Finite Group Databasebeta

← #15849

Subgroups — #15849

Order 256.

761 conjugacy classes of subgroups.

The subgroup lattice has 293 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#125C211yesmaximal, characteristic
2C2×#395C211yesmaximal, characteristic
3C2×#395C211yesmaximal, characteristic
4C2×#391C211yesmaximal, characteristic
5#1445C211yesmaximal, characteristic
6#1444C212yesmaximal
7#1442C212yesmaximal
8#1441C212yesmaximal
9#1436C212yesmaximal
10#1436C211yesmaximal, characteristic
11#1436C211yesmaximal, characteristic
12C23×Q8C2211yescharacteristic
13C2×#125—24no—
14C2×#125—24no—
15C2×#125—24no—
16C22×C4⋊C4C2211yescharacteristic
17C22×C4⋊C4C2211yescharacteristic
18C22×C4⋊C4C2211yescharacteristic
19C22×C4⋊C4C2212yes—
20C22×C4⋊C4C2212yes—
21C22×C4⋊C4C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23C22×C4⋊C4C2211yescharacteristic
24C22×C42C2211yescharacteristic
25C22×C42C2211yescharacteristic
26#395—24no—
27#395—24no—
28#391—24no—
29C2×#95C2212yes—
30C2×#95C2211yescharacteristic
31C2×#95C2211yescharacteristic
32C2×#95C2212yes—
33C2×#95C2212yes—
34C2×#95C2212yes—
35C2×#95C2212yes—
36C2×#95C2212yes—
37C2×#95C2212yes—
38C2×#95C2212yes—
39C2×#95C2212yes—
40C2×#95C2212yes—
41C22×Q8—22no—
42C22×Q8—22no—
43C22×Q8—22no—
44C22×Q8—24no—
45C22×Q8—42no—
46C23×C4C2311yescharacteristic
47C23×C4C2311yescharacteristic
48C23×C4C2311yescharacteristic
49C23×C4C2311yescharacteristic
50C23×C4C2311yescharacteristic
51C23×C4C2312yes—
52C23×C4C2312yes—
53C23×C4C2312yes—
54C23×C4C2312yes—
55C23×C4C2311yescharacteristic
56C23×C4C2311yescharacteristic
57#125—232no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—24no—
74C2×C4⋊C4—24no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4D414yes—
80C2×C4⋊C4D414yes—
81C2×C4⋊C4D414yes—
82C2×C42—22no—
83C2×C42—22no—
84C2×C42—22no—
85C2×C42—22no—
86C2×C42—42no—
87#95—24no—
88#95—24no—
89#95—22no—
90#95—22no—
91#95—24no—
92#95—24no—
93#95—24no—
94#95—24no—
95#95—24no—
96#95—42no—
97#95—42no—
98#95Q818yescomplemented
99C24C2411yescharacteristic
100C2×Q8—28no—
101C2×Q8—48no—
102C2×Q8—48no—
103C2×Q8—44no—
104C2×Q8—44no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4C2×D412yes—
155C22×C4C2×D412yes—
156C22×C4C2×D412yes—
157C22×C4C2×D412yes—
158C22×C4C2×D412yes—
159C22×C4C2×D412yes—
160C22×C4C2×D412yes—
161C22×C4C2×D412yes—
162C22×C4C2×D412yes—
163C22×C4C2×Q814yes—
164C22×C4C2×Q814yes—
165C22×C4C2×Q814yes—
166C22×C4#4114yes—
167C22×C4#4114yes—
168C4⋊C4—28no—
169C4⋊C4—28no—
170C4⋊C4—28no—
171C4⋊C4—28no—
172C4⋊C4—28no—
173C4⋊C4—28no—
174C4⋊C4—48no—
175C4⋊C4—48no—
176C42—28no—
177C42—44no—
178C23C22×D411yescharacteristic
179C23C22×D411yescharacteristic
180C23C22×D411yescharacteristic
181C23C22×Q812yes—
182C23C2×#4112yes—
183C23C2×#4112yes—
184C23H(5,𝔽2)12yes—
185C232−1+412yes—
186C232−1+412yes—
187Q8—416no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—24no—
193C2×C4—24no—
194C2×C4—24no—
195C2×C4—24no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—24no—
204C2×C4—22no—
205C2×C4—24no—
206C2×C4—24no—
207C2×C4—24no—
208C2×C4—24no—
209C2×C4—24no—
210C2×C4—24no—
211C2×C4—24no—
212C2×C4—24no—
213C2×C4—24no—
214C2×C4—24no—
215C2×C4—24no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—24no—
219C2×C4—42no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4#12014yes—
250C2×C4C4⋊Q814yes—
251C22C2×#12011yescharacteristic
252C22#52511yescharacteristic
253C22C2×C4⋊Q811yescharacteristic
254C22#53211yescharacteristic
255C22#53312yes—
256C22#53511yescharacteristic
257C22#53611yescharacteristic
258C22#53611yescharacteristic
259C22#53912yes—
260C22#54112yes—
261C22#54412yes—
262C22#54612yes—
263C22#54612yes—
264C22#54712yes—
265C22#54812yes—
266C22D4×Q812yes—
267C22D4×Q812yes—
268C22#55212yes—
269C22#55412yes—
270C22#55712yes—
271C22#56312yes—
272C4—24no—
273C4—24no—
274C4—42no—
275C4—42no—
276C4—44no—
277C4—42no—
278C4—42no—
279C4—42no—
280C4—44no—
281C4—44no—
282C4—44no—
283C2#240411yescharacteristic
284C2#243211yescharacteristic
285C2#243411yescharacteristic
286C2#246212yes—
287C2#255012yes—
288C2#255612yes—
289C2#269012yes—
290C2#271312yes—
291C2#272012yes—
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
98. #95Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#1252
C2×#3952
C2×#3912
#14452
#14442
#14422
#14412
#14362