Finite Group Databasebeta

← #13144509

Subgroups — #13144509

Order 1728.

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

normal subgroups only

This group has 803 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#11722670C211yesmaximal, characteristic, complemented
2H(3,𝔽3)×#124C211yesmaximal, characteristic, complemented
3#11721842C212yesmaximal, complemented
4#11720584C211yesmaximal, characteristic, complemented
5#11720522C212yesmaximal, complemented
6C3×#5062—34nomaximal
7C2×Q8×H(3,𝔽3)C2211yescharacteristic
8C2×D4×H(3,𝔽3)C2211yescharacteristic
9#88748—22no—
10Q8×#272—22no—
11#88740—22no—
12#88740C2212yes—
13C22⋊C4×H(3,𝔽3)—22no—
14C42×H(3,𝔽3)C2211yescharacteristic
15#88546—22no—
16#88545—22no—
17#88538C2212yes—
18C3×#1007—34no—
19C32×#124D314yescomplemented
20C3×#932—38no—
21C3×#836—34no—
22C3×#805—38no—
23C23×H(3,𝔽3)—21no—
24Q8×H(3,𝔽3)—22no—
25D4×H(3,𝔽3)—22no—
26C2×C4×H(3,𝔽3)—21no—
27C2×C4×H(3,𝔽3)D412yes—
28C2×C4×H(3,𝔽3)C2311yescharacteristic
29#6235—22no—
30C4⋊#272—22no—
31C4⋊#272D412yes—
32C4×#272—24no—
33C4×#272—22no—
34#6229—22no—
35#6229D412yes—
36#6180—22no—
37C3×#454—91no—
38#5062—34no—
39C2×C32×Q8C2×D314yes—
40C2×C32×D4C2×D314yes—
41C3×#239—68no—
42C3×D3×Q8—68no—
43C3×#235—38no—
44C3×#235—68no—
45C32×C22⋊C4—28no—
46C32×C42C2×D314yes—
47C3×#214—68no—
48C3×#213—68no—
49C3×#208—38no—
50C22×H(3,𝔽3)—42no—
51C22×H(3,𝔽3)—41no—
52C22×H(3,𝔽3)C2×D411yescharacteristic
53C2×#272—22no—
54C4×H(3,𝔽3)—22no—
55C4×H(3,𝔽3)—41no—
56C4×H(3,𝔽3)C2×D412yes—
57#1084—22no—
58C3×2−1+4—91no—
59#1007—34no—
60C3×AGL(1,ℤ/8)—92no—
61C3×#124—34no—
62C3×#124C3⋊D311yescharacteristic
63#932—38no—
64C3×#104—92no—
65C3×#101—91no—
66#836—34no—
67#805—38no—
68C23×C32—24no—
69C32×Q8—28no—
70C32×D4—28no—
71C2×C32×C4—24no—
72C2×C32×C4#6718yes—
73C2×C32×C4C22×D314yes—
74C3×#67—68no—
75C3×D12—38no—
76C3×D12—68no—
77C3×C4×D3—616no—
78C3×C4×D3—68no—
79C3×Dic6—38no—
80C3×Dic6—68no—
81C3×C3⋊C8—68no—
82#454—91noSylow
83C2×H(3,𝔽3)—21no—
84C2×H(3,𝔽3)—81no—
85C2×H(3,𝔽3)#12011yescharacteristic
86#272—42no—
87C3×#41—92no—
88C3×#41—182no—
89C3×#41—182no—
90C2×C3×Q8—34no—
91C2×C3×Q8—182no—
92C2×C3×Q8C2×C3⋊D311yescharacteristic
93C2×C3×D4—34no—
94C2×C3×D4C2×C3⋊D311yescharacteristic
95#239—68no—
96D3×Q8—68no—
97#235—38no—
98#235—68no—
99C3×SD16—182no—
100C3×D8—182no—
101C3×M16—92no—
102C3×C22⋊C4—22no—
103C3×C22⋊C4—68no—
104C3×C42—34no—
105C3×C42C2×C3⋊D311yescharacteristic
106#214—68no—
107#213—68no—
108#208—38no—
109C22×C32—48no—
110C22×C32—44no—
111C22×C32C2×#6714yes—
112C2×C3×D3—68no—
113C32×C4—28no—
114C32×C4—44no—
115C32×C4D3×D418yes—
116C3×Dic3—68no—
1172−1+4—91no—
118AGL(1,ℤ/8)—92no—
119#124#27211yescharacteristic, complemented
120#104—92no—
121#101—91no—
122H(3,𝔽3)#45411yescharacteristic, complemented, Sylow
123C23×C3—21no—
124C23×C3—64no—
125C3×Q8—22no—
126C3×Q8—68no—
127C3×Q8—92no—
128C3×Q8—182no—
129C3×D4—22no—
130C3×D4—68no—
131C3×D4—92no—
132C3×D4—182no—
133C3×D4—182no—
134C2×C3×C4—21no—
135C2×C3×C4—38no—
136C2×C3×C4—34no—
137C2×C3×C4—64no—
138C2×C3×C4—184no—
139C2×C3×C4—182no—
140C2×C3×C4#63812yes—
141C2×C3×C4C22×C3⋊D311yescharacteristic
142#67—68no—
143D12—38no—
144D12—68no—
145C4×D3—68no—
146C4×D3—616no—
147Dic6—38no—
148Dic6—68no—
149C3×C8—182no—
150C3⋊C8—68no—
151C2×C32—24no—
152C2×C32—84no—
153C2×C32#93714yes—
154C3×D3—128no—
155#41—92no—
156#41—182no—
157#41—182no—
158C2×Q8—182no—
159C2×Q8C2×#27211yescharacteristic
160C2×D4C2×#27211yescharacteristic
161SD16—182no—
162D8—182no—
163M16—92no—
164C22⋊C4—22no—
165C42C2×#27211yescharacteristic
166C22×C3—34no—
167C22×C3—42no—
168C22×C3—41no—
169C22×C3—124no—
170C22×C3—128no—
171C22×C3—182no—
172C22×C3C2×#63811yescharacteristic
173C2×D3—68no—
174C3×C4—22no—
175C3×C4—38no—
176C3×C4—41no—
177C3×C4—68no—
178C3×C4—124no—
179C3×C4—182no—
180C3×C4D4×C3⋊D312yes—
181Dic3—68no—
182C32#506214yescomplemented
183C23—21no—
184Q8—22no—
185Q8—92no—
186Q8—182no—
187D4—22no—
188D4—92no—
189D4—182no—
190D4—182no—
191C2×C4—21no—
192C2×C4—184no—
193C2×C4—182no—
194C2×C4#623512yes—
195C2×C4C22×#27211yescharacteristic
196C8—182no—
197C2×C3—21no—
198C2×C3—34no—
199C2×C3—64no—
200C2×C3—81no—
201C2×C3—244no—
202C2×C3—362no—
203C2×C3#6414611yescharacteristic
204D3—128no—
205C22—42no—
206C22—41no—
207C22—182no—
208C22C2×#623511yescharacteristic
209C4—22no—
210C4—41no—
211C4—182no—
212C4D4×#27212yes—
213C3—34no—
214C3#1059276111yescharacteristic
215C2—21no—
216C2—81no—
217C2—362no—
218C2#1172184711yescharacteristic
subgroupG/H
center, socleC2×C3#64146
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#124C2
hypercenterC3×#124C3⋊D3
hypocenterH(3,𝔽3)#454
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#124

Complements

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

NH ≅ G/Ntype
1. #11722670C2semidirect
2. H(3,𝔽3)×#124C2semidirect
3. #11721842C2semidirect
4. #11720584C2semidirect
5. #11720522C2semidirect
19. C32×#124D3semidirect
119. #124#272semidirect
122. H(3,𝔽3)#454semidirect
182. C32#5062semidirect

Maximal subgroups

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

subgroupindex
#117226702
H(3,𝔽3)×#1242
#117218422
#117205842
#117205222
C3×#50623