Finite Group Databasebeta

← #33754

Subgroups — #33754

Order 256.

323 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 323 conjugacy classes of subgroups — too many to draw. Only its 101 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
1#3050C211yesmaximal, characteristic, complemented
2C2×#466C211yesmaximal, characteristic, complemented
3#2236C211yesmaximal, characteristic, complemented
4#2234C212yesmaximal, complemented
5#2233C211yesmaximal, characteristic, complemented
6#2217C212yesmaximal, complemented
7#2215C212yesmaximal, complemented
8#2187C211yesmaximal, characteristic, complemented
9#2186C212yesmaximal, complemented
10#2184C211yesmaximal, characteristic, complemented
11C2×C22⋊C16C211yesmaximal, characteristic, complemented
12C2×#135C2211yescharacteristic, complemented
13C22×D8C2211yescharacteristic
14C23×C8C2211yescharacteristic
15#525—21no—
16C2×#121—21no—
17C2×SD32C2212yescomplemented
18C2×D16C2212yescomplemented
19C22×C16C2212yescomplemented
20#467C2211yescharacteristic
21#467C2211yescharacteristic, complemented
22#466—21no—
23#466—21no—
24#466C2211yescharacteristic
25#466C2212yes—
26#466C2212yes—
27#466C2211yescharacteristic, complemented
28#465—21no—
29#465—21no—
30#427C2211yescharacteristic, complemented
31C2×#107C2211yescharacteristic
32#416—21no—
33C2×#102—21no—
34#367C2212yescomplemented
35#366C2212yescomplemented
36#358C2212yescomplemented
37#357C2212yescomplemented
38#357C2212yescomplemented
39#357C2212yescomplemented
40C22⋊C16C2211yescharacteristic
41C22⋊C16C2211yescharacteristic
42C22⋊C16C2212yescomplemented
43C2×#41—21no—
44C22×D4—21no—
45C22×D4—41no—
46C23×C4D411yescharacteristic
47#135—42no—
48C2×Q16C2311yescharacteristic
49C2×SD16—21no—
50C2×D8—21no—
51C2×D8—42no—
52C2×D8C2312yes—
53C2×D8C2311yescharacteristic
54C22×C8—42no—
55C22×C8D411yescharacteristic
56C22×C8D411yescharacteristic
57C22×C8D411yescharacteristic
58C22×C8C2311yescharacteristic
59C22×C8C2311yescharacteristic
60C22×C8C2311yescharacteristic
61#123—41no—
62C22⋊Q8—21no—
63C22⋊Q8—21no—
64#121—22no—
65#121—22no—
66#121—21no—
67#121—21no—
68#121—41no—
69#121—41no—
70#120—41no—
71C4×D4—41no—
72C4×D4—41no—
73#117—21no—
74C2×C4⋊C4—21no—
75C2×C22⋊C4—41no—
76SD32—42no—
77D16—42no—
78C2×C16—42no—
79C2×C16C2312yes—
80C2×C16C2312yes—
81#107—21no—
82#107C2311yescharacteristic
83#107C2311yescharacteristic
84#107C2312yes—
85#106—21no—
86#103—21no—
87#103—21no—
88#102—22no—
89#102—22no—
90#102—21no—
91#102—21no—
92C24—41no—
93C24D811yescharacteristic
94#41—42no—
95#41—42no—
96C2×Q8—21no—
97C2×D4—22no—
98C2×D4—21no—
99C2×D4—42no—
100C2×D4—42no—
101C2×D4—41no—
102C2×D4—41no—
103C2×D4—41no—
104C2×D4—42no—
105C2×D4—42no—
106C2×D4—41no—
107C2×D4—84no—
108C22×C4—41no—
109C22×C4—41no—
110C22×C4—41no—
111C22×C4—41no—
112C22×C4D811yescharacteristic
113C22×C4D811yescharacteristic
114C22×C4D811yescharacteristic
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118Q16—22no—
119SD16—42no—
120D8—22no—
121D8—22no—
122D8—22no—
123D8—42no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—41no—
128C2×C8—41no—
129C2×C8—41no—
130C2×C8C2×D411yescharacteristic
131C2×C8C2×D411yescharacteristic
132C2×C8C2×D411yescharacteristic
133C2×C8C2411yescharacteristic
134C4⋊C4—22no—
135C4⋊C4—21no—
136C4⋊C4—21no—
137C4⋊C4—41no—
138C4⋊C4—41no—
139C4⋊C4—41no—
140C22⋊C4—41no—
141C22⋊C4—41no—
142C22⋊C4—41no—
143C22⋊C4—42no—
144C22⋊C4—42no—
145C22⋊C4—41no—
146C22⋊C4—41no—
147C42—41no—
148C16—22no—
149C16—22no—
150C23—22no—
151C23—22no—
152C23—41no—
153C23—41no—
154C23—42no—
155C23—82no—
156C23—82no—
157C23—82no—
158C23C2×D811yescharacteristic
159C23C2×D811yescharacteristic
160C23C2×D811yescharacteristic
161Q8—41no—
162Q8—41no—
163D4—42no—
164D4—41no—
165D4—42no—
166D4—41no—
167D4—82no—
168D4—84no—
169D4—82no—
170D4—82no—
171D4—82no—
172D4—84no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—41no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4C2×D811yescharacteristic
189C2×C4C2×D811yescharacteristic
190C2×C4C2×D811yescharacteristic
191C2×C4C22×D411yescharacteristic
192C8—21no—
193C8—21no—
194C8—21no—
195C8H(5,𝔽2)12yes—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—44no—
204C22—81no—
205C22—82no—
206C22—82no—
207C22—81no—
208C22—81no—
209C22—82no—
210C22—164no—
211C22#50812yes—
212C22#50912yes—
213C22C22×D811yescharacteristic
214C4—21no—
215C4—21no—
216C4—21no—
217C4—81no—
218C4—82no—
219C4—81no—
220C4—81no—
221C4#53411yescharacteristic
222C4#57511yescharacteristic
223C2—22no—
224C2—22no—
225C2—41no—
226C2—161no—
227C2—162no—
228C2#318611yescharacteristic
229C2C2×#50811yescharacteristic
230C2C2×#50911yescharacteristic
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. #3050C2semidirect
2. C2×#466C2semidirect
3. #2236C2semidirect
4. #2234C2semidirect
5. #2233C2semidirect
6. #2217C2semidirect
7. #2215C2semidirect
8. #2187C2semidirect
9. #2186C2semidirect
10. #2184C2semidirect
11. C2×C22⋊C16C2semidirect
12. C2×#135C22semidirect
17. C2×SD32C22semidirect
18. C2×D16C22semidirect
19. C22×C16C22semidirect
21. #467C22semidirect
27. #466C22semidirect
30. #427C22semidirect
34. #367C22semidirect
35. #366C22semidirect
36. #358C22semidirect
37. #357C22semidirect
38. #357C22semidirect
39. #357C22semidirect
42. C22⋊C16C22semidirect

Maximal subgroups

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

subgroupindex
#30502
C2×#4662
#22362
#22342
#22332
#22172
#22152
#21872
#21862
#21842
C2×C22⋊C162