Finite Group Databasebeta

← #27768

Subgroups — #27768

Order 256.

401 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 401 conjugacy classes of subgroups — too many to draw. Only its 131 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×#488C212yesmaximal
2#2552C211yesmaximal, characteristic, complemented
3#2547C211yesmaximal, characteristic
4#2042C212yesmaximal
5#2026C212yesmaximal, complemented
6#1981C211yesmaximal, characteristic, complemented
7#1956C211yesmaximal, characteristic, complemented
8#1933C212yesmaximal, complemented
9#1802C211yesmaximal, characteristic
10C8×C4⋊C4C211yesmaximal, characteristic, complemented
11C4×#102C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×#125C2212yes—
14C2×#124C2212yes—
15C2×C4×D4C2211yescharacteristic
16#488—28no—
17C2×C4⋊C8C2211yescharacteristic
18C2×#103C2212yes—
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×#103C2212yes—
22C2×#102C2212yes—
23C2×#102C2211yescharacteristic
24C2×#102C2211yescharacteristic
25C2×C4×C8C2212yes—
26C2×C4×C8C2211yescharacteristic
27#398—22no—
28#397—22no—
29#391C2212yes—
30#389—21no—
31#389C2212yes—
32#384—21no—
33#384C2211yescharacteristic
34#380—21no—
35#380—21no—
36C4×C4⋊C4C2211yescharacteristic
37C4×C4⋊C4C2211yescharacteristic
38#340C2212yes—
39#340C2211yescharacteristic
40#340C2211yescharacteristic
41#336C2212yes—
42C22×Q8C2312yes—
43C22×D4C2311yescharacteristic
44C23×C4—21no—
45C22×C8C2311yescharacteristic
46C22×C8C2312yes—
47C22×C8C2311yescharacteristic
48C4⋊Q8—22no—
49C4⋊Q8—22no—
50#125—24no—
51#125—42no—
52#124—24no—
53#124—42no—
54C4×D4—22no—
55C4×D4—22no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4D411yescharacteristic
64C2×C4⋊C4D411yescharacteristic
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2312yes—
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2311yescharacteristic
69C2×C22⋊C4—21no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—22no—
72C2×C42—21no—
73C2×C42C2312yes—
74C2×C42C2311yescharacteristic
75C4⋊C8—22no—
76#103—24no—
77#103—24no—
78#103—24no—
79#103—42no—
80#102—24no—
81#102—22no—
82#102D414yes—
83C4×C8—24no—
84C4×C8—22no—
85#95—21no—
86#95—22no—
87#95—22no—
88#95—22no—
89#95D412yes—
90C24—21no—
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×D4—21no—
98C2×D4—21no—
99C2×D4—21no—
100C2×D4—21no—
101C2×D4—21no—
102C2×D4C2×D412yes—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4—42no—
111C22×C4C2×D412yes—
112C22×C4C2×D412yes—
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D411yescharacteristic
115C22×C4C2411yescharacteristic
116C2×C8—21no—
117C2×C8—21no—
118C2×C8—22no—
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—21no—
124C2×C8C2×D412yes—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—22no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—22no—
133C4⋊C4—44no—
134C4⋊C4—44no—
135C4⋊C4—42no—
136C4⋊C4—42no—
137C4⋊C4—42no—
138C4⋊C4—81no—
139C4⋊C4—82no—
140C4⋊C4C2×D412yes—
141C4⋊C4#4112yes—
142C22⋊C4—44no—
143C22⋊C4—44no—
144C22⋊C4—42no—
145C42—22no—
146C42—22no—
147C42—42no—
148C42#4112yes—
149C42#4112yes—
150C23—22no—
151C23—41no—
152C23—41no—
153C23—41no—
154C23—41no—
155C23—41no—
156C23—41no—
157C23C22×D411yescharacteristic
158Q8—42no—
159Q8—42no—
160Q8—42no—
161Q8—42no—
162D4—22no—
163D4—22no—
164D4—41no—
165D4—41no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—84no—
190C2×C4#12412yes—
191C2×C4#12412yes—
192C2×C4#13512yes—
193C2×C4#13512yes—
194C2×C4C22×D411yescharacteristic
195C2×C4C2×#4111yescharacteristic
196C2×C4C2×#4111yescharacteristic
197C2×C4C2×#4111yescharacteristic
198C2×C4H(5,𝔽2)11yescharacteristic
199C2×C42−1+411yescharacteristic
200C8—22no—
201C8—42no—
202C8—41no—
203C22—42no—
204C22—42no—
205C22—42no—
206C22—81no—
207C22—81no—
208C22—81no—
209C22—81no—
210C22C2×#12411yescharacteristic
211C22D4211yescharacteristic
212C22#54811yescharacteristic
213C22C2×#13511yescharacteristic
214C22C2×#13511yescharacteristic
215C22#57711yescharacteristic
216C22#57811yescharacteristic
217C4—22no—
218C4—42no—
219C4—42no—
220C4—42no—
221C4—42no—
222C4—81no—
223C4—82no—
224C4—82no—
225C4#48814yescomplemented
226C4#53211yescharacteristic
227C4#54211yescharacteristic
228C4#54711yescharacteristic
229C4#55511yescharacteristic
230C2—82no—
231C2#249111yescharacteristic
232C2C2×#48811yescharacteristic
233C2#313711yescharacteristic
234C2#328311yescharacteristic
235C2#328411yescharacteristic
236C2#336011yescharacteristic
237C2#336411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2552C2semidirect
5. #2026C2semidirect
6. #1981C2semidirect
7. #1956C2semidirect
8. #1933C2semidirect
10. C8×C4⋊C4C2semidirect
225. C4#488semidirect

Maximal subgroups

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

subgroupindex
C2×#4882
#25522
#25472
#20422
#20262
#19812
#19562
#19332
#18022
C8×C4⋊C42
C4×#1022