Finite Group Databasebeta

← #14875

Subgroups — #14875

Order 256.

893 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Q8C211yesmaximal, characteristic, complemented
2C2×#399C211yesmaximal, characteristic
3C2×#393C211yesmaximal, characteristic
4C2×#388C211yesmaximal, characteristic
5#1446C211yesmaximal, characteristic, complemented
6#1445C211yesmaximal, characteristic, complemented
7#1444C212yesmaximal, complemented
8#1439C212yesmaximal
9#1437C212yesmaximal
10#1436C212yesmaximal, complemented
11#1434C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C2×C4×Q8—24no—
14C2×C4×Q8—24no—
15C2×C4×Q8—24no—
16C22×C4⋊C4C2212yes—
17C22×C4⋊C4C2212yes—
18C22×C4⋊C4C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic
20C22×C22⋊C4C2211yescharacteristic
21C22×C22⋊C4C2212yes—
22C22×C22⋊C4C2212yes—
23C22×C22⋊C4C2211yescharacteristic
24C22×C22⋊C4C2211yescharacteristic
25C22×C42C2212yes—
26C22×C42C2211yescharacteristic
27#399—24no—
28#393—24no—
29#388—24no—
30C2×#95C2211yescharacteristic
31C2×#95C2212yes—
32C2×#95C2211yescharacteristic
33C2×#95C2212yes—
34C2×#95C2212yes—
35C2×#95C2211yescharacteristic
36C2×#95C2212yes—
37C2×#95C2212yes—
38C2×#95C2211yescharacteristic
39C2×#95C2212yes—
40C2×#95C2212yes—
41C25C2311yescharacteristic
42C22×Q8—22no—
43C22×Q8—22no—
44C22×Q8—24no—
45C22×Q8—24no—
46C22×Q8D414yes—
47C23×C4C2311yescharacteristic
48C23×C4C2312yes—
49C23×C4C2311yescharacteristic
50C23×C4C2311yescharacteristic
51C23×C4C2312yes—
52C23×C4C2311yescharacteristic
53C23×C4C2312yes—
54C23×C4C2311yescharacteristic
55C23×C4C2311yescharacteristic
56C23×C4C2312yes—
57C4×Q8—232no—
58C2×C4⋊C4—24no—
59C2×C4⋊C4—24no—
60C2×C4⋊C4—24no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—24no—
65C2×C4⋊C4—24no—
66C2×C4⋊C4—42no—
67C2×C4⋊C4—42no—
68C2×C4⋊C4D414yes—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—24no—
75C2×C22⋊C4—24no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—24no—
78C2×C22⋊C4—24no—
79C2×C22⋊C4—24no—
80C2×C22⋊C4—42no—
81C2×C22⋊C4—42no—
82C2×C22⋊C4Q814yescomplemented
83C2×C42—24no—
84C2×C42—24no—
85C2×C42—24no—
86C2×C42—22no—
87C2×C42—22no—
88C2×C42D414yes—
89#95—22no—
90#95—22no—
91#95—22no—
92#95—24no—
93#95—22no—
94#95—24no—
95#95—24no—
96#95—24no—
97#95—24no—
98#95—42no—
99#95—42no—
100C24—21no—
101C24—21no—
102C24—21no—
103C24—21no—
104C24—22no—
105C24—22no—
106C24—22no—
107C24—22no—
108C24—21no—
109C24—21no—
110C24C2×Q812yes—
111C24C2411yescharacteristic
112C2×Q8—28no—
113C2×Q8—28no—
114C2×Q8—216no—
115C2×Q8—28no—
116C2×Q8—216no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—21no—
131C22×C4—22no—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
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—22no—
146C22×C4—22no—
147C22×C4—21no—
148C22×C4—22no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
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—21no—
162C22×C4—21no—
163C22×C4—22no—
164C22×C4C2×D414yes—
165C22×C4C2×D412yes—
166C22×C4C2×D412yes—
167C22×C4C2×D412yes—
168C22×C4C2×D412yes—
169C22×C4C2×D412yes—
170C22×C4C2×D414yes—
171C22×C4C2×Q812yes—
172C22×C4C2×Q812yes—
173C22×C4#4114yes—
174C22×C4#4112yes—
175C22×C4#4114yes—
176C22×C4#4114yes—
177C4⋊C4—216no—
178C4⋊C4—28no—
179C4⋊C4—48no—
180C4⋊C4—44no—
181C22⋊C4—28no—
182C22⋊C4—48no—
183C22⋊C4—44no—
184C22⋊C4—44no—
185C22⋊C4—48no—
186C42—216no—
187C42—28no—
188C23—22no—
189C23—24no—
190C23—22no—
191C23—22no—
192C23—24no—
193C23—41no—
194C23—42no—
195C23—41no—
196C23—41no—
197C23—42no—
198C23—42no—
199C23—41no—
200C23—41no—
201C23—41no—
202C23—42no—
203C23—42no—
204C23—42no—
205C23—41no—
206C23—41no—
207C23—42no—
208C23—42no—
209C23—42no—
210C23—42no—
211C23C22×D411yescharacteristic
212C23C22×D411yescharacteristic
213C23C22×D411yescharacteristic
214C23C22×Q811yescharacteristic
215C23C2×#4111yescharacteristic
216C23C2×#4112yes—
217C23C2×#4112yes—
218C23C2×#4112yes—
219C23H(5,𝔽2)12yes—
220C232−1+411yescharacteristic
221C232−1+411yescharacteristic
222Q8—232no—
223C2×C4—22no—
224C2×C4—24no—
225C2×C4—24no—
226C2×C4—24no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—24no—
230C2×C4—24no—
231C2×C4—24no—
232C2×C4—24no—
233C2×C4—24no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—24no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—24no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4#12014yes—
281C2×C4C22⋊Q814yes—
282C2×C4#12318yes—
283C22—42no—
284C22—42no—
285C22—42no—
286C22—44no—
287C22—44no—
288C22—81no—
289C22—81no—
290C22—81no—
291C22—81no—
292C22—82no—
293C22—82no—
294C22C2×#12011yescharacteristic
295C22C2×C22⋊Q811yescharacteristic
296C22C2×#12312yes—
297C22#52511yescharacteristic
298C22#52511yescharacteristic
299C22#52512yes—
300C22#52912yes—
301C22#53211yescharacteristic
302C22#53312yes—
303C22#53511yescharacteristic
304C22#53611yescharacteristic
305C22#54212yes—
306C22#54212yes—
307C22#54612yes—
308C22#54811yescharacteristic
309C22#54811yescharacteristic
310C22D4×Q811yescharacteristic
311C22D4×Q811yescharacteristic
312C22#55012yes—
313C22#55212yes—
314C22#55412yes—
315C22#55612yes—
316C22#55912yes—
317C4—24no—
318C4—28no—
319C4—24no—
320C4—42no—
321C4—42no—
322C4—44no—
323C4—44no—
324C4—81no—
325C4—81no—
326C4—82no—
327C2—82no—
328C2#239611yescharacteristic
329C2#240911yescharacteristic
330C2#242312yes—
331C2#243111yescharacteristic
332C2#244812yes—
333C2#245812yes—
334C2#246512yes—
335C2#258011yescharacteristic
336C2#260411yescharacteristic
337C2#263211yescharacteristic
338C2#284911yescharacteristic
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
5. #1446C2semidirect
6. #1445C2semidirect
7. #1444C2semidirect
10. #1436C2semidirect
11. #1434C2semidirect
82. C2×C22⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Q82
C2×#3992
C2×#3932
C2×#3882
#14462
#14452
#14442
#14392
#14372
#14362
#14342