Finite Group Databasebeta

← #28661

Subgroups — #28661

Order 256.

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

normal subgroups only

This group has 427 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×#501C211yesmaximal, characteristic, complemented
2C2×#490C211yesmaximal, characteristic
3#2858C211yesmaximal, characteristic
4C4×#125C211yesmaximal, characteristic, complemented
5#2067C212yesmaximal, complemented
6#2065C212yesmaximal, complemented
7#1993C211yesmaximal, characteristic, complemented
8#1961C211yesmaximal, characteristic
9#1926C212yesmaximal, complemented
10#1925C212yesmaximal, complemented
11#1767C211yesmaximal, characteristic, complemented
12C2×C4⋊Q8C2211yescharacteristic
13C2×#127C2211yescharacteristic
14C2×#125C2211yescharacteristic
15C2×#124—21no—
16C2×#124—21no—
17#501—24no—
18#490—24no—
19C2×#107C2212yes—
20C2×#106C2212yes—
21C2×C4⋊C8C2212yes—
22C2×#102C2212yes—
23C2×#102C2212yes—
24C2×#102C2212yes—
25C2×#102C2212yes—
26C2×#97C2211yescharacteristic
27C2×#97C2211yescharacteristic
28#390—21no—
29#390—21no—
30#390C2212yes—
31#390C2212yes—
32#389C2211yescharacteristic
33#389C2211yescharacteristic
34#387—21no—
35#387—21no—
36#382C2211yescharacteristic
37C4×C4⋊C4—21no—
38C4×C4⋊C4—21no—
39C4×C4⋊C4C2211yescharacteristic
40C4×C4⋊C4C2211yescharacteristic
41C43C2211yescharacteristic
42#340C2212yes—
43#340C2212yes—
44#336C2212yes—
45C22×Q8—21no—
46C22×D4—21no—
47C22×D4—21no—
48C22×D4C2312yes—
49C22×C8C2312yes—
50C22×C8C2312yes—
51C4⋊Q8—22no—
52C4⋊Q8—22no—
53#127—22no—
54#127—22no—
55#125—22no—
56#125D414yes—
57#124—44no—
58#124—44no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4C2312yes—
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2311yescharacteristic
69C2×C22⋊C4—22no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C42—21no—
74C2×C42—21no—
75C2×C42D411yescharacteristic
76C2×C42D411yescharacteristic
77C2×C42D411yescharacteristic
78C2×C42D411yescharacteristic
79C2×C42C2311yescharacteristic
80C2×C42C2311yescharacteristic
81C2×C42C2311yescharacteristic
82#107—24no—
83#106—24no—
84C4⋊C8—24no—
85#102—24no—
86#102—24no—
87#102—24no—
88#102—24no—
89#97—22no—
90#97—22no—
91#95—21no—
92#95—21no—
93#95—21no—
94#95—21no—
95#95—21no—
96#95—21no—
97C24—22no—
98C2×Q8—42no—
99C2×Q8—42no—
100C2×Q8—42no—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4—22no—
104C2×D4—22no—
105C2×D4—22no—
106C2×D4—22no—
107C2×D4—42no—
108C2×D4—42no—
109C2×D4—42no—
110C2×D4—42no—
111C2×D4—42no—
112C2×D4—42no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—22no—
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D411yescharacteristic
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D411yescharacteristic
124C22×C4C2411yescharacteristic
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—22no—
130C2×C8#4114yes—
131C4⋊C4—24no—
132C4⋊C4—24no—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—42no—
140C4⋊C4—42no—
141C4⋊C4—42no—
142C4⋊C4C2×D412yes—
143C4⋊C4C2×D412yes—
144C22⋊C4—44no—
145C22⋊C4—44no—
146C22⋊C4—44no—
147C22⋊C4—44no—
148C42—22no—
149C42—22no—
150C42—22no—
151C42—22no—
152C42—41no—
153C42—41no—
154C42—42no—
155C42—42no—
156C42C2×D412yes—
157C42#4112yes—
158C23—42no—
159C23—42no—
160C23—42no—
161C23—42no—
162C23—42no—
163C23—42no—
164C23—42no—
165C23C22×D411yescharacteristic
166Q8—84no—
167D4—42no—
168D4—42no—
169D4—42no—
170D4—42no—
171D4—84no—
172D4—84no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4#12112yes—
203C2×C4#12112yes—
204C2×C4AGL(1,ℤ/8)12yes—
205C2×C4AGL(1,ℤ/8)12yes—
206C2×C4C22×D411yescharacteristic
207C2×C4C2×#4112yes—
208C2×C4C2×#4111yescharacteristic
209C2×C42−1+412yes—
210C8—24no—
211C8—42no—
212C22—82no—
213C22—82no—
214C22—82no—
215C22—82no—
216C22—82no—
217C22—82no—
218C22—82no—
219C22C2×#12111yescharacteristic
220C22#53211yescharacteristic
221C22#53611yescharacteristic
222C22C2×AGL(1,ℤ/8)11yescharacteristic
223C22C2×AGL(1,ℤ/8)11yescharacteristic
224C22#57511yescharacteristic
225C22#57511yescharacteristic
226C4—22no—
227C4—22no—
228C4—22no—
229C4—22no—
230C4—41no—
231C4—41no—
232C4—42no—
233C4—42no—
234C4—81no—
235C4—81no—
236C4—82no—
237C4#54812yes—
238C4#55612yes—
239C2—162no—
240C2#254511yescharacteristic
241C2#303811yescharacteristic
242C2#304011yescharacteristic
243C2#317111yescharacteristic
244C2#318011yescharacteristic
245C2#323411yescharacteristic
246C2#323511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#501C2semidirect
4. C4×#125C2semidirect
5. #2067C2semidirect
6. #2065C2semidirect
7. #1993C2semidirect
9. #1926C2semidirect
10. #1925C2semidirect
11. #1767C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5012
C2×#4902
#28582
C4×#1252
#20672
#20652
#19932
#19612
#19262
#19252
#17672