Finite Group Databasebeta

← #13147772

Subgroups — #13147772

Order 1728.

810 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 810 conjugacy classes of subgroups — too many to draw. Only its 181 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
2C4×#634⋊C3C212yesmaximal
3#11721116C214yesmaximal
4#11721082C214yesmaximal
5#11721080C214yesmaximal
6#10593856—32nomaximal
7C2×#6229C2212yes—
8C2×#634⋊C3C2212yes—
9C2×#6196C2214yes—
10C42×H(3,𝔽3)C2211yescharacteristic
11C4⋊#1084C2214yes—
12C4⋊((C3⋊Dic3)⋊C3)C2212yes—
13#3750⋊C3C2214yes—
14C4×(C3⋊Dic3)⋊C3C2214yes—
15#88448C2218yes—
16#88447C2214yes—
17C4×#634D312yes—
18C3×#869—32no—
19C3×#869—61no—
20C3×C4×Dic6—32no—
21#63919—34no—
22#63918—34no—
23#63844—34no—
24#63843—34no—
25#63839—38no—
26C2×C4×H(3,𝔽3)C2311yescharacteristic
27C2×C4×H(3,𝔽3)C2312yes—
28C2×#1084C2314yes—
29#6229—42no—
30C2×(C3⋊Dic3)⋊C3C2314yes—
31C2×(C3⋊Dic3)⋊C3C2314yes—
32#634⋊C3Q818yescomplemented
33#6196—44no—
34#5722—91no—
35Q8×Dic6—92no—
36C2×#634C2×D312yes—
37C2×C3×Dic6—32no—
38C2×C3×Dic6—34no—
39C2×C3×Dic6—62no—
40C2×#627—38no—
41C32×C42—21no—
42C32×C42C2×D312yes—
43C4⋊(C3⋊Dic3)C2×D312yes—
44#3750C2×D314yes—
45C4×C3⋊Dic3C2×D314yes—
46C3×C4⋊Dic3—34no—
47C3×C4⋊Dic3—34no—
48C3×C4⋊Dic3—32no—
49C3×C4⋊Dic3—64no—
50C3×Dic3⋊C4—34no—
51C3×C4×Dic3—34no—
52#3724—38no—
53Dic3⋊Dic3—38no—
54Dic32—38no—
55C22×H(3,𝔽3)C2411yescharacteristic
56C4×H(3,𝔽3)—21no—
57C4×H(3,𝔽3)C2×Q814yes—
58#1084—24no—
59(C3⋊Dic3)⋊C3—24no—
60(C3⋊Dic3)⋊C3C2×Q818yes—
61C3×C4⋊Q8—91no—
62C3×C4×Q8—92no—
63Q8×Dic3—94no—
64#944—94no—
65#888—94no—
66#888—94no—
67#888—98no—
68#887—94no—
69#869—32no—
70#869—61no—
71#869—94no—
72C4×Dic6—32no—
73C4×Dic6—92no—
74C4×Dic6—94no—
75C2×C32×C4—22no—
76C2×C32×C4—21no—
77C2×C32×C4C22×D312yes—
78C2×C32×C4C22×D312yes—
79C2×C32×C4C22×D312yes—
80C2×C3⋊Dic3C22×D314yes—
81C2×C3⋊Dic3C22×D314yes—
82#634Dic618yescomplemented
83C2×C3×Dic3—34no—
84C2×C3×Dic3—34no—
85C2×C3×Dic3—38no—
86C2×C3×Dic3—64no—
87C3×Dic6—38no—
88C3×Dic6—124no—
89C3×Dic6—124no—
90#627—128no—
91Q82—271noSylow
92C2×H(3,𝔽3)C22×Q812yes—
93C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
94C2×C3×Q8—92no—
95C2×C3×Q8—92no—
96C2×Dic6—32no—
97C2×Dic6—34no—
98C2×Dic6—62no—
99C2×Dic6—92no—
100C2×Dic6—94no—
101C2×Dic6—98no—
102C3×C4⋊C4—94no—
103C3×C4⋊C4—92no—
104C3×C4⋊C4—94no—
105C3×C42—32no—
106C3×C42—61no—
107C3×C42—94no—
108C3×C42D3211yescharacteristic
109C4⋊Dic3—34no—
110C4⋊Dic3—34no—
111C4⋊Dic3—32no—
112C4⋊Dic3—64no—
113C4⋊Dic3—92no—
114C4⋊Dic3—98no—
115Dic3⋊C4—34no—
116Dic3⋊C4—94no—
117Dic3⋊C4—98no—
118Dic3⋊C4—98no—
119C4×Dic3—34no—
120C4×Dic3—94no—
121C4×Dic3—94no—
122C4×Dic3—98no—
123C22×C32—21no—
124C22×C32C23×D312yes—
125C32×C4—22no—
126C32×C4—22no—
127C32×C4—24no—
128C32×C4C2×Dic614yes—
129C32×C4D3×Q814yes—
130C3⋊Dic3—24no—
131C3⋊Dic3C2×Dic618yes—
132C3×Dic3—38no—
133C3×Dic3—64no—
134C3×Dic3—68no—
135C3×Dic3—124no—
136C4⋊Q8—91no—
137C4⋊Q8—274no—
138C4⋊Q8—274no—
139C4×Q8—92no—
140C4×Q8—274no—
141H(3,𝔽3)Q8211yescharacteristic, complemented, Sylow
142C3×Q8—98no—
143C3×Q8—362no—
144C2×C3×C4—32no—
145C2×C3×C4—32no—
146C2×C3×C4—32no—
147C2×C3×C4—62no—
148C2×C3×C4—61no—
149C2×C3×C4—94no—
150C2×C3×C4—94no—
151C2×C3×C4—94no—
152C2×C3×C4C2×D3211yescharacteristic
153C2×C3×C4C2×D3212yes—
154C2×Dic3—34no—
155C2×Dic3—38no—
156C2×Dic3—34no—
157C2×Dic3—64no—
158C2×Dic3—94no—
159C2×Dic3—94no—
160Dic6—38no—
161Dic6—98no—
162Dic6—124no—
163Dic6—124no—
164Dic6—364no—
165Dic6—368no—
166C2×C32—22no—
167C2×C32—21no—
168C2×C32C22×Dic612yes—
169C2×C32C2×D3×Q812yes—
170C2×C32#100912yes—
171C2×Q8—92no—
172C2×Q8—92no—
173C2×Q8—274no—
174C4⋊C4—94no—
175C4⋊C4—92no—
176C4⋊C4—94no—
177C4⋊C4—278no—
178C42—94no—
179C42—274no—
180C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
181C22×C3—32no—
182C22×C3—61no—
183C22×C3C22×D3211yescharacteristic
184C3×C4—21no—
185C3×C4—34no—
186C3×C4—34no—
187C3×C4—62no—
188C3×C4—62no—
189C3×C4—64no—
190C3×C4—98no—
191C3×C4—184no—
192C3×C4—184no—
193C3×C4D3×Dic614yes—
194Dic3—38no—
195Dic3—68no—
196Dic3—64no—
197Dic3—98no—
198Dic3—124no—
199Dic3—184no—
200C32—21no—
201C32Q8×Dic612yescomplemented
202Q8—98no—
203Q8—362no—
204Q8—1084no—
205C2×C4—94no—
206C2×C4—94no—
207C2×C4—94no—
208C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
209C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
210C2×C3—32no—
211C2×C3—32no—
212C2×C3—32no—
213C2×C3—62no—
214C2×C3—61no—
215C2×C3C2×D3×Dic612yes—
216C2×C3#6430811yescharacteristic
217C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
218C4—21no—
219C4—98no—
220C4—184no—
221C4—184no—
222C4((C3⋊D3)⋊C3)⋊Q814yes—
223C3—32no—
224C3—61no—
225C3Dic6211yescharacteristic
226C2C2×((C3⋊D3)⋊C3)⋊Q812yes—
227C2#1172249311yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, prime radical, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)Q82
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
32. #634⋊C3Q8semidirect
82. #634Dic6semidirect
141. H(3,𝔽3)Q82semidirect
201. C32Q8×Dic6semidirect

Maximal subgroups

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

subgroupindex
#117217152
C4×#634⋊C32
#117211162
#117210822
#117210802
#105938563