Finite Group Databasebeta

← #13143893

Subgroups — #13143893

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
2#11721864C211yesmaximal, characteristic, complemented
3#11721787C212yesmaximal, complemented
4H(3,𝔽3)×#101C211yesmaximal, characteristic, complemented
5#11720562C212yesmaximal, complemented
6C3×#4753—34nomaximal
7C2×Q8×H(3,𝔽3)C2211yescharacteristic
8#88748—22no—
9Q8×#272—22no—
10#88740—22no—
11#88740C2212yes—
12C2×C4⋊#272C2211yescharacteristic
13M16×H(3,𝔽3)C2212yes—
14#88542—22no—
15C4×#1084C2211yescharacteristic
16#88529—22no—
17#88528—22no—
18C3×#1007—34no—
19C3×#947—34no—
20C3×#908—38no—
21C32×#101D314yescomplemented
22C3×#825—38no—
23C22×#272—21no—
24Q8×H(3,𝔽3)—22no—
25C2×C4×H(3,𝔽3)D412yes—
26C2×C4×H(3,𝔽3)C2311yescharacteristic
27#6235—22no—
28C2×#1084—21no—
29C4⋊#272—22no—
30C4⋊#272—22no—
31C4⋊#272D412yes—
32C4×#272—24no—
33C4×#272—22no—
34#6229—22no—
35#6229D412yes—
36C8×H(3,𝔽3)—22no—
37C3×#454—91no—
38#4753—34no—
39C2×C32×Q8C2×D314yes—
40C3×#239—68no—
41C3×D3×Q8—68no—
42C3×#235—38no—
43C3×#235—68no—
44C2×C3×D12—34no—
45C32×M16C2×D318yes—
46C3×(C2×D3)⋊C4—68no—
47C3×C4×Dic3—34no—
48C3×D24—68no—
49C3×#204—68no—
50C22×H(3,𝔽3)C2×D411yescharacteristic
51C2×#272—22no—
52C2×#272—42no—
53C2×#272—41no—
54C4×H(3,𝔽3)—22no—
55C4×H(3,𝔽3)C2×D412yes—
56#1084—22no—
57#1084—41no—
58C3×2−1+4—91no—
59#1007—34no—
60C3×AGL(1,ℤ/8)—92no—
61C3×#124—91no—
62#947—34no—
63#908—38no—
64C3×#104—92no—
65C3×#101—34no—
66C3×#101C3⋊D311yescharacteristic
67#825—38no—
68C22×C3×D3—64no—
69C32×Q8—28no—
70C2×C32×C4D1218yes—
71C2×C32×C4C22×D314yes—
72C3×#67—68no—
73C2×C3×Dic3—64no—
74C3×D12—38no—
75C3×D12—68no—
76C3×D12—68no—
77C3×C4×D3—616no—
78C3×C4×D3—68no—
79C3×Dic6—38no—
80C3×Dic6—68no—
81C32×C8—28no—
82#454—91noSylow
83C2×H(3,𝔽3)—21no—
84C2×H(3,𝔽3)#12011yescharacteristic
85#272—42no—
86#272—81no—
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—91no—
94#239—68no—
95D3×Q8—68no—
96#235—38no—
97#235—68no—
98C2×D12—34no—
99C3×SD16—182no—
100C3×D8—182no—
101C3×M16—38no—
102C3×M16C2×C3⋊D312yes—
103C3×C22⋊C4—182no—
104C3×C42—91no—
105(C2×D3)⋊C4—68no—
106C4×Dic3—34no—
107D24—68no—
108#204—68no—
109C22×C32C2×D1214yes—
110C2×C3×D3—68no—
111C2×C3×D3—128no—
112C2×C3×D3—124no—
113C32×C4—28no—
114C32×C4D3×D418yes—
115C3×Dic3—68no—
116C3×Dic3—124no—
1172−1+4—91no—
118AGL(1,ℤ/8)—92no—
119#124—91no—
120#104—92no—
121#101#27211yescharacteristic, complemented
122H(3,𝔽3)#45411yescharacteristic, complemented, Sylow
123C23×C3—181no—
124C22×D3—64no—
125C3×Q8—22no—
126C3×Q8—68no—
127C3×Q8—92no—
128C3×Q8—182no—
129C3×D4—92no—
130C3×D4—182no—
131C3×D4—182no—
132C3×D4—182no—
133C2×C3×C4—38no—
134C2×C3×C4—34no—
135C2×C3×C4—184no—
136C2×C3×C4—182no—
137C2×C3×C4—181no—
138C2×C3×C4C4⋊(C3⋊D3)12yes—
139C2×C3×C4C22×C3⋊D311yescharacteristic
140#67—68no—
141C2×Dic3—64no—
142D12—38no—
143D12—68no—
144D12—68no—
145C4×D3—68no—
146C4×D3—616no—
147Dic6—38no—
148Dic6—68no—
149C3×C8—22no—
150C3×C8—68no—
151C2×C32—24no—
152C2×C32#88214yes—
153C3×D3—128no—
154C3×D3—244no—
155#41—92no—
156#41—182no—
157#41—182no—
158C2×Q8—182no—
159C2×Q8C2×#27211yescharacteristic
160C2×D4—91no—
161SD16—182no—
162D8—182no—
163M16C2×#27212yes—
164C22⋊C4—182no—
165C42—91no—
166C22×C3—34no—
167C22×C3—182no—
168C22×C3—362no—
169C22×C3—361no—
170C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
171C2×D3—68no—
172C2×D3—124no—
173C2×D3—128no—
174C3×C4—22no—
175C3×C4—38no—
176C3×C4—68no—
177C3×C4—182no—
178C3×C4—361no—
179C3×C4D4×C3⋊D312yes—
180Dic3—68no—
181Dic3—124no—
182C32#475314yescomplemented
183C23—181no—
184Q8—22no—
185Q8—92no—
186Q8—182no—
187D4—92no—
188D4—182no—
189D4—182no—
190D4—182no—
191C2×C4—184no—
192C2×C4—182no—
193C2×C4—181no—
194C2×C4C4⋊#27212yes—
195C2×C4C22×#27211yescharacteristic
196C8—22no—
197C2×C3—21no—
198C2×C3—34no—
199C2×C3—64no—
200C2×C3—362no—
201C2×C3—721no—
202C2×C3#6409111yescharacteristic
203D3—128no—
204D3—244no—
205C22—182no—
206C22—362no—
207C22—361no—
208C22C2×C4⋊#27211yescharacteristic
209C4—22no—
210C4—182no—
211C4—361no—
212C4D4×#27212yes—
213C3—34no—
214C3#1059245211yescharacteristic
215C2—21no—
216C2—362no—
217C2—721no—
218C2#1172173811yescharacteristic
subgroupG/H
center, socleC2×C3#64091
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#101C2
hypercenterC3×#101C3⋊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×#101

Complements

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

NH ≅ G/Ntype
1. #11722670C2semidirect
2. #11721864C2semidirect
3. #11721787C2semidirect
4. H(3,𝔽3)×#101C2semidirect
5. #11720562C2semidirect
21. C32×#101D3semidirect
121. #101#272semidirect
122. H(3,𝔽3)#454semidirect
182. C32#4753semidirect

Maximal subgroups

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

subgroupindex
#117226702
#117218642
#117217872
H(3,𝔽3)×#1012
#117205622
C3×#47533