Finite Group Databasebeta

← #32969

Subgroups — #32969

Order 256.

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

normal subgroups only

This group has 675 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#2542C211yesmaximal, characteristic, complemented
4#1859C214yesmaximal, complemented
5#1856C212yesmaximal, complemented
6#1806C212yesmaximal, complemented
7#1800C211yesmaximal, characteristic, complemented
8#1792C212yesmaximal
9C2×H(5,𝔽2)C2211yescharacteristic
10C2×C4⋊Q8C2211yescharacteristic, complemented
11C2×C22⋊Q8—22no—
12#520—22no—
13#520C2212yes—
14#518—22no—
15#518—22no—
16#518C2211yescharacteristic
17C4×#41—22no—
18C4×#41C2212yes—
19C2×C4×D4—22no—
20C2×C4×D4—22no—
21C2×C4×D4C2211yescharacteristic
22#477—22no—
23#476—22no—
24#475—22no—
25#474—22no—
26#428C2214yescomplemented
27#423C2212yescomplemented
28C2×#104C2214yes—
29#417C2214yes—
30C2×#100C2212yes—
31#399—22no—
32#390C2212yes—
33#384C2212yescomplemented
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—
51C22⋊Q8—48no—
52C4×Q8—22no—
53C4×Q8D412yes—
54C4×Q8D412yes—
55C4×D4—24no—
56C4×D4—24no—
57C4×D4—24no—
58C4×D4—24no—
59C4×D4—22no—
60C4×D4—22no—
61C4×D4—24no—
62C4×D4—22no—
63C4×D4—22no—
64C4×D4—22no—
65C4×D4—22no—
66C4×D4—22no—
67C4×D4D412yes—
68C4×D4D412yes—
69#117—24no—
70#117—22no—
71#117—22no—
72#117C2312yes—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4C2312yes—
79C2×C4⋊C4C2311yescharacteristic
80C2×C22⋊C4—22no—
81C2×C22⋊C4—24no—
82C2×C22⋊C4—22no—
83C2×C22⋊C4—24no—
84C2×C22⋊C4D412yes—
85C2×C42—22no—
86C2×C42—22no—
87C2×C42C2312yes—
88C2×C42C2311yescharacteristic
89#107—24no—
90#106—24no—
91C4⋊C8—22no—
92C4⋊C8—22no—
93#104—28no—
94#103—24no—
95#102—24no—
96#100—24no—
97#95—21no—
98C24—22no—
99C24C2×D412yes—
100#41—28no—
101#41—28no—
102#41—22no—
103#41—22no—
104#41C2×Q814yes—
105#41#4114yes—
106C2×Q8—44no—
107C2×Q8—41no—
108C2×Q8—41no—
109C2×Q8C2×D412yes—
110C2×D4—24no—
111C2×D4—28no—
112C2×D4—28no—
113C2×D4—28no—
114C2×D4—22no—
115C2×D4—28no—
116C2×D4—22no—
117C2×D4—22no—
118C2×D4C2×D412yes—
119C2×D4C2×Q812yes—
120C2×D4#4112yes—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—24no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—44no—
133C22×C4—84no—
134C22×C4C2×D411yescharacteristic
135C22×C4C2×D412yes—
136C22×C4C2×D411yescharacteristic
137C22×C4C2411yescharacteristic
138M16—24no—
139M16—24no—
140C2×C8—24no—
141C4⋊C4—24no—
142C4⋊C4—24no—
143C4⋊C4—22no—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C4⋊C4—24no—
147C4⋊C4—48no—
148C4⋊C4—42no—
149C4⋊C4C2×D412yes—
150C4⋊C4C2×D412yes—
151C22⋊C4—24no—
152C22⋊C4—24no—
153C22⋊C4—22no—
154C22⋊C4—22no—
155C22⋊C4—22no—
156C22⋊C4—24no—
157C22⋊C4—48no—
158C42—24no—
159C42—22no—
160C42—22no—
161C42—22no—
162C42—24no—
163C42C2×D412yes—
164C42C2×D412yes—
165C23—28no—
166C23—22no—
167C23—22no—
168C23—22no—
169C23—42no—
170C23—42no—
171C23—82no—
172C23—82no—
173C23C22⋊Q814yes—
174C23C22×D411yescharacteristic
175Q8—22no—
176Q8—82no—
177Q8—82no—
178Q8C22⋊Q814yescomplemented
179D4—216no—
180D4—24no—
181D4—24no—
182D4—24no—
183D4—22no—
184D4—24no—
185D4C22⋊Q814yescomplemented
186C2×C4—28no—
187C2×C4—22no—
188C2×C4—24no—
189C2×C4—24no—
190C2×C4—22no—
191C2×C4—24no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—44no—
199C2×C4—44no—
200C2×C4—42no—
201C2×C4—44no—
202C2×C4—42no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4—82no—
206C2×C4—84no—
207C2×C4—82no—
208C2×C4#12012yes—
209C2×C4#12011yescharacteristic
210C2×C4#12011yescharacteristic
211C2×C4C22×D412yes—
212C2×C4C22×Q811yescharacteristic
213C2×C4C2×#4112yes—
214C2×C4C2×#4111yescharacteristic
215C8—44no—
216C22—21no—
217C22—21no—
218C22—24no—
219C22—24no—
220C22—44no—
221C22—42no—
222C22—42no—
223C22—42no—
224C22—42no—
225C22—82no—
226C22—88no—
227C22C2×#12011yescharacteristic
228C22C2×C22⋊Q811yescharacteristic
229C22#52511yescharacteristic
230C4—24no—
231C4—21no—
232C4—22no—
233C4—44no—
234C4—42no—
235C4—42no—
236C4—42no—
237C4—82no—
238C4#45312yes—
239C4C2×C22⋊Q812yes—
240C4#52512yes—
241C2—21no—
242C2—21no—
243C2—44no—
244C2—44no—
245C2#284811yescharacteristic
246C2C2×#45311yescharacteristic
247C2#302211yescharacteristic
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. #2542C2semidirect
4. #1859C2semidirect
5. #1856C2semidirect
6. #1806C2semidirect
7. #1800C2semidirect
10. C2×C4⋊Q8C22semidirect
26. #428C22semidirect
27. #423C22semidirect
33. #384C22semidirect
34. C4×C4⋊C4C22semidirect
35. #343C22semidirect
36. #339C22semidirect
39. H(5,𝔽2)Q8semidirect
178. Q8C22⋊Q8semidirect
185. D4C22⋊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
#25422
#18592
#18562
#18062
#18002
#17922