Finite Group Databasebeta

← #14410

Subgroups — #14410

Order 256.

933 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
1C2×#388C212yesmaximal, complemented
2C2×#386C211yesmaximal, characteristic
3C2×#380C212yesmaximal, complemented
4#1442C212yesmaximal, complemented
5#1439C212yesmaximal, complemented
6#1438C211yesmaximal, characteristic
7#1438C211yesmaximal, characteristic
8#1437C212yesmaximal, complemented
9#1436C212yesmaximal, complemented
10C23×D4C2211yescharacteristic
11C24×C4C2211yescharacteristic
12C22×C4⋊C4C2212yes—
13C22×C4⋊C4C2212yes—
14C22×C22⋊C4C2212yes—
15C22×C22⋊C4C2212yes—
16C22×C22⋊C4C2211yescharacteristic
17C22×C22⋊C4C2212yes—
18C22×C22⋊C4C2212yes—
19C22×C22⋊C4C2211yescharacteristic
20C22×C22⋊C4C2211yescharacteristic
21C22×C42C2212yes—
22#388—28no—
23#386C418yescomplemented
24#380—28no—
25C2×#95C2212yes—
26C2×#95C2212yes—
27C2×#95C2211yescharacteristic
28C2×#95C2212yes—
29C2×#95C2212yes—
30C2×#95C2212yes—
31C2×#95C2211yescharacteristic
32C2×#95C2212yes—
33C2×#95C2212yes—
34C25C2311yescharacteristic
35C25C2311yescharacteristic
36C22×D4—24no—
37C22×D4—42no—
38C22×D4—41no—
39C22×D4—41no—
40C22×D4C2×C414yes—
41C23×C4—24no—
42C23×C4—24no—
43C23×C4—41no—
44C23×C4—41no—
45C23×C4C2×C414yes—
46C23×C4C2312yes—
47C23×C4C2312yes—
48C23×C4C2312yes—
49C23×C4C2311yescharacteristic
50C23×C4C2312yes—
51C23×C4C2312yes—
52C23×C4C2312yes—
53C2×C4⋊C4—24no—
54C2×C4⋊C4—24no—
55C2×C4⋊C4—24no—
56C2×C4⋊C4—24no—
57C2×C4⋊C4—42no—
58C2×C4⋊C4—42no—
59C2×C22⋊C4—24no—
60C2×C22⋊C4—24no—
61C2×C22⋊C4—24no—
62C2×C22⋊C4—24no—
63C2×C22⋊C4—24no—
64C2×C22⋊C4—24no—
65C2×C22⋊C4—24no—
66C2×C22⋊C4—24no—
67C2×C22⋊C4—42no—
68C2×C22⋊C4—41no—
69C2×C22⋊C4—42no—
70C2×C22⋊C4—42no—
71C2×C22⋊C4—42no—
72C2×C22⋊C4—42no—
73C2×C22⋊C4—42no—
74C2×C22⋊C4—41no—
75C2×C22⋊C4C2×C418yes—
76C2×C22⋊C4C2×C414yes—
77C2×C42—24no—
78C2×C42—42no—
79C2×C42—42no—
80#95—24no—
81#95—24no—
82#95—42no—
83#95—42no—
84#95—42no—
85#95—41no—
86#95—41no—
87#95C2×C418yes—
88#95D418yescomplemented
89C24—22no—
90C24—22no—
91C24—22no—
92C24—22no—
93C24—21no—
94C24—21no—
95C24—22no—
96C24—22no—
97C24—22no—
98C24—22no—
99C24—22no—
100C24—22no—
101C24—21no—
102C24—21no—
103C24—22no—
104C24C22×C412yes—
105C24C22×C412yes—
106C24#4114yes—
107C24C2411yescharacteristic
108C2×D4—28no—
109C2×D4—44no—
110C2×D4—44no—
111C2×D4—48no—
112C2×D4—48no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—21no—
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—216no—
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—44no—
155C22×C4—44no—
156C22×C4—48no—
157C22×C4—44no—
158C22×C4C22×C412yes—
159C22×C4C22×C414yes—
160C22×C4C22×C414yes—
161C22×C4C2×D414yes—
162C22×C4C2×D414yes—
163C22×C4C2×D414yes—
164C22×C4#4114yes—
165C4⋊C4—48no—
166C4⋊C4—48no—
167C22⋊C4—216no—
168C22⋊C4—44no—
169C22⋊C4—44no—
170C22⋊C4—48no—
171C22⋊C4—48no—
172C22⋊C4—48no—
173C22⋊C4—44no—
174C42—48no—
175C23—24no—
176C23—22no—
177C23—22no—
178C23—24no—
179C23—24no—
180C23—24no—
181C23—24no—
182C23—24no—
183C23—24no—
184C23—24no—
185C23—22no—
186C23—24no—
187C23—22no—
188C23—22no—
189C23—24no—
190C23—42no—
191C23—42no—
192C23—42no—
193C23—42no—
194C23—41no—
195C23—42no—
196C23—41no—
197C23—41no—
198C23—42no—
199C23—41no—
200C23—42no—
201C23—42no—
202C23—42no—
203C23—42no—
204C23—42no—
205C23—42no—
206C23—41no—
207C23—42no—
208C23—42no—
209C23—42no—
210C23—41no—
211C23—42no—
212C23—42no—
213C23—41no—
214C23—42no—
215C23—41no—
216C23#11714yes—
217C23C23×C411yescharacteristic
218C23C22×D412yes—
219C23C2×#4112yes—
220C23C2×#4112yes—
221C23H(5,𝔽2)12yes—
222C23H(5,𝔽2)11yescharacteristic
223C23H(5,𝔽2)11yescharacteristic
224C23H(5,𝔽2)12yes—
225C232−1+412yes—
226D4—416no—
227C2×C4—24no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—24no—
231C2×C4—24no—
232C2×C4—24no—
233C2×C4—22no—
234C2×C4—24no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—24no—
239C2×C4—24no—
240C2×C4—24no—
241C2×C4—24no—
242C2×C4—24no—
243C2×C4—24no—
244C2×C4—24no—
245C2×C4—416no—
246C2×C4—42no—
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—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4C4×D418yes—
275C22—24no—
276C22—28no—
277C22—44no—
278C22—42no—
279C22—42no—
280C22—44no—
281C22—44no—
282C22—44no—
283C22—44no—
284C22—42no—
285C22—44no—
286C22—44no—
287C22—42no—
288C22—42no—
289C22—82no—
290C22—82no—
291C22—82no—
292C22—82no—
293C22C2×#11711yescharacteristic
294C22C2×C4×D412yes—
295C22#51811yescharacteristic
296C22#51811yescharacteristic
297C22#52012yes—
298C22#53512yes—
299C22#53712yes—
300C22#53812yes—
301C22#53812yes—
302C22#54012yes—
303C22#54212yes—
304C22D4211yescharacteristic
305C22#54612yes—
306C22#54612yes—
307C22#54812yes—
308C22#55111yescharacteristic
309C22#55212yes—
310C22#55312yes—
311C22#56011yescharacteristic
312C22#56212yes—
313C22#56311yescharacteristic
314C4—28no—
315C4—42no—
316C4—44no—
317C4—44no—
318C4—44no—
319C4—82no—
320C4—82no—
321C2—44no—
322C2—82no—
323C2#231211yescharacteristic
324C2#232412yes—
325C2#235212yes—
326C2#235811yescharacteristic
327C2#237611yescharacteristic
328C2#267612yes—
329C2#270212yes—
330C2#270312yes—
331C2#274112yes—
subgroupG/H
center, Frattini, socle, cosocleC24C24
commutatorC23C23×C4
prime radicalC23×D4C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#388C2semidirect
3. C2×#380C2semidirect
4. #1442C2semidirect
5. #1439C2semidirect
8. #1437C2semidirect
9. #1436C2semidirect
23. #386C4semidirect
88. #95D4semidirect

Maximal subgroups

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

subgroupindex
C2×#3882
C2×#3862
C2×#3802
#14422
#14392
#14382
#14372
#14362