Finite Group Databasebeta

← #16682

Subgroups — #16682

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
1C22×#125C211yesmaximal, characteristic, complemented
2C2×#399C211yesmaximal, characteristic
3C2×#397C211yesmaximal, characteristic
4C2×#388C211yesmaximal, characteristic
5#1445C211yesmaximal, characteristic, complemented
6#1445C211yesmaximal, characteristic, complemented
7#1444C212yesmaximal, complemented
8#1441C212yesmaximal, complemented
9#1439C212yesmaximal
10#1439C212yesmaximal
11#1434C211yesmaximal, characteristic, complemented
12C2×#125—24no—
13C2×#125—24no—
14C2×#125—24no—
15C22×C4⋊C4C2212yes—
16C22×C4⋊C4C2212yes—
17C22×C4⋊C4C2212yes—
18C22×C4⋊C4C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic
20C22×C4⋊C4C2211yescharacteristic
21C22×C22⋊C4C2211yescharacteristic
22C22×C22⋊C4C2211yescharacteristic
23C22×C22⋊C4C2212yes—
24C22×C22⋊C4C2212yes—
25C22×C22⋊C4C2211yescharacteristic
26C22×C42C2211yescharacteristic
27#399—24no—
28#397—24no—
29#388—24no—
30C2×#95C2211yescharacteristic
31C2×#95C2212yes—
32C2×#95C2211yescharacteristic
33C2×#95C2212yes—
34C2×#95C2212yes—
35C2×#95C2211yescharacteristic
36C2×#95C2211yescharacteristic
37C2×#95C2212yes—
38C2×#95C2212yes—
39C2×#95C2212yes—
40C2×#95C2212yes—
41C25C2311yescharacteristic
42C23×C4C2311yescharacteristic
43C23×C4C2312yes—
44C23×C4C2311yescharacteristic
45C23×C4C2312yes—
46C23×C4C2312yes—
47C23×C4C2311yescharacteristic
48C23×C4C2311yescharacteristic
49C23×C4C2312yes—
50C23×C4C2311yescharacteristic
51C23×C4C2311yescharacteristic
52#125—232no—
53C2×C4⋊C4—24no—
54C2×C4⋊C4—24no—
55C2×C4⋊C4—24no—
56C2×C4⋊C4—24no—
57C2×C4⋊C4—24no—
58C2×C4⋊C4—24no—
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—24no—
66C2×C4⋊C4—42no—
67C2×C4⋊C4—42no—
68C2×C4⋊C4D414yes—
69C2×C4⋊C4D414yes—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—24no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—24no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—24no—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—24no—
81C2×C22⋊C4—42no—
82C2×C22⋊C4—42no—
83C2×C22⋊C4—42no—
84C2×C42—22no—
85C2×C42—22no—
86C2×C42D414yes—
87#95—22no—
88#95—22no—
89#95—24no—
90#95—22no—
91#95—22no—
92#95—24no—
93#95—24no—
94#95—42no—
95#95—42no—
96#95—42no—
97#95—42no—
98C24—21no—
99C24—21no—
100C24—21no—
101C24—21no—
102C24—22no—
103C24—22no—
104C24—21no—
105C24—22no—
106C24—21no—
107C24—22no—
108C24#4112yes—
109C24C2411yescharacteristic
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—21no—
113C22×C4—22no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—22no—
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—22no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
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—21no—
161C22×C4C2×D412yes—
162C22×C4C2×D414yes—
163C22×C4C2×D412yes—
164C22×C4C2×D414yes—
165C22×C4C2×D412yes—
166C22×C4C2×D412yes—
167C22×C4C2×D412yes—
168C22×C4#4112yes—
169C22×C4#4114yes—
170C4⋊C4—216no—
171C4⋊C4—216no—
172C4⋊C4—28no—
173C4⋊C4—28no—
174C4⋊C4—48no—
175C4⋊C4—44no—
176C22⋊C4—28no—
177C22⋊C4—48no—
178C22⋊C4—48no—
179C22⋊C4—44no—
180C22⋊C4—44no—
181C42—28no—
182C23—22no—
183C23—24no—
184C23—22no—
185C23—22no—
186C23—24no—
187C23—41no—
188C23—41no—
189C23—41no—
190C23—41no—
191C23—41no—
192C23—42no—
193C23—42no—
194C23—41no—
195C23—42no—
196C23—42no—
197C23—42no—
198C23—42no—
199C23—41no—
200C23—41no—
201C23—42no—
202C23—42no—
203C23—42no—
204C23—42no—
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—24no—
219C2×C4—24no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—24no—
223C2×C4—24no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—24no—
230C2×C4—24no—
231C2×C4—24no—
232C2×C4—24no—
233C2×C4—24no—
234C2×C4—24no—
235C2×C4—24no—
236C2×C4—24no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—41no—
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#12014yes—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—44no—
280C22—44no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22—81no—
285C22—82no—
286C22—82no—
287C22C2×#12011yescharacteristic
288C22#52511yescharacteristic
289C22#53211yescharacteristic
290C22#53511yescharacteristic
291C22#53611yescharacteristic
292C22#53611yescharacteristic
293C22#53612yes—
294C22#53612yes—
295C22#53711yescharacteristic
296C22#53912yes—
297C22#53912yes—
298C22#53912yes—
299C22#54112yes—
300C22#54611yescharacteristic
301C22#54612yes—
302C22#54711yescharacteristic
303C22#54811yescharacteristic
304C22#55011yescharacteristic
305C22#55012yes—
306C22#55212yes—
307C22#55212yes—
308C22#56212yes—
309C22#56312yes—
310C4—24no—
311C4—42no—
312C4—42no—
313C4—44no—
314C4—42no—
315C4—44no—
316C4—44no—
317C4—81no—
318C4—81no—
319C4—82no—
320C2—82no—
321C2#240711yescharacteristic
322C2#243111yescharacteristic
323C2#261111yescharacteristic
324C2#261411yescharacteristic
325C2#263311yescharacteristic
326C2#266411yescharacteristic
327C2#269012yes—
328C2#269412yes—
329C2#269412yes—
330C2#272112yes—
331C2#282011yescharacteristic
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. C22×#125C2semidirect
5. #1445C2semidirect
6. #1445C2semidirect
7. #1444C2semidirect
8. #1441C2semidirect
11. #1434C2semidirect

Maximal subgroups

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

subgroupindex
C22×#1252
C2×#3992
C2×#3972
C2×#3882
#14452
#14442
#14412
#14392
#14342