Finite Group Databasebeta

← #13145735

Subgroups — #13145735

Order 1728.

770 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4⋊(C3⋊Dic3)C212yesmaximal
2C22×C3×C3⋊(C3⋊C8)C211yesmaximal, characteristic
3#10589164—32nomaximal
4#10589163—31nomaximal
5C22×C33×C4C2211yescharacteristic
6C22×C3×C3⋊Dic3—22no—
7C3×C4⋊(C3⋊Dic3)—22no—
8C3×C4⋊(C3⋊Dic3)C414yescomplemented
9C2×C3×C3⋊(C3⋊C8)—22no—
10C2×C3×C3⋊(C3⋊C8)C412yescomplemented
11C2×C4⋊(C3⋊Dic3)D312yes—
12C22×C3⋊(C3⋊C8)D311yescharacteristic
13C2×C3×C4⋊Dic3—32no—
14C2×C3×C4⋊Dic3—32no—
15C2×C3×C4⋊Dic3—62no—
16C22×C3×C3⋊C8—32no—
17C22×C3×C3⋊C8—61no—
18C23×C33D411yescharacteristic
19C2×C33×C4C2×C411yescharacteristic
20C2×C33×C4C2×C412yes—
21C2×C33×C4D412yes—
22C2×C33×C4Q811yescharacteristic
23C2×C3×C3⋊Dic3—24no—
24C2×C3×C3⋊Dic3—42no—
25C2×C3×C3⋊Dic3—42no—
26C3×C3⋊(C3⋊C8)—22no—
27#4551—91no—
28#4529—92no—
29C22×C32×C4—21no—
30C22×C32×C4—22no—
31C22×C32×C4—41no—
32C22×C32×C4C2×D312yes—
33C22×C32×C4C2×D311yescharacteristic
34C22×C3⋊Dic3—22no—
35C22×C3×Dic3—64no—
36C22×C3×Dic3—62no—
37C22×C3×Dic3—62no—
38C4⋊(C3⋊Dic3)—22no—
39C4⋊(C3⋊Dic3)Dic314yescomplemented
40C2×C3⋊(C3⋊C8)—22no—
41C2×C3⋊(C3⋊C8)Dic312yescomplemented
42C3×C4⋊Dic3—34no—
43C3×C4⋊Dic3—34no—
44C3×C4⋊Dic3—64no—
45C3×C4⋊Dic3—64no—
46C3×C4⋊Dic3—62no—
47C3×C4⋊Dic3—62no—
48C2×C3×C3⋊C8—34no—
49C2×C3×C3⋊C8—62no—
50C2×C3×C3⋊C8—64no—
51C2×C3×C3⋊C8—62no—
52C2×C3×C3⋊C8—62no—
53C22×C33C22⋊C412yes—
54C22×C33C4⋊C411yescharacteristic
55C22×C33D811yescharacteristic
56C22×C33SD1612yes—
57C22×C33Q1611yescharacteristic
58C33×C4C4211yescharacteristic
59C33×C4C22⋊C411yescharacteristic
60C33×C4C4⋊C412yes—
61C3×C3⋊Dic3—44no—
62C22×C3×C8—91no—
63C2×C3×C4⋊C4—92no—
64C2×C4⋊Dic3—32no—
65C2×C4⋊Dic3—32no—
66C2×C4⋊Dic3—62no—
67C22×C3⋊C8—32no—
68C22×C3⋊C8—61no—
69C23×C32—21no—
70C23×C32—22no—
71C23×C32—41no—
72C23×C32D1211yescharacteristic
73C23×C32#6712yes—
74C2×C32×C4—22no—
75C2×C32×C4—22no—
76C2×C32×C4—22no—
77C2×C32×C4—21no—
78C2×C32×C4—22no—
79C2×C32×C4—22no—
80C2×C32×C4—22no—
81C2×C32×C4—22no—
82C2×C32×C4—21no—
83C2×C32×C4—22no—
84C2×C32×C4—41no—
85C2×C32×C4—42no—
86C2×C32×C4—42no—
87C2×C32×C4—41no—
88C2×C32×C4Dic611yescharacteristic
89C2×C32×C4Dic612yes—
90C2×C32×C4C4×D312yes—
91C2×C32×C4C4×D312yes—
92C2×C32×C4C4×D312yes—
93C2×C32×C4D1212yes—
94C2×C32×C4C2×Dic312yes—
95C2×C32×C4C2×Dic311yescharacteristic
96C2×C32×C4#6712yes—
97C2×C32×C4#6712yes—
98C2×C3⋊Dic3—24no—
99C2×C3⋊Dic3—42no—
100C2×C3⋊Dic3—42no—
101C2×C3×Dic3—68no—
102C2×C3×Dic3—64no—
103C2×C3×Dic3—64no—
104C2×C3×Dic3—124no—
105C2×C3×Dic3—124no—
106C2×C3×Dic3—122no—
107C2×C3×Dic3—122no—
108C2×C3×Dic3—122no—
109C2×C3×Dic3—122no—
110C3⋊(C3⋊C8)—22no—
111C3×C3⋊C8—64no—
112C3×C3⋊C8—64no—
113#340—271noSylow
114C2×C33#9511yescharacteristic
115C2×C33#10212yes—
116C2×C33#10312yes—
117C2×C33#10611yescharacteristic
118C2×C33#10711yescharacteristic
119C22×C3×C4—21no—
120C22×C3×C4—22no—
121C22×C3×C4—41no—
122C22×C3×C4—182no—
123C22×C3×C4D3212yes—
124C22×C3×C4D3211yescharacteristic
125C22×Dic3—64no—
126C22×Dic3—62no—
127C22×Dic3—62no—
128C2×C3×C8—92no—
129C2×C3×C8—182no—
130C3×C4⋊C4—94no—
131C3×C4⋊C4—182no—
132C4⋊Dic3—34no—
133C4⋊Dic3—34no—
134C4⋊Dic3—64no—
135C4⋊Dic3—64no—
136C4⋊Dic3—62no—
137C4⋊Dic3—62no—
138C2×C3⋊C8—34no—
139C2×C3⋊C8—62no—
140C2×C3⋊C8—64no—
141C2×C3⋊C8—62no—
142C2×C3⋊C8—62no—
143C22×C32—21no—
144C22×C32—22no—
145C22×C32—22no—
146C22×C32—22no—
147C22×C32—22no—
148C22×C32—21no—
149C22×C32—22no—
150C22×C32—22no—
151C22×C32—22no—
152C22×C32—22no—
153C22×C32—22no—
154C22×C32—21no—
155C22×C32—42no—
156C22×C32—41no—
157C22×C32—41no—
158C22×C32—42no—
159C22×C32—41no—
160C22×C32#20412yes—
161C22×C32D2411yescharacteristic
162C22×C32Dic1211yescharacteristic
163C22×C32Dic3⋊C412yes—
164C22×C32C4⋊Dic311yescharacteristic
165C22×C32(C2×D3)⋊C412yes—
166C22×C32(C2×D3)⋊C412yes—
167C22×C32#21312yes—
168C22×C32#21412yes—
169C22×C32#21512yes—
170C22×C32#21612yes—
171C22×C32#21712yes—
172C32×C4—21no—
173C32×C4—22no—
174C32×C4—22no—
175C32×C4—21no—
176C32×C4—22no—
177C32×C4—22no—
178C32×C4—22no—
179C32×C4—41no—
180C32×C4—41no—
181C32×C4—42no—
182C32×C4C4×Dic312yes—
183C32×C4C4×Dic311yescharacteristic
184C32×C4Dic3⋊C412yes—
185C32×C4Dic3⋊C412yes—
186C32×C4C4⋊Dic312yes—
187C32×C4(C2×D3)⋊C412yes—
188C32×C4#21711yescharacteristic
189C3⋊Dic3—44no—
190C3×Dic3—128no—
191C3×Dic3—124no—
192C3×Dic3—124no—
193C22×C8—91no—
194C2×C4⋊C4—92no—
195C33#34011yescharacteristic, complemented, Sylow
196C23×C3—21no—
197C23×C3—22no—
198C23×C3—41no—
199C23×C3#62511yescharacteristic
200C23×C3#62612yes—
201C2×C3×C4—22no—
202C2×C3×C4—22no—
203C2×C3×C4—22no—
204C2×C3×C4—21no—
205C2×C3×C4—22no—
206C2×C3×C4—22no—
207C2×C3×C4—22no—
208C2×C3×C4—22no—
209C2×C3×C4—21no—
210C2×C3×C4—22no—
211C2×C3×C4—41no—
212C2×C3×C4—42no—
213C2×C3×C4—42no—
214C2×C3×C4—41no—
215C2×C3×C4—184no—
216C2×C3×C4—362no—
217C2×C3×C4—362no—
218C2×C3×C4D3×Dic312yes—
219C2×C3×C4D3×Dic312yes—
220C2×C3×C4D3×Dic312yes—
221C2×C3×C4#62411yescharacteristic
222C2×C3×C4#62412yes—
223C2×C3×C4#62512yes—
224C2×C3×C4#62612yes—
225C2×C3×C4#62612yes—
226C2×C3×C4#62711yescharacteristic
227C2×C3×C4#62712yes—
228C2×Dic3—68no—
229C2×Dic3—64no—
230C2×Dic3—64no—
231C2×Dic3—124no—
232C2×Dic3—124no—
233C2×Dic3—122no—
234C2×Dic3—122no—
235C2×Dic3—122no—
236C2×Dic3—122no—
237C3×C8—182no—
238C3⋊C8—64no—
239C3⋊C8—64no—
240C2×C32—22no—
241C2×C32—22no—
242C2×C32—22no—
243C2×C32—22no—
244C2×C32—22no—
245C2×C32—22no—
246C2×C32—22no—
247C2×C32—21no—
248C2×C32—22no—
249C2×C32—21no—
250C2×C32—21no—
251C2×C32—22no—
252C2×C32—41no—
253C2×C32—42no—
254C2×C32—42no—
255C2×C32—41no—
256C2×C32—41no—
257C2×C32#80712yes—
258C2×C32#80812yes—
259C2×C32#80912yes—
260C2×C32#81012yes—
261C2×C32#81612yes—
262C2×C32#81711yescharacteristic
263C2×C32#81811yescharacteristic
264C2×C32#82112yes—
265C2×C32C3⋊#9512yes—
266C2×C32C3⋊#9511yescharacteristic
267C2×C32#83212yes—
268C2×C32#83512yes—
269C22×C4—182no—
270C22×C4#111311yescharacteristic
271C2×C8—92no—
272C2×C8—182no—
273C4⋊C4—94no—
274C4⋊C4—182no—
275C22×C3—22no—
276C22×C3—21no—
277C22×C3—22no—
278C22×C3—22no—
279C22×C3—22no—
280C22×C3—21no—
281C22×C3—22no—
282C22×C3—22no—
283C22×C3—22no—
284C22×C3—22no—
285C22×C3—22no—
286C22×C3—21no—
287C22×C3—41no—
288C22×C3—42no—
289C22×C3—41no—
290C22×C3—42no—
291C22×C3—41no—
292C22×C3#371311yescharacteristic
293C22×C3#371412yes—
294C22×C3#371512yes—
295C22×C3#371612yes—
296C22×C3#371712yes—
297C22×C3#371811yescharacteristic
298C22×C3#371912yes—
299C22×C3(C2×D3)⋊Dic312yes—
300C22×C3(C2×D3)⋊Dic312yes—
301C22×C3#372212yes—
302C22×C3Dic3⋊Dic312yes—
303C22×C3#372411yescharacteristic
304C3×C4—21no—
305C3×C4—22no—
306C3×C4—21no—
307C3×C4—22no—
308C3×C4—22no—
309C3×C4—22no—
310C3×C4—22no—
311C3×C4—41no—
312C3×C4—41no—
313C3×C4—42no—
314C3×C4—364no—
315C3×C4Dic3211yescharacteristic
316C3×C4Dic3212yes—
317C3×C4(C2×D3)⋊Dic312yes—
318C3×C4#372211yescharacteristic
319C3×C4Dic3⋊Dic312yes—
320C3×C4Dic3⋊Dic312yes—
321C3×C4#372412yes—
322Dic3—128no—
323Dic3—124no—
324Dic3—124no—
325C32—21no—
326C32—22no—
327C32—41no—
328C32#452912yescomplemented
329C32#455111yescharacteristic, complemented
330C23#629511yescharacteristic
331C2×C4—184no—
332C2×C4—362no—
333C2×C4—362no—
334C2×C4#629411yescharacteristic
335C2×C4#629412yes—
336C2×C4#629512yes—
337C2×C4#629611yescharacteristic
338C8—182no—
339C2×C3—22no—
340C2×C3—22no—
341C2×C3—22no—
342C2×C3—22no—
343C2×C3—22no—
344C2×C3—22no—
345C2×C3—22no—
346C2×C3—21no—
347C2×C3—22no—
348C2×C3—21no—
349C2×C3—22no—
350C2×C3—21no—
351C2×C3—42no—
352C2×C3—42no—
353C2×C3—41no—
354C2×C3—41no—
355C2×C3—41no—
356C2×C3#6356212yes—
357C2×C3#6356312yes—
358C2×C3#6356412yes—
359C2×C3#6356512yes—
360C2×C3#6356612yes—
361C2×C3#6356712yes—
362C2×C3#6357312yes—
363C2×C3#6357411yescharacteristic
364C2×C3#6357512yes—
365C2×C3#6357611yescharacteristic
366C2×C3#6357911yescharacteristic
367C2×C3#6357912yes—
368C22#8881011yescharacteristic
369C22#8881112yes—
370C22#8881211yescharacteristic
371C22#8881412yes—
372C22#8881511yescharacteristic
373C4—364no—
374C4#8881311yescharacteristic
375C4#8881411yescharacteristic
376C4#8881512yes—
377C3—22no—
378C3—21no—
379C3—41no—
380C3#1058916311yescharacteristic, complemented
381C3#1058916412yescomplemented
382C2#1172207512yes—
383C2#1172207612yes—
384C2#1172207911yescharacteristic
385C2#1172208011yescharacteristic
386C2#1172208211yescharacteristic
subgroupG/H
centerC23#6295
commutatorC33×C4C42
Frattini, hypercenterC22×C4#1113
Fitting, cosocleC22×C33×C4C22
socle, prime radicalC23×C33D4
hypocenterC33#340
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
8. C3×C4⋊(C3⋊Dic3)C4semidirect
10. C2×C3×C3⋊(C3⋊C8)C4semidirect
39. C4⋊(C3⋊Dic3)Dic3semidirect
41. C2×C3⋊(C3⋊C8)Dic3semidirect
195. C33#340semidirect
328. C32#4529semidirect
329. C32#4551semidirect
380. C3#10589163semidirect
381. C3#10589164semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊(C3⋊Dic3)2
C22×C3×C3⋊(C3⋊C8)2
#105891643
#105891633