Finite Group Databasebeta

← #13161828

Subgroups — #13161828

Order 1728.

738 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64135C212yesmaximal
2C22×C3×C3⋊(C3⋊C8)C211yesmaximal, characteristic
3#11722080C212yesmaximal, complemented
4#11722079C212yesmaximal, complemented
5#10591138—31nomaximal
6#10591122—32nomaximal
7C22×C33×C4C2211yescharacteristic
8C3×#3757—22no—
9C3×C4⋊(C3⋊Dic3)C2214yes—
10C3×C4×C3⋊Dic3—22no—
11C2×C3×C3⋊(C3⋊C8)C414yescomplemented
12C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
13C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
14#64135D312yes—
15C22×C3⋊(C3⋊C8)D311yescharacteristic
16C3×#926—32no—
17C3×#926—32no—
18C3×#926—62no—
19C22×C3×C3⋊C8—32no—
20C22×C3×C3⋊C8—61no—
21#63576—32no—
22#63575—34no—
23#63574—32no—
24#63573—34no—
25C23×C33D411yescharacteristic
26C2×C33×C4C2×C412yes—
27C2×C33×C4D411yescharacteristic
28C2×C33×C4Q812yes—
29C2×C33×C4C2311yescharacteristic
30C2×C3×C3⋊Dic3—24no—
31C3×C3⋊(C3⋊C8)C2×C412yes—
32C3×C3⋊(C3⋊C8)C2×C412yes—
33#5106—91no—
34#4998—92no—
35C22×C32×C4—21no—
36C22×C32×C4—22no—
37C22×C32×C4—41no—
38C22×C32×C4C2×D312yes—
39C22×C32×C4C2×D311yescharacteristic
40#3757—22no—
41C4⋊(C3⋊Dic3)C2×D314yes—
42C4×C3⋊Dic3—22no—
43C2×C3⋊(C3⋊C8)Dic314yescomplemented
44C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
45C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
46C3×#217—64no—
47C3×#217—62no—
48C3×#217—62no—
49C3×C4⋊Dic3—34no—
50C3×C4⋊Dic3—34no—
51C3×C4⋊Dic3—64no—
52C3×C4×Dic3—64no—
53C3×C4×Dic3—62no—
54C3×C4×Dic3—62no—
55C2×C3×C3⋊C8—38no—
56C2×C3×C3⋊C8—32no—
57C2×C3×C3⋊C8—32no—
58C2×C3×C3⋊C8—64no—
59C2×C3×C3⋊C8—61no—
60C2×C3×C3⋊C8—61no—
61C22×C33—22no—
62C22×C33C4⋊C412yes—
63C22×C33C2×D411yescharacteristic
64C33×C4C4⋊C412yes—
65C33×C4C22×C411yescharacteristic
66C33×C4C2×Q811yescharacteristic
67C3×C3⋊Dic3—44no—
68C22×C3×C8—91no—
69C3×#117—92no—
70#926—32no—
71#926—32no—
72#926—62no—
73C22×C3⋊C8—32no—
74C22×C3⋊C8—61no—
75#818—92no—
76#817—92no—
77#808—94no—
78#807—94no—
79C23×C32—21no—
80C23×C32—22no—
81C23×C32—41no—
82C23×C32D1211yescharacteristic
83C23×C32#6712yes—
84C2×C32×C4—24no—
85C2×C32×C4—22no—
86C2×C32×C4—21no—
87C2×C32×C4—22no—
88C2×C32×C4—24no—
89C2×C32×C4—22no—
90C2×C32×C4—21no—
91C2×C32×C4—22no—
92C2×C32×C4—42no—
93C2×C32×C4—41no—
94C2×C32×C4—42no—
95C2×C32×C4—41no—
96C2×C32×C4Dic614yes—
97C2×C32×C4Dic612yes—
98C2×C32×C4C4×D314yes—
99C2×C32×C4D1211yescharacteristic
100C2×C32×C4C2×Dic312yes—
101C2×C32×C4#6712yes—
102C2×C32×C4C22×D312yes—
103C2×C32×C4C22×D311yescharacteristic
104C2×C3⋊Dic3—24no—
105C2×C3×Dic3—68no—
106C2×C3×Dic3—64no—
107C2×C3×Dic3—64no—
108C3⋊(C3⋊C8)C2×Dic312yes—
109C3⋊(C3⋊C8)C2×Dic312yes—
110C3×C3⋊C8—34no—
111C3×C3⋊C8—34no—
112C3×C3⋊C8—62no—
113C3×C3⋊C8—62no—
114#427—271noSylow
115C2×C33—22no—
116C2×C33C2×C4⋊C411yescharacteristic
117C2×C33#13512yes—
118C22×C3×C4—21no—
119C22×C3×C4—22no—
120C22×C3×C4—41no—
121C22×C3×C4D3212yes—
122C22×C3×C4D3211yescharacteristic
123C2×C3×C8—94no—
124C2×C3×C8—91no—
125C2×C3×C8—91no—
126C3×C4⋊C4—94no—
127C3×C22⋊C4—182no—
128C3×C42—182no—
129#217—64no—
130#217—62no—
131#217—62no—
132C4⋊Dic3—34no—
133C4⋊Dic3—34no—
134C4⋊Dic3—64no—
135C4×Dic3—64no—
136C4×Dic3—62no—
137C4×Dic3—62no—
138C2×C3⋊C8—38no—
139C2×C3⋊C8—32no—
140C2×C3⋊C8—32no—
141C2×C3⋊C8—64no—
142C2×C3⋊C8—61no—
143C2×C3⋊C8—61no—
144C22×C32—22no—
145C22×C32—24no—
146C22×C32—24no—
147C22×C32—24no—
148C22×C32—22no—
149C22×C32—21no—
150C22×C32—22no—
151C22×C32—42no—
152C22×C32—41no—
153C22×C32—44no—
154C22×C32—44no—
155C22×C32Dic3⋊C414yes—
156C22×C32C4⋊Dic312yes—
157C22×C32C2×D1211yescharacteristic
158C22×C32C2×#6712yes—
159C32×C4—24no—
160C32×C4—22no—
161C32×C4—22no—
162C32×C4—21no—
163C32×C4—22no—
164C32×C4—21no—
165C32×C4—42no—
166C32×C4—41no—
167C32×C4—41no—
168C32×C4Dic3⋊C414yes—
169C32×C4C4⋊Dic312yes—
170C32×C4C2×Dic612yes—
171C32×C4C2×Dic611yescharacteristic
172C32×C4C2×C4×D312yes—
173C32×C4C22×Dic311yescharacteristic
174C3⋊Dic3—44no—
175C3×Dic3—128no—
176C3×Dic3—124no—
177C3×Dic3—124no—
178C22×C8—91no—
179#117—92no—
180#107—272no—
181#106—272no—
182C33#42711yescharacteristic, complemented, Sylow
183C23×C3—21no—
184C23×C3—22no—
185C23×C3—41no—
186C23×C3#62511yescharacteristic
187C23×C3#62612yes—
188C2×C3×C4—24no—
189C2×C3×C4—22no—
190C2×C3×C4—21no—
191C2×C3×C4—22no—
192C2×C3×C4—24no—
193C2×C3×C4—22no—
194C2×C3×C4—21no—
195C2×C3×C4—22no—
196C2×C3×C4—42no—
197C2×C3×C4—41no—
198C2×C3×C4—42no—
199C2×C3×C4—41no—
200C2×C3×C4—184no—
201C2×C3×C4D3×Dic314yes—
202C2×C3×C4#62412yes—
203C2×C3×C4#62511yescharacteristic
204C2×C3×C4#62612yes—
205C2×C3×C4#62714yes—
206C2×C3×C4#62712yes—
207C2×C3×C4C2×D3212yes—
208C2×C3×C4C2×D3211yescharacteristic
209C2×Dic3—68no—
210C2×Dic3—64no—
211C2×Dic3—64no—
212C3×C8—92no—
213C3×C8—92no—
214C3⋊C8—34no—
215C3⋊C8—34no—
216C3⋊C8—62no—
217C3⋊C8—62no—
218C2×C32—22no—
219C2×C32—24no—
220C2×C32—24no—
221C2×C32—22no—
222C2×C32—24no—
223C2×C32—22no—
224C2×C32—21no—
225C2×C32—41no—
226C2×C32—42no—
227C2×C32—44no—
228C2×C32—44no—
229C2×C32#90412yes—
230C2×C32C2×Dic3⋊C412yes—
231C2×C32C2×C4⋊Dic311yescharacteristic
232C2×C32#95014yes—
233C22×C4#111311yescharacteristic
234C2×C8—94no—
235C2×C8—91no—
236C2×C8—91no—
237C4⋊C4—94no—
238C22⋊C4—182no—
239C42—182no—
240C22×C3—24no—
241C22×C3—22no—
242C22×C3—24no—
243C22×C3—22no—
244C22×C3—21no—
245C22×C3—22no—
246C22×C3—24no—
247C22×C3—44no—
248C22×C3—44no—
249C22×C3—41no—
250C22×C3—42no—
251C22×C3Dic3⋊Dic314yes—
252C22×C3#372412yes—
253C22×C3C2×#62511yescharacteristic
254C22×C3C2×#62612yes—
255C3×C4—21no—
256C3×C4—22no—
257C3×C4—22no—
258C3×C4—21no—
259C3×C4—24no—
260C3×C4—22no—
261C3×C4—41no—
262C3×C4—41no—
263C3×C4—42no—
264C3×C4—364no—
265C3×C4Dic3⋊Dic314yes—
266C3×C4#372412yes—
267C3×C4C2×D3×Dic312yes—
268C3×C4C2×#62411yescharacteristic
269C3×C4C2×#62712yes—
270C3×C4C2×#62711yescharacteristic
271Dic3—128no—
272Dic3—124no—
273Dic3—124no—
274C32—22no—
275C32—21no—
276C32—41no—
277C32#499812yescomplemented
278C32#510611yescharacteristic, complemented
279C23#629511yescharacteristic
280C2×C4—184no—
281C2×C4#629412yes—
282C2×C4#629511yescharacteristic
283C2×C4#629612yes—
284C2×C4C2×#111311yescharacteristic
285C8—92no—
286C8—92no—
287C2×C3—24no—
288C2×C3—24no—
289C2×C3—22no—
290C2×C3—24no—
291C2×C3—22no—
292C2×C3—22no—
293C2×C3—21no—
294C2×C3—44no—
295C2×C3—44no—
296C2×C3—42no—
297C2×C3—41no—
298C2×C3#6382212yes—
299C2×C3#6382914yes—
300C2×C3C2×Dic3⋊Dic312yes—
301C2×C3C2×#372411yescharacteristic
302C22—22no—
303C22#8881512yes—
304C22C2×#629511yescharacteristic
305C4—364no—
306C4#8881512yes—
307C4C2×#629411yescharacteristic
308C4C2×#629611yescharacteristic
309C3—21no—
310C3—22no—
311C3—41no—
312C3#1059112212yescomplemented
313C3#1059113811yescharacteristic, complemented
314C2—22no—
315C2#1172366012yes—
316C2C2×#8881511yescharacteristic
subgroupG/H
centerC2×C4#6295
commutatorC33×C4C22×C4
FrattiniC2×C4C2×#1113
FittingC22×C33×C4C22
socleC22×C33C2×D4
prime radicalC23×C33D4
cosocleC2×C33×C4C23
hypercenterC22×C4#1113
hypocenterC33#427
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. #11722080C2semidirect
4. #11722079C2semidirect
11. C2×C3×C3⋊(C3⋊C8)C4semidirect
43. C2×C3⋊(C3⋊C8)Dic3semidirect
182. C33#427semidirect
277. C32#4998semidirect
278. C32#5106semidirect
312. C3#10591122semidirect
313. C3#10591138semidirect

Maximal subgroups

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

subgroupindex
C3×#641352
C22×C3×C3⋊(C3⋊C8)2
#117220802
#117220792
#105911383
#105911223