Finite Group Databasebeta

← #13154232

Subgroups — #13154232

Order 1728.

1056 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 205 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
1H(3,𝔽3)×#125C211yesmaximal, characteristic, complemented
2#11721766C214yesmaximal
3#11721765C212yesmaximal
4#11721758C214yesmaximal, complemented
5#11721755C212yesmaximal, complemented
6#11721718C211yesmaximal, characteristic
7#11721715C211yesmaximal, characteristic, complemented
8C3×#5698—34nomaximal
9C2×C4×#272C2211yescharacteristic
10C2×#6229C2212yes—
11C4⋊C4×H(3,𝔽3)C2212yes—
12C4⋊C4×H(3,𝔽3)C2214yes—
13C42×H(3,𝔽3)C2211yescharacteristic
14#88542C2214yes—
15#88542C2212yes—
16C4⋊#1084C2214yes—
17C4⋊#1084C2214yes—
18#88540C2214yes—
19#88540C2212yes—
20C4×#1084C2211yescharacteristic
21C4×#1084C2214yes—
22C32×#125D314yescomplemented
23C3×#898—316no—
24C3×#897—38no—
25C3×#890—316no—
26C3×#887—38no—
27C3×#872—34no—
28C3×#869—34no—
29C22×#272C2311yescharacteristic
30C2×C4×H(3,𝔽3)C2314yes—
31C2×C4×H(3,𝔽3)C2312yes—
32C2×C4×H(3,𝔽3)C2311yescharacteristic
33C2×#1084C2314yes—
34C2×#1084C2311yescharacteristic
35C2×#1084C2312yes—
36C4×#272—22no—
37#6229—24no—
38C3×#541—91no—
39#5698—34no—
40C2×C3×C4×D3—34no—
41C2×C3×Dic6—38no—
42C32×C4⋊C4C2×D318yes—
43C32×C4⋊C4C2×D3116yes—
44C32×C42C2×D314yes—
45C3×(C2×D3)⋊C4—316no—
46C3×(C2×D3)⋊C4—38no—
47C3×C4⋊Dic3—316no—
48C3×C4⋊Dic3—316no—
49C3×Dic3⋊C4—316no—
50C3×Dic3⋊C4—38no—
51C3×C4×Dic3—34no—
52C3×C4×Dic3—316no—
53C22×H(3,𝔽3)C2411yescharacteristic
54C2×#272—21no—
55C2×#272—22no—
56C4×H(3,𝔽3)—24no—
57C4×H(3,𝔽3)—22no—
58C4×H(3,𝔽3)#4112yes—
59#1084—24no—
60#1084—21no—
61#1084—22no—
62C3×C4⋊Q8—91no—
63C3×#126—94no—
64C3×#125—34no—
65C3×#125—94no—
66C3×#125C3⋊D311yescharacteristic
67C3×C22⋊Q8—92no—
68C3×C4×Q8—92no—
69C3×#117—91no—
70#898—316no—
71#897—38no—
72#890—316no—
73#887—38no—
74#872—34no—
75#869—34no—
76C22×C3×D3—34no—
77C2×C32×C4C22×D3116yes—
78C2×C32×C4C22×D318yes—
79C2×C32×C4C22×D314yes—
80C2×C3×Dic3—38no—
81C2×C3×Dic3—316no—
82C2×C3×Dic3—34no—
83C3×C4×D3—68no—
84C3×Dic6—616no—
85#541—91noSylow
86C2×H(3,𝔽3)C2×#4111yescharacteristic
87C2×H(3,𝔽3)2−1+412yes—
88#272—41no—
89C2×C3×Q8—92no—
90C22×C3×C4—91no—
91C2×C4×D3—34no—
92C2×Dic6—38no—
93C3×C4⋊C4—38no—
94C3×C4⋊C4—316no—
95C3×C4⋊C4—92no—
96C3×C4⋊C4—94no—
97C3×C4⋊C4—94no—
98C3×C4⋊C4—94no—
99C3×C4⋊C4C2×C3⋊D312yes—
100C3×C4⋊C4C2×C3⋊D314yes—
101C3×C22⋊C4—92no—
102C3×C22⋊C4—94no—
103C3×C42—34no—
104C3×C42—91no—
105C3×C42—94no—
106C3×C42C2×C3⋊D311yescharacteristic
107(C2×D3)⋊C4—38no—
108(C2×D3)⋊C4—316no—
109C4⋊Dic3—316no—
110C4⋊Dic3—316no—
111Dic3⋊C4—38no—
112Dic3⋊C4—316no—
113C4×Dic3—34no—
114C4×Dic3—316no—
115C22×C32C23×D314yes—
116C2×C3×D3—64no—
117C2×C3×D3—68no—
118C32×C4—216no—
119C32×C4—28no—
120C32×C4#23918yes—
121C3×Dic3—616no—
122C3×Dic3—64no—
123C3×Dic3—68no—
124C4⋊Q8—91no—
125#126—94no—
126#125—94no—
127#125#27211yescharacteristic, complemented
128C22⋊Q8—92no—
129C4×Q8—92no—
130#117—91no—
131H(3,𝔽3)#54111yescharacteristic, complemented, Sylow
132C23×C3—91no—
133C22×D3—34no—
134C3×Q8—184no—
135C2×C3×C4—34no—
136C2×C3×C4—316no—
137C2×C3×C4—38no—
138C2×C3×C4—92no—
139C2×C3×C4—94no—
140C2×C3×C4—91no—
141C2×C3×C4—182no—
142C2×C3×C4C22×C3⋊D314yes—
143C2×C3×C4C22×C3⋊D312yes—
144C2×C3×C4C22×C3⋊D311yescharacteristic
145C2×Dic3—38no—
146C2×Dic3—316no—
147C2×Dic3—34no—
148C4×D3—68no—
149Dic6—616no—
150C2×C32C2×#23914yes—
151C2×C32#101018yes—
152C3×D3—124no—
153C2×Q8—92no—
154C22×C4—91no—
155C4⋊C4—92no—
156C4⋊C4—94no—
157C4⋊C4—94no—
158C4⋊C4—94no—
159C4⋊C4C2×#27212yes—
160C4⋊C4C2×#27214yes—
161C22⋊C4—92no—
162C22⋊C4—94no—
163C42—91no—
164C42—94no—
165C42C2×#27211yescharacteristic
166C22×C3—34no—
167C22×C3—181no—
168C22×C3—182no—
169C22×C3C23×C3⋊D311yescharacteristic
170C2×D3—64no—
171C2×D3—68no—
172C3×C4—24no—
173C3×C4—22no—
174C3×C4—38no—
175C3×C4—616no—
176C3×C4—68no—
177C3×C4—181no—
178C3×C4—184no—
179C3×C4—182no—
180C3×C4#383212yes—
181Dic3—616no—
182Dic3—64no—
183Dic3—68no—
184C32#569814yescomplemented
185C23—91no—
186Q8—184no—
187C2×C4—92no—
188C2×C4—94no—
189C2×C4—91no—
190C2×C4—182no—
191C2×C4C22×#27214yes—
192C2×C4C22×#27212yes—
193C2×C4C22×#27211yescharacteristic
194C2×C3—34no—
195C2×C3—38no—
196C2×C3—361no—
197C2×C3C2×#383211yescharacteristic
198C2×C3#6436712yes—
199D3—124no—
200C22—181no—
201C22—182no—
202C22C23×#27211yescharacteristic
203C4—24no—
204C4—22no—
205C4—181no—
206C4—184no—
207C4—182no—
208C4#8874812yes—
209C3—34no—
210C3#1059616911yescharacteristic
211C2—361no—
212C2C2×#8874811yescharacteristic
213C2#1172267612yes—
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FittingH(3,𝔽3)×#125C2
prime radicalC22×#272C23
hypercenterC3×#125C3⋊D3
hypocenterH(3,𝔽3)#541
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×#125

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#125C2semidirect
4. #11721758C2semidirect
5. #11721755C2semidirect
7. #11721715C2semidirect
22. C32×#125D3semidirect
127. #125#272semidirect
131. H(3,𝔽3)#541semidirect
184. C32#5698semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1252
#117217662
#117217652
#117217582
#117217552
#117217182
#117217152
C3×#56983