Finite Group Databasebeta

← #35013

Subgroups — #35013

Order 256.

845 conjugacy classes of subgroups.

The subgroup lattice has 285 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2855C212yesmaximal
2#2851C212yesmaximal
3#2659C211yesmaximal, characteristic
4#2659C211yesmaximal, characteristic
5#2627C212yesmaximal
6#2586C214yesmaximal, complemented
7#2585C214yesmaximal, complemented
8#2520C214yesmaximal, complemented
9#2462C214yesmaximal, complemented
10#2329C212yesmaximal
11#2313C212yesmaximal
12C4×C22⋊Q8C212yesmaximal
13C2×C4×C4⋊C4C211yesmaximal, characteristic
14C2×#125C2212yes—
15C2×C22⋊Q8C2212yes—
16C2×C22⋊Q8C2212yes—
17C2×C4×Q8C2212yes—
18C22×C4⋊C4C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic
20C22×C42C2211yescharacteristic
21C22×C42C2211yescharacteristic
22C22×C42C2211yescharacteristic
23#400C2214yes—
24#398C2214yes—
25#398C2214yes—
26#398C2214yes—
27#398C2214yes—
28#395C2214yes—
29#395C2214yes—
30#395C2214yes—
31#391C2214yes—
32#391C2212yes—
33#391C2212yes—
34#391C2212yes—
35#389C2214yes—
36#389C2212yes—
37#389C2212yes—
38#389C2212yes—
39#389C2212yes—
40#389C2212yes—
41#387C2214yes—
42#387C2214yes—
43#387C2214yes—
44#387C2214yes—
45#387C2214yes—
46#387C2214yes—
47#387C2214yes—
48#387C2214yes—
49#387C2214yes—
50#387C2214yes—
51#387C2214yes—
52#385C2214yes—
53#385C2214yes—
54#382C2212yes—
55#381C2214yes—
56#380C2212yes—
57#380C2212yes—
58#380C2212yes—
59#380C2212yes—
60C4×C4⋊C4C2214yes—
61C4×C4⋊C4C2214yes—
62C4×C4⋊C4C2214yes—
63C4×C4⋊C4C2212yes—
64C4×C22⋊C4C2214yes—
65C2×#95C2211yescharacteristic
66C2×#95C2211yescharacteristic
67C22×Q8C2312yes—
68C22×Q8C2312yes—
69C23×C4C2311yescharacteristic
70C23×C4C2311yescharacteristic
71C23×C4C2311yescharacteristic
72C23×C4C2311yescharacteristic
73C23×C4C2311yescharacteristic
74C23×C4C2311yescharacteristic
75C23×C4C2311yescharacteristic
76#125—44no—
77C22⋊Q8—28no—
78C22⋊Q8—28no—
79C4×Q8—44no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4Q818yescomplemented
82C2×C4⋊C4C2314yes—
83C2×C4⋊C4C2314yes—
84C2×C4⋊C4C2314yes—
85C2×C4⋊C4C2312yes—
86C2×C4⋊C4C2314yes—
87C2×C4⋊C4C2312yes—
88C2×C4⋊C4C2314yes—
89C2×C4⋊C4C2312yes—
90C2×C4⋊C4C2312yes—
91C2×C4⋊C4C2312yes—
92C2×C4⋊C4C2314yes—
93C2×C4⋊C4C2312yes—
94C2×C4⋊C4C2312yes—
95C2×C4⋊C4C2312yes—
96C2×C4⋊C4C2312yes—
97C2×C4⋊C4C2312yes—
98C2×C4⋊C4C2314yes—
99C2×C4⋊C4C2314yes—
100C2×C22⋊C4C2314yes—
101C2×C22⋊C4C2314yes—
102C2×C42—24no—
103C2×C42—42no—
104C2×C42—42no—
105C2×C42C2312yes—
106C2×C42C2312yes—
107C2×C42C2312yes—
108C2×C42C2312yes—
109C2×C42C2314yes—
110C2×C42C2314yes—
111C2×C42C2312yes—
112C2×C42C2312yes—
113#95C2314yes—
114#95C2314yes—
115#95C2314yes—
116#95C2314yes—
117#95C2314yes—
118#95C2314yes—
119#95C2314yes—
120#95C2314yes—
121#95C2314yes—
122#95C2314yes—
123#95C2314yes—
124#95C2314yes—
125#95C2314yes—
126#95C2314yes—
127#95C2314yes—
128#95C2314yes—
129C24C2411yescharacteristic
130C2×Q8—24no—
131C2×Q8—24no—
132C2×Q8—42no—
133C2×Q8—42no—
134C2×Q8—42no—
135C2×Q8—42no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4C2×Q814yes—
153C22×C4C2×Q814yes—
154C22×C4C2×Q814yes—
155C22×C4#4114yes—
156C22×C4#4114yes—
157C22×C4C2414yes—
158C22×C4C2414yes—
159C22×C4C2412yes—
160C22×C4C2414yes—
161C22×C4C2414yes—
162C22×C4C2412yes—
163C22×C4C2412yes—
164C22×C4C2412yes—
165C22×C4C2412yes—
166C22×C4C2412yes—
167C22×C4C2411yescharacteristic
168C22×C4C2411yescharacteristic
169C4⋊C4—28no—
170C4⋊C4—24no—
171C4⋊C4—24no—
172C4⋊C4—28no—
173C4⋊C4—28no—
174C4⋊C4—28no—
175C4⋊C4—28no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—44no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—44no—
186C4⋊C4C2×Q8116yes—
187C22⋊C4—28no—
188C22⋊C4—28no—
189C42—24no—
190C42—24no—
191C42—24no—
192C42—24no—
193C42—42no—
194C42—42no—
195C42—44no—
196C42—44no—
197C23—21no—
198C23—21no—
199C23—21no—
200C23—21no—
201C23C22×Q812yes—
202C23C2×#4112yes—
203C23C2×#4112yes—
204C23C2511yescharacteristic
205Q8—48no—
206Q8—84no—
207C2×C4—28no—
208C2×C4—28no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—24no—
224C2×C4—24no—
225C2×C4—24no—
226C2×C4—24no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—24no—
230C2×C4—24no—
231C2×C4—24no—
232C2×C4—24no—
233C2×C4—24no—
234C2×C4—24no—
235C2×C4—22no—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4C22×Q814yes—
244C2×C4C22×Q814yes—
245C2×C4C22×Q814yes—
246C2×C4C2×#4114yes—
247C2×C4C2×#4114yes—
248C2×C4H(5,𝔽2)12yes—
249C2×C42−1+412yes—
250C22—22no—
251C22—22no—
252C22—22no—
253C22—22no—
254C22—22no—
255C22—22no—
256C22#53314yes—
257C22C23×Q811yescharacteristic
258C22C22×#4111yescharacteristic
259C22C22×#4111yescharacteristic
260C22C2×H(5,𝔽2)11yescharacteristic
261C22C2×2−1+411yescharacteristic
262C22#58511yescharacteristic
263C22#58511yescharacteristic
264C4—24no—
265C4—24no—
266C4—24no—
267C4—22no—
268C4—22no—
269C4—24no—
270C4—24no—
271C4—42no—
272C4—44no—
273C4—44no—
274C4—44no—
275C4—44no—
276C2—24no—
277C2C2×#53311yescharacteristic
278C2#345911yescharacteristic
279C2#346111yescharacteristic
280C2#349111yescharacteristic
281C2#349211yescharacteristic
282C2#350111yescharacteristic
283C2#350211yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C25
prime radicalC24C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #2586C2semidirect
7. #2585C2semidirect
8. #2520C2semidirect
9. #2462C2semidirect
81. C2×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#28552
#28512
#26592
#26272
#25862
#25852
#25202
#24622
#23292
#23132
C4×C22⋊Q82
C2×C4×C4⋊C42