Finite Group Databasebeta

← #15071

Subgroups — #15071

Order 256.

739 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#125C211yesmaximal, characteristic
2C2×#387C212yesmaximal
3C2×C4×C4⋊C4C211yesmaximal, characteristic
4#1444C211yesmaximal, characteristic
5#1444C212yesmaximal
6#1442C211yesmaximal, characteristic
7#1442C212yesmaximal
8#1441C211yesmaximal, characteristic
9#1441C211yesmaximal, characteristic
10#1436C212yesmaximal
11#1434C211yesmaximal, characteristic
12C2×#125—28no—
13C2×#125—24no—
14C22×C4⋊C4C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic
16C22×C4⋊C4C2212yes—
17C22×C4⋊C4C2211yescharacteristic
18C22×C4⋊C4C2212yes—
19C22×C4⋊C4C2212yes—
20C22×C4⋊C4C2211yescharacteristic
21C22×C42C2211yescharacteristic
22C22×C42C2211yescharacteristic
23C22×C42C2211yescharacteristic
24#387—28no—
25C4×C4⋊C4—24no—
26C2×#95C2212yes—
27C2×#95C2211yescharacteristic
28C2×#95C2211yescharacteristic
29C2×#95C2212yes—
30C2×#95C2212yes—
31C2×#95C2212yes—
32C2×#95C2212yes—
33C2×#95C2212yes—
34C2×#95C2212yes—
35C2×#95C2212yes—
36C2×#95C2211yescharacteristic
37C2×#95C2211yescharacteristic
38C2×#95C2212yes—
39C23×C4C2312yes—
40C23×C4C2312yes—
41C23×C4C2312yes—
42C23×C4C2311yescharacteristic
43C23×C4C2311yescharacteristic
44C23×C4C2311yescharacteristic
45C23×C4C2312yes—
46C23×C4C2311yescharacteristic
47C23×C4C2311yescharacteristic
48C23×C4C2311yescharacteristic
49C23×C4C2311yescharacteristic
50#125—232no—
51C2×C4⋊C4—24no—
52C2×C4⋊C4—24no—
53C2×C4⋊C4—24no—
54C2×C4⋊C4—24no—
55C2×C4⋊C4—24no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—24no—
59C2×C4⋊C4—24no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—24no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—24no—
65C2×C4⋊C4—24no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4D418yes—
69C2×C4⋊C4Q814yes—
70C2×C42—24no—
71C2×C42—22no—
72C2×C42—22no—
73C2×C42—22no—
74C2×C42—22no—
75C2×C42—24no—
76C2×C42D414yes—
77#95—22no—
78#95—22no—
79#95—24no—
80#95—24no—
81#95—24no—
82#95—24no—
83#95—24no—
84#95—22no—
85#95—24no—
86#95—24no—
87#95—24no—
88#95—24no—
89#95—22no—
90C24C2411yescharacteristic
91C22×C4—22no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—21no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—22no—
106C22×C4—22no—
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—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—22no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—21no—
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×C4C2×D414yes—
142C22×C4C2×D412yes—
143C22×C4C2×D414yes—
144C22×C4C2×D414yes—
145C22×C4C2×D412yes—
146C22×C4C2×D412yes—
147C22×C4C2×Q812yes—
148C22×C4C2×Q812yes—
149C22×C4C2×Q812yes—
150C22×C4#4112yes—
151C22×C4#4114yes—
152C22×C4#4112yes—
153C22×C4#4112yes—
154C22×C4#4112yes—
155C22×C4#4112yes—
156C4⋊C4—28no—
157C4⋊C4—216no—
158C4⋊C4—28no—
159C4⋊C4—28no—
160C4⋊C4—216no—
161C4⋊C4—216no—
162C4⋊C4—28no—
163C42—28no—
164C42—28no—
165C42—44no—
166C23C22×D411yescharacteristic
167C23C22×D412yes—
168C23C22×Q811yescharacteristic
169C23C2×#4111yescharacteristic
170C23C2×#4111yescharacteristic
171C23C2×#4111yescharacteristic
172C23C2×#4111yescharacteristic
173C23C2×#4111yescharacteristic
174C23C2×#4112yes—
175C232−1+412yes—
176C232−1+412yes—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—24no—
189C2×C4—24no—
190C2×C4—24no—
191C2×C4—22no—
192C2×C4—24no—
193C2×C4—24no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—24no—
198C2×C4—22no—
199C2×C4—24no—
200C2×C4—24no—
201C2×C4—22no—
202C2×C4—24no—
203C2×C4—24no—
204C2×C4—24no—
205C2×C4—24no—
206C2×C4—24no—
207C2×C4—24no—
208C2×C4—24no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—41no—
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#12014yes—
240C2×C4C22⋊Q814yes—
241C2×C4#12514yescomplemented
242C22C2×#12011yescharacteristic
243C22C2×C22⋊Q811yescharacteristic
244C22#52511yescharacteristic
245C22#52511yescharacteristic
246C22#52512yes—
247C22C2×#12511yescharacteristic
248C22#52911yescharacteristic
249C22#52911yescharacteristic
250C22#53211yescharacteristic
251C22#53212yes—
252C22#53311yescharacteristic
253C22#53611yescharacteristic
254C22#53611yescharacteristic
255C22#54112yes—
256C22#54112yes—
257C22#54712yes—
258C22#54712yes—
259C22#54812yes—
260C22D4×Q812yes—
261C22#55212yes—
262C22#55212yes—
263C22#55212yes—
264C22#55712yes—
265C4—24no—
266C4—24no—
267C4—24no—
268C4—42no—
269C4—42no—
270C4—44no—
271C4—42no—
272C4—44no—
273C4—42no—
274C4—44no—
275C4—82no—
276C2#238811yescharacteristic
277C2#240611yescharacteristic
278C2#240911yescharacteristic
279C2#246212yes—
280C2#247512yes—
281C2#250612yes—
282C2#251912yes—
283C2#256511yescharacteristic
284C2#261111yescharacteristic
285C2#263411yescharacteristic
286C2#285111yescharacteristic
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
241. 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×#3872
C2×C4×C4⋊C42
#14442
#14422
#14412
#14362
#14342