Finite Group Databasebeta

← #13147369

Subgroups — #13147369

Order 1728.

283 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 283 conjugacy classes of subgroups — too many to draw. Only its 47 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#11722451C211yesmaximal, characteristic, complemented
2#11722449C211yesmaximal, characteristic, complemented
3#11721802C211yesmaximal, characteristic
4#11720791C211yesmaximal, characteristic, complemented
5#11720789C211yesmaximal, characteristic, complemented
6#11720776C211yesmaximal, characteristic, complemented
7C8×#1088C211yesmaximal, characteristic, complemented
8#88748C2211yescharacteristic
9#88744C2211yescharacteristic
10#88631—21no—
11C2×#6188—21no—
12C4×#1088C2211yescharacteristic
13#88627—21no—
14#88627C2211yescharacteristic
15#88627C2211yescharacteristic
16#88626—21no—
17#88626C2211yescharacteristic
18SD16×H(3,𝔽3)C411yescharacteristic, complemented
19#88550C411yescharacteristic, complemented
20#88545C411yescharacteristic, complemented
21#88528C411yescharacteristic, complemented
22C8×#272C2211yescharacteristic
23C3×#916—62no—
24C2×#1088—21no—
25Q8×H(3,𝔽3)C2×C411yescharacteristic
26D4×H(3,𝔽3)C2×C411yescharacteristic
27#6235—21no—
28C2×#1084—21no—
29C4⋊#272—21no—
30C4⋊#272C2×C411yescharacteristic
31C4×#272—21no—
32C4×#272C2311yescharacteristic
33#6229C2×C411yescharacteristic
34#6188—21no—
35#6188—21no—
36#6188D411yescharacteristic
37#6188D411yescharacteristic
38C8×H(3,𝔽3)C2×C411yescharacteristic
39#6180C2×C411yescharacteristic
40C3×#443—91nomaximal
41C3×#239—62no—
42C3×#237—62no—
43C32×SD16—22no—
44C3×#216—62no—
45C3×#213—62no—
46C3×#204—62no—
47C3×D3×C8—62no—
48C22×H(3,𝔽3)—21no—
49C2×#272—21no—
50C2×#272#4111yescharacteristic
51#1088—22no—
52C4×H(3,𝔽3)—21no—
53C4×H(3,𝔽3)C22×C411yescharacteristic
54#1084—21no—
55#1084C2×D411yescharacteristic
56C3×#135—91no—
57C3×#131—91no—
58C3×#131—91no—
59#916—62no—
60C3×#108—91no—
61C3×#104—91no—
62C3×#104—91no—
63C3×C4×C8—91no—
64C32×Q8—22no—
65C32×D4—22no—
66C3×#67—122no—
67C2×C3×Dic3—122no—
68C3×D12—62no—
69C3×D12—122no—
70C3×C4×D3—62no—
71C3×C4×D3—122no—
72C3×Dic6—62no—
73C32×C8—22no—
74C3×C3⋊C8—62no—
75#443—91noSylow
76C2×H(3,𝔽3)—41no—
77C2×H(3,𝔽3)C4×D411yescharacteristic
78#272—21no—
79#272—41no—
80C3×#41—91no—
81C3×#41—91no—
82#239—62no—
83#237—62no—
84C3×Q16—91no—
85C3×SD16—62no—
86C3×SD16—91no—
87C3×SD16#15711yescharacteristic
88C3×D8—91no—
89C3×M16—91no—
90C3×M16—91no—
91C3×M16—181no—
92C3×M16—181no—
93C2×C3×C8—91no—
94C2×C3×C8—91no—
95C2×C3×C8—181no—
96C2×C3×C8—181no—
97C3×C42—91no—
98#216—62no—
99#213—62no—
100#204—62no—
101D3×C8—62no—
102C22×C32—42no—
103C2×C3×D3—62no—
104C2×C3×D3—122no—
105C32×C4—22no—
106C32×C4—42no—
107C3×Dic3—62no—
108C3×Dic3—122no—
109#135—91no—
110#131—91no—
111#131—91no—
112#108—91no—
113#104—91no—
114#104—91no—
115C4×C8—91no—
116H(3,𝔽3)#44311yescharacteristic, complemented, Sylow
117C3×Q8—62no—
118C3×Q8—91no—
119C3×Q8C2×#15711yescharacteristic
120C3×D4—62no—
121C3×D4—91no—
122C3×D4—181no—
123C3×D4—181no—
124C3×D4C2×#15711yescharacteristic
125C2×C3×C4—91no—
126C2×C3×C4—181no—
127C2×C3×C4—181no—
128C2×C3×C4—181no—
129#67—122no—
130C2×Dic3—122no—
131D12—62no—
132D12—122no—
133C4×D3—62no—
134C4×D3—122no—
135Dic6—62no—
136C3×C8—62no—
137C3×C8—91no—
138C3×C8—91no—
139C3×C8—91no—
140C3×C8—181no—
141C3×C8—181no—
142C3×C8C2×#15711yescharacteristic
143C3⋊C8—62no—
144C2×C32—22no—
145C2×C32—82no—
146C3×D3—122no—
147C3×D3—242no—
148#41—91no—
149#41—91no—
150Q16—91no—
151SD16—91no—
152SD16#108811yescharacteristic, complemented
153D8—91no—
154M16—91no—
155M16—91no—
156M16—181no—
157M16—181no—
158C2×C8—91no—
159C2×C8—91no—
160C2×C8—181no—
161C2×C8—181no—
162C42—91no—
163C22×C3—21no—
164C22×C3—91no—
165C22×C3—122no—
166C22×C3—181no—
167C2×D3—62no—
168C2×D3—122no—
169C3×C4—21no—
170C3×C4—62no—
171C3×C4—91no—
172C3×C4—122no—
173C3×C4—182no—
174C3×C4—181no—
175C3×C4C22×#15711yescharacteristic
176Dic3—62no—
177Dic3—122no—
178C32—22no—
179Q8—91no—
180Q8C2×#108811yescharacteristic
181D4—91no—
182D4—181no—
183D4—181no—
184D4C2×#108811yescharacteristic
185C2×C4—91no—
186C2×C4—181no—
187C2×C4—181no—
188C2×C4—181no—
189C8—91no—
190C8—91no—
191C8—91no—
192C8—181no—
193C8—181no—
194C8C2×#108811yescharacteristic
195C2×C3—41no—
196C2×C3—62no—
197C2×C3—181no—
198C2×C3—242no—
199C2×C3—361no—
200C2×C3D4×#15711yescharacteristic
201D3—122no—
202D3—242no—
203C22—21no—
204C22—91no—
205C22—181no—
206C4—21no—
207C4—91no—
208C4—182no—
209C4—181no—
210C4C22×#108811yescharacteristic
211C3—62no—
212C3#1059378311yescharacteristic
213C2—41no—
214C2—181no—
215C2—361no—
216C2D4×#108811yescharacteristic
subgroupG/H
center, socleC2×C3D4×#157
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC3×C4C22×#157
FittingSD16×H(3,𝔽3)C4
prime radical#11721802C2
cosocleC4×#272C23
hypercenterC3×SD16#157
hypocenterH(3,𝔽3)#443
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×SD16

Complements

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

NH ≅ G/Ntype
1. #11722451C2semidirect
2. #11722449C2semidirect
4. #11720791C2semidirect
5. #11720789C2semidirect
6. #11720776C2semidirect
7. C8×#1088C2semidirect
18. SD16×H(3,𝔽3)C4semidirect
19. #88550C4semidirect
20. #88545C4semidirect
21. #88528C4semidirect
116. H(3,𝔽3)#443semidirect
152. SD16#1088semidirect

Maximal subgroups

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

subgroupindex
#117224512
#117224492
#117218022
#117207912
#117207892
#117207762
C8×#10882
C3×#4439