Finite Group Databasebeta

← #13144872

Subgroups — #13144872

Order 1728.

836 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 123 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)×H(5,𝔽2)C211yesmaximal, characteristic
2#11721872C212yesmaximal, complemented
3#11721846C211yesmaximal, characteristic, complemented
4#11720586C212yesmaximal, complemented
5#11720579C211yesmaximal, characteristic, complemented
6C3×#5242—34nomaximal
7#41×H(3,𝔽3)—22no—
8#41×H(3,𝔽3)C2212yes—
9C2×D4×H(3,𝔽3)—22no—
10C2×D4×H(3,𝔽3)—22no—
11C2×D4×H(3,𝔽3)C2211yescharacteristic
12C2×#6229C2211yescharacteristic, complemented
13#88552—22no—
14#88550—22no—
15#88546—22no—
16C4×#1084C2211yescharacteristic, complemented
17#88538C2212yescomplemented
18C32×H(5,𝔽2)D314yes—
19C3×#951—38no—
20C3×#936—34no—
21C3×#837—38no—
22C3×#833—34no—
23C23×H(3,𝔽3)—22no—
24C23×H(3,𝔽3)D412yes—
25Q8×H(3,𝔽3)D412yes—
26D4×H(3,𝔽3)—24no—
27D4×H(3,𝔽3)—22no—
28D4×H(3,𝔽3)—22no—
29D4×H(3,𝔽3)D412yes—
30C2×C4×H(3,𝔽3)—22no—
31C2×C4×H(3,𝔽3)—22no—
32C2×C4×H(3,𝔽3)C2311yescharacteristic
33C2×#1084—21no—
34C2×#1084—21no—
35#6229—22no—
36#6180—22no—
37C3×#455—91no—
38#5242—34no—
39C32×#41—28no—
40C32×#41C2×D318yes—
41C2×C32×D4—28no—
42C2×C32×D4—28no—
43C2×C32×D4C2×D314yes—
44C2×C3×Dic6—34no—
45C3×#217—68no—
46C3×#216—68no—
47C3×#214—68no—
48C3×C4×Dic3—34no—
49C3×#208—38no—
50C22×H(3,𝔽3)—22no—
51C22×H(3,𝔽3)—22no—
52C22×H(3,𝔽3)—42no—
53C22×H(3,𝔽3)—82no—
54C22×H(3,𝔽3)C2×D411yescharacteristic
55C4×H(3,𝔽3)—22no—
56C4×H(3,𝔽3)C2×D412yes—
57#1084—41no—
58#1084—41no—
59C3×H(5,𝔽2)—34no—
60C3×H(5,𝔽2)C3⋊D311yescharacteristic
61C3×#137—92no—
62C3×#124—91no—
63#951—38no—
64#936—34no—
65C3×#104—92no—
66C3×#100—91no—
67#837—38no—
68#833—34no—
69C23×C32—28no—
70C23×C32#6718yes—
71C32×Q8#6718yes—
72C32×D4—216no—
73C32×D4—28no—
74C32×D4—28no—
75C32×D4#6718yes—
76C2×C32×C4—28no—
77C2×C32×C4—28no—
78C2×C32×C4C22×D314yes—
79C2×C3×Dic3—64no—
80C2×C3×Dic3—64no—
81C3×Dic6—68no—
82C3×C3⋊C8—68no—
83#455—91noSylow
84C2×H(3,𝔽3)—21no—
85C2×H(3,𝔽3)—42no—
86C2×H(3,𝔽3)—42no—
87C2×H(3,𝔽3)#12011yescharacteristic
88C3×#41—22no—
89C3×#41—38no—
90C3×#41—68no—
91C3×#41C2×C3⋊D312yes—
92C2×C3×Q8—91no—
93C2×C3×D4—22no—
94C2×C3×D4—22no—
95C2×C3×D4—34no—
96C2×C3×D4—68no—
97C2×C3×D4—68no—
98C2×C3×D4C2×C3⋊D311yescharacteristic
99C2×Dic6—34no—
100C3×Q16—182no—
101C3×SD16—182no—
102C3×M16—92no—
103C3×C22⋊C4—182no—
104C3×C42—91no—
105#217—68no—
106#216—68no—
107#214—68no—
108C4×Dic3—34no—
109#208—38no—
110C22×C32—28no—
111C22×C32—28no—
112C22×C32—48no—
113C22×C32—88no—
114C22×C32C2×#6714yes—
115C32×C4—28no—
116C32×C4C2×#6718yes—
117C3×Dic3—124no—
118C3×Dic3—124no—
119H(5,𝔽2)#27211yescharacteristic
120#137—92no—
121#124—91no—
122#104—92no—
123#100—91no—
124H(3,𝔽3)#45511yescharacteristic, complemented, Sylow
125C23×C3—22no—
126C23×C3—38no—
127C23×C3—68no—
128C23×C3#63812yes—
129C3×Q8—38no—
130C3×Q8—182no—
131C3×Q8#63812yes—
132C3×D4—24no—
133C3×D4—22no—
134C3×D4—22no—
135C3×D4—38no—
136C3×D4—68no—
137C3×D4—68no—
138C3×D4—616no—
139C3×D4#63812yes—
140C2×C3×C4—22no—
141C2×C3×C4—22no—
142C2×C3×C4—34no—
143C2×C3×C4—68no—
144C2×C3×C4—68no—
145C2×C3×C4—181no—
146C2×C3×C4—181no—
147C2×C3×C4C22×C3⋊D311yescharacteristic
148C2×Dic3—64no—
149C2×Dic3—64no—
150Dic6—68no—
151C3×C8—182no—
152C3⋊C8—68no—
153C2×C32—24no—
154C2×C32—48no—
155C2×C32—48no—
156C2×C32#95314yes—
157#41—22no—
158#41C2×#27212yes—
159C2×Q8—91no—
160C2×D4—22no—
161C2×D4—22no—
162C2×D4C2×#27211yescharacteristic
163Q16—182no—
164SD16—182no—
165M16—92no—
166C22⋊C4—182no—
167C42—91no—
168C22×C3—22no—
169C22×C3—22no—
170C22×C3—34no—
171C22×C3—42no—
172C22×C3—68no—
173C22×C3—68no—
174C22×C3—82no—
175C22×C3—128no—
176C22×C3—248no—
177C22×C3C2×#63811yescharacteristic
178C3×C4—22no—
179C3×C4—38no—
180C3×C4—68no—
181C3×C4—361no—
182C3×C4—361no—
183C3×C4C2×#63812yes—
184Dic3—124no—
185Dic3—124no—
186C32#524214yescomplemented
187C23—22no—
188C23#623512yes—
189Q8—182no—
190Q8#623512yes—
191D4—24no—
192D4—22no—
193D4—22no—
194D4#623512yes—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—181no—
198C2×C4—181no—
199C2×C4C22×#27211yescharacteristic
200C8—182no—
201C2×C3—21no—
202C2×C3—34no—
203C2×C3—42no—
204C2×C3—42no—
205C2×C3—64no—
206C2×C3—128no—
207C2×C3—128no—
208C2×C3#6416211yescharacteristic
209C22—22no—
210C22—22no—
211C22—42no—
212C22—82no—
213C22C2×#623511yescharacteristic
214C4—22no—
215C4—361no—
216C4—361no—
217C4C2×#623512yes—
218C3—34no—
219C3#1059294111yescharacteristic
220C2—21no—
221C2—42no—
222C2—42no—
223C2#1172187611yescharacteristic
subgroupG/H
center, socleC2×C3#64162
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
Fitting, prime radicalH(3,𝔽3)×H(5,𝔽2)C2
hypercenterC3×H(5,𝔽2)C3⋊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×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
2. #11721872C2semidirect
3. #11721846C2semidirect
4. #11720586C2semidirect
5. #11720579C2semidirect
12. C2×#6229C22semidirect
16. C4×#1084C22semidirect
17. #88538C22semidirect
124. H(3,𝔽3)#455semidirect
186. C32#5242semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×H(5,𝔽2)2
#117218722
#117218462
#117205862
#117205792
C3×#52423