Finite Group Databasebeta

← #33606

Subgroups — #33606

Order 256.

343 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 343 conjugacy classes of subgroups — too many to draw. Only its 121 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×#510C211yesmaximal, characteristic, complemented
2C22×M32C211yesmaximal, characteristic, complemented
3#3050C211yesmaximal, characteristic
4C2×#467C211yesmaximal, characteristic, complemented
5#2220C212yesmaximal, complemented
6#2219C212yesmaximal, complemented
7#2217C212yesmaximal, complemented
8#2216C212yesmaximal, complemented
9#2159C211yesmaximal, characteristic, complemented
10#2141C211yesmaximal, characteristic, complemented
11#2139C211yesmaximal, characteristic, complemented
12C2×#135C2211yescharacteristic
13C22×Q16C2211yescharacteristic
14C23×C8C2211yescharacteristic
15#525—21no—
16C2×C22⋊Q8—21no—
17#510—24no—
18C2×Q32C2212yescomplemented
19C2×SD32C2212yescomplemented
20C2×M32—24no—
21C2×M32C2211yescharacteristic
22C2×M32C2212yes—
23C2×M32C2211yescharacteristic
24C22×C16C2212yes—
25#467—21no—
26#467—21no—
27#467C2211yescharacteristic
28#467C2212yes—
29#467C2211yescharacteristic
30#467C2212yes—
31#466C2211yescharacteristic
32#466C2211yescharacteristic
33#465—21no—
34#465—21no—
35#427C2211yescharacteristic
36C2×#107C2211yescharacteristic
37#416—21no—
38C2×#103—21no—
39#367C2212yescomplemented
40#366C2212yescomplemented
41#358C2212yescomplemented
42#358C2212yescomplemented
43#358C2212yescomplemented
44#357C2212yescomplemented
45C2×#41—21no—
46C22×Q8—21no—
47C23×C4D411yescharacteristic
48#135—22no—
49#135—22no—
50C2×Q16—22no—
51C2×Q16—21no—
52C2×Q16—22no—
53C2×Q16C2312yes—
54C2×Q16C2311yescharacteristic
55C2×SD16—21no—
56C2×D8C2311yescharacteristic
57C22×C8—22no—
58C22×C8—22no—
59C22×C8D411yescharacteristic
60C22×C8D411yescharacteristic
61C22×C8D411yescharacteristic
62C22×C8C2311yescharacteristic
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65#123—41no—
66C22⋊Q8—22no—
67C22⋊Q8—22no—
68C22⋊Q8—21no—
69C22⋊Q8—21no—
70C22⋊Q8—41no—
71C22⋊Q8—41no—
72#121—21no—
73#121—21no—
74#120—41no—
75C4×D4—41no—
76C4×D4—41no—
77#117—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—41no—
80C2×C22⋊C4—41no—
81Q32—24no—
82SD32—24no—
83M32—22no—
84M32—24no—
85M32D414yescomplemented
86C2×C16—24no—
87C2×C16C2312yes—
88C2×C16C2312yes—
89#107—21no—
90#107C2312yes—
91#107C2311yescharacteristic
92#107C2311yescharacteristic
93#106—21no—
94#103—22no—
95#103—22no—
96#103—21no—
97#103—21no—
98#102—21no—
99#102—21no—
100C24D811yescharacteristic
101#41—42no—
102#41—42no—
103C2×Q8—21no—
104C2×Q8—22no—
105C2×Q8—41no—
106C2×Q8—42no—
107C2×Q8—42no—
108C2×D4—21no—
109C2×D4—41no—
110C2×D4—41no—
111C2×D4—41no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—41no—
115C22×C4—41no—
116C22×C4—41no—
117C22×C4D811yescharacteristic
118C22×C4D811yescharacteristic
119C22×C4D811yescharacteristic
120C22×C4C2×D411yescharacteristic
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D411yescharacteristic
123Q16—22no—
124Q16—21no—
125Q16—22no—
126Q16—21no—
127Q16—42no—
128SD16—42no—
129D8—21no—
130D8—21no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8C2×D411yescharacteristic
139C2×C8C2×D411yescharacteristic
140C2×C8C2×D411yescharacteristic
141C2×C8C2×D412yes—
142C2×C8#4112yes—
143C2×C8C2411yescharacteristic
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—22no—
147C4⋊C4—41no—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C4⋊C4—41no—
151C4⋊C4—41no—
152C22⋊C4—42no—
153C22⋊C4—42no—
154C22⋊C4—41no—
155C22⋊C4—41no—
156C22⋊C4—41no—
157C22⋊C4—41no—
158C22⋊C4—41no—
159C42—41no—
160C16—22no—
161C16C2×D414yes—
162C23—22no—
163C23—22no—
164C23—41no—
165C23—41no—
166C23C2×D811yescharacteristic
167C23C2×D811yescharacteristic
168C23C2×D811yescharacteristic
169Q8—41no—
170Q8—42no—
171Q8—42no—
172Q8—41no—
173Q8—82no—
174D4—41no—
175D4—41no—
176D4—82no—
177D4—82no—
178D4—82no—
179C2×C4—22no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4—42no—
191C2×C4—82no—
192C2×C4—82no—
193C2×C4—82no—
194C2×C4#12112yes—
195C2×C4C2×D811yescharacteristic
196C2×C4C2×D811yescharacteristic
197C2×C4C2×D811yescharacteristic
198C2×C4#13512yes—
199C2×C4C22×D411yescharacteristic
200C8—21no—
201C8—21no—
202C8#12112yes—
203C8C22×D411yescharacteristic
204C8C2×#4111yescharacteristic
205C22—21no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—44no—
213C22—81no—
214C22—81no—
215C22—81no—
216C22#46612yes—
217C22#51012yescomplemented
218C22C22×D811yescharacteristic
219C4—21no—
220C4—21no—
221C4—81no—
222C4—82no—
223C4—81no—
224C4—81no—
225C4—82no—
226C4#46612yes—
227C4C2×#12111yescharacteristic
228C4C2×#13511yescharacteristic
229C2—22no—
230C2—22no—
231C2—41no—
232C2—161no—
233C2C2×#46611yescharacteristic
234C2C2×#51011yescharacteristic
235C2#341411yescharacteristic
subgroupG/H
center, socleC22C22×D8
commutator, Frattini, cosocleC2×C8C24
prime radical#3050C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#510C2semidirect
2. C22×M32C2semidirect
4. C2×#467C2semidirect
5. #2220C2semidirect
6. #2219C2semidirect
7. #2217C2semidirect
8. #2216C2semidirect
9. #2159C2semidirect
10. #2141C2semidirect
11. #2139C2semidirect
18. C2×Q32C22semidirect
19. C2×SD32C22semidirect
39. #367C22semidirect
40. #366C22semidirect
41. #358C22semidirect
42. #358C22semidirect
43. #358C22semidirect
44. #357C22semidirect
85. M32D4semidirect
217. C22#510semidirect

Maximal subgroups

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

subgroupindex
C2×#5102
C22×M322
#30502
C2×#4672
#22202
#22192
#22172
#22162
#21592
#21412
#21392