Finite Group Databasebeta

← #13143932

Subgroups — #13143932

Order 1728.

704 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 101 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#11721850C211yesmaximal, characteristic
2C2×#88546C211yesmaximal, characteristic, complemented
3C2×#88528C211yesmaximal, characteristic, complemented
4#11721755C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#102C211yesmaximal, characteristic, complemented
6#11720547C211yesmaximal, characteristic, complemented
7#11720531C211yesmaximal, characteristic, complemented
8C3×#4762—34nomaximal
9C2×D4×H(3,𝔽3)C2211yescharacteristic
10C2×#6235—21no—
11C2×#6235—21no—
12C2×C4⋊#272C2211yescharacteristic
13C2×#6229C2211yescharacteristic
14C2×C8×H(3,𝔽3)C2211yescharacteristic
15C4⋊C4×H(3,𝔽3)C2211yescharacteristic
16#88546—22no—
17#88540—21no—
18C4×#1084—21no—
19C4×#1084C2211yescharacteristic
20C2×#6180C2211yescharacteristic
21#88528—22no—
22C3×#940—34no—
23C2×C3×#214—34no—
24C2×C3×#204—34no—
25C3×#887—34no—
26C32×#102D314yescomplemented
27C3×Dic3⋊C8—34no—
28C3×#809—34no—
29C23×H(3,𝔽3)—21no—
30C22×#272—21no—
31D4×H(3,𝔽3)—21no—
32D4×H(3,𝔽3)—21no—
33C2×C4×H(3,𝔽3)—21no—
34C2×C4×H(3,𝔽3)C2311yescharacteristic
35#6235—42no—
36#6235—42no—
37C2×#1084—21no—
38C2×#1084D411yescharacteristic
39C2×#1084D411yescharacteristic
40C4⋊#272—21no—
41C4⋊#272—21no—
42#6229—21no—
43#6229D412yescomplemented
44C8×H(3,𝔽3)—21no—
45#6180—21no—
46C3×#460—91no—
47#4762—34no—
48C2×C32×D4C2×D314yes—
49C2×C3×#67—64no—
50C2×C3×#67—64no—
51C2×C3×D12—34no—
52C2×C3×Dic6—34no—
53C2×C32×C8C2×D314yes—
54C32×C4⋊C4C2×D314yes—
55C3×#214—68no—
56C3×Dic3⋊C4—64no—
57C3×C4×Dic3—34no—
58C3×C4×Dic3—64no—
59C2×C3×C3⋊C8—34no—
60C3×#204—68no—
61C22×H(3,𝔽3)—41no—
62C22×H(3,𝔽3)—41no—
63C22×H(3,𝔽3)—41no—
64C22×H(3,𝔽3)C2×D411yescharacteristic
65C2×#272—41no—
66C2×#272—41no—
67C2×#272—41no—
68C4×H(3,𝔽3)—41no—
69C4×H(3,𝔽3)C2×D411yescharacteristic
70C4×H(3,𝔽3)#4111yescharacteristic
71#1084—22no—
72#1084—21no—
73#1084SD1612yescomplemented
74C2×C3×SD16—91no—
75C2×C3×SD16—91no—
76C3×#127—91no—
77C3×C4×Q8—91no—
78#940—34no—
79C2×#214—34no—
80C2×#204—34no—
81#887—34no—
82C3×C4⋊C8—91no—
83C3×#102—34no—
84C3×#102—91no—
85C3×#102C3⋊D311yescharacteristic
86Dic3⋊C8—34no—
87#809—34no—
88C23×C32—24no—
89C22×C3×D3—64no—
90C32×D4—24no—
91C32×D4—24no—
92C2×C32×C4—24no—
93C2×C32×C4C22×D314yes—
94C3×#67—128no—
95C3×#67—128no—
96C2×C3×Dic3—34no—
97C2×C3×Dic3—34no—
98C2×C3×Dic3—64no—
99C3×D12—64no—
100C3×D12—64no—
101C3×Dic6—38no—
102C3×Dic6—64no—
103C32×C8—24no—
104C3×C3⋊C8—64no—
105#460—91noSylow
106C2×H(3,𝔽3)—81no—
107C2×H(3,𝔽3)#12111yescharacteristic
108C2×H(3,𝔽3)C2×SD1611yescharacteristic
109C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
110#272—81no—
111C2×C3×Q8—91no—
112C2×C3×D4—34no—
113C2×C3×D4—91no—
114C2×C3×D4—181no—
115C2×C3×D4—181no—
116C2×C3×D4C2×C3⋊D311yescharacteristic
117C2×#67—64no—
118C2×#67—64no—
119C2×D12—34no—
120C2×Dic6—34no—
121C3×SD16—182no—
122C3×SD16—182no—
123C2×C3×C8—34no—
124C2×C3×C8—91no—
125C2×C3×C8C2×C3⋊D311yescharacteristic
126C3×C4⋊C4—34no—
127C3×C4⋊C4—181no—
128C3×C4⋊C4C2×C3⋊D311yescharacteristic
129C3×C42—91no—
130C3×C42—181no—
131#214—68no—
132Dic3⋊C4—64no—
133C4×Dic3—34no—
134C4×Dic3—64no—
135C2×C3⋊C8—34no—
136#204—68no—
137C22×C32—44no—
138C22×C32—44no—
139C22×C32—44no—
140C22×C32D3×D414yes—
141C2×C3×D3—124no—
142C2×C3×D3—124no—
143C2×C3×D3—124no—
144C32×C4—44no—
145C32×C4#23514yes—
146C32×C4D3×D414yes—
147C3×Dic3—38no—
148C3×Dic3—68no—
149C3×Dic3—64no—
150C2×SD16—91no—
151C2×SD16—91no—
152#127—91no—
153C4×Q8—91no—
154C4⋊C8—91no—
155#102—91no—
156#102#27211yescharacteristic, complemented
157H(3,𝔽3)#46011yescharacteristic, complemented, Sylow
158C23×C3—21no—
159C23×C3—64no—
160C23×C3—181no—
161C22×D3—64no—
162C3×Q8—92no—
163C3×Q8—181no—
164C3×D4—21no—
165C3×D4—21no—
166C3×D4—64no—
167C3×D4—64no—
168C3×D4—181no—
169C3×D4—181no—
170C3×D4—362no—
171C3×D4—362no—
172C2×C3×C4—21no—
173C2×C3×C4—34no—
174C2×C3×C4—64no—
175C2×C3×C4—91no—
176C2×C3×C4—91no—
177C2×C3×C4—181no—
178C2×C3×C4C22×C3⋊D311yescharacteristic
179#67—128no—
180#67—128no—
181C2×Dic3—34no—
182C2×Dic3—34no—
183C2×Dic3—64no—
184D12—64no—
185D12—64no—
186Dic6—38no—
187Dic6—64no—
188C3×C8—21no—
189C3×C8—64no—
190C3×C8—181no—
191C3⋊C8—64no—
192C2×C32—84no—
193C2×C32#88414yes—
194C2×C32#91114yes—
195C2×C32D3×SD1614yes—
196C3×D3—244no—
197C2×Q8—91no—
198C2×D4—91no—
199C2×D4—181no—
200C2×D4—181no—
201C2×D4C2×#27211yescharacteristic
202SD16—182no—
203SD16—182no—
204C2×C8—91no—
205C2×C8C2×#27211yescharacteristic
206C4⋊C4—181no—
207C4⋊C4C2×#27211yescharacteristic
208C42—91no—
209C42—181no—
210C22×C3—34no—
211C22×C3—41no—
212C22×C3—41no—
213C22×C3—41no—
214C22×C3—124no—
215C22×C3—124no—
216C22×C3—124no—
217C22×C3—361no—
218C22×C3—361no—
219C22×C3—361no—
220C22×C3D4×C3⋊D311yescharacteristic
221C2×D3—124no—
222C2×D3—124no—
223C2×D3—124no—
224C3×C4—34no—
225C3×C4—34no—
226C3×C4—41no—
227C3×C4—92no—
228C3×C4—124no—
229C3×C4—182no—
230C3×C4—181no—
231C3×C4#382811yescharacteristic
232C3×C4D4×C3⋊D311yescharacteristic
233Dic3—38no—
234Dic3—68no—
235Dic3—64no—
236C32#476214yescomplemented
237C23—21no—
238C23—181no—
239Q8—92no—
240Q8—181no—
241D4—21no—
242D4—21no—
243D4—181no—
244D4—181no—
245D4—362no—
246D4—362no—
247C2×C4—21no—
248C2×C4—91no—
249C2×C4—91no—
250C2×C4—181no—
251C2×C4C22×#27211yescharacteristic
252C8—21no—
253C8—181no—
254C2×C3—34no—
255C2×C3—34no—
256C2×C3—34no—
257C2×C3—81no—
258C2×C3—244no—
259C2×C3—721no—
260C2×C3#6409311yescharacteristic
261C2×C3#6412011yescharacteristic
262C2×C3SD16×C3⋊D311yescharacteristic
263D3—244no—
264C22—41no—
265C22—41no—
266C22—41no—
267C22—361no—
268C22—361no—
269C22—361no—
270C22D4×#27211yescharacteristic
271C4—41no—
272C4—92no—
273C4—182no—
274C4—181no—
275C4#8874011yescharacteristic
276C4D4×#27211yescharacteristic
277C3—34no—
278C3#1059246111yescharacteristic
279C2—81no—
280C2—721no—
281C2#1172174011yescharacteristic
282C2#1172179311yescharacteristic
283C2SD16×#27211yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#102C2
prime radical#11721850C2
hypercenterC3×#102C3⋊D3
hypocenterH(3,𝔽3)#460
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#102

Complements

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

NH ≅ G/Ntype
2. C2×#88546C2semidirect
3. C2×#88528C2semidirect
4. #11721755C2semidirect
5. H(3,𝔽3)×#102C2semidirect
6. #11720547C2semidirect
7. #11720531C2semidirect
26. C32×#102D3semidirect
43. #6229D4semidirect
73. #1084SD16semidirect
156. #102#272semidirect
157. H(3,𝔽3)#460semidirect
236. C32#4762semidirect

Maximal subgroups

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

subgroupindex
#117218502
C2×#885462
C2×#885282
#117217552
H(3,𝔽3)×#1022
#117205472
#117205312
C3×#47623