Finite Group Databasebeta

← #13173385

Subgroups — #13173385

Order 1728.

368 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 67 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#11724517C212yesmaximal, complemented
2C3×#64128C211yesmaximal, characteristic
3#11723110C212yesmaximal, complemented
4#11723095C211yesmaximal, characteristic, complemented
5#11723092C211yesmaximal, characteristic, complemented
6#10593788—31nomaximal
7C3×#3832C2212yes—
8C4×C3⋊#157C2211yescharacteristic
9#88989—22no—
10#88989C2212yes—
11#88988—22no—
12#88988C2211yescharacteristic
13C33×Q16C411yescharacteristic, complemented
14C3×#3755C412yescomplemented
15C3×#3745C411yescharacteristic, complemented
16C3×C8×C3⋊D3C2211yescharacteristic
17#64289—32no—
18#64128D311yescharacteristic
19C3×#919—62no—
20#63785—32no—
21#63770—31no—
22#63767—31no—
23C2×C3⋊#157—21no—
24C33×Q8C2×C412yes—
25C3×C4⋊(C3⋊D3)—22no—
26C3×C4⋊(C3⋊D3)C2×C412yes—
27C3×C4×C3⋊D3—22no—
28C3×C4×C3⋊D3C2311yescharacteristic
29C3⋊#622—22no—
30C3⋊#622D412yes—
31C33×C8C2×C411yescharacteristic
32C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
33#5197—91nomaximal
34#3832C2×D312yes—
35C3×#239—64no—
36C4×#157—31no—
37#3788—32no—
38#3788—62no—
39#3787—31no—
40#3787—62no—
41C32×Q16—22no—
42C32×Q16—42no—
43C32×Q16Dic311yescharacteristic, complemented
44#3755Dic312yescomplemented
45#3745Dic311yescharacteristic, complemented
46C8×C3⋊D3C2×D311yescharacteristic
47C3×#215—64no—
48C3×D24—62no—
49C3×D3×C8—62no—
50C2×C3×C3⋊D3—22no—
51C2×C3×C3⋊D3#4111yescharacteristic
52C3⋊#157—41no—
53C33×C4—22no—
54C33×C4C22×C411yescharacteristic
55C3×C3⋊Dic3C2×D411yescharacteristic
56C3×#135—91no—
57#948—92no—
58#919—62no—
59#837—92no—
60#819—91no—
61#815—91no—
62C2×#157—61no—
63C32×Q8—24no—
64C32×Q8—44no—
65C32×Q8C2×Dic312yes—
66C4⋊(C3⋊D3)—22no—
67C4⋊(C3⋊D3)C2×Dic312yes—
68C4×C3⋊D3—22no—
69C4×C3⋊D3C22×D311yescharacteristic
70C3×D12—64no—
71C3×D12—124no—
72C3×C4×D3—62no—
73C3×C4×D3—124no—
74#622—32no—
75#622—62no—
76C32×C8—22no—
77C32×C8—42no—
78C32×C8C2×Dic311yescharacteristic
79C3⋊(C3⋊C8)C2×Dic311yescharacteristic
80C3×C3⋊C8—62no—
81#444—271noSylow
82C2×C33C4×D411yescharacteristic
83C3×C3⋊D3—21no—
84C3×C3⋊D3—42no—
85C3×#41—92no—
86#239—64no—
87C3×Q16—22no—
88C3×Q16—42no—
89C3×Q16#15711yescharacteristic, complemented
90C3×SD16—92no—
91C3×D8—91no—
92C2×C3×C8—91no—
93#215—64no—
94C4×Dic3—91no—
95#208—92no—
96#208—182no—
97C2×C3⋊C8—91no—
98C2×C3⋊C8—182no—
99D24—62no—
100D3×C8—62no—
101C2×C3⋊D3—22no—
102C2×C3⋊D3#23711yescharacteristic
103C2×C3×D3—62no—
104C2×C3×D3—124no—
105#157—121no—
106C32×C4—22no—
107C32×C4—22no—
108C32×C4—42no—
109C32×C4—44no—
110C32×C4—84no—
111C32×C4C22×Dic311yescharacteristic
112C3⋊Dic3D3×D411yescharacteristic
113C3×Dic3—62no—
114#135—91no—
115#131—272no—
116#108—271no—
117#104—272no—
118#97—271no—
119C33#44411yescharacteristic, complemented, Sylow
120C3×Q8—24no—
121C3×Q8—44no—
122C3×Q8C2×#15712yes—
123C3×D4—92no—
124C3×D4—182no—
125C2×C3×C4—91no—
126C2×C3×C4—182no—
127C2×Dic3—181no—
128D12—64no—
129D12—124no—
130C4×D3—62no—
131C4×D3—124no—
132C3×C8—22no—
133C3×C8—42no—
134C3×C8—91no—
135C3×C8C2×#15711yescharacteristic
136C3⋊C8—62no—
137C3⋊C8—92no—
138C3⋊C8—182no—
139C2×C32—22no—
140C2×C32—42no—
141C2×C32D4×Dic311yescharacteristic
142C3⋊D3—21no—
143C3⋊D3—42no—
144C3×D3—122no—
145C3×D3—244no—
146#41—92no—
147Q16C3⋊#15711yescharacteristic, complemented
148SD16—92no—
149D8—91no—
150M16—272no—
151M16—542no—
152C2×C8—91no—
153C2×C8—271no—
154C2×C8—542no—
155C42—271no—
156C22×C3—91no—
157C22×C3—182no—
158C2×D3—62no—
159C2×D3—124no—
160C3×C4—22no—
161C3×C4—22no—
162C3×C4—44no—
163C3×C4—42no—
164C3×C4—84no—
165C3×C4—91no—
166C3×C4C22×#15711yescharacteristic
167Dic3—62no—
168Dic3—361no—
169C32—22no—
170C32—42no—
171C32#519711yescharacteristic, complemented
172Q8C2×C3⋊#15712yes—
173D4—92no—
174D4—182no—
175C2×C4—91no—
176C2×C4—182no—
177C2×C4—541no—
178C8—91no—
179C8—272no—
180C8—542no—
181C8C2×C3⋊#15711yescharacteristic
182C2×C3—22no—
183C2×C3—42no—
184C2×C3—181no—
185C2×C3—362no—
186C2×C3D4×#15711yescharacteristic
187D3—122no—
188D3—244no—
189C22—91no—
190C22—182no—
191C4—22no—
192C4—91no—
193C4—1081no—
194C4C22×C3⋊#15711yescharacteristic
195C3—22no—
196C3—42no—
197C3#1059378811yescharacteristic, complemented
198C2—181no—
199C2—362no—
200C2D4×C3⋊#15711yescharacteristic
subgroupG/H
centerC2D4×C3⋊#157
commutatorC33×C4C22×C4
FrattiniC4C22×C3⋊#157
FittingC33×Q16C4
socleC2×C33C4×D4
prime radicalC3×#64128C2
cosocleC3×C4×C3⋊D3C23
hypercenterQ16C3⋊#157
hypocenterC33#444
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ Q16

Complements

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

NH ≅ G/Ntype
1. #11724517C2semidirect
3. #11723110C2semidirect
4. #11723095C2semidirect
5. #11723092C2semidirect
13. C33×Q16C4semidirect
14. C3×#3755C4semidirect
15. C3×#3745C4semidirect
43. C32×Q16Dic3semidirect
44. #3755Dic3semidirect
45. #3745Dic3semidirect
89. C3×Q16#157semidirect
119. C33#444semidirect
147. Q16C3⋊#157semidirect
171. C32#5197semidirect
197. C3#10593788semidirect

Maximal subgroups

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

subgroupindex
#117245172
C3×#641282
#117231102
#117230952
#117230922
#105937883
#51979