Finite Group Databasebeta

← #15927

Subgroups — #15927

Order 256.

783 conjugacy classes of subgroups.

The subgroup lattice has 321 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#126C211yesmaximal, characteristic
2C2×#399C211yesmaximal, characteristic
3C2×#398C211yesmaximal, characteristic, complemented
4C2×#382C211yesmaximal, characteristic, complemented
5#1445C211yesmaximal, characteristic, complemented
6#1444C212yesmaximal, complemented
7#1440C212yesmaximal
8#1439C212yesmaximal
9#1437C211yesmaximal, characteristic
10#1436C212yesmaximal, complemented
11#1436C211yesmaximal, characteristic, complemented
12C2×#126—24no—
13C2×#126—24no—
14C2×#126—24no—
15C22×C4⋊C4C2211yescharacteristic
16C22×C4⋊C4C2211yescharacteristic
17C22×C4⋊C4C2212yes—
18C22×C4⋊C4C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic
20C22×C4⋊C4C2212yes—
21C22×C22⋊C4C2211yescharacteristic
22C22×C22⋊C4C2211yescharacteristic
23C22×C22⋊C4C2212yes—
24C22×C22⋊C4C2212yes—
25C22×C22⋊C4C2211yescharacteristic
26C22×C42C2211yescharacteristic
27C22×C42C2211yescharacteristic
28#399—24no—
29#398—24no—
30#382—24no—
31C2×#95C2212yes—
32C2×#95C2211yescharacteristic
33C2×#95C2211yescharacteristic
34C2×#95C2212yes—
35C2×#95C2212yes—
36C2×#95C2212yes—
37C2×#95C2212yes—
38C2×#95C2212yes—
39C2×#95C2212yes—
40C2×#95C2212yes—
41C25C2311yescharacteristic
42C23×C4C2312yes—
43C23×C4C2311yescharacteristic
44C23×C4C2311yescharacteristic
45C23×C4C2311yescharacteristic
46C23×C4C2311yescharacteristic
47C23×C4C2311yescharacteristic
48C23×C4C2312yes—
49C23×C4C2312yes—
50C23×C4C2312yes—
51C23×C4C2311yescharacteristic
52#126—232no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—24no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—24no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—24no—
66C2×C4⋊C4—42no—
67C2×C4⋊C4—42no—
68C2×C4⋊C4—42no—
69C2×C4⋊C4D414yes—
70C2×C4⋊C4D414yes—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—22no—
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—24no—
83C2×C22⋊C4—42no—
84C2×C22⋊C4—42no—
85C2×C22⋊C4D414yes—
86C2×C42—22no—
87C2×C42—22no—
88C2×C42—22no—
89C2×C42—22no—
90C2×C42—24no—
91#95—24no—
92#95—24no—
93#95—22no—
94#95—24no—
95#95—22no—
96#95—24no—
97#95—24no—
98#95—24no—
99#95—24no—
100#95—42no—
101C24—21no—
102C24—21no—
103C24—22no—
104C24—22no—
105C24—22no—
106C24—22no—
107C24—22no—
108C24—22no—
109C24C2×D412yes—
110C24C2411yescharacteristic
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—22no—
127C22×C4—22no—
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—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
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×C4C2×D412yes—
159C22×C4C2×D412yes—
160C22×C4C2×D412yes—
161C22×C4C2×D412yes—
162C22×C4C2×D412yes—
163C22×C4C2×D412yes—
164C22×C4C2×D412yes—
165C22×C4C2×D412yes—
166C22×C4#4114yes—
167C22×C4#4114yes—
168C22×C4#4114yes—
169C4⋊C4—28no—
170C4⋊C4—28no—
171C4⋊C4—28no—
172C4⋊C4—48no—
173C4⋊C4—48no—
174C4⋊C4—44no—
175C22⋊C4—28no—
176C22⋊C4—28no—
177C22⋊C4—28no—
178C22⋊C4—48no—
179C22⋊C4—48no—
180C42—28no—
181C42—28no—
182C23—22no—
183C23—22no—
184C23—22no—
185C23—24no—
186C23—24no—
187C23—41no—
188C23—42no—
189C23—41no—
190C23—41no—
191C23—41no—
192C23—42no—
193C23—42no—
194C23—42no—
195C23—42no—
196C23—42no—
197C23—42no—
198C23—42no—
199C23—42no—
200C23—42no—
201C23—42no—
202C23—42no—
203C23C22×D411yescharacteristic
204C23C22×D411yescharacteristic
205C23C22×D411yescharacteristic
206C23C2×#4112yes—
207C23C2×#4112yes—
208C23C2×#4112yes—
209C23H(5,𝔽2)12yes—
210C232−1+412yes—
211C232−1+412yes—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—24no—
215C2×C4—24no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—24no—
222C2×C4—24no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—24no—
226C2×C4—24no—
227C2×C4—24no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—24no—
231C2×C4—24no—
232C2×C4—24no—
233C2×C4—24no—
234C2×C4—24no—
235C2×C4—24no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
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—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
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#12014yes—
270C22—42no—
271C22—42no—
272C22—42no—
273C22—44no—
274C22—44no—
275C22—82no—
276C22—82no—
277C22—82no—
278C22—82no—
279C22C2×#12011yescharacteristic
280C22#52511yescharacteristic
281C22#52912yes—
282C22#53211yescharacteristic
283C22#53211yescharacteristic
284C22#53511yescharacteristic
285C22#53611yescharacteristic
286C22#53611yescharacteristic
287C22#53912yes—
288C22#54112yes—
289C22#54212yes—
290C22#54612yes—
291C22#54712yes—
292C22#54712yes—
293C22#54712yes—
294C22#54812yes—
295C22#55012yes—
296C22#55112yes—
297C22#55212yes—
298C22#55212yes—
299C22#56312yes—
300C4—24no—
301C4—42no—
302C4—42no—
303C4—42no—
304C4—42no—
305C4—42no—
306C4—44no—
307C4—44no—
308C4—44no—
309C4—82no—
310C2—82no—
311C2#240611yescharacteristic
312C2#243111yescharacteristic
313C2#243211yescharacteristic
314C2#247912yes—
315C2#251912yes—
316C2#255312yes—
317C2#269812yes—
318C2#272112yes—
319C2#272312yes—
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
3. C2×#398C2semidirect
4. C2×#382C2semidirect
5. #1445C2semidirect
6. #1444C2semidirect
10. #1436C2semidirect
11. #1436C2semidirect

Maximal subgroups

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

subgroupindex
C22×#1262
C2×#3992
C2×#3982
C2×#3822
#14452
#14442
#14402
#14392
#14372
#14362