Finite Group Databasebeta

← #10595351⋊C3 #13166147

Subgroups — #10595351⋊C3 #13166147

Order 1728.

348 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 56 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
1SL(2,𝔽3)×C3⋊Dic3C211yesmaximal, characteristic
2C3×Dic3×SL(2,𝔽3)C212yesmaximal
3#10595351C311yesmaximal, characteristic, complemented
4#5569⋊C3—32nomaximal
5C2×C32×SL(2,𝔽3)C2211yescharacteristic
6C3×#3724—41nomaximal
7Q8×C3⋊Dic3C2×C311yescharacteristic
8C3×Q8×Dic3C2×C312yes—
9C3×C4×SL(2,𝔽3)—32no—
10#63919—31no—
11C4×#627—31no—
12#63869—32no—
13Dic3×SL(2,𝔽3)—22no—
14Dic3×SL(2,𝔽3)—32no—
15Dic3×SL(2,𝔽3)—61no—
16Dic3×SL(2,𝔽3)D312yes—
17C2×C3×C3⋊Dic3—41no—
18C2×C32×Dic3—42no—
19C32×SL(2,𝔽3)Q811yescharacteristic, complemented
20#5569—32no—
21#557⋊C3—91no—
22C2×C32×Q8C22×C311yescharacteristic
23C2×C3×SL(2,𝔽3)—22no—
24C2×C3×SL(2,𝔽3)—21no—
25C2×C3×SL(2,𝔽3)—41no—
26C2×C3×SL(2,𝔽3)C2×D312yes—
27C2×#627—31no—
28C4⋊(C3⋊Dic3)—31no—
29C4×C3⋊Dic3—31no—
30C3×C4⋊Dic3—32no—
31C3×Dic3⋊C4—122no—
32C3×C4×Dic3—32no—
33#3724—32no—
34#3724A411yescharacteristic
35Dic3⋊Dic3—32no—
36C22×C33—41no—
37C3×C3⋊Dic3—42no—
38C32×Dic3—82no—
39C3×C4×Q8—32no—
40Q8×Dic3—32no—
41Q8×Dic3—61no—
42Q8×Dic3C3×D312yes—
43#890—92no—
44#888—92no—
45#870—92no—
46C4×Dic6—92no—
47C4×SL(2,𝔽3)—32no—
48C4×SL(2,𝔽3)—62no—
49C4×SL(2,𝔽3)—91no—
50C32×Q8—31no—
51C32×Q8C3×Q811yescharacteristic
52C2×C32×C4—31no—
53C2×C32×C4—122no—
54C2×C3⋊Dic3—31no—
55C2×C3⋊Dic3C2×A411yescharacteristic
56C2×C3×Dic3—32no—
57C2×C3×Dic3—42no—
58C2×C3×Dic3—82no—
59C2×C3×Dic3—122no—
60C2×C3×Dic3—241no—
61C2×C3×Dic3C2×A412yes—
62C3×SL(2,𝔽3)—22no—
63C3×SL(2,𝔽3)—21no—
64C3×SL(2,𝔽3)—41no—
65C3×SL(2,𝔽3)Dic612yescomplemented
66#627—62no—
67#557—91noSylow
68C2×C33—41no—
69C2×C33—41no—
70C2×C33—41no—
71C2×C3×Q8—21no—
72C2×C3×Q8C2×C3×D312yes—
73C2×Dic6—92no—
74C2×SL(2,𝔽3)—22no—
75C2×SL(2,𝔽3)—41no—
76C2×SL(2,𝔽3)D3211yescharacteristic
77C3×C4⋊C4—92no—
78C3×C4⋊C4—361no—
79C3×C42—92no—
80C4⋊Dic3—32no—
81C4⋊Dic3—92no—
82C4⋊Dic3—92no—
83C4⋊Dic3—181no—
84Dic3⋊C4—32no—
85Dic3⋊C4—92no—
86Dic3⋊C4—94no—
87C4×Dic3—32no—
88C4×Dic3—92no—
89C4×Dic3—181no—
90C22×C32—42no—
91C22×C32—81no—
92C22×C32—82no—
93C22×C32—161no—
94C22×C32C22×A411yescharacteristic
95C32×C4—31no—
96C32×C4—31no—
97C32×C4—242no—
98C3⋊Dic3—61no—
99C3⋊Dic3#41⋊C312yes—
100C3×Dic3—22no—
101C3×Dic3—62no—
102C3×Dic3—82no—
103C3×Dic3—84no—
104C3×Dic3—124no—
105C3×Dic3—242no—
106C4⋊Q8—271no—
107#125—272no—
108C4×Q8—32no—
109C4×Q8—91no—
110C4×Q8—271no—
111C33—41noSylow
112C3×Q8—21no—
113C3×Q8—32no—
114C3×Q8—61no—
115C3×Q8C3×Dic612yes—
116C2×C3×C4—32no—
117C2×C3×C4—32no—
118C2×C3×C4—61no—
119C2×C3×C4—92no—
120C2×C3×C4—122no—
121C2×C3×C4—242no—
122C2×C3×C4—361no—
123C2×Dic3—32no—
124C2×Dic3—32no—
125C2×Dic3—61no—
126C2×Dic3—92no—
127C2×Dic3—181no—
128C2×Dic3D3×A412yes—
129Dic6—184no—
130SL(2,𝔽3)—22no—
131SL(2,𝔽3)—41no—
132SL(2,𝔽3)#62711yescharacteristic, complemented
133C2×C32—42no—
134C2×C32—42no—
135C2×C32—42no—
136C2×C32—81no—
137C2×C32—81no—
138C2×C32—81no—
139C2×C32—82no—
140C2×C32—82no—
141C2×C32—82no—
142C2×C32—161no—
143C2×C32—161no—
144C2×C32—161no—
145C2×C32Q8×A411yescharacteristic
146C2×C32C2×#41⋊C311yescharacteristic
147C2×C322−1+4⋊C311yescharacteristic
148C2×Q8—271no—
149C2×Q8C3×D3211yescharacteristic
150C4⋊C4—91no—
151C4⋊C4—92no—
152C4⋊C4—272no—
153C4⋊C4—271no—
154C4⋊C4—272no—
155C42—92no—
156C42—271no—
157C22×C3—21no—
158C22×C3—41no—
159C22×C3—82no—
160C22×C3—161no—
161C22×C3C2×D3×A412yes—
162C3×C4—32no—
163C3×C4—32no—
164C3×C4—61no—
165C3×C4—61no—
166C3×C4—62no—
167C3×C4—182no—
168C3×C4—242no—
169C3×C4—244no—
170C3×C4—362no—
171Dic3—22no—
172Dic3—34no—
173Dic3—62no—
174Dic3—62no—
175Dic3—182no—
176Dic3—361no—
177C32—42no—
178C32—81no—
179C32—82no—
180C32—161no—
181C32#557⋊C311yescharacteristic, complemented
182Q8—31no—
183Q8—542no—
184Q8C3×#62711yescharacteristic
185C2×C4—31no—
186C2×C4—32no—
187C2×C4—91no—
188C2×C4—92no—
189C2×C4—271no—
190C2×C3—21no—
191C2×C3—21no—
192C2×C3—21no—
193C2×C3—41no—
194C2×C3—41no—
195C2×C3—41no—
196C2×C3—82no—
197C2×C3—82no—
198C2×C3—82no—
199C2×C3—161no—
200C2×C3—161no—
201C2×C3—161no—
202C2×C3A4×Dic612yes—
203C2×C3#1007⋊C312yes—
204C2×C3D3×#41⋊C312yes—
205C22D32×A411yescharacteristic
206C4—31no—
207C4—31no—
208C4—62no—
209C4—92no—
210C4—182no—
211C4—541no—
212C3—21no—
213C3—41no—
214C3—82no—
215C3—161no—
216C3#5569⋊C312yescomplemented
217C2#64319⋊C311yescharacteristic
218C2#64316⋊C311yescharacteristic
219C2A4×#62711yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D32×A4
commutator, Fitting, cosocleC2×C32×Q8C22×C3
socleC22×C32C22×A4
prime radicalC2×C32×SL(2,𝔽3)C22
hypocenterC32×Q8C3×Q8
LCSG ▷ C2×C32×Q8 ▷ C32×Q8
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
3. #10595351C3semidirect
19. C32×SL(2,𝔽3)Q8semidirect
65. C3×SL(2,𝔽3)Dic6semidirect
132. SL(2,𝔽3)#627semidirect
181. C32#557⋊C3semidirect
216. C3#5569⋊C3semidirect

Maximal subgroups

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

subgroupindex
SL(2,𝔽3)×C3⋊Dic32
C3×Dic3×SL(2,𝔽3)2
#105953513
#5569⋊C33
C3×#37244