Finite Group Databasebeta

← #13143880

Subgroups — #13143880

Order 1728.

836 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 99 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#11722664C211yesmaximal, characteristic
2#11721846C211yesmaximal, characteristic, complemented
3#11721788C212yesmaximal, complemented
4H(3,𝔽3)×#100C211yesmaximal, characteristic, complemented
5#11720562C212yesmaximal, complemented
6C3×#4746—34nomaximal
7C2×D4×H(3,𝔽3)C2211yescharacteristic
8C2×#6235—22no—
9#88744—22no—
10D4×#272—22no—
11#88740C2212yes—
12C2×#6229C2211yescharacteristic, complemented
13M16×H(3,𝔽3)C2212yescomplemented
14#88552—22no—
15C4×#1084C2211yescharacteristic, complemented
16#88530—22no—
17#88528—22no—
18C3×#1004—34no—
19C3×#936—34no—
20C3×#909—38no—
21C32×#100D314yescomplemented
22C3×#825—38no—
23C23×H(3,𝔽3)D412yes—
24C22×#272—22no—
25D4×H(3,𝔽3)—22no—
26C2×C4×H(3,𝔽3)C2311yescharacteristic
27#6235—24no—
28#6235—22no—
29C2×#1084—22no—
30C2×#1084—21no—
31C2×#1084—21no—
32C4⋊#272D412yes—
33C4×#272—22no—
34#6229—22no—
35#6229D412yes—
36C8×H(3,𝔽3)—22no—
37C3×#455—91no—
38#4746—34no—
39C2×C32×D4C2×D314yes—
40C2×C3×#67—68no—
41C3×#237—68no—
42C3×D3×D4—68no—
43C3×#235—38no—
44C2×C3×Dic6—34no—
45C32×M16C2×D318yescomplemented
46C3×#217—68no—
47C3×C4×Dic3—34no—
48C3×Dic12—68no—
49C3×#204—68no—
50C22×H(3,𝔽3)—22no—
51C22×H(3,𝔽3)—42no—
52C22×H(3,𝔽3)C2×D411yescharacteristic
53C2×#272—22no—
54C2×#272—82no—
55C4×H(3,𝔽3)C2×D412yes—
56#1084—22no—
57#1084—41no—
58#1084—41no—
59C3×H(5,𝔽2)—91no—
60#1004—34no—
61C3×#137—92no—
62C3×#124—91no—
63#936—34no—
64#909—38no—
65C3×#104—92no—
66C3×#100—34no—
67C3×#100C3⋊D311yescharacteristic
68#825—38no—
69C23×C32D1218yes—
70C22×C3×D3—68no—
71C32×D4—28no—
72C2×C32×C4C22×D314yes—
73C3×#67—616no—
74C3×#67—68no—
75C2×C3×Dic3—68no—
76C2×C3×Dic3—64no—
77C2×C3×Dic3—64no—
78C3×D12—38no—
79C3×C4×D3—68no—
80C3×Dic6—38no—
81C3×Dic6—68no—
82C32×C8—28no—
83#455—91noSylow
84C2×H(3,𝔽3)—21no—
85C2×H(3,𝔽3)—42no—
86C2×H(3,𝔽3)#12011yescharacteristic
87#272—42no—
88C3×#41—92no—
89C3×#41—182no—
90C2×C3×Q8—91no—
91C2×C3×D4—34no—
92C2×C3×D4—182no—
93C2×C3×D4—182no—
94C2×C3×D4C2×C3⋊D311yescharacteristic
95C2×#67—68no—
96#237—68no—
97D3×D4—68no—
98#235—38no—
99C2×Dic6—34no—
100C3×Q16—182no—
101C3×SD16—182no—
102C3×M16—38no—
103C3×M16C2×C3⋊D312yes—
104C3×C22⋊C4—182no—
105C3×C42—91no—
106#217—68no—
107C4×Dic3—34no—
108Dic12—68no—
109#204—68no—
110C22×C32—28no—
111C22×C32—48no—
112C22×C32C2×D1214yes—
113C2×C3×D3—68no—
114C2×C3×D3—248no—
115C32×C4D3×D418yes—
116C3×Dic3—68no—
117C3×Dic3—124no—
118C3×Dic3—124no—
119H(5,𝔽2)—91no—
120#137—92no—
121#124—91no—
122#104—92no—
123#100#27211yescharacteristic, complemented
124H(3,𝔽3)#45511yescharacteristic, complemented, Sylow
125C23×C3—38no—
126C23×C3—182no—
127C23×C3C4⋊(C3⋊D3)12yes—
128C22×D3—68no—
129C3×Q8—92no—
130C3×Q8—182no—
131C3×D4—22no—
132C3×D4—68no—
133C3×D4—92no—
134C3×D4—184no—
135C3×D4—182no—
136C2×C3×C4—34no—
137C2×C3×C4—182no—
138C2×C3×C4—182no—
139C2×C3×C4—181no—
140C2×C3×C4—181no—
141C2×C3×C4C22×C3⋊D311yescharacteristic
142#67—68no—
143#67—616no—
144C2×Dic3—68no—
145C2×Dic3—64no—
146C2×Dic3—64no—
147D12—38no—
148C4×D3—68no—
149Dic6—38no—
150Dic6—68no—
151C3×C8—22no—
152C3×C8—68no—
153C2×C32—24no—
154C2×C32—48no—
155C2×C32#88214yes—
156C3×D3—128no—
157#41—92no—
158#41—182no—
159C2×Q8—91no—
160C2×D4—182no—
161C2×D4—182no—
162C2×D4C2×#27211yescharacteristic
163Q16—182no—
164SD16—182no—
165M16C2×#27212yescomplemented
166C22⋊C4—182no—
167C42—91no—
168C22×C3—22no—
169C22×C3—34no—
170C22×C3—42no—
171C22×C3—68no—
172C22×C3—128no—
173C22×C3—182no—
174C22×C3—722no—
175C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
176C2×D3—68no—
177C2×D3—248no—
178C3×C4—38no—
179C3×C4—182no—
180C3×C4—361no—
181C3×C4—361no—
182C3×C4D4×C3⋊D312yes—
183Dic3—68no—
184Dic3—124no—
185Dic3—124no—
186C32#474614yescomplemented
187C23—182no—
188C23C4⋊#27212yes—
189Q8—92no—
190Q8—182no—
191D4—22no—
192D4—92no—
193D4—184no—
194D4—182no—
195C2×C4—182no—
196C2×C4—182no—
197C2×C4—181no—
198C2×C4—181no—
199C2×C4C22×#27211yescharacteristic
200C8—22no—
201C2×C3—21no—
202C2×C3—34no—
203C2×C3—42no—
204C2×C3—64no—
205C2×C3—128no—
206C2×C3—362no—
207C2×C3#6409111yescharacteristic
208D3—128no—
209C22—22no—
210C22—42no—
211C22—182no—
212C22—722no—
213C22C2×C4⋊#27211yescharacteristic
214C4—182no—
215C4—361no—
216C4—361no—
217C4D4×#27212yes—
218C3—34no—
219C3#1059244511yescharacteristic
220C2—21no—
221C2—42no—
222C2—362no—
223C2#1172173811yescharacteristic
subgroupG/H
center, socleC2×C3#64091
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#100C2
prime radical#11722664C2
hypercenterC3×#100C3⋊D3
hypocenterH(3,𝔽3)#455
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#100

Complements

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

NH ≅ G/Ntype
2. #11721846C2semidirect
3. #11721788C2semidirect
4. H(3,𝔽3)×#100C2semidirect
5. #11720562C2semidirect
12. C2×#6229C22semidirect
13. M16×H(3,𝔽3)C22semidirect
15. C4×#1084C22semidirect
21. C32×#100D3semidirect
45. C32×M16C2×D3semidirect
123. #100#272semidirect
124. H(3,𝔽3)#455semidirect
165. M16C2×#272semidirect
186. C32#4746semidirect

Maximal subgroups

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

subgroupindex
#117226642
#117218462
#117217882
H(3,𝔽3)×#1002
#117205622
C3×#47463