Finite Group Databasebeta

← #23856

Subgroups — #23856

Order 256.

867 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 867 conjugacy classes of subgroups — too many to draw. Only its 167 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
1C23×Q16C211yesmaximal, characteristic, complemented
2C2×D4×Q8C211yesmaximal, characteristic
3C2×#451C212yesmaximal, complemented
4C2×#451C211yesmaximal, characteristic, complemented
5#2409C211yesmaximal, characteristic
6#2016C212yesmaximal, complemented
7#2001C212yesmaximal, complemented
8#1892C211yesmaximal, characteristic, complemented
9#1878C212yesmaximal, complemented
10#1869C211yesmaximal, characteristic
11#1787C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C23×Q8C2211yescharacteristic
14C23×Q8C2211yescharacteristic
15C22×Q16—24no—
16C22×Q16—28no—
17C22×Q16C2212yes—
18C22×Q16C2212yes—
19C22×Q16C2212yes—
20C23×C8C2211yescharacteristic
21D4×Q8—24no—
22D4×Q8—24no—
23C2×C4⋊Q8—21no—
24C2×C4⋊Q8C2211yescharacteristic
25C2×C22⋊Q8—21no—
26C2×C22⋊Q8—21no—
27C2×C22⋊Q8C2211yescharacteristic
28C2×C22⋊Q8C2212yes—
29C2×C22⋊Q8C2211yescharacteristic
30C2×C4×Q8C2211yescharacteristic
31C2×C4×D4—21no—
32C2×C4×D4C2211yescharacteristic
33#451—28no—
34#451—24no—
35C2×#103C2212yes—
36C2×#103C2212yes—
37C2×#103C2212yes—
38C2×#103C2212yes—
39C2×C22⋊C8C2212yes—
40C2×C22⋊C8C2211yescharacteristic
41#395—21no—
42#393—21no—
43#391C2212yes—
44#386—21no—
45#385—21no—
46#380C2211yescharacteristic
47#340C2212yes—
48#336C2212yes—
49C22×Q8—22no—
50C22×Q8—22no—
51C22×Q8—21no—
52C22×Q8—21no—
53C22×Q8—22no—
54C22×Q8—21no—
55C22×Q8—24no—
56C22×Q8—24no—
57C22×Q8—24no—
58C22×Q8—22no—
59C22×Q8—22no—
60C22×Q8—42no—
61C22×Q8—42no—
62C22×Q8—42no—
63C22×Q8D414yes—
64C22×Q8C2311yescharacteristic
65C22×Q8C2311yescharacteristic
66C22×Q8C2312yes—
67C22×Q8C2311yescharacteristic
68C22×D4D411yescharacteristic
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4C2311yescharacteristic
73C23×C4C2311yescharacteristic
74C2×Q16—232no—
75C2×Q16—24no—
76C2×Q16—24no—
77C2×Q16—24no—
78C2×Q16—24no—
79C2×Q16—24no—
80C2×Q16—24no—
81C2×Q16—24no—
82C2×Q16—42no—
83C2×Q16D418yescomplemented
84C22×C8—22no—
85C22×C8—24no—
86C22×C8C2312yes—
87C22×C8C2312yes—
88C4⋊Q8—28no—
89C4⋊Q8—22no—
90C4⋊Q8—22no—
91C22⋊Q8—28no—
92C22⋊Q8—28no—
93C22⋊Q8—22no—
94C22⋊Q8—22no—
95C22⋊Q8—22no—
96C22⋊Q8—24no—
97C22⋊Q8—42no—
98C22⋊Q8D414yes—
99C4×Q8—22no—
100C4×Q8—22no—
101C4×D4—28no—
102C4×D4—22no—
103C4×D4—22no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4D411yescharacteristic
111C2×C4⋊C4C2312yes—
112C2×C4⋊C4C2311yescharacteristic
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4D411yescharacteristic
118C2×C22⋊C4D411yescharacteristic
119C2×C42—21no—
120C2×C42C2311yescharacteristic
121#103—24no—
122#103—24no—
123#103—24no—
124#103—42no—
125C22⋊C8—22no—
126C22⋊C8—24no—
127#95—21no—
128#95—22no—
129#95—22no—
130C24C2×D411yescharacteristic
131C24C2×D411yescharacteristic
132C2×Q8—24no—
133C2×Q8—24no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—22no—
138C2×Q8—22no—
139C2×Q8—28no—
140C2×Q8—28no—
141C2×Q8—22no—
142C2×Q8—22no—
143C2×Q8—22no—
144C2×Q8—22no—
145C2×Q8—22no—
146C2×Q8—22no—
147C2×Q8—22no—
148C2×Q8—22no—
149C2×Q8—44no—
150C2×Q8—44no—
151C2×Q8—44no—
152C2×Q8—44no—
153C2×Q8—48no—
154C2×Q8—48no—
155C2×Q8—44no—
156C2×Q8—48no—
157C2×Q8—44no—
158C2×Q8—44no—
159C2×Q8—42no—
160C2×Q8C2×D412yes—
161C2×Q8C2×D414yes—
162C2×Q8C2×D412yes—
163C2×Q8C2×D414yes—
164C2×Q8C2×D412yes—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4Q1614yescomplemented
168C22×C4—21no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—24no—
172C22×C4—24no—
173C22×C4—22no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—22no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—44no—
183C22×C4—44no—
184C22×C4—44no—
185C22×C4—42no—
186C22×C4C2×D411yescharacteristic
187C22×C4C2×D411yescharacteristic
188C22×C4C2×D411yescharacteristic
189C22×C4C2×D412yes—
190C22×C4C2×D412yes—
191C22×C4C2×D411yescharacteristic
192C22×C4C2411yescharacteristic
193Q16—216no—
194Q16—216no—
195Q16—48no—
196C2×C8—28no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8—22no—
202C2×C8C2×D414yes—
203C4⋊C4—24no—
204C4⋊C4—24no—
205C4⋊C4—24no—
206C4⋊C4—24no—
207C4⋊C4—24no—
208C4⋊C4—22no—
209C4⋊C4—21no—
210C4⋊C4—22no—
211C4⋊C4—22no—
212C4⋊C4—44no—
213C4⋊C4C2×D412yes—
214C22⋊C4—24no—
215C22⋊C4—24no—
216C22⋊C4—22no—
217C22⋊C4—22no—
218C22⋊C4—44no—
219C22⋊C4—42no—
220C42—24no—
221C42—21no—
222C42—21no—
223C23—22no—
224C23—22no—
225C23—21no—
226C23—22no—
227C23—21no—
228C23—22no—
229C23#12012yes—
230C23C2×Q1612yes—
231C23C2×Q1612yescomplemented
232C23#13712yes—
233C23C22×D411yescharacteristic
234Q8—22no—
235Q8—24no—
236Q8—22no—
237Q8—24no—
238Q8—22no—
239Q8—24no—
240Q8—24no—
241Q8—48no—
242Q8—48no—
243Q8—48no—
244Q8—41no—
245Q8—41no—
246Q8—42no—
247Q8—42no—
248Q8—42no—
249Q8#12014yes—
250D4—28no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—24no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—24no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—24no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—48no—
268C2×C4—48no—
269C2×C4—48no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4#12012yes—
282C2×C4C2×Q1612yescomplemented
283C2×C4C22×D411yescharacteristic
284C2×C4C22×D412yes—
285C2×C4C22×D411yescharacteristic
286C2×C4H(5,𝔽2)12yes—
287C8—24no—
288C8—42no—
289C22—22no—
290C22—22no—
291C22—24no—
292C22—24no—
293C22—22no—
294C22—24no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22#45114yescomplemented
299C22C2×#12011yescharacteristic
300C22D4212yes—
301C22C22×Q1611yescharacteristic
302C22C2×#13711yescharacteristic
303C22#57812yes—
304C4—22no—
305C4—24no—
306C4—22no—
307C4—41no—
308C4—42no—
309C4—42no—
310C4—44no—
311C4—41no—
312C4—42no—
313C4—82no—
314C4C2×#12011yescharacteristic
315C4#53411yescharacteristic
316C4D4212yes—
317C2—24no—
318C2—42no—
319C2#240311yescharacteristic
320C2C2×#45111yescharacteristic
321C2#301111yescharacteristic
322C2#327012yes—
323C2D4×Q1612yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C23×Q16C2semidirect
3. C2×#451C2semidirect
4. C2×#451C2semidirect
6. #2016C2semidirect
7. #2001C2semidirect
8. #1892C2semidirect
9. #1878C2semidirect
11. #1787C2semidirect
83. C2×Q16D4semidirect
167. C2×D4Q16semidirect
231. C23C2×Q16semidirect
282. C2×C4C2×Q16semidirect
298. C22#451semidirect

Maximal subgroups

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

subgroupindex
C23×Q162
C2×D4×Q82
C2×#4512
#24092
#20162
#20012
#18922
#18782
#18692
#17872