Finite Group Databasebeta

← #16821

Subgroups — #16821

Order 256.

711 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#398C211yesmaximal, characteristic, complemented
2C2×#397C211yesmaximal, characteristic
3C2×#396C212yesmaximal
4C2×#389C211yesmaximal, characteristic, complemented
5C2×#388C211yesmaximal, characteristic
6#1442C212yesmaximal, complemented
7#1441C212yesmaximal, complemented
8#1441C211yesmaximal, characteristic, complemented
9#1440C212yesmaximal
10#1439C211yesmaximal, characteristic
11C4⋊#95C211yesmaximal, characteristic, complemented
12C22×C4⋊C4C2212yes—
13C22×C4⋊C4C2211yescharacteristic
14C22×C4⋊C4C2212yes—
15C22×C4⋊C4C2212yes—
16C22×C4⋊C4C2211yescharacteristic
17C22×C4⋊C4C2211yescharacteristic
18C22×C22⋊C4C2211yescharacteristic
19C22×C22⋊C4C2212yes—
20C22×C22⋊C4C2212yes—
21C22×C22⋊C4C2211yescharacteristic
22C22×C22⋊C4C2211yescharacteristic
23C22×C42C2211yescharacteristic
24#398—24no—
25#397—24no—
26#396—28no—
27#389—24no—
28#388—24no—
29C2×#95C2212yes—
30C2×#95C2211yescharacteristic
31C2×#95C2212yes—
32C2×#95C2211yescharacteristic
33C2×#95C2212yes—
34C2×#95C2212yes—
35C2×#95C2212yes—
36C2×#95C2211yescharacteristic
37C2×#95C2211yescharacteristic
38C2×#95C2212yes—
39C2×#95C2212yes—
40C25C2311yescharacteristic
41C23×C4C2312yes—
42C23×C4C2312yes—
43C23×C4C2311yescharacteristic
44C23×C4C2311yescharacteristic
45C23×C4C2311yescharacteristic
46C23×C4C2312yes—
47C23×C4C2311yescharacteristic
48C23×C4C2312yes—
49C23×C4C2311yescharacteristic
50C23×C4C2311yescharacteristic
51C2×C4⋊C4—22no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—24no—
56C2×C4⋊C4—24no—
57C2×C4⋊C4—24no—
58C2×C4⋊C4—24no—
59C2×C4⋊C4—24no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—24no—
62C2×C4⋊C4—24no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—42no—
65C2×C4⋊C4—42no—
66C2×C4⋊C4—42no—
67C2×C4⋊C4D414yes—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—24no—
71C2×C22⋊C4—24no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—24no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—24no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—42no—
79C2×C22⋊C4—42no—
80C2×C22⋊C4—42no—
81C2×C22⋊C4D414yes—
82C2×C42—42no—
83C2×C42D414yes—
84#95—22no—
85#95—22no—
86#95—24no—
87#95—24no—
88#95—24no—
89#95—22no—
90#95—22no—
91#95—24no—
92#95—24no—
93#95—42no—
94#95—42no—
95C24—21no—
96C24—21no—
97C24—21no—
98C24—22no—
99C24—21no—
100C24—21no—
101C24—22no—
102C24—22no—
103C24—22no—
104C24—21no—
105C24C2×D412yes—
106C24C2411yescharacteristic
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—21no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—22no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—21no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—21no—
127C22×C4—22no—
128C22×C4—21no—
129C22×C4—22no—
130C22×C4—21no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—22no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—21no—
144C22×C4—22no—
145C22×C4—21no—
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×C4C2×D412yes—
158C22×C4C2×D414yes—
159C22×C4C2×D412yes—
160C22×C4C2×D412yes—
161C22×C4C2×D412yes—
162C22×C4C2×D412yes—
163C22×C4C2×D412yes—
164C22×C4#4114yes—
165C22×C4#4112yes—
166C22×C4#4112yes—
167C4⋊C4—28no—
168C4⋊C4—48no—
169C4⋊C4—48no—
170C4⋊C4—44no—
171C4⋊C4—48no—
172C4⋊C4—44no—
173C22⋊C4—28no—
174C22⋊C4—48no—
175C22⋊C4—48no—
176C22⋊C4—44no—
177C22⋊C4—44no—
178C42—44no—
179C23—22no—
180C23—24no—
181C23—22no—
182C23—24no—
183C23—22no—
184C23—41no—
185C23—41no—
186C23—41no—
187C23—41no—
188C23—41no—
189C23—42no—
190C23—41no—
191C23—42no—
192C23—42no—
193C23—42no—
194C23—42no—
195C23—41no—
196C23—42no—
197C23—42no—
198C23—41no—
199C23—42no—
200C23—42no—
201C23—42no—
202C23C22×D411yescharacteristic
203C23C22×D411yescharacteristic
204C23C22×D411yescharacteristic
205C23C2×#4111yescharacteristic
206C23C2×#4111yescharacteristic
207C23C2×#4112yes—
208C23H(5,𝔽2)12yes—
209C23H(5,𝔽2)11yescharacteristic
210C232−1+412yes—
211C232−1+412yes—
212C232−1+411yescharacteristic
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—24no—
218C2×C4—24no—
219C2×C4—24no—
220C2×C4—24no—
221C2×C4—22no—
222C2×C4—24no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—24no—
227C2×C4—24no—
228C2×C4—22no—
229C2×C4—24no—
230C2×C4—22no—
231C2×C4—24no—
232C2×C4—24no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—41no—
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#12114yes—
272C2×C4#12114yes—
273C22—42no—
274C22—44no—
275C22—44no—
276C22—42no—
277C22—42no—
278C22—81no—
279C22—81no—
280C22—82no—
281C22—82no—
282C22—81no—
283C22—81no—
284C22C2×#12111yescharacteristic
285C22C2×#12111yescharacteristic
286C22#53211yescharacteristic
287C22#53411yescharacteristic
288C22#53611yescharacteristic
289C22#53612yes—
290C22#53611yescharacteristic
291C22#53712yes—
292C22#53711yescharacteristic
293C22#53912yes—
294C22#53912yes—
295C22#54112yes—
296C22#54112yes—
297C22D4211yescharacteristic
298C22#54612yes—
299C22#54711yescharacteristic
300C22#54711yescharacteristic
301C22#54812yes—
302C22#55312yes—
303C22#55511yescharacteristic
304C22#55612yes—
305C22#56212yes—
306C22#56312yes—
307C4—24no—
308C4—24no—
309C4—44no—
310C4—42no—
311C4—44no—
312C4—42no—
313C4—81no—
314C4—82no—
315C4—82no—
316C4—81no—
317C2—82no—
318C2#253711yescharacteristic
319C2#254311yescharacteristic
320C2#254511yescharacteristic
321C2#261811yescharacteristic
322C2#262111yescharacteristic
323C2#266411yescharacteristic
324C2#268112yes—
325C2#268912yes—
326C2#272512yes—
327C2#278312yes—
328C2#281411yescharacteristic
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
4. C2×#389C2semidirect
6. #1442C2semidirect
7. #1441C2semidirect
8. #1441C2semidirect
11. C4⋊#95C2semidirect

Maximal subgroups

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

subgroupindex
C2×#3982
C2×#3972
C2×#3962
C2×#3892
C2×#3882
#14422
#14412
#14402
#14392
C4⋊#952