Finite Group Databasebeta

← #24580

Subgroups — #24580

Order 256.

421 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 421 conjugacy classes of subgroups — too many to draw. Only its 139 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×#483C211yesmaximal, characteristic
2C22×#106C211yesmaximal, characteristic
3#2502C211yesmaximal, characteristic
4#2344C211yesmaximal, characteristic
5#2084C212yesmaximal, complemented
6#2066C212yesmaximal, complemented
7#1948C212yesmaximal, complemented
8#1927C212yesmaximal, complemented
9#1892C211yesmaximal, characteristic
10#1850C211yesmaximal, characteristic
11#1824C211yesmaximal, characteristic
12C23×C8C2211yescharacteristic
13C2×C22⋊Q8C2211yescharacteristic
14C2×#117—21no—
15C22×C4⋊C4C2211yescharacteristic
16C22×C4⋊C4C2211yescharacteristic
17#483—24no—
18C2×#106—24no—
19C2×#106C2211yescharacteristic
20C2×#106C2211yescharacteristic
21C2×#106C2212yes—
22C2×#106C2212yes—
23C2×C4⋊C8C2212yes—
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2212yes—
27C2×C22⋊C8C2211yescharacteristic
28#398—22no—
29#391C2212yes—
30#389C2212yes—
31#387—22no—
32#385—21no—
33#385—21no—
34#384C2212yes—
35#381—21no—
36#380C2211yescharacteristic
37C4×C4⋊C4C2212yes—
38C4×C22⋊C4C2211yescharacteristic
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C22×Q8C2311yescharacteristic
44C23×C4—21no—
45C23×C4—21no—
46C23×C4C2311yescharacteristic
47C22×C8—22no—
48C22×C8—24no—
49C22×C8C2311yescharacteristic
50C22×C8C2311yescharacteristic
51C22×C8C2312yes—
52C22⋊Q8—22no—
53C22⋊Q8—22no—
54#117—44no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4D414yes—
65C2×C4⋊C4C2312yes—
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2312yes—
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2311yescharacteristic
70C2×C22⋊C4—21no—
71C2×C22⋊C4—21no—
72C2×C22⋊C4D411yescharacteristic
73C2×C22⋊C4D411yescharacteristic
74C2×C42—22no—
75C2×C42C2312yes—
76#106—24no—
77#106—24no—
78#106—22no—
79#106—22no—
80C4⋊C8—24no—
81#103—24no—
82#103—22no—
83#103—22no—
84C22⋊C8—22no—
85#95—22no—
86#95—21no—
87#95—22no—
88#95—22no—
89#95—22no—
90#95D412yes—
91C24C2×D411yescharacteristic
92C2×Q8—21no—
93C2×Q8—22no—
94C2×Q8—21no—
95C2×Q8—22no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—24no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—21no—
105C22×C4—42no—
106C22×C4—42no—
107C22×C4—42no—
108C22×C4—82no—
109C22×C4C2×D412yes—
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D412yes—
112C22×C4C2×D412yes—
113C22×C4#4112yes—
114C22×C4C2411yescharacteristic
115C2×C8—28no—
116C2×C8—22no—
117C2×C8—22no—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—22no—
121C2×C8—22no—
122C2×C8#4114yes—
123C4⋊C4—24no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—22no—
127C4⋊C4—21no—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—21no—
131C4⋊C4—22no—
132C4⋊C4—42no—
133C4⋊C4—42no—
134C4⋊C4—44no—
135C4⋊C4—82no—
136C4⋊C4—82no—
137C4⋊C4C2×D412yes—
138C4⋊C4C2×D412yes—
139C22⋊C4—22no—
140C22⋊C4—42no—
141C22⋊C4—42no—
142C22⋊C4SD1614yes—
143C42—22no—
144C42—22no—
145C42—44no—
146C23—22no—
147C23—22no—
148C23#12112yes—
149C23C2×SD1612yes—
150C23#13712yes—
151C23C22×D411yescharacteristic
152Q8—41no—
153Q8—41no—
154Q8—42no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—24no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—48no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—84no—
182C2×C4#12112yes—
183C2×C4C2×SD1614yes—
184C2×C4C22×D411yescharacteristic
185C2×C4C2×#4112yes—
186C2×C4C2×#4111yescharacteristic
187C2×C42−1+412yes—
188C8—24no—
189C8—41no—
190C8—41no—
191C22—24no—
192C22—24no—
193C22—22no—
194C22—22no—
195C22#46114yes—
196C22C2×#12111yescharacteristic
197C22#54712yes—
198C22C22×SD1611yescharacteristic
199C22C2×#13711yescharacteristic
200C22#57712yes—
201C4—24no—
202C4—22no—
203C4—42no—
204C4—42no—
205C4—42no—
206C4—82no—
207C4—81no—
208C4—81no—
209C4—82no—
210C4#52511yescharacteristic
211C4#53611yescharacteristic
212C4#55212yes—
213C2—24no—
214C2#241311yescharacteristic
215C2C2×#46111yescharacteristic
216C2#304311yescharacteristic
217C2#333512yes—
218C2#334412yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #2084C2semidirect
6. #2066C2semidirect
7. #1948C2semidirect
8. #1927C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4832
C22×#1062
#25022
#23442
#20842
#20662
#19482
#19272
#18922
#18502
#18242