Finite Group Databasebeta

← #15175

Subgroups — #15175

Order 256.

709 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#400C211yesmaximal, characteristic, complemented
2C2×#399C211yesmaximal, characteristic
3C2×#388C211yesmaximal, characteristic
4C2×C4×C4⋊C4C211yesmaximal, characteristic, complemented
5#1442C212yesmaximal, complemented
6#1441C211yesmaximal, characteristic, complemented
7#1440C212yesmaximal
8#1439C211yesmaximal, characteristic
9#1437C212yesmaximal
10#1436C212yesmaximal, complemented
11#1434C211yesmaximal, characteristic, complemented
12C22×C4⋊C4C2211yescharacteristic
13C22×C4⋊C4C2211yescharacteristic
14C22×C4⋊C4C2212yes—
15C22×C4⋊C4C2211yescharacteristic
16C22×C22⋊C4C2211yescharacteristic
17C22×C22⋊C4C2211yescharacteristic
18C22×C22⋊C4C2211yescharacteristic
19C22×C22⋊C4C2212yes—
20C22×C22⋊C4C2212yes—
21C22×C42C2211yescharacteristic
22C22×C42C2212yes—
23#400—24no—
24#399—24no—
25#388—24no—
26C4×C4⋊C4—24no—
27C2×#95C2211yescharacteristic
28C2×#95C2211yescharacteristic
29C2×#95C2212yes—
30C2×#95C2212yes—
31C2×#95C2212yes—
32C2×#95C2212yes—
33C2×#95C2211yescharacteristic
34C2×#95C2212yes—
35C2×#95C2212yes—
36C2×#95C2212yes—
37C2×#95C2211yescharacteristic
38C2×#95C2212yes—
39C25C2311yescharacteristic
40C23×C4C2311yescharacteristic
41C23×C4C2311yescharacteristic
42C23×C4C2312yes—
43C23×C4C2312yes—
44C23×C4C2311yescharacteristic
45C23×C4C2311yescharacteristic
46C23×C4C2312yes—
47C23×C4C2312yes—
48C23×C4C2311yescharacteristic
49C23×C4C2311yescharacteristic
50C2×C4⋊C4—22no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4—24no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—24no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—24no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—42no—
60C2×C4⋊C4D414yes—
61C2×C22⋊C4—22no—
62C2×C22⋊C4—22no—
63C2×C22⋊C4—22no—
64C2×C22⋊C4—24no—
65C2×C22⋊C4—22no—
66C2×C22⋊C4—22no—
67C2×C22⋊C4—24no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—24no—
70C2×C22⋊C4—24no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—24no—
73C2×C22⋊C4—42no—
74C2×C22⋊C4—42no—
75C2×C42—24no—
76C2×C42—24no—
77C2×C42—42no—
78C2×C42—42no—
79C2×C42D414yes—
80#95—24no—
81#95—22no—
82#95—22no—
83#95—22no—
84#95—24no—
85#95—22no—
86#95—24no—
87#95—24no—
88#95—24no—
89#95—24no—
90#95—24no—
91#95—42no—
92C24—21no—
93C24—21no—
94C24—21no—
95C24—22no—
96C24—22no—
97C24—22no—
98C24—21no—
99C24—22no—
100C24—21no—
101C24—21no—
102C24#4112yes—
103C24C2411yescharacteristic
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—21no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
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—21no—
135C22×C4—21no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—22no—
143C22×C4—22no—
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—22no—
154C22×C4C2×D412yes—
155C22×C4C2×D414yes—
156C22×C4C2×D412yes—
157C22×C4C2×D412yes—
158C22×C4C2×D412yes—
159C22×C4#4112yes—
160C22×C4#4114yes—
161C22×C4#4114yes—
162C22×C4#4112yes—
163C22×C4#4114yes—
164C4⋊C4—28no—
165C4⋊C4—28no—
166C4⋊C4—48no—
167C4⋊C4—44no—
168C22⋊C4—28no—
169C22⋊C4—48no—
170C22⋊C4—48no—
171C22⋊C4—44no—
172C22⋊C4—44no—
173C42—28no—
174C42—48no—
175C23—22no—
176C23—24no—
177C23—22no—
178C23—24no—
179C23—22no—
180C23—41no—
181C23—41no—
182C23—41no—
183C23—42no—
184C23—42no—
185C23—42no—
186C23—41no—
187C23—41no—
188C23—41no—
189C23—42no—
190C23—42no—
191C23—42no—
192C23—41no—
193C23—42no—
194C23—41no—
195C23—42no—
196C23—42no—
197C23—42no—
198C23C22×D411yescharacteristic
199C23C22×D411yescharacteristic
200C23C2×#4111yescharacteristic
201C23C2×#4112yes—
202C23C2×#4111yescharacteristic
203C23C2×#4111yescharacteristic
204C23C2×#4112yes—
205C23C2×#4112yes—
206C23H(5,𝔽2)12yes—
207C232−1+411yescharacteristic
208C232−1+411yescharacteristic
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—24no—
212C2×C4—24no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—24no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—24no—
222C2×C4—24no—
223C2×C4—22no—
224C2×C4—24no—
225C2×C4—24no—
226C2×C4—24no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—22no—
230C2×C4—24no—
231C2×C4—24no—
232C2×C4—41no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
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#12114yes—
268C2×C4#12314yes—
269C22—44no—
270C22—44no—
271C22—42no—
272C22—42no—
273C22—42no—
274C22—82no—
275C22—81no—
276C22—81no—
277C22—81no—
278C22—81no—
279C22—82no—
280C22C2×#12111yescharacteristic
281C22C2×#12311yescharacteristic
282C22#52511yescharacteristic
283C22#52511yescharacteristic
284C22#52912yes—
285C22#52912yes—
286C22#52912yes—
287C22#53212yes—
288C22#53211yescharacteristic
289C22#53511yescharacteristic
290C22#53611yescharacteristic
291C22#53912yes—
292C22#53912yes—
293C22#54612yes—
294C22#54811yescharacteristic
295C22#54811yescharacteristic
296C22#55112yes—
297C22#55211yescharacteristic
298C22#55212yes—
299C22#55211yescharacteristic
300C22#55312yes—
301C22#55612yes—
302C22#55912yes—
303C4—24no—
304C4—24no—
305C4—44no—
306C4—42no—
307C4—42no—
308C4—44no—
309C4—44no—
310C4—82no—
311C4—81no—
312C4—81no—
313C2—82no—
314C2#239511yescharacteristic
315C2#253611yescharacteristic
316C2#254411yescharacteristic
317C2#255812yes—
318C2#256012yes—
319C2#257512yes—
320C2#259612yes—
321C2#259811yescharacteristic
322C2#260411yescharacteristic
323C2#262311yescharacteristic
324C2#285411yescharacteristic
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
4. C2×C4×C4⋊C4C2semidirect
5. #1442C2semidirect
6. #1441C2semidirect
10. #1436C2semidirect
11. #1434C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4002
C2×#3992
C2×#3882
C2×C4×C4⋊C42
#14422
#14412
#14402
#14392
#14372
#14362
#14342