Finite Group Databasebeta

← #13173186

Subgroups — #13173186

Order 1728.

644 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C2×C32×#41⋊C3C211yesmaximal, characteristic, complemented
2C2×C3⋊D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3SL(2,𝔽3)×C3⋊Dic3C211yesmaximal, characteristic, complemented
4#10596302C311yesmaximal, characteristic, complemented
5#5831⋊C3—34nomaximal
6C32×#41⋊C3—21no—
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C3⋊D3×SL(2,𝔽3)—21no—
9C3×#3752—41nomaximal
10C2×C32×#41C2×C311yescharacteristic, complemented
11C2×Q8×C3⋊D3C2×C311yescharacteristic
12C2×C3×#41⋊C3—24no—
13C2×C3×#41⋊C3D314yescomplemented
14C2×D3×SL(2,𝔽3)—34no—
15Q8×C3⋊Dic3C2×C311yescharacteristic, complemented
16#64147—31no—
17#64145—31no—
18C4×#638—31no—
19#64133—31no—
20Dic3×SL(2,𝔽3)—34no—
21C22×C3×C3⋊D3—41no—
22C2×C33×C4—41no—
23C2×C3×C3⋊Dic3—41no—
24C32×SL(2,𝔽3)D411yescharacteristic, complemented
25#5831—34no—
26#548⋊C3—91no—
27C32×#41—21no—
28C32×#41—61no—
29C2×C32×Q8C22×C311yescharacteristic
30C2×C32×D4—31no—
31C22×C32×C4—31no—
32C2×#638—31no—
33Q8×C3⋊D3—21no—
34Q8×C3⋊D3—61no—
35C2×C4×C3⋊D3—31no—
36C2×#634—31no—
37C3×#41⋊C3—28no—
38C3×#41⋊C3—24no—
39C2×C3×SL(2,𝔽3)—24no—
40C2×C3×SL(2,𝔽3)C2×D314yes—
41D3×SL(2,𝔽3)—64no—
42#3757—32no—
43#3757—31no—
44#3752A411yescharacteristic
45C4⋊(C3⋊Dic3)—31no—
46#3750—31no—
47C4×C3⋊Dic3—31no—
48C3×(C2×D3)⋊C4—124no—
49C22×C33—41no—
50C2×C3×C3⋊D3—42no—
51C2×C3×C3⋊D3—81no—
52C2×C3×C3⋊D3—81no—
53C33×C4—81no—
54C3×C3⋊Dic3—81no—
55C2×C3×#41C3×D314yescomplemented
56C2×D3×Q8—34no—
57C2×#41⋊C3—24no—
58C2×#41⋊C3C3⋊D311yescharacteristic, complemented
59C22×SL(2,𝔽3)—91no—
60Q8×Dic3—34no—
61#938—94no—
62#936—94no—
63C4×#67—94no—
64#924—94no—
65C4×SL(2,𝔽3)—91no—
66C23×C32—31no—
67C22×C3⋊D3C2×A411yescharacteristic
68C22×C3×D3—124no—
69C32×Q8—31no—
70C32×Q8C3×D411yescharacteristic
71C32×D4—61no—
72C32×D4—61no—
73C2×C32×C4—31no—
74C2×C32×C4—44no—
75C2×C32×C4—61no—
76C2×C32×C4—61no—
77C2×C32×C4—84no—
78C2×C32×C4C2×A411yescharacteristic
79#638—62no—
80C2×C3⋊Dic3—31no—
81C2×C3⋊Dic3—31no—
82C2×C3⋊Dic3C2×A411yescharacteristic
83C4×C3⋊D3—61no—
84C4×C3⋊D3—61no—
85#634—61no—
86#634—61no—
87C2×C3×Dic3—124no—
88C3×SL(2,𝔽3)—24no—
89C3×SL(2,𝔽3)#6714yescomplemented
90#548—91noSylow
91C2×C33—41no—
92C2×C33—41no—
93C2×C33—41no—
94C3×C3⋊D3—82no—
95C3×#41—24no—
96C3×#41—64no—
97C2×C3×Q8C2×C3×D314yes—
98C2×C3×D4—34no—
99C22×C3×C4—34no—
100C2×#67—94no—
101D3×Q8—64no—
102D3×Q8—184no—
103C2×C4×D3—94no—
104C2×Dic6—94no—
105#41⋊C3—28no—
106#41⋊C3—21no—
107C2×SL(2,𝔽3)—24no—
108C2×SL(2,𝔽3)—181no—
109C2×SL(2,𝔽3)C2×C3⋊D311yescharacteristic
110C3×C22⋊C4—361no—
111#217—94no—
112#217—98no—
113(C2×D3)⋊C4—34no—
114C4⋊Dic3—94no—
115Dic3⋊C4—94no—
116C4×Dic3—94no—
117C22×C32—44no—
118C22×C32—61no—
119C22×C32—61no—
120C22×C32—61no—
121C22×C32—84no—
122C22×C32C22×A411yescharacteristic
123C2×C3⋊D3—21no—
124C2×C3⋊D3—21no—
125C2×C3⋊D3#41⋊C312yescomplemented
126C2×C3×D3—128no—
127C2×C3×D3—244no—
128C2×C3×D3—244no—
129C32×C4—21no—
130C32×C4—31no—
131C32×C4—31no—
132C32×C4—84no—
133C32×C4—88no—
134C3⋊Dic3—21no—
135C3⋊Dic3—61no—
136C3⋊Dic3—61no—
137C3×Dic3—244no—
138C2×#41C3×C3⋊D311yescharacteristic, complemented
139C22×Q8—91no—
140#124—271no—
141C22⋊Q8—271no—
142#121—271no—
143C4×Q8—91no—
144C4×D4—271no—
145C33—41noSylow
146C23×C3—34no—
147C23×C3—361no—
148C22×D3—34no—
149C3×Q8—34no—
150C3×Q8C3×#6714yes—
151C3×D4—64no—
152C3×D4—64no—
153C2×C3×C4—34no—
154C2×C3×C4—41no—
155C2×C3×C4—64no—
156C2×C3×C4—64no—
157C2×C3×C4—84no—
158C2×C3×C4—361no—
159C2×C3×C4D3×A414yes—
160#67—188no—
161C2×Dic3—34no—
162C2×Dic3—94no—
163C2×Dic3—94no—
164C4×D3—184no—
165C4×D3—184no—
166Dic6—184no—
167Dic6—184no—
168SL(2,𝔽3)—24no—
169SL(2,𝔽3)#63811yescharacteristic, complemented
170C2×C32—44no—
171C2×C32—44no—
172C2×C32—44no—
173C2×C32—84no—
174C2×C32—84no—
175C2×C32—84no—
176C2×C32—121no—
177C2×C32D4×A411yescharacteristic
178C2×C32C2×#41⋊C311yescharacteristic
179C2×C322−1+4⋊C311yescharacteristic
180C3⋊D3—22no—
181C3×D3—248no—
182#41—21no—
183#41—61no—
184C2×Q8—181no—
185C2×Q8—271no—
186C2×Q8—541no—
187C2×Q8C2×C3×C3⋊D311yescharacteristic
188C2×D4—31no—
189C2×D4—271no—
190C22×C4—31no—
191C22×C4—271no—
192C4⋊C4—271no—
193C4⋊C4—271no—
194C22⋊C4—91no—
195C22⋊C4—271no—
196C22⋊C4—272no—
197C42—271no—
198C22×C3—41no—
199C22×C3—64no—
200C22×C3—64no—
201C22×C3—64no—
202C22×C3—84no—
203C22×C3—362no—
204C22×C3—721no—
205C22×C3—721no—
206C22×C3C2×D3×A414yes—
207C2×D3—38no—
208C2×D3—64no—
209C2×D3—64no—
210C3×C4—24no—
211C3×C4—34no—
212C3×C4—34no—
213C3×C4—81no—
214C3×C4—88no—
215C3×C4—721no—
216Dic3—64no—
217Dic3—184no—
218Dic3—184no—
219C32—44no—
220C32—84no—
221C32#548⋊C311yescharacteristic, complemented
222C23—31no—
223C23—91no—
224Q8—31no—
225Q8—541no—
226Q8—541no—
227Q8C3×#63811yescharacteristic
228D4—61no—
229D4—61no—
230D4—542no—
231C2×C4—31no—
232C2×C4—61no—
233C2×C4—61no—
234C2×C4—91no—
235C2×C4—271no—
236C2×C4—271no—
237C2×C4—541no—
238C2×C4—541no—
239C2×C4A4×C3⋊D311yescharacteristic
240C2×C3—41no—
241C2×C3—41no—
242C2×C3—41no—
243C2×C3—84no—
244C2×C3—84no—
245C2×C3—84no—
246C2×C3—124no—
247C2×C3—722no—
248C2×C3D3×#41⋊C314yes—
249C2×C3#1010⋊C314yes—
250C2×C3A4×#6714yes—
251D3—68no—
252C22—61no—
253C22—61no—
254C22—61no—
255C22—92no—
256C22—181no—
257C22—181no—
258C22C2×A4×C3⋊D311yescharacteristic
259C4—21no—
260C4—31no—
261C4—31no—
262C4—181no—
263C4—541no—
264C4—541no—
265C3—41no—
266C3—84no—
267C3#5831⋊C314yescomplemented
268C2—121no—
269C2—182no—
270C2C3⋊D3×#41⋊C311yescharacteristic
271C2#1172449311yescharacteristic
272C2A4×#63811yescharacteristic
subgroupG/H
center, FrattiniC22C2×A4×C3⋊D3
commutator, cosocleC2×C32×Q8C22×C3
FittingC2×C32×#41C2×C3
socleC22×C32C22×A4
hypercenterC2×C4A4×C3⋊D3
hypocenterC32×Q8C3×D4
LCSG ▷ C2×C32×Q8 ▷ C32×Q8
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C32×#41⋊C3C2semidirect
2. C2×C3⋊D3×SL(2,𝔽3)C2semidirect
3. SL(2,𝔽3)×C3⋊Dic3C2semidirect
4. #10596302C3semidirect
10. C2×C32×#41C2×C3semidirect
13. C2×C3×#41⋊C3D3semidirect
15. Q8×C3⋊Dic3C2×C3semidirect
24. C32×SL(2,𝔽3)D4semidirect
55. C2×C3×#41C3×D3semidirect
58. C2×#41⋊C3C3⋊D3semidirect
89. C3×SL(2,𝔽3)#67semidirect
125. C2×C3⋊D3#41⋊C3semidirect
138. C2×#41C3×C3⋊D3semidirect
169. SL(2,𝔽3)#638semidirect
221. C32#548⋊C3semidirect
267. C3#5831⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#41⋊C32
C2×C3⋊D3×SL(2,𝔽3)2
SL(2,𝔽3)×C3⋊Dic32
#105963023
#5831⋊C33
C3×#37524