Finite Group Databasebeta

← #13143939

Subgroups — #13143939

Order 1728.

528 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 97 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#11721846C211yesmaximal, characteristic
2#11721758C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×#102C211yesmaximal, characteristic
4#11720578C211yesmaximal, characteristic
5#11720549C211yesmaximal, characteristic, complemented
6C8×#1084C211yesmaximal, characteristic, complemented
7#11720532C211yesmaximal, characteristic, complemented
8C3×#4768—34nomaximal
9C2×D4×H(3,𝔽3)C2211yescharacteristic
10C2×#6229C2211yescharacteristic
11C2×C8×H(3,𝔽3)C2211yescharacteristic
12C4⋊C4×H(3,𝔽3)C2211yescharacteristic
13#88552—21no—
14#88552—21no—
15C4⋊#1084—21no—
16C4⋊#1084C2211yescharacteristic
17#88540—21no—
18C4×#1084C2211yescharacteristic
19C2×#6180C2211yescharacteristic
20C3×#936—34no—
21C3×#890—34no—
22C32×#102D314yes—
23C3×#832—34no—
24C3×#816—34no—
25C3×C8×Dic3—34no—
26C3×#810—34no—
27C23×H(3,𝔽3)—21no—
28D4×H(3,𝔽3)—21no—
29D4×H(3,𝔽3)—21no—
30C2×C4×H(3,𝔽3)—21no—
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C2×#1084—21no—
33C2×#1084—21no—
34C2×#1084D411yescharacteristic
35C2×#1084D411yescharacteristic
36#6229—21no—
37#6229—21no—
38C8×H(3,𝔽3)—21no—
39#6180—21no—
40C3×#488—91no—
41#4768—34no—
42C2×C32×D4C2×D314yes—
43C2×C3×Dic6—34no—
44C2×C32×C8C2×D314yes—
45C32×C4⋊C4C2×D314yes—
46C3×#217—64no—
47C3×#217—64no—
48C3×C4⋊Dic3—34no—
49C3×C4⋊Dic3—64no—
50C3×Dic3⋊C4—64no—
51C3×C4×Dic3—34no—
52C2×C3×C3⋊C8—34no—
53C22×H(3,𝔽3)—41no—
54C22×H(3,𝔽3)—41no—
55C22×H(3,𝔽3)—41no—
56C22×H(3,𝔽3)C2×D411yescharacteristic
57C4×H(3,𝔽3)—41no—
58C4×H(3,𝔽3)#4111yescharacteristic
59C4×H(3,𝔽3)#4111yescharacteristic
60#1084—21no—
61#1084—21no—
62#1084—41no—
63#1084—41no—
64C3×#125—91no—
65C3×#124—91no—
66#936—34no—
67#890—34no—
68C3×#103—91no—
69C3×#103—91no—
70C3×#102—34no—
71C3×#102—91no—
72C3×#102C3⋊D311yescharacteristic
73C3×C4×C8—91no—
74#832—34no—
75#816—34no—
76C8×Dic3—34no—
77#810—34no—
78C23×C32—24no—
79C32×D4—24no—
80C32×D4—24no—
81C2×C32×C4—24no—
82C2×C32×C4C22×D314yes—
83C2×C3×Dic3—34no—
84C2×C3×Dic3—34no—
85C2×C3×Dic3—64no—
86C2×C3×Dic3—64no—
87C3×Dic6—64no—
88C3×Dic6—64no—
89C32×C8—24no—
90C3×C3⋊C8—64no—
91#488—91noSylow
92C2×H(3,𝔽3)—81no—
93C2×H(3,𝔽3)#12411yescharacteristic
94C2×H(3,𝔽3)#13511yescharacteristic
95C2×H(3,𝔽3)#13511yescharacteristic
96C2×C3×Q8—91no—
97C2×C3×D4—34no—
98C2×C3×D4C2×C3⋊D311yescharacteristic
99C2×Dic6—34no—
100C2×C3×C8—34no—
101C2×C3×C8—91no—
102C2×C3×C8C2×C3⋊D311yescharacteristic
103C3×C4⋊C4—34no—
104C3×C4⋊C4—91no—
105C3×C4⋊C4—181no—
106C3×C4⋊C4—181no—
107C3×C4⋊C4C2×C3⋊D311yescharacteristic
108C3×C22⋊C4—181no—
109C3×C22⋊C4—181no—
110C3×C42—91no—
111#217—64no—
112#217—64no—
113C4⋊Dic3—34no—
114C4⋊Dic3—64no—
115Dic3⋊C4—64no—
116C4×Dic3—34no—
117C2×C3⋊C8—34no—
118C22×C32—44no—
119C22×C32—44no—
120C22×C32—44no—
121C22×C32D3×D414yes—
122C32×C4—44no—
123C32×C4#23514yes—
124C32×C4#23714yes—
125C3×Dic3—64no—
126C3×Dic3—64no—
127C3×Dic3—124no—
128C3×Dic3—124no—
129#125—91no—
130#124—91no—
131#103—91no—
132#103—91no—
133#102—91no—
134#102#27211yescharacteristic
135C4×C8—91no—
136H(3,𝔽3)#48811yescharacteristic, complemented, Sylow
137C23×C3—21no—
138C23×C3—64no—
139C3×Q8—181no—
140C3×Q8—181no—
141C3×D4—21no—
142C3×D4—21no—
143C3×D4—64no—
144C3×D4—64no—
145C2×C3×C4—21no—
146C2×C3×C4—34no—
147C2×C3×C4—64no—
148C2×C3×C4—91no—
149C2×C3×C4—91no—
150C2×C3×C4—181no—
151C2×C3×C4—181no—
152C2×C3×C4C22×C3⋊D311yescharacteristic
153C2×Dic3—34no—
154C2×Dic3—34no—
155C2×Dic3—64no—
156C2×Dic3—64no—
157Dic6—64no—
158Dic6—64no—
159C3×C8—21no—
160C3×C8—64no—
161C3×C8—181no—
162C3⋊C8—64no—
163C2×C32—84no—
164C2×C32#88514yes—
165C2×C32#91214yes—
166C2×C32#91614yes—
167C2×Q8—91no—
168C2×D4C2×#27211yescharacteristic
169C2×C8—91no—
170C2×C8C2×#27211yescharacteristic
171C4⋊C4—91no—
172C4⋊C4—181no—
173C4⋊C4—181no—
174C4⋊C4C2×#27211yescharacteristic
175C22⋊C4—181no—
176C22⋊C4—181no—
177C42—91no—
178C22×C3—34no—
179C22×C3—41no—
180C22×C3—41no—
181C22×C3—41no—
182C22×C3—124no—
183C22×C3—124no—
184C22×C3—124no—
185C22×C3D4×C3⋊D311yescharacteristic
186C3×C4—34no—
187C3×C4—34no—
188C3×C4—41no—
189C3×C4—124no—
190C3×C4—181no—
191C3×C4—181no—
192C3×C4—361no—
193C3×C4—361no—
194C3×C4#382811yescharacteristic
195C3×C4#383011yescharacteristic
196Dic3—64no—
197Dic3—64no—
198Dic3—124no—
199Dic3—124no—
200C32#476814yescomplemented
201C23—21no—
202Q8—181no—
203Q8—181no—
204D4—21no—
205D4—21no—
206C2×C4—21no—
207C2×C4—91no—
208C2×C4—91no—
209C2×C4—181no—
210C2×C4—181no—
211C2×C4C22×#27211yescharacteristic
212C8—21no—
213C8—181no—
214C2×C3—34no—
215C2×C3—34no—
216C2×C3—34no—
217C2×C3—81no—
218C2×C3—244no—
219C2×C3#6409411yescharacteristic
220C2×C3#6412111yescharacteristic
221C2×C3#6412511yescharacteristic
222C22—41no—
223C22—41no—
224C22—41no—
225C22D4×#27211yescharacteristic
226C4—41no—
227C4—181no—
228C4—181no—
229C4—361no—
230C4—361no—
231C4#8874011yescharacteristic
232C4#8874411yescharacteristic
233C3—34no—
234C3#1059246711yescharacteristic
235C2—81no—
236C2#1172174111yescharacteristic
237C2#1172179411yescharacteristic
238C2#1172180211yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#102C2
prime radicalC2×D4×H(3,𝔽3)C22
hypercenterC3×#102C3⋊D3
hypocenterH(3,𝔽3)#488
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#102

Complements

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

NH ≅ G/Ntype
2. #11721758C2semidirect
5. #11720549C2semidirect
6. C8×#1084C2semidirect
7. #11720532C2semidirect
136. H(3,𝔽3)#488semidirect
200. C32#4768semidirect

Maximal subgroups

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

subgroupindex
#117218462
#117217582
H(3,𝔽3)×#1022
#117205782
#117205492
C8×#10842
#117205322
C3×#47683