Finite Group Databasebeta

← #13814

Subgroups — #13814

Order 256.

1149 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#385C212yesmaximal
2C2×C4×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊C4C212yesmaximal
4C2×C4×C22⋊C4C211yesmaximal, characteristic
5C22×#95C211yesmaximal, characteristic
6#1436C212yesmaximal, complemented
7#1436C212yesmaximal, complemented
8C4⋊#95C211yesmaximal, characteristic, complemented
9#1434C211yesmaximal, characteristic, complemented
10C22⋊#95C211yesmaximal, characteristic
11C4×#95C211yesmaximal, characteristic, complemented
12C24×C4C2211yescharacteristic
13C24×C4C2212yes—
14C22×C4⋊C4C2212yes—
15C22×C4⋊C4C2212yes—
16C22×C22⋊C4C2212yes—
17C22×C22⋊C4C2211yescharacteristic
18C22×C22⋊C4C2211yescharacteristic
19C22×C42C2212yes—
20C22×C42C2212yes—
21C22×C42C2211yescharacteristic
22C22×C42C2211yescharacteristic
23#385C4116yescomplemented
24C4×C4⋊C4—24no—
25C4×C22⋊C4—28no—
26C4×C22⋊C4C418yescomplemented
27C2×#95—24no—
28C2×#95C2211yescharacteristic
29C2×#95C2211yescharacteristic
30C2×#95C2211yescharacteristic
31C2×#95C2212yes—
32C2×#95C2212yes—
33C2×#95C2212yes—
34C2×#95C2212yes—
35C2×#95C2212yes—
36C2×#95C2212yes—
37C2×#95C2211yescharacteristic
38C2×#95C2211yescharacteristic
39C2×#95C2211yescharacteristic
40C25C2311yescharacteristic
41C23×C4—24no—
42C23×C4—24no—
43C23×C4—24no—
44C23×C4—22no—
45C23×C4—24no—
46C23×C4—22no—
47C23×C4—24no—
48C23×C4—42no—
49C23×C4—42no—
50C23×C4—42no—
51C23×C4C2×C414yes—
52C23×C4C2×C418yes—
53C23×C4C2312yes—
54C23×C4C2312yes—
55C23×C4C2312yes—
56C23×C4C2311yescharacteristic
57C23×C4C2311yescharacteristic
58C23×C4C2311yescharacteristic
59C23×C4C2311yescharacteristic
60C23×C4C2311yescharacteristic
61C23×C4C2312yes—
62C23×C4C2311yescharacteristic
63C2×C4⋊C4—24no—
64C2×C4⋊C4—24no—
65C2×C4⋊C4—42no—
66C2×C4⋊C4—42no—
67C2×C4⋊C4C2×C418yescomplemented
68C2×C4⋊C4C2×C418yescomplemented
69C2×C22⋊C4—24no—
70C2×C22⋊C4—24no—
71C2×C22⋊C4C2×C418yes—
72C2×C22⋊C4C2×C414yes—
73C2×C22⋊C4C2×C418yes—
74C2×C22⋊C4C2×C414yes—
75C2×C22⋊C4D414yes—
76C2×C22⋊C4Q814yes—
77C2×C42—22no—
78C2×C42—22no—
79C2×C42—24no—
80C2×C42—24no—
81C2×C42—24no—
82C2×C42—24no—
83C2×C42—42no—
84C2×C42—42no—
85C2×C42—42no—
86C2×C42C2×C414yescomplemented
87C2×C42C2×C414yescomplemented
88#95—22no—
89#95—24no—
90#95—24no—
91#95—24no—
92#95—22no—
93#95—22no—
94#95—42no—
95#95C2×C414yescomplemented
96#95C2×C414yescomplemented
97#95C2×C418yescomplemented
98#95C2×C418yescomplemented
99#95D414yes—
100C24—21no—
101C24—22no—
102C24—21no—
103C24—21no—
104C24—22no—
105C24—21no—
106C24C22×C412yes—
107C24C22×C414yes—
108C24C2×D412yes—
109C24C2×Q812yes—
110C24#4114yes—
111C24C2411yescharacteristic
112C22×C4—21no—
113C22×C4—22no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—21no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—28no—
131C22×C4—216no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—21no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—21no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—28no—
155C22×C4—216no—
156C22×C4—44no—
157C22×C4—48no—
158C22×C4—48no—
159C22×C4—48no—
160C22×C4—48no—
161C22×C4—48no—
162C22×C4—48no—
163C22×C4—48no—
164C22×C4C22×C414yes—
165C22×C4C22×C412yes—
166C22×C4C22×C414yes—
167C22×C4C22×C414yes—
168C22×C4C22×C412yes—
169C22×C4C22×C412yes—
170C22×C4C22×C414yes—
171C22×C4C22×C414yes—
172C22×C4C22×C412yes—
173C22×C4C22×C412yes—
174C22×C4C22×C414yes—
175C22×C4C22×C412yes—
176C22×C4C2×D412yes—
177C22×C4C2×D412yes—
178C22×C4C2×D412yes—
179C22×C4C2×D414yes—
180C22×C4C2×Q812yes—
181C22×C4C2×Q812yes—
182C22×C4#4112yes—
183C22×C4#4114yes—
184C4⋊C4—216no—
185C4⋊C4—48no—
186C22⋊C4—216no—
187C22⋊C4C42116yescomplemented
188C22⋊C4C4⋊C4116yescomplemented
189C42—28no—
190C42—44no—
191C42—48no—
192C42—48no—
193C23—22no—
194C23—24no—
195C23—22no—
196C23—24no—
197C23—22no—
198C23—24no—
199C23—22no—
200C23—24no—
201C23—22no—
202C23—24no—
203C23—24no—
204C23—22no—
205C23—22no—
206C23—24no—
207C23—24no—
208C23—24no—
209C23—22no—
210C23—24no—
211C23C2×C4214yes—
212C23C2×C4⋊C414yes—
213C23#11714yes—
214C23C4×D418yes—
215C23C4×Q818yes—
216C23C23×C411yescharacteristic
217C23C23×C412yes—
218C23C22×D411yescharacteristic
219C23C22×D411yescharacteristic
220C23C22×Q811yescharacteristic
221C23C2×#4112yes—
222C23C2×#4112yes—
223C23C2×#4111yescharacteristic
224C23H(5,𝔽2)11yescharacteristic
225C23H(5,𝔽2)12yes—
226C232−1+411yescharacteristic
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—24no—
231C2×C4—24no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—24no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—24no—
239C2×C4—24no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—24no—
243C2×C4—22no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—24no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—24no—
252C2×C4—24no—
253C2×C4—22no—
254C2×C4—24no—
255C2×C4—22no—
256C2×C4—24no—
257C2×C4—24no—
258C2×C4—24no—
259C2×C4—22no—
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—22no—
266C2×C4—24no—
267C2×C4—416no—
268C2×C4—432no—
269C2×C4—42no—
270C2×C4—42no—
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—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4C2×C4214yes—
284C2×C4C2×C4214yes—
285C2×C4C2×C4⋊C414yes—
286C2×C4C2×C4⋊C414yes—
287C2×C4C4×D414yes—
288C2×C4C4×D418yes—
289C22—24no—
290C22—24no—
291C22—24no—
292C22—24no—
293C22—28no—
294C22—28no—
295C22—28no—
296C22—28no—
297C22—24no—
298C22—24no—
299C22C4×C4⋊C418yescomplemented
300C22C22×C4211yescharacteristic
301C22C22×C4⋊C411yescharacteristic
302C22C2×#11711yescharacteristic
303C22C2×C4×D411yescharacteristic
304C22C2×C4×D412yes—
305C22C2×C4×D412yes—
306C22C2×C4×Q812yes—
307C22C4×#4111yescharacteristic
308C22C4×#4112yes—
309C22#51811yescharacteristic
310C22#51812yes—
311C22#52011yescharacteristic
312C22#52012yes—
313C22D4211yescharacteristic
314C22#54612yes—
315C22#54612yes—
316C22#54711yescharacteristic
317C22D4×Q811yescharacteristic
318C22#55112yes—
319C22#55112yes—
320C22#55212yes—
321C22#55411yescharacteristic
322C22#55412yes—
323C4—24no—
324C4—24no—
325C4—24no—
326C4—28no—
327C4—24no—
328C4—24no—
329C4—42no—
330C4—44no—
331C4—44no—
332C4—44no—
333C2—28no—
334C2C2×C4×C4⋊C411yescharacteristic
335C2C42×D411yescharacteristic
336C2#227511yescharacteristic
337C2#234211yescharacteristic
338C2#234311yescharacteristic
339C2D4×C4⋊C411yescharacteristic
340C2#234911yescharacteristic
341C2#235812yes—
342C2#235912yes—
343C2#236012yes—
344C2#236812yes—
subgroupG/H
center, Frattini, socle, cosocleC24C24
commutatorC22C22×C42
prime radicalC25C23
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×C4⋊C4C2semidirect
6. #1436C2semidirect
7. #1436C2semidirect
8. C4⋊#95C2semidirect
9. #1434C2semidirect
11. C4×#95C2semidirect
23. #385C4semidirect
26. C4×C22⋊C4C4semidirect
67. C2×C4⋊C4C2×C4semidirect
68. C2×C4⋊C4C2×C4semidirect
86. C2×C42C2×C4semidirect
87. C2×C42C2×C4semidirect
95. #95C2×C4semidirect
96. #95C2×C4semidirect
97. #95C2×C4semidirect
98. #95C2×C4semidirect
187. C22⋊C4C42semidirect
188. C22⋊C4C4⋊C4semidirect
299. C22C4×C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#3852
C2×C4×C4⋊C42
C2×C4×C22⋊C42
C22×#952
#14362
C4⋊#952
#14342
C22⋊#952
C4×#952