Finite Group Databasebeta

← #32417

Subgroups — #32417

Order 256.

312 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2496C211yesmaximal, characteristic
2#2494C211yesmaximal, characteristic
3#2088C211yesmaximal, characteristic, complemented
4#2086C211yesmaximal, characteristic, complemented
5#2085C211yesmaximal, characteristic, complemented
6#2083C211yesmaximal, characteristic, complemented
7#2077C211yesmaximal, characteristic
8#2076C211yesmaximal, characteristic
9#1949C211yesmaximal, characteristic, complemented
10#1948C211yesmaximal, characteristic, complemented
11#1921C211yesmaximal, characteristic, complemented
12#1919C211yesmaximal, characteristic, complemented
13#1851C211yesmaximal, characteristic
14#1809C211yesmaximal, characteristic
15#1809C211yesmaximal, characteristic
16C22×M16C2211yescharacteristic
17C2×#126—21no—
18C2×#107C2211yescharacteristic
19C2×#107C2211yescharacteristic
20C2×#107C2211yescharacteristic
21C2×#106C2211yescharacteristic
22C2×#106C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26C2×C22⋊C8C2211yescharacteristic
27#401—21no—
28#400—21no—
29#397—21no—
30#397—21no—
31#389C2211yescharacteristic
32#389C2211yescharacteristic
33#389C2211yescharacteristic
34#389C2211yescharacteristic
35#388—21no—
36#387—21no—
37#387—21no—
38#384C2211yescharacteristic
39#382C2211yescharacteristic
40#380C2211yescharacteristic
41#380C2211yescharacteristic
42#380C2211yescharacteristic
43#380C2211yescharacteristic
44C4×C4⋊C4C2211yescharacteristic
45C4×C4⋊C4C2211yescharacteristic
46C4×C22⋊C4C2211yescharacteristic
47#340C2211yescharacteristic
48#340C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C23×C4C2311yescharacteristic
60C2×M16—22no—
61C2×M16—22no—
62C2×M16—41no—
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67#126—44no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4C2311yescharacteristic
75C2×C4⋊C4C2311yescharacteristic
76C2×C4⋊C4C2311yescharacteristic
77C2×C4⋊C4C2311yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C22⋊C4—21no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4D411yescharacteristic
87C2×C22⋊C4D411yescharacteristic
88C2×C42—21no—
89C2×C42—21no—
90C2×C42C2311yescharacteristic
91C2×C42C2311yescharacteristic
92#107—22no—
93#107—41no—
94#107—41no—
95#106—22no—
96#106—41no—
97#106—41no—
98C4⋊C8—22no—
99C4⋊C8—22no—
100C22⋊C8—41no—
101#95—21no—
102#95—21no—
103#95—21no—
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
115C24C2×D411yescharacteristic
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4C2411yescharacteristic
136M16—44no—
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—21no—
147C2×C8#4112yes—
148C2×C8#4112yes—
149C4⋊C4—21no—
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—21no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—81no—
171C22⋊C4—22no—
172C22⋊C4—41no—
173C22⋊C4—42no—
174C22⋊C4—42no—
175C22⋊C4—42no—
176C22⋊C4—81no—
177C42—21no—
178C42—21no—
179C42—21no—
180C42—21no—
181C42—42no—
182C42—81no—
183C23—21no—
184C23—21no—
185C23—21no—
186C23—21no—
187C23—21no—
188C23—21no—
189C23—21no—
190C23C22×D411yescharacteristic
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
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—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
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×C4AGL(1,ℤ/8)12yes—
234C2×C4C2×#4111yescharacteristic
235C2×C4C2×#4111yescharacteristic
236C2×C4H(5,𝔽2)11yescharacteristic
237C2×C42−1+411yescharacteristic
238C2×C42−1+411yescharacteristic
239C2×C42−1+411yescharacteristic
240C8—22no—
241C8—22no—
242C8—41no—
243C8—41no—
244C22—41no—
245C22—41no—
246C22—41no—
247C22—41no—
248C22—41no—
249C22—41no—
250C22—41no—
251C22#53711yescharacteristic
252C22#54711yescharacteristic
253C22#54711yescharacteristic
254C22C2×AGL(1,ℤ/8)11yescharacteristic
255C22#57511yescharacteristic
256C22#57711yescharacteristic
257C22#57811yescharacteristic
258C4—22no—
259C4—41no—
260C4—41no—
261C4—41no—
262C4—41no—
263C4—81no—
264C4—81no—
265C4—81no—
266C4—81no—
267C4—81no—
268C4—81no—
269C4—81no—
270C4—81no—
271C4#54111yescharacteristic
272C4#54111yescharacteristic
273C4#55111yescharacteristic
274C4#56311yescharacteristic
275C2—81no—
276C2#272211yescharacteristic
277C2#325111yescharacteristic
278C2#325811yescharacteristic
279C2#334511yescharacteristic
280C2#335011yescharacteristic
281C2#335211yescharacteristic
282C2#335311yescharacteristic
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
3. #2088C2semidirect
4. #2086C2semidirect
5. #2085C2semidirect
6. #2083C2semidirect
9. #1949C2semidirect
10. #1948C2semidirect
11. #1921C2semidirect
12. #1919C2semidirect

Maximal subgroups

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

subgroupindex
#24962
#24942
#20882
#20862
#20852
#20832
#20772
#20762
#19492
#19482
#19212
#19192
#18512
#18092