Finite Group Databasebeta

← #27649

Subgroups — #27649

Order 256.

405 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 405 conjugacy classes of subgroups — too many to draw. Only its 135 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×#485C212yesmaximal
2C4×C22⋊Q8C211yesmaximal, characteristic
3#2285C211yesmaximal, characteristic
4#2085C212yesmaximal, complemented
5#2074C212yesmaximal, complemented
6#1948C211yesmaximal, characteristic, complemented
7#1947C211yesmaximal, characteristic, complemented
8#1825C212yesmaximal
9#1803C211yesmaximal, characteristic, complemented
10#1803C211yesmaximal, characteristic, complemented
11#1797C211yesmaximal, characteristic
12C2×C22⋊Q8C2211yescharacteristic
13C2×C4×Q8C2211yescharacteristic
14C2×#117C2212yes—
15C22×C42C2211yescharacteristic
16#485—28no—
17C2×#107C2212yes—
18C2×#106C2212yes—
19C2×C4⋊C8C2211yescharacteristic
20C2×C4⋊C8C2211yescharacteristic
21C2×#103C2212yes—
22C2×#103C2212yes—
23C2×C22⋊C8C2212yes—
24#391C2211yescharacteristic
25#389C2211yescharacteristic
26#387—21no—
27#385—21no—
28#384C2211yescharacteristic
29#384C2211yescharacteristic
30#382C2212yes—
31#380—21no—
32#380—21no—
33#380C2211yescharacteristic
34C4×C4⋊C4—21no—
35C4×C4⋊C4—21no—
36C4×C4⋊C4—21no—
37C4×C4⋊C4C2211yescharacteristic
38C4×C22⋊C4—21no—
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#340C2212yes—
43#336C2212yes—
44C22×Q8C2311yescharacteristic
45C23×C4D411yescharacteristic
46C23×C4D411yescharacteristic
47C23×C4C2311yescharacteristic
48C22×C8C2312yes—
49C22×C8C2312yes—
50C22⋊Q8—22no—
51C22⋊Q8D414yes—
52C4×Q8—22no—
53C4×Q8—22no—
54#117—24no—
55#117—24no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4C2312yes—
64C2×C4⋊C4C2312yes—
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2311yescharacteristic
68C2×C22⋊C4—21no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—21no—
71C2×C42—22no—
72C2×C42—22no—
73C2×C42—22no—
74C2×C42—21no—
75C2×C42—21no—
76C2×C42D411yescharacteristic
77C2×C42D411yescharacteristic
78C2×C42C2311yescharacteristic
79C2×C42C2311yescharacteristic
80#107—24no—
81#106—24no—
82C4⋊C8—22no—
83C4⋊C8—22no—
84#103—24no—
85#103—24no—
86C22⋊C8—24no—
87#95—21no—
88#95—21no—
89#95—22no—
90#95—22no—
91#95—21no—
92#95—21no—
93#95—21no—
94C24C2×D411yescharacteristic
95C2×Q8—21no—
96C2×Q8—21no—
97C2×Q8—21no—
98C2×Q8—21no—
99C2×Q8—21no—
100C2×Q8C2×D412yes—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—21no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—21no—
115C22×C4—41no—
116C22×C4—41no—
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D412yes—
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D411yescharacteristic
123C22×C4#4112yes—
124C22×C4C2411yescharacteristic
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—22no—
130C2×C8—22no—
131C4⋊C4—24no—
132C4⋊C4—21no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4C2×D412yes—
148C22⋊C4—24no—
149C22⋊C4—42no—
150C22⋊C4—44no—
151C42—22no—
152C42—22no—
153C42—21no—
154C42—21no—
155C42—44no—
156C42—42no—
157C42—42no—
158C42#4112yes—
159C42#4112yes—
160C23—21no—
161C23—21no—
162C23—21no—
163C23—21no—
164C23—21no—
165C23—21no—
166C23#12112yes—
167C23C22×D411yescharacteristic
168Q8—22no—
169Q8—22no—
170Q8—41no—
171Q8—41no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—44no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—44no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4#12112yes—
207C2×C4#12312yes—
208C2×C4#12312yes—
209C2×C4#13512yes—
210C2×C4#13712yes—
211C2×C4C22×D411yescharacteristic
212C2×C4C2×#4111yescharacteristic
213C2×C4C2×#4111yescharacteristic
214C2×C4C2×#4111yescharacteristic
215C2×C42−1+411yescharacteristic
216C2×C42−1+411yescharacteristic
217C8—42no—
218C8—42no—
219C22—22no—
220C22—22no—
221C22—22no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22C2×#12111yescharacteristic
227C22C2×#12311yescharacteristic
228C22#53611yescharacteristic
229C22C2×#13511yescharacteristic
230C22C2×#13711yescharacteristic
231C22#57511yescharacteristic
232C22#57511yescharacteristic
233C4—22no—
234C4—22no—
235C4—41no—
236C4—41no—
237C4—42no—
238C4—42no—
239C4—81no—
240C4—81no—
241C4—82no—
242C4—82no—
243C4#48514yescomplemented
244C4#54711yescharacteristic
245C4#54711yescharacteristic
246C4#55211yescharacteristic
247C4#55211yescharacteristic
248C2—42no—
249C2#248611yescharacteristic
250C2#303511yescharacteristic
251C2#303711yescharacteristic
252C2C2×#48511yescharacteristic
253C2#309311yescharacteristic
254C2#323311yescharacteristic
255C2#323711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
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
4. #2085C2semidirect
5. #2074C2semidirect
6. #1948C2semidirect
7. #1947C2semidirect
9. #1803C2semidirect
10. #1803C2semidirect
243. C4#485semidirect

Maximal subgroups

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

subgroupindex
C2×#4852
C4×C22⋊Q82
#22852
#20852
#20742
#19482
#19472
#18252
#18032
#17972