Finite Group Databasebeta

← #13140975

Subgroups — #13140975

Order 1728.

495 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 495 conjugacy classes of subgroups — too many to draw. Only its 79 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
2C2×#3756⋊C3C212yesmaximal
3#11721099C211yesmaximal, characteristic
4C2×#88439C212yesmaximal
5#11720302C211yesmaximal, characteristic
6#10591626—32nomaximal
7C2×Q8×H(3,𝔽3)C2211yescharacteristic
8C2×#6229C2211yescharacteristic
9C2×#634⋊C3C2212yes—
10#88540—22no—
11C4×#1084C2211yescharacteristic
12#3756⋊C3—24no—
13C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
14#88449—22no—
15#88439—24no—
16C2×#3756D312yes—
17C3×#944—32no—
18C3×#944—61no—
19C2×C3×#216—32no—
20#63866—32no—
21C2×#3719—32no—
22C2×#3718—32no—
23Dic3×C3⋊C8—32no—
24Q8×H(3,𝔽3)—22no—
25C2×C4×H(3,𝔽3)—21no—
26C2×C4×H(3,𝔽3)C2311yescharacteristic
27C2×#1084—21no—
28C2×#1084D412yes—
29#6229—22no—
30C2×(C3⋊Dic3)⋊C3—22no—
31#634⋊C3—22no—
32#634⋊C3—22no—
33(C3⋊(C3⋊C8))⋊C3D414yes—
34#5183—92no—
35#5092—91no—
36C2×C32×Q8—21no—
37C2×C32×Q8C2×D312yes—
38C2×#634C2×D312yes—
39C2×C3×Dic6—32no—
40C2×C3×Dic6—32no—
41C2×C3×Dic6—61no—
42#3756—24no—
43C2×C3⋊(C3⋊C8)C2×D312yes—
44C3×#216—64no—
45C3×Dic3⋊C4—64no—
46C3×Dic3⋊C4—64no—
47C3×C4×Dic3—32no—
48C3×C4×Dic3—61no—
49C2×C3×C3⋊C8—32no—
50Dic3⋊Dic3—64no—
51#3719—64no—
52#3718—64no—
53C22×H(3,𝔽3)C2×D411yescharacteristic
54C4×H(3,𝔽3)—41no—
55C4×H(3,𝔽3)C2×D412yes—
56#1084—41no—
57#1084Q1614yescomplemented
58(C3⋊Dic3)⋊C3—42no—
59C2×C3×Q16—92no—
60C3×C4⋊Q8—91no—
61#944—32no—
62#944—61no—
63#944—92no—
64C2×#216—32no—
65C2×#216—92no—
66C2×#216—92no—
67C2×#216—92no—
68C2×Dic12—92no—
69#869—91no—
70C8×Dic3—92no—
71C4×C3⋊C8—91no—
72C32×Q8—22no—
73C32×Q8—22no—
74C32×Q8—42no—
75C2×C32×C4—22no—
76C2×C32×C4—22no—
77C2×C32×C4—21no—
78C2×C32×C4C22×D312yes—
79C2×C3⋊Dic3—22no—
80#634—22no—
81#634—22no—
82C2×C3×Dic3—34no—
83C2×C3×Dic3—62no—
84C2×C3×Dic3—62no—
85C2×C3×Dic3—62no—
86C2×C3×Dic3—62no—
87C3×Dic6—62no—
88C3×Dic6—62no—
89C3×Dic6—62no—
90C3×Dic6—62no—
91C3×Dic6—122no—
92C3⋊(C3⋊C8)#6714yes—
93C3×C3⋊C8—34no—
94#494—271noSylow
95C2×H(3,𝔽3)#12711yescharacteristic
96C2×H(3,𝔽3)C2×Q1612yes—
97C2×C3×Q8—32no—
98C2×C3×Q8—61no—
99C2×C3×Q8—92no—
100C2×C3×Q8—91no—
101C2×C3×Q8D3211yescharacteristic
102C2×Dic6—32no—
103C2×Dic6—32no—
104C2×Dic6—61no—
105C2×Dic6—92no—
106C3×Q16—184no—
107C2×C3×C8—92no—
108C3×C4⋊C4—182no—
109C3×C42—91no—
110#216—64no—
111#216—184no—
112#216—184no—
113#216—184no—
114C4⋊Dic3—182no—
115Dic3⋊C4—64no—
116Dic3⋊C4—64no—
117Dic3⋊C4—184no—
118C4×Dic3—32no—
119C4×Dic3—61no—
120C2×C3⋊C8—32no—
121C2×C3⋊C8—92no—
122Dic12—184no—
123C22×C32—21no—
124C22×C32D3×D412yes—
125C32×C4—22no—
126C32×C4—42no—
127C32×C4—42no—
128C32×C4D3×D412yes—
129C32×C4C2×#6712yes—
130C3⋊Dic3—42no—
131C3×Dic3—38no—
132C3×Dic3—64no—
133C3×Dic3—122no—
134C3×Dic3—122no—
135C3×Dic3—122no—
136C2×Q16—92no—
137C2×Q16—272no—
138C4⋊Q8—91no—
139C4⋊Q8—271no—
140C4×C8—271no—
141H(3,𝔽3)#49411yescharacteristic, complemented, Sylow
142C3×Q8—22no—
143C3×Q8—62no—
144C3×Q8—62no—
145C3×Q8—122no—
146C3×Q8—182no—
147C3×Q8—182no—
148C3×Q8—182no—
149C2×C3×C4—21no—
150C2×C3×C4—32no—
151C2×C3×C4—62no—
152C2×C3×C4—62no—
153C2×C3×C4—61no—
154C2×C3×C4—92no—
155C2×C3×C4—182no—
156C2×C3×C4—181no—
157C2×C3×C4C2×D3211yescharacteristic
158C2×Dic3—34no—
159C2×Dic3—62no—
160C2×Dic3—62no—
161C2×Dic3—62no—
162C2×Dic3—62no—
163C2×Dic3—182no—
164Dic6—62no—
165Dic6—62no—
166Dic6—62no—
167Dic6—62no—
168Dic6—122no—
169Dic6—182no—
170Dic6—182no—
171C3×C8—94no—
172C3⋊C8—34no—
173C3⋊C8—94no—
174C2×C32—22no—
175C2×C32—21no—
176C2×C32D3×Q1614yes—
177C2×C32#94012yes—
178C2×Q8—91no—
179C2×Q8—92no—
180C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
181Q16—184no—
182Q16—544no—
183C2×C8—92no—
184C4⋊C4—182no—
185C4⋊C4—542no—
186C42—91no—
187C22×C3—32no—
188C22×C3—61no—
189C22×C3#381111yescharacteristic
190C3×C4—32no—
191C3×C4—32no—
192C3×C4—41no—
193C3×C4—62no—
194C3×C4—94no—
195C3×C4—122no—
196C3×C4—122no—
197C3×C4—362no—
198C3×C4—361no—
199C3×C4D3×#6712yes—
200Dic3—38no—
201Dic3—64no—
202Dic3—122no—
203Dic3—122no—
204Dic3—122no—
205Dic3—362no—
206C32—21no—
207C32#518312yescomplemented
208Q8—22no—
209Q8—182no—
210Q8—182no—
211Q8—182no—
212C2×C4—21no—
213C2×C4—92no—
214C2×C4—181no—
215C2×C4—182no—
216C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
217C8—94no—
218C2×C3—34no—
219C2×C3—32no—
220C2×C3—62no—
221C2×C3—61no—
222C2×C3#6394412yes—
223C2×C3#6397911yescharacteristic
224C22#8867711yescharacteristic
225C4—41no—
226C4—94no—
227C4—361no—
228C4—362no—
229C4#8867612yes—
230C3—32no—
231C3—61no—
232C3#1059162811yescharacteristic
233C2#1172122612yes—
234C2#1172131711yescharacteristic
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)#494
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
57. #1084Q16semidirect
141. H(3,𝔽3)#494semidirect
207. C32#5183semidirect

Maximal subgroups

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

subgroupindex
#117218612
C2×#3756⋊C32
#117210992
C2×#884392
#117203022
#105916263