Finite Group Databasebeta

← #15047

Subgroups — #15047

Order 256.

681 conjugacy classes of subgroups.

The subgroup lattice has 323 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#389C211yesmaximal, characteristic, complemented
2C2×#388C211yesmaximal, characteristic
3#1442C212yesmaximal, complemented
4#1441C211yesmaximal, characteristic, complemented
5#1441C211yesmaximal, characteristic, complemented
6#1439C211yesmaximal, characteristic
7#1439C211yesmaximal, characteristic
8#1439C212yesmaximal
9#1437C212yesmaximal
10#1434C212yesmaximal, complemented
11C4×#95C211yesmaximal, characteristic, complemented
12C22×C4⋊C4C2211yescharacteristic
13C22×C4⋊C4C2211yescharacteristic
14C22×C4⋊C4C2211yescharacteristic
15C22×C4⋊C4C2212yes—
16C22×C22⋊C4C2211yescharacteristic
17C22×C22⋊C4C2212yes—
18C22×C22⋊C4C2211yescharacteristic
19C22×C22⋊C4C2212yes—
20C22×C22⋊C4C2211yescharacteristic
21C22×C42C2212yes—
22C22×C42C2211yescharacteristic
23#389—24no—
24#388—24no—
25C2×#95C2211yescharacteristic
26C2×#95C2211yescharacteristic
27C2×#95C2212yes—
28C2×#95C2211yescharacteristic
29C2×#95C2212yes—
30C2×#95C2212yes—
31C2×#95C2212yes—
32C2×#95C2212yes—
33C2×#95C2211yescharacteristic
34C2×#95C2212yes—
35C2×#95C2212yes—
36C2×#95C2212yes—
37C25C2311yescharacteristic
38C23×C4C2312yes—
39C23×C4C2311yescharacteristic
40C23×C4C2311yescharacteristic
41C23×C4C2311yescharacteristic
42C23×C4C2311yescharacteristic
43C23×C4C2312yes—
44C23×C4C2312yes—
45C23×C4C2312yes—
46C23×C4C2311yescharacteristic
47C23×C4C2311yescharacteristic
48C2×C4⋊C4—22no—
49C2×C4⋊C4—24no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4—24no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—42no—
56C2×C4⋊C4—42no—
57C2×C4⋊C4—42no—
58C2×C22⋊C4—22no—
59C2×C22⋊C4—22no—
60C2×C22⋊C4—22no—
61C2×C22⋊C4—22no—
62C2×C22⋊C4—24no—
63C2×C22⋊C4—24no—
64C2×C22⋊C4—24no—
65C2×C22⋊C4—22no—
66C2×C22⋊C4—22no—
67C2×C22⋊C4—42no—
68C2×C22⋊C4—42no—
69C2×C22⋊C4—42no—
70C2×C22⋊C4—42no—
71C2×C22⋊C4Q814yes—
72C2×C42—24no—
73C2×C42—24no—
74C2×C42—42no—
75C2×C42—42no—
76C2×C42D414yes—
77#95—22no—
78#95—22no—
79#95—24no—
80#95—24no—
81#95—22no—
82#95—22no—
83#95—24no—
84#95—24no—
85#95—24no—
86#95—42no—
87#95—42no—
88#95—42no—
89C24—21no—
90C24—21no—
91C24—22no—
92C24—22no—
93C24—21no—
94C24—21no—
95C24—22no—
96C24—22no—
97C24—21no—
98C24—21no—
99C24C2×Q812yes—
100C24C2411yescharacteristic
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—21no—
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—21no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—22no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4C2×D412yes—
151C22×C4C2×D412yes—
152C22×C4C2×D412yes—
153C22×C4C2×Q812yes—
154C22×C4C2×Q812yes—
155C22×C4#4114yes—
156C22×C4#4114yes—
157C22×C4#4114yes—
158C22×C4#4112yes—
159C22×C4#4112yes—
160C22×C4#4112yes—
161C4⋊C4—28no—
162C4⋊C4—44no—
163C4⋊C4—48no—
164C4⋊C4—44no—
165C22⋊C4—28no—
166C22⋊C4—48no—
167C22⋊C4—44no—
168C22⋊C4—48no—
169C22⋊C4—44no—
170C42—44no—
171C42—48no—
172C23—22no—
173C23—24no—
174C23—22no—
175C23—22no—
176C23—24no—
177C23—41no—
178C23—41no—
179C23—42no—
180C23—42no—
181C23—42no—
182C23—41no—
183C23—41no—
184C23—41no—
185C23—41no—
186C23—42no—
187C23—42no—
188C23—42no—
189C23—42no—
190C23—42no—
191C23—42no—
192C23—42no—
193C23—41no—
194C23—41no—
195C23C22×D411yescharacteristic
196C23C22×Q811yescharacteristic
197C23C2×#4111yescharacteristic
198C23C2×#4112yes—
199C23C2×#4111yescharacteristic
200C23C2×#4111yescharacteristic
201C23C2×#4112yes—
202C23C2×#4112yes—
203C23H(5,𝔽2)12yes—
204C23H(5,𝔽2)11yescharacteristic
205C23H(5,𝔽2)11yescharacteristic
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—24no—
211C2×C4—24no—
212C2×C4—24no—
213C2×C4—24no—
214C2×C4—22no—
215C2×C4—24no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—24no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—24no—
223C2×C4—24no—
224C2×C4—24no—
225C2×C4—24no—
226C2×C4—24no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—41no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—41no—
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—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—41no—
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×C4C22⋊Q814yes—
265C2×C4#12414yes—
266C22—42no—
267C22—42no—
268C22—44no—
269C22—42no—
270C22—44no—
271C22—81no—
272C22—81no—
273C22—81no—
274C22—81no—
275C22—82no—
276C22—82no—
277C22C2×C22⋊Q811yescharacteristic
278C22#52511yescharacteristic
279C22C2×#12411yescharacteristic
280C22#52912yes—
281C22#52912yes—
282C22#52912yes—
283C22#53211yescharacteristic
284C22#53211yescharacteristic
285C22#53312yes—
286C22#53411yescharacteristic
287C22#53511yescharacteristic
288C22#53812yes—
289C22#54012yes—
290C22#55112yes—
291C22#55112yes—
292C22#55312yes—
293C22#55312yes—
294C22#55312yes—
295C22#55411yescharacteristic
296C22#55411yescharacteristic
297C22#55412yes—
298C22#55511yescharacteristic
299C22#55511yescharacteristic
300C4—24no—
301C4—24no—
302C4—42no—
303C4—44no—
304C4—42no—
305C4—44no—
306C4—44no—
307C4—81no—
308C4—81no—
309C4—82no—
310C2—82no—
311C2#239611yescharacteristic
312C2#251812yes—
313C2#252212yes—
314C2#252612yes—
315C2#256311yescharacteristic
316C2#256411yescharacteristic
317C2#258712yes—
318C2#259111yescharacteristic
319C2#260311yescharacteristic
320C2#266111yescharacteristic
321C2#286111yescharacteristic
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
1. C2×#389C2semidirect
3. #1442C2semidirect
4. #1441C2semidirect
5. #1441C2semidirect
10. #1434C2semidirect
11. C4×#95C2semidirect

Maximal subgroups

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

subgroupindex
C2×#3892
C2×#3882
#14422
#14412
#14392
#14372
#14342
C4×#952