Finite Group Databasebeta

← #14866

Subgroups — #14866

Order 256.

767 conjugacy classes of subgroups.

The subgroup lattice has 333 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#393C211yesmaximal, characteristic
2C2×#391C211yesmaximal, characteristic, complemented
3C2×#388C211yesmaximal, characteristic
4C2×C4×C4⋊C4C211yesmaximal, characteristic, complemented
5#1446C211yesmaximal, characteristic, complemented
6#1442C211yesmaximal, characteristic, complemented
7#1441C212yesmaximal, complemented
8#1439C211yesmaximal, characteristic
9#1439C212yesmaximal
10#1437C212yesmaximal
11#1434C212yesmaximal, complemented
12C23×Q8C2211yescharacteristic
13C22×C4⋊C4C2212yes—
14C22×C4⋊C4C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic
16C22×C22⋊C4C2211yescharacteristic
17C22×C22⋊C4C2211yescharacteristic
18C22×C22⋊C4C2212yes—
19C22×C22⋊C4C2211yescharacteristic
20C22×C22⋊C4C2212yes—
21C22×C42C2212yes—
22C22×C42C2211yescharacteristic
23#393—24no—
24#391—24no—
25#388—24no—
26C4×C4⋊C4—24no—
27C2×#95C2212yes—
28C2×#95C2211yescharacteristic
29C2×#95C2212yes—
30C2×#95C2212yes—
31C2×#95C2212yes—
32C2×#95C2212yes—
33C2×#95C2211yescharacteristic
34C2×#95C2211yescharacteristic
35C2×#95C2212yes—
36C2×#95C2212yes—
37C2×#95C2212yes—
38C2×#95C2211yescharacteristic
39C25C2311yescharacteristic
40C22×Q8—22no—
41C22×Q8—22no—
42C22×Q8—24no—
43C22×Q8—22no—
44C22×Q8—42no—
45C23×C4C2311yescharacteristic
46C23×C4C2312yes—
47C23×C4C2311yescharacteristic
48C23×C4C2312yes—
49C23×C4C2311yescharacteristic
50C23×C4C2312yes—
51C23×C4C2312yes—
52C23×C4C2311yescharacteristic
53C23×C4C2311yescharacteristic
54C23×C4C2311yescharacteristic
55C2×C4⋊C4—24no—
56C2×C4⋊C4—24no—
57C2×C4⋊C4—24no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—24no—
61C2×C4⋊C4—42no—
62C2×C22⋊C4—22no—
63C2×C22⋊C4—22no—
64C2×C22⋊C4—22no—
65C2×C22⋊C4—22no—
66C2×C22⋊C4—24no—
67C2×C22⋊C4—22no—
68C2×C22⋊C4—24no—
69C2×C22⋊C4—24no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—24no—
72C2×C22⋊C4—42no—
73C2×C22⋊C4—42no—
74C2×C22⋊C4—42no—
75C2×C22⋊C4Q814yescomplemented
76C2×C42—22no—
77C2×C42—22no—
78C2×C42—24no—
79C2×C42—24no—
80C2×C42—42no—
81C2×C42D414yes—
82#95—22no—
83#95—22no—
84#95—22no—
85#95—24no—
86#95—24no—
87#95—24no—
88#95—24no—
89#95—24no—
90#95—42no—
91#95—42no—
92#95—42no—
93#95D414yes—
94C24—21no—
95C24—21no—
96C24—21no—
97C24—22no—
98C24—22no—
99C24—21no—
100C24—22no—
101C24—22no—
102C24—21no—
103C24—21no—
104C24C2×Q812yes—
105C24C2411yescharacteristic
106C2×Q8—28no—
107C2×Q8—48no—
108C2×Q8—48no—
109C2×Q8—44no—
110C2×Q8—44no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—21no—
118C22×C4—22no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—21no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—21no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
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—21no—
143C22×C4—21no—
144C22×C4—22no—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—21no—
159C22×C4C2×D414yes—
160C22×C4C2×D412yes—
161C22×C4C2×D412yes—
162C22×C4C2×D412yes—
163C22×C4C2×D412yes—
164C22×C4C2×Q812yes—
165C22×C4C2×Q812yes—
166C22×C4#4114yes—
167C22×C4#4114yes—
168C22×C4#4112yes—
169C22×C4#4112yes—
170C22×C4#4114yes—
171C4⋊C4—216no—
172C4⋊C4—44no—
173C4⋊C4—44no—
174C22⋊C4—28no—
175C22⋊C4—48no—
176C22⋊C4—48no—
177C22⋊C4—44no—
178C22⋊C4—44no—
179C42—28no—
180C42—48no—
181C23—22no—
182C23—24no—
183C23—22no—
184C23—24no—
185C23—22no—
186C23—41no—
187C23—41no—
188C23—42no—
189C23—41no—
190C23—42no—
191C23—42no—
192C23—41no—
193C23—41no—
194C23—41no—
195C23—42no—
196C23—42no—
197C23—42no—
198C23—41no—
199C23—42no—
200C23—41no—
201C23—42no—
202C23—42no—
203C23—42no—
204C23C22×D411yescharacteristic
205C23C22×D411yescharacteristic
206C23C22×Q811yescharacteristic
207C23C2×#4111yescharacteristic
208C23C2×#4111yescharacteristic
209C23C2×#4112yes—
210C23C2×#4112yes—
211C23C2×#4112yes—
212C23H(5,𝔽2)12yes—
213C23H(5,𝔽2)11yescharacteristic
214C232−1+411yescharacteristic
215Q8—416no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—24no—
219C2×C4—22no—
220C2×C4—24no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—24no—
226C2×C4—22no—
227C2×C4—24no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
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—24no—
241C2×C4—42no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4#12114yes—
274C2×C4C22⋊Q814yes—
275C2×C4#12414yes—
276C22—42no—
277C22—44no—
278C22—42no—
279C22—42no—
280C22—44no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22—81no—
285C22—82no—
286C22—82no—
287C22C2×#12111yescharacteristic
288C22C2×C22⋊Q811yescharacteristic
289C22#52511yescharacteristic
290C22#52512yes—
291C22C2×#12411yescharacteristic
292C22#52912yes—
293C22#52912yes—
294C22#53211yescharacteristic
295C22#53312yes—
296C22#53511yescharacteristic
297C22#53711yescharacteristic
298C22#53812yes—
299C22#54212yes—
300C22#54612yes—
301C22#54611yescharacteristic
302C22D4×Q811yescharacteristic
303C22#55112yes—
304C22#55211yescharacteristic
305C22#55412yes—
306C22#55411yescharacteristic
307C22#55512yes—
308C22#55612yes—
309C22#55912yes—
310C4—24no—
311C4—24no—
312C4—24no—
313C4—42no—
314C4—44no—
315C4—44no—
316C4—44no—
317C4—81no—
318C4—81no—
319C4—82no—
320C2—82no—
321C2#239611yescharacteristic
322C2#241111yescharacteristic
323C2#241711yescharacteristic
324C2#245912yes—
325C2#246512yes—
326C2#253212yes—
327C2#256311yescharacteristic
328C2#258912yes—
329C2#259211yescharacteristic
330C2#266211yescharacteristic
331C2#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
2. C2×#391C2semidirect
4. C2×C4×C4⋊C4C2semidirect
5. #1446C2semidirect
6. #1442C2semidirect
7. #1441C2semidirect
11. #1434C2semidirect
75. C2×C22⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#3932
C2×#3912
C2×#3882
C2×C4×C4⋊C42
#14462
#14422
#14412
#14392
#14372
#14342