Finite Group Databasebeta

← #24592

Subgroups — #24592

Order 256.

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

normal subgroups only

This group has 431 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×#482C211yesmaximal, characteristic
2C2×#428C211yesmaximal, characteristic, complemented
3#2501C211yesmaximal, characteristic
4#2347C211yesmaximal, characteristic, complemented
5#2083C212yesmaximal, complemented
6#2067C212yesmaximal, complemented
7#1948C212yesmaximal, complemented
8#1926C212yesmaximal, complemented
9#1895C211yesmaximal, characteristic
10#1851C211yesmaximal, characteristic, complemented
11#1824C211yesmaximal, characteristic, complemented
12C22×M16C2211yescharacteristic
13C2×#123—22no—
14C2×#121C2211yescharacteristic
15C2×#117C2211yescharacteristic
16C22×C4⋊C4C2211yescharacteristic
17#482—24no—
18#428—24no—
19C2×#107C2211yescharacteristic
20C2×#107C2212yes—
21C2×#106C2211yescharacteristic
22C2×#106C2212yes—
23C2×C4⋊C8C2212yes—
24C2×#102C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#102C2212yes—
27C2×C22⋊C8C2211yescharacteristic
28#399—21no—
29#399—21no—
30#390C2212yes—
31#389C2212yes—
32#388—22no—
33#384C2212yes—
34#383—21no—
35#380C2211yescharacteristic
36C4×C4⋊C4—21no—
37C4×C4⋊C4C2212yes—
38C4×C22⋊C4C2211yescharacteristic
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C22×D4—21no—
44C22×D4C2311yescharacteristic
45C23×C4—21no—
46C23×C4C2311yescharacteristic
47C2×M16—22no—
48C2×M16—24no—
49C22×C8C2311yescharacteristic
50C22×C8C2311yescharacteristic
51C22×C8C2312yes—
52#123—48no—
53#121—22no—
54#121—22no—
55#117—22no—
56#117D414yes—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4C2312yes—
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2312yes—
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2311yescharacteristic
68C2×C22⋊C4—21no—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4D412yes—
74C2×C42—21no—
75C2×C42—22no—
76C2×C42C2312yes—
77#107—24no—
78#107—22no—
79#106—24no—
80#106—22no—
81C4⋊C8—24no—
82#102—24no—
83#102—22no—
84#102—22no—
85C22⋊C8—22no—
86#95—21no—
87#95—22no—
88#95—22no—
89#95D411yescharacteristic
90#95D411yescharacteristic
91C24—21no—
92C24C2×D411yescharacteristic
93C2×D4—21no—
94C2×D4—22no—
95C2×D4—21no—
96C2×D4—22no—
97C2×D4—44no—
98C2×D4—42no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—21no—
106C22×C4—44no—
107C22×C4—42no—
108C22×C4C2×D412yes—
109C22×C4C2×D411yescharacteristic
110C22×C4C2×D412yes—
111C22×C4C2×D412yes—
112C22×C4#4112yes—
113C22×C4C2411yescharacteristic
114M16—28no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8#4114yes—
122C4⋊C4—21no—
123C4⋊C4—21no—
124C4⋊C4—22no—
125C4⋊C4—21no—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—21no—
129C4⋊C4—22no—
130C4⋊C4—44no—
131C4⋊C4—42no—
132C4⋊C4—44no—
133C4⋊C4—44no—
134C4⋊C4C2×D412yes—
135C4⋊C4C2×D412yes—
136C22⋊C4—24no—
137C22⋊C4—24no—
138C22⋊C4—44no—
139C22⋊C4—42no—
140C22⋊C4—44no—
141C22⋊C4—81no—
142C42—24no—
143C42—22no—
144C42—22no—
145C42—82no—
146C23—22no—
147C23—22no—
148C23—21no—
149C23—21no—
150C23—41no—
151C23—42no—
152C23—42no—
153C23—41no—
154C23—41no—
155C23#12112yes—
156C23C22×D411yescharacteristic
157D4—41no—
158D4—41no—
159D4—42no—
160D4—84no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—24no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—42no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—84no—
184C2×C4#12112yes—
185C2×C4AGL(1,ℤ/8)14yes—
186C2×C4C22×D411yescharacteristic
187C2×C4C2×#4112yes—
188C2×C4C2×#4111yescharacteristic
189C2×C42−1+412yes—
190C8—24no—
191C8—41no—
192C8—41no—
193C22—22no—
194C22—22no—
195C22—22no—
196C22—42no—
197C22—42no—
198C22—82no—
199C22—82no—
200C22—81no—
201C22—81no—
202C22—81no—
203C22C2×#12111yescharacteristic
204C22#54712yes—
205C22C2×AGL(1,ℤ/8)12yes—
206C22#57711yescharacteristic
207C22#57811yescharacteristic
208C4—24no—
209C4—22no—
210C4—42no—
211C4—42no—
212C4—42no—
213C4—82no—
214C4—82no—
215C4—81no—
216C4#52511yescharacteristic
217C4#53611yescharacteristic
218C4#55212yes—
219C2—42no—
220C2—161no—
221C2#241311yescharacteristic
222C2#303811yescharacteristic
223C2#304611yescharacteristic
224C2#334512yes—
225C2#335312yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
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
2. C2×#428C2semidirect
4. #2347C2semidirect
5. #2083C2semidirect
6. #2067C2semidirect
7. #1948C2semidirect
8. #1926C2semidirect
10. #1851C2semidirect
11. #1824C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4822
C2×#4282
#25012
#23472
#20832
#20672
#19482
#19262
#18952
#18512
#18242