Finite Group Databasebeta

← #24496

Subgroups — #24496

Order 256.

462 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 127 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×#485C211yesmaximal, characteristic
2C2×#419C211yesmaximal, characteristic
3#2434C211yesmaximal, characteristic
4#2347C211yesmaximal, characteristic
5#2066C212yesmaximal, complemented
6#2043C212yesmaximal, complemented
7#1981C212yesmaximal, complemented
8#1928C212yesmaximal, complemented
9#1894C211yesmaximal, characteristic
10#1851C211yesmaximal, characteristic
11#1787C211yesmaximal, characteristic
12C23×Q8C2211yescharacteristic
13C22×M16C2211yescharacteristic
14C2×C4⋊Q8C2212yes—
15C2×C22⋊Q8C2211yescharacteristic
16C2×#117C2211yescharacteristic
17#485—24no—
18C2×#107C2211yescharacteristic
19C2×#106C2211yescharacteristic
20C2×C4⋊C8C2212yes—
21#419—24no—
22C2×#103C2211yescharacteristic
23C2×#103C2211yescharacteristic
24C2×#103C2212yes—
25C2×#103C2212yes—
26C2×#103C2212yes—
27C2×#103C2212yes—
28C2×#103C2212yes—
29C2×C22⋊C8C2211yescharacteristic
30#395—21no—
31#395—21no—
32#391C2212yes—
33#389C2212yes—
34#388—22no—
35#385—22no—
36#383—21no—
37#380C2211yescharacteristic
38C4×C4⋊C4—21no—
39C4×C4⋊C4C2212yes—
40C4×C22⋊C4C2211yescharacteristic
41#340C2212yes—
42#336C2212yes—
43C22×Q8—21no—
44C22×Q8—24no—
45C22×Q8—22no—
46C22×Q8—22no—
47C22×Q8—42no—
48C22×Q8C2311yescharacteristic
49C22×Q8C2312yes—
50C23×C4—21no—
51C23×C4C2311yescharacteristic
52C2×M16—22no—
53C2×M16—42no—
54C22×C8C2312yes—
55C22×C8C2311yescharacteristic
56C22×C8C2311yescharacteristic
57C4⋊Q8—24no—
58C4⋊Q8—42no—
59C22⋊Q8—22no—
60C22⋊Q8—22no—
61#117—22no—
62#117D414yes—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2312yes—
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2311yescharacteristic
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—21no—
74C2×C22⋊C4D412yes—
75C2×C42—21no—
76C2×C42—22no—
77C2×C42C2312yes—
78#107—22no—
79#106—22no—
80C4⋊C8—42no—
81#103—24no—
82#103—24no—
83#103—24no—
84#103—22no—
85#103—22no—
86#103—42no—
87#103—42no—
88C22⋊C8—22no—
89#95—21no—
90#95—22no—
91#95—22no—
92#95—22no—
93#95—22no—
94#95D411yescharacteristic
95#95D411yescharacteristic
96C24C2×D411yescharacteristic
97C2×Q8—22no—
98C2×Q8—28no—
99C2×Q8—22no—
100C2×Q8—21no—
101C2×Q8—22no—
102C2×Q8—22no—
103C2×Q8—21no—
104C2×Q8—22no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—44no—
108C2×Q8—48no—
109C2×Q8—44no—
110C2×Q8—42no—
111C2×Q8#4114yes—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—22no—
117C22×C4—21no—
118C22×C4—22no—
119C22×C4—21no—
120C22×C4—44no—
121C22×C4—42no—
122C22×C4C2×D412yes—
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D412yes—
125C22×C4C2×D412yes—
126C22×C4#4112yes—
127C22×C4C2411yescharacteristic
128M16—44no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—22no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—22no—
140C4⋊C4—21no—
141C4⋊C4—22no—
142C4⋊C4—44no—
143C4⋊C4—44no—
144C4⋊C4—42no—
145C4⋊C4—44no—
146C4⋊C4C2×D412yes—
147C4⋊C4C2×D412yes—
148C22⋊C4—24no—
149C22⋊C4—24no—
150C22⋊C4—42no—
151C22⋊C4—81no—
152C42—24no—
153C42—22no—
154C42—22no—
155C42—82no—
156C23—22no—
157C23—22no—
158C23—21no—
159C23—21no—
160C23#12112yes—
161C23C22×D411yescharacteristic
162Q8—24no—
163Q8—24no—
164Q8—48no—
165Q8—42no—
166Q8—42no—
167Q8—41no—
168Q8—41no—
169Q8—42no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—24no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—24no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—48no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4#12112yes—
197C2×C4#13714yes—
198C2×C4C22×D411yescharacteristic
199C2×C4C2×#4112yes—
200C2×C4C2×#4111yescharacteristic
201C2×C4H(5,𝔽2)12yes—
202C8—41no—
203C8—41no—
204C8—42no—
205C22—22no—
206C22—22no—
207C22—22no—
208C22—42no—
209C22—42no—
210C22C2×#12111yescharacteristic
211C22#54612yes—
212C22C2×#13712yes—
213C22#57711yescharacteristic
214C22#57811yescharacteristic
215C4—22no—
216C4—24no—
217C4—42no—
218C4—44no—
219C4—42no—
220C4—42no—
221C4—81no—
222C4—81no—
223C4—82no—
224C4#52511yescharacteristic
225C4#53511yescharacteristic
226C4#55112yes—
227C2—42no—
228C2#241111yescharacteristic
229C2#303911yescharacteristic
230C2#304611yescharacteristic
231C2#332012yes—
232C2#332812yes—
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. #2066C2semidirect
6. #2043C2semidirect
7. #1981C2semidirect
8. #1928C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4852
C2×#4192
#24342
#23472
#20662
#20432
#19812
#19282
#18942
#18512
#17872