Finite Group Databasebeta

← #24850

Subgroups — #24850

Order 256.

394 conjugacy classes of subgroups.

The subgroup lattice has 313 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#479C211yesmaximal, characteristic, complemented
2C2×#424C211yesmaximal, characteristic
3#2629C211yesmaximal, characteristic
4#2323C211yesmaximal, characteristic
5#2088C211yesmaximal, characteristic, complemented
6#2086C211yesmaximal, characteristic, complemented
7#2068C211yesmaximal, characteristic, complemented
8#2068C211yesmaximal, characteristic, complemented
9#1989C211yesmaximal, characteristic, complemented
10#1947C211yesmaximal, characteristic, complemented
11#1894C211yesmaximal, characteristic
12#1892C211yesmaximal, characteristic
13#1809C211yesmaximal, characteristic
14#1808C211yesmaximal, characteristic
15#1803C211yesmaximal, characteristic, complemented
16C22×M16C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×#125C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20C2×C4×Q8C2211yescharacteristic
21C2×#117C2211yescharacteristic
22#479—24no—
23C2×#107C2211yescharacteristic
24C2×#107C2211yescharacteristic
25C2×#107C2211yescharacteristic
26C2×#106C2211yescharacteristic
27#424—24no—
28C2×C4⋊C8C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×C4⋊C8C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36#398—21no—
37#398—21no—
38#396—21no—
39#396—21no—
40#391C2211yescharacteristic
41#389—21no—
42#389C2211yescharacteristic
43#389C2211yescharacteristic
44#389C2211yescharacteristic
45#388—21no—
46#388—21no—
47#387—21no—
48#384C2211yescharacteristic
49#380C2211yescharacteristic
50#380C2211yescharacteristic
51#380C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C23×C4C2311yescharacteristic
63C2×M16—22no—
64C2×M16—22no—
65C2×M16—41no—
66C22×C8—22no—
67C22×C8—22no—
68C22×C8—41no—
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73#125—22no—
74#125—22no—
75C22⋊Q8—22no—
76C22⋊Q8Q814yes—
77C4×Q8—22no—
78C4×Q8—22no—
79#117—22no—
80#117—22no—
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⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4D411yescharacteristic
91C2×C4⋊C4D411yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4D411yescharacteristic
104C2×C22⋊C4D411yescharacteristic
105C2×C42—21no—
106C2×C42—21no—
107C2×C42C2311yescharacteristic
108C2×C42C2311yescharacteristic
109#107—22no—
110#107—41no—
111#107—41no—
112#106—22no—
113C4⋊C8—22no—
114C4⋊C8—22no—
115C4⋊C8—22no—
116C4⋊C8D414yes—
117#103—22no—
118#103—22no—
119#103—41no—
120#103—41no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130C24C2×D411yescharacteristic
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8C2×Q812yes—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×Q812yes—
155C22×C4#4112yes—
156C22×C4C2411yescharacteristic
157M16—44no—
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—21no—
167C2×C8—21no—
168C2×C8—44no—
169C2×C8C2×D412yes—
170C2×C8C2×D412yes—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—24no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—81no—
191C4⋊C4—81no—
192C4⋊C4—81no—
193C4⋊C4—81no—
194C4⋊C4—81no—
195C4⋊C4C2×Q812yes—
196C22⋊C4—24no—
197C22⋊C4—22no—
198C22⋊C4—22no—
199C22⋊C4—81no—
200C22⋊C4—81no—
201C22⋊C4—81no—
202C42—21no—
203C42—21no—
204C42—42no—
205C42—81no—
206C42C2×D412yes—
207C42#4112yes—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23C22⋊Q812yes—
215C23C22×D411yescharacteristic
216Q8—22no—
217Q8—22no—
218Q8—41no—
219Q8—41no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
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—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4#12112yes—
261C2×C4#12112yes—
262C2×C4C22⋊Q812yes—
263C2×C4C22×D411yescharacteristic
264C2×C4C22×Q811yescharacteristic
265C2×C4C2×#4111yescharacteristic
266C2×C4C2×#4111yescharacteristic
267C2×C4H(5,𝔽2)11yescharacteristic
268C2×C42−1+411yescharacteristic
269C8—22no—
270C8—22no—
271C8—41no—
272C8—41no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22C2×#12111yescharacteristic
281C22C2×C22⋊Q811yescharacteristic
282C22#53711yescharacteristic
283C22#57611yescharacteristic
284C22#57711yescharacteristic
285C22#57811yescharacteristic
286C22#57811yescharacteristic
287C4—22no—
288C4—22no—
289C4—22no—
290C4—42no—
291C4—42no—
292C4—41no—
293C4—41no—
294C4—81no—
295C4—81no—
296C4—81no—
297C4—81no—
298C4—81no—
299C4—81no—
300C4#54611yescharacteristic
301C4D4×Q811yescharacteristic
302C4#55211yescharacteristic
303C4#55411yescharacteristic
304C2—42no—
305C2#241711yescharacteristic
306C2#306111yescharacteristic
307C2#306311yescharacteristic
308C2#308211yescharacteristic
309C2#308311yescharacteristic
310C2#325711yescharacteristic
311C2#326111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#479C2semidirect
5. #2088C2semidirect
6. #2086C2semidirect
7. #2068C2semidirect
8. #2068C2semidirect
9. #1989C2semidirect
10. #1947C2semidirect
15. #1803C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4792
C2×#4242
#26292
#23232
#20882
#20862
#20682
#19892
#19472
#18942
#18922
#18092
#18082
#18032