Finite Group Databasebeta

← #13156482

Subgroups — #13156482

Order 1728.

580 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723426C211yesmaximal, characteristic
2C3×#63841C211yesmaximal, characteristic
3#11722079C211yesmaximal, characteristic, complemented
4#11722042C211yesmaximal, characteristic, complemented
5#11722041C211yesmaximal, characteristic, complemented
6C3×#63575C211yesmaximal, characteristic, complemented
7C3×(C2×D3)⋊(C3⋊C8)C211yesmaximal, characteristic
8#10591134—31nomaximal
9#10590708—31nomaximal
10#10590429—31nomaximal
11C2×#6308C2211yescharacteristic
12C2×C32×C4×D3C2211yescharacteristic
13C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88807—21no—
19(C2×D3)⋊(C3⋊Dic3)—21no—
20C3×(C2×D3)⋊Dic3—21no—
21C3×Dic32—21no—
22#64133—31no—
23C3×#926—31no—
24C3×#892—31no—
25#63857—31no—
26#63857—31no—
27#63857—61no—
28#63841D311yescharacteristic
29#63643—31no—
30#63637—31no—
31C3×#830—31no—
32C3×(C2×D3)⋊C8—31no—
33C3×#818—31no—
34C3×#807—31no—
35#63575D311yescharacteristic, complemented
36#63574—31no—
37#63573—31no—
38#63573—31no—
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×C3⋊(C3⋊C8)—21no—
58C32×C3⋊C8—21no—
59#4996—91no—
60#4888—91no—
61#4723—91no—
62C22×C32×C4—31no—
63C2×#634—31no—
64C2×#634—31no—
65C2×#634—31no—
66C2×#634—61no—
67C2×#634—61no—
68C2×#634—61no—
69C2×#634—121no—
70C2×C3×C4×D3—21no—
71C2×C3×C4×D3C2×D311yescharacteristic
72C2×C3×C4×D3C2×D311yescharacteristic
73C2×C32×C8—31no—
74C32×C4⋊C4—31no—
75#3757—61no—
76C4⋊(C3⋊Dic3)C2×D311yescharacteristic
77C4⋊(C3⋊Dic3)C2×D311yescharacteristic
78#3750—61no—
79C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
80C3×#217—61no—
81C3×(C2×D3)⋊C4—61no—
82C3×C4⋊Dic3—21no—
83C3×C4⋊Dic3—31no—
84C3×C4⋊Dic3—31no—
85C3×C4⋊Dic3—31no—
86C3×C4⋊Dic3—31no—
87C3×C4⋊Dic3—61no—
88C3×C4⋊Dic3—61no—
89C3×C4⋊Dic3C2×D311yescharacteristic
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×C4×Dic3—61no—
92C3×C4×Dic3—61no—
93C2×C3×C3⋊C8—21no—
94C2×C3×C3⋊C8—31no—
95C2×C3×C3⋊C8—31no—
96C2×C3×C3⋊C8—61no—
97C2×C3×C3⋊C8C2×D311yescharacteristic
98C2×C3×C3⋊C8C2×D311yescharacteristic
99Dic3⋊Dic3—62no—
100Dic3⋊Dic3—61no—
101Dic3⋊Dic3—61no—
102(C2×D3)⋊Dic3—21no—
103(C2×D3)⋊Dic3—62no—
104(C2×D3)⋊Dic3—61no—
105(C2×D3)⋊Dic3—61no—
106Dic32—21no—
107C22×C33C2×D411yescharacteristic
108C2×C32×D3—21no—
109C2×C32×D3—21no—
110C2×C32×D3—21no—
111C33×C4#4111yescharacteristic
112C33×C4#4111yescharacteristic
113C3⋊(C3⋊Dic3)—41no—
114C3×C3⋊Dic3—41no—
115C3×C3⋊Dic3—41no—
116C32×Dic3—21no—
117C32×Dic3—41no—
118C3×#117—91no—
119#926—31no—
120#924—91no—
121#924—91no—
122#924—181no—
123#897—91no—
124#892—31no—
125C3×#107—91no—
126C3×C22⋊C8—91no—
127#830—31no—
128(C2×D3)⋊C8—31no—
129#818—31no—
130#817—91no—
131#816—91no—
132#816—91no—
133#816—91no—
134#816—181no—
135#810—91no—
136#810—91no—
137#810—91no—
138#810—181no—
139#808—91no—
140#808—91no—
141#807—31no—
142C23×C32—31no—
143C22×C3×D3—21no—
144C22×C3×D3D1211yescharacteristic
145C22×C3×D3#6711yescharacteristic
146C2×C32×C4—21no—
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4—31no—
150C2×C32×C4—41no—
151C2×C32×C4—61no—
152C2×C32×C4—61no—
153C2×C32×C4—61no—
154C2×C32×C4C22×D311yescharacteristic
155C2×C32×C4C22×D311yescharacteristic
156C2×C32×C4C22×D311yescharacteristic
157C2×C3⋊Dic3—21no—
158C2×C3⋊Dic3—21no—
159C2×C3⋊Dic3—61no—
160C2×C3⋊Dic3—62no—
161C2×C3⋊Dic3—61no—
162C2×C3⋊Dic3—61no—
163C2×C3⋊Dic3—121no—
164C2×C3⋊Dic3—121no—
165C2×C3⋊Dic3—122no—
166#634—61no—
167#634—61no—
168#634—62no—
169#634—62no—
170#634—61no—
171#634—61no—
172#634—61no—
173#634—61no—
174#634—121no—
175#634—121no—
176#634—121no—
177#634—121no—
178#634—122no—
179#634—122no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3—41no—
184C2×C3×Dic3—62no—
185C2×C3×Dic3—61no—
186C2×C3×Dic3—61no—
187C2×C3×Dic3—61no—
188C2×C3×Dic3—61no—
189C2×C3×Dic3—121no—
190C2×C3×Dic3D1211yescharacteristic
191C2×C3×Dic3#6711yescharacteristic
192C3×C4×D3—21no—
193C3×C4×D3—21no—
194C3×C4×D3—21no—
195C3×C4×D3—21no—
196C3×C4×D3—41no—
197C3×C4×D3—41no—
198C32×C8—61no—
199C3⋊(C3⋊C8)—21no—
200C3×C3⋊C8—21no—
201C3×C3⋊C8—21no—
202C3×C3⋊C8—41no—
203C3×C3⋊C8—61no—
204C3×C3⋊C8—61no—
205C3×C3⋊C8—121no—
206#485—271noSylow
207C2×C33#12311yescharacteristic
208C2×C33#13511yescharacteristic
209C2×C33#13711yescharacteristic
210C32×D3—41no—
211C22×C3×C4—31no—
212C22×C3×C4—31no—
213C22×C3×C4—61no—
214C2×C4×D3D3211yescharacteristic
215C2×Dic6—91no—
216C2×Dic6—91no—
217C2×Dic6—91no—
218C2×Dic6—181no—
219C2×Dic6—181no—
220C2×Dic6—181no—
221C2×Dic6—361no—
222C2×C3×C8—31no—
223C2×C3×C8—31no—
224C2×C3×C8—61no—
225C2×C3×C8—91no—
226C3×C4⋊C4—31no—
227C3×C4⋊C4—31no—
228C3×C4⋊C4—61no—
229C3×C4⋊C4—91no—
230C3×C4⋊C4—91no—
231C3×C22⋊C4—181no—
232C3×C42—181no—
233#217—61no—
234#217—181no—
235#217—181no—
236#217—182no—
237(C2×D3)⋊C4—61no—
238(C2×D3)⋊C4—181no—
239C4⋊Dic3—31no—
240C4⋊Dic3—31no—
241C4⋊Dic3—31no—
242C4⋊Dic3—31no—
243C4⋊Dic3—61no—
244C4⋊Dic3—61no—
245C4⋊Dic3—181no—
246C4⋊Dic3D3211yescharacteristic
247Dic3⋊C4—181no—
248Dic3⋊C4—181no—
249Dic3⋊C4—182no—
250C4×Dic3—61no—
251C4×Dic3—61no—
252C2×C3⋊C8—31no—
253C2×C3⋊C8—31no—
254C2×C3⋊C8—61no—
255C2×C3⋊C8D3211yescharacteristic
256C22×C32—21no—
257C22×C32—21no—
258C22×C32—21no—
259C22×C32—41no—
260C22×C32—61no—
261C22×C32—61no—
262C22×C32—61no—
263C22×C32C2×D1211yescharacteristic
264C22×C32D3×D411yescharacteristic
265C22×C32C2×#6711yescharacteristic
266C2×C3×D3—22no—
267C2×C3×D3—21no—
268C2×C3×D3—21no—
269C2×C3×D3—21no—
270C2×C3×D3—21no—
271C2×C3×D3—21no—
272C2×C3×D3—21no—
273C2×C3×D3—41no—
274C2×C3×D3—41no—
275C32×C4—21no—
276C32×C4—21no—
277C32×C4—21no—
278C32×C4—21no—
279C32×C4—21no—
280C32×C4—21no—
281C32×C4—41no—
282C32×C4—41no—
283C32×C4—61no—
284C32×C4—121no—
285C32×C4#23511yescharacteristic
286C32×C4#23511yescharacteristic
287C32×C4#23511yescharacteristic
288C32×C4#23711yescharacteristic
289C32×C4#23711yescharacteristic
290C32×C4#23911yescharacteristic
291C3⋊Dic3—41no—
292C3⋊Dic3—41no—
293C3⋊Dic3—122no—
294C3⋊Dic3—122no—
295C3⋊Dic3—122no—
296C3⋊Dic3—124no—
297C3⋊Dic3—121no—
298C3⋊Dic3—121no—
299C3⋊Dic3—121no—
300C3×Dic3—21no—
301C3×Dic3—21no—
302C3×Dic3—41no—
303C3×Dic3—41no—
304C3×Dic3—41no—
305C3×Dic3—81no—
306C3×Dic3—122no—
307C3×Dic3—121no—
308C3×Dic3—121no—
309C3×Dic3—121no—
310C3×Dic3—121no—
311C3×Dic3—241no—
312C22⋊Q8—271no—
313#117—91no—
314#107—91no—
315#106—271no—
316#103—271no—
317#103—271no—
318C22⋊C8—91no—
319C33#48511yescharacteristic, complemented, Sylow
320C23×C3—31no—
321C23×C3—31no—
322C23×C3—61no—
323C22×D3#62611yescharacteristic
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—31no—
328C2×C3×C4—31no—
329C2×C3×C4—41no—
330C2×C3×C4—61no—
331C2×C3×C4—61no—
332C2×C3×C4—61no—
333C2×C3×C4—61no—
334C2×C3×C4—61no—
335C2×C3×C4—61no—
336C2×C3×C4—61no—
337C2×C3×C4—121no—
338C2×C3×C4—121no—
339C2×C3×C4—121no—
340C2×C3×C4—181no—
341C2×C3×C4—181no—
342C2×C3×C4C2×D3211yescharacteristic
343C2×C3×C4C2×D3211yescharacteristic
344C2×C3×C4C2×D3211yescharacteristic
345C2×Dic3—21no—
346C2×Dic3—62no—
347C2×Dic3—61no—
348C2×Dic3—61no—
349C2×Dic3—61no—
350C2×Dic3—61no—
351C2×Dic3—121no—
352C2×Dic3—182no—
353C2×Dic3—181no—
354C2×Dic3—181no—
355C2×Dic3—181no—
356C2×Dic3—361no—
357C2×Dic3—361no—
358C2×Dic3—362no—
359C2×Dic3#62611yescharacteristic
360C4×D3—21no—
361C4×D3—21no—
362Dic6—182no—
363Dic6—182no—
364Dic6—181no—
365Dic6—181no—
366Dic6—181no—
367Dic6—181no—
368Dic6—181no—
369Dic6—181no—
370Dic6—361no—
371Dic6—361no—
372Dic6—361no—
373Dic6—361no—
374Dic6—362no—
375Dic6—362no—
376C3×C8—61no—
377C3×C8—61no—
378C3×C8—121no—
379C3×C8—181no—
380C3⋊C8—21no—
381C3⋊C8—61no—
382C3⋊C8—61no—
383C3⋊C8—121no—
384C2×C32—21no—
385C2×C32—21no—
386C2×C32—21no—
387C2×C32—21no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—21no—
391C2×C32—21no—
392C2×C32—21no—
393C2×C32—41no—
394C2×C32—41no—
395C2×C32—41no—
396C2×C32—121no—
397C2×C32#88611yescharacteristic
398C2×C32#89411yescharacteristic
399C2×C32#90411yescharacteristic
400C2×C32#90911yescharacteristic
401C2×C32#91211yescharacteristic
402C2×C32#91811yescharacteristic
403C2×C32#92911yescharacteristic
404C2×C32#95011yescharacteristic
405C2×C32#95111yescharacteristic
406C3×D3—42no—
407C3×D3—41no—
408C3×D3—41no—
409C2×Q8—271no—
410C22×C4—31no—
411C2×C8—31no—
412C2×C8—91no—
413C4⋊C4—31no—
414C4⋊C4—91no—
415C4⋊C4—91no—
416C4⋊C4—541no—
417C22⋊C4—181no—
418C22⋊C4—541no—
419C42—181no—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—41no—
424C22×C3—62no—
425C22×C3—61no—
426C22×C3—61no—
427C22×C3—61no—
428C22×C3—61no—
429C22×C3—61no—
430C22×C3—61no—
431C22×C3—121no—
432C22×C3—121no—
433C22×C3D3×D1211yescharacteristic
434C22×C3C2×#62611yescharacteristic
435C22×C3D3×#6711yescharacteristic
436C2×D3—21no—
437C2×D3—21no—
438C2×D3—21no—
439C3×C4—21no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—21no—
444C3×C4—21no—
445C3×C4—41no—
446C3×C4—41no—
447C3×C4—61no—
448C3×C4—61no—
449C3×C4—121no—
450C3×C4—121no—
451C3×C4—121no—
452C3×C4—241no—
453C3×C4—361no—
454C3×C4—361no—
455C3×C4#379611yescharacteristic
456C3×C4#379811yescharacteristic
457C3×C4#379911yescharacteristic
458C3×C4#380411yescharacteristic
459C3×C4#380411yescharacteristic
460C3×C4#380411yescharacteristic
461Dic3—21no—
462Dic3—41no—
463Dic3—122no—
464Dic3—121no—
465Dic3—121no—
466Dic3—121no—
467Dic3—121no—
468Dic3—241no—
469Dic3—362no—
470Dic3—362no—
471Dic3—362no—
472Dic3—364no—
473Dic3—361no—
474Dic3—361no—
475Dic3—361no—
476C32—21no—
477C32—21no—
478C32—21no—
479C32—41no—
480C32#472311yescharacteristic, complemented
481C32#488811yescharacteristic, complemented
482C32#499611yescharacteristic, complemented
483C23—31no—
484Q8—541no—
485Q8—541no—
486C2×C4—31no—
487C2×C4—61no—
488C2×C4—61no—
489C2×C4—61no—
490C2×C4—181no—
491C2×C4—181no—
492C2×C4—541no—
493C2×C4D3311yescharacteristic
494C8—61no—
495C8—181no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—41no—
506C2×C3—41no—
507C2×C3—41no—
508C2×C3—122no—
509C2×C3—121no—
510C2×C3—121no—
511C2×C3#6380111yescharacteristic
512C2×C3#6380711yescharacteristic
513C2×C3#6382911yescharacteristic
514C2×C3#6383111yescharacteristic
515C2×C3#6385011yescharacteristic
516C2×C3#6389011yescharacteristic
517C2×C3#6393311yescharacteristic
518C2×C3#6394311yescharacteristic
519C2×C3#6396611yescharacteristic
520D3—41no—
521C22—61no—
522C22—61no—
523C22—61no—
524C22D3×#62611yescharacteristic
525C4—61no—
526C4—121no—
527C4—361no—
528C4—361no—
529C4—1081no—
530C4#8895911yescharacteristic
531C4#8896211yescharacteristic
532C3—21no—
533C3—21no—
534C3—21no—
535C3—41no—
536C3#1059042911yescharacteristic, complemented
537C3#1059070811yescharacteristic, complemented
538C3#1059113411yescharacteristic, complemented
539C2—121no—
540C2#1172285411yescharacteristic
541C2#1172288711yescharacteristic
542C2#1172298711yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC22×C32×D3D4
hypocenterC33#485
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×#63575C2semidirect
35. #63575D3semidirect
319. C33#485semidirect
480. C32#4723semidirect
481. C32#4888semidirect
482. C32#4996semidirect
536. C3#10590429semidirect
537. C3#10590708semidirect
538. C3#10591134semidirect

Maximal subgroups

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

subgroupindex
#117234262
C3×#638412
#117220792
#117220422
#117220412
C3×#635752
C3×(C2×D3)⋊(C3⋊C8)2
#105911343
#105907083
#105904293