Finite Group Databasebeta

← #17015

Subgroups — #17015

Order 256.

841 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#398C211yesmaximal, characteristic, complemented
2C2×#396C212yesmaximal, complemented
3C2×#394C211yesmaximal, characteristic
4C2×#388C212yesmaximal, complemented
5#1445C212yesmaximal, complemented
6#1439C212yesmaximal, complemented
7#1439C212yesmaximal, complemented
8#1438C212yesmaximal
9C4⋊#95C211yesmaximal, characteristic, complemented
10C23×D4C2211yescharacteristic
11C22×C4⋊C4C2212yes—
12C22×C4⋊C4C2212yes—
13C22×C4⋊C4C2211yescharacteristic
14C22×C4⋊C4C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic
16C22×C22⋊C4C2212yes—
17C22×C22⋊C4C2212yes—
18C22×C22⋊C4C2212yes—
19C22×C22⋊C4C2212yes—
20C22×C22⋊C4C2212yes—
21C22×C22⋊C4C2212yes—
22C22×C42C2211yescharacteristic
23#398—24no—
24#396—28no—
25#394—24no—
26#388—28no—
27C2×#95C2211yescharacteristic
28C2×#95C2212yes—
29C2×#95C2212yes—
30C2×#95C2211yescharacteristic
31C2×#95C2212yes—
32C2×#95C2212yes—
33C2×#95C2212yes—
34C2×#95C2212yes—
35C25C2312yes—
36C22×D4—24no—
37C22×D4—22no—
38C22×D4—42no—
39C22×D4—41no—
40C22×D4—41no—
41C23×C4C2312yes—
42C23×C4C2312yes—
43C23×C4C2311yescharacteristic
44C23×C4C2312yes—
45C23×C4C2312yes—
46C23×C4C2311yescharacteristic
47C23×C4C2312yes—
48C23×C4C2311yescharacteristic
49C2×C4⋊C4—24no—
50C2×C4⋊C4—24no—
51C2×C4⋊C4—24no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—24no—
54C2×C4⋊C4—24no—
55C2×C4⋊C4—41no—
56C2×C4⋊C4—41no—
57C2×C4⋊C4—41no—
58C2×C4⋊C4—41no—
59C2×C4⋊C4—42no—
60C2×C4⋊C4—41no—
61C2×C4⋊C4—41no—
62C2×C4⋊C4D414yes—
63C2×C4⋊C4D414yescomplemented
64C2×C22⋊C4—24no—
65C2×C22⋊C4—24no—
66C2×C22⋊C4—24no—
67C2×C22⋊C4—24no—
68C2×C22⋊C4—24no—
69C2×C22⋊C4—24no—
70C2×C22⋊C4—24no—
71C2×C22⋊C4—24no—
72C2×C22⋊C4—24no—
73C2×C22⋊C4—24no—
74C2×C22⋊C4—24no—
75C2×C22⋊C4—24no—
76C2×C22⋊C4—42no—
77C2×C22⋊C4—42no—
78C2×C22⋊C4—42no—
79C2×C22⋊C4—42no—
80C2×C22⋊C4—42no—
81C2×C22⋊C4—42no—
82C2×C42—41no—
83C2×C42—41no—
84C2×C42D414yescomplemented
85#95—24no—
86#95—24no—
87#95—22no—
88#95—24no—
89#95—22no—
90#95—24no—
91#95—42no—
92#95—42no—
93C24—22no—
94C24—22no—
95C24—22no—
96C24—22no—
97C24—22no—
98C24—22no—
99C24—22no—
100C24—22no—
101C24—22no—
102C24—22no—
103C24—22no—
104C24—22no—
105C24—22no—
106C24—22no—
107C24#4114yes—
108C24C2411yescharacteristic
109C2×D4—48no—
110C2×D4—44no—
111C2×D4—44no—
112C2×D4—48no—
113C2×D4—44no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—22no—
143C22×C4—22no—
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×C4C2×D412yes—
159C22×C4C2×D414yes—
160C22×C4C2×D412yes—
161C22×C4C2×D414yes—
162C22×C4C2×D412yes—
163C22×C4C2×D412yes—
164C22×C4C2×D412yes—
165C22×C4#4114yes—
166C4⋊C4—44no—
167C4⋊C4—44no—
168C4⋊C4—48no—
169C4⋊C4—44no—
170C4⋊C4—48no—
171C22⋊C4—216no—
172C22⋊C4—48no—
173C22⋊C4—48no—
174C22⋊C4—48no—
175C22⋊C4—48no—
176C22⋊C4—48no—
177C42—44no—
178C23—24no—
179C23—24no—
180C23—24no—
181C23—24no—
182C23—24no—
183C23—24no—
184C23—24no—
185C23—42no—
186C23—42no—
187C23—42no—
188C23—42no—
189C23—42no—
190C23—42no—
191C23—42no—
192C23—42no—
193C23—42no—
194C23—42no—
195C23—42no—
196C23—42no—
197C23—42no—
198C23—42no—
199C23—42no—
200C23—42no—
201C23—42no—
202C23—42no—
203C23—42no—
204C23—42no—
205C23—42no—
206C23—42no—
207C23—42no—
208C23—42no—
209C23—42no—
210C23—42no—
211C23—42no—
212C23—42no—
213C23C22×D411yescharacteristic
214C23C22×D411yescharacteristic
215C23C22×D411yescharacteristic
216C23C2×#4112yes—
217C23C2×#4112yes—
218C23H(5,𝔽2)11yescharacteristic
219C23H(5,𝔽2)11yescharacteristic
220C23H(5,𝔽2)11yescharacteristic
221C23H(5,𝔽2)12yes—
222C232−1+411yescharacteristic
223C232−1+412yes—
224D4—416no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—24no—
232C2×C4—24no—
233C2×C4—24no—
234C2×C4—24no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—24no—
238C2×C4—24no—
239C2×C4—24no—
240C2×C4—24no—
241C2×C4—24no—
242C2×C4—24no—
243C2×C4—24no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4#12714yes—
277C22—44no—
278C22—44no—
279C22—44no—
280C22—44no—
281C22—44no—
282C22—44no—
283C22—44no—
284C22—82no—
285C22—82no—
286C22—82no—
287C22—82no—
288C22—82no—
289C22—82no—
290C22—82no—
291C22—82no—
292C22#52512yes—
293C22C2×#12711yescharacteristic
294C22#53411yescharacteristic
295C22#53411yescharacteristic
296C22#53411yescharacteristic
297C22#53711yescharacteristic
298C22#53712yes—
299C22#53711yescharacteristic
300C22#53711yescharacteristic
301C22#53812yes—
302C22#53912yes—
303C22#54012yes—
304C22#54212yes—
305C22#54612yes—
306C22#54612yes—
307C22#54812yes—
308C22#54812yes—
309C22#55111yescharacteristic
310C22#55212yes—
311C22#55511yescharacteristic
312C22#56011yescharacteristic
313C22#56312yes—
314C22#56311yescharacteristic
315C4—24no—
316C4—44no—
317C4—42no—
318C4—42no—
319C4—44no—
320C4—44no—
321C4—82no—
322C4—82no—
323C2—84no—
324C2#254012yes—
325C2#260911yescharacteristic
326C2#263012yes—
327C2#265611yescharacteristic
328C2#269711yescharacteristic
329C2#269911yescharacteristic
330C2#270212yes—
331C2#270712yes—
332C2#272811yescharacteristic
333C2#273111yescharacteristic
334C2#282311yescharacteristic
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×#398C2semidirect
2. C2×#396C2semidirect
4. C2×#388C2semidirect
5. #1445C2semidirect
6. #1439C2semidirect
7. #1439C2semidirect
9. C4⋊#95C2semidirect
63. C2×C4⋊C4D4semidirect
84. C2×C42D4semidirect

Maximal subgroups

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

subgroupindex
C2×#3982
C2×#3962
C2×#3942
C2×#3882
#14452
#14392
#14382
C4⋊#952