Finite Group Databasebeta

← #32375

Subgroups — #32375

Order 256.

314 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2570C211yesmaximal, characteristic, complemented
2#2512C211yesmaximal, characteristic
3#2092C211yesmaximal, characteristic
4#2092C211yesmaximal, characteristic
5#2085C211yesmaximal, characteristic, complemented
6#2083C211yesmaximal, characteristic, complemented
7#2071C211yesmaximal, characteristic
8#2065C211yesmaximal, characteristic
9#1949C211yesmaximal, characteristic, complemented
10#1934C211yesmaximal, characteristic
11#1926C211yesmaximal, characteristic
12#1920C211yesmaximal, characteristic, complemented
13#1919C211yesmaximal, characteristic, complemented
14#1916C211yesmaximal, characteristic, complemented
15#1844C211yesmaximal, characteristic, complemented
16C2×#126—21no—
17C2×#107C2211yescharacteristic
18C2×#107C2211yescharacteristic
19C2×#106C2211yescharacteristic
20C2×#106C2211yescharacteristic
21C2×#106C2211yescharacteristic
22C2×#106C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×#102C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×C4×C8C2211yescharacteristic
29#400—21no—
30#398—21no—
31#397—21no—
32#396—21no—
33#390C2211yescharacteristic
34#390C2211yescharacteristic
35#390C2211yescharacteristic
36#389C2211yescharacteristic
37#389C2211yescharacteristic
38#389C2211yescharacteristic
39#388—21no—
40#387—21no—
41#387—21no—
42#384C2211yescharacteristic
43#384C2211yescharacteristic
44#382C2211yescharacteristic
45#382C2211yescharacteristic
46C4×C4⋊C4C2211yescharacteristic
47C4×C4⋊C4C2211yescharacteristic
48C4×C4⋊C4C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×D4C2311yescharacteristic
60C22×C8C2311yescharacteristic
61C22×C8C2311yescharacteristic
62C22×C8C2311yescharacteristic
63C22×C8C2311yescharacteristic
64#126—44no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4D411yescharacteristic
73C2×C4⋊C4D411yescharacteristic
74C2×C4⋊C4C2311yescharacteristic
75C2×C4⋊C4C2311yescharacteristic
76C2×C4⋊C4C2311yescharacteristic
77C2×C4⋊C4C2311yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C22⋊C4—21no—
82C2×C22⋊C4—21no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C42—21no—
88C2×C42—21no—
89C2×C42C2311yescharacteristic
90C2×C42C2311yescharacteristic
91C2×C42C2311yescharacteristic
92#107—41no—
93#107—41no—
94#106—22no—
95#106—22no—
96#106—41no—
97#106—41no—
98C4⋊C8—22no—
99#102—22no—
100#102—22no—
101#102—41no—
102#102—41no—
103C4×C8—22no—
104#95—21no—
105#95—21no—
106#95—21no—
107#95—21no—
108#95—21no—
109#95—21no—
110#95—21no—
111#95—21no—
112#95—21no—
113#95D411yescharacteristic
114#95D411yescharacteristic
115C24—21no—
116C2×D4—21no—
117C2×D4—21no—
118C2×D4—21no—
119C2×D4—21no—
120C2×D4—21no—
121C2×D4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4C2411yescharacteristic
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8#4112yes—
147C2×C8#4112yes—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—81no—
169C4⋊C4—81no—
170C4⋊C4—81no—
171C22⋊C4—42no—
172C22⋊C4—42no—
173C22⋊C4—42no—
174C22⋊C4—42no—
175C22⋊C4—81no—
176C22⋊C4—81no—
177C42—21no—
178C42—21no—
179C42—21no—
180C42—21no—
181C42—21no—
182C42—21no—
183C42—42no—
184C42—81no—
185C23—41no—
186C23—41no—
187C23—41no—
188C23—41no—
189C23—41no—
190C23—41no—
191C23—41no—
192C23C22×D411yescharacteristic
193D4—41no—
194D4—41no—
195D4—41no—
196D4—41no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4#13512yes—
235C2×C4AGL(1,ℤ/8)12yes—
236C2×C4C2×#4111yescharacteristic
237C2×C4C2×#4111yescharacteristic
238C2×C4H(5,𝔽2)11yescharacteristic
239C2×C42−1+411yescharacteristic
240C2×C42−1+411yescharacteristic
241C2×C42−1+411yescharacteristic
242C8—22no—
243C8—22no—
244C8—41no—
245C8—41no—
246C22—81no—
247C22—81no—
248C22—81no—
249C22—81no—
250C22—81no—
251C22—81no—
252C22—81no—
253C22#53611yescharacteristic
254C22#54711yescharacteristic
255C22#55011yescharacteristic
256C22C2×#13511yescharacteristic
257C22C2×AGL(1,ℤ/8)11yescharacteristic
258C22#57711yescharacteristic
259C22#57711yescharacteristic
260C4—22no—
261C4—22no—
262C4—41no—
263C4—41no—
264C4—41no—
265C4—41no—
266C4—81no—
267C4—81no—
268C4—81no—
269C4—81no—
270C4—81no—
271C4—81no—
272C4—81no—
273C4#53911yescharacteristic
274C4#54111yescharacteristic
275C4#55211yescharacteristic
276C4#56311yescharacteristic
277C2—161no—
278C2#272111yescharacteristic
279C2#322811yescharacteristic
280C2#324411yescharacteristic
281C2#333711yescharacteristic
282C2#334511yescharacteristic
283C2#339411yescharacteristic
284C2#339711yescharacteristic
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
1. #2570C2semidirect
5. #2085C2semidirect
6. #2083C2semidirect
9. #1949C2semidirect
12. #1920C2semidirect
13. #1919C2semidirect
14. #1916C2semidirect
15. #1844C2semidirect

Maximal subgroups

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

subgroupindex
#25702
#25122
#20922
#20852
#20832
#20712
#20652
#19492
#19342
#19262
#19202
#19192
#19162
#18442