Finite Group Databasebeta

← #13156448

Subgroups — #13156448

Order 1728.

612 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723426C211yesmaximal, characteristic
2C3×D3×C4⋊Dic3C211yesmaximal, characteristic
3#11722079C211yesmaximal, characteristic, complemented
4#11722042C211yesmaximal, characteristic, complemented
5#11722041C211yesmaximal, characteristic, complemented
6C3×#63573C211yesmaximal, characteristic, complemented
7C3×(C2×D3)⋊(C3⋊C8)C211yesmaximal, characteristic
8#10591124—31nomaximal
9#10590674—31nomaximal
10#10590426—31nomaximal
11C2×#6308C2211yescharacteristic
12C2×C32×C4×D3C2211yescharacteristic
13C2×C3×D3×Dic3—21no—
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88807—21no—
20(C2×D3)⋊(C3⋊Dic3)—21no—
21C3×Dic3⋊Dic3—21no—
22#64133—31no—
23C2×C3×C4⋊Dic3—31no—
24C3×D3×C4⋊C4—31no—
25D3×C4⋊Dic3D311yescharacteristic
26#63857—31no—
27#63857—31no—
28#63857—61no—
29#63643—31no—
30#63637—31no—
31C3×#830—31no—
32C3×(C2×D3)⋊C8—31no—
33C3×#817—31no—
34C3×#808—31no—
35#63574—31no—
36#63573—31no—
37#63573—31no—
38#63573D311yescharacteristic, complemented
39#63567—31no—
40#63567—31no—
41#63567—31no—
42#63567—31no—
43#63567—61no—
44#63567—61no—
45(C2×D3)⋊(C3⋊C8)D311yescharacteristic
46C22×C32×D3D411yescharacteristic
47C2×C33×C4C2311yescharacteristic
48C2×C3⋊(C3⋊Dic3)—21no—
49#6308—21no—
50#6308—21no—
51C2×C3×C3⋊Dic3—21no—
52C2×C3×C3⋊Dic3—21no—
53C2×C32×Dic3—21no—
54C2×C32×Dic3D411yescharacteristic
55C32×C4×D3—21no—
56C32×C4×D3—21no—
57C3×D3×Dic3—42no—
58C3×C3⋊(C3⋊C8)—21no—
59C32×C3⋊C8—21no—
60#4971—91no—
61#4862—91no—
62#4721—91no—
63C22×C32×C4—31no—
64C2×#634—31no—
65C2×#634—31no—
66C2×#634—31no—
67C2×#634—61no—
68C2×#634—61no—
69C2×#634—61no—
70C2×#634—121no—
71C22×C3×Dic3—61no—
72C2×C3×C4×D3—21no—
73C2×C3×C4×D3—61no—
74C2×C3×C4×D3C2×D311yescharacteristic
75C2×C3×C4×D3C2×D311yescharacteristic
76C2×D3×Dic3—21no—
77C2×C32×C8—31no—
78C32×C4⋊C4—31no—
79#3757—61no—
80C4⋊(C3⋊Dic3)C2×D311yescharacteristic
81C4⋊(C3⋊Dic3)C2×D311yescharacteristic
82#3750—61no—
83C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
84C3×C4⋊Dic3—21no—
85C3×C4⋊Dic3—31no—
86C3×C4⋊Dic3—31no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—31no—
89C3×C4⋊Dic3—61no—
90C3×C4⋊Dic3—61no—
91C3×C4⋊Dic3—61no—
92C3×C4⋊Dic3C2×D311yescharacteristic
93C3×C4⋊Dic3C2×D311yescharacteristic
94C3×Dic3⋊C4—61no—
95C2×C3×C3⋊C8—21no—
96C2×C3×C3⋊C8—31no—
97C2×C3×C3⋊C8—31no—
98C2×C3×C3⋊C8—61no—
99C2×C3×C3⋊C8C2×D311yescharacteristic
100C2×C3×C3⋊C8C2×D311yescharacteristic
101Dic3⋊Dic3—21no—
102Dic3⋊Dic3—62no—
103Dic3⋊Dic3—61no—
104Dic3⋊Dic3—61no—
105(C2×D3)⋊Dic3—62no—
106(C2×D3)⋊Dic3—61no—
107(C2×D3)⋊Dic3—61no—
108C22×C33C2×D411yescharacteristic
109C2×C32×D3—21no—
110C2×C32×D3—21no—
111C2×C32×D3SD1612yes—
112C33×C4#4111yescharacteristic
113C33×C4#4111yescharacteristic
114C3⋊(C3⋊Dic3)—41no—
115C3×C3⋊Dic3—41no—
116C3×C3⋊Dic3—41no—
117C32×Dic3—21no—
118C32×Dic3—41no—
119C2×C3×C4⋊C4—91no—
120C2×C4⋊Dic3—31no—
121#924—91no—
122#924—91no—
123#924—181no—
124#897—91no—
125D3×C4⋊C4—31no—
126C3×#106—91no—
127C3×C22⋊C8—91no—
128#830—31no—
129(C2×D3)⋊C8—31no—
130#817—31no—
131#817—91no—
132#816—91no—
133#816—91no—
134#816—91no—
135#816—181no—
136#810—91no—
137#810—91no—
138#810—91no—
139#810—181no—
140#808—31no—
141#808—91no—
142#808—91no—
143C23×C32—31no—
144C22×C3×D3—21no—
145C22×C3×D3D1211yescharacteristic
146C22×C3×D3#6711yescharacteristic
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4—21no—
150C2×C32×C4—31no—
151C2×C32×C4—41no—
152C2×C32×C4—61no—
153C2×C32×C4—61no—
154C2×C32×C4—61no—
155C2×C32×C4C22×D311yescharacteristic
156C2×C32×C4C22×D311yescharacteristic
157C2×C32×C4C22×D311yescharacteristic
158C2×C3⋊Dic3—21no—
159C2×C3⋊Dic3—21no—
160C2×C3⋊Dic3—61no—
161C2×C3⋊Dic3—62no—
162C2×C3⋊Dic3—61no—
163C2×C3⋊Dic3—61no—
164C2×C3⋊Dic3—121no—
165C2×C3⋊Dic3—121no—
166C2×C3⋊Dic3—122no—
167#634—61no—
168#634—61no—
169#634—62no—
170#634—62no—
171#634—61no—
172#634—61no—
173#634—61no—
174#634—61no—
175#634—121no—
176#634—121no—
177#634—121no—
178#634—121no—
179#634—122no—
180#634—122no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3—21no—
184C2×C3×Dic3—41no—
185C2×C3×Dic3—62no—
186C2×C3×Dic3—61no—
187C2×C3×Dic3—61no—
188C2×C3×Dic3—61no—
189C2×C3×Dic3—61no—
190C2×C3×Dic3—121no—
191C2×C3×Dic3—122no—
192C2×C3×Dic3D1211yescharacteristic
193C2×C3×Dic3#6711yescharacteristic
194C3×C4×D3—21no—
195C3×C4×D3—21no—
196C3×C4×D3—21no—
197C3×C4×D3—21no—
198C3×C4×D3—41no—
199C3×C4×D3—41no—
200C3×C4×D3—122no—
201D3×Dic3—42no—
202C32×C8—61no—
203C3⋊(C3⋊C8)—21no—
204C3×C3⋊C8—21no—
205C3×C3⋊C8—21no—
206C3×C3⋊C8—41no—
207C3×C3⋊C8—61no—
208C3×C3⋊C8—61no—
209C3×C3⋊C8—121no—
210#483—271noSylow
211C2×C33#12311yescharacteristic
212C2×C33C2×SD1611yescharacteristic
213C2×C33#13711yescharacteristic
214C32×D3—22no—
215C22×C3×C4—31no—
216C22×C3×C4—31no—
217C22×C3×C4—61no—
218C22×C3×C4—181no—
219C22×Dic3—61no—
220C2×C4×D3—61no—
221C2×C4×D3D3211yescharacteristic
222C2×Dic6—91no—
223C2×Dic6—91no—
224C2×Dic6—91no—
225C2×Dic6—181no—
226C2×Dic6—181no—
227C2×Dic6—181no—
228C2×Dic6—361no—
229C2×C3×C8—31no—
230C2×C3×C8—31no—
231C2×C3×C8—61no—
232C2×C3×C8—91no—
233C3×C4⋊C4—31no—
234C3×C4⋊C4—31no—
235C3×C4⋊C4—61no—
236C3×C4⋊C4—91no—
237C3×C4⋊C4—91no—
238C3×C4⋊C4—181no—
239#217—181no—
240#217—181no—
241#217—182no—
242(C2×D3)⋊C4—181no—
243C4⋊Dic3—31no—
244C4⋊Dic3—31no—
245C4⋊Dic3—31no—
246C4⋊Dic3—31no—
247C4⋊Dic3—61no—
248C4⋊Dic3—61no—
249C4⋊Dic3—61no—
250C4⋊Dic3—181no—
251C4⋊Dic3D3211yescharacteristic
252Dic3⋊C4—61no—
253Dic3⋊C4—181no—
254Dic3⋊C4—181no—
255Dic3⋊C4—182no—
256C2×C3⋊C8—31no—
257C2×C3⋊C8—31no—
258C2×C3⋊C8—61no—
259C2×C3⋊C8D3211yescharacteristic
260C22×C32—21no—
261C22×C32—21no—
262C22×C32—21no—
263C22×C32—32no—
264C22×C32—41no—
265C22×C32—61no—
266C22×C32—61no—
267C22×C32C2×D1211yescharacteristic
268C22×C32D3×D411yescharacteristic
269C22×C32C2×#6711yescharacteristic
270C2×C3×D3—21no—
271C2×C3×D3—21no—
272C2×C3×D3—21no—
273C2×C3×D3—21no—
274C2×C3×D3—22no—
275C2×C3×D3—41no—
276C2×C3×D3—41no—
277C2×C3×D3#20412yes—
278C2×C3×D3#21412yes—
279C32×C4—21no—
280C32×C4—21no—
281C32×C4—21no—
282C32×C4—21no—
283C32×C4—21no—
284C32×C4—21no—
285C32×C4—41no—
286C32×C4—41no—
287C32×C4—61no—
288C32×C4—121no—
289C32×C4#23511yescharacteristic
290C32×C4#23511yescharacteristic
291C32×C4#23511yescharacteristic
292C32×C4#23711yescharacteristic
293C32×C4#23711yescharacteristic
294C32×C4#23911yescharacteristic
295C3⋊Dic3—41no—
296C3⋊Dic3—41no—
297C3⋊Dic3—122no—
298C3⋊Dic3—122no—
299C3⋊Dic3—122no—
300C3⋊Dic3—124no—
301C3⋊Dic3—121no—
302C3⋊Dic3—121no—
303C3⋊Dic3—121no—
304C3×Dic3—21no—
305C3×Dic3—21no—
306C3×Dic3—41no—
307C3×Dic3—41no—
308C3×Dic3—41no—
309C3×Dic3—81no—
310C3×Dic3—122no—
311C3×Dic3—121no—
312C3×Dic3—121no—
313C3×Dic3—121no—
314C3×Dic3—121no—
315C3×Dic3—241no—
316C22⋊Q8—271no—
317C2×C4⋊C4—91no—
318#106—91no—
319#106—271no—
320#103—271no—
321#103—271no—
322C22⋊C8—91no—
323C33#48311yescharacteristic, complemented, Sylow
324C23×C3—31no—
325C23×C3—31no—
326C23×C3—61no—
327C22×D3#62611yescharacteristic
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—31no—
332C2×C3×C4—31no—
333C2×C3×C4—41no—
334C2×C3×C4—61no—
335C2×C3×C4—61no—
336C2×C3×C4—61no—
337C2×C3×C4—61no—
338C2×C3×C4—61no—
339C2×C3×C4—61no—
340C2×C3×C4—61no—
341C2×C3×C4—121no—
342C2×C3×C4—121no—
343C2×C3×C4—121no—
344C2×C3×C4—181no—
345C2×C3×C4—181no—
346C2×C3×C4—362no—
347C2×C3×C4C2×D3211yescharacteristic
348C2×C3×C4C2×D3211yescharacteristic
349C2×C3×C4C2×D3211yescharacteristic
350C2×Dic3—21no—
351C2×Dic3—62no—
352C2×Dic3—61no—
353C2×Dic3—61no—
354C2×Dic3—61no—
355C2×Dic3—61no—
356C2×Dic3—121no—
357C2×Dic3—122no—
358C2×Dic3—182no—
359C2×Dic3—181no—
360C2×Dic3—181no—
361C2×Dic3—181no—
362C2×Dic3—361no—
363C2×Dic3—361no—
364C2×Dic3—362no—
365C2×Dic3#62611yescharacteristic
366C4×D3—21no—
367C4×D3—21no—
368C4×D3—122no—
369Dic6—182no—
370Dic6—182no—
371Dic6—181no—
372Dic6—181no—
373Dic6—181no—
374Dic6—181no—
375Dic6—181no—
376Dic6—181no—
377Dic6—361no—
378Dic6—361no—
379Dic6—361no—
380Dic6—361no—
381Dic6—362no—
382Dic6—362no—
383C3×C8—61no—
384C3×C8—61no—
385C3×C8—121no—
386C3×C8—181no—
387C3⋊C8—21no—
388C3⋊C8—61no—
389C3⋊C8—61no—
390C3⋊C8—121no—
391C2×C32—21no—
392C2×C32—21no—
393C2×C32—21no—
394C2×C32—21no—
395C2×C32—21no—
396C2×C32—21no—
397C2×C32—21no—
398C2×C32—21no—
399C2×C32—21no—
400C2×C32—41no—
401C2×C32—41no—
402C2×C32—41no—
403C2×C32—62no—
404C2×C32#88611yescharacteristic
405C2×C32#89411yescharacteristic
406C2×C32C2×#20411yescharacteristic
407C2×C32#90911yescharacteristic
408C2×C32D3×SD1611yescharacteristic
409C2×C32#91511yescharacteristic
410C2×C32#92911yescharacteristic
411C2×C32C2×#21411yescharacteristic
412C2×C32#94211yescharacteristic
413C3×D3—22no—
414C3×D3—22no—
415C3×D3—42no—
416C2×Q8—271no—
417C22×C4—31no—
418C22×C4—181no—
419C2×C8—31no—
420C2×C8—91no—
421C4⋊C4—31no—
422C4⋊C4—91no—
423C4⋊C4—91no—
424C4⋊C4—181no—
425C4⋊C4—541no—
426C22⋊C4—541no—
427C22×C3—21no—
428C22×C3—21no—
429C22×C3—21no—
430C22×C3—32no—
431C22×C3—32no—
432C22×C3—41no—
433C22×C3—62no—
434C22×C3—61no—
435C22×C3—61no—
436C22×C3—61no—
437C22×C3—61no—
438C22×C3—121no—
439C22×C3—121no—
440C22×C3D3×D1211yescharacteristic
441C22×C3C2×#62611yescharacteristic
442C22×C3D3×#6711yescharacteristic
443C2×D3—21no—
444C2×D3—21no—
445C2×D3#371612yes—
446C3×C4—21no—
447C3×C4—21no—
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—21no—
452C3×C4—41no—
453C3×C4—41no—
454C3×C4—61no—
455C3×C4—61no—
456C3×C4—121no—
457C3×C4—121no—
458C3×C4—121no—
459C3×C4—241no—
460C3×C4—361no—
461C3×C4—361no—
462C3×C4#379611yescharacteristic
463C3×C4#379811yescharacteristic
464C3×C4#379911yescharacteristic
465C3×C4#380411yescharacteristic
466C3×C4#380411yescharacteristic
467C3×C4#380411yescharacteristic
468Dic3—21no—
469Dic3—41no—
470Dic3—122no—
471Dic3—121no—
472Dic3—121no—
473Dic3—121no—
474Dic3—121no—
475Dic3—241no—
476Dic3—362no—
477Dic3—362no—
478Dic3—362no—
479Dic3—364no—
480Dic3—361no—
481Dic3—361no—
482Dic3—361no—
483C32—21no—
484C32—21no—
485C32—21no—
486C32—41no—
487C32#472111yescharacteristic, complemented
488C32#486211yescharacteristic, complemented
489C32#497111yescharacteristic, complemented
490C23—31no—
491Q8—541no—
492Q8—541no—
493C2×C4—31no—
494C2×C4—61no—
495C2×C4—61no—
496C2×C4—61no—
497C2×C4—181no—
498C2×C4—181no—
499C2×C4—362no—
500C2×C4—541no—
501C2×C4D3311yescharacteristic
502C8—61no—
503C8—181no—
504C2×C3—21no—
505C2×C3—21no—
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—21no—
510C2×C3—21no—
511C2×C3—21no—
512C2×C3—21no—
513C2×C3—41no—
514C2×C3—41no—
515C2×C3—41no—
516C2×C3—62no—
517C2×C3—62no—
518C2×C3—122no—
519C2×C3D3×#20411yescharacteristic
520C2×C3#6380011yescharacteristic
521C2×C3C2×#371611yescharacteristic
522C2×C3#6383311yescharacteristic
523C2×C3#6385011yescharacteristic
524C2×C3#6389011yescharacteristic
525C2×C3D3×#21411yescharacteristic
526C2×C3#6394611yescharacteristic
527C2×C3#6396611yescharacteristic
528D3—22no—
529C22—32no—
530C22—61no—
531C22—61no—
532C22D3×#62611yescharacteristic
533C4—61no—
534C4—121no—
535C4—361no—
536C4—361no—
537C4—1081no—
538C4#8895911yescharacteristic
539C4#8896211yescharacteristic
540C3—21no—
541C3—21no—
542C3—21no—
543C3—41no—
544C3#1059042611yescharacteristic, complemented
545C3#1059067411yescharacteristic, complemented
546C3#1059112411yescharacteristic, complemented
547C2—62no—
548C2D3×#371611yescharacteristic
549C2#1172288411yescharacteristic
550C2#1172298711yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC22×C32×D3D4
hypocenterC33#483
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. #11722079C2semidirect
4. #11722042C2semidirect
5. #11722041C2semidirect
6. C3×#63573C2semidirect
38. #63573D3semidirect
323. C33#483semidirect
487. C32#4721semidirect
488. C32#4862semidirect
489. C32#4971semidirect
544. C3#10590426semidirect
545. C3#10590674semidirect
546. C3#10591124semidirect

Maximal subgroups

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

subgroupindex
#117234262
C3×D3×C4⋊Dic32
#117220792
#117220422
#117220412
C3×#635732
C3×(C2×D3)⋊(C3⋊C8)2
#105911243
#105906743
#105904263