Finite Group Databasebeta

← #16399

Subgroups — #16399

Order 256.

769 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#398C211yesmaximal, characteristic, complemented
2C2×#397C212yesmaximal
3C2×#385C211yesmaximal, characteristic
4C2×#380C211yesmaximal, characteristic
5C2×#380C211yesmaximal, characteristic
6#1445C211yesmaximal, characteristic, complemented
7#1442C211yesmaximal, characteristic, complemented
8#1441C211yesmaximal, characteristic, complemented
9#1441C212yesmaximal, complemented
10#1441C212yesmaximal, complemented
11#1440C212yesmaximal
12C24×C4C2211yescharacteristic
13C22×C4⋊C4C2211yescharacteristic
14C22×C4⋊C4C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic
16C22×C4⋊C4C2212yes—
17C22×C4⋊C4C2212yes—
18C22×C4⋊C4C2211yescharacteristic
19C22×C22⋊C4C2212yes—
20C22×C22⋊C4C2212yes—
21C22×C22⋊C4C2211yescharacteristic
22C22×C22⋊C4C2211yescharacteristic
23#398—24no—
24#397—28no—
25#385—24no—
26#380—24no—
27#380—24no—
28C2×#95C2212yes—
29C2×#95C2212yes—
30C2×#95C2212yes—
31C2×#95C2212yes—
32C2×#95C2212yes—
33C2×#95C2211yescharacteristic
34C2×#95C2211yescharacteristic
35C2×#95C2211yescharacteristic
36C2×#95C2212yes—
37C2×#95C2212yes—
38C2×#95C2212yes—
39C2×#95C2211yescharacteristic
40C25C2311yescharacteristic
41C23×C4—22no—
42C23×C4—22no—
43C23×C4—24no—
44C23×C4—42no—
45C23×C4D414yes—
46C23×C4C2311yescharacteristic
47C23×C4C2312yes—
48C23×C4C2312yes—
49C23×C4C2312yes—
50C23×C4C2311yescharacteristic
51C23×C4C2311yescharacteristic
52C23×C4C2312yes—
53C23×C4C2311yescharacteristic
54C23×C4C2311yescharacteristic
55C23×C4C2311yescharacteristic
56C2×C4⋊C4—24no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—24no—
62C2×C4⋊C4—24no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—24no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—24no—
69C2×C4⋊C4—42no—
70C2×C4⋊C4—42no—
71C2×C4⋊C4—42no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—24no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—24no—
77C2×C22⋊C4—24no—
78C2×C22⋊C4—24no—
79C2×C22⋊C4—24no—
80C2×C22⋊C4—42no—
81C2×C22⋊C4—42no—
82C2×C22⋊C4D414yes—
83#95—22no—
84#95—24no—
85#95—22no—
86#95—22no—
87#95—24no—
88#95—24no—
89#95—24no—
90#95—24no—
91#95—42no—
92#95—42no—
93#95—42no—
94#95D414yes—
95C24—21no—
96C24—21no—
97C24—22no—
98C24—21no—
99C24—22no—
100C24—22no—
101C24—22no—
102C24—21no—
103C24C2×D412yes—
104C24C2×D412yes—
105C24#4112yes—
106C24C2411yescharacteristic
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—22no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—22no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—21no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—21no—
128C22×C4—28no—
129C22×C4—28no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—21no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—21no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—21no—
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×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—21no—
161C22×C4—44no—
162C22×C4—48no—
163C22×C4—48no—
164C22×C4C2×D412yes—
165C22×C4C2×D412yes—
166C22×C4C2×D414yes—
167C22×C4C2×D412yes—
168C22×C4C2×D412yes—
169C22×C4C2×D412yes—
170C22×C4#4112yes—
171C22×C4#4114yes—
172C4⋊C4—48no—
173C4⋊C4—48no—
174C4⋊C4—44no—
175C4⋊C4—44no—
176C4⋊C4—44no—
177C4⋊C4—44no—
178C22⋊C4—28no—
179C22⋊C4—48no—
180C22⋊C4—44no—
181C22⋊C4—48no—
182C23—22no—
183C23—22no—
184C23—22no—
185C23—22no—
186C23—22no—
187C23—24no—
188C23—22no—
189C23—24no—
190C23—22no—
191C23—22no—
192C23—24no—
193C23—22no—
194C23—24no—
195C23—22no—
196C23—42no—
197C23—42no—
198C23—42no—
199C23—42no—
200C23—42no—
201C23—42no—
202C23—42no—
203C23—42no—
204C23#12114yes—
205C23C22×D411yescharacteristic
206C23C22×D411yescharacteristic
207C23C22×D411yescharacteristic
208C23C2×#4111yescharacteristic
209C23C2×#4111yescharacteristic
210C23C2×#4112yes—
211C23H(5,𝔽2)12yes—
212C23H(5,𝔽2)11yescharacteristic
213C232−1+412yes—
214C232−1+412yes—
215C232−1+411yescharacteristic
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—24no—
221C2×C4—24no—
222C2×C4—24no—
223C2×C4—24no—
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—416no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
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—42no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—41no—
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—42no—
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—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4#12114yes—
276C2×C4#12314yes—
277C22—24no—
278C22—24no—
279C22—24no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—44no—
284C22—44no—
285C22—44no—
286C22—44no—
287C22—42no—
288C22C2×#12111yescharacteristic
289C22C2×#12111yescharacteristic
290C22C2×#12311yescharacteristic
291C22#53411yescharacteristic
292C22#53611yescharacteristic
293C22#53612yes—
294C22#53611yescharacteristic
295C22#53712yes—
296C22#53711yescharacteristic
297C22#53912yes—
298C22#54112yes—
299C22#54112yes—
300C22#54212yes—
301C22D4211yescharacteristic
302C22#54612yes—
303C22#54711yescharacteristic
304C22#54712yes—
305C22#54811yescharacteristic
306C22#55112yes—
307C22#55212yes—
308C22#55311yescharacteristic
309C22#56212yes—
310C22#56312yes—
311C4—24no—
312C4—24no—
313C4—42no—
314C4—44no—
315C4—42no—
316C4—82no—
317C4—81no—
318C4—82no—
319C4—82no—
320C4—81no—
321C2—44no—
322C2#241211yescharacteristic
323C2#241611yescharacteristic
324C2#248611yescharacteristic
325C2#261811yescharacteristic
326C2#262111yescharacteristic
327C2#262511yescharacteristic
328C2#267212yes—
329C2#269012yes—
330C2#272212yes—
331C2#274712yes—
332C2#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
6. #1445C2semidirect
7. #1442C2semidirect
8. #1441C2semidirect
9. #1441C2semidirect
10. #1441C2semidirect

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×#3852
C2×#3802
#14452
#14422
#14412
#14402