Finite Group Databasebeta

← #32971

Subgroups — #32971

Order 256.

647 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 647 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C4×H(5,𝔽2)C211yesmaximal, characteristic
2#3079C212yesmaximal, complemented
3#2563C211yesmaximal, characteristic, complemented
4#1860C214yesmaximal, complemented
5#1856C212yesmaximal, complemented
6#1806C212yesmaximal, complemented
7#1800C211yesmaximal, characteristic, complemented
8#1792C212yesmaximal
9C2×H(5,𝔽2)C2211yescharacteristic
10C2×C4⋊Q8C2211yescharacteristic, complemented
11#520—22no—
12#520C2212yes—
13#518—22no—
14#518—22no—
15#518C2211yescharacteristic
16C4×#41—22no—
17C4×#41C2212yes—
18C2×C4×D4—22no—
19C2×C4×D4—22no—
20C2×C4×D4C2211yescharacteristic
21#477—22no—
22#476—22no—
23#475—22no—
24#474—22no—
25#428C2214yescomplemented
26#423C2212yescomplemented
27C2×#104C2214yes—
28#417C2214yes—
29C2×#100C2212yes—
30#393—22no—
31#390C2212yes—
32#382C2212yescomplemented
33#380—22no—
34C4×C4⋊C4C2211yescharacteristic, complemented
35#343C2212yescomplemented
36#339C2214yescomplemented
37H(5,𝔽2)—24no—
38H(5,𝔽2)—22no—
39H(5,𝔽2)Q814yescomplemented
40C2×#41—22no—
41C2×#41C2312yes—
42C22×Q8—21no—
43C22×D4—22no—
44C22×D4—22no—
45C22×D4C2311yescharacteristic
46C23×C4—22no—
47C23×C4D412yes—
48C2×M16C2314yes—
49C4⋊Q8—22no—
50C4⋊Q8—22no—
51C4×Q8—22no—
52C4×Q8D412yes—
53C4×Q8D412yes—
54C4×D4—24no—
55C4×D4—24no—
56C4×D4—24no—
57C4×D4—24no—
58C4×D4—22no—
59C4×D4—22no—
60C4×D4—24no—
61C4×D4—22no—
62C4×D4—22no—
63C4×D4—22no—
64C4×D4—22no—
65C4×D4—22no—
66C4×D4D412yes—
67C4×D4D412yes—
68#117—24no—
69#117—22no—
70#117—22no—
71#117C2312yes—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4C2312yes—
77C2×C4⋊C4C2311yescharacteristic
78C2×C22⋊C4—22no—
79C2×C22⋊C4—24no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—24no—
82C2×C22⋊C4D412yes—
83C2×C42—22no—
84C2×C42—22no—
85C2×C42C2312yes—
86C2×C42C2311yescharacteristic
87#107—24no—
88#106—24no—
89C4⋊C8—22no—
90C4⋊C8—22no—
91#104—28no—
92#103—24no—
93#102—24no—
94#100—24no—
95#95—21no—
96#95—24no—
97C24—22no—
98C24C2×D412yes—
99#41—28no—
100#41—28no—
101#41—22no—
102#41—22no—
103#41C2×Q814yes—
104#41#4114yes—
105C2×Q8—42no—
106C2×Q8—82no—
107C2×Q8C2×D412yes—
108C2×D4—24no—
109C2×D4—28no—
110C2×D4—28no—
111C2×D4—28no—
112C2×D4—22no—
113C2×D4—28no—
114C2×D4—22no—
115C2×D4—22no—
116C2×D4C2×D412yes—
117C2×D4C2×Q812yes—
118C2×D4#4112yes—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—24no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—44no—
131C22×C4—84no—
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D412yes—
134C22×C4C2×D411yescharacteristic
135C22×C4C2411yescharacteristic
136M16—24no—
137M16—24no—
138C2×C8—24no—
139C4⋊C4—24no—
140C4⋊C4—22no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—24no—
144C4⋊C4—42no—
145C4⋊C4—41no—
146C4⋊C4—41no—
147C4⋊C4C2×D412yes—
148C4⋊C4C2×D412yes—
149C22⋊C4—24no—
150C22⋊C4—24no—
151C22⋊C4—22no—
152C22⋊C4—22no—
153C22⋊C4—22no—
154C22⋊C4—24no—
155C22⋊C4—84no—
156C42—24no—
157C42—22no—
158C42—22no—
159C42—22no—
160C42—24no—
161C42C2×D412yes—
162C42C2×D412yes—
163C23—28no—
164C23—22no—
165C23—22no—
166C23—22no—
167C23—42no—
168C23—42no—
169C23—82no—
170C23—82no—
171C23C22⋊Q814yes—
172C23C22×D411yescharacteristic
173Q8—22no—
174Q8—84no—
175Q8C22⋊Q814yes—
176D4—216no—
177D4—24no—
178D4—24no—
179D4—24no—
180D4—22no—
181D4—24no—
182D4C22⋊Q814yes—
183C2×C4—28no—
184C2×C4—22no—
185C2×C4—24no—
186C2×C4—24no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—41no—
192C2×C4—44no—
193C2×C4—44no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—44no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4—84no—
204C2×C4—82no—
205C2×C4#12012yes—
206C2×C4#12011yescharacteristic
207C2×C4#12011yescharacteristic
208C2×C4C22×D412yes—
209C2×C4C22×Q811yescharacteristic
210C2×C4C2×#4112yes—
211C2×C4C2×#4111yescharacteristic
212C8—44no—
213C22—21no—
214C22—21no—
215C22—24no—
216C22—24no—
217C22—44no—
218C22—42no—
219C22—42no—
220C22—42no—
221C22—42no—
222C22—82no—
223C22—88no—
224C22C2×#12011yescharacteristic
225C22C2×C22⋊Q811yescharacteristic
226C22#52511yescharacteristic
227C4—24no—
228C4—21no—
229C4—22no—
230C4—42no—
231C4—42no—
232C4—42no—
233C4—82no—
234C4—82no—
235C4#45512yes—
236C4C2×C22⋊Q812yes—
237C4#52512yes—
238C2—21no—
239C2—21no—
240C2—44no—
241C2—44no—
242C2#284811yescharacteristic
243C2C2×#45511yescharacteristic
244C2#302111yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×H(5,𝔽2)C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3079C2semidirect
3. #2563C2semidirect
4. #1860C2semidirect
5. #1856C2semidirect
6. #1806C2semidirect
7. #1800C2semidirect
10. C2×C4⋊Q8C22semidirect
25. #428C22semidirect
26. #423C22semidirect
32. #382C22semidirect
34. C4×C4⋊C4C22semidirect
35. #343C22semidirect
36. #339C22semidirect
39. H(5,𝔽2)Q8semidirect

Maximal subgroups

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

subgroupindex
C4×H(5,𝔽2)2
#30792
#25632
#18602
#18562
#18062
#18002
#17922