Finite Group Databasebeta

← #14826

Subgroups — #14826

Order 256.

897 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#400C211yesmaximal, characteristic, complemented
2C2×#385C211yesmaximal, characteristic
3C2×#380C211yesmaximal, characteristic
4C22×#95C211yesmaximal, characteristic
5#1445C211yesmaximal, characteristic, complemented
6#1443C212yesmaximal, complemented
7#1442C211yesmaximal, characteristic, complemented
8#1442C212yesmaximal, complemented
9#1441C211yesmaximal, characteristic, complemented
10C22⋊#95C212yesmaximal
11C22⋊#95C212yesmaximal
12C24×C4C2212yes—
13C24×C4C2211yescharacteristic
14C22×C4⋊C4C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic
16C22×C4⋊C4C2211yescharacteristic
17C22×C4⋊C4C2211yescharacteristic
18C22×C22⋊C4C2212yes—
19C22×C22⋊C4C2211yescharacteristic
20C22×C22⋊C4C2211yescharacteristic
21#400—24no—
22#385—24no—
23#380—24no—
24C2×#95—24no—
25C2×#95C2212yes—
26C2×#95C2212yes—
27C2×#95C2212yes—
28C2×#95C2212yes—
29C2×#95C2211yescharacteristic
30C2×#95C2212yes—
31C2×#95C2212yes—
32C2×#95C2211yescharacteristic
33C2×#95C2211yescharacteristic
34C2×#95C2212yes—
35C2×#95C2212yes—
36C2×#95C2212yes—
37C2×#95C2212yes—
38C2×#95C2211yescharacteristic
39C25C2311yescharacteristic
40C23×C4—24no—
41C23×C4—24no—
42C23×C4—24no—
43C23×C4—22no—
44C23×C4—24no—
45C23×C4—22no—
46C23×C4—24no—
47C23×C4—24no—
48C23×C4—42no—
49C23×C4—42no—
50C23×C4—42no—
51C23×C4D414yes—
52C23×C4C2311yescharacteristic
53C23×C4C2312yes—
54C23×C4C2312yes—
55C23×C4C2311yescharacteristic
56C23×C4C2311yescharacteristic
57C23×C4C2312yes—
58C23×C4C2311yescharacteristic
59C23×C4C2311yescharacteristic
60C23×C4C2312yes—
61C23×C4C2311yescharacteristic
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—42no—
71C2×C4⋊C4—42no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—24no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—24no—
77C2×C22⋊C4—24no—
78C2×C22⋊C4—24no—
79C2×C22⋊C4Q814yes—
80#95—24no—
81#95—22no—
82#95—24no—
83#95—22no—
84#95—22no—
85#95—24no—
86#95—24no—
87#95—24no—
88#95—24no—
89#95—42no—
90#95—42no—
91#95—42no—
92#95—42no—
93#95D414yes—
94C24—21no—
95C24—21no—
96C24—22no—
97C24—22no—
98C24—21no—
99C24—21no—
100C24C2×D412yes—
101C24C2×Q812yes—
102C24#4112yes—
103C24#4114yes—
104C24#4114yes—
105C24C2411yescharacteristic
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—216no—
109C22×C4—28no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—22no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—22no—
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—21no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—21no—
136C22×C4—28no—
137C22×C4—21no—
138C22×C4—22no—
139C22×C4—21no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—21no—
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—21no—
154C22×C4—22no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—21no—
161C22×C4—44no—
162C22×C4—48no—
163C22×C4—48no—
164C22×C4—48no—
165C22×C4—48no—
166C22×C4—48no—
167C22×C4—48no—
168C22×C4—48no—
169C22×C4—48no—
170C22×C4C2×D414yes—
171C22×C4C2×D412yes—
172C22×C4C2×D412yes—
173C22×C4C2×D412yes—
174C22×C4C2×Q812yes—
175C22×C4C2×Q812yes—
176C22×C4#4114yes—
177C22×C4#4112yes—
178C4⋊C4—44no—
179C4⋊C4—44no—
180C4⋊C4—44no—
181C4⋊C4—44no—
182C22⋊C4—28no—
183C22⋊C4—44no—
184C22⋊C4—48no—
185C23—22no—
186C23—24no—
187C23—24no—
188C23—22no—
189C23—22no—
190C23—22no—
191C23—24no—
192C23—24no—
193C23—24no—
194C23—22no—
195C23—22no—
196C23—24no—
197C23—24no—
198C23—24no—
199C23—22no—
200C23—24no—
201C23—24no—
202C23—22no—
203C23C22⋊Q814yes—
204C23#12314yes—
205C23#12414yes—
206C23#12518yes—
207C23#12618yes—
208C23C22×D411yescharacteristic
209C23C22×D411yescharacteristic
210C23C22×Q811yescharacteristic
211C23C2×#4111yescharacteristic
212C23C2×#4111yescharacteristic
213C23C2×#4112yes—
214C23C2×#4112yes—
215C23C2×#4112yes—
216C23H(5,𝔽2)12yes—
217C23H(5,𝔽2)11yescharacteristic
218C232−1+411yescharacteristic
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—24no—
222C2×C4—24no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—24no—
228C2×C4—24no—
229C2×C4—22no—
230C2×C4—24no—
231C2×C4—24no—
232C2×C4—22no—
233C2×C4—24no—
234C2×C4—24no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—416no—
238C2×C4—432no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4#12114yes—
281C22—24no—
282C22—24no—
283C22—24no—
284C22—24no—
285C22—24no—
286C22—28no—
287C22—28no—
288C22—28no—
289C22—24no—
290C22—28no—
291C22#40018yescomplemented
292C22C2×#12111yescharacteristic
293C22C2×C22⋊Q811yescharacteristic
294C22C2×#12311yescharacteristic
295C22#52512yes—
296C22C2×#12411yescharacteristic
297C22C2×#12512yes—
298C22C2×#12612yes—
299C22#52912yes—
300C22#53211yescharacteristic
301C22#53411yescharacteristic
302C22#53711yescharacteristic
303C22#53812yes—
304C22#53912yes—
305C22#54612yes—
306C22#54611yescharacteristic
307C22D4×Q811yescharacteristic
308C22#55112yes—
309C22#55112yes—
310C22#55112yes—
311C22#55211yescharacteristic
312C22#55212yes—
313C22#55412yes—
314C22#55411yescharacteristic
315C4—24no—
316C4—42no—
317C4—42no—
318C4—42no—
319C4—44no—
320C4—44no—
321C4—82no—
322C4—82no—
323C4—81no—
324C4—81no—
325C2—28no—
326C2C2×#40011yescharacteristic
327C2#241211yescharacteristic
328C2#241711yescharacteristic
329C2#247312yes—
330C2#248012yes—
331C2#249312yes—
332C2#249811yescharacteristic
333C2#250512yes—
334C2#256411yescharacteristic
335C2#263911yescharacteristic
336C2#285011yescharacteristic
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×#400C2semidirect
5. #1445C2semidirect
6. #1443C2semidirect
7. #1442C2semidirect
8. #1442C2semidirect
9. #1441C2semidirect
291. C22#400semidirect

Maximal subgroups

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

subgroupindex
C2×#4002
C2×#3852
C2×#3802
C22×#952
#14452
#14432
#14422
#14412
C22⋊#952