Finite Group Databasebeta

← #13141347

Subgroups — #13141347

Order 1728.

727 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 727 conjugacy classes of subgroups — too many to draw. Only its 138 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×#3752⋊C3C213yesmaximal
2C2×#3750⋊C3C213yesmaximal, complemented
3H(3,𝔽3)×#95C211yesmaximal, characteristic, complemented
4#10592375C311yesmaximal, characteristic, complemented
5C3×#4676—31nomaximal
6C23×(C3⋊D3)⋊C3C2211yescharacteristic
7C22×C4×H(3,𝔽3)C2213yes—
8C22×(C3⋊Dic3)⋊C3C2213yes—
9#3752⋊C3—26no—
10#3750⋊C3—26no—
11C2×#3752C2×C313yes—
12C2×#3750C2×C313yescomplemented
13C2×C3×(C2×D3)⋊C4—33no—
14C2×C3×Dic3⋊C4—33no—
15C32×#95—21no—
16C32×#95D311yescharacteristic, complemented
17C32×#95C2×C311yescharacteristic, complemented
18C23×H(3,𝔽3)C2311yescharacteristic
19C22×(C3⋊D3)⋊C3—23no—
20C22×(C3⋊D3)⋊C3—23no—
21C22×(C3⋊D3)⋊C3Q812yes—
22C2×C4×H(3,𝔽3)—23no—
23C2×C4×H(3,𝔽3)—23no—
24C2×C4×H(3,𝔽3)—23no—
25C2×(C3⋊Dic3)⋊C3—23no—
26C2×(C3⋊Dic3)⋊C3—23no—
27C2×(C3⋊Dic3)⋊C3D416yes—
28C3×#396—91no—
29#4676—31no—
30#4676—91no—
31C23×C3⋊D3C22×C311yescharacteristic
32C23×C3×D3—31no—
33C22×C32×C4—23no—
34C22×C32×C4C2×D313yes—
35C22×C32×C4C22×C313yes—
36C22×C3⋊Dic3C22×C313yes—
37C22×C3×Dic3—33no—
38#3752—26no—
39#3750—26no—
40C3×(C2×D3)⋊C4—66no—
41C3×Dic3⋊C4—66no—
42C22×H(3,𝔽3)C2×D413yes—
43C22×H(3,𝔽3)C2×Q811yescharacteristic
44C22×H(3,𝔽3)#4113yes—
45C2×(C3⋊D3)⋊C3—26no—
46C2×(C3⋊D3)⋊C3—43no—
47C2×(C3⋊D3)⋊C3—41no—
48C4×H(3,𝔽3)—43no—
49(C3⋊Dic3)⋊C3—26no—
50C2×C3×C4⋊C4—93no—
51C2×C3×C22⋊C4—93no—
52C2×(C2×D3)⋊C4—33no—
53C2×(C2×D3)⋊C4—93no—
54C2×Dic3⋊C4—33no—
55C2×Dic3⋊C4—93no—
56C3×#95—31no—
57C3×#95—31no—
58C3×#95—61no—
59C3×#95C3×D311yescharacteristic
60C23×C32—21no—
61C23×C32C22×D311yescharacteristic
62C23×C32C23×C311yescharacteristic
63C22×C3⋊D3—23no—
64C22×C3⋊D3—23no—
65C22×C3⋊D3C3×Q812yes—
66C22×C3×D3—32no—
67C22×C3×D3—63no—
68C22×C3×D3—63no—
69C2×C32×C4—26no—
70C2×C32×C4—23no—
71C2×C32×C4—23no—
72C2×C32×C4—23no—
73C2×C32×C4—23no—
74C2×C32×C4—23no—
75C2×C32×C4—23no—
76C2×C32×C4—43no—
77C2×C32×C4—43no—
78C2×C3⋊Dic3—23no—
79C2×C3⋊Dic3—23no—
80C2×C3⋊Dic3C3×D416yes—
81C2×C3×Dic3—36no—
82C2×C3×Dic3—63no—
83C2×C3×Dic3—63no—
84#396—91noSylow
85C2×H(3,𝔽3)#12113yes—
86C2×H(3,𝔽3)C22⋊Q813yes—
87C2×H(3,𝔽3)#12611yescharacteristic
88(C3⋊D3)⋊C3—42no—
89C24×C3—91no—
90C23×D3—31no—
91C23×D3—91no—
92C22×C3×C4—33no—
93C22×C3×C4—33no—
94C22×C3×C4—63no—
95C22×C3×C4—93no—
96C22×C3×C4C2×C3×D313yes—
97C22×Dic3—33no—
98C22×Dic3—93no—
99C3×C4⋊C4—186no—
100C3×C22⋊C4—186no—
101(C2×D3)⋊C4—66no—
102(C2×D3)⋊C4—186no—
103Dic3⋊C4—66no—
104Dic3⋊C4—186no—
105C22×C32—23no—
106C22×C32—23no—
107C22×C32—21no—
108C22×C32#23513yes—
109C22×C32D3×D413yes—
110C22×C32D3×Q811yescharacteristic
111C22×C32C2×C3×D413yes—
112C22×C32C2×C3×Q811yescharacteristic
113C22×C32C3×#4113yes—
114C2×C3⋊D3—26no—
115C2×C3⋊D3—43no—
116C2×C3⋊D3—41no—
117C2×C3×D3—66no—
118C2×C3×D3—121no—
119C2×C3×D3—123no—
120C32×C4—46no—
121C32×C4—43no—
122C32×C4—43no—
123C3⋊Dic3—26no—
124C3×Dic3—66no—
125C2×C4⋊C4—93no—
126C2×C22⋊C4—93no—
127#95(C3⋊D3)⋊C311yescharacteristic, complemented
128H(3,𝔽3)#39611yescharacteristic, complemented, Sylow
129C23×C3—31no—
130C23×C3—31no—
131C23×C3—61no—
132C23×C3—92no—
133C23×C3—183no—
134C23×C3—183no—
135C23×C3C22×C3×D311yescharacteristic
136C22×D3—32no—
137C22×D3—63no—
138C22×D3—63no—
139C22×D3—92no—
140C22×D3—183no—
141C22×D3—183no—
142C2×C3×C4—23no—
143C2×C3×C4—23no—
144C2×C3×C4—23no—
145C2×C3×C4—63no—
146C2×C3×C4—63no—
147C2×C3×C4—63no—
148C2×C3×C4—63no—
149C2×C3×C4—63no—
150C2×C3×C4—63no—
151C2×C3×C4—66no—
152C2×C3×C4—96no—
153C2×C3×C4—123no—
154C2×C3×C4—123no—
155C2×C3×C4—183no—
156C2×C3×C4—183no—
157C2×Dic3—36no—
158C2×Dic3—63no—
159C2×Dic3—63no—
160C2×Dic3—96no—
161C2×Dic3—183no—
162C2×Dic3—183no—
163C2×C32—23no—
164C2×C32—23no—
165C2×C32—21no—
166C2×C32#87611yescharacteristic
167C2×C32#88413yes—
168C2×C32#89613yes—
169C2×C32C3×#12113yes—
170C2×C32C3×C22⋊Q813yes—
171C2×C32C3×#12611yescharacteristic
172C3⋊D3—42no—
173C3×D3—122no—
174C24—91no—
175C22×C4—93no—
176C22×C4C2×(C3⋊D3)⋊C313yes—
177C4⋊C4—186no—
178C22⋊C4—186no—
179C22×C3—33no—
180C22×C3—33no—
181C22×C3—33no—
182C22×C3—33no—
183C22×C3—31no—
184C22×C3—31no—
185C22×C3—63no—
186C22×C3—63no—
187C22×C3—61no—
188C22×C3—186no—
189C22×C3—361no—
190C22×C3—363no—
191C22×C3C3×#23513yes—
192C22×C3C3×D3×D413yes—
193C22×C3C3×D3×Q811yescharacteristic
194C2×D3—66no—
195C2×D3—123no—
196C2×D3—121no—
197C2×D3—186no—
198C2×D3—361no—
199C2×D3—363no—
200C3×C4—43no—
201C3×C4—123no—
202C3×C4—123no—
203C3×C4—126no—
204C3×C4—186no—
205Dic3—66no—
206Dic3—186no—
207C32—21no—
208C32#467611yescharacteristic, complemented
209C32C3×#39611yescharacteristic, complemented
210C23—92no—
211C23—183no—
212C23—183no—
213C23C22×(C3⋊D3)⋊C311yescharacteristic
214C2×C4—23no—
215C2×C4—23no—
216C2×C4—23no—
217C2×C4—96no—
218C2×C4—183no—
219C2×C4—183no—
220C2×C3—31no—
221C2×C3—31no—
222C2×C3—33no—
223C2×C3—33no—
224C2×C3—33no—
225C2×C3—33no—
226C2×C3—61no—
227C2×C3—63no—
228C2×C3—63no—
229C2×C3—362no—
230C2×C3C3×#87611yescharacteristic
231C2×C3C3×#88413yes—
232C2×C3C3×#89613yes—
233D3—122no—
234D3—362no—
235C22—186no—
236C22—361no—
237C22—363no—
238C22#3828⋊C313yes—
239C22D4×(C3⋊D3)⋊C313yes—
240C22Q8×(C3⋊D3)⋊C311yescharacteristic
241C4—43no—
242C4—186no—
243C3—31no—
244C3—31no—
245C3—61no—
246C3C3×#467611yescharacteristic
247C2—362no—
248C2#1172138711yescharacteristic
249C2#64093⋊C313yes—
250C2#64105⋊C313yes—
subgroupG/H
centerC23C22×(C3⋊D3)⋊C3
commutator, cosocleC23×C32C23×C3
Frattini, socleC23×C3C22×C3×D3
FittingH(3,𝔽3)×#95C2
prime radicalC23×(C3⋊D3)⋊C3C22
hypercenter#95(C3⋊D3)⋊C3
hypocenterC32C3×#396
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
2. C2×#3750⋊C3C2semidirect
3. H(3,𝔽3)×#95C2semidirect
4. #10592375C3semidirect
12. C2×#3750C2×C3semidirect
16. C32×#95D3semidirect
17. C32×#95C2×C3semidirect
127. #95(C3⋊D3)⋊C3semidirect
128. H(3,𝔽3)#396semidirect
208. C32#4676semidirect
209. C32C3×#396semidirect

Maximal subgroups

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

subgroupindex
C2×#3752⋊C32
C2×#3750⋊C32
H(3,𝔽3)×#952
#105923753
C3×#46763