Finite Group Databasebeta

← #16321

Subgroups — #16321

Order 256.

791 conjugacy classes of subgroups.

The subgroup lattice has 309 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#399C211yesmaximal, characteristic
2C2×#398C212yesmaximal, complemented
3C2×#385C212yesmaximal
4C2×#385C211yesmaximal, characteristic
5#1444C212yesmaximal, complemented
6#1444C212yesmaximal, complemented
7#1443C212yesmaximal, complemented
8#1440C212yesmaximal
9#1439C211yesmaximal, characteristic
10C24×C4C2211yescharacteristic
11C22×C4⋊C4C2211yescharacteristic
12C22×C4⋊C4C2212yes—
13C22×C4⋊C4C2212yes—
14C22×C4⋊C4C2212yes—
15C22×C4⋊C4C2211yescharacteristic
16C22×C4⋊C4C2212yes—
17C22×C22⋊C4C2212yes—
18C22×C22⋊C4C2212yes—
19C22×C22⋊C4C2211yescharacteristic
20C22×C22⋊C4C2211yescharacteristic
21#399—24no—
22#398—28no—
23#385—24no—
24#385—28no—
25C2×#95C2212yes—
26C2×#95C2212yes—
27C2×#95C2211yescharacteristic
28C2×#95C2212yes—
29C2×#95C2212yes—
30C2×#95C2212yes—
31C2×#95C2212yes—
32C2×#95C2211yescharacteristic
33C2×#95C2212yes—
34C2×#95C2212yes—
35C25C2311yescharacteristic
36C23×C4—22no—
37C23×C4—24no—
38C23×C4—41no—
39C23×C4—42no—
40C23×C4—41no—
41C23×C4C2312yes—
42C23×C4C2311yescharacteristic
43C23×C4C2312yes—
44C23×C4C2312yes—
45C23×C4C2311yescharacteristic
46C23×C4C2312yes—
47C23×C4C2312yes—
48C23×C4C2312yes—
49C2×C4⋊C4—24no—
50C2×C4⋊C4—24no—
51C2×C4⋊C4—24no—
52C2×C4⋊C4—24no—
53C2×C4⋊C4—24no—
54C2×C4⋊C4—24no—
55C2×C4⋊C4—24no—
56C2×C4⋊C4—24no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—42no—
59C2×C4⋊C4—41no—
60C2×C4⋊C4—42no—
61C2×C4⋊C4—42no—
62C2×C4⋊C4—42no—
63C2×C4⋊C4—42no—
64C2×C4⋊C4—41no—
65C2×C4⋊C4—41no—
66C2×C22⋊C4—22no—
67C2×C22⋊C4—24no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—24no—
70C2×C22⋊C4—24no—
71C2×C22⋊C4—24no—
72C2×C22⋊C4—24no—
73C2×C22⋊C4—41no—
74C2×C22⋊C4—41no—
75C2×C22⋊C4—42no—
76C2×C22⋊C4D414yes—
77C2×C22⋊C4Q814yes—
78#95—24no—
79#95—24no—
80#95—24no—
81#95—24no—
82#95—22no—
83#95—24no—
84#95—24no—
85#95—41no—
86#95—42no—
87#95D418yes—
88C24—22no—
89C24—22no—
90C24—21no—
91C24—21no—
92C24—22no—
93C24—21no—
94C24—21no—
95C24—22no—
96C24C2×D412yes—
97C24C2×Q812yes—
98C24#4112yes—
99C24C2411yescharacteristic
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—21no—
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—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—44no—
146C22×C4—48no—
147C22×C4—48no—
148C22×C4—44no—
149C22×C4—44no—
150C22×C4C2×D412yes—
151C22×C4C2×D414yes—
152C22×C4C2×D414yes—
153C22×C4C2×D414yes—
154C22×C4C2×D412yes—
155C22×C4C2×Q814yes—
156C22×C4#4114yes—
157C4⋊C4—48no—
158C4⋊C4—44no—
159C4⋊C4—48no—
160C4⋊C4—48no—
161C4⋊C4—44no—
162C4⋊C4—48no—
163C22⋊C4—216no—
164C22⋊C4—28no—
165C22⋊C4—44no—
166C22⋊C4—48no—
167C23—24no—
168C23—22no—
169C23—22no—
170C23—22no—
171C23—22no—
172C23—24no—
173C23—22no—
174C23—22no—
175C23—24no—
176C23—24no—
177C23—24no—
178C23—24no—
179C23—42no—
180C23—42no—
181C23—42no—
182C23—42no—
183C23—41no—
184C23—42no—
185C23—41no—
186C23—41no—
187C23—41no—
188C23—42no—
189C23C22⋊Q814yes—
190C23C22×D411yescharacteristic
191C23C22×D412yes—
192C23C22×Q811yescharacteristic
193C23C2×#4111yescharacteristic
194C23C2×#4112yes—
195C23H(5,𝔽2)11yescharacteristic
196C23H(5,𝔽2)11yescharacteristic
197C23H(5,𝔽2)12yes—
198C232−1+411yescharacteristic
199C232−1+412yes—
200C232−1+411yescharacteristic
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—24no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—24no—
207C2×C4—24no—
208C2×C4—24no—
209C2×C4—24no—
210C2×C4—24no—
211C2×C4—24no—
212C2×C4—24no—
213C2×C4—24no—
214C2×C4—24no—
215C2×C4—24no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—24no—
219C2×C4—24no—
220C2×C4—416no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C22—24no—
255C22—28no—
256C22—42no—
257C22—44no—
258C22—42no—
259C22—42no—
260C22—44no—
261C22—42no—
262C22—44no—
263C22—42no—
264C22—42no—
265C22C2×C22⋊Q811yescharacteristic
266C22#52512yes—
267C22#53411yescharacteristic
268C22#53512yes—
269C22#53611yescharacteristic
270C22#53612yes—
271C22#53911yescharacteristic
272C22#53912yes—
273C22#53911yescharacteristic
274C22#53912yes—
275C22#53912yes—
276C22#54411yescharacteristic
277C22#54411yescharacteristic
278C22D4211yescharacteristic
279C22D4212yes—
280C22#54712yes—
281C22#54712yes—
282C22D4×Q812yes—
283C22#55412yes—
284C22#55911yescharacteristic
285C22#56011yescharacteristic
286C22#56311yescharacteristic
287C22#56312yes—
288C4—44no—
289C4—42no—
290C4—44no—
291C4—44no—
292C4—44no—
293C4—44no—
294C4—42no—
295C4—82no—
296C2—44no—
297C2#244912yes—
298C2#245511yescharacteristic
299C2#262711yescharacteristic
300C2#263312yes—
301C2#266911yescharacteristic
302C2#267711yescharacteristic
303C2#268012yes—
304C2#272012yes—
305C2#272411yescharacteristic
306C2#280711yescharacteristic
307C2#283411yescharacteristic
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×#398C2semidirect
5. #1444C2semidirect
6. #1444C2semidirect
7. #1443C2semidirect

Maximal subgroups

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

subgroupindex
C2×#3992
C2×#3982
C2×#3852
#14442
#14432
#14402
#14392