Finite Group Databasebeta

← #13138880

Subgroups — #13138880

Order 1728.

429 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 429 conjugacy classes of subgroups — too many to draw. Only its 83 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#11721084C212yesmaximal
2C2×#88433C211yesmaximal, characteristic
3C8×#1084C211yesmaximal, characteristic
4#63643⋊C3C212yesmaximal
5#11720315C211yesmaximal, characteristic
6#10590793—32nomaximal
7C2×#634⋊C3C2212yes—
8C2×C8×H(3,𝔽3)C2211yescharacteristic
9C4×#1084C2211yescharacteristic
10C2×#6180C2211yescharacteristic
11C4⋊((C3⋊Dic3)⋊C3)C2212yes—
12#88448—22no—
13#88447—22no—
14#88433C414yescomplemented
15#63844—32no—
16Dic3×Dic6—32no—
17C2×#3719—32no—
18#63643D312yes—
19C3×#816—32no—
20C3×C8×Dic3—32no—
21C3×C8×Dic3—61no—
22#63575—32no—
23C2×C4×H(3,𝔽3)C2311yescharacteristic
24C2×#1084D411yescharacteristic
25C2×#1084D411yescharacteristic
26C2×(C3⋊Dic3)⋊C3—22no—
27C2×(C3⋊Dic3)⋊C3—22no—
28#634⋊C3—22no—
29#634⋊C3C2×C414yes—
30C8×H(3,𝔽3)—21no—
31#6180C2×C412yes—
32#4790—92no—
33C4×Dic12—91no—
34C2×#634C2×D312yes—
35C2×C3×Dic6—32no—
36C2×C32×C8—21no—
37C2×C32×C8C2×D312yes—
38C4⋊(C3⋊Dic3)C2×D312yes—
39C3×C4⋊Dic3—32no—
40C3×C4×Dic3—32no—
41C3×C4×Dic3—61no—
42C2×C3×C3⋊C8—32no—
43C2×C3×C3⋊C8—61no—
44#3724—62no—
45Dic3⋊Dic3—62no—
46Dic32—62no—
47Dic32—62no—
48#3719—38no—
49C22×H(3,𝔽3)C2×D411yescharacteristic
50C4×H(3,𝔽3)C22×C411yescharacteristic
51C4×H(3,𝔽3)#4111yescharacteristic
52#1084—21no—
53#1084Q1612yescomplemented
54(C3⋊Dic3)⋊C3—24no—
55(C3⋊Dic3)⋊C3—42no—
56Q8×Dic3—92no—
57C2×#216—92no—
58C2×Dic12—91no—
59#887—92no—
60C4×Dic6—92no—
61C3×#103—92no—
62C3×C4×C8—91no—
63#818—91no—
64#816—32no—
65#816—92no—
66C8×Dic3—32no—
67C8×Dic3—61no—
68#807—92no—
69C2×C32×C4—21no—
70C2×C32×C4C22×D312yes—
71C2×C3⋊Dic3—22no—
72C2×C3⋊Dic3—22no—
73#634—22no—
74#634C4×D314yes—
75C2×C3×Dic3—32no—
76C2×C3×Dic3—32no—
77C2×C3×Dic3—61no—
78C2×C3×Dic3—61no—
79C2×C3×Dic3—62no—
80C2×C3×Dic3—62no—
81C3×Dic6—34no—
82C3×Dic6—62no—
83C32×C8—22no—
84C32×C8—22no—
85C3×C3⋊C8—34no—
86C3×C3⋊C8—62no—
87C4×Q16—271noSylow
88C2×H(3,𝔽3)C4×D411yescharacteristic
89C2×H(3,𝔽3)C2×Q1611yescharacteristic
90C2×H(3,𝔽3)#13511yescharacteristic
91C2×C3×Q8—92no—
92C2×Dic6—32no—
93C2×Dic6—92no—
94C2×C3×C8—32no—
95C2×C3×C8—61no—
96C2×C3×C8—91no—
97C2×C3×C8D3211yescharacteristic
98C3×C4⋊C4—92no—
99C3×C42—91no—
100#216—98no—
101C4⋊Dic3—32no—
102C4⋊Dic3—92no—
103C4⋊Dic3—182no—
104Dic3⋊C4—182no—
105Dic3⋊C4—182no—
106C4×Dic3—32no—
107C4×Dic3—61no—
108C4×Dic3—182no—
109C4×Dic3—182no—
110C4×Dic3—182no—
111C2×C3⋊C8—32no—
112C2×C3⋊C8—61no—
113Dic12—94no—
114C22×C32—21no—
115C22×C32D3×D412yes—
116C32×C4—21no—
117C32×C4—21no—
118C32×C4C2×C4×D312yes—
119C32×C4#23712yes—
120C3⋊Dic3—24no—
121C3⋊Dic3—42no—
122C3×Dic3—34no—
123C3×Dic3—62no—
124C3×Dic3—62no—
125C3×Dic3—64no—
126C3×Dic3—62no—
127C3×Dic3—122no—
128C2×Q16—271no—
129C4×Q8—272no—
130#107—271no—
131#103—92no—
132C4×C8—91no—
133H(3,𝔽3)C4×Q1611yescharacteristic, complemented, Sylow
134C3×Q8—94no—
135C3×Q8—182no—
136C2×C3×C4—32no—
137C2×C3×C4—61no—
138C2×C3×C4—91no—
139C2×C3×C4—91no—
140C2×C3×C4—182no—
141C2×C3×C4—182no—
142C2×C3×C4C2×D3211yescharacteristic
143C2×Dic3—32no—
144C2×Dic3—32no—
145C2×Dic3—61no—
146C2×Dic3—61no—
147C2×Dic3—62no—
148C2×Dic3—62no—
149C2×Dic3—182no—
150C2×Dic3—182no—
151Dic6—34no—
152Dic6—62no—
153Dic6—94no—
154Dic6—182no—
155C3×C8—21no—
156C3×C8—62no—
157C3×C8—62no—
158C3×C8—92no—
159C3⋊C8—34no—
160C3⋊C8—62no—
161C2×C32—21no—
162C2×C32—21no—
163C2×C32—21no—
164C2×C32#67⋊C412yes—
165C2×C32#91612yes—
166C2×C32D3×Q1612yes—
167C2×Q8—92no—
168Q16—274no—
169C2×C8—91no—
170C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
171C4⋊C4—92no—
172C4⋊C4—542no—
173C42—91no—
174C42—542no—
175C22×C3—32no—
176C22×C3—61no—
177C22×C3#380211yescharacteristic
178C3×C4—32no—
179C3×C4—32no—
180C3×C4—61no—
181C3×C4—61no—
182C3×C4—92no—
183C3×C4—184no—
184C3×C4—181no—
185C3×C4—362no—
186C3×C4C4×D3211yescharacteristic
187C3×C4#380511yescharacteristic
188Dic3—34no—
189Dic3—62no—
190Dic3—62no—
191Dic3—62no—
192Dic3—64no—
193Dic3—122no—
194Dic3—184no—
195Dic3—362no—
196C32—21no—
197C32#479012yescomplemented
198Q8—94no—
199Q8—182no—
200C2×C4—91no—
201C2×C4—91no—
202C2×C4—182no—
203C2×C4—182no—
204C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
205C8—21no—
206C8—92no—
207C2×C3—32no—
208C2×C3—32no—
209C2×C3—32no—
210C2×C3—61no—
211C2×C3—61no—
212C2×C3—61no—
213C2×C3#6380211yescharacteristic
214C2×C3#6380911yescharacteristic
215C2×C3#6390211yescharacteristic
216C22#8865511yescharacteristic
217C4—92no—
218C4—181no—
219C4—184no—
220C4—362no—
221C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
222C4#8867211yescharacteristic
223C3—32no—
224C3—61no—
225C3#1059079011yescharacteristic
226C2#1172083311yescharacteristic
227C2#1172084311yescharacteristic
228C2#1172113811yescharacteristic
subgroupG/H
centerC22#88655
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
prime radicalC22×H(3,𝔽3)C2×D4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C4×Q16
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
14. #88433C4semidirect
53. #1084Q16semidirect
133. H(3,𝔽3)C4×Q16semidirect
197. C32#4790semidirect

Maximal subgroups

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

subgroupindex
#117210842
C2×#884332
C8×#10842
#63643⋊C32
#117203152
#105907933