Finite Group Databasebeta

← #15850

Subgroups — #15850

Order 256.

729 conjugacy classes of subgroups.

The subgroup lattice has 289 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#125C211yesmaximal, characteristic
2C2×#400C211yesmaximal, characteristic
3C2×#398C211yesmaximal, characteristic
4C2×#384C211yesmaximal, characteristic
5#1445C211yesmaximal, characteristic
6#1444C212yesmaximal
7#1444C212yesmaximal
8#1441C212yesmaximal
9#1436C212yesmaximal
10#1436C211yesmaximal, characteristic
11#1436C211yesmaximal, characteristic
12C2×#125—24no—
13C2×#125—24no—
14C2×#125—24no—
15C22×C4⋊C4C2211yescharacteristic
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×C4⋊C4C2212yes—
25C22×C42C2211yescharacteristic
26C22×C42C2211yescharacteristic
27#400—24no—
28#398—24no—
29#384—24no—
30C2×#95C2212yes—
31C2×#95C2211yescharacteristic
32C2×#95C2212yes—
33C2×#95C2211yescharacteristic
34C2×#95C2212yes—
35C2×#95C2212yes—
36C2×#95C2212yes—
37C2×#95C2212yes—
38C2×#95C2212yes—
39C2×#95C2212yes—
40C2×#95C2212yes—
41C23×C4C2312yes—
42C23×C4C2312yes—
43C23×C4C2311yescharacteristic
44C23×C4C2311yescharacteristic
45C23×C4C2311yescharacteristic
46C23×C4C2311yescharacteristic
47C23×C4C2311yescharacteristic
48C23×C4C2312yes—
49C23×C4C2311yescharacteristic
50C23×C4C2311yescharacteristic
51C23×C4C2312yes—
52#125—232no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—22no—
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—24no—
69C2×C4⋊C4—24no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—24no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—24no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4D414yes—
81C2×C4⋊C4D414yes—
82C2×C4⋊C4D414yes—
83C2×C42—22no—
84C2×C42—22no—
85C2×C42—22no—
86C2×C42—22no—
87C2×C42—24no—
88#95—22no—
89#95—24no—
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#95Q818yes—
99C24C2411yescharacteristic
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
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×C4C2×D412yes—
150C22×C4C2×D412yes—
151C22×C4C2×D412yes—
152C22×C4C2×D412yes—
153C22×C4C2×D412yes—
154C22×C4C2×D412yes—
155C22×C4C2×D412yes—
156C22×C4C2×D412yes—
157C22×C4C2×D412yes—
158C22×C4C2×Q814yes—
159C22×C4C2×Q814yes—
160C22×C4C2×Q814yes—
161C22×C4#4114yes—
162C22×C4#4114yes—
163C4⋊C4—28no—
164C4⋊C4—28no—
165C4⋊C4—28no—
166C4⋊C4—28no—
167C4⋊C4—28no—
168C4⋊C4—28no—
169C4⋊C4—48no—
170C4⋊C4—48no—
171C4⋊C4—48no—
172C4⋊C4—44no—
173C42—28no—
174C42—28no—
175C23C22×D411yescharacteristic
176C23C22×D411yescharacteristic
177C23C22×D411yescharacteristic
178C23C22×Q812yes—
179C23C2×#4112yes—
180C23C2×#4112yes—
181C23H(5,𝔽2)12yes—
182C232−1+412yes—
183C232−1+412yes—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—24no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—24no—
190C2×C4—24no—
191C2×C4—24no—
192C2×C4—24no—
193C2×C4—24no—
194C2×C4—24no—
195C2×C4—24no—
196C2×C4—24no—
197C2×C4—24no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
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—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
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#12014yes—
245C2×C4#12518yescomplemented
246C2×C4C4⋊Q814yes—
247C22C2×#12011yescharacteristic
248C22#52511yescharacteristic
249C22C2×#12512yes—
250C22C2×C4⋊Q811yescharacteristic
251C22#53211yescharacteristic
252C22#53511yescharacteristic
253C22#53611yescharacteristic
254C22#53611yescharacteristic
255C22#53912yes—
256C22#54112yes—
257C22#54412yes—
258C22#54612yes—
259C22#54712yes—
260C22#54712yes—
261C22D4×Q812yes—
262C22D4×Q812yes—
263C22#55012yes—
264C22#55212yes—
265C22#55412yes—
266C22#55712yes—
267C22#56312yes—
268C4—24no—
269C4—24no—
270C4—28no—
271C4—42no—
272C4—42no—
273C4—42no—
274C4—42no—
275C4—42no—
276C4—44no—
277C4—82no—
278C4—82no—
279C2#240611yescharacteristic
280C2#243111yescharacteristic
281C2#243411yescharacteristic
282C2#247312yes—
283C2#250612yes—
284C2#255212yes—
285C2#271312yes—
286C2#272012yes—
287C2#272112yes—
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
245. C2×C4#125semidirect

Maximal subgroups

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

subgroupindex
C22×#1252
C2×#4002
C2×#3982
C2×#3842
#14452
#14442
#14412
#14362