Finite Group Databasebeta

← #24035

Subgroups — #24035

Order 256.

682 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 143 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×#548C211yesmaximal, characteristic
2C2×#480C211yesmaximal, characteristic, complemented
3C22×#103C211yesmaximal, characteristic, complemented
4#2486C211yesmaximal, characteristic, complemented
5#2042C212yesmaximal, complemented
6#2036C212yesmaximal, complemented
7#1893C211yesmaximal, characteristic, complemented
8#1869C211yesmaximal, characteristic, complemented
9#1863C212yesmaximal, complemented
10#1788C212yesmaximal, complemented
11#1787C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C23×Q8C2211yescharacteristic
14C23×C8C2211yescharacteristic
15#548—24no—
16#548—24no—
17C2×#124—21no—
18C2×#124C2211yescharacteristic
19C2×#123—21no—
20C2×C22⋊Q8—21no—
21C2×C22⋊Q8C2211yescharacteristic
22C2×#121—21no—
23C2×#121C2211yescharacteristic
24C2×C4×Q8C2211yescharacteristic
25C2×C4×D4—21no—
26C2×C4×D4C2211yescharacteristic
27C22×C4⋊C4C2211yescharacteristic
28#480—24no—
29C2×#107C2212yes—
30C2×#103—24no—
31C2×#103C2212yes—
32C2×#103C2212yes—
33C2×#103C2212yes—
34C2×#103C2212yes—
35C2×#102C2212yes—
36C2×#102C2212yes—
37C2×C22⋊C8C2211yescharacteristic
38C2×C22⋊C8C2212yes—
39#398—21no—
40#397—21no—
41#389C2212yes—
42#381—21no—
43#380C2211yescharacteristic
44#380C2212yes—
45#340C2212yes—
46#336C2212yes—
47C2×#41—22no—
48C2×#41—22no—
49C2×#41—24no—
50C2×#41—42no—
51C22×Q8—21no—
52C22×Q8—22no—
53C22×Q8—24no—
54C22×Q8—42no—
55C22×Q8D414yes—
56C22×Q8C2311yescharacteristic
57C22×Q8C2311yescharacteristic
58C22×D4—21no—
59C22×D4—21no—
60C22×D4D411yescharacteristic
61C22×D4C2311yescharacteristic
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4C2311yescharacteristic
66C23×C4C2311yescharacteristic
67C22×C8—22no—
68C22×C8—42no—
69C22×C8C2312yes—
70C22×C8C2312yes—
71#124—28no—
72#124—22no—
73#124—22no—
74#123—44no—
75C22⋊Q8—28no—
76C22⋊Q8—22no—
77C22⋊Q8—22no—
78#121—28no—
79#121—22no—
80#121D414yes—
81C4×Q8—22no—
82C4×Q8—22no—
83C4×D4—28no—
84C4×D4—22no—
85C4×D4—22no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4D411yescharacteristic
93C2×C4⋊C4C2312yes—
94C2×C4⋊C4C2312yes—
95C2×C4⋊C4C2311yescharacteristic
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4D411yescharacteristic
103C2×C22⋊C4D411yescharacteristic
104C2×C42—21no—
105C2×C42C2311yescharacteristic
106#107—24no—
107#103—24no—
108#103—24no—
109#103—24no—
110#103—42no—
111#102—24no—
112#102—42no—
113C22⋊C8—22no—
114C22⋊C8—42no—
115#95—22no—
116#95—21no—
117#95—22no—
118C24—21no—
119C24C2×D411yescharacteristic
120C24C2×D411yescharacteristic
121#41—48no—
122#41—44no—
123#41—44no—
124C2×Q8—24no—
125C2×Q8—21no—
126C2×Q8—21no—
127C2×Q8—22no—
128C2×Q8—28no—
129C2×Q8—22no—
130C2×Q8—22no—
131C2×Q8—44no—
132C2×Q8—44no—
133C2×Q8—48no—
134C2×Q8C2×D412yes—
135C2×Q8C2×D412yes—
136C2×Q8#4114yes—
137C2×D4—24no—
138C2×D4—22no—
139C2×D4—24no—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4—21no—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4—44no—
146C2×D4—42no—
147C2×D4—42no—
148C2×D4C2×D412yes—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—24no—
153C22×C4—24no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—44no—
163C22×C4—44no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D411yescharacteristic
171C22×C4C2×D412yes—
172C22×C4C2×D412yes—
173C22×C4C2411yescharacteristic
174C2×C8—22no—
175C2×C8—22no—
176C2×C8—22no—
177C2×C8—22no—
178C2×C8—22no—
179C2×C8—22no—
180C2×C8—44no—
181C4⋊C4—24no—
182C4⋊C4—24no—
183C4⋊C4—22no—
184C4⋊C4—21no—
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—44no—
190C4⋊C4—42no—
191C4⋊C4C2×D412yes—
192C22⋊C4—24no—
193C22⋊C4—24no—
194C22⋊C4—24no—
195C22⋊C4—22no—
196C22⋊C4—24no—
197C22⋊C4—22no—
198C22⋊C4—44no—
199C22⋊C4—42no—
200C42—24no—
201C42—21no—
202C42—21no—
203C23—22no—
204C23—22no—
205C23—21no—
206C23—22no—
207C23—22no—
208C23—21no—
209C23—21no—
210C23—22no—
211C23—42no—
212C23—41no—
213C23—41no—
214C23—42no—
215C23#12012yes—
216C23#13512yes—
217C23AGL(1,ℤ/8)12yes—
218C23C22×D411yescharacteristic
219Q8—22no—
220Q8—24no—
221Q8—22no—
222Q8—22no—
223Q8—48no—
224Q8—42no—
225Q8#12314yescomplemented
226D4—24no—
227D4—48no—
228D4—44no—
229D4—48no—
230D4—41no—
231D4—41no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—24no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—24no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—48no—
248C2×C4—48no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—84no—
262C2×C4#12012yes—
263C2×C4C22×D411yescharacteristic
264C2×C4C22×D411yescharacteristic
265C2×C4C2×#4112yes—
266C2×C4H(5,𝔽2)12yes—
267C8—42no—
268C8—42no—
269C22—22no—
270C22—22no—
271C22—24no—
272C22—24no—
273C22—42no—
274C22—41no—
275C22—41no—
276C22—42no—
277C22—41no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—82no—
282C22—82no—
283C22#44914yes—
284C22C2×#12011yescharacteristic
285C22#54612yes—
286C22C2×#13511yescharacteristic
287C22C2×AGL(1,ℤ/8)11yescharacteristic
288C22#57812yes—
289C4—22no—
290C4—24no—
291C4—41no—
292C4—42no—
293C4—41no—
294C4—42no—
295C4—41no—
296C4—82no—
297C4—82no—
298C4C2×#12311yescharacteristic
299C4#53411yescharacteristic
300C4#54612yes—
301C2—24no—
302C2—81no—
303C2—82no—
304C2#240511yescharacteristic
305C2C2×#44911yescharacteristic
306C2#301111yescharacteristic
307C2#330012yes—
308C2#330812yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#548C2
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
2. C2×#480C2semidirect
3. C22×#103C2semidirect
4. #2486C2semidirect
5. #2042C2semidirect
6. #2036C2semidirect
7. #1893C2semidirect
8. #1869C2semidirect
9. #1863C2semidirect
10. #1788C2semidirect
11. #1787C2semidirect
225. Q8#123semidirect

Maximal subgroups

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

subgroupindex
C2×#5482
C2×#4802
C22×#1032
#24862
#20422
#20362
#18932
#18692
#18632
#17882
#17872