Finite Group Databasebeta

← #14906

Subgroups — #14906

Order 256.

893 conjugacy classes of subgroups.

The subgroup lattice has 337 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#117C211yesmaximal, characteristic
2C2×#400C211yesmaximal, characteristic, complemented
3C2×#398C211yesmaximal, characteristic, complemented
4C2×#385C211yesmaximal, characteristic
5#1445C211yesmaximal, characteristic, complemented
6#1445C211yesmaximal, characteristic, complemented
7#1440C212yesmaximal
8#1439C212yesmaximal
9#1436C212yesmaximal, complemented
10#1436C212yesmaximal, complemented
11C22⋊#95C211yesmaximal, characteristic
12C24×C4C2211yescharacteristic
13C2×#117—24no—
14C2×#117—24no—
15C2×#117—24no—
16C22×C4⋊C4C2211yescharacteristic
17C22×C4⋊C4C2211yescharacteristic
18C22×C4⋊C4C2212yes—
19C22×C4⋊C4C2211yescharacteristic
20C22×C4⋊C4C2211yescharacteristic
21C22×C22⋊C4C2212yes—
22C22×C22⋊C4C2211yescharacteristic
23C22×C22⋊C4C2211yescharacteristic
24C22×C22⋊C4C2212yes—
25C22×C42C2212yes—
26#400—24no—
27#398—24no—
28#385—24no—
29C2×#95C2211yescharacteristic
30C2×#95C2212yes—
31C2×#95C2211yescharacteristic
32C2×#95C2212yes—
33C2×#95C2212yes—
34C2×#95C2212yes—
35C2×#95C2211yescharacteristic
36C2×#95C2212yes—
37C2×#95C2212yes—
38C2×#95C2212yes—
39C2×#95C2211yescharacteristic
40C2×#95C2212yes—
41C25C2311yescharacteristic
42C23×C4—24no—
43C23×C4—22no—
44C23×C4—22no—
45C23×C4—24no—
46C23×C4D414yes—
47C23×C4C2311yescharacteristic
48C23×C4C2312yes—
49C23×C4C2311yescharacteristic
50C23×C4C2312yes—
51C23×C4C2312yes—
52C23×C4C2311yescharacteristic
53C23×C4C2312yes—
54C23×C4C2311yescharacteristic
55C23×C4C2311yescharacteristic
56C23×C4C2311yescharacteristic
57#117—232no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—24no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—24no—
68C2×C4⋊C4—24no—
69C2×C4⋊C4—42no—
70C2×C4⋊C4D414yes—
71C2×C4⋊C4D414yes—
72C2×C22⋊C4—24no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—24no—
75C2×C22⋊C4—24no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—24no—
80C2×C22⋊C4—24no—
81C2×C22⋊C4—24no—
82C2×C22⋊C4—42no—
83C2×C42—24no—
84C2×C42—24no—
85C2×C42—24no—
86#95—22no—
87#95—22no—
88#95—22no—
89#95—24no—
90#95—24no—
91#95—22no—
92#95—24no—
93#95—24no—
94#95—24no—
95#95—24no—
96#95—24no—
97#95—42no—
98C24—21no—
99C24—21no—
100C24—21no—
101C24—21no—
102C24—22no—
103C24—22no—
104C24—21no—
105C24—22no—
106C24—21no—
107C24C2×D412yes—
108C24#4114yes—
109C24C2411yescharacteristic
110C22×C4—216no—
111C22×C4—216no—
112C22×C4—28no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—28no—
119C22×C4—28no—
120C22×C4—21no—
121C22×C4—22no—
122C22×C4—21no—
123C22×C4—21no—
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—21no—
135C22×C4—21no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—21no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—21no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4C2×D414yes—
166C22×C4C2×D414yes—
167C22×C4C2×D412yes—
168C22×C4C2×D412yes—
169C22×C4C2×D412yes—
170C22×C4C2×D412yes—
171C22×C4#4112yes—
172C22×C4#4114yes—
173C22×C4#4112yes—
174C22×C4#4114yes—
175C4⋊C4—28no—
176C4⋊C4—28no—
177C4⋊C4—48no—
178C4⋊C4—44no—
179C4⋊C4—44no—
180C22⋊C4—28no—
181C22⋊C4—216no—
182C22⋊C4—48no—
183C22⋊C4—44no—
184C42—216no—
185C23—22no—
186C23—22no—
187C23—22no—
188C23—24no—
189C23—24no—
190C23—22no—
191C23—22no—
192C23—22no—
193C23—24no—
194C23—24no—
195C23—24no—
196C23—24no—
197C23—42no—
198C23—42no—
199C23—42no—
200C23—42no—
201C23—41no—
202C23—41no—
203C23—42no—
204C23—41no—
205C23—42no—
206C23—41no—
207C23#12314yes—
208C23C22×D411yescharacteristic
209C23C22×D411yescharacteristic
210C23C22×D411yescharacteristic
211C23C2×#4112yes—
212C23C2×#4112yes—
213C23C2×#4111yescharacteristic
214C23C2×#4111yescharacteristic
215C23C2×#4112yes—
216C23H(5,𝔽2)11yescharacteristic
217C232−1+412yes—
218C232−1+411yescharacteristic
219C2×C4—232no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—24no—
225C2×C4—24no—
226C2×C4—24no—
227C2×C4—24no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—24no—
233C2×C4—24no—
234C2×C4—22no—
235C2×C4—24no—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—24no—
239C2×C4—22no—
240C2×C4—24no—
241C2×C4—24no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4#12014yes—
279C2×C4#12314yes—
280C22—24no—
281C22—24no—
282C22—24no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—44no—
288C22—44no—
289C22—44no—
290C22—44no—
291C22C2×#12011yescharacteristic
292C22C2×#12311yescharacteristic
293C22C2×#12311yescharacteristic
294C22#52512yes—
295C22#52512yes—
296C22#52511yescharacteristic
297C22#52912yes—
298C22#52912yes—
299C22#53211yescharacteristic
300C22#53611yescharacteristic
301C22#53711yescharacteristic
302C22#54112yes—
303C22#54212yes—
304C22#54611yescharacteristic
305C22#54711yescharacteristic
306C22#54712yes—
307C22#54811yescharacteristic
308C22#54812yes—
309C22#55011yescharacteristic
310C22#55112yes—
311C22#55212yes—
312C22#55212yes—
313C22#55212yes—
314C4—24no—
315C4—24no—
316C4—42no—
317C4—44no—
318C4—42no—
319C4—44no—
320C4—44no—
321C4—81no—
322C4—82no—
323C4—81no—
324C2—44no—
325C2#239511yescharacteristic
326C2#240711yescharacteristic
327C2#243211yescharacteristic
328C2#247512yes—
329C2#247512yes—
330C2#247712yes—
331C2#247912yes—
332C2#248411yescharacteristic
333C2#249811yescharacteristic
334C2#262511yescharacteristic
335C2#284911yescharacteristic
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×#400C2semidirect
3. C2×#398C2semidirect
5. #1445C2semidirect
6. #1445C2semidirect
9. #1436C2semidirect
10. #1436C2semidirect

Maximal subgroups

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

subgroupindex
C22×#1172
C2×#4002
C2×#3982
C2×#3852
#14452
#14402
#14392
#14362
C22⋊#952