Finite Group Databasebeta

← #13137875

Subgroups — #13137875

Order 1728.

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

normal subgroups only

This group has 411 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#11721756C211yesmaximal, characteristic
2#11721099C211yesmaximal, characteristic
3#63637⋊C3C212yesmaximal
4#11720309C212yesmaximal
5#11720302C211yesmaximal, characteristic
6#10590560—32nomaximal
7C2×#6229—21no—
8C2×#634⋊C3C2212yes—
9C4⋊C4×H(3,𝔽3)C2211yescharacteristic
10C4⋊#1084C2211yescharacteristic
11#88540—21no—
12C4×#1084C2211yescharacteristic
13C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
14#88449—21no—
15#88449—21no—
16C3×#888—32no—
17C3×#888—61no—
18#63866—32no—
19#63637D312yes—
20C3×#810—32no—
21#63567—32no—
22#63565—32no—
23Dic3×C3⋊C8—32no—
24C2×C4×H(3,𝔽3)—21no—
25C2×C4×H(3,𝔽3)C2311yescharacteristic
26C2×#1084—21no—
27C2×#1084D411yescharacteristic
28C2×#1084D411yescharacteristic
29#6229—42no—
30C2×(C3⋊Dic3)⋊C3—22no—
31#634⋊C3—22no—
32#634⋊C3—22no—
33(C3⋊(C3⋊C8))⋊C3—22no—
34#5093—91no—
35#4792—92no—
36C2×#634C2×D312yes—
37C2×C3×Dic6—32no—
38C2×C3×Dic6—62no—
39C2×C3×Dic6—62no—
40C32×C4⋊C4—21no—
41C32×C4⋊C4C2×D312yes—
42C2×C3⋊(C3⋊C8)C2×D312yes—
43C3×C4⋊Dic3—32no—
44C3×C4⋊Dic3—61no—
45C3×Dic3⋊C4—62no—
46C3×Dic3⋊C4—62no—
47C3×C4×Dic3—32no—
48C3×C4×Dic3—61no—
49C2×C3×C3⋊C8—32no—
50Dic3⋊Dic3—62no—
51Dic3⋊Dic3—62no—
52C22×H(3,𝔽3)C2×D411yescharacteristic
53C4×H(3,𝔽3)—41no—
54C4×H(3,𝔽3)#4112yes—
55#1084—41no—
56#1084SD1612yes—
57#1084Q1612yes—
58(C3⋊Dic3)⋊C3—42no—
59C3×C4⋊Q8—91no—
60#944—92no—
61#888—32no—
62#888—61no—
63#869—91no—
64C3×#103—92no—
65#835—92no—
66#816—92no—
67C8×Dic3—92no—
68#810—32no—
69#810—92no—
70#810—92no—
71C4×C3⋊C8—91no—
72C2×C32×C4—22no—
73C2×C32×C4—22no—
74C2×C32×C4—21no—
75C2×C32×C4C22×D312yes—
76C2×C3⋊Dic3—22no—
77#634—22no—
78#634—22no—
79C2×C3×Dic3—32no—
80C2×C3×Dic3—32no—
81C2×C3×Dic3—61no—
82C2×C3×Dic3—61no—
83C2×C3×Dic3—62no—
84C2×C3×Dic3—62no—
85C2×C3×Dic3—62no—
86C3×Dic6—62no—
87C3×Dic6—62no—
88C3×Dic6—124no—
89C3×Dic6—124no—
90C3⋊(C3⋊C8)—22no—
91C3×C3⋊C8—62no—
92#487—271noSylow
93C2×H(3,𝔽3)#12411yescharacteristic
94C2×H(3,𝔽3)C2×SD1611yescharacteristic
95C2×H(3,𝔽3)C2×Q1611yescharacteristic
96C2×C3×Q8—92no—
97C2×C3×Q8—181no—
98C2×Dic6—32no—
99C2×Dic6—62no—
100C2×Dic6—62no—
101C2×Dic6—92no—
102C2×C3×C8—92no—
103C3×C4⋊C4—32no—
104C3×C4⋊C4—61no—
105C3×C4⋊C4—91no—
106C3×C4⋊C4—181no—
107C3×C4⋊C4D3211yescharacteristic
108C3×C42—91no—
109C4⋊Dic3—32no—
110C4⋊Dic3—61no—
111C4⋊Dic3—181no—
112C4⋊Dic3—181no—
113Dic3⋊C4—62no—
114Dic3⋊C4—62no—
115Dic3⋊C4—182no—
116Dic3⋊C4—182no—
117C4×Dic3—32no—
118C4×Dic3—61no—
119C2×C3⋊C8—32no—
120C2×C3⋊C8—92no—
121C22×C32—21no—
122C22×C32D3×D412yes—
123C32×C4—22no—
124C32×C4—42no—
125C32×C4—42no—
126C32×C4#23512yes—
127C32×C4#23712yes—
128C3⋊Dic3—42no—
129C3×Dic3—34no—
130C3×Dic3—34no—
131C3×Dic3—62no—
132C3×Dic3—62no—
133C3×Dic3—122no—
134C3×Dic3—122no—
135C3×Dic3—122no—
136C4⋊Q8—91no—
137C4⋊Q8—271no—
138#103—92no—
139#103—272no—
140C4×C8—271no—
141H(3,𝔽3)#48711yescharacteristic, complemented, Sylow
142C3×Q8—182no—
143C3×Q8—182no—
144C3×Q8—362no—
145C2×C3×C4—21no—
146C2×C3×C4—32no—
147C2×C3×C4—62no—
148C2×C3×C4—62no—
149C2×C3×C4—61no—
150C2×C3×C4—91no—
151C2×C3×C4—91no—
152C2×C3×C4—182no—
153C2×C3×C4—181no—
154C2×C3×C4C2×D3211yescharacteristic
155C2×Dic3—32no—
156C2×Dic3—32no—
157C2×Dic3—61no—
158C2×Dic3—61no—
159C2×Dic3—62no—
160C2×Dic3—62no—
161C2×Dic3—62no—
162C2×Dic3—182no—
163Dic6—62no—
164Dic6—62no—
165Dic6—124no—
166Dic6—124no—
167Dic6—182no—
168Dic6—182no—
169C3×C8—182no—
170C3⋊C8—62no—
171C3⋊C8—182no—
172C2×C32—21no—
173C2×C32—21no—
174C2×C32—21no—
175C2×C32#88512yes—
176C2×C32D3×SD1612yes—
177C2×C32D3×Q1612yes—
178C2×Q8—92no—
179C2×Q8—181no—
180C2×C8—92no—
181C4⋊C4—91no—
182C4⋊C4—181no—
183C4⋊C4—541no—
184C4⋊C4—541no—
185C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
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—92no—
195C3×C4—92no—
196C3×C4—122no—
197C3×C4—122no—
198C3×C4—362no—
199C3×C4—361no—
200C3×C4#379512yes—
201Dic3—34no—
202Dic3—34no—
203Dic3—62no—
204Dic3—62no—
205Dic3—122no—
206Dic3—122no—
207Dic3—122no—
208Dic3—362no—
209C32—21no—
210C32#479212yescomplemented
211Q8—182no—
212Q8—182no—
213Q8—362no—
214C2×C4—21no—
215C2×C4—91no—
216C2×C4—91no—
217C2×C4—181no—
218C2×C4—182no—
219C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
220C8—182no—
221C2×C3—32no—
222C2×C3—32no—
223C2×C3—32no—
224C2×C3—61no—
225C2×C3—61no—
226C2×C3—61no—
227C2×C3#6391311yescharacteristic
228C2×C3#6393011yescharacteristic
229C2×C3#6394411yescharacteristic
230C22#8867711yescharacteristic
231C4—41no—
232C4—92no—
233C4—92no—
234C4—362no—
235C4—361no—
236C4#8864812yes—
237C3—32no—
238C3—61no—
239C3#1059056711yescharacteristic
240C2#1172114311yescharacteristic
241C2#1172118811yescharacteristic
242C2#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)#487
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
141. H(3,𝔽3)#487semidirect
210. C32#4792semidirect

Maximal subgroups

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

subgroupindex
#117217562
#117210992
#63637⋊C32
#117203092
#117203022
#105905603