Finite Group Databasebeta

← #15952

Subgroups — #15952

Order 256.

931 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 931 conjugacy classes of subgroups — too many to draw. Only its 175 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#386C212yesmaximal
2C2×#381C212yesmaximal, complemented
3#1445C211yesmaximal, characteristic, complemented
4#1443C212yesmaximal, complemented
5#1442C211yesmaximal, characteristic, complemented
6#1439C212yesmaximal, complemented
7#1439C212yesmaximal, complemented
8#1438C211yesmaximal, characteristic
9C22⋊#95C212yesmaximal, complemented
10C23×D4C2211yescharacteristic
11C24×C4C2212yes—
12C22×C4⋊C4C2211yescharacteristic
13C22×C4⋊C4C2211yescharacteristic
14C22×C22⋊C4C2212yes—
15C22×C22⋊C4C2212yes—
16C22×C22⋊C4C2212yes—
17C22×C22⋊C4C2212yes—
18C22×C22⋊C4C2212yes—
19#386—28no—
20#381—28no—
21C2×#95C2212yes—
22C2×#95C2212yes—
23C2×#95C2212yes—
24C2×#95C2212yes—
25C2×#95C2212yes—
26C2×#95C2212yes—
27C2×#95C2211yescharacteristic
28C2×#95C2212yes—
29C2×#95C2211yescharacteristic
30C2×#95C2211yescharacteristic
31C2×#95C2212yes—
32C2×#95C2211yescharacteristic
33C25C2312yes—
34C22×D4—24no—
35C22×D4—24no—
36C22×D4—41no—
37C22×D4—41no—
38C22×D4D414yes—
39C23×C4—24no—
40C23×C4—24no—
41C23×C4—24no—
42C23×C4—24no—
43C23×C4—24no—
44C23×C4—42no—
45C23×C4—42no—
46C23×C4C2312yes—
47C23×C4C2312yes—
48C23×C4C2312yes—
49C23×C4C2311yescharacteristic
50C23×C4C2312yes—
51C23×C4C2311yescharacteristic
52C23×C4C2311yescharacteristic
53C23×C4C2312yes—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—41no—
56C2×C4⋊C4—41no—
57C2×C4⋊C4—41no—
58C2×C4⋊C4—42no—
59C2×C22⋊C4—24no—
60C2×C22⋊C4—24no—
61C2×C22⋊C4—24no—
62C2×C22⋊C4—24no—
63C2×C22⋊C4—24no—
64C2×C22⋊C4—24no—
65C2×C22⋊C4—24no—
66C2×C22⋊C4—24no—
67C2×C22⋊C4—24no—
68C2×C22⋊C4—42no—
69C2×C22⋊C4—42no—
70C2×C22⋊C4—42no—
71C2×C22⋊C4—42no—
72C2×C22⋊C4—42no—
73C2×C22⋊C4—42no—
74#95—24no—
75#95—24no—
76#95—24no—
77#95—24no—
78#95—24no—
79#95—24no—
80#95—42no—
81#95—41no—
82#95—41no—
83#95—41no—
84#95—42no—
85#95D414yescomplemented
86C24—22no—
87C24—22no—
88C24—22no—
89C24—22no—
90C24—22no—
91C24—22no—
92C24—22no—
93C24—22no—
94C24—22no—
95C24—22no—
96C24—22no—
97C24—22no—
98C24C2×D414yes—
99C24#4114yes—
100C24#4114yes—
101C24C2411yescharacteristic
102C2×D4—216no—
103C2×D4—44no—
104C2×D4—44no—
105C2×D4—44no—
106C2×D4—48no—
107C22×C4—216no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
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—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—48no—
154C22×C4—48no—
155C22×C4—48no—
156C22×C4—48no—
157C22×C4—48no—
158C22×C4—48no—
159C22×C4C2×D412yes—
160C22×C4C2×D412yes—
161C22×C4C2×D414yes—
162C22×C4#4114yes—
163C22×C4#4112yes—
164C4⋊C4—44no—
165C4⋊C4—44no—
166C22⋊C4—48no—
167C22⋊C4—48no—
168C22⋊C4—48no—
169C22⋊C4—48no—
170C22⋊C4—48no—
171C23—24no—
172C23—24no—
173C23—24no—
174C23—24no—
175C23—24no—
176C23—24no—
177C23—24no—
178C23—24no—
179C23—24no—
180C23—24no—
181C23—24no—
182C23—24no—
183C23—24no—
184C23—24no—
185C23—24no—
186C23—24no—
187C23—24no—
188C23—24no—
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#12318yes—
206C23C22×D411yescharacteristic
207C23C22×D411yescharacteristic
208C23C2×#4112yes—
209C23C2×#4112yes—
210C23C2×#4112yes—
211C23C2×#4111yescharacteristic
212C23H(5,𝔽2)11yescharacteristic
213C23H(5,𝔽2)11yescharacteristic
214C23H(5,𝔽2)11yescharacteristic
215C23H(5,𝔽2)12yes—
216C232−1+411yescharacteristic
217D4—416no—
218C2×C4—24no—
219C2×C4—24no—
220C2×C4—24no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—24no—
225C2×C4—24no—
226C2×C4—24no—
227C2×C4—24no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—432no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—41no—
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—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C22—28no—
270C22—28no—
271C22—28no—
272C22—44no—
273C22—44no—
274C22—44no—
275C22—44no—
276C22—44no—
277C22—44no—
278C22—44no—
279C22—44no—
280C22—44no—
281C22—44no—
282C22—44no—
283C22—44no—
284C22C2×#12312yes—
285C22#52512yes—
286C22#52912yes—
287C22#53411yescharacteristic
288C22#53711yescharacteristic
289C22#53811yescharacteristic
290C22#53812yes—
291C22#53912yes—
292C22#54012yes—
293C22D4211yescharacteristic
294C22#54612yes—
295C22#54612yes—
296C22#54711yescharacteristic
297C22#55011yescharacteristic
298C22#55112yes—
299C22#55111yescharacteristic
300C22#55111yescharacteristic
301C22#55112yes—
302C22#55112yes—
303C22#55212yes—
304C22#55511yescharacteristic
305C22#56011yescharacteristic
306C22#56211yescharacteristic
307C4—44no—
308C4—44no—
309C4—42no—
310C4—42no—
311C4—42no—
312C4—82no—
313C4—82no—
314C4—82no—
315C2—48no—
316C2#241812yes—
317C2#241912yes—
318C2#247112yes—
319C2#248311yescharacteristic
320C2#257211yescharacteristic
321C2#260911yescharacteristic
322C2#268211yescharacteristic
323C2#269311yescharacteristic
324C2#269911yescharacteristic
325C2#274312yes—
326C2#276011yescharacteristic
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
2. C2×#381C2semidirect
3. #1445C2semidirect
4. #1443C2semidirect
5. #1442C2semidirect
6. #1439C2semidirect
7. #1439C2semidirect
9. C22⋊#95C2semidirect
85. #95D4semidirect

Maximal subgroups

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

subgroupindex
C2×#3862
C2×#3812
#14452
#14432
#14422
#14392
#14382
C22⋊#952