Finite Group Databasebeta

← #23937

Subgroups — #23937

Order 256.

652 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 652 conjugacy classes of subgroups — too many to draw. Only its 141 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×#529C211yesmaximal, characteristic
2C2×#451C211yesmaximal, characteristic, complemented
3C2×#448C211yesmaximal, characteristic
4C2×#391C211yesmaximal, characteristic, complemented
5#2026C212yesmaximal, complemented
6#2015C212yesmaximal, complemented
7#1878C212yesmaximal, complemented
8#1877C212yesmaximal, complemented
9#1874C211yesmaximal, characteristic
10#1872C211yesmaximal, characteristic, complemented
11#1817C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C22×Q16C2212yes—
14C22×SD16C2212yes—
15#529—24no—
16#529—24no—
17C2×#126—21no—
18C2×#125C2211yescharacteristic
19C2×#124C2211yescharacteristic
20C2×#123—21no—
21C2×C22⋊Q8C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×C4×Q8C2211yescharacteristic
24C2×C4×D4—21no—
25C2×C4×D4C2211yescharacteristic
26C2×#117—21no—
27C22×C4⋊C4C2211yescharacteristic
28C22×C42C2211yescharacteristic
29#451—24no—
30#448—24no—
31C2×#103C2212yes—
32C2×#103C2212yes—
33C2×#103C2212yes—
34C2×#102C2212yes—
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37#391—21no—
38#391—21no—
39#391C2212yes—
40#391C2212yes—
41C2×#95—21no—
42C2×#95—21no—
43#340C2212yes—
44#340C2212yes—
45#336C2212yes—
46#336C2212yes—
47C22×Q8—24no—
48C22×Q8—21no—
49C22×Q8—22no—
50C22×Q8—22no—
51C22×Q8—42no—
52C22×Q8C2311yescharacteristic
53C22×Q8C2312yes—
54C22×D4—21no—
55C22×D4C2311yescharacteristic
56C23×C4—21no—
57C23×C4—21no—
58C23×C4—21no—
59C23×C4D411yescharacteristic
60C23×C4D411yescharacteristic
61C23×C4C2311yescharacteristic
62C2×Q16—24no—
63C2×Q16—24no—
64C2×Q16—42no—
65C2×SD16—24no—
66C2×SD16—24no—
67C2×SD16—42no—
68C22×C8C2312yes—
69C22×C8C2312yes—
70#126—28no—
71#125—22no—
72#125—22no—
73#124—22no—
74#124—22no—
75#123—28no—
76C22⋊Q8—22no—
77C22⋊Q8D414yes—
78#121—22no—
79#121D414yes—
80C4×Q8—22no—
81C4×Q8—22no—
82C4×D4—28no—
83C4×D4—22no—
84C4×D4—22no—
85#117—28no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4C2312yes—
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C42—21no—
101C2×C42—22no—
102C2×C42—22no—
103C2×C42D411yescharacteristic
104C2×C42D411yescharacteristic
105C2×C42C2311yescharacteristic
106C2×C42C2311yescharacteristic
107#103—24no—
108#103—24no—
109#103—42no—
110#102—42no—
111C22⋊C8—22no—
112C22⋊C8—22no—
113#95—22no—
114#95—22no—
115#95—22no—
116#95—22no—
117C24—21no—
118C24C2×D411yescharacteristic
119C2×Q8—24no—
120C2×Q8—21no—
121C2×Q8—22no—
122C2×Q8—22no—
123C2×Q8—22no—
124C2×Q8—22no—
125C2×Q8—22no—
126C2×Q8—22no—
127C2×Q8—22no—
128C2×Q8—48no—
129C2×Q8—48no—
130C2×Q8—42no—
131C2×Q8—42no—
132C2×Q8C2×D412yes—
133C2×Q8#4114yes—
134C2×D4—24no—
135C2×D4—21no—
136C2×D4—22no—
137C2×D4—22no—
138C2×D4—42no—
139C2×D4—42no—
140C2×D4C2×D412yes—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—24no—
144C22×C4—24no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—41no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167C22×C4C2×D412yes—
168C22×C4C2×D412yes—
169C22×C4C2×D411yescharacteristic
170C22×C4C2411yescharacteristic
171Q16—48no—
172SD16—48no—
173C2×C8—22no—
174C2×C8—22no—
175C2×C8—22no—
176C2×C8—22no—
177C2×C8—22no—
178C2×C8—22no—
179C4⋊C4—24no—
180C4⋊C4—24no—
181C4⋊C4—24no—
182C4⋊C4—24no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4—42no—
188C4⋊C4C2×D412yes—
189C4⋊C4C2×D412yes—
190C22⋊C4—24no—
191C22⋊C4—24no—
192C22⋊C4—24no—
193C22⋊C4—24no—
194C22⋊C4—24no—
195C42—24no—
196C42—22no—
197C42—22no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—21no—
202C23—21no—
203C23—21no—
204C23—21no—
205C23—21no—
206C23—22no—
207C23—41no—
208C23—41no—
209C23—41no—
210C23—41no—
211C23—41no—
212C23—41no—
213C23#12012yes—
214C23#13512yes—
215C23#13712yes—
216C23C22×D411yescharacteristic
217Q8—24no—
218Q8—24no—
219Q8—24no—
220Q8—48no—
221Q8—42no—
222Q8—42no—
223Q8—42no—
224D4—24no—
225D4—48no—
226D4—42no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—24no—
232C2×C4—24no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—24no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—44no—
247C2×C4—48no—
248C2×C4—48no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—44no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—84no—
262C2×C4#12012yes—
263C2×C4#13712yes—
264C2×C4C22×D411yescharacteristic
265C2×C4C22×D411yescharacteristic
266C2×C4C2×#4112yes—
267C2×C4H(5,𝔽2)12yes—
268C8—42no—
269C8—42no—
270C22—22no—
271C22—22no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—81no—
280C22—81no—
281C22—81no—
282C22—81no—
283C22#45214yes—
284C22C2×#12011yescharacteristic
285C22#52511yescharacteristic
286C22#53511yescharacteristic
287C22C2×#13511yescharacteristic
288C22C2×#13711yescharacteristic
289C22C2×#13711yescharacteristic
290C22#57511yescharacteristic
291C4—22no—
292C4—22no—
293C4—44no—
294C4—42no—
295C4—41no—
296C4—41no—
297C4—42no—
298C4—41no—
299C4—42no—
300C4—82no—
301C4#54612yes—
302C4#54612yes—
303C2—24no—
304C2—82no—
305C2#240411yescharacteristic
306C2C2×#45211yescharacteristic
307C2#300711yescharacteristic
308C2#310811yescharacteristic
309C2#311711yescharacteristic
310C2#321111yescharacteristic
311C2#321811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#451C2semidirect
4. C2×#391C2semidirect
5. #2026C2semidirect
6. #2015C2semidirect
7. #1878C2semidirect
8. #1877C2semidirect
10. #1872C2semidirect
11. #1817C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5292
C2×#4512
C2×#4482
C2×#3912
#20262
#20152
#18782
#18772
#18742
#18722
#18172