187C2×C4C2×#4111yescharacteristic
188C2×C4C2×#4111yescharacteristic
189C2×C4C2×#4111yescharacteristic
190C2×C4H(5,𝔽2)11yescharacteristic
191C2×C42−1+411yescharacteristic
192C8—22no—
193C8—41no—
194C8—41no—
195C8#12414yes—
196C22—22no—
197C22—22no—
198C22—22no—
199C22—41no—
200C22—41no—
201C22—41no—
202C22—41no—
203C22#53211yescharacteristic
204C22D4211yescharacteristic
205C22#54711yescharacteristic
206C22C2×#13511yescharacteristic
207C22C2×#13511yescharacteristic
208C22#57811yescharacteristic
209C22#57811yescharacteristic
210C4—22no—
211C4—41no—
212C4—41no—
213C4—41no—
214C4—44no—
215C4—41no—
216C4—82no—
217C4—82no—
218C4—82no—
219C4C2×#12411yescharacteristic
220C4#54211yescharacteristic
221C4#54811yescharacteristic
222C4#55511yescharacteristic
223C2—42no—
224C2#249111yescharacteristic
225C2#316811yescharacteristic
226C2#317711yescharacteristic
227C2#328311yescharacteristic
228C2#328511yescharacteristic
229C2#333611yescharacteristic
230C2#333611yescharacteristic
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×#499C2semidirect
5. #2026C2semidirect
6. #1969C2semidirect
7. #1969C2semidirect
8. #1927C2semidirect
10. C4×#107C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4992
C2×#4672
#24892
#20352
#20262
#19692
#19272
#18082
C4×#1072
C8×C22⋊C42