Finite Group Databasebeta

← #15265

Subgroups — #15265

Order 256.

1093 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊Q8C211yesmaximal, characteristic, complemented
2C22×C22⋊Q8C211yesmaximal, characteristic
3C2×#395C211yesmaximal, characteristic, complemented
4C2×#393C211yesmaximal, characteristic
5C2×#385C212yesmaximal
6#1446C211yesmaximal, characteristic, complemented
7#1444C212yesmaximal, complemented
8#1441C212yesmaximal, complemented
9#1439C212yesmaximal
10#1437C211yesmaximal, characteristic
11#1436C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C23×Q8C2211yescharacteristic
14C24×C4C2211yescharacteristic
15C2×C4⋊Q8—24no—
16C2×C4⋊Q8—28no—
17C2×C22⋊Q8—24no—
18C2×C22⋊Q8—24no—
19C2×C22⋊Q8—24no—
20C22×C4⋊C4C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic
22C22×C4⋊C4C2212yes—
23C22×C4⋊C4C2212yes—
24C22×C4⋊C4C2211yescharacteristic
25C22×C4⋊C4C2212yes—
26C22×C22⋊C4C2211yescharacteristic
27C22×C22⋊C4C2212yes—
28C22×C22⋊C4C2211yescharacteristic
29C22×C22⋊C4C2212yes—
30C22×C42C2211yescharacteristic
31#395—24no—
32#393—24no—
33#385—28no—
34C2×#95C2211yescharacteristic
35C2×#95C2212yes—
36C2×#95C2211yescharacteristic
37C2×#95C2212yes—
38C2×#95C2212yes—
39C2×#95C2212yes—
40C2×#95C2212yes—
41C2×#95C2212yes—
42C2×#95C2212yes—
43C25C2311yescharacteristic
44C22×Q8—22no—
45C22×Q8—22no—
46C22×Q8—24no—
47C22×Q8—24no—
48C22×Q8—22no—
49C22×Q8—22no—
50C22×Q8—22no—
51C22×Q8—42no—
52C22×Q8—42no—
53C22×Q8D414yes—
54C23×C4—22no—
55C23×C4—22no—
56C23×C4—22no—
57C23×C4—24no—
58C23×C4—42no—
59C23×C4C2312yes—
60C23×C4C2312yes—
61C23×C4C2311yescharacteristic
62C23×C4C2311yescharacteristic
63C23×C4C2311yescharacteristic
64C23×C4C2311yescharacteristic
65C23×C4C2312yes—
66C23×C4C2311yescharacteristic
67C23×C4C2311yescharacteristic
68C23×C4C2312yes—
69C4⋊Q8—232no—
70C22⋊Q8—232no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—24no—
84C2×C4⋊C4—24no—
85C2×C4⋊C4—24no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4D414yes—
88C2×C4⋊C4D418yes—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—22no—
94C2×C22⋊C4—24no—
95C2×C22⋊C4—24no—
96C2×C22⋊C4—24no—
97C2×C22⋊C4—24no—
98C2×C22⋊C4—42no—
99C2×C22⋊C4D414yes—
100C2×C22⋊C4Q818yescomplemented
101C2×C42—22no—
102C2×C42—24no—
103#95—24no—
104#95—24no—
105#95—24no—
106#95—22no—
107#95—22no—
108#95—24no—
109#95—24no—
110#95—24no—
111#95—42no—
112C24—21no—
113C24—21no—
114C24—22no—
115C24—22no—
116C24—22no—
117C24—22no—
118C24—22no—
119C24C2×D412yes—
120C24C2×Q814yes—
121C24C2411yescharacteristic
122C2×Q8—28no—
123C2×Q8—28no—
124C2×Q8—28no—
125C2×Q8—44no—
126C2×Q8—48no—
127C2×Q8—48no—
128C2×Q8—48no—
129C2×Q8—48no—
130C2×Q8—44no—
131C2×Q8—44no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—28no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
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×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—48no—
178C22×C4—48no—
179C22×C4—44no—
180C22×C4—44no—
181C22×C4C2×D412yes—
182C22×C4C2×D412yes—
183C22×C4C2×D412yes—
184C22×C4C2×D414yes—
185C22×C4C2×D412yes—
186C22×C4C2×D412yes—
187C22×C4C2×D412yes—
188C22×C4C2×D414yes—
189C22×C4C2×D412yes—
190C22×C4C2×D412yes—
191C22×C4C2×D414yes—
192C22×C4C2×Q814yes—
193C22×C4C2×Q814yes—
194C22×C4#4114yes—
195C4⋊C4—28no—
196C4⋊C4—28no—
197C4⋊C4—216no—
198C4⋊C4—28no—
199C4⋊C4—216no—
200C4⋊C4—48no—
201C22⋊C4—28no—
202C22⋊C4—28no—
203C22⋊C4—216no—
204C22⋊C4—48no—
205C42—28no—
206C23—22no—
207C23—24no—
208C23—22no—
209C23—24no—
210C23—24no—
211C23—24no—
212C23—24no—
213C23—24no—
214C23—24no—
215C23—24no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23—42no—
220C23—41no—
221C23—42no—
222C23—42no—
223C23—42no—
224C23—42no—
225C23—42no—
226C23C4⋊Q814yescomplemented
227C23C22×D411yescharacteristic
228C23C22×D412yes—
229C23C22×D411yescharacteristic
230C23C22×D411yescharacteristic
231C23C22×Q812yes—
232C23C2×#4112yes—
233C23H(5,𝔽2)12yes—
234C232−1+412yes—
235C232−1+412yes—
236Q8—416no—
237Q8—416no—
238C2×C4—24no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—24no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—24no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—24no—
258C2×C4—24no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—416no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—41no—
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#12014yes—
295C2×C4#12014yes—
296C2×C4C22⋊Q818yescomplemented
297C22—24no—
298C22—28no—
299C22—42no—
300C22—44no—
301C22—44no—
302C22—44no—
303C22—42no—
304C22—44no—
305C22—44no—
306C22C2×#12011yescharacteristic
307C22C2×#12011yescharacteristic
308C22C2×C22⋊Q812yes—
309C22C2×C4⋊Q811yescharacteristic
310C22#53211yescharacteristic
311C22#53411yescharacteristic
312C22#53612yes—
313C22#53611yescharacteristic
314C22#53611yescharacteristic
315C22#54212yes—
316C22#54412yes—
317C22D4212yes—
318C22D4212yes—
319C22#54712yes—
320C22#54812yes—
321C22#54812yes—
322C22D4×Q812yes—
323C22D4×Q812yes—
324C22D4×Q812yes—
325C22#55412yes—
326C22#56312yes—
327C4—24no—
328C4—24no—
329C4—28no—
330C4—42no—
331C4—42no—
332C4—44no—
333C4—42no—
334C4—42no—
335C4—44no—
336C4—82no—
337C2—44no—
338C2#240311yescharacteristic
339C2#240912yes—
340C2#243211yescharacteristic
341C2#243411yescharacteristic
342C2#244012yes—
343C2#245212yes—
344C2#267412yes—
345C2#267712yes—
346C2#272912yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C24
prime radicalC25C23
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×C4⋊Q8C2semidirect
3. C2×#395C2semidirect
6. #1446C2semidirect
7. #1444C2semidirect
8. #1441C2semidirect
11. #1436C2semidirect
100. C2×C22⋊C4Q8semidirect
226. C23C4⋊Q8semidirect
296. C2×C4C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊Q82
C22×C22⋊Q82
C2×#3952
C2×#3932
C2×#3852
#14462
#14442
#14412
#14392
#14372
#14362