Finite Group Databasebeta

← #15882

Subgroups — #15882

Order 256.

913 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#125C211yesmaximal, characteristic, complemented
2C2×#394C211yesmaximal, characteristic
3C2×#394C211yesmaximal, characteristic
4C2×#390C211yesmaximal, characteristic
5#1445C211yesmaximal, characteristic, complemented
6#1439C212yesmaximal, complemented
7#1439C212yesmaximal, complemented
8#1439C212yesmaximal, complemented
9#1437C212yesmaximal, complemented
10#1436C211yesmaximal, characteristic, complemented
11#1436C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C2×#125—24no—
14C2×#125—24no—
15C2×#125—24no—
16C22×C4⋊C4C2211yescharacteristic
17C22×C4⋊C4C2211yescharacteristic
18C22×C4⋊C4C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic
20C22×C4⋊C4C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic
22C22×C22⋊C4C2212yes—
23C22×C22⋊C4C2212yes—
24C22×C22⋊C4C2212yes—
25C22×C22⋊C4C2212yes—
26C22×C22⋊C4C2212yes—
27C22×C22⋊C4C2212yes—
28C22×C42C2211yescharacteristic
29C22×C42C2211yescharacteristic
30#394—24no—
31#394—24no—
32#390—24no—
33C2×#95C2211yescharacteristic
34C2×#95C2211yescharacteristic
35C2×#95C2212yes—
36C2×#95C2212yes—
37C2×#95C2212yes—
38C2×#95C2212yes—
39C2×#95C2212yes—
40C2×#95C2212yes—
41C25C2312yes—
42C22×D4—24no—
43C22×D4—22no—
44C22×D4—22no—
45C22×D4—22no—
46C22×D4—42no—
47C23×C4C2312yes—
48C23×C4C2312yes—
49C23×C4C2311yescharacteristic
50C23×C4C2311yescharacteristic
51C23×C4C2311yescharacteristic
52C23×C4C2311yescharacteristic
53C23×C4C2311yescharacteristic
54C23×C4C2312yes—
55C23×C4C2311yescharacteristic
56C23×C4C2311yescharacteristic
57#125—232no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4D414yes—
74C2×C4⋊C4D414yes—
75C2×C4⋊C4D414yes—
76C2×C22⋊C4—24no—
77C2×C22⋊C4—24no—
78C2×C22⋊C4—24no—
79C2×C22⋊C4—24no—
80C2×C22⋊C4—24no—
81C2×C22⋊C4—24no—
82C2×C22⋊C4—24no—
83C2×C22⋊C4—24no—
84C2×C22⋊C4—24no—
85C2×C22⋊C4—24no—
86C2×C22⋊C4—24no—
87C2×C22⋊C4—24no—
88C2×C22⋊C4—42no—
89C2×C22⋊C4—42no—
90C2×C22⋊C4—42no—
91C2×C22⋊C4—42no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4—42no—
94C2×C42—22no—
95C2×C42—22no—
96C2×C42—22no—
97C2×C42—22no—
98C2×C42—42no—
99#95—22no—
100#95—22no—
101#95—24no—
102#95—24no—
103#95—42no—
104#95—42no—
105#95—42no—
106#95—42no—
107C24—22no—
108C24—22no—
109C24—22no—
110C24—22no—
111C24—22no—
112C24—22no—
113C24—22no—
114C24—22no—
115C24—22no—
116C24—22no—
117C24—22no—
118C24—22no—
119C24—22no—
120C24—22no—
121C24#4114yes—
122C24C2411yescharacteristic
123C2×D4—28no—
124C2×D4—48no—
125C2×D4—48no—
126C2×D4—44no—
127C2×D4—44no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
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—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
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×C4C2×D412yes—
174C22×C4C2×D412yes—
175C22×C4C2×D412yes—
176C22×C4C2×D412yes—
177C22×C4C2×D412yes—
178C22×C4C2×D412yes—
179C22×C4C2×D412yes—
180C22×C4C2×D412yes—
181C22×C4C2×D412yes—
182C22×C4#4114yes—
183C22×C4#4114yes—
184C4⋊C4—28no—
185C4⋊C4—28no—
186C4⋊C4—28no—
187C4⋊C4—28no—
188C4⋊C4—28no—
189C4⋊C4—28no—
190C22⋊C4—48no—
191C22⋊C4—48no—
192C22⋊C4—48no—
193C22⋊C4—48no—
194C22⋊C4—48no—
195C22⋊C4—48no—
196C42—28no—
197C42—44no—
198C23—24no—
199C23—24no—
200C23—24no—
201C23—24no—
202C23—24no—
203C23—24no—
204C23—24no—
205C23—42no—
206C23—42no—
207C23—42no—
208C23—42no—
209C23—42no—
210C23—42no—
211C23—42no—
212C23—42no—
213C23—42no—
214C23—42no—
215C23—42no—
216C23—42no—
217C23—42no—
218C23—42no—
219C23—42no—
220C23—42no—
221C23—42no—
222C23—42no—
223C23—42no—
224C23—42no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23C22×D411yescharacteristic
234C23C22×D411yescharacteristic
235C23C22×D411yescharacteristic
236C23C2×#4112yes—
237C23C2×#4112yes—
238C23C2×#4112yes—
239C23H(5,𝔽2)12yes—
240C23H(5,𝔽2)12yes—
241C232−1+412yes—
242D4—416no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—24no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—24no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
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—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4#12014yes—
298C2×C4#12414yes—
299C22—44no—
300C22—44no—
301C22—44no—
302C22—44no—
303C22—44no—
304C22—44no—
305C22—44no—
306C22—82no—
307C22—82no—
308C22—82no—
309C22—82no—
310C22—82no—
311C22—82no—
312C22—82no—
313C22—82no—
314C22C2×#12011yescharacteristic
315C22#52511yescharacteristic
316C22C2×#12411yescharacteristic
317C22#52912yes—
318C22#53211yescharacteristic
319C22#53511yescharacteristic
320C22#53511yescharacteristic
321C22#53611yescharacteristic
322C22#53912yes—
323C22#53912yes—
324C22#54012yes—
325C22#54612yes—
326C22#54612yes—
327C22#54612yes—
328C22#54812yes—
329C22#54812yes—
330C22#55012yes—
331C22#55212yes—
332C22#55312yes—
333C22#55312yes—
334C22#56212yes—
335C4—24no—
336C4—24no—
337C4—42no—
338C4—42no—
339C4—42no—
340C4—42no—
341C4—42no—
342C4—44no—
343C4—82no—
344C4—82no—
345C2—84no—
346C2#240411yescharacteristic
347C2#243111yescharacteristic
348C2#243611yescharacteristic
349C2#246012yes—
350C2#249012yes—
351C2#255812yes—
352C2#269412yes—
353C2#271112yes—
354C2#271112yes—
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. C22×#125C2semidirect
5. #1445C2semidirect
6. #1439C2semidirect
7. #1439C2semidirect
8. #1439C2semidirect
9. #1437C2semidirect
10. #1436C2semidirect
11. #1436C2semidirect

Maximal subgroups

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

subgroupindex
C22×#1252
C2×#3942
C2×#3902
#14452
#14392
#14372
#14362