Finite Group Databasebeta

← #13140886

Subgroups — #13140886

Order 1728.

383 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 383 conjugacy classes of subgroups — too many to draw. Only its 67 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#11721861C211yesmaximal, characteristic
2#11721095C211yesmaximal, characteristic
3#63662⋊C3C212yesmaximal
4#11720326C212yesmaximal
5#11720303C211yesmaximal, characteristic
6#10591613—32nomaximal
7C2×Q8×H(3,𝔽3)C2211yescharacteristic
8C2×#6229C2211yescharacteristic
9#88540—22no—
10C4×#1084C2211yescharacteristic
11C4⋊((C3⋊Dic3)⋊C3)C2212yes—
12C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
13#88448—22no—
14C3×#944—32no—
15C3×#944—61no—
16#63869—32no—
17#63662D312yes—
18C3×#835—32no—
19#63566—32no—
20#63565—32no—
21#63554—32no—
22Q8×H(3,𝔽3)—22no—
23C2×C4×H(3,𝔽3)—21no—
24C2×C4×H(3,𝔽3)C2311yescharacteristic
25C2×#1084—21no—
26C2×#1084D412yes—
27#6229—22no—
28C2×(C3⋊Dic3)⋊C3—22no—
29(C3⋊(C3⋊C8))⋊C3—22no—
30#5082—91no—
31#4797—92no—
32C2×C32×Q8—21no—
33C2×C32×Q8C2×D312yes—
34C2×C3×Dic6—32no—
35C2×C3×Dic6—61no—
36C4⋊(C3⋊Dic3)C2×D312yes—
37C2×C3⋊(C3⋊C8)C2×D312yes—
38C3×C4⋊Dic3—32no—
39C3×Dic3⋊C4—62no—
40C3×Dic3⋊C4—62no—
41C3×Dic3⋊C4—64no—
42C3×C4×Dic3—32no—
43C3×C4×Dic3—61no—
44C2×C3×C3⋊C8—32no—
45#3724—62no—
46Dic3⋊Dic3—62no—
47C22×H(3,𝔽3)C2×D411yescharacteristic
48C4×H(3,𝔽3)—41no—
49C4×H(3,𝔽3)#4112yes—
50#1084—22no—
51#1084—41no—
52(C3⋊Dic3)⋊C3—42no—
53C3×C4⋊Q8—91no—
54#944—32no—
55#944—61no—
56#890—92no—
57#870—91no—
58C3×#103—92no—
59#835—32no—
60#835—92no—
61#835—92no—
62#816—92no—
63#815—92no—
64#810—92no—
65#803—91no—
66C32×Q8—22no—
67C32×Q8—22no—
68C32×Q8—42no—
69C2×C32×C4—22no—
70C2×C32×C4—22no—
71C2×C32×C4—21no—
72C2×C32×C4C22×D312yes—
73C2×C3⋊Dic3—22no—
74C2×C3×Dic3—32no—
75C2×C3×Dic3—32no—
76C2×C3×Dic3—62no—
77C2×C3×Dic3—62no—
78C2×C3×Dic3—62no—
79C2×C3×Dic3—62no—
80C3×Dic6—62no—
81C3×Dic6—62no—
82C3×Dic6—122no—
83C3⋊(C3⋊C8)—22no—
84C3×C3⋊C8—62no—
85#491—271noSylow
86C2×H(3,𝔽3)#12411yescharacteristic
87C2×H(3,𝔽3)#13712yes—
88C2×C3×Q8—32no—
89C2×C3×Q8—61no—
90C2×C3×Q8—91no—
91C2×C3×Q8D3211yescharacteristic
92C2×Dic6—32no—
93C2×Dic6—61no—
94C2×C3×C8—92no—
95C3×C4⋊C4—92no—
96C3×C4⋊C4—182no—
97C3×C42—91no—
98C4⋊Dic3—32no—
99C4⋊Dic3—92no—
100C4⋊Dic3—182no—
101Dic3⋊C4—62no—
102Dic3⋊C4—62no—
103Dic3⋊C4—64no—
104Dic3⋊C4—182no—
105Dic3⋊C4—182no—
106C4×Dic3—32no—
107C4×Dic3—61no—
108C2×C3⋊C8—32no—
109C2×C3⋊C8—92no—
110C22×C32—21no—
111C22×C32D3×D412yes—
112C32×C4—22no—
113C32×C4—42no—
114C32×C4—42no—
115C32×C4#23512yes—
116C32×C4#23712yes—
117C3⋊Dic3—42no—
118C3×Dic3—64no—
119C3×Dic3—62no—
120C3×Dic3—62no—
121C3×Dic3—122no—
122C3×Dic3—122no—
123C3×Dic3—122no—
124C4⋊Q8—91no—
125#125—271no—
126#103—92no—
127#103—272no—
128#97—271no—
129H(3,𝔽3)#49111yescharacteristic, complemented, Sylow
130C3×Q8—22no—
131C3×Q8—62no—
132C3×Q8—62no—
133C3×Q8—122no—
134C3×Q8—182no—
135C2×C3×C4—21no—
136C2×C3×C4—32no—
137C2×C3×C4—62no—
138C2×C3×C4—62no—
139C2×C3×C4—61no—
140C2×C3×C4—92no—
141C2×C3×C4—182no—
142C2×C3×C4—181no—
143C2×C3×C4C2×D3211yescharacteristic
144C2×Dic3—32no—
145C2×Dic3—32no—
146C2×Dic3—62no—
147C2×Dic3—62no—
148C2×Dic3—62no—
149C2×Dic3—62no—
150C2×Dic3—182no—
151Dic6—62no—
152Dic6—62no—
153Dic6—122no—
154C3×C8—182no—
155C3⋊C8—62no—
156C3⋊C8—182no—
157C2×C32—22no—
158C2×C32—21no—
159C2×C32#88512yes—
160C2×C32#91512yes—
161C2×C32#91812yes—
162C2×Q8—91no—
163C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
164C2×C8—92no—
165C4⋊C4—92no—
166C4⋊C4—182no—
167C4⋊C4—542no—
168C42—91no—
169C22×C3—32no—
170C22×C3—61no—
171C22×C3#381111yescharacteristic
172C3×C4—32no—
173C3×C4—32no—
174C3×C4—41no—
175C3×C4—62no—
176C3×C4—122no—
177C3×C4—122no—
178C3×C4—182no—
179C3×C4—362no—
180C3×C4—361no—
181C3×C4#380412yes—
182Dic3—62no—
183Dic3—62no—
184Dic3—64no—
185Dic3—122no—
186Dic3—122no—
187Dic3—122no—
188Dic3—362no—
189C32—21no—
190C32#479712yescomplemented
191Q8—22no—
192Q8—182no—
193C2×C4—21no—
194C2×C4—92no—
195C2×C4—182no—
196C2×C4—181no—
197C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
198C8—182no—
199C2×C3—32no—
200C2×C3—32no—
201C2×C3—32no—
202C2×C3—62no—
203C2×C3—61no—
204C2×C3#6395112yes—
205C2×C3#6395311yescharacteristic
206C22#8867711yescharacteristic
207C4—41no—
208C4—182no—
209C4—362no—
210C4—361no—
211C4#8867112yes—
212C3—32no—
213C3—61no—
214C3#1059161611yescharacteristic
215C2#1172123112yes—
216C2#1172129411yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC2×Q8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#491
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
129. H(3,𝔽3)#491semidirect
190. C32#4797semidirect

Maximal subgroups

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

subgroupindex
#117218612
#117210952
#63662⋊C32
#117203262
#117203032
#105916133