Finite Group Databasebeta

← #13135777

Subgroups — #13135777

Order 1728.

473 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 473 conjugacy classes of subgroups — too many to draw. Only its 61 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#11721850C211yesmaximal, characteristic
2H(3,𝔽3)×#99C211yesmaximal, characteristic, complemented
3#11720580C211yesmaximal, characteristic, complemented
4C3×#4474—34nomaximal
5C2×D4×H(3,𝔽3)C2211yescharacteristic
6C2×#6235—21no—
7C2×#6235—21no—
8C2×C4⋊#272C411yescharacteristic, complemented
9C22⋊C4×H(3,𝔽3)—21no—
10#88552—21no—
11C4×#1084C411yescharacteristic, complemented
12C3×#940—34no—
13C32×#99D314yescomplemented
14C3×#834—34no—
15C23×H(3,𝔽3)D411yescharacteristic
16C23×H(3,𝔽3)D411yescharacteristic
17C22×#272—21no—
18D4×H(3,𝔽3)—41no—
19C2×C4×H(3,𝔽3)—41no—
20C2×C4×H(3,𝔽3)C2×C411yescharacteristic
21#6235—42no—
22#6235—42no—
23C2×#1084—21no—
24C2×#1084—41no—
25C4⋊#272—41no—
26C3×#353—91no—
27#4474—34no—
28C2×C32×D4C2×D314yes—
29C2×C3×#67—64no—
30C2×C3×#67—64no—
31C2×C3×D12—34no—
32C32×C22⋊C4—24no—
33C3×#217—64no—
34C3×C4×Dic3—34no—
35C22×H(3,𝔽3)—21no—
36C22×H(3,𝔽3)—21no—
37C22×H(3,𝔽3)—41no—
38C22×H(3,𝔽3)—41no—
39C22×H(3,𝔽3)C22⋊C411yescharacteristic
40C2×#272—41no—
41C2×#272—41no—
42C2×#272—41no—
43C4×H(3,𝔽3)—21no—
44C4×H(3,𝔽3)—81no—
45#1084—22no—
46#1084—81no—
47C3×#127—91no—
48#940—34no—
49C3×#99—34no—
50C3×#99—91no—
51C3×#99C3⋊D311yescharacteristic
52#834—34no—
53C23×C32D1214yes—
54C23×C32#6714yes—
55C22×C3×D3—64no—
56C32×D4—44no—
57C2×C32×C4—44no—
58C2×C32×C4C4×D314yes—
59C3×#67—128no—
60C3×#67—128no—
61C2×C3×Dic3—64no—
62C2×C3×Dic3—124no—
63C3×D12—124no—
64#353—91noSylow
65C2×H(3,𝔽3)—21no—
66C2×H(3,𝔽3)—41no—
67C2×H(3,𝔽3)—41no—
68C2×H(3,𝔽3)#9911yescharacteristic
69#272—81no—
70C2×C3×D4—34no—
71C2×C3×D4—91no—
72C2×C3×D4—181no—
73C2×C3×D4—181no—
74C2×C3×D4C2×C3⋊D311yescharacteristic
75C2×#67—64no—
76C2×#67—64no—
77C2×D12—34no—
78C3×C22⋊C4—21no—
79C3×C22⋊C4—64no—
80C3×C22⋊C4—181no—
81C3×C42—91no—
82#217—64no—
83C4×Dic3—34no—
84C22×C32—24no—
85C22×C32—24no—
86C22×C32—44no—
87C22×C32—44no—
88C22×C32(C2×D3)⋊C414yes—
89C2×C3×D3—124no—
90C2×C3×D3—124no—
91C2×C3×D3—124no—
92C32×C4—24no—
93C32×C4—84no—
94C3×Dic3—68no—
95C3×Dic3—244no—
96#127—91no—
97#99—91no—
98#99#27211yescharacteristic, complemented
99H(3,𝔽3)#35311yescharacteristic, complemented, Sylow
100C23×C3—34no—
101C23×C3—34no—
102C23×C3—181no—
103C23×C3C4⋊(C3⋊D3)11yescharacteristic
104C23×C3#63811yescharacteristic
105C22×D3—64no—
106C3×D4—41no—
107C3×D4—124no—
108C3×D4—361no—
109C3×D4—362no—
110C3×D4—362no—
111C2×C3×C4—34no—
112C2×C3×C4—41no—
113C2×C3×C4—124no—
114C2×C3×C4—181no—
115C2×C3×C4—361no—
116C2×C3×C4C4×C3⋊D311yescharacteristic
117#67—128no—
118#67—128no—
119C2×Dic3—64no—
120C2×Dic3—124no—
121D12—124no—
122C2×C32—24no—
123C2×C32—44no—
124C2×C32—44no—
125C2×C32#80614yes—
126C3×D3—244no—
127C2×D4—91no—
128C2×D4—181no—
129C2×D4—181no—
130C2×D4C2×#27211yescharacteristic
131C22⋊C4—21no—
132C22⋊C4—181no—
133C42—91no—
134C22×C3—21no—
135C22×C3—21no—
136C22×C3—34no—
137C22×C3—41no—
138C22×C3—41no—
139C22×C3—64no—
140C22×C3—64no—
141C22×C3—124no—
142C22×C3—124no—
143C22×C3—361no—
144C22×C3—361no—
145C22×C3—361no—
146C22×C3#375211yescharacteristic
147C2×D3—124no—
148C2×D3—124no—
149C2×D3—124no—
150C3×C4—21no—
151C3×C4—64no—
152C3×C4—81no—
153C3×C4—182no—
154C3×C4—244no—
155C3×C4—721no—
156Dic3—68no—
157Dic3—244no—
158C32#447414yescomplemented
159C23—181no—
160C23C4⋊#27211yescharacteristic
161C23#623511yescharacteristic
162D4—41no—
163D4—361no—
164D4—362no—
165D4—362no—
166C2×C4—41no—
167C2×C4—181no—
168C2×C4—361no—
169C2×C4C4×#27211yescharacteristic
170C2×C3—21no—
171C2×C3—34no—
172C2×C3—41no—
173C2×C3—41no—
174C2×C3—64no—
175C2×C3—124no—
176C2×C3—124no—
177C2×C3—721no—
178C2×C3#6363311yescharacteristic
179D3—244no—
180C22—21no—
181C22—21no—
182C22—41no—
183C22—41no—
184C22—361no—
185C22—361no—
186C22—361no—
187C22#8854211yescharacteristic
188C4—21no—
189C4—81no—
190C4—182no—
191C4—721no—
192C3—34no—
193C3#1058934011yescharacteristic
194C2—21no—
195C2—41no—
196C2—41no—
197C2—721no—
198C2#1172052411yescharacteristic
subgroupG/H
center, socleC2×C3#63633
commutatorC2×C4×H(3,𝔽3)C2×C4
FrattiniC2×C3×D4C2×C3⋊D3
FittingH(3,𝔽3)×#99C2
prime radical#11721850C2
cosocleC2×D4×H(3,𝔽3)C22
hypercenterC3×#99C3⋊D3
hypocenterH(3,𝔽3)#353
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×D4 ◁ C3×#99

Complements

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

NH ≅ G/Ntype
2. H(3,𝔽3)×#99C2semidirect
3. #11720580C2semidirect
8. C2×C4⋊#272C4semidirect
11. C4×#1084C4semidirect
13. C32×#99D3semidirect
98. #99#272semidirect
99. H(3,𝔽3)#353semidirect
158. C32#4474semidirect

Maximal subgroups

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

subgroupindex
#117218502
H(3,𝔽3)×#992
#117205802
C3×#44743