Finite Group Databasebeta

← #13137435

Subgroups — #13137435

Order 1728.

565 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 565 conjugacy classes of subgroups — too many to draw. Only its 103 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#11721715C211yesmaximal, characteristic, complemented
2#11721385C211yesmaximal, characteristic
3C2×#88438C214yesmaximal, complemented
4C4×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic, complemented
5#10590375—31nomaximal
6#10590373—31nomaximal
7C2×#6229C2212yes—
8C2×C4⋊((C3⋊D3)⋊C3)—22no—
9C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
10C42×H(3,𝔽3)C2211yescharacteristic
11C4⋊#1084—22no—
12C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
13#88438—28no—
14#64083D311yescharacteristic
15C3×#874—31no—
16C3×#869—31no—
17C3×#869—31no—
18C3×#869—61no—
19C2×#3717—34no—
20C2×#3715—34no—
21C4×C3⋊(C3⋊C8)D311yescharacteristic, complemented
22C3×C4×C3⋊C8—31no—
23C22×(C3⋊D3)⋊C3—22no—
24C2×C4×H(3,𝔽3)D412yes—
25C2×C4×H(3,𝔽3)C2311yescharacteristic
26C2×#1084—22no—
27#6229—24no—
28C4⋊((C3⋊D3)⋊C3)—24no—
29C4⋊((C3⋊D3)⋊C3)—44no—
30(C3⋊(C3⋊C8))⋊C3D414yescomplemented
31#5085—91no—
32#5076—91no—
33#4693—91no—
34C2×C4⋊(C3⋊D3)—22no—
35C2×C4⋊(C3⋊D3)C2×D312yes—
36C2×C3×D12—32no—
37C2×C3×D12—62no—
38C2×C3×Dic6—32no—
39C2×C3×Dic6—32no—
40C2×C3×Dic6—62no—
41C32×C42—21no—
42C32×C42C2×D311yescharacteristic
43C32×C42C2×D311yescharacteristic
44C2×C3⋊(C3⋊C8)C2×D312yes—
45C3×C4⋊Dic3—64no—
46C3×C4⋊Dic3—62no—
47C3×C4⋊Dic3—62no—
48C2×C3×C3⋊C8—32no—
49#3717—68no—
50#3715—68no—
51C22×H(3,𝔽3)C2×D411yescharacteristic
52C2×(C3⋊D3)⋊C3—42no—
53C2×(C3⋊D3)⋊C3—44no—
54C4×H(3,𝔽3)SD1614yescomplemented
55C4×H(3,𝔽3)C2×D412yes—
56#1084—42no—
57C3×C4⋊Q8—91no—
58C3×#127—91no—
59C2×#215—94no—
60C2×#214—94no—
61C2×#204—94no—
62#874—31no—
63#874—91no—
64#869—31no—
65#869—31no—
66#869—61no—
67C3×C4×C8—91no—
68C4×C3⋊C8—31no—
69C4×C3⋊C8—91no—
70C22×C3⋊D3—22no—
71C22×C3×D3—62no—
72C2×C32×C4—22no—
73C2×C32×C4—21no—
74C2×C32×C4D1212yes—
75C2×C32×C4#6712yes—
76C2×C32×C4C22×D311yescharacteristic
77C2×C32×C4C22×D311yescharacteristic
78C4⋊(C3⋊D3)—24no—
79C4⋊(C3⋊D3)—44no—
80C2×C3×Dic3—64no—
81C2×C3×Dic3—62no—
82C2×C3×Dic3—62no—
83C3×D12—64no—
84C3×D12—124no—
85C3×Dic6—64no—
86C3×Dic6—64no—
87C3×Dic6—124no—
88C3⋊(C3⋊C8)D1214yescomplemented
89C3×C3⋊C8—34no—
90#492—271noSylow
91C2×H(3,𝔽3)#12711yescharacteristic
92C2×H(3,𝔽3)C2×SD1612yes—
93(C3⋊D3)⋊C3—82no—
94C2×C3×Q8—92no—
95C2×C3×D4—92no—
96C2×C3×D4—182no—
97C2×D12—32no—
98C2×D12—62no—
99C2×D12—92no—
100C2×D12—182no—
101C2×Dic6—32no—
102C2×Dic6—32no—
103C2×Dic6—62no—
104C2×C3×C8—92no—
105C3×C4⋊C4—182no—
106C3×C42—31no—
107C3×C42—31no—
108C3×C42—61no—
109C3×C42D3211yescharacteristic
110#215—188no—
111#214—188no—
112C4⋊Dic3—62no—
113C4⋊Dic3—62no—
114C4⋊Dic3—64no—
115C2×C3⋊C8—32no—
116C2×C3⋊C8—92no—
117#204—188no—
118C22×C32—21no—
119C22×C32C2×D1211yescharacteristic
120C22×C32C2×#6711yescharacteristic
121C2×C3⋊D3—44no—
122C2×C3⋊D3—42no—
123C2×C3×D3—122no—
124C2×C3×D3—124no—
125C32×C4—24no—
126C32×C4—22no—
127C32×C4#20414yescomplemented
128C32×C4#21414yescomplemented
129C32×C4C2×D1212yes—
130C32×C4D3×D412yes—
131C3×Dic3—124no—
132C3×Dic3—122no—
133C3×Dic3—122no—
134C2×SD16—274no—
135C4⋊Q8—91no—
136#127—91no—
137C4×C8—91no—
138H(3,𝔽3)#49211yescharacteristic, complemented, Sylow
139C23×C3—182no—
140C22×D3—62no—
141C22×D3—182no—
142C3×Q8—184no—
143C3×D4—184no—
144C3×D4—364no—
145C2×C3×C4—32no—
146C2×C3×C4—32no—
147C2×C3×C4—31no—
148C2×C3×C4—31no—
149C2×C3×C4—62no—
150C2×C3×C4—61no—
151C2×C3×C4—182no—
152C2×C3×C4#62612yes—
153C2×C3×C4C2×D3211yescharacteristic
154C2×Dic3—62no—
155C2×Dic3—62no—
156C2×Dic3—64no—
157D12—64no—
158D12—124no—
159D12—184no—
160D12—364no—
161Dic6—64no—
162Dic6—64no—
163Dic6—124no—
164C3×C8—94no—
165C3⋊C8—34no—
166C3⋊C8—94no—
167C2×C32—22no—
168C2×C32—21no—
169C2×C32#87411yescharacteristic
170C2×C32C2×#20412yes—
171C2×C32C2×#21412yes—
172C2×C32#94011yescharacteristic
173C3⋊D3—82no—
174C3×D3—242no—
175C2×Q8—92no—
176C2×D4—92no—
177C2×D4—182no—
178SD16—548no—
179C2×C8—92no—
180C4⋊C4—182no—
181C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
182C22×C3—31no—
183C22×C3—31no—
184C22×C3—61no—
185C22×C3—362no—
186C22×C3—364no—
187C22×C3C2×#62611yescharacteristic
188C2×D3—124no—
189C2×D3—122no—
190C2×D3—362no—
191C2×D3—364no—
192C3×C4—32no—
193C3×C4—32no—
194C3×C4—34no—
195C3×C4—34no—
196C3×C4—62no—
197C3×C4—64no—
198C3×C4—362no—
199C3×C4#371614yes—
200C3×C4D3×D1212yes—
201Dic3—122no—
202Dic3—122no—
203Dic3—124no—
204C32—21no—
205C32#469311yescharacteristic, complemented
206C32#507611yescharacteristic, complemented
207C23—182no—
208Q8—184no—
209D4—184no—
210D4—364no—
211C2×C4—182no—
212C2×C4#620012yes—
213C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
214C8—94no—
215C2×C3—32no—
216C2×C3—32no—
217C2×C3—31no—
218C2×C3—31no—
219C2×C3—62no—
220C2×C3—61no—
221C2×C3—722no—
222C2×C3C2×#371612yes—
223C2×C3#6391111yescharacteristic
224D3—242no—
225D3—722no—
226C22—364no—
227C22—362no—
228C22C2×#620011yescharacteristic
229C4—362no—
230C4#8843814yescomplemented
231C4#8865412yes—
232C3—31no—
233C3—31no—
234C3—61no—
235C3#1059036911yescharacteristic
236C2—722no—
237C2C2×#8843812yes—
238C2#1172115811yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radical#11721385C2
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#492
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11721715C2semidirect
3. C2×#88438C2semidirect
4. C4×(C3⋊(C3⋊C8))⋊C3C2semidirect
21. C4×C3⋊(C3⋊C8)D3semidirect
30. (C3⋊(C3⋊C8))⋊C3D4semidirect
54. C4×H(3,𝔽3)SD16semidirect
88. C3⋊(C3⋊C8)D12semidirect
127. C32×C4#204semidirect
128. C32×C4#214semidirect
138. H(3,𝔽3)#492semidirect
205. C32#4693semidirect
206. C32#5076semidirect
230. C4#88438semidirect

Maximal subgroups

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

subgroupindex
#117217152
#117213852
C2×#884382
C4×(C3⋊(C3⋊C8))⋊C32
#105903753
#105903733