Finite Group Databasebeta

← #13143603

Subgroups — #13143603

Order 1728.

863 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 863 conjugacy classes of subgroups — too many to draw. Only its 156 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
1C22×Q8×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#3752⋊C3C213yesmaximal
3#63658⋊C3C213yesmaximal, complemented
4#10592930C311yesmaximal, characteristic, complemented
5C3×#5231—31nomaximal
6C23×(C3⋊D3)⋊C3C2211yescharacteristic
7C2×Q8×H(3,𝔽3)—26no—
8C22×C4×H(3,𝔽3)C2213yes—
9C22×(C3⋊Dic3)⋊C3C2213yes—
10#3752⋊C3—26no—
11C22×C32×Q8—21no—
12C22×C32×Q8D311yescharacteristic, complemented
13C22×C32×Q8C2×C311yescharacteristic, complemented
14C2×#3752C2×C313yes—
15C2×C3×(C2×D3)⋊C4—33no—
16#63658C2×C313yescomplemented
17C3×C3⋊#95—33no—
18C23×H(3,𝔽3)C2311yescharacteristic
19C22×(C3⋊D3)⋊C3—23no—
20C22×(C3⋊D3)⋊C3—23no—
21C22×(C3⋊D3)⋊C3Q812yescomplemented
22Q8×H(3,𝔽3)—28no—
23C2×C4×H(3,𝔽3)—26no—
24C2×C4×H(3,𝔽3)D416yes—
25C2×(C3⋊Dic3)⋊C3—23no—
26C2×(C3⋊Dic3)⋊C3—26no—
27C3×#393—91no—
28#5231—31no—
29#5231—91no—
30C23×C3⋊D3C22×C311yescharacteristic
31C23×C3×D3—31no—
32C2×C32×Q8—212no—
33C2×C32×Q8—26no—
34C2×C32×Q8—26no—
35C22×C32×C4—23no—
36C22×C32×C4C2×D313yes—
37C22×C32×C4C22×C313yes—
38C22×C3⋊Dic3C22×C313yes—
39C22×C3×Dic3—33no—
40#3752—26no—
41C3×(C2×D3)⋊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—41no—
47C2×(C3⋊D3)⋊C3—43no—
48C4×H(3,𝔽3)—26no—
49(C3⋊Dic3)⋊C3—43no—
50C22×C3×Q8—31no—
51C22×C3×Q8—31no—
52C22×C3×Q8—61no—
53C22×C3×Q8C3×D311yescharacteristic
54C2×C3×C22⋊C4—93no—
55C2×(C2×D3)⋊C4—33no—
56C2×(C2×D3)⋊C4—93no—
57C3×#95—93no—
58C3⋊#95—33no—
59C3⋊#95—93no—
60C23×C32—21no—
61C23×C32C22×D311yescharacteristic
62C23×C32C23×C311yescharacteristic
63C22×C3⋊D3—23no—
64C22×C3⋊D3—23no—
65C22×C3⋊D3C3×Q812yescomplemented
66C22×C3×D3—32no—
67C22×C3×D3—63no—
68C22×C3×D3—63no—
69C32×Q8—216no—
70C32×Q8—28no—
71C32×Q8—28no—
72C2×C32×C4—212no—
73C2×C32×C4—26no—
74C2×C32×C4—26no—
75C2×C32×C4—26no—
76C2×C32×C4#6716yes—
77C2×C32×C4C3×D416yes—
78C2×C3⋊Dic3—23no—
79C2×C3⋊Dic3—26no—
80C2×C3×Dic3—66no—
81C2×C3×Dic3—63no—
82#393—91noSylow
83C2×H(3,𝔽3)#12011yescharacteristic
84C2×H(3,𝔽3)C22⋊Q813yes—
85C2×H(3,𝔽3)#12413yes—
86(C3⋊D3)⋊C3—42no—
87C24×C3—91no—
88C23×D3—31no—
89C23×D3—91no—
90C2×C3×Q8—26no—
91C2×C3×Q8—66no—
92C2×C3×Q8—66no—
93C2×C3×Q8—612no—
94C22×C3×C4—33no—
95C22×C3×C4—33no—
96C22×C3×C4—63no—
97C22×C3×C4—93no—
98C22×C3×C4C2×C3×D313yes—
99C22×Dic3—33no—
100C22×Dic3—93no—
101C3×C22⋊C4—186no—
102(C2×D3)⋊C4—66no—
103(C2×D3)⋊C4—186no—
104C22×C32—21no—
105C22×C32—23no—
106C22×C32—23no—
107C22×C32D3×Q811yescharacteristic
108C22×C32#23913yes—
109C22×C32C2×#6713yes—
110C22×C32C2×C3×D413yes—
111C22×C32C2×C3×Q811yescharacteristic
112C22×C32C3×#4113yes—
113C2×C3⋊D3—26no—
114C2×C3⋊D3—41no—
115C2×C3⋊D3—43no—
116C2×C3×D3—66no—
117C2×C3×D3—123no—
118C2×C3×D3—121no—
119C32×C4—212no—
120C32×C4—26no—
121C32×C4—26no—
122C3⋊Dic3—43no—
123C3×Dic3—123no—
124C22×Q8(C3⋊D3)⋊C311yescharacteristic, complemented
125C2×C22⋊C4—93no—
126#95—93no—
127H(3,𝔽3)#39311yescharacteristic, complemented, Sylow
128C23×C3—31no—
129C23×C3—31no—
130C23×C3—61no—
131C23×C3—92no—
132C23×C3—183no—
133C23×C3—183no—
134C23×C3C22×C3×D311yescharacteristic
135C22×D3—32no—
136C22×D3—63no—
137C22×D3—63no—
138C22×D3—92no—
139C22×D3—183no—
140C22×D3—183no—
141C3×Q8—28no—
142C3×Q8—68no—
143C3×Q8—68no—
144C3×Q8—616no—
145C2×C3×C4—26no—
146C2×C3×C4—36no—
147C2×C3×C4—36no—
148C2×C3×C4—66no—
149C2×C3×C4—66no—
150C2×C3×C4—612no—
151C2×C3×C4—66no—
152C2×C3×C4—183no—
153C2×C3×C4—186no—
154C2×C3×C4C3×#6716yes—
155C2×Dic3—66no—
156C2×Dic3—63no—
157C2×Dic3—183no—
158C2×Dic3—186no—
159C2×C32—23no—
160C2×C32—23no—
161C2×C32—21no—
162C2×C32#94613yes—
163C2×C32#94713yes—
164C2×C32#95311yescharacteristic
165C2×C32C3×#12011yescharacteristic
166C2×C32C3×C22⋊Q813yes—
167C2×C32C3×#12413yes—
168C3⋊D3—42no—
169C3×D3—122no—
170C24—91no—
171C2×Q8—26no—
172C22×C4—93no—
173C22×C4C2×(C3⋊D3)⋊C313yes—
174C22⋊C4—186no—
175C22×C3—31no—
176C22×C3—31no—
177C22×C3—33no—
178C22×C3—33no—
179C22×C3—33no—
180C22×C3—33no—
181C22×C3—61no—
182C22×C3—63no—
183C22×C3—63no—
184C22×C3—186no—
185C22×C3—363no—
186C22×C3—361no—
187C22×C3C3×D3×Q811yescharacteristic
188C22×C3C3×#23913yes—
189C22×C3C2×C3×#6713yes—
190C2×D3—66no—
191C2×D3—123no—
192C2×D3—121no—
193C2×D3—186no—
194C2×D3—363no—
195C2×D3—361no—
196C3×C4—26no—
197C3×C4—66no—
198C3×C4—66no—
199C3×C4—612no—
200C3×C4—363no—
201Dic3—123no—
202Dic3—363no—
203C32—21no—
204C32#523111yescharacteristic, complemented
205C32C3×#39311yescharacteristic, complemented
206C23—92no—
207C23—183no—
208C23—183no—
209C23C22×(C3⋊D3)⋊C311yescharacteristic
210Q8—28no—
211C2×C4—26no—
212C2×C4—186no—
213C2×C4—183no—
214C2×C4#638⋊C316yes—
215C2×C3—33no—
216C2×C3—33no—
217C2×C3—33no—
218C2×C3—33no—
219C2×C3—31no—
220C2×C3—31no—
221C2×C3—63no—
222C2×C3—63no—
223C2×C3—61no—
224C2×C3—362no—
225C2×C3C3×#94613yes—
226C2×C3C3×#94713yes—
227C2×C3C3×#95311yescharacteristic
228D3—122no—
229D3—362no—
230C22—186no—
231C22—363no—
232C22—361no—
233C22Q8×(C3⋊D3)⋊C311yescharacteristic
234C22#8872213yes—
235C22C2×#638⋊C313yes—
236C4—26no—
237C4—363no—
238C3—31no—
239C3—31no—
240C3—61no—
241C3C3×#523111yescharacteristic
242C2—362no—
243C2#64155⋊C313yes—
244C2#64156⋊C313yes—
245C2#1172170211yescharacteristic
subgroupG/H
centerC23C22×(C3⋊D3)⋊C3
commutator, cosocleC23×C32C23×C3
Frattini, socleC23×C3C22×C3×D3
FittingC22×Q8×H(3,𝔽3)C2
prime radicalC23×(C3⋊D3)⋊C3C22
hypercenterC22×Q8(C3⋊D3)⋊C3
hypocenterC32C3×#393
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
1. C22×Q8×H(3,𝔽3)C2semidirect
3. #63658⋊C3C2semidirect
4. #10592930C3semidirect
12. C22×C32×Q8D3semidirect
13. C22×C32×Q8C2×C3semidirect
16. #63658C2×C3semidirect
21. C22×(C3⋊D3)⋊C3Q8semidirect
65. C22×C3⋊D3C3×Q8semidirect
124. C22×Q8(C3⋊D3)⋊C3semidirect
127. H(3,𝔽3)#393semidirect
204. C32#5231semidirect
205. C32C3×#393semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×H(3,𝔽3)2
C2×#3752⋊C32
#63658⋊C32
#105929303
C3×#52313