Finite Group Databasebeta

← #14735

Subgroups — #14735

Order 256.

1055 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×#397C211yesmaximal, characteristic, complemented
2C2×#386C211yesmaximal, characteristic
3C2×#386C212yesmaximal
4C2×#380C211yesmaximal, characteristic, complemented
5C22×#95C211yesmaximal, characteristic, complemented
6#1445C211yesmaximal, characteristic, complemented
7#1443C212yesmaximal, complemented
8#1442C211yesmaximal, characteristic, complemented
9#1440C212yesmaximal, complemented
10#1439C211yesmaximal, characteristic, complemented
11C22⋊#95C212yesmaximal, complemented
12C23×D4C2211yescharacteristic
13C24×C4C2212yes—
14C24×C4C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic
16C22×C4⋊C4C2211yescharacteristic
17C22×C22⋊C4C2211yescharacteristic
18C22×C22⋊C4C2211yescharacteristic
19C22×C22⋊C4C2212yes—
20C22×C22⋊C4C2212yes—
21C22×C22⋊C4C2212yes—
22C22×C22⋊C4C2211yescharacteristic
23#397—24no—
24#386—28no—
25#386—24no—
26#380—24no—
27C2×#95—24no—
28C2×#95C2212yes—
29C2×#95C2212yes—
30C2×#95C2211yescharacteristic
31C2×#95C2212yes—
32C2×#95C2212yes—
33C2×#95C2212yes—
34C2×#95C2211yescharacteristic
35C2×#95C2211yescharacteristic
36C2×#95C2212yes—
37C2×#95C2212yes—
38C2×#95C2212yes—
39C2×#95C2211yescharacteristic
40C25C2311yescharacteristic
41C25C2311yescharacteristic
42C22×D4—22no—
43C22×D4—22no—
44C22×D4—24no—
45C22×D4—24no—
46C22×D4D414yes—
47C23×C4—24no—
48C23×C4—24no—
49C23×C4—24no—
50C23×C4—22no—
51C23×C4—24no—
52C23×C4—22no—
53C23×C4—24no—
54C23×C4—24no—
55C23×C4—42no—
56C23×C4—42no—
57C23×C4—42no—
58C23×C4D414yes—
59C23×C4C2312yes—
60C23×C4C2312yes—
61C23×C4C2311yescharacteristic
62C23×C4C2311yescharacteristic
63C23×C4C2311yescharacteristic
64C23×C4C2312yes—
65C23×C4C2311yescharacteristic
66C23×C4C2312yes—
67C23×C4C2311yescharacteristic
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—42no—
73C2×C22⋊C4—24no—
74C2×C22⋊C4—24no—
75C2×C22⋊C4—24no—
76C2×C22⋊C4—24no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—24no—
79C2×C22⋊C4—24no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4—24no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—24no—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—42no—
87C2×C22⋊C4—42no—
88C2×C22⋊C4—42no—
89#95—24no—
90#95—22no—
91#95—24no—
92#95—22no—
93#95—22no—
94#95—24no—
95#95—24no—
96#95—24no—
97#95—24no—
98#95—42no—
99#95—42no—
100#95D414yescomplemented
101C24—21no—
102C24—21no—
103C24—22no—
104C24—22no—
105C24—21no—
106C24—21no—
107C24—21no—
108C24—21no—
109C24—22no—
110C24—21no—
111C24—21no—
112C24—22no—
113C24—22no—
114C24—21no—
115C24—22no—
116C24—21no—
117C24C2×D412yes—
118C24C2×D412yes—
119C24C2×D412yes—
120C24#4112yes—
121C24#4114yes—
122C24#4114yes—
123C24C2411yescharacteristic
124C2×D4—28no—
125C2×D4—216no—
126C2×D4—44no—
127C2×D4—44no—
128C2×D4—48no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—216no—
132C22×C4—28no—
133C22×C4—21no—
134C22×C4—22no—
135C22×C4—21no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—21no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—28no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—21no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—21no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—21no—
177C22×C4—22no—
178C22×C4—44no—
179C22×C4—48no—
180C22×C4—48no—
181C22×C4—48no—
182C22×C4—48no—
183C22×C4—48no—
184C22×C4—48no—
185C22×C4—48no—
186C22×C4—48no—
187C22×C4C2×D412yes—
188C22×C4C2×D412yes—
189C22×C4C2×D414yes—
190C22×C4C2×D412yes—
191C22×C4C2×D412yes—
192C22×C4#4114yes—
193C22×C4#4112yes—
194C4⋊C4—44no—
195C4⋊C4—44no—
196C22⋊C4—48no—
197C22⋊C4—48no—
198C22⋊C4—44no—
199C22⋊C4—48no—
200C22⋊C4—44no—
201C22⋊C4—44no—
202C23—22no—
203C23—24no—
204C23—24no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—24no—
209C23—24no—
210C23—22no—
211C23—24no—
212C23—24no—
213C23—22no—
214C23—22no—
215C23—24no—
216C23—24no—
217C23—24no—
218C23—22no—
219C23—24no—
220C23—22no—
221C23—24no—
222C23—22no—
223C23—24no—
224C23—22no—
225C23—41no—
226C23—41no—
227C23—42no—
228C23—41no—
229C23—42no—
230C23—42no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—42no—
235C23—42no—
236C23—42no—
237C23—41no—
238C23—42no—
239C23—42no—
240C23—42no—
241C23—41no—
242C23—42no—
243C23#12114yes—
244C23#12314yes—
245C23#12318yes—
246C23#12414yes—
247C23#12618yes—
248C23C22×D411yescharacteristic
249C23C22×D411yescharacteristic
250C23C22×D411yescharacteristic
251C23C2×#4111yescharacteristic
252C23C2×#4111yescharacteristic
253C23C2×#4112yes—
254C23C2×#4112yes—
255C23C2×#4112yes—
256C23H(5,𝔽2)12yes—
257C23H(5,𝔽2)11yescharacteristic
258C232−1+411yescharacteristic
259D4—416no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—22no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—416no—
276C2×C4—432no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—41no—
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#12114yes—
315C22—24no—
316C22—24no—
317C22—24no—
318C22—24no—
319C22—28no—
320C22—28no—
321C22—28no—
322C22—24no—
323C22—28no—
324C22—24no—
325C22—42no—
326C22—42no—
327C22—44no—
328C22—44no—
329C22—42no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22—82no—
334C22—81no—
335C22—82no—
336C22#39718yescomplemented
337C22C2×#12111yescharacteristic
338C22C2×#12111yescharacteristic
339C22C2×#12311yescharacteristic
340C22C2×#12312yes—
341C22#52512yes—
342C22C2×#12411yescharacteristic
343C22C2×#12612yes—
344C22#52912yes—
345C22#53211yescharacteristic
346C22#53411yescharacteristic
347C22#53711yescharacteristic
348C22#53812yes—
349C22#53912yes—
350C22D4211yescharacteristic
351C22#54612yes—
352C22#54612yes—
353C22#54711yescharacteristic
354C22#54811yescharacteristic
355C22#55112yes—
356C22#55112yes—
357C22#55212yes—
358C22#55311yescharacteristic
359C22#55312yes—
360C4—24no—
361C4—42no—
362C4—42no—
363C4—44no—
364C4—44no—
365C4—81no—
366C4—81no—
367C4—82no—
368C4—82no—
369C2—28no—
370C2—82no—
371C2C2×#39711yescharacteristic
372C2#241211yescharacteristic
373C2#241611yescharacteristic
374C2#241812yes—
375C2#241912yes—
376C2#248112yes—
377C2#249811yescharacteristic
378C2#251012yes—
379C2#254311yescharacteristic
380C2#263911yescharacteristic
381C2#285011yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C24
prime radicalC23×D4C22
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×#397C2semidirect
4. C2×#380C2semidirect
5. C22×#95C2semidirect
6. #1445C2semidirect
7. #1443C2semidirect
8. #1442C2semidirect
9. #1440C2semidirect
10. #1439C2semidirect
11. C22⋊#95C2semidirect
100. #95D4semidirect
336. C22#397semidirect

Maximal subgroups

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

subgroupindex
C2×#3972
C2×#3862
C2×#3802
C22×#952
#14452
#14432
#14422
#14402
#14392
C22⋊#952