Finite Group Databasebeta

← #15284

Subgroups — #15284

Order 256.

1417 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1417 conjugacy classes of subgroups — too many to draw. Only its 187 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#399C211yesmaximal, characteristic, complemented
2C2×#388C212yesmaximal, complemented
3C2×#379C211yesmaximal, characteristic
4C2×#379C212yesmaximal
5#1440C212yesmaximal, complemented
6#1440C212yesmaximal, complemented
7#1439C212yesmaximal, complemented
8#1439C212yesmaximal, complemented
9#1437C211yesmaximal, characteristic, complemented
10C26C2211yescharacteristic
11C22×C4⋊C4C2212yescomplemented
12C22×C4⋊C4C2211yescharacteristic, complemented
13C22×C22⋊C4C2211yescharacteristic
14C22×C22⋊C4C2212yes—
15C22×C22⋊C4C2212yes—
16C22×C22⋊C4C2212yes—
17C22×C22⋊C4C2212yes—
18C22×C22⋊C4C2212yes—
19C22×C22⋊C4C2212yes—
20C22×C22⋊C4C2212yes—
21C22×C22⋊C4C2212yes—
22C22×C22⋊C4C2211yescharacteristic
23C22×C42C2211yescharacteristic, complemented
24#399—24no—
25#388—28no—
26#379—28no—
27#379—24no—
28C2×#95C2211yescharacteristic, complemented
29C2×#95C2212yescomplemented
30C2×#95C2212yescomplemented
31C2×#95C2211yescharacteristic, complemented
32C2×#95C2212yescomplemented
33C2×#95C2212yescomplemented
34C2×#95C2212yescomplemented
35C25—24no—
36C25—22no—
37C25—24no—
38C25—24no—
39C25—24no—
40C25—41no—
41C25—41no—
42C25—42no—
43C25—42no—
44C25C2312yes—
45C25C2311yescharacteristic
46C23×C4C2312yes—
47C23×C4C2312yes—
48C23×C4C2312yes—
49C23×C4C2311yescharacteristic
50C23×C4C2312yes—
51C23×C4C2311yescharacteristic
52C23×C4C2312yes—
53C2×C4⋊C4—24no—
54C2×C4⋊C4—24no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—42no—
57C2×C4⋊C4—41no—
58C2×C4⋊C4—41no—
59C2×C22⋊C4—24no—
60C2×C22⋊C4—24no—
61C2×C22⋊C4—24no—
62C2×C22⋊C4—22no—
63C2×C22⋊C4—24no—
64C2×C22⋊C4—24no—
65C2×C22⋊C4—24no—
66C2×C22⋊C4—24no—
67C2×C22⋊C4—24no—
68C2×C22⋊C4—24no—
69C2×C22⋊C4—24no—
70C2×C22⋊C4—24no—
71C2×C22⋊C4—24no—
72C2×C22⋊C4—24no—
73C2×C22⋊C4—24no—
74C2×C22⋊C4—24no—
75C2×C22⋊C4—24no—
76C2×C22⋊C4—41no—
77C2×C22⋊C4—41no—
78C2×C22⋊C4—42no—
79C2×C22⋊C4—42no—
80C2×C22⋊C4—42no—
81C2×C22⋊C4—42no—
82C2×C22⋊C4—42no—
83C2×C22⋊C4—42no—
84C2×C22⋊C4—42no—
85C2×C22⋊C4—42no—
86C2×C22⋊C4D418yes—
87C2×C22⋊C4D414yes—
88C2×C42—24no—
89C2×C42—41no—
90#95—24no—
91#95—24no—
92#95—22no—
93#95—42no—
94#95—41no—
95#95—42no—
96#95—42no—
97C24—22no—
98C24—22no—
99C24—22no—
100C24—22no—
101C24—22no—
102C24—22no—
103C24—22no—
104C24—22no—
105C24—22no—
106C24—21no—
107C24—21no—
108C24—22no—
109C24—22no—
110C24—22no—
111C24—22no—
112C24—22no—
113C24—22no—
114C24—22no—
115C24—22no—
116C24—28no—
117C24—216no—
118C24—44no—
119C24—48no—
120C24—48no—
121C24—48no—
122C24—48no—
123C24—48no—
124C24—48no—
125C24—48no—
126C24—48no—
127C24—44no—
128C24—44no—
129C24—48no—
130C24—48no—
131C24—48no—
132C24—48no—
133C24—44no—
134C24—44no—
135C24—48no—
136C24—44no—
137C24C2×D414yes—
138C24C2×D412yes—
139C24#4114yes—
140C24#4114yes—
141C24#4114yes—
142C24C2411yescharacteristic
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—21no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—21no—
158C22×C4—22no—
159C22×C4—21no—
160C22×C4—21no—
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×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4C2×D414yes—
185C22×C4C2×D414yes—
186C22×C4C2×D412yes—
187C22×C4C2×D412yes—
188C4⋊C4—48no—
189C4⋊C4—44no—
190C22⋊C4—216no—
191C22⋊C4—44no—
192C22⋊C4—48no—
193C22⋊C4—48no—
194C22⋊C4—48no—
195C22⋊C4—48no—
196C22⋊C4—48no—
197C22⋊C4—48no—
198C22⋊C4—48no—
199C22⋊C4—44no—
200C42—44no—
201C23—22no—
202C23—24no—
203C23—24no—
204C23—24no—
205C23—24no—
206C23—22no—
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—24no—
228C23—24no—
229C23—416no—
230C23—432no—
231C23—432no—
232C23—432no—
233C23—432no—
234C23—416no—
235C23—432no—
236C23—416no—
237C23—432no—
238C23—42no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23—42no—
250C23—41no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23—41no—
259C23—42no—
260C23—41no—
261C23—42no—
262C23—42no—
263C23—41no—
264C23#12318yes—
265C23#12414yes—
266C23C22×D412yes—
267C23C22×D411yescharacteristic
268C23C2×#4112yes—
269C23C2×#4112yes—
270C23C2×#4112yes—
271C23H(5,𝔽2)11yescharacteristic
272C23H(5,𝔽2)11yescharacteristic
273C23H(5,𝔽2)12yes—
274C23H(5,𝔽2)12yes—
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—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
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—
318C2×C4—42no—
319C2×C4—42no—
320C22—24no—
321C22—28no—
322C22—28no—
323C22—28no—
324C22—28no—
325C22—464no—
326C22—44no—
327C22—42no—
328C22—44no—
329C22—44no—
330C22—44no—
331C22—44no—
332C22—44no—
333C22—44no—
334C22—44no—
335C22—44no—
336C22—42no—
337C22—44no—
338C22—44no—
339C22—44no—
340C22—44no—
341C22—44no—
342C22—44no—
343C22—44no—
344C22—44no—
345C22C2×#12312yes—
346C22#52511yescharacteristic
347C22#52512yes—
348C22C2×#12411yescharacteristic
349C22#53411yescharacteristic
350C22#53512yes—
351C22#53812yes—
352C22#53812yes—
353C22#53812yes—
354C22D4211yescharacteristic
355C22D4212yes—
356C22#54612yes—
357C22#54612yes—
358C22#54612yes—
359C22#55112yes—
360C22#55112yes—
361C22#55112yes—
362C22#55111yescharacteristic
363C22#55112yes—
364C22#56011yescharacteristic
365C22#56011yescharacteristic
366C4—42no—
367C4—44no—
368C4—44no—
369C4—42no—
370C4—82no—
371C4—82no—
372C4—82no—
373C2—44no—
374C2—48no—
375C2#241812yes—
376C2#241812yes—
377C2#247212yes—
378C2#248212yes—
379C2#256111yescharacteristic
380C2#266911yescharacteristic
381C2#268512yes—
382C2#269711yescharacteristic
383C2#273512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C24
prime radicalC26C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#399C2semidirect
2. C2×#388C2semidirect
5. #1440C2semidirect
6. #1440C2semidirect
7. #1439C2semidirect
8. #1439C2semidirect
9. #1437C2semidirect
11. C22×C4⋊C4C22semidirect
12. C22×C4⋊C4C22semidirect
23. C22×C42C22semidirect
28. C2×#95C22semidirect
29. C2×#95C22semidirect
30. C2×#95C22semidirect
31. C2×#95C22semidirect
32. C2×#95C22semidirect
33. C2×#95C22semidirect
34. C2×#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#3992
C2×#3882
C2×#3792
#14402
#14392
#14372