Finite Group Databasebeta

← #13137874

Subgroups — #13137874

Order 1728.

448 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 71 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#11721755C211yesmaximal, characteristic
2#11721099C211yesmaximal, characteristic
3C2×#88439C212yesmaximal
4#63637⋊C3C212yesmaximal
5#11720304C211yesmaximal, characteristic
6#10590561—32nomaximal
7C2×#6229C2211yescharacteristic
8C2×#634⋊C3C2212yes—
9C4⋊C4×H(3,𝔽3)C2211yescharacteristic
10#88540—21no—
11C4×#1084—21no—
12C4×#1084C2211yescharacteristic
13C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
14#88449—21no—
15#88449—21no—
16#88439—24no—
17C3×#887—32no—
18C3×#887—61no—
19#63866—32no—
20C2×#3719—32no—
21C2×#3718—32no—
22#63637D312yes—
23C3×#810—32no—
24Dic3⋊(C3⋊C8)—32no—
25C2×C4×H(3,𝔽3)—21no—
26C2×C4×H(3,𝔽3)C2311yescharacteristic
27C2×#1084—21no—
28C2×#1084D411yescharacteristic
29C2×#1084D411yescharacteristic
30#6229—21no—
31#6229D412yes—
32C2×(C3⋊Dic3)⋊C3—22no—
33#634⋊C3—22no—
34#634⋊C3—22no—
35(C3⋊(C3⋊C8))⋊C3—22no—
36#5031—91no—
37#4795—92no—
38C2×#634C2×D312yes—
39C2×C3×Dic6—32no—
40C2×C3×Dic6—32no—
41C2×C3×Dic6—61no—
42C32×C4⋊C4—21no—
43C32×C4⋊C4C2×D312yes—
44C2×C3⋊(C3⋊C8)C2×D312yes—
45C3×Dic3⋊C4—62no—
46C3×Dic3⋊C4—62no—
47C3×C4×Dic3—32no—
48C3×C4×Dic3—62no—
49C3×C4×Dic3—61no—
50C3×C4×Dic3—62no—
51C2×C3×C3⋊C8—32no—
52Dic3⋊Dic3—62no—
53Dic3⋊Dic3—62no—
54#3719—64no—
55#3718—64no—
56C22×H(3,𝔽3)C2×D411yescharacteristic
57C4×H(3,𝔽3)—41no—
58C4×H(3,𝔽3)C2×D411yescharacteristic
59C4×H(3,𝔽3)#4111yescharacteristic
60#1084—22no—
61#1084—21no—
62#1084Q1612yescomplemented
63(C3⋊Dic3)⋊C3—42no—
64C3×C4×Q8—91no—
65#944—92no—
66C2×#216—92no—
67C2×#216—92no—
68C2×Dic12—92no—
69#887—32no—
70#887—61no—
71#869—91no—
72C3×#103—92no—
73Dic3⋊C8—92no—
74#810—32no—
75#810—92no—
76C4⋊(C3⋊C8)—91no—
77C2×C32×C4—22no—
78C2×C32×C4—22no—
79C2×C32×C4—21no—
80C2×C32×C4C22×D312yes—
81C2×C3⋊Dic3—22no—
82#634—22no—
83#634—22no—
84C2×C3×Dic3—32no—
85C2×C3×Dic3—32no—
86C2×C3×Dic3—61no—
87C2×C3×Dic3—61no—
88C2×C3×Dic3—62no—
89C2×C3×Dic3—62no—
90C2×C3×Dic3—62no—
91C3×Dic6—34no—
92C3×Dic6—62no—
93C3×Dic6—62no—
94C3×Dic6—62no—
95C3×Dic6—62no—
96C3×Dic6—62no—
97C3⋊(C3⋊C8)—22no—
98C3×C3⋊C8—62no—
99#462—271noSylow
100C2×H(3,𝔽3)#12111yescharacteristic
101C2×H(3,𝔽3)C2×Q1611yescharacteristic
102C2×H(3,𝔽3)#13711yescharacteristic
103C2×C3×Q8—92no—
104C2×C3×Q8—91no—
105C2×Dic6—32no—
106C2×Dic6—32no—
107C2×Dic6—61no—
108C2×Dic6—92no—
109C2×C3×C8—92no—
110C3×C4⋊C4—32no—
111C3×C4⋊C4—61no—
112C3×C4⋊C4—181no—
113C3×C4⋊C4D3211yescharacteristic
114C3×C42—91no—
115C3×C42—181no—
116#216—184no—
117#216—184no—
118C4⋊Dic3—181no—
119C4⋊Dic3—181no—
120Dic3⋊C4—62no—
121Dic3⋊C4—62no—
122Dic3⋊C4—182no—
123Dic3⋊C4—182no—
124C4×Dic3—32no—
125C4×Dic3—61no—
126C4×Dic3—62no—
127C4×Dic3—62no—
128C2×C3⋊C8—32no—
129C2×C3⋊C8—92no—
130Dic12—184no—
131C22×C32—21no—
132C22×C32D3×D412yes—
133C32×C4—21no—
134C32×C4—21no—
135C32×C4—42no—
136C32×C4—42no—
137C32×C4#23512yes—
138C32×C4D3×D412yes—
139C3⋊Dic3—42no—
140C3×Dic3—34no—
141C3×Dic3—62no—
142C3×Dic3—64no—
143C3×Dic3—64no—
144C3×Dic3—62no—
145C3×Dic3—121no—
146C3×Dic3—122no—
147C2×Q16—272no—
148C4⋊Q8—271no—
149C4×Q8—91no—
150C4⋊C8—271no—
151#103—92no—
152H(3,𝔽3)#46211yescharacteristic, complemented, Sylow
153C3×Q8—92no—
154C3×Q8—182no—
155C3×Q8—182no—
156C3×Q8—181no—
157C2×C3×C4—21no—
158C2×C3×C4—32no—
159C2×C3×C4—62no—
160C2×C3×C4—62no—
161C2×C3×C4—61no—
162C2×C3×C4—91no—
163C2×C3×C4—91no—
164C2×C3×C4—181no—
165C2×C3×C4—182no—
166C2×C3×C4C2×D3211yescharacteristic
167C2×Dic3—32no—
168C2×Dic3—32no—
169C2×Dic3—61no—
170C2×Dic3—61no—
171C2×Dic3—62no—
172C2×Dic3—62no—
173C2×Dic3—62no—
174C2×Dic3—182no—
175Dic6—34no—
176Dic6—62no—
177Dic6—62no—
178Dic6—62no—
179Dic6—62no—
180Dic6—62no—
181Dic6—182no—
182Dic6—182no—
183C3×C8—182no—
184C3⋊C8—62no—
185C3⋊C8—182no—
186C2×C32—21no—
187C2×C32—21no—
188C2×C32—21no—
189C2×C32#88412yes—
190C2×C32#91512yes—
191C2×C32D3×Q1612yes—
192C2×Q8—91no—
193C2×Q8—92no—
194Q16—544no—
195C2×C8—92no—
196C4⋊C4—181no—
197C4⋊C4—541no—
198C4⋊C4—541no—
199C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
200C42—91no—
201C42—181no—
202C22×C3—32no—
203C22×C3—61no—
204C22×C3#381111yescharacteristic
205C3×C4—32no—
206C3×C4—32no—
207C3×C4—41no—
208C3×C4—61no—
209C3×C4—61no—
210C3×C4—92no—
211C3×C4—122no—
212C3×C4—122no—
213C3×C4—182no—
214C3×C4—181no—
215C3×C4—362no—
216C3×C4#379811yescharacteristic
217C3×C4#380211yescharacteristic
218Dic3—34no—
219Dic3—62no—
220Dic3—64no—
221Dic3—62no—
222Dic3—64no—
223Dic3—121no—
224Dic3—122no—
225Dic3—362no—
226C32—21no—
227C32#479512yescomplemented
228Q8—92no—
229Q8—181no—
230Q8—182no—
231Q8—182no—
232C2×C4—21no—
233C2×C4—91no—
234C2×C4—91no—
235C2×C4—181no—
236C2×C4—182no—
237C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
238C8—182no—
239C2×C3—32no—
240C2×C3—32no—
241C2×C3—32no—
242C2×C3—61no—
243C2×C3—61no—
244C2×C3—61no—
245C2×C3#6391211yescharacteristic
246C2×C3#6393111yescharacteristic
247C2×C3#6394411yescharacteristic
248C22#8867711yescharacteristic
249C4—41no—
250C4—92no—
251C4—182no—
252C4—181no—
253C4—362no—
254C4#8865011yescharacteristic
255C4#8865511yescharacteristic
256C3—32no—
257C3—61no—
258C3#1059056611yescharacteristic
259C2#1172114211yescharacteristic
260C2#1172118911yescharacteristic
261C2#1172122611yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#462
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
62. #1084Q16semidirect
152. H(3,𝔽3)#462semidirect
227. C32#4795semidirect

Maximal subgroups

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

subgroupindex
#117217552
#117210992
C2×#884392
#63637⋊C32
#117203042
#105905613