Finite Group Databasebeta

← #13143602

Subgroups — #13143602

Order 1728.

793 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 793 conjugacy classes of subgroups — too many to draw. Only its 164 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
2C2×#3750⋊C3C213yesmaximal
3#63658⋊C3C213yesmaximal
4#10592929C311yesmaximal, characteristic, complemented
5C3×#5230—31nomaximal
6C2×Q8×H(3,𝔽3)—26no—
7C22×C4×H(3,𝔽3)C2213yes—
8C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
9C22×(C3⋊Dic3)⋊C3C2213yes—
10#3750⋊C3—26no—
11C22×C32×Q8—21no—
12C22×C32×Q8D311yescharacteristic
13C22×C32×Q8C2×C311yescharacteristic
14C2×#3750C2×C313yes—
15C2×C3×Dic3⋊C4—33no—
16#63658C2×C313yes—
17C3×C3⋊#95—33no—
18C23×H(3,𝔽3)C2311yescharacteristic
19Q8×H(3,𝔽3)—28no—
20C2×C4×H(3,𝔽3)—26no—
21C2×C4×H(3,𝔽3)D416yes—
22C2×(C3⋊Dic3)⋊C3—23no—
23C2×(C3⋊Dic3)⋊C3—23no—
24C2×(C3⋊Dic3)⋊C3—23no—
25C2×(C3⋊Dic3)⋊C3Q816yescomplemented
26C3×#395—91no—
27#5230—31no—
28#5230—91no—
29C2×C32×Q8—212no—
30C2×C32×Q8—26no—
31C2×C32×Q8—26no—
32C22×C32×C4—23no—
33C22×C32×C4C2×D313yes—
34C22×C32×C4C22×C313yes—
35C22×C3⋊Dic3C22×C311yescharacteristic
36C22×C3⋊Dic3C22×C313yes—
37C22×C3×Dic3—33no—
38C22×C3×Dic3—31no—
39#3750—26no—
40C3×Dic3⋊C4—66no—
41C22×H(3,𝔽3)C2×D413yes—
42C22×H(3,𝔽3)C2×Q813yes—
43C22×H(3,𝔽3)#4111yescharacteristic
44C4×H(3,𝔽3)—26no—
45(C3⋊Dic3)⋊C3—26no—
46(C3⋊Dic3)⋊C3—41no—
47C22×C3×Q8—31no—
48C22×C3×Q8—31no—
49C22×C3×Q8—61no—
50C22×C3×Q8C3×D311yescharacteristic
51C2×C3×C4⋊C4—93no—
52C2×Dic3⋊C4—33no—
53C2×Dic3⋊C4—93no—
54C3×#95—93no—
55C3⋊#95—33no—
56C3⋊#95—93no—
57C23×C32—21no—
58C23×C32C22×D311yescharacteristic
59C23×C32C23×C311yescharacteristic
60C32×Q8—216no—
61C32×Q8—28no—
62C32×Q8—28no—
63C2×C32×C4—212no—
64C2×C32×C4—26no—
65C2×C32×C4—26no—
66C2×C32×C4—26no—
67C2×C32×C4#6716yes—
68C2×C32×C4C3×D416yes—
69C2×C3⋊Dic3—23no—
70C2×C3⋊Dic3—23no—
71C2×C3⋊Dic3—23no—
72C2×C3⋊Dic3C3×Q816yescomplemented
73C2×C3×Dic3—36no—
74C2×C3×Dic3—63no—
75C2×C3×Dic3—63no—
76C2×C3×Dic3—63no—
77#395—91noSylow
78C2×H(3,𝔽3)#12011yescharacteristic
79C2×H(3,𝔽3)C22⋊Q813yes—
80C2×H(3,𝔽3)C4⋊Q813yes—
81C2×C3×Q8—26no—
82C2×C3×Q8—66no—
83C2×C3×Q8—66no—
84C2×C3×Q8—612no—
85C22×C3×C4—33no—
86C22×C3×C4—33no—
87C22×C3×C4—63no—
88C22×C3×C4—93no—
89C22×C3×C4—91no—
90C22×C3×C4C2×C3×D313yes—
91C22×Dic3—33no—
92C22×Dic3—31no—
93C22×Dic3—93no—
94C22×Dic3—91no—
95C3×C4⋊C4—186no—
96Dic3⋊C4—66no—
97Dic3⋊C4—186no—
98C22×C32—23no—
99C22×C32—21no—
100C22×C32—23no—
101C22×C32D3×Q813yes—
102C22×C32#23911yescharacteristic
103C22×C32C2×#6713yes—
104C22×C32C2×C3×D413yes—
105C22×C32C2×C3×Q813yes—
106C22×C32C3×#4111yescharacteristic
107C32×C4—212no—
108C32×C4—26no—
109C32×C4—26no—
110C3⋊Dic3—26no—
111C3⋊Dic3—41no—
112C3×Dic3—66no—
113C3×Dic3—121no—
114C22×Q8(C3⋊D3)⋊C311yescharacteristic
115C2×C4⋊C4—93no—
116#95—93no—
117H(3,𝔽3)#39511yescharacteristic, complemented, Sylow
118C23×C3—31no—
119C23×C3—31no—
120C23×C3—61no—
121C23×C3C22×C3×D311yescharacteristic
122C3×Q8—28no—
123C3×Q8—68no—
124C3×Q8—68no—
125C3×Q8—616no—
126C2×C3×C4—26no—
127C2×C3×C4—36no—
128C2×C3×C4—36no—
129C2×C3×C4—66no—
130C2×C3×C4—66no—
131C2×C3×C4—612no—
132C2×C3×C4—66no—
133C2×C3×C4—96no—
134C2×C3×C4—183no—
135C2×C3×C4—183no—
136C2×C3×C4—183no—
137C2×C3×C4C3×#6716yes—
138C2×Dic3—36no—
139C2×Dic3—63no—
140C2×Dic3—63no—
141C2×Dic3—63no—
142C2×Dic3—96no—
143C2×Dic3—183no—
144C2×Dic3—183no—
145C2×Dic3—183no—
146C2×C32—23no—
147C2×C32—23no—
148C2×C32—21no—
149C2×C32#94413yes—
150C2×C32#94613yes—
151C2×C32#95311yescharacteristic
152C2×C32C3×#12011yescharacteristic
153C2×C32C3×C22⋊Q813yes—
154C2×C32C3×C4⋊Q813yes—
155C2×Q8—26no—
156C22×C4—93no—
157C22×C4—91no—
158C22×C4C2×(C3⋊D3)⋊C313yes—
159C4⋊C4—186no—
160C22×C3—33no—
161C22×C3—33no—
162C22×C3—31no—
163C22×C3—31no—
164C22×C3—33no—
165C22×C3—33no—
166C22×C3—63no—
167C22×C3—61no—
168C22×C3—63no—
169C22×C3C3×D3×Q813yes—
170C22×C3C3×#23911yescharacteristic
171C22×C3C2×C3×#6713yes—
172C3×C4—26no—
173C3×C4—66no—
174C3×C4—66no—
175C3×C4—612no—
176C3×C4—186no—
177C3×C4—361no—
178Dic3—66no—
179Dic3—121no—
180Dic3—186no—
181Dic3—361no—
182C32—21no—
183C32#523011yescharacteristic, complemented
184C32C3×#39511yescharacteristic, complemented
185C23C22×(C3⋊D3)⋊C311yescharacteristic
186Q8—28no—
187C2×C4—26no—
188C2×C4—96no—
189C2×C4—183no—
190C2×C4—183no—
191C2×C4—183no—
192C2×C4#638⋊C316yes—
193C2×C3—33no—
194C2×C3—33no—
195C2×C3—33no—
196C2×C3—33no—
197C2×C3—31no—
198C2×C3—31no—
199C2×C3—63no—
200C2×C3—63no—
201C2×C3—61no—
202C2×C3C3×#94413yes—
203C2×C3C3×#94613yes—
204C2×C3C3×#95311yescharacteristic
205C22Q8×(C3⋊D3)⋊C313yes—
206C22#8872211yescharacteristic
207C22C2×#638⋊C313yes—
208C4—26no—
209C4—186no—
210C4—361no—
211C3—31no—
212C3—31no—
213C3—61no—
214C3C3×#523011yescharacteristic
215C2#64153⋊C313yes—
216C2#64155⋊C313yes—
217C2#1172170211yescharacteristic
subgroupG/H
centerC23C22×(C3⋊D3)⋊C3
commutator, cosocleC23×C32C23×C3
Frattini, socleC23×C3C22×C3×D3
FittingC22×Q8×H(3,𝔽3)C2
prime radicalC23×H(3,𝔽3)C23
hypercenterC22×Q8(C3⋊D3)⋊C3
hypocenterC32C3×#395
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
4. #10592929C3semidirect
25. C2×(C3⋊Dic3)⋊C3Q8semidirect
72. C2×C3⋊Dic3C3×Q8semidirect
117. H(3,𝔽3)#395semidirect
183. C32#5230semidirect
184. C32C3×#395semidirect

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×#3750⋊C32
#63658⋊C32
#105929293
C3×#52303