Finite Group Databasebeta

← #13154272

Subgroups — #13154272

Order 1728.

2354 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 2354 conjugacy classes of subgroups — too many to draw. Only its 235 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×D4×#272C214yesmaximal, complemented
2H(3,𝔽3)×#127C211yesmaximal, characteristic, complemented
3#11721848C214yesmaximal, complemented
4#11721847C214yesmaximal, complemented
5C4×C4⋊#272C212yesmaximal, complemented
6C3×#5717—34nomaximal
7C23×#272C2214yescomplemented
8C2×D4×H(3,𝔽3)C2214yescomplemented
9C2×D4×H(3,𝔽3)C2212yescomplemented
10C2×#6235C2218yescomplemented
11D4×#272—216no—
12C2×C4⋊#272C2212yes—
13C2×C4×#272C2214yescomplemented
14C42×H(3,𝔽3)C2211yescharacteristic, complemented
15#88552C2214yescomplemented
16#88542C2214yescomplemented
17C4⋊#1084C2212yescomplemented
18C2×C3×D3×D4—316no—
19C32×#127D314yescomplemented
20C3×#938—316no—
21C3×#937—316no—
22C3×C4×D12—38no—
23C23×H(3,𝔽3)C2314yes—
24C22×#272—216no—
25C22×#272—44no—
26C22×#272C2314yes—
27D4×H(3,𝔽3)—28no—
28D4×H(3,𝔽3)—42no—
29C2×C4×H(3,𝔽3)C2311yescharacteristic
30C2×C4×H(3,𝔽3)C2312yes—
31#6235—216no—
32C2×#1084C2314yes—
33C4⋊#272D418yescomplemented
34C4×#272—28no—
35C3×D42—91no—
36#5717—34no—
37C23×C3×D3—316no—
38C2×C32×D4C2×D3116yescomplemented
39C2×C32×D4C2×D318yescomplemented
40C2×C3×#67—332no—
41C3×D3×D4—664no—
42C2×C3×D12—38no—
43C2×C3×C4×D3—316no—
44C32×C42C2×D314yescomplemented
45C3×#217—316no—
46C3×(C2×D3)⋊C4—316no—
47C3×C4⋊Dic3—38no—
48C22×H(3,𝔽3)—28no—
49C22×H(3,𝔽3)—24no—
50C22×H(3,𝔽3)C2411yescharacteristic
51C2×#272—24no—
52C2×#272—24no—
53C2×#272—416no—
54C2×#272C2×D418yescomplemented
55C4×H(3,𝔽3)—21no—
56C4×H(3,𝔽3)C2×D414yescomplemented
57#1084—24no—
58C22×C3×D4—94no—
59C2×D3×D4—316no—
60C3×#127—34no—
61C3×#127C3⋊D311yescharacteristic
62C3×#121—94no—
63C3×#120—94no—
64C3×C4×D4—92no—
65#938—316no—
66#937—316no—
67C4×D12—38no—
68C23×C32C22×D3116yes—
69C22×C3×D3—316no—
70C22×C3×D3—664no—
71C22×C3×D3—1216no—
72C32×D4—232no—
73C32×D4—48no—
74C2×C32×C4C22×D314yes—
75C2×C32×C4C22×D318yes—
76C3×#67—664no—
77C2×C3×Dic3—316no—
78C3×D12—332no—
79C3×C4×D3—632no—
80D42—91noSylow
81C2×H(3,𝔽3)—44no—
82C2×H(3,𝔽3)C22×D412yes—
83C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
84#272—28no—
85C24×C3—94no—
86C23×D3—316no—
87C2×C3×D4—316no—
88C2×C3×D4—38no—
89C2×C3×D4—98no—
90C2×C3×D4—92no—
91C2×C3×D4—1816no—
92C2×C3×D4C2×C3⋊D314yes—
93C2×C3×D4C2×C3⋊D312yes—
94C22×C3×C4—94no—
95C2×#67—332no—
96D3×D4—664no—
97C2×D12—38no—
98C2×C4×D3—316no—
99C3×C4⋊C4—92no—
100C3×C22⋊C4—94no—
101C3×C22⋊C4—94no—
102C3×C42—34no—
103C3×C42C2×C3⋊D311yescharacteristic
104#217—316no—
105(C2×D3)⋊C4—316no—
106C4⋊Dic3—38no—
107C22×C32—232no—
108C22×C32—216no—
109C22×C32C23×D314yes—
110C2×C3×D3—332no—
111C2×C3×D3—616no—
112C2×C3×D3—616no—
113C2×C3×D3—1264no—
114C32×C4—24no—
115C32×C4D3×D4116yescomplemented
116C3×Dic3—616no—
117C22×D4—94no—
118#127#27211yescharacteristic, complemented
119#121—94no—
120#120—94no—
121C4×D4—92no—
122H(3,𝔽3)D4211yescharacteristic, complemented, Sylow
123C23×C3—316no—
124C23×C3—94no—
125C23×C3—1816no—
126C23×C3—364no—
127C23×C3C22×C3⋊D314yes—
128C22×D3—316no—
129C22×D3—664no—
130C22×D3—1216no—
131C3×D4—28no—
132C3×D4—42no—
133C3×D4—632no—
134C3×D4—98no—
135C3×D4—128no—
136C3×D4—1816no—
137C2×C3×C4—38no—
138C2×C3×C4—34no—
139C2×C3×C4—94no—
140C2×C3×C4—188no—
141C2×C3×C4C22×C3⋊D311yescharacteristic
142C2×C3×C4C22×C3⋊D312yes—
143#67—664no—
144C2×Dic3—316no—
145D12—332no—
146C4×D3—632no—
147C2×C32—416no—
148C2×C32C2×D3×D418yes—
149C2×C32#100414yes—
150C3×D3—632no—
151C24—94no—
152C2×D4—98no—
153C2×D4—92no—
154C2×D4—1816no—
155C2×D4C2×#27214yescomplemented
156C2×D4C2×#27212yescomplemented
157C22×C4—94no—
158C4⋊C4—92no—
159C22⋊C4—94no—
160C22⋊C4—94no—
161C42C2×#27211yescharacteristic, complemented
162C22×C3—28no—
163C22×C3—24no—
164C22×C3—34no—
165C22×C3—632no—
166C22×C3—616no—
167C22×C3—98no—
168C22×C3—184no—
169C22×C3—184no—
170C22×C3—3616no—
171C22×C3C23×C3⋊D311yescharacteristic
172C2×D3—332no—
173C2×D3—616no—
174C2×D3—616no—
175C2×D3—1264no—
176C3×C4—21no—
177C3×C4—316no—
178C3×C4—64no—
179C3×C4—184no—
180C3×C4D4×C3⋊D314yes—
181Dic3—616no—
182C32#571714yescomplemented
183C23—94no—
184C23—1816no—
185C23—364no—
186C23C22×#27214yes—
187D4—28no—
188D4—42no—
189D4—98no—
190D4—1816no—
191C2×C4—94no—
192C2×C4—188no—
193C2×C4C22×#27211yescharacteristic
194C2×C4C22×#27212yes—
195C2×C3—38no—
196C2×C3—34no—
197C2×C3—44no—
198C2×C3—1216no—
199C2×C3—188no—
200C2×C3C2×D4×C3⋊D312yes—
201C2×C3#6436111yescharacteristic
202D3—632no—
203C22—28no—
204C22—24no—
205C22—98no—
206C22—184no—
207C22—184no—
208C22—3616no—
209C22C23×#27211yescharacteristic
210C4—21no—
211C4—184no—
212C4D4×#27214yescomplemented
213C3—34no—
214C3#1059618811yescharacteristic
215C2—44no—
216C2—188no—
217C2C2×D4×#27212yes—
218C2#1172266411yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FittingH(3,𝔽3)×#127C2
hypercenterC3×#127C3⋊D3
hypocenterH(3,𝔽3)D42
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×#127

Complements

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

NH ≅ G/Ntype
1. C2×D4×#272C2semidirect
2. H(3,𝔽3)×#127C2semidirect
3. #11721848C2semidirect
4. #11721847C2semidirect
5. C4×C4⋊#272C2semidirect
7. C23×#272C22semidirect
8. C2×D4×H(3,𝔽3)C22semidirect
9. C2×D4×H(3,𝔽3)C22semidirect
10. C2×#6235C22semidirect
13. C2×C4×#272C22semidirect
14. C42×H(3,𝔽3)C22semidirect
15. #88552C22semidirect
16. #88542C22semidirect
17. C4⋊#1084C22semidirect
19. C32×#127D3semidirect
33. C4⋊#272D4semidirect
38. C2×C32×D4C2×D3semidirect
39. C2×C32×D4C2×D3semidirect
44. C32×C42C2×D3semidirect
54. C2×#272C2×D4semidirect
56. C4×H(3,𝔽3)C2×D4semidirect
115. C32×C4D3×D4semidirect
118. #127#272semidirect
122. H(3,𝔽3)D42semidirect
155. C2×D4C2×#272semidirect
156. C2×D4C2×#272semidirect
161. C42C2×#272semidirect
182. C32#5717semidirect
212. C4D4×#272semidirect

Maximal subgroups

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

subgroupindex
C2×D4×#2722
H(3,𝔽3)×#1272
#117218482
#117218472
C4×C4⋊#2722
C3×#57173