Finite Group Databasebeta

← #13156744

Subgroups — #13156744

Order 1728.

1074 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 1074 conjugacy classes of subgroups — too many to draw. Only its 222 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×#88813C211yesmaximal, characteristic
2C2×#88807C213yesmaximal
3C2×C3×#3724C213yesmaximal
4#10591244—33nomaximal
5C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
6C22×C3×C3⋊Dic3C2213yes—
7C22×C32×Dic3C2213yes—
8#88813C414yescomplemented
9#88807—26no—
10C3×#3724—26no—
11C2×#3750—33no—
12C2×C3×Dic3⋊C4—36no—
13C2×#3724D313yes—
14C2×Dic3⋊Dic3—36no—
15C2×Dic3⋊Dic3—63no—
16C2×Dic32—33no—
17C23×C33C2311yescharacteristic
18C2×C3⋊(C3⋊Dic3)—23no—
19C2×C3×C3⋊Dic3C2×C416yes—
20C2×C3×C3⋊Dic3D416yes—
21C2×C3×C3⋊Dic3Q816yes—
22C2×C32×Dic3—23no—
23C2×C32×Dic3—26no—
24#4975—93no—
25C22×C32×C4—33no—
26C22×C3⋊Dic3—33no—
27C22×C3⋊Dic3—63no—
28C22×C3⋊Dic3—121no—
29C22×C3⋊Dic3C2×D313yes—
30C22×C3×Dic3—23no—
31C22×C3×Dic3—36no—
32C22×C3×Dic3—63no—
33C22×C3×Dic3C2×D316yes—
34#3750—66no—
35C3×Dic3⋊C4—612no—
36#3724—26no—
37Dic3⋊Dic3—612no—
38Dic3⋊Dic3—612no—
39Dic32—312no—
40C22×C33C22×C411yescharacteristic
41C22×C33C2×D413yes—
42C22×C33C2×Q813yes—
43C3⋊(C3⋊Dic3)—41no—
44C3×C3⋊Dic3C4⋊C4112yescomplemented
45C32×Dic3—43no—
46C2×C3×C4⋊C4—93no—
47C2×C4⋊Dic3—93no—
48C2×Dic3⋊C4—36no—
49C2×Dic3⋊C4—96no—
50C2×Dic3⋊C4—183no—
51C2×C4×Dic3—93no—
52C23×C32—23no—
53C23×C32—41no—
54C23×C32C22×D313yes—
55C2×C32×C4—63no—
56C2×C32×C4—66no—
57C2×C3⋊Dic3—63no—
58C2×C3⋊Dic3—66no—
59C2×C3⋊Dic3—66no—
60C2×C3⋊Dic3—126no—
61C2×C3⋊Dic3—126no—
62C2×C3⋊Dic3Dic616yes—
63C2×C3⋊Dic3C4×D316yes—
64C2×C3⋊Dic3#6716yes—
65C2×C3×Dic3—26no—
66C2×C3×Dic3—26no—
67C2×C3×Dic3—26no—
68C2×C3×Dic3—26no—
69C2×C3×Dic3—312no—
70C2×C3×Dic3—312no—
71C2×C3×Dic3—312no—
72C2×C3×Dic3—46no—
73C2×C3×Dic3—66no—
74C2×C3×Dic3—66no—
75C2×C3×Dic3—66no—
76#384—271noSylow
77C2×C33C2×C4⋊C413yes—
78C2×C33#12711yescharacteristic
79C2×C33C4⋊Q813yes—
80C22×C3×C4—36no—
81C22×C3×C4—63no—
82C22×C3×C4—93no—
83C22×Dic3—36no—
84C22×Dic3—63no—
85C22×Dic3—93no—
86C22×Dic3—183no—
87C22×Dic3—361no—
88C22×Dic3D3213yes—
89C3×C4⋊C4—186no—
90C4⋊Dic3—186no—
91Dic3⋊C4—612no—
92Dic3⋊C4—1812no—
93Dic3⋊C4—1812no—
94C4×Dic3—912no—
95C22×C32—26no—
96C22×C32—23no—
97C22×C32—23no—
98C22×C32—26no—
99C22×C32—23no—
100C22×C32—41no—
101C22×C32—43no—
102C22×C32—43no—
103C22×C32C2×Dic613yes—
104C22×C32C2×C4×D313yes—
105C22×C32D3×D416yes—
106C22×C32D3×Q816yes—
107C22×C32C2×#6713yes—
108C32×C4—123no—
109C3⋊Dic3—126no—
110C3⋊Dic3—124no—
111C3⋊Dic3—123no—
112C3⋊Dic3Dic3⋊C4112yescomplemented
113C3×Dic3—324no—
114C3×Dic3—46no—
115C3×Dic3—46no—
116C3×Dic3—612no—
117C2×C4⋊C4—93no—
118C2×C4⋊C4—273no—
119C2×C42—271no—
120C33#38411yescharacteristic, complemented, Sylow
121C23×C3—23no—
122C23×C3—41no—
123C23×C3C2×D3213yes—
124C2×C3×C4—66no—
125C2×C3×C4—66no—
126C2×C3×C4—66no—
127C2×C3×C4—66no—
128C2×C3×C4—96no—
129C2×C3×C4—96no—
130C2×C3×C4—96no—
131C2×C3×C4—126no—
132C2×Dic3—26no—
133C2×Dic3—23no—
134C2×Dic3—312no—
135C2×Dic3—312no—
136C2×Dic3—312no—
137C2×Dic3—66no—
138C2×Dic3—66no—
139C2×Dic3—66no—
140C2×Dic3—186no—
141C2×Dic3—186no—
142C2×Dic3—183no—
143C2×Dic3—366no—
144C2×Dic3—366no—
145C2×C32—26no—
146C2×C32—23no—
147C2×C32—26no—
148C2×C32—23no—
149C2×C32—23no—
150C2×C32—43no—
151C2×C32—41no—
152C2×C32—43no—
153C2×C32#88816yes—
154C2×C32D3×C4⋊C416yes—
155C2×C32C2×Dic3⋊C413yes—
156C2×C32#94013yes—
157C2×C32#94413yes—
158C22×C4—33no—
159C22×C4—93no—
160C22×C4—271no—
161C4⋊C4—186no—
162C4⋊C4—546no—
163C42—274no—
164C22×C3—23no—
165C22×C3—23no—
166C22×C3—26no—
167C22×C3—23no—
168C22×C3—26no—
169C22×C3—41no—
170C22×C3—43no—
171C22×C3—43no—
172C22×C3D3×Dic616yes—
173C22×C3#379713yes—
174C22×C3C4×D3213yes—
175C22×C3D3×#6716yes—
176C22×C3#381113yes—
177C3×C4—912no—
178C3×C4—126no—
179C3×C4—126no—
180Dic3—324no—
181Dic3—43no—
182Dic3—612no—
183Dic3—366no—
184Dic3—364no—
185Dic3—363no—
186C32—23no—
187C32—41no—
188C32#497513yescomplemented
189C23D3311yescharacteristic
190C2×C4—66no—
191C2×C4—63no—
192C2×C4—96no—
193C2×C4—96no—
194C2×C4—96no—
195C2×C4—543no—
196C2×C3—23no—
197C2×C3—26no—
198C2×C3—23no—
199C2×C3—23no—
200C2×C3—26no—
201C2×C3—41no—
202C2×C3—43no—
203C2×C3—43no—
204C2×C3#6383916yes—
205C2×C3#6384013yes—
206C2×C3D3×Dic3⋊C416yes—
207C2×C3#6388313yes—
208C2×C3#6397913yes—
209C22#8894911yescharacteristic
210C22#8895213yes—
211C22#8895713yes—
212C4—912no—
213C4—123no—
214C4—1081no—
215C3—23no—
216C3—41no—
217C3#1059124413yescomplemented
218C2#1172291013yes—
219C2#1172303111yescharacteristic
220C2#1172303413yes—
subgroupG/H
center, Frattini, hypercenterC23D33
commutatorC22×C33C22×C4
Fitting, socle, prime radical, cosocleC23×C33C23
hypocenterC33#384
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
8. #88813C4semidirect
44. C3×C3⋊Dic3C4⋊C4semidirect
112. C3⋊Dic3Dic3⋊C4semidirect
120. C33#384semidirect
188. C32#4975semidirect
217. C3#10591244semidirect

Maximal subgroups

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

subgroupindex
C2×#888132
C2×#888072
C2×C3×#37242
#105912443