Finite Group Databasebeta

← #13173384

Subgroups — #13173384

Order 1728.

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

normal subgroups only

This group has 372 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#11723096C211yesmaximal, characteristic, complemented
5C8×C3⋊#157C211yesmaximal, characteristic, complemented
6#10593787—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#63771—31no—
22C8×#157—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⋊#622D411yescharacteristic
31C3⋊#622D411yescharacteristic
32C33×C8C2×C411yescharacteristic
33C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
34#5196—91nomaximal
35#3832C2×D312yes—
36C3×#239—64no—
37C4×#157—31no—
38#3788—32no—
39#3788—62no—
40#3787—31no—
41#3787—62no—
42C32×Q16—22no—
43C32×Q16—42no—
44C32×Q16Dic311yescharacteristic, complemented
45#3755Dic312yescomplemented
46#3745Dic311yescharacteristic, complemented
47C8×C3⋊D3C2×D311yescharacteristic
48C3×#215—64no—
49C3×D24—62no—
50C3×D3×C8—62no—
51C2×C3×C3⋊D3—22no—
52C2×C3×C3⋊D3#4111yescharacteristic
53C3⋊#157—22no—
54C33×C4—22no—
55C33×C4C22×C411yescharacteristic
56C3×C3⋊Dic3C2×D411yescharacteristic
57C3×#135—91no—
58#948—92no—
59#919—62no—
60#837—92no—
61#819—91no—
62C8×Dic3—91no—
63C2×#157—61no—
64C32×Q8—24no—
65C32×Q8—44no—
66C32×Q8C2×Dic312yes—
67C4⋊(C3⋊D3)—22no—
68C4⋊(C3⋊D3)C2×Dic312yes—
69C4×C3⋊D3—22no—
70C4×C3⋊D3C22×D311yescharacteristic
71C3×D12—64no—
72C3×D12—124no—
73C3×C4×D3—62no—
74C3×C4×D3—124no—
75#622—31no—
76#622—31no—
77#622—62no—
78C32×C8—22no—
79C32×C8—42no—
80C32×C8C2×Dic311yescharacteristic
81C3⋊(C3⋊C8)C2×Dic311yescharacteristic
82C3×C3⋊C8—62no—
83#443—271noSylow
84C2×C33C4×D411yescharacteristic
85C3×C3⋊D3—21no—
86C3×C3⋊D3—42no—
87C3×#41—92no—
88#239—64no—
89C3×Q16—22no—
90C3×Q16—42no—
91C3×Q16#15711yescharacteristic, complemented
92C3×SD16—92no—
93C3×D8—91no—
94C2×C3×C8—91no—
95#215—64no—
96C4×Dic3—91no—
97#208—92no—
98#208—182no—
99C2×C3⋊C8—91no—
100C2×C3⋊C8—182no—
101D24—62no—
102D3×C8—62no—
103C2×C3⋊D3—22no—
104C2×C3⋊D3#23711yescharacteristic
105C2×C3×D3—62no—
106C2×C3×D3—124no—
107#157—62no—
108C32×C4—22no—
109C32×C4—22no—
110C32×C4—42no—
111C32×C4—44no—
112C32×C4—84no—
113C32×C4C22×Dic311yescharacteristic
114C3⋊Dic3D3×D411yescharacteristic
115C3×Dic3—62no—
116#135—91no—
117#131—272no—
118#108—271no—
119#104—272no—
120C4×C8—271no—
121C33#44311yescharacteristic, complemented, Sylow
122C3×Q8—24no—
123C3×Q8—44no—
124C3×Q8C2×#15712yes—
125C3×D4—92no—
126C3×D4—182no—
127C2×C3×C4—91no—
128C2×C3×C4—182no—
129C2×Dic3—181no—
130D12—64no—
131D12—124no—
132C4×D3—62no—
133C4×D3—124no—
134C3×C8—22no—
135C3×C8—42no—
136C3×C8—91no—
137C3×C8C2×#15711yescharacteristic
138C3⋊C8—62no—
139C3⋊C8—91no—
140C3⋊C8—91no—
141C3⋊C8—182no—
142C2×C32—22no—
143C2×C32—42no—
144C2×C32D4×Dic311yescharacteristic
145C3⋊D3—21no—
146C3⋊D3—42no—
147C3×D3—122no—
148C3×D3—244no—
149#41—92no—
150Q16C3⋊#15711yescharacteristic, complemented
151SD16—92no—
152D8—91no—
153M16—272no—
154M16—542no—
155C2×C8—91no—
156C2×C8—271no—
157C2×C8—542no—
158C42—271no—
159C22×C3—91no—
160C22×C3—182no—
161C2×D3—62no—
162C2×D3—124no—
163C3×C4—22no—
164C3×C4—22no—
165C3×C4—44no—
166C3×C4—42no—
167C3×C4—84no—
168C3×C4—91no—
169C3×C4C22×#15711yescharacteristic
170Dic3—62no—
171Dic3—182no—
172C32—22no—
173C32—42no—
174C32#519611yescharacteristic, complemented
175Q8C2×C3⋊#15712yes—
176D4—92no—
177D4—182no—
178C2×C4—91no—
179C2×C4—182no—
180C2×C4—541no—
181C8—91no—
182C8—271no—
183C8—271no—
184C8—542no—
185C8C2×C3⋊#15711yescharacteristic
186C2×C3—22no—
187C2×C3—42no—
188C2×C3—181no—
189C2×C3—362no—
190C2×C3D4×#15711yescharacteristic
191D3—122no—
192D3—244no—
193C22—91no—
194C22—182no—
195C4—22no—
196C4—91no—
197C4—542no—
198C4C22×C3⋊#15711yescharacteristic
199C3—22no—
200C3—42no—
201C3#1059378711yescharacteristic, complemented
202C2—181no—
203C2—362no—
204C2D4×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#443
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. #11723096C2semidirect
5. C8×C3⋊#157C2semidirect
13. C33×Q16C4semidirect
14. C3×#3755C4semidirect
15. C3×#3745C4semidirect
44. C32×Q16Dic3semidirect
45. #3755Dic3semidirect
46. #3745Dic3semidirect
91. C3×Q16#157semidirect
121. C33#443semidirect
150. Q16C3⋊#157semidirect
174. C32#5196semidirect
201. C3#10593787semidirect

Maximal subgroups

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

subgroupindex
#117245172
C3×#641282
#117231102
#117230962
C8×C3⋊#1572
#105937873
#51969