Finite Group Databasebeta

← #29852

Subgroups — #29852

Order 256.

393 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 393 conjugacy classes of subgroups — too many to draw. Only its 123 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×#491C211yesmaximal, characteristic, complemented
2C2×#432C211yesmaximal, characteristic
3#2857C211yesmaximal, characteristic
4#2851C211yesmaximal, characteristic
5#2068C212yesmaximal, complemented
6#2066C212yesmaximal, complemented
7#2036C212yesmaximal
8#2035C212yesmaximal
9#1992C211yesmaximal, characteristic, complemented
10#1960C211yesmaximal, characteristic, complemented
11#1847C211yesmaximal, characteristic, complemented
12C2×C4⋊Q8C2211yescharacteristic
13C2×#125C2211yescharacteristic
14C2×C22⋊Q8C2212yes—
15C22×C42C2211yescharacteristic
16#491—24no—
17#432—24no—
18C2×#107C2212yes—
19C2×#106C2212yes—
20C2×C4⋊C8C2212yes—
21C2×#103C2212yes—
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×C22⋊C8C2212yes—
26C2×#97C2211yescharacteristic
27C2×#97C2211yescharacteristic
28#391C2212yes—
29#389—21no—
30#389—21no—
31#389C2211yescharacteristic
32#389C2211yescharacteristic
33#387—21no—
34#387—21no—
35#385—21no—
36#385—21no—
37#384C2211yescharacteristic
38#382C2211yescharacteristic
39#380—21no—
40#380—21no—
41#380C2211yescharacteristic
42#380C2211yescharacteristic
43#340C2212yes—
44#340C2212yes—
45C22×Q8C2312yes—
46C23×C4D411yescharacteristic
47C23×C4D411yescharacteristic
48C23×C4C2311yescharacteristic
49C22×C8C2312yes—
50C22×C8C2312yes—
51C4⋊Q8—22no—
52C4⋊Q8—22no—
53#125—22no—
54#125—22no—
55C22⋊Q8—42no—
56C22⋊Q8—42no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4C2312yes—
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2311yescharacteristic
71C2×C22⋊C4—22no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—21no—
74C2×C42—22no—
75C2×C42—22no—
76C2×C42D411yescharacteristic
77C2×C42D411yescharacteristic
78C2×C42C2311yescharacteristic
79C2×C42C2311yescharacteristic
80#107—42no—
81#106—42no—
82C4⋊C8—24no—
83#103—24no—
84#103—24no—
85#103—42no—
86#103—42no—
87C22⋊C8—24no—
88#97—22no—
89#97D414yes—
90#95—22no—
91#95—22no—
92#95—21no—
93#95—21no—
94#95—21no—
95#95—21no—
96#95—21no—
97#95—21no—
98C24C2×D411yescharacteristic
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8—22no—
102C2×Q8—22no—
103C2×Q8—22no—
104C2×Q8—22no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—41no—
120C22×C4—41no—
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D411yescharacteristic
126C22×C4C2411yescharacteristic
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8C2×D414yes—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—21no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4—44no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C22⋊C4—42no—
151C22⋊C4—44no—
152C22⋊C4—42no—
153C42—22no—
154C42—22no—
155C42—21no—
156C42—21no—
157C42C2×D412yes—
158C42#4112yes—
159C23—21no—
160C23—21no—
161C23—21no—
162C23—21no—
163C23—21no—
164C23—21no—
165C23#13712yes—
166C23C22×D411yescharacteristic
167Q8—42no—
168Q8—42no—
169Q8—42no—
170Q8—42no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—44no—
184C2×C4—44no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—42no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4#12112yes—
206C2×C4#12112yes—
207C2×C4#13712yes—
208C2×C4C22×D411yescharacteristic
209C2×C4C2×#4111yescharacteristic
210C2×C4H(5,𝔽2)12yes—
211C2×C42−1+412yes—
212C8—24no—
213C8—42no—
214C22—22no—
215C22—22no—
216C22—22no—
217C22—41no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22C2×#12111yescharacteristic
222C22#53511yescharacteristic
223C22#53611yescharacteristic
224C22C2×#13711yescharacteristic
225C22C2×#13711yescharacteristic
226C22#57511yescharacteristic
227C22#57511yescharacteristic
228C4—22no—
229C4—22no—
230C4—22no—
231C4—41no—
232C4—41no—
233C4—82no—
234C4—81no—
235C4—82no—
236C4—81no—
237C4—81no—
238C4—81no—
239C4#53712yes—
240C4#54212yes—
241C2—42no—
242C2#262011yescharacteristic
243C2#305611yescharacteristic
244C2#305611yescharacteristic
245C2#321811yescharacteristic
246C2#321911yescharacteristic
247C2#323611yescharacteristic
248C2#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
1. C2×#491C2semidirect
5. #2068C2semidirect
6. #2066C2semidirect
9. #1992C2semidirect
10. #1960C2semidirect
11. #1847C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4912
C2×#4322
#28572
#28512
#20682
#20662
#20362
#20352
#19922
#19602
#18472