Finite Group Databasebeta

← #26036

Subgroups — #26036

Order 256.

372 conjugacy classes of subgroups.

The subgroup lattice has 301 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2551C211yesmaximal, characteristic, complemented
2#2357C211yesmaximal, characteristic
3#2067C211yesmaximal, characteristic
4#2065C211yesmaximal, characteristic
5#2023C211yesmaximal, characteristic
6#2023C211yesmaximal, characteristic
7#1988C211yesmaximal, characteristic, complemented
8#1987C211yesmaximal, characteristic, complemented
9#1949C211yesmaximal, characteristic, complemented
10#1940C211yesmaximal, characteristic, complemented
11#1921C211yesmaximal, characteristic, complemented
12#1919C211yesmaximal, characteristic, complemented
13#1836C211yesmaximal, characteristic, complemented
14#1802C211yesmaximal, characteristic
15#1762C211yesmaximal, characteristic
16C2×C4⋊Q8—21no—
17C2×C4⋊Q8C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C2×#107C2211yescharacteristic
21C2×#106C2211yescharacteristic
22C2×C4⋊C8C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#97C2211yescharacteristic
30#395—21no—
31#395—21no—
32#390C2211yescharacteristic
33#390C2211yescharacteristic
34#389—21no—
35#389C2211yescharacteristic
36#389C2211yescharacteristic
37#388—21no—
38#388—21no—
39#384C2211yescharacteristic
40#382C2211yescharacteristic
41#381—21no—
42C4×C4⋊C4C2211yescharacteristic
43C4×C4⋊C4C2211yescharacteristic
44C4×C4⋊C4C2211yescharacteristic
45C4×C4⋊C4C2211yescharacteristic
46C4×C22⋊C4—21no—
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×Q8—21no—
60C22×Q8—21no—
61C22×D4C2311yescharacteristic
62C23×C4—21no—
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C4⋊Q8—22no—
68C4⋊Q8—22no—
69C4⋊Q8—41no—
70C4⋊Q8—41no—
71C4⋊Q8—44no—
72C4×D4—22no—
73C4×D4Q814yescomplemented
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4D411yescharacteristic
81C2×C4⋊C4D411yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42C2311yescharacteristic
98C2×C42C2311yescharacteristic
99C2×C42C2311yescharacteristic
100#107—22no—
101#106—22no—
102C4⋊C8—22no—
103C4⋊C8—41no—
104C4⋊C8—41no—
105#102—22no—
106#102—22no—
107#102—41no—
108#102—41no—
109#97—22no—
110#95—21no—
111#95—21no—
112#95—21no—
113#95—21no—
114#95—21no—
115#95—21no—
116#95—21no—
117#95D411yescharacteristic
118#95D411yescharacteristic
119C24—21no—
120C2×Q8—42no—
121C2×Q8—42no—
122C2×Q8—42no—
123C2×Q8—42no—
124C2×Q8—81no—
125C2×Q8—81no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4C2×Q812yes—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—24no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—81no—
141C22×C4—81no—
142C22×C4C2×D411yescharacteristic
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
148C22×C4C2411yescharacteristic
149C2×C8—21no—
150C2×C8—21no—
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—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—21no—
164C4⋊C4—21no—
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—42no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—81no—
180C4⋊C4C2×Q812yes—
181C4⋊C4#4112yes—
182C4⋊C4#4112yes—
183C22⋊C4—24no—
184C22⋊C4—42no—
185C22⋊C4—42no—
186C22⋊C4—81no—
187C22⋊C4—81no—
188C42—21no—
189C42—21no—
190C42—21no—
191C42—21no—
192C42—42no—
193C42—81no—
194C42—81no—
195C42C2×Q812yes—
196C42#4112yes—
197C23—22no—
198C23—41no—
199C23—41no—
200C23—41no—
201C23—41no—
202C23—41no—
203C23—41no—
204C23C22×D411yescharacteristic
205Q8—84no—
206Q8—84no—
207D4—22no—
208D4—22no—
209D4—41no—
210D4—41no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
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—84no—
247C2×C4C22⋊Q812yes—
248C2×C4C22⋊Q812yes—
249C2×C4AGL(1,ℤ/8)12yes—
250C2×C4#13712yes—
251C2×C4C22×Q811yescharacteristic
252C2×C4C2×#4111yescharacteristic
253C2×C4C2×#4111yescharacteristic
254C2×C4C2×#4111yescharacteristic
255C2×C4H(5,𝔽2)11yescharacteristic
256C2×C42−1+411yescharacteristic
257C8—41no—
258C8—41no—
259C8—41no—
260C8—41no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—81no—
265C22—81no—
266C22—81no—
267C22—81no—
268C22C2×C22⋊Q811yescharacteristic
269C22#54611yescharacteristic
270C22#54811yescharacteristic
271C22C2×AGL(1,ℤ/8)11yescharacteristic
272C22C2×#13711yescharacteristic
273C22#57611yescharacteristic
274C22#57711yescharacteristic
275C4—22no—
276C4—22no—
277C4—22no—
278C4—22no—
279C4—42no—
280C4—42no—
281C4—41no—
282C4—42no—
283C4—41no—
284C4—81no—
285C4—81no—
286C4—81no—
287C4—81no—
288C4#53311yescharacteristic
289C4#54411yescharacteristic
290C4#55511yescharacteristic
291C4#55611yescharacteristic
292C2—82no—
293C2#243911yescharacteristic
294C2#307911yescharacteristic
295C2#308111yescharacteristic
296C2#331811yescharacteristic
297C2#332011yescharacteristic
298C2#337311yescharacteristic
299C2#337511yescharacteristic
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. #2551C2semidirect
7. #1988C2semidirect
8. #1987C2semidirect
9. #1949C2semidirect
10. #1940C2semidirect
11. #1921C2semidirect
12. #1919C2semidirect
13. #1836C2semidirect
73. C4×D4Q8semidirect

Maximal subgroups

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

subgroupindex
#25512
#23572
#20672
#20652
#20232
#19882
#19872
#19492
#19402
#19212
#19192
#18362
#18022
#17622