Finite Group Databasebeta

← #33605

Subgroups — #33605

Order 256.

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

normal subgroups only

This group has 369 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×#509C211yesmaximal, characteristic, complemented
2C22×M32C211yesmaximal, characteristic, complemented
3#3050C211yesmaximal, characteristic, complemented
4C2×#466C211yesmaximal, characteristic, complemented
5#2220C212yesmaximal, complemented
6#2218C212yesmaximal, complemented
7#2217C212yesmaximal, complemented
8#2215C212yesmaximal, complemented
9#2159C211yesmaximal, characteristic, complemented
10#2140C211yesmaximal, characteristic, complemented
11#2139C211yesmaximal, characteristic, complemented
12C2×#135C2211yescharacteristic
13C22×D8C2211yescharacteristic
14C23×C8C2211yescharacteristic
15#525—21no—
16C2×#121—21no—
17#509—24no—
18C2×SD32C2212yescomplemented
19C2×D16C2212yescomplemented
20C2×M32—24no—
21C2×M32C2211yescharacteristic
22C2×M32C2211yescharacteristic
23C2×M32C2212yescomplemented
24C22×C16C2212yescomplemented
25#467C2211yescharacteristic, complemented
26#467C2211yescharacteristic, complemented
27#466—21no—
28#466—21no—
29#466C2212yes—
30#466C2212yes—
31#466C2211yescharacteristic, complemented
32#466C2211yescharacteristic, complemented
33#465—21no—
34#465—21no—
35#427C2211yescharacteristic
36C2×#107C2211yescharacteristic
37#416—21no—
38C2×#102—21no—
39#367C2212yescomplemented
40#366C2212yescomplemented
41#358C2212yescomplemented
42#357C2212yescomplemented
43#357C2212yescomplemented
44#357C2212yescomplemented
45C2×#41—21no—
46C22×D4—21no—
47C22×D4—41no—
48C23×C4D411yescharacteristic
49#135—22no—
50#135—22no—
51C2×Q16C2311yescharacteristic
52C2×SD16—21no—
53C2×D8—22no—
54C2×D8—21no—
55C2×D8—22no—
56C2×D8C2312yes—
57C2×D8C2311yescharacteristic
58C22×C8—22no—
59C22×C8—22no—
60C22×C8D411yescharacteristic
61C22×C8D411yescharacteristic
62C22×C8D411yescharacteristic
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66#123—41no—
67C22⋊Q8—21no—
68C22⋊Q8—21no—
69#121—22no—
70#121—22no—
71#121—21no—
72#121—21no—
73#121—41no—
74#121—41no—
75#120—41no—
76C4×D4—41no—
77C4×D4—41no—
78#117—21no—
79C2×C4⋊C4—21no—
80C2×C22⋊C4—41no—
81SD32—24no—
82D16—24no—
83M32—22no—
84M32—24no—
85M32D414yescomplemented
86C2×C16—24no—
87C2×C16C2312yes—
88C2×C16C2312yes—
89#107—21no—
90#107C2311yescharacteristic
91#107C2311yescharacteristic
92#107C2312yes—
93#106—21no—
94#103—21no—
95#103—21no—
96#102—22no—
97#102—22no—
98#102—21no—
99#102—21no—
100C24—41no—
101C24D811yescharacteristic
102#41—42no—
103#41—42no—
104C2×Q8—21no—
105C2×D4—22no—
106C2×D4—21no—
107C2×D4—41no—
108C2×D4—41no—
109C2×D4—41no—
110C2×D4—42no—
111C2×D4—42no—
112C2×D4—42no—
113C2×D4—41no—
114C2×D4—42no—
115C2×D4—84no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—41no—
119C22×C4—41no—
120C22×C4D811yescharacteristic
121C22×C4D811yescharacteristic
122C22×C4D811yescharacteristic
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D411yescharacteristic
126Q16—21no—
127Q16—21no—
128SD16—42no—
129D8—21no—
130D8—22no—
131D8—22no—
132D8—21no—
133D8—42no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8C2×D411yescharacteristic
142C2×C8C2×D411yescharacteristic
143C2×C8C2×D411yescharacteristic
144C2×C8C2×D412yes—
145C2×C8#4112yes—
146C2×C8C2411yescharacteristic
147C4⋊C4—22no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—41no—
151C4⋊C4—41no—
152C4⋊C4—41no—
153C22⋊C4—41no—
154C22⋊C4—41no—
155C22⋊C4—41no—
156C22⋊C4—42no—
157C22⋊C4—42no—
158C22⋊C4—41no—
159C22⋊C4—41no—
160C42—41no—
161C16—22no—
162C16C2×D414yescomplemented
163C23—22no—
164C23—22no—
165C23—41no—
166C23—41no—
167C23—42no—
168C23—82no—
169C23—82no—
170C23—82no—
171C23C2×D811yescharacteristic
172C23C2×D811yescharacteristic
173C23C2×D811yescharacteristic
174Q8—41no—
175Q8—41no—
176D4—42no—
177D4—41no—
178D4—42no—
179D4—41no—
180D4—82no—
181D4—84no—
182D4—82no—
183D4—82no—
184D4—84no—
185D4—82no—
186C2×C4—22no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—41no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4#12112yes—
200C2×C4C2×D811yescharacteristic
201C2×C4C2×D811yescharacteristic
202C2×C4C2×D811yescharacteristic
203C2×C4#13512yes—
204C2×C4C22×D411yescharacteristic
205C8—21no—
206C8—21no—
207C8#12112yes—
208C8C22×D411yescharacteristic
209C8C2×#4111yescharacteristic
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—21no—
217C22—44no—
218C22—82no—
219C22—81no—
220C22—81no—
221C22—82no—
222C22—81no—
223C22—82no—
224C22—164no—
225C22#46612yes—
226C22#50912yescomplemented
227C22C22×D811yescharacteristic
228C4—21no—
229C4—21no—
230C4—82no—
231C4—81no—
232C4—81no—
233C4—81no—
234C4#46612yes—
235C4C2×#12111yescharacteristic
236C4C2×#13511yescharacteristic
237C2—22no—
238C2—22no—
239C2—41no—
240C2—161no—
241C2—162no—
242C2C2×#46611yescharacteristic
243C2C2×#50911yescharacteristic
244C2#341411yescharacteristic
subgroupG/H
center, socleC22C22×D8
commutator, Frattini, cosocleC2×C8C24
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×#509C2semidirect
2. C22×M32C2semidirect
3. #3050C2semidirect
4. C2×#466C2semidirect
5. #2220C2semidirect
6. #2218C2semidirect
7. #2217C2semidirect
8. #2215C2semidirect
9. #2159C2semidirect
10. #2140C2semidirect
11. #2139C2semidirect
18. C2×SD32C22semidirect
19. C2×D16C22semidirect
23. C2×M32C22semidirect
24. C22×C16C22semidirect
25. #467C22semidirect
26. #467C22semidirect
31. #466C22semidirect
32. #466C22semidirect
39. #367C22semidirect
40. #366C22semidirect
41. #358C22semidirect
42. #357C22semidirect
43. #357C22semidirect
44. #357C22semidirect
85. M32D4semidirect
162. C16C2×D4semidirect
226. C22#509semidirect

Maximal subgroups

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

subgroupindex
C2×#5092
C22×M322
#30502
C2×#4662
#22202
#22182
#22172
#22152
#21592
#21402
#21392