Finite Group Databasebeta

← #15910

Subgroups — #15910

Order 256.

815 conjugacy classes of subgroups.

The subgroup lattice has 325 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#126C211yesmaximal, characteristic
2C2×#395C211yesmaximal, characteristic, complemented
3C2×#393C211yesmaximal, characteristic
4C2×#391C211yesmaximal, characteristic, complemented
5#1445C211yesmaximal, characteristic, complemented
6#1442C212yesmaximal, complemented
7#1440C212yesmaximal
8#1439C212yesmaximal
9#1437C211yesmaximal, characteristic
10#1436C212yesmaximal, complemented
11#1436C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C2×#126—24no—
14C2×#126—24no—
15C2×#126—24no—
16C22×C4⋊C4C2211yescharacteristic
17C22×C4⋊C4C2212yes—
18C22×C4⋊C4C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic
20C22×C22⋊C4C2211yescharacteristic
21C22×C22⋊C4C2211yescharacteristic
22C22×C22⋊C4C2212yes—
23C22×C22⋊C4C2212yes—
24C22×C22⋊C4C2211yescharacteristic
25C22×C42C2211yescharacteristic
26C22×C42C2211yescharacteristic
27#395—24no—
28#393—24no—
29#391—24no—
30C2×#95C2212yes—
31C2×#95C2211yescharacteristic
32C2×#95C2211yescharacteristic
33C2×#95C2212yes—
34C2×#95C2212yes—
35C2×#95C2212yes—
36C2×#95C2212yes—
37C2×#95C2212yes—
38C2×#95C2212yes—
39C2×#95C2212yes—
40C2×#95C2212yes—
41C25C2311yescharacteristic
42C22×Q8—22no—
43C22×Q8—22no—
44C22×Q8—22no—
45C22×Q8—24no—
46C22×Q8—42no—
47C23×C4C2312yes—
48C23×C4C2311yescharacteristic
49C23×C4C2311yescharacteristic
50C23×C4C2311yescharacteristic
51C23×C4C2311yescharacteristic
52C23×C4C2311yescharacteristic
53C23×C4C2312yes—
54C23×C4C2312yes—
55C23×C4C2312yes—
56C23×C4C2311yescharacteristic
57#126—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—24no—
66C2×C4⋊C4—24no—
67C2×C4⋊C4—42no—
68C2×C4⋊C4D414yes—
69C2×C4⋊C4D414yes—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—24no—
79C2×C22⋊C4—24no—
80C2×C22⋊C4—24no—
81C2×C22⋊C4—24no—
82C2×C22⋊C4—42no—
83C2×C22⋊C4—42no—
84C2×C22⋊C4D414yes—
85C2×C42—22no—
86C2×C42—22no—
87C2×C42—22no—
88C2×C42—22no—
89C2×C42—42no—
90#95—24no—
91#95—24no—
92#95—24no—
93#95—22no—
94#95—22no—
95#95—24no—
96#95—24no—
97#95—24no—
98#95—24no—
99#95—24no—
100#95—42no—
101C24—21no—
102C24—21no—
103C24—22no—
104C24—22no—
105C24—22no—
106C24—22no—
107C24—22no—
108C24—22no—
109C24C2×D412yes—
110C24C2411yescharacteristic
111C2×Q8—28no—
112C2×Q8—48no—
113C2×Q8—48no—
114C2×Q8—44no—
115C2×Q8—44no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
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×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4C2×D412yes—
164C22×C4C2×D412yes—
165C22×C4C2×D412yes—
166C22×C4C2×D412yes—
167C22×C4C2×D412yes—
168C22×C4C2×D412yes—
169C22×C4C2×D412yes—
170C22×C4C2×D412yes—
171C22×C4#4114yes—
172C22×C4#4114yes—
173C22×C4#4114yes—
174C4⋊C4—28no—
175C4⋊C4—28no—
176C4⋊C4—28no—
177C4⋊C4—48no—
178C22⋊C4—28no—
179C22⋊C4—28no—
180C22⋊C4—28no—
181C22⋊C4—48no—
182C22⋊C4—48no—
183C42—28no—
184C42—44no—
185C23—22no—
186C23—22no—
187C23—22no—
188C23—24no—
189C23—24no—
190C23—41no—
191C23—42no—
192C23—41no—
193C23—41no—
194C23—41no—
195C23—42no—
196C23—42no—
197C23—42no—
198C23—42no—
199C23—42no—
200C23—42no—
201C23—42no—
202C23—42no—
203C23—42no—
204C23—42no—
205C23—42no—
206C23C22×D411yescharacteristic
207C23C22×D411yescharacteristic
208C23C22×D411yescharacteristic
209C23C2×#4112yes—
210C23C2×#4112yes—
211C23C2×#4112yes—
212C23H(5,𝔽2)12yes—
213C232−1+412yes—
214C232−1+412yes—
215Q8—416no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—24no—
219C2×C4—24no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
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—24no—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—24no—
239C2×C4—24no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4#12014yes—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—44no—
278C22—44no—
279C22—82no—
280C22—82no—
281C22—82no—
282C22—82no—
283C22C2×#12011yescharacteristic
284C22#52511yescharacteristic
285C22#52912yes—
286C22#53211yescharacteristic
287C22#53211yescharacteristic
288C22#53511yescharacteristic
289C22#53611yescharacteristic
290C22#53611yescharacteristic
291C22#53912yes—
292C22#54112yes—
293C22#54212yes—
294C22#54612yes—
295C22#54612yes—
296C22#54712yes—
297C22#54712yes—
298C22#54812yes—
299C22#54812yes—
300C22#55112yes—
301C22#55612yes—
302C22#55612yes—
303C22#56312yes—
304C4—24no—
305C4—44no—
306C4—42no—
307C4—44no—
308C4—42no—
309C4—42no—
310C4—42no—
311C4—42no—
312C4—44no—
313C4—82no—
314C2—82no—
315C2#240411yescharacteristic
316C2#243211yescharacteristic
317C2#243211yescharacteristic
318C2#246912yes—
319C2#252112yes—
320C2#256012yes—
321C2#268912yes—
322C2#271612yes—
323C2#272312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C24
prime radicalC25C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#395C2semidirect
4. C2×#391C2semidirect
5. #1445C2semidirect
6. #1442C2semidirect
10. #1436C2semidirect
11. #1436C2semidirect

Maximal subgroups

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

subgroupindex
C22×#1262
C2×#3952
C2×#3932
C2×#3912
#14452
#14422
#14402
#14392
#14372
#14362