Finite Group Databasebeta

← #14766

Subgroups — #14766

Order 256.

969 conjugacy classes of subgroups.

The subgroup lattice has 345 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#398C211yesmaximal, characteristic, complemented
2C2×#385C211yesmaximal, characteristic
3C2×#380C211yesmaximal, characteristic
4C2×#380C212yesmaximal
5C22×#95C211yesmaximal, characteristic
6#1445C211yesmaximal, characteristic, complemented
7#1444C211yesmaximal, characteristic, complemented
8#1442C212yesmaximal, complemented
9#1441C211yesmaximal, characteristic, complemented
10#1441C212yesmaximal, complemented
11C22⋊#95C212yesmaximal
12C24×C4C2211yescharacteristic
13C24×C4C2212yes—
14C22×C4⋊C4C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic
16C22×C4⋊C4C2212yes—
17C22×C4⋊C4C2212yes—
18C22×C4⋊C4C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic
20C22×C22⋊C4C2211yescharacteristic
21C22×C22⋊C4C2212yes—
22C22×C22⋊C4C2211yescharacteristic
23#398—24no—
24#385—24no—
25#380—24no—
26#380—28no—
27C2×#95—24no—
28C2×#95C2212yes—
29C2×#95C2212yes—
30C2×#95C2212yes—
31C2×#95C2212yes—
32C2×#95C2212yes—
33C2×#95C2211yescharacteristic
34C2×#95C2212yes—
35C2×#95C2211yescharacteristic
36C2×#95C2211yescharacteristic
37C2×#95C2212yes—
38C2×#95C2212yes—
39C2×#95C2211yescharacteristic
40C25C2311yescharacteristic
41C23×C4—24no—
42C23×C4—24no—
43C23×C4—24no—
44C23×C4—22no—
45C23×C4—24no—
46C23×C4—22no—
47C23×C4—24no—
48C23×C4—42no—
49C23×C4—42no—
50C23×C4—42no—
51C23×C4D414yes—
52C23×C4Q818yes—
53C23×C4C2311yescharacteristic
54C23×C4C2312yes—
55C23×C4C2312yes—
56C23×C4C2311yescharacteristic
57C23×C4C2311yescharacteristic
58C23×C4C2311yescharacteristic
59C23×C4C2311yescharacteristic
60C23×C4C2312yes—
61C23×C4C2311yescharacteristic
62C23×C4C2312yes—
63C2×C4⋊C4—24no—
64C2×C4⋊C4—24no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—24no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—24no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—42no—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—24no—
82C2×C22⋊C4—24no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—24no—
85C2×C22⋊C4—24no—
86C2×C22⋊C4D414yes—
87#95—22no—
88#95—24no—
89#95—24no—
90#95—24no—
91#95—24no—
92#95—22no—
93#95—22no—
94#95—24no—
95#95—24no—
96#95—42no—
97#95—42no—
98#95D414yes—
99C24—21no—
100C24—21no—
101C24—22no—
102C24—22no—
103C24—21no—
104C24—21no—
105C24C2×D412yes—
106C24C2×D412yes—
107C24C2×Q814yes—
108C24#4112yes—
109C24#4114yes—
110C24C2411yescharacteristic
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—216no—
118C22×C4—28no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—22no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—21no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—21no—
139C22×C4—28no—
140C22×C4—216no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4—21no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—21no—
165C22×C4—44no—
166C22×C4—48no—
167C22×C4—48no—
168C22×C4—48no—
169C22×C4—48no—
170C22×C4—48no—
171C22×C4—48no—
172C22×C4—48no—
173C22×C4C2×D412yes—
174C22×C4C2×D412yes—
175C22×C4C2×D412yes—
176C22×C4C2×D412yes—
177C22×C4C2×D412yes—
178C22×C4C2×D414yes—
179C22×C4C2×Q814yes—
180C22×C4C2×Q814yes—
181C22×C4#4112yes—
182C22×C4#4114yes—
183C4⋊C4—48no—
184C4⋊C4—48no—
185C4⋊C4—44no—
186C4⋊C4—44no—
187C4⋊C4—44no—
188C4⋊C4—44no—
189C22⋊C4—28no—
190C22⋊C4—48no—
191C22⋊C4—44no—
192C23—22no—
193C23—24no—
194C23—22no—
195C23—24no—
196C23—22no—
197C23—22no—
198C23—24no—
199C23—24no—
200C23—22no—
201C23—24no—
202C23—24no—
203C23—22no—
204C23—22no—
205C23—24no—
206C23—24no—
207C23—24no—
208C23—22no—
209C23—24no—
210C23#12114yes—
211C23C22⋊Q818yes—
212C23#12314yes—
213C23#12518yes—
214C23C4⋊Q814yes—
215C23C22×D411yescharacteristic
216C23C22×D411yescharacteristic
217C23C22×D411yescharacteristic
218C23C22×Q812yes—
219C23C2×#4111yescharacteristic
220C23C2×#4111yescharacteristic
221C23C2×#4112yes—
222C23C2×#4112yes—
223C23H(5,𝔽2)12yes—
224C23H(5,𝔽2)11yescharacteristic
225C232−1+411yescharacteristic
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—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—24no—
240C2×C4—24no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—24no—
244C2×C4—22no—
245C2×C4—24no—
246C2×C4—22no—
247C2×C4—416no—
248C2×C4—432no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4#12114yes—
287C2×C4C22⋊Q818yes—
288C22—28no—
289C22—24no—
290C22—24no—
291C22—24no—
292C22—24no—
293C22—28no—
294C22—28no—
295C22—28no—
296C22—24no—
297C22—24no—
298C22#39818yescomplemented
299C22C2×#12111yescharacteristic
300C22C2×#12111yescharacteristic
301C22C2×C22⋊Q812yes—
302C22C2×C22⋊Q812yes—
303C22C2×#12311yescharacteristic
304C22C2×#12512yes—
305C22C2×C4⋊Q811yescharacteristic
306C22#53211yescharacteristic
307C22#53312yes—
308C22#53411yescharacteristic
309C22#53711yescharacteristic
310C22#54312yes—
311C22#54412yes—
312C22D4211yescharacteristic
313C22#54612yes—
314C22#54612yes—
315C22#54711yescharacteristic
316C22#54811yescharacteristic
317C22#55112yes—
318C22#55112yes—
319C22#55212yes—
320C22#55311yescharacteristic
321C22#55312yes—
322C4—24no—
323C4—28no—
324C4—42no—
325C4—42no—
326C4—44no—
327C4—42no—
328C4—81no—
329C4—82no—
330C4—81no—
331C4—82no—
332C2—28no—
333C2C2×#39811yescharacteristic
334C2#241211yescharacteristic
335C2#241611yescharacteristic
336C2#243712yes—
337C2#243812yes—
338C2#244212yes—
339C2#249811yescharacteristic
340C2#254311yescharacteristic
341C2#256912yes—
342C2#265911yescharacteristic
343C2#285711yescharacteristic
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×#398C2semidirect
6. #1445C2semidirect
7. #1444C2semidirect
8. #1442C2semidirect
9. #1441C2semidirect
10. #1441C2semidirect
298. C22#398semidirect

Maximal subgroups

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

subgroupindex
C2×#3982
C2×#3852
C2×#3802
C22×#952
#14452
#14442
#14422
#14412
C22⋊#952