Finite Group Databasebeta

← #13156739

Subgroups — #13156739

Order 1728.

866 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Dic3×C3⋊Dic3C211yesmaximal, characteristic
2#11722082C211yesmaximal, characteristic
3#11722061C212yesmaximal
4C3×#63579C212yesmaximal
5C3×#63579C211yesmaximal, characteristic
6#10591248—32nomaximal
7#10591229—31nomaximal
8C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
9C22×C3×C3⋊Dic3C2212yes—
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C22×C32×Dic3C2212yes—
12C22×C32×Dic3C2211yescharacteristic
13Dic3×C3⋊Dic3C414yescomplemented
14C2×C4×C3⋊Dic3—31no—
15C2×Dic32—32no—
16C2×Dic32—61no—
17#63658—32no—
18C3×C3⋊#95—32no—
19C3×C3⋊#95—32no—
20C3×C3⋊#95—32no—
21#63579—32no—
22#63579—31no—
23#63579—32no—
24#63579—32no—
25#63579—62no—
26#63579D312yes—
27#63579D311yescharacteristic
28C23×C33C2311yescharacteristic
29C2×C3⋊(C3⋊Dic3)—21no—
30C2×C3⋊(C3⋊Dic3)—21no—
31C2×C3⋊(C3⋊Dic3)C2×C412yes—
32C2×C3×C3⋊Dic3—22no—
33C2×C3×C3⋊Dic3—22no—
34C2×C3×C3⋊Dic3—22no—
35C2×C3×C3⋊Dic3—22no—
36C2×C3×C3⋊Dic3C2×C412yes—
37C2×C32×Dic3—22no—
38C2×C32×Dic3—22no—
39C2×C32×Dic3—22no—
40C2×C32×Dic3—22no—
41C2×C32×Dic3C2×C412yes—
42#4651—91no—
43#4651—92no—
44C22×C32×C4—31no—
45C22×C32×C4—32no—
46C22×C3⋊Dic3—31no—
47C22×C3⋊Dic3—32no—
48C22×C3⋊Dic3—61no—
49C22×C3⋊Dic3—62no—
50C22×C3⋊Dic3—121no—
51C22×C3⋊Dic3C2×D312yes—
52C22×C3⋊Dic3C2×D311yescharacteristic
53C22×C3×Dic3—21no—
54C22×C3×Dic3—22no—
55C22×C3×Dic3—32no—
56C22×C3×Dic3—32no—
57C22×C3×Dic3—32no—
58C22×C3×Dic3—61no—
59C22×C3×Dic3—62no—
60C22×C3×Dic3C2×D312yes—
61C22×C3×Dic3C2×D312yes—
62C22×C3×Dic3C2×D312yes—
63C4×C3⋊Dic3—34no—
64Dic32—38no—
65Dic32—64no—
66C22×C33C22×C411yescharacteristic
67C22×C33#4111yescharacteristic
68C22×C33#4111yescharacteristic
69C22×C33#4112yes—
70C22×C33#4112yes—
71C3⋊(C3⋊Dic3)—22no—
72C3×C3⋊Dic3—22no—
73C3×C3⋊Dic3—42no—
74C32×Dic3—22no—
75C32×Dic3—42no—
76C2×C4×Dic3—92no—
77C2×C4×Dic3—91no—
78C2×C4×Dic3—181no—
79C3×#95—92no—
80C3×#95—91no—
81C3⋊#95—32no—
82C3⋊#95—32no—
83C3⋊#95—32no—
84C3⋊#95—92no—
85C3⋊#95—91no—
86C3⋊#95—92no—
87C3⋊#95—92no—
88C3⋊#95—92no—
89C3⋊#95—182no—
90C23×C32—22no—
91C23×C32—21no—
92C23×C32—41no—
93C23×C32C22×D311yescharacteristic
94C23×C32C22×D312yes—
95C2×C32×C4—32no—
96C2×C32×C4—62no—
97C2×C32×C4—62no—
98C2×C32×C4—62no—
99C2×C32×C4—62no—
100C2×C3⋊Dic3—22no—
101C2×C3⋊Dic3—22no—
102C2×C3⋊Dic3—22no—
103C2×C3⋊Dic3—22no—
104C2×C3⋊Dic3—34no—
105C2×C3⋊Dic3—32no—
106C2×C3⋊Dic3—62no—
107C2×C3⋊Dic3—64no—
108C2×C3⋊Dic3—61no—
109C2×C3⋊Dic3—61no—
110C2×C3⋊Dic3—62no—
111C2×C3⋊Dic3—62no—
112C2×C3⋊Dic3—62no—
113C2×C3⋊Dic3—62no—
114C2×C3⋊Dic3—122no—
115C2×C3⋊Dic3—122no—
116C2×C3⋊Dic3—122no—
117C2×C3⋊Dic3—122no—
118C2×C3⋊Dic3—122no—
119C2×C3⋊Dic3C4×D312yes—
120C2×C3×Dic3—24no—
121C2×C3×Dic3—22no—
122C2×C3×Dic3—22no—
123C2×C3×Dic3—24no—
124C2×C3×Dic3—22no—
125C2×C3×Dic3—22no—
126C2×C3×Dic3—22no—
127C2×C3×Dic3—22no—
128C2×C3×Dic3—22no—
129C2×C3×Dic3—22no—
130C2×C3×Dic3—22no—
131C2×C3×Dic3—34no—
132C2×C3×Dic3—42no—
133C2×C3×Dic3—42no—
134C2×C3×Dic3—62no—
135C2×C3×Dic3—64no—
136C2×C3×Dic3—62no—
137C2×C3×Dic3—62no—
138C2×C3×Dic3—64no—
139C2×C3×Dic3—62no—
140C2×C3×Dic3—62no—
141C2×C3×Dic3—62no—
142C2×C3×Dic3—62no—
143C2×C3×Dic3—62no—
144C2×C3×Dic3—62no—
145C2×C3×Dic3—122no—
146C2×C3×Dic3—122no—
147C2×C3×Dic3C4×D314yes—
148#383—271noSylow
149C2×C33#11711yescharacteristic
150C2×C33#11711yescharacteristic
151C2×C33#11711yescharacteristic
152C2×C33#12612yes—
153C2×C33#12612yes—
154C22×C3×C4—32no—
155C22×C3×C4—32no—
156C22×C3×C4—32no—
157C22×C3×C4—61no—
158C22×C3×C4—62no—
159C22×C3×C4—91no—
160C22×C3×C4—92no—
161C22×Dic3—32no—
162C22×Dic3—32no—
163C22×Dic3—32no—
164C22×Dic3—61no—
165C22×Dic3—62no—
166C22×Dic3—92no—
167C22×Dic3—91no—
168C22×Dic3—181no—
169C22×Dic3—182no—
170C22×Dic3—361no—
171C22×Dic3D3212yes—
172C22×Dic3D3211yescharacteristic
173C4×Dic3—98no—
174C4×Dic3—94no—
175C4×Dic3—184no—
176C22×C32—22no—
177C22×C32—22no—
178C22×C32—22no—
179C22×C32—21no—
180C22×C32—21no—
181C22×C32—22no—
182C22×C32—22no—
183C22×C32—22no—
184C22×C32—22no—
185C22×C32—22no—
186C22×C32—22no—
187C22×C32—21no—
188C22×C32—41no—
189C22×C32—41no—
190C22×C32—41no—
191C22×C32—42no—
192C22×C32—42no—
193C22×C32C2×C4×D311yescharacteristic
194C22×C32C2×C4×D312yes—
195C22×C32#23512yes—
196C22×C32#23512yes—
197C22×C32#23512yes—
198C22×C32#23712yes—
199C22×C32#23712yes—
200C22×C32#23712yes—
201C22×C32#23711yescharacteristic
202C22×C32#23712yes—
203C22×C32#23912yes—
204C22×C32#23911yescharacteristic
205C32×C4—62no—
206C32×C4—122no—
207C3⋊Dic3—22no—
208C3⋊Dic3—42no—
209C3⋊Dic3—62no—
210C3⋊Dic3—64no—
211C3⋊Dic3—64no—
212C3⋊Dic3—124no—
213C3⋊Dic3—124no—
214C3×Dic3—24no—
215C3×Dic3—24no—
216C3×Dic3—44no—
217C3×Dic3—42no—
218C3×Dic3—42no—
219C3×Dic3—64no—
220C3×Dic3—64no—
221C3×Dic3—124no—
222C3×Dic3—122no—
223C3×Dic3—122no—
224C2×C42—271no—
225#95—92no—
226#95—91no—
227#95—272no—
228#95—271no—
229C33#38311yescharacteristic, complemented, Sylow
230C23×C3—21no—
231C23×C3—22no—
232C23×C3—41no—
233C23×C3C2×D3212yes—
234C23×C3C2×D3211yescharacteristic
235C2×C3×C4—34no—
236C2×C3×C4—62no—
237C2×C3×C4—64no—
238C2×C3×C4—62no—
239C2×C3×C4—62no—
240C2×C3×C4—64no—
241C2×C3×C4—62no—
242C2×C3×C4—62no—
243C2×C3×C4—62no—
244C2×C3×C4—62no—
245C2×C3×C4—62no—
246C2×C3×C4—62no—
247C2×C3×C4—92no—
248C2×C3×C4—122no—
249C2×C3×C4—122no—
250C2×C3×C4—182no—
251C2×C3×C4—182no—
252C2×C3×C4—182no—
253C2×C3×C4—182no—
254C2×Dic3—22no—
255C2×Dic3—22no—
256C2×Dic3—22no—
257C2×Dic3—22no—
258C2×Dic3—34no—
259C2×Dic3—62no—
260C2×Dic3—64no—
261C2×Dic3—62no—
262C2×Dic3—62no—
263C2×Dic3—64no—
264C2×Dic3—62no—
265C2×Dic3—62no—
266C2×Dic3—62no—
267C2×Dic3—62no—
268C2×Dic3—62no—
269C2×Dic3—62no—
270C2×Dic3—94no—
271C2×Dic3—92no—
272C2×Dic3—122no—
273C2×Dic3—122no—
274C2×Dic3—182no—
275C2×Dic3—184no—
276C2×Dic3—182no—
277C2×Dic3—182no—
278C2×Dic3—181no—
279C2×Dic3—181no—
280C2×Dic3—182no—
281C2×Dic3—182no—
282C2×Dic3—362no—
283C2×Dic3—362no—
284C2×Dic3—362no—
285C2×Dic3—362no—
286C2×Dic3—362no—
287C2×Dic3#62412yes—
288C2×C32—22no—
289C2×C32—22no—
290C2×C32—21no—
291C2×C32—22no—
292C2×C32—22no—
293C2×C32—22no—
294C2×C32—22no—
295C2×C32—21no—
296C2×C32—22no—
297C2×C32—22no—
298C2×C32—22no—
299C2×C32—21no—
300C2×C32—41no—
301C2×C32—41no—
302C2×C32—41no—
303C2×C32—42no—
304C2×C32—42no—
305C2×C32#87212yes—
306C2×C32#87211yescharacteristic
307C2×C32#87712yes—
308C2×C32#87711yescharacteristic
309C2×C32#87912yes—
310C2×C32#87912yes—
311C2×C32#87912yes—
312C2×C32#89211yescharacteristic
313C2×C32#89212yes—
314C2×C32#89812yes—
315C2×C32#89812yes—
316C2×C32#89812yes—
317C22×C4—31no—
318C22×C4—32no—
319C22×C4—91no—
320C22×C4—92no—
321C22×C4—271no—
322C42—274no—
323C22×C3—21no—
324C22×C3—22no—
325C22×C3—21no—
326C22×C3—22no—
327C22×C3—22no—
328C22×C3—22no—
329C22×C3—22no—
330C22×C3—22no—
331C22×C3—22no—
332C22×C3—22no—
333C22×C3—22no—
334C22×C3—21no—
335C22×C3—41no—
336C22×C3—41no—
337C22×C3—42no—
338C22×C3—42no—
339C22×C3—41no—
340C22×C3#379512yes—
341C22×C3#379612yes—
342C22×C3#379612yes—
343C22×C3#379811yescharacteristic
344C22×C3#379811yescharacteristic
345C22×C3#379912yes—
346C22×C3C4×D3212yes—
347C22×C3#380412yes—
348C22×C3#380412yes—
349C22×C3#380512yes—
350C22×C3#380512yes—
351C22×C3C2×#62411yescharacteristic
352C3×C4—64no—
353C3×C4—64no—
354C3×C4—124no—
355C3×C4—122no—
356C3×C4—122no—
357C3×C4—182no—
358C3×C4—362no—
359Dic3—22no—
360Dic3—42no—
361Dic3—64no—
362Dic3—64no—
363Dic3—124no—
364Dic3—122no—
365Dic3—122no—
366Dic3—184no—
367Dic3—184no—
368Dic3—182no—
369Dic3—364no—
370Dic3—364no—
371C32—22no—
372C32—21no—
373C32—41no—
374C32#465111yescharacteristic, complemented
375C32#465112yescomplemented
376C23D3311yescharacteristic
377C2×C4—32no—
378C2×C4—62no—
379C2×C4—62no—
380C2×C4—62no—
381C2×C4—62no—
382C2×C4—92no—
383C2×C4—182no—
384C2×C4—182no—
385C2×C4—182no—
386C2×C4—182no—
387C2×C4—272no—
388C2×C4—541no—
389C2×C4—541no—
390C2×C3—22no—
391C2×C3—22no—
392C2×C3—22no—
393C2×C3—22no—
394C2×C3—22no—
395C2×C3—22no—
396C2×C3—21no—
397C2×C3—22no—
398C2×C3—21no—
399C2×C3—22no—
400C2×C3—21no—
401C2×C3—22no—
402C2×C3—42no—
403C2×C3—42no—
404C2×C3—41no—
405C2×C3—41no—
406C2×C3—41no—
407C2×C3#6383612yes—
408C2×C3#6384812yes—
409C2×C3#6384911yescharacteristic
410C2×C3#6385812yes—
411C2×C3#6386112yes—
412C2×C3#6386212yes—
413C2×C3#6386212yes—
414C2×C3#6387411yescharacteristic
415C2×C3#6387712yes—
416C2×C3#6387812yes—
417C2×C3#6395612yes—
418C2×C3#6395711yescharacteristic
419C22D3×#62411yescharacteristic
420C22#8895911yescharacteristic
421C22#8896112yes—
422C22#8896412yes—
423C22#8896511yescharacteristic
424C4—62no—
425C4—122no—
426C4—182no—
427C4—362no—
428C4—542no—
429C3—22no—
430C3—21no—
431C3—41no—
432C3#1059122911yescharacteristic, complemented
433C3#1059124812yescomplemented
434C2#1172291611yescharacteristic
435C2#1172291811yescharacteristic
436C2#1172292111yescharacteristic
437C2#1172301912yes—
438C2#1172302712yes—
subgroupG/H
center, Frattini, hypercenterC23D33
commutatorC22×C33C22×C4
Fitting, socle, prime radical, cosocleC23×C33C23
hypocenterC33#383
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
13. Dic3×C3⋊Dic3C4semidirect
229. C33#383semidirect
374. C32#4651semidirect
375. C32#4651semidirect
432. C3#10591229semidirect
433. C3#10591248semidirect

Maximal subgroups

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

subgroupindex
C2×Dic3×C3⋊Dic32
#117220822
#117220612
C3×#635792
#105912483
#105912293