Finite Group Databasebeta

← #29759

Subgroups — #29759

Order 256.

406 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 125 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
2#2857C211yesmaximal, characteristic
3#2296C211yesmaximal, characteristic
4#2084C211yesmaximal, characteristic, complemented
5#2083C211yesmaximal, characteristic, complemented
6#2068C212yesmaximal, complemented
7#2066C212yesmaximal, complemented
8#2043C211yesmaximal, characteristic, complemented
9#2043C211yesmaximal, characteristic, complemented
10#1873C212yesmaximal
11#1797C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×C22⋊Q8C2212yes—
14C2×#117C2211yescharacteristic
15C2×#117C2211yescharacteristic
16C22×C42C2211yescharacteristic
17#485—28no—
18C2×#107C2212yes—
19C2×#106C2212yes—
20C2×C4⋊C8C2212yes—
21C2×#103C2212yes—
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×C22⋊C8C2212yes—
26#391C2212yes—
27#389—21no—
28#389—21no—
29#389C2211yescharacteristic
30#389C2211yescharacteristic
31#389C2211yescharacteristic
32#389C2211yescharacteristic
33#385—21no—
34#385—21no—
35#384C2211yescharacteristic
36#380—21no—
37#380—21no—
38C4×C4⋊C4—21no—
39C4×C4⋊C4—21no—
40#340C2212yes—
41#340C2212yes—
42#336C2211yescharacteristic
43#336C2211yescharacteristic
44C22×Q8C2312yes—
45C23×C4D411yescharacteristic
46C23×C4D411yescharacteristic
47C23×C4C2311yescharacteristic
48C22×C8C2312yes—
49C22×C8C2312yes—
50C4⋊Q8—42no—
51C22⋊Q8—24no—
52C22⋊Q8—24no—
53#117—22no—
54#117—42no—
55#117D414yes—
56C2×C4⋊C4—22no—
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⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2312yes—
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2311yescharacteristic
70C2×C22⋊C4—21no—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—22no—
73C2×C42—21no—
74C2×C42—21no—
75C2×C42—42no—
76C2×C42D411yescharacteristic
77C2×C42D411yescharacteristic
78C2×C42C2311yescharacteristic
79C2×C42C2311yescharacteristic
80#107—24no—
81#106—24no—
82C4⋊C8—42no—
83#103—24no—
84#103—24no—
85#103—42no—
86#103—42no—
87C22⋊C8—24no—
88#95—21no—
89#95—21no—
90#95—21no—
91#95—21no—
92#95—22no—
93#95—22no—
94C24C2×D411yescharacteristic
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×Q8—22no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—41no—
113C22×C4—41no—
114C22×C4—41no—
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D412yes—
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D411yescharacteristic
121C22×C4#4112yes—
122C22×C4C2411yescharacteristic
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—22no—
128C2×C8—22no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—42no—
136C4⋊C4—42no—
137C4⋊C4—42no—
138C4⋊C4—42no—
139C4⋊C4—44no—
140C4⋊C4—42no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4C2×D412yes—
145C4⋊C4C2×D412yes—
146C22⋊C4—24no—
147C22⋊C4—42no—
148C22⋊C4—44no—
149C42—24no—
150C42—22no—
151C42—22no—
152C42—41no—
153C42—41no—
154C42—42no—
155C23—21no—
156C23—21no—
157C23—21no—
158C23—21no—
159C23—21no—
160C23—21no—
161C23#12112yes—
162C23C22×D411yescharacteristic
163Q8—42no—
164Q8—42no—
165Q8—42no—
166Q8—42no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—24no—
176C2×C4—24no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—44no—
183C2×C4—44no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4#12112yes—
201C2×C4#13512yes—
202C2×C4#13712yes—
203C2×C4#13712yes—
204C2×C4C22×D411yescharacteristic
205C2×C4C2×#4111yescharacteristic
206C2×C4H(5,𝔽2)12yes—
207C2×C42−1+412yes—
208C8—42no—
209C8—42no—
210C22—22no—
211C22—22no—
212C22—22no—
213C22—41no—
214C22—41no—
215C22—41no—
216C22—41no—
217C22C2×#12111yescharacteristic
218C22#53411yescharacteristic
219C22#53611yescharacteristic
220C22C2×#13511yescharacteristic
221C22C2×#13711yescharacteristic
222C22C2×#13711yescharacteristic
223C22#57511yescharacteristic
224C4—22no—
225C4—22no—
226C4—22no—
227C4—22no—
228C4—41no—
229C4—42no—
230C4—42no—
231C4—82no—
232C4—81no—
233C4—81no—
234C4—82no—
235C4#53712yes—
236C4#53912yes—
237C2—42no—
238C2#261811yescharacteristic
239C2#303511yescharacteristic
240C2#303911yescharacteristic
241C2#319011yescharacteristic
242C2#319611yescharacteristic
243C2#323111yescharacteristic
244C2#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. #2084C2semidirect
5. #2083C2semidirect
6. #2068C2semidirect
7. #2066C2semidirect
8. #2043C2semidirect
9. #2043C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4852
#28572
#22962
#20842
#20832
#20682
#20662
#20432
#18732
#17972