Finite Group Databasebeta

← #15833

Subgroups — #15833

Order 256.

893 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 893 conjugacy classes of subgroups — too many to draw. Only its 179 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×#394C211yesmaximal, characteristic
2C2×#393C212yesmaximal, complemented
3C2×#391C211yesmaximal, characteristic, complemented
4C2×#388C212yesmaximal, complemented
5#1445C212yesmaximal, complemented
6#1439C212yesmaximal, complemented
7#1438C212yesmaximal
8#1437C212yesmaximal, complemented
9#1436C211yesmaximal, characteristic, complemented
10C23×Q8C2211yescharacteristic
11C23×D4C2211yescharacteristic
12C22×C4⋊C4C2212yes—
13C22×C4⋊C4C2211yescharacteristic
14C22×C22⋊C4C2212yes—
15C22×C22⋊C4C2212yes—
16C22×C22⋊C4C2212yes—
17C22×C22⋊C4C2212yes—
18C22×C22⋊C4C2212yes—
19C22×C22⋊C4C2212yes—
20C22×C42C2211yescharacteristic
21C22×C42C2211yescharacteristic
22#394—24no—
23#393—28no—
24#391—24no—
25#388—28no—
26C2×#95C2212yes—
27C2×#95C2212yes—
28C2×#95C2211yescharacteristic
29C2×#95C2212yes—
30C2×#95C2212yes—
31C2×#95C2212yes—
32C2×#95C2212yes—
33C2×#95C2211yescharacteristic
34C2×#95C2212yes—
35C25C2312yes—
36C22×Q8—22no—
37C22×Q8—24no—
38C22×Q8—24no—
39C22×Q8—41no—
40C22×Q8—41no—
41C22×D4—24no—
42C22×D4—22no—
43C22×D4—42no—
44C22×D4—41no—
45C22×D4—41no—
46C23×C4C2312yes—
47C23×C4C2312yes—
48C23×C4C2311yescharacteristic
49C23×C4C2312yes—
50C23×C4C2311yescharacteristic
51C23×C4C2312yes—
52C23×C4C2312yes—
53C23×C4C2311yescharacteristic
54C2×C4⋊C4—24no—
55C2×C4⋊C4—24no—
56C2×C4⋊C4—41no—
57C2×C4⋊C4—41no—
58C2×C4⋊C4—42no—
59C2×C4⋊C4D414yes—
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—24no—
69C2×C22⋊C4—24no—
70C2×C22⋊C4—24no—
71C2×C22⋊C4—24no—
72C2×C22⋊C4—24no—
73C2×C22⋊C4—24no—
74C2×C22⋊C4—42no—
75C2×C22⋊C4—42no—
76C2×C22⋊C4—42no—
77C2×C22⋊C4—42no—
78C2×C42—24no—
79C2×C42—22no—
80C2×C42—42no—
81C2×C42—41no—
82#95—24no—
83#95—24no—
84#95—24no—
85#95—24no—
86#95—24no—
87#95—22no—
88#95—42no—
89#95—41no—
90#95D418yescomplemented
91C24—22no—
92C24—22no—
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#4114yes—
106C24C2411yescharacteristic
107C2×Q8—28no—
108C2×Q8—48no—
109C2×Q8—48no—
110C2×Q8—44no—
111C2×Q8—44no—
112C2×D4—48no—
113C2×D4—44no—
114C2×D4—44no—
115C2×D4—48no—
116C2×D4—44no—
117C22×C4—22no—
118C22×C4—21no—
119C22×C4—21no—
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—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—21no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—21no—
149C22×C4—21no—
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×C4—22no—
159C22×C4C2×D412yes—
160C22×C4C2×D414yes—
161C22×C4C2×D412yes—
162C22×C4C2×D414yes—
163C22×C4C2×D414yes—
164C22×C4C2×D412yes—
165C22×C4#4114yes—
166C22×C4#4114yes—
167C4⋊C4—44no—
168C4⋊C4—48no—
169C22⋊C4—216no—
170C22⋊C4—48no—
171C22⋊C4—48no—
172C22⋊C4—48no—
173C22⋊C4—48no—
174C22⋊C4—48no—
175C42—44no—
176C42—44no—
177C23—24no—
178C23—24no—
179C23—24no—
180C23—24no—
181C23—24no—
182C23—24no—
183C23—24no—
184C23—42no—
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—
212C23C22×D411yescharacteristic
213C23C22×D412yes—
214C23C2×#4112yes—
215C23C2×#4112yes—
216C23C2×#4112yes—
217C23H(5,𝔽2)11yescharacteristic
218C23H(5,𝔽2)11yescharacteristic
219C23H(5,𝔽2)12yes—
220C232−1+412yes—
221Q8—416no—
222D4—416no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—24no—
226C2×C4—24no—
227C2×C4—24no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—24no—
233C2×C4—24no—
234C2×C4—24no—
235C2×C4—24no—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—24no—
239C2×C4—24no—
240C2×C4—24no—
241C2×C4—24no—
242C2×C4—24no—
243C2×C4—42no—
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—42no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4#12714yes—
273C22—44no—
274C22—44no—
275C22—44no—
276C22—44no—
277C22—44no—
278C22—44no—
279C22—44no—
280C22—82no—
281C22—82no—
282C22—82no—
283C22—82no—
284C22—82no—
285C22—82no—
286C22—82no—
287C22—82no—
288C22#52511yescharacteristic
289C22#52512yes—
290C22#52912yes—
291C22C2×#12711yescharacteristic
292C22#53411yescharacteristic
293C22#53712yes—
294C22#53812yes—
295C22#54212yes—
296C22#54212yes—
297C22#54612yes—
298C22#54612yes—
299C22#54612yes—
300C22#54612yes—
301C22#54812yes—
302C22#54812yes—
303C22#55111yescharacteristic
304C22#55212yes—
305C22#55511yescharacteristic
306C22#55912yes—
307C22H(3,𝔽4)11yescharacteristic
308C22#56311yescharacteristic
309C4—24no—
310C4—42no—
311C4—42no—
312C4—44no—
313C4—44no—
314C4—44no—
315C4—44no—
316C4—82no—
317C2—84no—
318C2#245812yes—
319C2#245912yes—
320C2#247712yes—
321C2#253911yescharacteristic
322C2#254012yes—
323C2#268411yescharacteristic
324C2#270212yes—
325C2#271212yes—
326C2#273111yescharacteristic
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×#393C2semidirect
3. C2×#391C2semidirect
4. C2×#388C2semidirect
5. #1445C2semidirect
6. #1439C2semidirect
8. #1437C2semidirect
9. #1436C2semidirect
90. #95D4semidirect

Maximal subgroups

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

subgroupindex
C2×#3942
C2×#3932
C2×#3912
C2×#3882
#14452
#14392
#14382
#14372
#14362