Finite Group Databasebeta

← #26417

Subgroups — #26417

Order 256.

369 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2441C211yesmaximal, characteristic
2#2362C211yesmaximal, characteristic
3#2100C211yesmaximal, characteristic, complemented
4#2086C211yesmaximal, characteristic, complemented
5#2068C211yesmaximal, characteristic, complemented
6#2026C211yesmaximal, characteristic, complemented
7#1989C211yesmaximal, characteristic, complemented
8#1919C211yesmaximal, characteristic, complemented
9#1892C211yesmaximal, characteristic
10#1873C211yesmaximal, characteristic
11#1865C211yesmaximal, characteristic, complemented
12#1854C211yesmaximal, characteristic
13#1850C211yesmaximal, characteristic
14#1803C211yesmaximal, characteristic, complemented
15#1735C211yesmaximal, characteristic
16C23×C8C2211yescharacteristic
17C2×#125C2211yescharacteristic
18C2×C22⋊Q8C2211yescharacteristic
19C2×C4×Q8C2211yescharacteristic
20C2×#117C2211yescharacteristic
21C2×#117C2211yescharacteristic
22C2×#107C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×C22⋊C8C2211yescharacteristic
31#399—21no—
32#396—21no—
33#391C2211yescharacteristic
34#389—21no—
35#389C2211yescharacteristic
36#389C2211yescharacteristic
37#389C2211yescharacteristic
38#388—21no—
39#388—21no—
40#387—21no—
41#387—21no—
42#384C2211yescharacteristic
43#380C2211yescharacteristic
44C4×C4⋊C4—21no—
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
59C22×Q8C2311yescharacteristic
60C23×C4C2311yescharacteristic
61C22×C8—22no—
62C22×C8—41no—
63C22×C8—41no—
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68#125—22no—
69#125—41no—
70C22⋊Q8—22no—
71C22⋊Q8Q814yes—
72C4×Q8—22no—
73C4×Q8—41no—
74#117—22no—
75#117—22no—
76#117—41no—
77#117—41no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4D411yescharacteristic
99C2×C22⋊C4D411yescharacteristic
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42C2311yescharacteristic
105C2×C42C2311yescharacteristic
106#107—41no—
107#106—22no—
108C4⋊C8—41no—
109C4⋊C8—41no—
110#103—22no—
111#103—41no—
112#103—41no—
113#103—41no—
114C22⋊C8—22no—
115#95—21no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95D411yescharacteristic
123#95D411yescharacteristic
124C24C2×D411yescharacteristic
125C2×Q8—21no—
126C2×Q8—21no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8C2×Q812yes—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
148C22×C4C2×Q812yes—
149C22×C4#4112yes—
150C22×C4C2411yescharacteristic
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—44no—
164C4⋊C4—24no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—81no—
183C4⋊C4—81no—
184C4⋊C4C2×Q812yes—
185C4⋊C4#4112yes—
186C4⋊C4#4112yes—
187C22⋊C4—24no—
188C22⋊C4—22no—
189C22⋊C4—41no—
190C22⋊C4—42no—
191C22⋊C4—42no—
192C22⋊C4—81no—
193C42—21no—
194C42—21no—
195C42—21no—
196C42—21no—
197C42—42no—
198C42—42no—
199C42—42no—
200C42—42no—
201C23—21no—
202C23—21no—
203C23—21no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23C22⋊Q812yes—
208C23C22×D411yescharacteristic
209Q8—22no—
210Q8—22no—
211Q8—41no—
212Q8—41no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4C22⋊Q812yes—
255C2×C4#13512yes—
256C2×C4C22×Q811yescharacteristic
257C2×C4C2×#4111yescharacteristic
258C2×C4C2×#4111yescharacteristic
259C2×C4C2×#4111yescharacteristic
260C2×C4H(5,𝔽2)11yescharacteristic
261C2×C4H(5,𝔽2)11yescharacteristic
262C8—41no—
263C8—41no—
264C8—41no—
265C8—41no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22C2×C22⋊Q811yescharacteristic
274C22#54611yescharacteristic
275C22#54611yescharacteristic
276C22C2×#13511yescharacteristic
277C22#57511yescharacteristic
278C22#57711yescharacteristic
279C22#57811yescharacteristic
280C4—22no—
281C4—22no—
282C4—42no—
283C4—42no—
284C4—41no—
285C4—42no—
286C4—41no—
287C4—42no—
288C4—41no—
289C4—81no—
290C4—81no—
291C4—81no—
292C4—81no—
293C4#52911yescharacteristic
294C4#53811yescharacteristic
295C4#55411yescharacteristic
296C4#55411yescharacteristic
297C2—42no—
298C2#245011yescharacteristic
299C2#307611yescharacteristic
300C2#308211yescharacteristic
301C2#330111yescharacteristic
302C2#330211yescharacteristic
303C2#331511yescharacteristic
304C2#331711yescharacteristic
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. #2100C2semidirect
4. #2086C2semidirect
5. #2068C2semidirect
6. #2026C2semidirect
7. #1989C2semidirect
8. #1919C2semidirect
11. #1865C2semidirect
14. #1803C2semidirect

Maximal subgroups

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

subgroupindex
#24412
#23622
#21002
#20862
#20682
#20262
#19892
#19192
#18922
#18732
#18652
#18542
#18502
#18032
#17352