Finite Group Databasebeta

← #13161960

Subgroups — #13161960

Order 1728.

656 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
1C3×D4×C3⋊Dic3C211yesmaximal, characteristic
2#11723680C211yesmaximal, characteristic, complemented
3C3×#63659C211yesmaximal, characteristic
4C3×#63659C211yesmaximal, characteristic
5#11722080C211yesmaximal, characteristic, complemented
6#11722079C211yesmaximal, characteristic, complemented
7#11722078C211yesmaximal, characteristic, complemented
8#10591479—31nomaximal
9#10591411—31nomaximal
10#10591409—31nomaximal
11C2×C33×D4C2211yescharacteristic
12C22×C3×C3⋊Dic3—21no—
13C3×#3757—21no—
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
20#88815—21no—
21#88815—21no—
22D4×C3⋊Dic3D311yescharacteristic
23C3×D4×Dic3—31no—
24C3×D4×Dic3—31no—
25C3×D4×Dic3—61no—
26#63872—31no—
27#63869—31no—
28#63869—31no—
29#63659D311yescharacteristic
30#63659D311yescharacteristic
31C3×#832—31no—
32C3×#832—31no—
33C3×#832—31no—
34C3×#832—31no—
35C3×#832—61no—
36C3×#832—61no—
37#63576—31no—
38#63575—31no—
39#63575—31no—
40#63574—31no—
41#63573—31no—
42#63573—31no—
43#63572—31no—
44Dic3⋊(C3⋊C8)—31no—
45Dic3⋊(C3⋊C8)—31no—
46C23×C33—21no—
47C33×D4—21no—
48C33×D4Q812yes—
49C2×C33×C4C2311yescharacteristic
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3—21no—
52C2×C3×C3⋊Dic3—21no—
53C2×C3×C3⋊Dic3—42no—
54C2×C3×C3⋊Dic3D411yescharacteristic
55C2×C3×C3⋊Dic3D411yescharacteristic
56C3×C3⋊(C3⋊C8)—21no—
57C3×C3⋊(C3⋊C8)—21no—
58#5009—91no—
59#4766—91no—
60#4763—91no—
61C2×C32×D4—21no—
62C2×C32×D4—21no—
63C2×C32×D4—21no—
64C2×C32×D4—41no—
65C2×C32×D4C2×D311yescharacteristic
66C2×C32×D4C2×D311yescharacteristic
67C2×C32×D4C2×D311yescharacteristic
68C22×C3⋊Dic3—21no—
69C22×C3×Dic3—61no—
70C22×C3×Dic3—61no—
71C22×C3×Dic3—62no—
72#3757—21no—
73C4⋊(C3⋊Dic3)C2×D311yescharacteristic
74C4⋊(C3⋊Dic3)C2×D311yescharacteristic
75C4⋊(C3⋊Dic3)C2×D311yescharacteristic
76C4×C3⋊Dic3C2×D311yescharacteristic
77C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
78C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
79C3×#217—61no—
80C3×#217—61no—
81C3×#217—62no—
82C3×C4⋊Dic3—31no—
83C3×C4⋊Dic3—31no—
84C3×C4⋊Dic3—31no—
85C3×C4⋊Dic3—31no—
86C3×C4⋊Dic3—31no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—61no—
89C3×C4⋊Dic3—61no—
90C3×C4⋊Dic3—61no—
91C3×C4×Dic3—31no—
92C3×C4×Dic3—31no—
93C3×C4×Dic3—61no—
94C2×C3×C3⋊C8—31no—
95C2×C3×C3⋊C8—31no—
96C2×C3×C3⋊C8—31no—
97C2×C3×C3⋊C8—31no—
98C2×C3×C3⋊C8—61no—
99C2×C3×C3⋊C8—61no—
100#3724—61no—
101#3724—61no—
102Dic3⋊Dic3—61no—
103Dic3⋊Dic3—61no—
104Dic3⋊Dic3—61no—
105Dic3⋊Dic3—61no—
106C22×C33—22no—
107C22×C33—41no—
108C22×C33—41no—
109C22×C33C2×D411yescharacteristic
110C33×C4C2×Q811yescharacteristic
111C33×C4#4111yescharacteristic
112C3×C3⋊Dic3—21no—
113C3×C3⋊Dic3—21no—
114C3×C3⋊Dic3—41no—
115C3×C3⋊Dic3—41no—
116C3×C3⋊Dic3—41no—
117C3×C4×D4—91no—
118D4×Dic3—31no—
119D4×Dic3—31no—
120D4×Dic3—61no—
121#890—91no—
122#890—91no—
123#870—91no—
124C3×#102—91no—
125C3×#102—91no—
126#832—31no—
127#832—31no—
128#832—31no—
129#832—31no—
130#832—61no—
131#832—61no—
132#818—91no—
133#817—91no—
134Dic3⋊C8—91no—
135Dic3⋊C8—91no—
136#808—91no—
137#808—91no—
138#807—91no—
139#807—91no—
140C4⋊(C3⋊C8)—91no—
141C23×C32—21no—
142C23×C32—22no—
143C23×C32—21no—
144C23×C32—21no—
145C23×C32—41no—
146C23×C32—41no—
147C23×C32—42no—
148C32×D4—21no—
149C32×D4—22no—
150C32×D4—21no—
151C32×D4—21no—
152C32×D4—22no—
153C32×D4—22no—
154C32×D4—22no—
155C32×D4—42no—
156C32×D4—41no—
157C32×D4—41no—
158C32×D4—42no—
159C32×D4Dic612yes—
160C32×D4Dic612yes—
161C32×D4Dic612yes—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—41no—
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C22×D311yescharacteristic
168C2×C32×C4C22×D311yescharacteristic
169C2×C3⋊Dic3—21no—
170C2×C3⋊Dic3—21no—
171C2×C3⋊Dic3—21no—
172C2×C3⋊Dic3—42no—
173C2×C3⋊Dic3#6711yescharacteristic
174C2×C3⋊Dic3#6711yescharacteristic
175C2×C3×Dic3—31no—
176C2×C3×Dic3—31no—
177C2×C3×Dic3—31no—
178C2×C3×Dic3—31no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—62no—
182C2×C3×Dic3—61no—
183C2×C3×Dic3—61no—
184C2×C3×Dic3—61no—
185C2×C3×Dic3—61no—
186C2×C3×Dic3—61no—
187C2×C3×Dic3—61no—
188C2×C3×Dic3—121no—
189C2×C3×Dic3—121no—
190C2×C3×Dic3—124no—
191C2×C3×Dic3—122no—
192C2×C3×Dic3—122no—
193C3⋊(C3⋊C8)—21no—
194C3⋊(C3⋊C8)—21no—
195C3×C3⋊C8—61no—
196C3×C3⋊C8—61no—
197C3×C3⋊C8—61no—
198C3×C3⋊C8—61no—
199C3×C3⋊C8—121no—
200C3×C3⋊C8—121no—
201#478—271noSylow
202C2×C33—42no—
203C2×C33C22⋊Q811yescharacteristic
204C2×C33#13511yescharacteristic
205C2×C33AGL(1,ℤ/8)11yescharacteristic
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—41no—
210C2×C3×D4D3211yescharacteristic
211C2×C3×D4D3211yescharacteristic
212C2×C3×D4D3211yescharacteristic
213C22×C3×C4—181no—
214C22×Dic3—61no—
215C22×Dic3—61no—
216C22×Dic3—62no—
217C2×C3×C8—91no—
218C2×C3×C8—91no—
219C3×C4⋊C4—91no—
220C3×C4⋊C4—91no—
221C3×C4⋊C4—91no—
222C3×C22⋊C4—181no—
223C3×C42—91no—
224#217—61no—
225#217—61no—
226#217—62no—
227C4⋊Dic3—31no—
228C4⋊Dic3—31no—
229C4⋊Dic3—31no—
230C4⋊Dic3—31no—
231C4⋊Dic3—31no—
232C4⋊Dic3—31no—
233C4⋊Dic3—61no—
234C4⋊Dic3—61no—
235C4⋊Dic3—61no—
236C4⋊Dic3—181no—
237C4⋊Dic3—181no—
238Dic3⋊C4—181no—
239Dic3⋊C4—181no—
240Dic3⋊C4—181no—
241Dic3⋊C4—181no—
242C4×Dic3—31no—
243C4×Dic3—31no—
244C4×Dic3—61no—
245C2×C3⋊C8—31no—
246C2×C3⋊C8—31no—
247C2×C3⋊C8—31no—
248C2×C3⋊C8—31no—
249C2×C3⋊C8—61no—
250C2×C3⋊C8—61no—
251C22×C32—22no—
252C22×C32—24no—
253C22×C32—22no—
254C22×C32—22no—
255C22×C32—21no—
256C22×C32—21no—
257C22×C32—21no—
258C22×C32—41no—
259C22×C32—42no—
260C22×C32—42no—
261C22×C32—44no—
262C22×C32—41no—
263C22×C32—41no—
264C22×C32—42no—
265C22×C32—42no—
266C22×C32—41no—
267C22×C32—41no—
268C22×C32—41no—
269C22×C32—41no—
270C22×C32—81no—
271C22×C32—81no—
272C22×C32—81no—
273C22×C32—81no—
274C22×C32—82no—
275C22×C32—82no—
276C22×C32D3×D411yescharacteristic
277C22×C32D3×D411yescharacteristic
278C22×C32C2×#6711yescharacteristic
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×C4C2×Dic611yescharacteristic
288C32×C4C2×Dic611yescharacteristic
289C32×C4C2×Dic611yescharacteristic
290C32×C4#23511yescharacteristic
291C32×C4#23711yescharacteristic
292C32×C4#23711yescharacteristic
293C3⋊Dic3—21no—
294C3⋊Dic3—21no—
295C3⋊Dic3—41no—
296C3⋊Dic3—41no—
297C3⋊Dic3—41no—
298C3×Dic3—62no—
299C3×Dic3—61no—
300C3×Dic3—61no—
301C3×Dic3—61no—
302C3×Dic3—61no—
303C3×Dic3—121no—
304C3×Dic3—122no—
305C3×Dic3—121no—
306C3×Dic3—121no—
307C3×Dic3—121no—
308C3×Dic3—121no—
309C3×Dic3—121no—
310C3×Dic3—121no—
311C3×Dic3—241no—
312C3×Dic3—241no—
313#125—271no—
314C4×D4—91no—
315#107—271no—
316#106—271no—
317C4⋊C8—271no—
318#102—91no—
319#102—91no—
320C33#47811yescharacteristic, complemented, Sylow
321C23×C3—22no—
322C23×C3—21no—
323C23×C3—21no—
324C23×C3—21no—
325C23×C3—41no—
326C23×C3—41no—
327C23×C3—42no—
328C3×D4—22no—
329C3×D4—21no—
330C3×D4—21no—
331C3×D4—21no—
332C3×D4—22no—
333C3×D4—22no—
334C3×D4—22no—
335C3×D4—41no—
336C3×D4—41no—
337C3×D4—42no—
338C3×D4—42no—
339C3×D4#62712yes—
340C3×D4#62712yes—
341C3×D4#62712yes—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—41no—
346C2×C3×C4—91no—
347C2×C3×C4—91no—
348C2×C3×C4—181no—
349C2×C3×C4—181no—
350C2×C3×C4—181no—
351C2×C3×C4—362no—
352C2×C3×C4C2×D3211yescharacteristic
353C2×C3×C4C2×D3211yescharacteristic
354C2×C3×C4C2×D3211yescharacteristic
355C2×Dic3—31no—
356C2×Dic3—31no—
357C2×Dic3—31no—
358C2×Dic3—31no—
359C2×Dic3—61no—
360C2×Dic3—61no—
361C2×Dic3—62no—
362C2×Dic3—61no—
363C2×Dic3—61no—
364C2×Dic3—61no—
365C2×Dic3—61no—
366C2×Dic3—61no—
367C2×Dic3—61no—
368C2×Dic3—121no—
369C2×Dic3—121no—
370C2×Dic3—124no—
371C2×Dic3—122no—
372C2×Dic3—122no—
373C3×C8—181no—
374C3×C8—181no—
375C3⋊C8—61no—
376C3⋊C8—61no—
377C3⋊C8—61no—
378C3⋊C8—61no—
379C3⋊C8—121no—
380C3⋊C8—121no—
381C2×C32—21no—
382C2×C32—21no—
383C2×C32—21no—
384C2×C32—21no—
385C2×C32—21no—
386C2×C32—21no—
387C2×C32—21no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—41no—
391C2×C32—41no—
392C2×C32—41no—
393C2×C32—42no—
394C2×C32—44no—
395C2×C32—42no—
396C2×C32—42no—
397C2×C32—82no—
398C2×C32—82no—
399C2×C32—84no—
400C2×C32#87811yescharacteristic
401C2×C32#87811yescharacteristic
402C2×C32#91111yescharacteristic
403C2×C32#91211yescharacteristic
404C2×C32#91411yescharacteristic
405C2×C32#91611yescharacteristic
406C2×C32#92411yescharacteristic
407C2×C32#93211yescharacteristic
408C2×C32#95011yescharacteristic
409C2×D4#111311yescharacteristic
410C22×C4—181no—
411C2×C8—91no—
412C2×C8—91no—
413C4⋊C4—91no—
414C4⋊C4—91no—
415C4⋊C4—91no—
416C4⋊C4—541no—
417C4⋊C4—541no—
418C22⋊C4—181no—
419C42—91no—
420C22×C3—24no—
421C22×C3—22no—
422C22×C3—22no—
423C22×C3—22no—
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—21no—
427C22×C3—42no—
428C22×C3—42no—
429C22×C3—44no—
430C22×C3—42no—
431C22×C3—42no—
432C22×C3—41no—
433C22×C3—41no—
434C22×C3—41no—
435C22×C3—41no—
436C22×C3—41no—
437C22×C3—41no—
438C22×C3—41no—
439C22×C3—81no—
440C22×C3—81no—
441C22×C3—81no—
442C22×C3—81no—
443C22×C3—82no—
444C22×C3—82no—
445C22×C3D3×#6711yescharacteristic
446C22×C3D3×#6711yescharacteristic
447C22×C3#381111yescharacteristic
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—21no—
452C3×C4—21no—
453C3×C4—21no—
454C3×C4—41no—
455C3×C4—41no—
456C3×C4—181no—
457C3×C4—181no—
458C3×C4—361no—
459C3×C4—361no—
460C3×C4—361no—
461C3×C4#380411yescharacteristic
462C3×C4#380411yescharacteristic
463C3×C4#380511yescharacteristic
464C3×C4C2×#62711yescharacteristic
465C3×C4C2×#62711yescharacteristic
466C3×C4C2×#62711yescharacteristic
467Dic3—62no—
468Dic3—61no—
469Dic3—61no—
470Dic3—61no—
471Dic3—61no—
472Dic3—121no—
473Dic3—122no—
474Dic3—121no—
475Dic3—121no—
476Dic3—121no—
477Dic3—121no—
478Dic3—121no—
479Dic3—121no—
480Dic3—241no—
481Dic3—241no—
482C32—21no—
483C32—21no—
484C32—21no—
485C32—41no—
486C32#476311yescharacteristic, complemented
487C32#476611yescharacteristic, complemented
488C32#500911yescharacteristic, complemented
489C23—21no—
490D4—21no—
491D4#629612yes—
492C2×C4—91no—
493C2×C4—91no—
494C2×C4—181no—
495C2×C4—181no—
496C2×C4—181no—
497C2×C4—362no—
498C2×C4C2×#111311yescharacteristic
499C8—181no—
500C8—181no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—21no—
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—21no—
510C2×C3—44no—
511C2×C3—42no—
512C2×C3—42no—
513C2×C3—42no—
514C2×C3—41no—
515C2×C3—41no—
516C2×C3—41no—
517C2×C3—82no—
518C2×C3—82no—
519C2×C3—84no—
520C2×C3#6393211yescharacteristic
521C2×C3#6393311yescharacteristic
522C2×C3#6393411yescharacteristic
523C2×C3#6393511yescharacteristic
524C2×C3#6393611yescharacteristic
525C2×C3#6393711yescharacteristic
526C2×C3#6396411yescharacteristic
527C2×C3#6396411yescharacteristic
528C2×C3#6398211yescharacteristic
529C22—22no—
530C22—41no—
531C22—41no—
532C22#8904911yescharacteristic
533C4—181no—
534C4—181no—
535C4—361no—
536C4—361no—
537C4—361no—
538C4#8904611yescharacteristic
539C4C2×#629611yescharacteristic
540C3—21no—
541C3—21no—
542C3—21no—
543C3—41no—
544C3#1059140911yescharacteristic, complemented
545C3#1059141111yescharacteristic, complemented
546C3#1059147911yescharacteristic, complemented
547C2—42no—
548C2#1172370611yescharacteristic
549C2#1172370711yescharacteristic
550C2#1172372111yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
Fitting, prime radicalC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×D4#1113
hypocenterC33#478
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11723680C2semidirect
5. #11722080C2semidirect
6. #11722079C2semidirect
7. #11722078C2semidirect
320. C33#478semidirect
486. C32#4763semidirect
487. C32#4766semidirect
488. C32#5009semidirect
544. C3#10591409semidirect
545. C3#10591411semidirect
546. C3#10591479semidirect

Maximal subgroups

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

subgroupindex
C3×D4×C3⋊Dic32
#117236802
C3×#636592
#117220802
#117220792
#117220782
#105914793
#105914113
#105914093