Finite Group Databasebeta

← #14365

Subgroups — #14365

Order 256.

1427 conjugacy classes of subgroups.

The subgroup lattice has 383 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#388C212yesmaximal, complemented
2C2×#379C212yesmaximal
3C2×#379C211yesmaximal, characteristic
4#1440C212yesmaximal, complemented
5#1439C212yesmaximal, complemented
6#1439C212yesmaximal, complemented
7#1437C211yesmaximal, characteristic, complemented
8#1437C211yesmaximal, characteristic, complemented
9#1437C212yesmaximal, complemented
10C26C2211yescharacteristic
11C22×C4⋊C4C2212yescomplemented
12C22×C22⋊C4C2212yes—
13C22×C22⋊C4C2212yes—
14C22×C22⋊C4C2212yes—
15C22×C22⋊C4C2212yes—
16C22×C22⋊C4C2212yes—
17C22×C22⋊C4C2212yes—
18C22×C22⋊C4C2211yescharacteristic
19C22×C22⋊C4C2212yes—
20C22×C22⋊C4C2211yescharacteristic
21C22×C22⋊C4C2212yes—
22C22×C42C2211yescharacteristic, complemented
23C22×C42C2211yescharacteristic, complemented
24#388—28no—
25#379—28no—
26#379C418yescomplemented
27C2×#95C2212yescomplemented
28C2×#95C2212yescomplemented
29C2×#95C2212yescomplemented
30C2×#95C2211yescharacteristic, complemented
31C2×#95C2212yescomplemented
32C2×#95C2211yescharacteristic, complemented
33C2×#95C2212yescomplemented
34C25—24no—
35C25—24no—
36C25—24no—
37C25—41no—
38C25—41no—
39C25—42no—
40C25—42no—
41C25—42no—
42C25C2×C414yes—
43C25C2312yes—
44C25C2311yescharacteristic
45C23×C4C2312yes—
46C23×C4C2312yes—
47C23×C4C2312yes—
48C23×C4C2311yescharacteristic
49C23×C4C2311yescharacteristic
50C23×C4C2312yes—
51C23×C4C2312yes—
52C2×C4⋊C4—24no—
53C2×C4⋊C4—42no—
54C2×C4⋊C4—42no—
55C2×C22⋊C4—24no—
56C2×C22⋊C4—24no—
57C2×C22⋊C4—24no—
58C2×C22⋊C4—24no—
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—42no—
67C2×C22⋊C4—41no—
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—42no—
75C2×C22⋊C4—42no—
76C2×C22⋊C4—42no—
77C2×C22⋊C4—42no—
78C2×C22⋊C4—42no—
79C2×C22⋊C4C2×C418yes—
80C2×C22⋊C4C2×C414yes—
81C2×C22⋊C4C2×C414yes—
82C2×C22⋊C4C2×C418yes—
83C2×C22⋊C4D418yes—
84C2×C42—24no—
85C2×C42—41no—
86C2×C42—42no—
87C2×C42—41no—
88#95—24no—
89#95—24no—
90#95—42no—
91#95—42no—
92#95—41no—
93#95—41no—
94#95—42no—
95C24—22no—
96C24—22no—
97C24—22no—
98C24—22no—
99C24—22no—
100C24—22no—
101C24—22no—
102C24—21no—
103C24—21no—
104C24—22no—
105C24—22no—
106C24—22no—
107C24—22no—
108C24—22no—
109C24—22no—
110C24—22no—
111C24—22no—
112C24—22no—
113C24—22no—
114C24—28no—
115C24—216no—
116C24—44no—
117C24—48no—
118C24—44no—
119C24—44no—
120C24—48no—
121C24—48no—
122C24—48no—
123C24—48no—
124C24—48no—
125C24—44no—
126C24—48no—
127C24—48no—
128C24—48no—
129C24—48no—
130C24—48no—
131C24—48no—
132C24—48no—
133C24—44no—
134C24—44no—
135C24C22×C412yes—
136C24C22×C414yes—
137C24C2×D414yes—
138C24#4114yes—
139C24#4114yes—
140C24C2411yescharacteristic
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—21no—
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4—21no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4C22×C412yes—
182C22×C4C22×C414yes—
183C22×C4C22×C412yes—
184C22×C4C2×D414yes—
185C22×C4C2×D414yes—
186C4⋊C4—48no—
187C22⋊C4—216no—
188C22⋊C4—44no—
189C22⋊C4—48no—
190C22⋊C4—44no—
191C22⋊C4—48no—
192C22⋊C4—48no—
193C22⋊C4—48no—
194C22⋊C4—48no—
195C22⋊C4—48no—
196C22⋊C4—48no—
197C42—44no—
198C42—44no—
199C23—24no—
200C23—24no—
201C23—24no—
202C23—22no—
203C23—22no—
204C23—24no—
205C23—24no—
206C23—24no—
207C23—24no—
208C23—24no—
209C23—24no—
210C23—24no—
211C23—24no—
212C23—24no—
213C23—24no—
214C23—24no—
215C23—24no—
216C23—24no—
217C23—24no—
218C23—24no—
219C23—24no—
220C23—24no—
221C23—24no—
222C23—24no—
223C23—24no—
224C23—24no—
225C23—24no—
226C23—24no—
227C23—432no—
228C23—416no—
229C23—416no—
230C23—432no—
231C23—416no—
232C23—432no—
233C23—432no—
234C23—432no—
235C23—432no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—41no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—41no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—41no—
260C23—42no—
261C23—41no—
262C23#11714yes—
263C23C4×D418yes—
264C23C23×C411yescharacteristic
265C23C22×D412yes—
266C23C2×#4112yes—
267C23C2×#4112yes—
268C23H(5,𝔽2)12yes—
269C23H(5,𝔽2)12yes—
270C23H(5,𝔽2)12yes—
271C23H(5,𝔽2)11yescharacteristic
272C23H(5,𝔽2)11yescharacteristic
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C22—24no—
319C22—28no—
320C22—28no—
321C22—28no—
322C22—28no—
323C22—464no—
324C22—44no—
325C22—44no—
326C22—44no—
327C22—44no—
328C22—42no—
329C22—42no—
330C22—44no—
331C22—44no—
332C22—44no—
333C22—44no—
334C22—44no—
335C22—44no—
336C22—44no—
337C22—44no—
338C22—44no—
339C22—44no—
340C22—44no—
341C22—44no—
342C22—44no—
343C22C2×#11711yescharacteristic
344C22C2×C4×D412yes—
345C22#51811yescharacteristic
346C22#51811yescharacteristic
347C22#51812yes—
348C22#53412yes—
349C22#53512yes—
350C22#53812yes—
351C22#53812yes—
352C22#53812yes—
353C22#53812yes—
354C22D4211yescharacteristic
355C22#54612yes—
356C22#54612yes—
357C22#54612yes—
358C22#55112yes—
359C22#55112yes—
360C22#55111yescharacteristic
361C22#56012yes—
362C22#56011yescharacteristic
363C22H(3,𝔽4)11yescharacteristic
364C4—44no—
365C4—42no—
366C4—42no—
367C4—44no—
368C4—44no—
369C4—82no—
370C4—82no—
371C2—44no—
372C2—48no—
373C2#231211yescharacteristic
374C2#232112yes—
375C2#234212yes—
376C2#235811yescharacteristic
377C2#237511yescharacteristic
378C2#267812yes—
379C2#268412yes—
380C2#268512yes—
381C2#273512yes—
subgroupG/H
center, Frattini, socle, cosocleC24C24
commutatorC23C23×C4
prime radicalC26C22
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
4. #1440C2semidirect
5. #1439C2semidirect
6. #1439C2semidirect
7. #1437C2semidirect
8. #1437C2semidirect
9. #1437C2semidirect
11. C22×C4⋊C4C22semidirect
22. C22×C42C22semidirect
23. C22×C42C22semidirect
26. #379C4semidirect
27. C2×#95C22semidirect
28. C2×#95C22semidirect
29. C2×#95C22semidirect
30. C2×#95C22semidirect
31. C2×#95C22semidirect
32. C2×#95C22semidirect
33. C2×#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#3882
C2×#3792
#14402
#14392
#14372