Finite Group Databasebeta

← #26542

Subgroups — #26542

Order 256.

376 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2552C211yesmaximal, characteristic, complemented
2#2367C211yesmaximal, characteristic
3#2095C211yesmaximal, characteristic, complemented
4#2093C211yesmaximal, characteristic
5#2068C211yesmaximal, characteristic, complemented
6#2067C211yesmaximal, characteristic
7#1983C211yesmaximal, characteristic
8#1959C211yesmaximal, characteristic
9#1947C211yesmaximal, characteristic, complemented
10#1940C211yesmaximal, characteristic, complemented
11#1865C211yesmaximal, characteristic, complemented
12#1864C211yesmaximal, characteristic
13#1803C211yesmaximal, characteristic, complemented
14#1802C211yesmaximal, characteristic
15#1776C211yesmaximal, characteristic, complemented
16C2×#125—21no—
17C2×#124C2211yescharacteristic
18C2×C4×Q8C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C2×#107C2211yescharacteristic
21C2×#107C2211yescharacteristic
22C2×#107C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26C2×C4⋊C8C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#97C2211yescharacteristic
36#398—21no—
37#398—21no—
38#391C2211yescharacteristic
39#390C2211yescharacteristic
40#389—21no—
41#389C2211yescharacteristic
42#389C2211yescharacteristic
43#388—21no—
44#388—21no—
45#385—21no—
46#384C2211yescharacteristic
47#384C2211yescharacteristic
48#384C2211yescharacteristic
49C4×C4⋊C4C2211yescharacteristic
50C4×C4⋊C4C2211yescharacteristic
51C4×C4⋊C4C2211yescharacteristic
52C4×C22⋊C4—21no—
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×D4C2311yescharacteristic
61C23×C4—21no—
62C22×C8C2311yescharacteristic
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66#125—44no—
67#124—22no—
68#124Q814yes—
69C4×Q8—22no—
70C4×Q8—41no—
71C4×D4—22no—
72C4×D4—41no—
73C2×C4⋊C4—21no—
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⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4D411yescharacteristic
83C2×C4⋊C4D411yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42—21no—
98C2×C42C2311yescharacteristic
99C2×C42C2311yescharacteristic
100C2×C42C2311yescharacteristic
101#107—22no—
102#107—41no—
103#107—41no—
104#106—22no—
105C4⋊C8—22no—
106C4⋊C8—41no—
107C4⋊C8—41no—
108#103—22no—
109#103—41no—
110#103—41no—
111#103—41no—
112#102—22no—
113#102—41no—
114#102—41no—
115#102—41no—
116#97—22no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95D411yescharacteristic
124#95D411yescharacteristic
125C24—21no—
126C2×Q8—21no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8C2×Q812yes—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4C2×Q812yes—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—81no—
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2411yescharacteristic
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—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8#4112yes—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—81no—
187C4⋊C4—81no—
188C4⋊C4—81no—
189C4⋊C4#4112yes—
190C22⋊C4—24no—
191C22⋊C4—24no—
192C22⋊C4—42no—
193C22⋊C4—42no—
194C22⋊C4—81no—
195C42—21no—
196C42—21no—
197C42—21no—
198C42—21no—
199C42—42no—
200C42—42no—
201C42—81no—
202C42C2×Q812yes—
203C42#4112yes—
204C23—22no—
205C23—41no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23—41no—
210C23—41no—
211C23C22×D411yescharacteristic
212Q8—22no—
213Q8—22no—
214Q8—41no—
215Q8—41no—
216D4—22no—
217D4—22no—
218D4—41no—
219D4—41no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
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×C4—41no—
255C2×C4—84no—
256C2×C4C22⋊Q812yes—
257C2×C4C22⋊Q812yes—
258C2×C4AGL(1,ℤ/8)12yes—
259C2×C4#13712yes—
260C2×C4C22×Q811yescharacteristic
261C2×C4C2×#4111yescharacteristic
262C2×C4C2×#4111yescharacteristic
263C2×C4C2×#4111yescharacteristic
264C2×C4H(5,𝔽2)11yescharacteristic
265C2×C4H(5,𝔽2)11yescharacteristic
266C8—22no—
267C8—41no—
268C8—41no—
269C8—41no—
270C22—42no—
271C22—42no—
272C22—42no—
273C22—81no—
274C22—81no—
275C22—81no—
276C22—81no—
277C22C2×C22⋊Q811yescharacteristic
278C22#54611yescharacteristic
279C22#55011yescharacteristic
280C22C2×AGL(1,ℤ/8)11yescharacteristic
281C22C2×#13711yescharacteristic
282C22#57611yescharacteristic
283C22#57811yescharacteristic
284C4—22no—
285C4—22no—
286C4—22no—
287C4—22no—
288C4—42no—
289C4—41no—
290C4—41no—
291C4—42no—
292C4—81no—
293C4—81no—
294C4—81no—
295C4—81no—
296C4—81no—
297C4#52911yescharacteristic
298C4#54011yescharacteristic
299C4#55411yescharacteristic
300C4#55411yescharacteristic
301C2—82no—
302C2#245411yescharacteristic
303C2#307911yescharacteristic
304C2#308311yescharacteristic
305C2#330811yescharacteristic
306C2#331411yescharacteristic
307C2#339811yescharacteristic
308C2#339911yescharacteristic
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. #2552C2semidirect
3. #2095C2semidirect
5. #2068C2semidirect
9. #1947C2semidirect
10. #1940C2semidirect
11. #1865C2semidirect
13. #1803C2semidirect
15. #1776C2semidirect

Maximal subgroups

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

subgroupindex
#25522
#23672
#20952
#20932
#20682
#20672
#19832
#19592
#19472
#19402
#18652
#18642
#18032
#18022
#17762