Finite Group Databasebeta

← #13161818

Subgroups — #13161818

Order 1728.

638 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64135C211yesmaximal, characteristic
2C3×#64133C211yesmaximal, characteristic
3C3×#63657C211yesmaximal, characteristic
4#11722080C211yesmaximal, characteristic, complemented
5#11722079C211yesmaximal, characteristic, complemented
6#11722076C211yesmaximal, characteristic, complemented
7#11722076C211yesmaximal, characteristic, complemented
8#10591134—31nomaximal
9#10591127—31nomaximal
10#10591102—31nomaximal
11C22×C33×C4C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C3×#3757—21no—
14C3×#3757—21no—
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C3×#3750—21no—
19C3×C4×C3⋊Dic3—21no—
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22#64135D311yescharacteristic
23#64133D311yescharacteristic
24C3×#926—31no—
25C3×#926—31no—
26C3×#926—61no—
27C3×#924—31no—
28C3×#924—31no—
29C3×#924—61no—
30#63657D311yescharacteristic
31C3×#830—31no—
32C3×#830—31no—
33C3×#830—61no—
34#63576—31no—
35#63575—31no—
36#63575—31no—
37#63574—31no—
38#63573—31no—
39#63573—31no—
40#63567—31no—
41#63567—31no—
42#63566—31no—
43#63566—31no—
44#63565—31no—
45#63565—31no—
46C23×C33D411yescharacteristic
47C2×C33×C4—21no—
48C2×C33×C4—21no—
49C2×C33×C4D411yescharacteristic
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3—21no—
52C2×C3×C3⋊Dic3—21no—
53C2×C3×C3⋊Dic3—21no—
54C2×C3×C3⋊Dic3—21no—
55C3×#634—21no—
56C3×#634—21no—
57C3×C3⋊(C3⋊C8)—21no—
58C3×C3⋊(C3⋊C8)—21no—
59#5045—91no—
60#4996—91no—
61#4723—91no—
62C22×C32×C4—21no—
63C22×C32×C4—21no—
64C22×C32×C4—21no—
65C22×C32×C4—41no—
66C22×C32×C4C2×D311yescharacteristic
67C22×C32×C4C2×D311yescharacteristic
68C22×C32×C4C2×D311yescharacteristic
69C2×#634C2×D311yescharacteristic
70C2×C3×Dic6—31no—
71C2×C3×Dic6—31no—
72C2×C3×Dic6—61no—
73#3757—21no—
74#3757—21no—
75C4⋊(C3⋊Dic3)C2×D311yescharacteristic
76C4⋊(C3⋊Dic3)C2×D311yescharacteristic
77C4⋊(C3⋊Dic3)C2×D311yescharacteristic
78#3750—21no—
79C4×C3⋊Dic3—21no—
80C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
81C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
82C3×#217—62no—
83C3×#217—62no—
84C3×#217—61no—
85C3×#217—61no—
86C3×#217—61no—
87C3×#217—61no—
88C3×C4⋊Dic3—31no—
89C3×C4⋊Dic3—31no—
90C3×C4⋊Dic3—31no—
91C3×C4⋊Dic3—31no—
92C3×C4⋊Dic3—31no—
93C3×C4⋊Dic3—31no—
94C3×C4⋊Dic3—61no—
95C3×C4⋊Dic3—61no—
96C3×C4⋊Dic3—61no—
97C3×Dic3⋊C4—62no—
98C3×Dic3⋊C4—61no—
99C3×Dic3⋊C4—61no—
100C3×C4×Dic3—62no—
101C3×C4×Dic3—61no—
102C3×C4×Dic3—61no—
103C2×C3×C3⋊C8—31no—
104C2×C3×C3⋊C8—31no—
105C2×C3×C3⋊C8—31no—
106C2×C3×C3⋊C8—31no—
107C2×C3×C3⋊C8—61no—
108C2×C3×C3⋊C8—61no—
109C22×C33—21no—
110C22×C33—21no—
111C22×C33—21no—
112C22×C33C2×D411yescharacteristic
113C33×C4—21no—
114C33×C4#4111yescharacteristic
115C33×C4#4111yescharacteristic
116C3×C3⋊Dic3—41no—
117C3×C3⋊Dic3—41no—
118C3×C3⋊Dic3—41no—
119C3×C3⋊Dic3—41no—
120C3×C22⋊Q8—91no—
121C3×#117—91no—
122#926—31no—
123#926—31no—
124#926—61no—
125#924—31no—
126#924—31no—
127#924—61no—
128C3×C22⋊C8—91no—
129#835—91no—
130#835—91no—
131#830—31no—
132#830—31no—
133#830—61no—
134#818—91no—
135#817—91no—
136#816—91no—
137#816—91no—
138#810—91no—
139#810—91no—
140#808—91no—
141#808—91no—
142#807—91no—
143#807—91no—
144C23×C32—21no—
145C23×C32—21no—
146C23×C32—21no—
147C23×C32—41no—
148C23×C32D1211yescharacteristic
149C23×C32#6711yescharacteristic
150C23×C32#6711yescharacteristic
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—22no—
154C2×C32×C4—22no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—41no—
166C2×C32×C4—41no—
167C2×C32×C4—41no—
168C2×C32×C4—41no—
169C2×C32×C4—41no—
170C2×C32×C4—41no—
171C2×C32×C4—42no—
172C2×C32×C4—42no—
173C2×C32×C4D1211yescharacteristic
174C2×C32×C4#6711yescharacteristic
175C2×C32×C4#6711yescharacteristic
176C2×C32×C4C22×D311yescharacteristic
177C2×C32×C4C22×D311yescharacteristic
178C2×C32×C4C22×D311yescharacteristic
179C2×C3⋊Dic3—21no—
180C2×C3⋊Dic3—21no—
181C2×C3⋊Dic3—21no—
182C2×C3⋊Dic3—21no—
183#634—21no—
184#634—21no—
185C2×C3×Dic3—62no—
186C2×C3×Dic3—62no—
187C2×C3×Dic3—62no—
188C2×C3×Dic3—62no—
189C2×C3×Dic3—61no—
190C2×C3×Dic3—61no—
191C2×C3×Dic3—61no—
192C2×C3×Dic3—61no—
193C2×C3×Dic3—61no—
194C2×C3×Dic3—61no—
195C2×C3×Dic3—61no—
196C2×C3×Dic3—61no—
197C3×Dic6—61no—
198C3×Dic6—61no—
199C3×Dic6—61no—
200C3×Dic6—61no—
201C3×Dic6—121no—
202C3×Dic6—121no—
203C3⋊(C3⋊C8)—21no—
204C3⋊(C3⋊C8)—21no—
205C3×C3⋊C8—61no—
206C3×C3⋊C8—61no—
207C3×C3⋊C8—61no—
208C3×C3⋊C8—61no—
209C3×C3⋊C8—121no—
210C3×C3⋊C8—121no—
211#485—271noSylow
212C2×C33—41no—
213C2×C33#12311yescharacteristic
214C2×C33#13511yescharacteristic
215C2×C33#13711yescharacteristic
216C2×C3×Q8—91no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—41no—
221C22×C3×C4D3211yescharacteristic
222C22×C3×C4D3211yescharacteristic
223C22×C3×C4D3211yescharacteristic
224C2×Dic6—31no—
225C2×Dic6—31no—
226C2×Dic6—61no—
227C2×C3×C8—91no—
228C2×C3×C8—91no—
229C3×C4⋊C4—91no—
230C3×C4⋊C4—91no—
231C3×C4⋊C4—91no—
232C3×C4⋊C4—181no—
233C3×C22⋊C4—181no—
234C3×C22⋊C4—181no—
235C3×C42—181no—
236#217—62no—
237#217—62no—
238#217—61no—
239#217—61no—
240#217—61no—
241#217—61no—
242C4⋊Dic3—31no—
243C4⋊Dic3—31no—
244C4⋊Dic3—31no—
245C4⋊Dic3—31no—
246C4⋊Dic3—31no—
247C4⋊Dic3—31no—
248C4⋊Dic3—61no—
249C4⋊Dic3—61no—
250C4⋊Dic3—61no—
251Dic3⋊C4—62no—
252Dic3⋊C4—61no—
253Dic3⋊C4—61no—
254C4×Dic3—62no—
255C4×Dic3—61no—
256C4×Dic3—61no—
257C2×C3⋊C8—31no—
258C2×C3⋊C8—31no—
259C2×C3⋊C8—31no—
260C2×C3⋊C8—31no—
261C2×C3⋊C8—61no—
262C2×C3⋊C8—61no—
263C22×C32—21no—
264C22×C32—21no—
265C22×C32—21no—
266C22×C32—22no—
267C22×C32—22no—
268C22×C32—22no—
269C22×C32—21no—
270C22×C32—21no—
271C22×C32—21no—
272C22×C32—21no—
273C22×C32—21no—
274C22×C32—21no—
275C22×C32—21no—
276C22×C32—21no—
277C22×C32—21no—
278C22×C32—41no—
279C22×C32—41no—
280C22×C32—41no—
281C22×C32—41no—
282C22×C32—41no—
283C22×C32—41no—
284C22×C32—41no—
285C22×C32—42no—
286C22×C32—42no—
287C22×C32—42no—
288C22×C32C2×D1211yescharacteristic
289C22×C32C2×#6711yescharacteristic
290C22×C32C2×#6711yescharacteristic
291C32×C4—21no—
292C32×C4—22no—
293C32×C4—21no—
294C32×C4—21no—
295C32×C4—21no—
296C32×C4—21no—
297C32×C4—21no—
298C32×C4—21no—
299C32×C4—21no—
300C32×C4—21no—
301C32×C4—41no—
302C32×C4—41no—
303C32×C4—41no—
304C32×C4—41no—
305C32×C4—42no—
306C32×C4#23511yescharacteristic
307C32×C4#23511yescharacteristic
308C32×C4#23711yescharacteristic
309C32×C4#23711yescharacteristic
310C32×C4#23711yescharacteristic
311C32×C4#23711yescharacteristic
312C3⋊Dic3—41no—
313C3⋊Dic3—41no—
314C3⋊Dic3—41no—
315C3⋊Dic3—41no—
316C3×Dic3—122no—
317C3×Dic3—122no—
318C3×Dic3—122no—
319C3×Dic3—122no—
320C3×Dic3—121no—
321C3×Dic3—121no—
322C3×Dic3—121no—
323C3×Dic3—121no—
324C3×Dic3—121no—
325C3×Dic3—121no—
326C3×Dic3—121no—
327C3×Dic3—121no—
328C22⋊Q8—91no—
329#117—91no—
330#107—271no—
331#106—271no—
332#103—271no—
333#103—271no—
334C22⋊C8—91no—
335C33#48511yescharacteristic, complemented, Sylow
336C23×C3—21no—
337C23×C3—21no—
338C23×C3—21no—
339C23×C3—41no—
340C23×C3#62511yescharacteristic
341C23×C3#62611yescharacteristic
342C23×C3#62611yescharacteristic
343C3×Q8—181no—
344C3×Q8—181no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—42no—
364C2×C3×C4—42no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—181no—
368C2×C3×C4—181no—
369C2×C3×C4—181no—
370C2×C3×C4—181no—
371C2×C3×C4#62511yescharacteristic
372C2×C3×C4#62611yescharacteristic
373C2×C3×C4#62611yescharacteristic
374C2×C3×C4C2×D3211yescharacteristic
375C2×C3×C4C2×D3211yescharacteristic
376C2×C3×C4C2×D3211yescharacteristic
377C2×Dic3—62no—
378C2×Dic3—62no—
379C2×Dic3—62no—
380C2×Dic3—62no—
381C2×Dic3—61no—
382C2×Dic3—61no—
383C2×Dic3—61no—
384C2×Dic3—61no—
385C2×Dic3—61no—
386C2×Dic3—61no—
387C2×Dic3—61no—
388C2×Dic3—61no—
389Dic6—61no—
390Dic6—61no—
391Dic6—61no—
392Dic6—61no—
393Dic6—121no—
394Dic6—121no—
395C3×C8—181no—
396C3×C8—181no—
397C3⋊C8—61no—
398C3⋊C8—61no—
399C3⋊C8—61no—
400C3⋊C8—61no—
401C3⋊C8—121no—
402C3⋊C8—121no—
403C2×C32—21no—
404C2×C32—21no—
405C2×C32—21no—
406C2×C32—21no—
407C2×C32—21no—
408C2×C32—21no—
409C2×C32—21no—
410C2×C32—21no—
411C2×C32—21no—
412C2×C32—41no—
413C2×C32—41no—
414C2×C32—41no—
415C2×C32—42no—
416C2×C32—42no—
417C2×C32—42no—
418C2×C32—44no—
419C2×C32—41no—
420C2×C32—41no—
421C2×C32—41no—
422C2×C32#88611yescharacteristic
423C2×C32#90411yescharacteristic
424C2×C32#90911yescharacteristic
425C2×C32#92911yescharacteristic
426C2×C32#93511yescharacteristic
427C2×C32#94211yescharacteristic
428C2×C32#95011yescharacteristic
429C2×C32#95011yescharacteristic
430C2×C32#95111yescharacteristic
431C2×Q8—91no—
432C22×C4#111311yescharacteristic
433C2×C8—91no—
434C2×C8—91no—
435C4⋊C4—91no—
436C4⋊C4—91no—
437C4⋊C4—91no—
438C4⋊C4—181no—
439C22⋊C4—181no—
440C22⋊C4—181no—
441C42—181no—
442C22×C3—22no—
443C22×C3—22no—
444C22×C3—22no—
445C22×C3—21no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—21no—
451C22×C3—21no—
452C22×C3—21no—
453C22×C3—21no—
454C22×C3—21no—
455C22×C3—21no—
456C22×C3—21no—
457C22×C3—41no—
458C22×C3—41no—
459C22×C3—41no—
460C22×C3—41no—
461C22×C3—41no—
462C22×C3—41no—
463C22×C3—42no—
464C22×C3—42no—
465C22×C3—42no—
466C22×C3—41no—
467C22×C3C2×#62511yescharacteristic
468C22×C3C2×#62611yescharacteristic
469C22×C3C2×#62611yescharacteristic
470C3×C4—22no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—21no—
475C3×C4—21no—
476C3×C4—21no—
477C3×C4—21no—
478C3×C4—21no—
479C3×C4—21no—
480C3×C4—41no—
481C3×C4—41no—
482C3×C4—42no—
483C3×C4—41no—
484C3×C4—41no—
485C3×C4—361no—
486C3×C4—361no—
487C3×C4—361no—
488C3×C4—361no—
489C3×C4#380411yescharacteristic
490C3×C4#380411yescharacteristic
491C3×C4#380411yescharacteristic
492C3×C4#380411yescharacteristic
493C3×C4#380511yescharacteristic
494C3×C4#380511yescharacteristic
495Dic3—122no—
496Dic3—122no—
497Dic3—122no—
498Dic3—122no—
499Dic3—121no—
500Dic3—121no—
501Dic3—121no—
502Dic3—121no—
503Dic3—121no—
504Dic3—121no—
505Dic3—121no—
506Dic3—121no—
507C32—21no—
508C32—21no—
509C32—21no—
510C32—41no—
511C32#472311yescharacteristic, complemented
512C32#499611yescharacteristic, complemented
513C32#504511yescharacteristic, complemented
514C23#629511yescharacteristic
515Q8—181no—
516Q8—181no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—181no—
520C2×C4—181no—
521C2×C4—181no—
522C2×C4—181no—
523C2×C4#629511yescharacteristic
524C2×C4C2×#111311yescharacteristic
525C8—181no—
526C8—181no—
527C2×C3—21no—
528C2×C3—21no—
529C2×C3—21no—
530C2×C3—21no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—21no—
535C2×C3—21no—
536C2×C3—42no—
537C2×C3—42no—
538C2×C3—42no—
539C2×C3—44no—
540C2×C3—41no—
541C2×C3—41no—
542C2×C3—41no—
543C2×C3—41no—
544C2×C3—41no—
545C2×C3—41no—
546C2×C3#6382211yescharacteristic
547C2×C3#6382711yescharacteristic
548C2×C3#6382911yescharacteristic
549C2×C3#6382911yescharacteristic
550C2×C3#6383111yescharacteristic
551C2×C3#6383311yescharacteristic
552C2×C3#6396111yescharacteristic
553C2×C3#6396211yescharacteristic
554C2×C3#6396611yescharacteristic
555C22—21no—
556C22—21no—
557C22—21no—
558C22C2×#629511yescharacteristic
559C4—21no—
560C4—361no—
561C4—361no—
562C4—361no—
563C4—361no—
564C4#8904611yescharacteristic
565C4#8904611yescharacteristic
566C3—21no—
567C3—21no—
568C3—21no—
569C3—41no—
570C3#1059110211yescharacteristic, complemented
571C3#1059112711yescharacteristic, complemented
572C3#1059113411yescharacteristic, complemented
573C2—41no—
574C2#1172366011yescharacteristic
575C2#1172366311yescharacteristic
576C2#1172372011yescharacteristic
subgroupG/H
centerC22C2×#6295
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC22×C33×C4C22
socleC22×C33C2×D4
prime radicalC23×C33D4
hypercenterC22×C4#1113
hypocenterC33#485
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. #11722080C2semidirect
5. #11722079C2semidirect
6. #11722076C2semidirect
7. #11722076C2semidirect
335. C33#485semidirect
511. C32#4723semidirect
512. C32#4996semidirect
513. C32#5045semidirect
570. C3#10591102semidirect
571. C3#10591127semidirect
572. C3#10591134semidirect

Maximal subgroups

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

subgroupindex
C3×#641352
C3×#641332
C3×#636572
#117220802
#117220792
#117220762
#105911343
#105911273
#105911023