Finite Group Databasebeta

← #13143993

Subgroups — #13143993

Order 1728.

528 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 113 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
1Q8×#1084C211yesmaximal, characteristic
2#11721756C211yesmaximal, characteristic
3H(3,𝔽3)×#103C211yesmaximal, characteristic
4#11720583C211yesmaximal, characteristic
5#11720550C211yesmaximal, characteristic
6#11720547C211yesmaximal, characteristic
7#11720530C211yesmaximal, characteristic
8C3×#4791—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10C2×#6229—21no—
11C2×C8×H(3,𝔽3)C2211yescharacteristic
12C4⋊C4×H(3,𝔽3)C2211yescharacteristic
13C4⋊#1084—21no—
14C4⋊#1084C2211yescharacteristic
15C4⋊#1084C2211yescharacteristic
16#88540—21no—
17C4×#1084—21no—
18C4×#1084C2211yescharacteristic
19C2×#6180C2211yescharacteristic
20C3×Q8×Dic3—34no—
21C3×#888—34no—
22C32×#103D314yes—
23C3×#835—34no—
24C3×#817—34no—
25C3×Dic3⋊C8—34no—
26C3×#808—34no—
27Q8×H(3,𝔽3)—21no—
28Q8×H(3,𝔽3)Q812yescomplemented
29C2×C4×H(3,𝔽3)—21no—
30C2×C4×H(3,𝔽3)—21no—
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C2×#1084—21no—
33C2×#1084—21no—
34C2×#1084D411yescharacteristic
35C2×#1084D411yescharacteristic
36#6229—42no—
37C8×H(3,𝔽3)—21no—
38#6180—21no—
39C3×#475—91no—
40#4791—34no—
41C2×C32×Q8C2×D314yes—
42C2×C3×Dic6—64no—
43C2×C32×C8C2×D314yes—
44C32×C4⋊C4C2×D314yes—
45C3×C4⋊Dic3—34no—
46C3×C4⋊Dic3—34no—
47C3×C4⋊Dic3—64no—
48C3×Dic3⋊C4—64no—
49C3×C4×Dic3—34no—
50C3×C4×Dic3—64no—
51C2×C3×C3⋊C8—34no—
52C22×H(3,𝔽3)C2×D411yescharacteristic
53C4×H(3,𝔽3)—22no—
54C4×H(3,𝔽3)—41no—
55C4×H(3,𝔽3)C2×Q811yescharacteristic
56C4×H(3,𝔽3)#4111yescharacteristic
57#1084—21no—
58#1084—41no—
59#1084—41no—
60#1084SD1612yes—
61C3×C4⋊Q8—91no—
62C3×C4×Q8—91no—
63Q8×Dic3—34no—
64#888—34no—
65C3×#106—91no—
66C3×#106—91no—
67C3×C4⋊C8—91no—
68C3×#103—34no—
69C3×#103—91no—
70C3×#103C3⋊D311yescharacteristic
71#835—34no—
72#817—34no—
73Dic3⋊C8—34no—
74#808—34no—
75C32×Q8—24no—
76C32×Q8Dic618yescomplemented
77C2×C32×C4—24no—
78C2×C32×C4—24no—
79C2×C32×C4C22×D314yes—
80C2×C3×Dic3—34no—
81C2×C3×Dic3—34no—
82C2×C3×Dic3—64no—
83C2×C3×Dic3—64no—
84C3×Dic6—128no—
85C32×C8—24no—
86C3×C3⋊C8—64no—
87#475—91noSylow
88C2×H(3,𝔽3)C22⋊Q811yescharacteristic
89C2×H(3,𝔽3)C2×SD1611yescharacteristic
90C2×H(3,𝔽3)#13711yescharacteristic
91C2×C3×Q8—34no—
92C2×C3×Q8—181no—
93C2×C3×Q8C2×C3⋊D311yescharacteristic
94C2×Dic6—64no—
95C2×C3×C8—34no—
96C2×C3×C8—91no—
97C2×C3×C8C2×C3⋊D311yescharacteristic
98C3×C4⋊C4—34no—
99C3×C4⋊C4—91no—
100C3×C4⋊C4—91no—
101C3×C4⋊C4—181no—
102C3×C4⋊C4—181no—
103C3×C4⋊C4C2×C3⋊D311yescharacteristic
104C3×C42—91no—
105C3×C42—181no—
106C4⋊Dic3—34no—
107C4⋊Dic3—34no—
108C4⋊Dic3—64no—
109Dic3⋊C4—64no—
110C4×Dic3—34no—
111C4×Dic3—64no—
112C2×C3⋊C8—34no—
113C22×C32D3×D414yes—
114C32×C4—28no—
115C32×C4—44no—
116C32×C4C2×Dic614yes—
117C32×C4#23714yes—
118C3×Dic3—38no—
119C3×Dic3—64no—
120C3×Dic3—124no—
121C3×Dic3—124no—
122C4⋊Q8—91no—
123C4×Q8—91no—
124#106—91no—
125#106—91no—
126C4⋊C8—91no—
127#103—91no—
128#103#27211yescharacteristic
129H(3,𝔽3)#47511yescharacteristic, complemented, Sylow
130C3×Q8—21no—
131C3×Q8—38no—
132C3×Q8—64no—
133C3×Q8—362no—
134C3×Q8#63412yes—
135C2×C3×C4—21no—
136C2×C3×C4—21no—
137C2×C3×C4—34no—
138C2×C3×C4—64no—
139C2×C3×C4—64no—
140C2×C3×C4—91no—
141C2×C3×C4—91no—
142C2×C3×C4—181no—
143C2×C3×C4—181no—
144C2×C3×C4C22×C3⋊D311yescharacteristic
145C2×Dic3—34no—
146C2×Dic3—34no—
147C2×Dic3—64no—
148C2×Dic3—64no—
149Dic6—128no—
150C3×C8—21no—
151C3×C8—64no—
152C3×C8—181no—
153C3⋊C8—64no—
154C2×C32#87814yes—
155C2×C32D3×SD1614yes—
156C2×C32#91814yes—
157C2×Q8—181no—
158C2×Q8C2×#27211yescharacteristic
159C2×C8—91no—
160C2×C8C2×#27211yescharacteristic
161C4⋊C4—91no—
162C4⋊C4—91no—
163C4⋊C4—181no—
164C4⋊C4—181no—
165C4⋊C4C2×#27211yescharacteristic
166C42—91no—
167C42—181no—
168C22×C3—34no—
169C22×C3D4×C3⋊D311yescharacteristic
170C3×C4—22no—
171C3×C4—34no—
172C3×C4—34no—
173C3×C4—41no—
174C3×C4—68no—
175C3×C4—92no—
176C3×C4—124no—
177C3×C4—181no—
178C3×C4—361no—
179C3×C4—361no—
180C3×C4C2×#63411yescharacteristic
181C3×C4#383011yescharacteristic
182Dic3—38no—
183Dic3—64no—
184Dic3—124no—
185Dic3—124no—
186C32#479114yescomplemented
187Q8—21no—
188Q8—362no—
189Q8#622912yescomplemented
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—91no—
193C2×C4—91no—
194C2×C4—181no—
195C2×C4—181no—
196C2×C4C22×#27211yescharacteristic
197C8—21no—
198C8—181no—
199C2×C3—34no—
200C2×C3—34no—
201C2×C3—34no—
202C2×C3#6408711yescharacteristic
203C2×C3SD16×C3⋊D311yescharacteristic
204C2×C3#6412711yescharacteristic
205C22D4×#27211yescharacteristic
206C4—22no—
207C4—41no—
208C4—92no—
209C4—181no—
210C4—361no—
211C4—361no—
212C4C2×#622911yescharacteristic
213C4#8874411yescharacteristic
214C3—34no—
215C3#1059249011yescharacteristic
216C2#1172173411yescharacteristic
217C2SD16×#27211yescharacteristic
218C2#1172180711yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#103C2
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC3×#103C3⋊D3
hypocenterH(3,𝔽3)#475
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#103

Complements

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

NH ≅ G/Ntype
28. Q8×H(3,𝔽3)Q8semidirect
76. C32×Q8Dic6semidirect
129. H(3,𝔽3)#475semidirect
186. C32#4791semidirect
189. Q8#6229semidirect

Maximal subgroups

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

subgroupindex
Q8×#10842
#117217562
H(3,𝔽3)×#1032
#117205832
#117205502
#117205472
#117205302
C3×#47913