Finite Group Databasebeta

← #13143928

Subgroups — #13143928

Order 1728.

594 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 113 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
1D4×#1084C211yesmaximal, characteristic
2#11721756C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×#102C211yesmaximal, characteristic
4#11720578C211yesmaximal, characteristic
5#11720551C211yesmaximal, characteristic, complemented
6#11720547C211yesmaximal, characteristic, complemented
7#11720529C211yesmaximal, characteristic, complemented
8C3×#4759—34nomaximal
9C2×D4×H(3,𝔽3)C2211yescharacteristic
10C22×#1084—21no—
11C2×#6229—21no—
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14#88552—21no—
15C4⋊#1084C2211yescharacteristic
16C4⋊#1084C2211yescharacteristic
17#88540—21no—
18C4×#1084C2211yescharacteristic
19C2×#6180C2211yescharacteristic
20C3×D4×Dic3—34no—
21C3×#888—34no—
22C32×#102D314yes—
23C3×#832—34no—
24C3×#818—34no—
25C3×Dic3⋊C8—34no—
26C3×#807—34no—
27C23×H(3,𝔽3)—21no—
28D4×H(3,𝔽3)—21no—
29D4×H(3,𝔽3)Q812yescomplemented
30C2×C4×H(3,𝔽3)—21no—
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C2×#1084—21no—
33C2×#1084—21no—
34C2×#1084—42no—
35C2×#1084D411yescharacteristic
36C2×#1084D411yescharacteristic
37#6229—42no—
38C8×H(3,𝔽3)—21no—
39#6180—21no—
40C3×#474—91no—
41#4759—34no—
42C2×C32×D4C2×D314yes—
43C22×C3×Dic3—64no—
44C2×C3×Dic6—64no—
45C2×C32×C8C2×D314yes—
46C32×C4⋊C4C2×D314yes—
47C3×#217—64no—
48C3×C4⋊Dic3—34no—
49C3×C4⋊Dic3—34no—
50C3×Dic3⋊C4—64no—
51C3×C4×Dic3—34no—
52C2×C3×C3⋊C8—34no—
53C22×H(3,𝔽3)—22no—
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)C2×Q811yescharacteristic
59C4×H(3,𝔽3)#4111yescharacteristic
60#1084—21no—
61#1084—41no—
62#1084—41no—
63#1084D812yes—
64C3×C4⋊Q8—91no—
65C3×C4×D4—91no—
66D4×Dic3—34no—
67#888—34no—
68C3×#107—91no—
69C3×#107—91no—
70C3×C4⋊C8—91no—
71C3×#102—34no—
72C3×#102—91no—
73C3×#102C3⋊D311yescharacteristic
74#832—34no—
75#818—34no—
76Dic3⋊C8—34no—
77#807—34no—
78C23×C32—24no—
79C32×D4—24no—
80C32×D4Dic618yescomplemented
81C2×C32×C4—24no—
82C2×C32×C4C22×D314yes—
83C2×C3×Dic3—34no—
84C2×C3×Dic3—34no—
85C2×C3×Dic3—64no—
86C2×C3×Dic3—64no—
87C2×C3×Dic3—128no—
88C3×Dic6—128no—
89C32×C8—24no—
90C3×C3⋊C8—64no—
91#474—91noSylow
92C2×H(3,𝔽3)—42no—
93C2×H(3,𝔽3)C22⋊Q811yescharacteristic
94C2×H(3,𝔽3)C2×D811yescharacteristic
95C2×H(3,𝔽3)#13711yescharacteristic
96C2×C3×Q8—181no—
97C2×C3×D4—34no—
98C2×C3×D4C2×C3⋊D311yescharacteristic
99C22×C3×C4—181no—
100C22×Dic3—64no—
101C2×Dic6—64no—
102C2×C3×C8—34no—
103C2×C3×C8—91no—
104C2×C3×C8C2×C3⋊D311yescharacteristic
105C3×C4⋊C4—34no—
106C3×C4⋊C4—91no—
107C3×C4⋊C4—91no—
108C3×C4⋊C4—181no—
109C3×C4⋊C4C2×C3⋊D311yescharacteristic
110C3×C22⋊C4—181no—
111C3×C42—91no—
112#217—64no—
113C4⋊Dic3—34no—
114C4⋊Dic3—34no—
115Dic3⋊C4—64no—
116C4×Dic3—34no—
117C2×C3⋊C8—34no—
118C22×C32—28no—
119C22×C32—44no—
120C22×C32—44no—
121C22×C32D3×D414yes—
122C32×C4—44no—
123C32×C4C2×Dic614yes—
124C32×C4#23714yes—
125C3×Dic3—38no—
126C3×Dic3—64no—
127C3×Dic3—124no—
128C3×Dic3—124no—
129C4⋊Q8—91no—
130C4×D4—91no—
131#107—91no—
132#107—91no—
133C4⋊C8—91no—
134#102—91no—
135#102#27211yescharacteristic
136H(3,𝔽3)#47411yescharacteristic, complemented, Sylow
137C23×C3—21no—
138C23×C3—64no—
139C3×Q8—362no—
140C3×D4—21no—
141C3×D4—38no—
142C3×D4—64no—
143C3×D4#63412yes—
144C2×C3×C4—21no—
145C2×C3×C4—34no—
146C2×C3×C4—64no—
147C2×C3×C4—91no—
148C2×C3×C4—91no—
149C2×C3×C4—181no—
150C2×C3×C4—181no—
151C2×C3×C4—362no—
152C2×C3×C4C22×C3⋊D311yescharacteristic
153C2×Dic3—34no—
154C2×Dic3—34no—
155C2×Dic3—64no—
156C2×Dic3—64no—
157C2×Dic3—128no—
158Dic6—128no—
159C3×C8—21no—
160C3×C8—64no—
161C3×C8—181no—
162C3⋊C8—64no—
163C2×C32—48no—
164C2×C32#87814yes—
165C2×C32D3×D814yes—
166C2×C32#91514yes—
167C2×Q8—181no—
168C2×D4C2×#27211yescharacteristic
169C22×C4—181no—
170C2×C8—91no—
171C2×C8C2×#27211yescharacteristic
172C4⋊C4—91no—
173C4⋊C4—91no—
174C4⋊C4—181no—
175C4⋊C4C2×#27211yescharacteristic
176C22⋊C4—181no—
177C42—91no—
178C22×C3—22no—
179C22×C3—34no—
180C22×C3—41no—
181C22×C3—41no—
182C22×C3—68no—
183C22×C3—124no—
184C22×C3—124no—
185C22×C3D4×C3⋊D311yescharacteristic
186C3×C4—34no—
187C3×C4—34no—
188C3×C4—41no—
189C3×C4—92no—
190C3×C4—124no—
191C3×C4—181no—
192C3×C4—361no—
193C3×C4—361no—
194C3×C4C2×#63411yescharacteristic
195C3×C4#383011yescharacteristic
196Dic3—38no—
197Dic3—64no—
198Dic3—124no—
199Dic3—124no—
200C32#475914yescomplemented
201C23—21no—
202Q8—362no—
203D4—21no—
204D4#622912yescomplemented
205C2×C4—21no—
206C2×C4—91no—
207C2×C4—91no—
208C2×C4—181no—
209C2×C4—181no—
210C2×C4—362no—
211C2×C4C22×#27211yescharacteristic
212C8—21no—
213C8—181no—
214C2×C3—34no—
215C2×C3—34no—
216C2×C3—34no—
217C2×C3—42no—
218C2×C3—128no—
219C2×C3#6408711yescharacteristic
220C2×C3D8×C3⋊D311yescharacteristic
221C2×C3#6412411yescharacteristic
222C22—22no—
223C22—41no—
224C22—41no—
225C22D4×#27211yescharacteristic
226C4—41no—
227C4—92no—
228C4—181no—
229C4—361no—
230C4—361no—
231C4C2×#622911yescharacteristic
232C4#8874411yescharacteristic
233C3—34no—
234C3#1059245811yescharacteristic
235C2—42no—
236C2#1172173411yescharacteristic
237C2D8×#27211yescharacteristic
238C2#1172180111yescharacteristic
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)#474
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. #11721756C2semidirect
5. #11720551C2semidirect
6. #11720547C2semidirect
7. #11720529C2semidirect
29. D4×H(3,𝔽3)Q8semidirect
80. C32×D4Dic6semidirect
136. H(3,𝔽3)#474semidirect
200. C32#4759semidirect
204. D4#6229semidirect

Maximal subgroups

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

subgroupindex
D4×#10842
#117217562
H(3,𝔽3)×#1022
#117205782
#117205512
#117205472
#117205292
C3×#47593