Finite Group Databasebeta

← #13147774

Subgroups — #13147774

Order 1728.

1001 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×((C3⋊D3)⋊C3)⋊Q8C211yesmaximal, characteristic
2C4×#6229C211yesmaximal, characteristic, complemented
3#64081⋊C3C211yesmaximal, characteristic
4C4×#634⋊C3C211yesmaximal, characteristic, complemented
5#11721121C212yesmaximal
6#11721120C212yesmaximal
7#11721083C211yesmaximal, characteristic, complemented
8#11721082C211yesmaximal, characteristic, complemented
9#11721081C211yesmaximal, characteristic
10#11721078C212yesmaximal, complemented
11#11721077C212yesmaximal, complemented
12#10593885—31nomaximal
13#10593848—31nomaximal
14C2×#6229C2211yescharacteristic
15C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
16C2×#634⋊C3C2211yescharacteristic
17C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
18C2×#6196C2212yes—
19((C3⋊D3)⋊C3)⋊Q8C418yescomplemented
20C42×H(3,𝔽3)C2211yescharacteristic
21C4⋊#1084C2211yescharacteristic
22#88540C2212yes—
23C4×#1084C2212yes—
24#3752⋊C3C2211yescharacteristic
25#3752⋊C3C2211yescharacteristic
26C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
27#3750⋊C3C2211yescharacteristic
28#3750⋊C3C2211yescharacteristic
29#3750⋊C3C2212yes—
30C4×(C3⋊Dic3)⋊C3C2212yes—
31C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
32#88450C2212yes—
33#88449C2212yes—
34#88448C2212yes—
35#88447C2212yes—
36#88447C2212yes—
37#88447C2212yes—
38C2×#3797—31no—
39C2×D3×Dic6—31no—
40#64081D311yescharacteristic
41C4×#634D311yescharacteristic, complemented
42C3×#872—31no—
43C3×C4×Dic6—31no—
44C3×C4×Dic6—31no—
45C3×C4×Dic6—31no—
46C3×C4×Dic6—61no—
47#63878—32no—
48#63877—32no—
49#63849—31no—
50#63844—31no—
51#63843—31no—
52#63843—31no—
53Dic3×Dic6—31no—
54#63841—31no—
55#63838—32no—
56#63837—32no—
57#63837—32no—
58#63837—32no—
59#63836—32no—
60#63836—32no—
61C22×(C3⋊D3)⋊C3C2311yescharacteristic
62C2×C4×H(3,𝔽3)C2311yescharacteristic
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×C4×H(3,𝔽3)C2311yescharacteristic
65C2×#1084C2312yes—
66C2×#1084C2312yes—
67#6229C2×C414yes—
68C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
69C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
70C2×(C3⋊Dic3)⋊C3C2312yes—
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C2×(C3⋊Dic3)⋊C3C2312yes—
73C4×(C3⋊D3)⋊C3C2×C414yes—
74#634⋊C3C2×C414yes—
75((C3⋊D3)⋊C3)⋊C4C2×C418yes—
76#6196C2×C418yes—
77#5572—91no—
78#5572—91no—
79#5516—91no—
80C2×C4×C3⋊D3C2×D311yescharacteristic
81C2×#634C2×D311yescharacteristic
82C2×C3×C4×D3—31no—
83C2×C3×Dic6—31no—
84C2×C3×Dic6—31no—
85C2×C3×Dic6—31no—
86C2×C3×Dic6—61no—
87C2×#627—32no—
88C2×#627—32no—
89C2×#624—32no—
90C2×D3×Dic3—32no—
91#3797—38no—
92D3×Dic6—38no—
93C32×C42—21no—
94C32×C42C2×D311yescharacteristic
95C32×C42C2×D311yescharacteristic
96#3752C2×D311yescharacteristic
97#3752C2×D311yescharacteristic
98C4⋊(C3⋊Dic3)C2×D311yescharacteristic
99#3750C2×D312yes—
100#3750C2×D311yescharacteristic
101#3750C2×D311yescharacteristic
102C4×C3⋊Dic3C2×D312yes—
103C4×C3⋊Dic3C2×D311yescharacteristic
104C3×(C2×D3)⋊C4—31no—
105C3×(C2×D3)⋊C4—31no—
106C3×C4⋊Dic3—31no—
107C3×C4⋊Dic3—31no—
108C3×C4⋊Dic3—31no—
109C3×C4⋊Dic3—61no—
110C3×Dic3⋊C4—32no—
111C3×Dic3⋊C4—32no—
112C3×Dic3⋊C4—31no—
113C3×Dic3⋊C4—31no—
114C3×Dic3⋊C4—32no—
115C3×Dic3⋊C4—62no—
116C3×C4×Dic3—32no—
117C3×C4×Dic3—32no—
118C3×C4×Dic3—32no—
119C3×C4×Dic3—31no—
120C3×C4×Dic3—62no—
121#3724—32no—
122Dic3⋊Dic3—32no—
123Dic3⋊Dic3—32no—
124Dic3⋊Dic3—32no—
125#3722—32no—
126(C2×D3)⋊Dic3—32no—
127Dic32—32no—
128Dic32—32no—
129Dic32—32no—
130Dic32—32no—
131Dic32—32no—
132Dic32—32no—
133C22×H(3,𝔽3)C2411yescharacteristic
134C2×(C3⋊D3)⋊C3—21no—
135C2×(C3⋊D3)⋊C3—21no—
136C2×(C3⋊D3)⋊C3C22×C412yes—
137C4×H(3,𝔽3)—21no—
138C4×H(3,𝔽3)—21no—
139C4×H(3,𝔽3)C22×C412yes—
140#1084—22no—
141#1084C22×C414yes—
142(C3⋊Dic3)⋊C3—21no—
143(C3⋊Dic3)⋊C3—21no—
144(C3⋊Dic3)⋊C3—22no—
145(C3⋊Dic3)⋊C3C22×C412yes—
146(C3⋊Dic3)⋊C3C22×C414yes—
147C2×D3×Q8—91no—
148C2×D3×Q8—91no—
149C22×Dic6—91no—
150C3×C4×Q8—91no—
151C3×C4×Q8—91no—
152C3×#117—91no—
153Q8×Dic3—91no—
154Q8×Dic3—91no—
155#926—91no—
156#892—92no—
157#892—92no—
158#892—91no—
159#892—91no—
160#887—92no—
161#887—91no—
162#887—92no—
163#887—92no—
164#887—92no—
165#887—91no—
166#877—92no—
167#877—92no—
168#872—31no—
169#872—91no—
170#872—92no—
171#872—92no—
172C4×Dic6—31no—
173C4×Dic6—31no—
174C4×Dic6—31no—
175C4×Dic6—61no—
176C4×Dic6—91no—
177C4×Dic6—92no—
178C4×Dic6—92no—
179C4×Dic6—91no—
180C4×Dic6—91no—
181C22×C3⋊D3C22×D311yescharacteristic
182C22×C3×D3—31no—
183C2×C32×C4—21no—
184C2×C32×C4—21no—
185C2×C32×C4—21no—
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C22×D311yescharacteristic
188C2×C32×C4C22×D311yescharacteristic
189C2×C32×C4C22×D311yescharacteristic
190C2×C32×C4C22×D311yescharacteristic
191C2×C32×C4C22×D311yescharacteristic
192C2×C3⋊Dic3C22×D312yes—
193C2×C3⋊Dic3C22×D311yescharacteristic
194C2×C3⋊Dic3C22×D311yescharacteristic
195C2×C3⋊Dic3C22×D311yescharacteristic
196C2×C3⋊Dic3C22×D312yes—
197C4×C3⋊D3C4×D314yes—
198#634C4×D314yes—
199C2×C3×Dic3—31no—
200C2×C3×Dic3—32no—
201C2×C3×Dic3—32no—
202C2×C3×Dic3—32no—
203C2×C3×Dic3—32no—
204C2×C3×Dic3—31no—
205C2×C3×Dic3—31no—
206C2×C3×Dic3—32no—
207C2×C3×Dic3—32no—
208C2×C3×Dic3—62no—
209C2×C3×Dic3—62no—
210C3×C4×D3—34no—
211C3×Dic6—34no—
212C3×Dic6—34no—
213C3×Dic6—34no—
214C3×Dic6—64no—
215#627—38no—
216#627—38no—
217#624—38no—
218D3×Dic3—38no—
219#519—271noSylow
220C2×H(3,𝔽3)C23×C411yescharacteristic
221C2×H(3,𝔽3)2−1+411yescharacteristic
222C2×H(3,𝔽3)2−1+411yescharacteristic
223(C3⋊D3)⋊C3—22no—
224C2×C3×Q8—91no—
225C2×C3×Q8—91no—
226C22×C3×C4—91no—
227C22×Dic3—92no—
228D3×Q8—98no—
229D3×Q8—98no—
230C2×C4×D3—31no—
231C2×C4×D3—91no—
232C2×C4×D3—92no—
233C2×C4×D3—92no—
234C2×Dic6—31no—
235C2×Dic6—31no—
236C2×Dic6—31no—
237C2×Dic6—61no—
238C2×Dic6—91no—
239C2×Dic6—98no—
240C2×Dic6—92no—
241C2×Dic6—92no—
242C2×Dic6—92no—
243C3×C4⋊C4—91no—
244C3×C4⋊C4—91no—
245C3×C4⋊C4—92no—
246C3×C4⋊C4—91no—
247C3×C4⋊C4—92no—
248C3×C4⋊C4—91no—
249C3×C22⋊C4—91no—
250C3×C22⋊C4—91no—
251C3×C42—31no—
252C3×C42—31no—
253C3×C42—61no—
254C3×C42—91no—
255C3×C42—92no—
256C3×C42—92no—
257C3×C42D3211yescharacteristic
258#217—92no—
259(C2×D3)⋊C4—31no—
260(C2×D3)⋊C4—31no—
261(C2×D3)⋊C4—91no—
262(C2×D3)⋊C4—91no—
263(C2×D3)⋊C4—92no—
264(C2×D3)⋊C4—92no—
265C4⋊Dic3—31no—
266C4⋊Dic3—31no—
267C4⋊Dic3—31no—
268C4⋊Dic3—61no—
269C4⋊Dic3—91no—
270C4⋊Dic3—92no—
271C4⋊Dic3—92no—
272Dic3⋊C4—32no—
273Dic3⋊C4—32no—
274Dic3⋊C4—32no—
275Dic3⋊C4—31no—
276Dic3⋊C4—31no—
277Dic3⋊C4—62no—
278Dic3⋊C4—91no—
279Dic3⋊C4—92no—
280Dic3⋊C4—91no—
281Dic3⋊C4—92no—
282Dic3⋊C4—92no—
283Dic3⋊C4—92no—
284Dic3⋊C4—92no—
285C4×Dic3—32no—
286C4×Dic3—32no—
287C4×Dic3—32no—
288C4×Dic3—31no—
289C4×Dic3—62no—
290C4×Dic3—92no—
291C4×Dic3—91no—
292C4×Dic3—92no—
293C4×Dic3—92no—
294C4×Dic3—92no—
295C4×Dic3—92no—
296C4×Dic3—92no—
297C4×Dic3—92no—
298C4×Dic3—92no—
299C4×Dic3—92no—
300C4×Dic3—92no—
301C22×C32—21no—
302C22×C32C23×D311yescharacteristic
303C22×C32C23×D311yescharacteristic
304C2×C3⋊D3—21no—
305C2×C3⋊D3—21no—
306C2×C3⋊D3C2×C4×D312yes—
307C2×C3×D3—32no—
308C2×C3×D3—61no—
309C2×C3×D3—61no—
310C32×C4—22no—
311C32×C4—21no—
312C32×C4—21no—
313C32×C4—21no—
314C32×C4—21no—
315C32×C4—22no—
316C32×C4—41no—
317C32×C4C2×C4×D312yes—
318C32×C4C2×C4×D312yes—
319C3⋊Dic3—21no—
320C3⋊Dic3—21no—
321C3⋊Dic3—22no—
322C3⋊Dic3C2×C4×D314yes—
323C3⋊Dic3C2×C4×D312yes—
324C3×Dic3—34no—
325C3×Dic3—32no—
326C3×Dic3—34no—
327C3×Dic3—34no—
328C3×Dic3—64no—
329C3×Dic3—61no—
330C3×Dic3—62no—
331C3×Dic3—62no—
332C3×Dic3—61no—
333C3×Dic3—62no—
334C3×Dic3—122no—
335C22×Q8—271no—
336C4×Q8—91no—
337C4×Q8—91no—
338C4×Q8—272no—
339C4×Q8—272no—
340C4×Q8—271no—
341C4×Q8—271no—
342#117—91no—
343#117—272no—
344#117—272no—
345#117—271no—
346H(3,𝔽3)#51911yescharacteristic, complemented, Sylow
347C23×C3—91no—
348C22×D3—31no—
349C22×D3—91no—
350C3×Q8—94no—
351C3×Q8—94no—
352C2×C3×C4—31no—
353C2×C3×C4—31no—
354C2×C3×C4—31no—
355C2×C3×C4—31no—
356C2×C3×C4—31no—
357C2×C3×C4—31no—
358C2×C3×C4—61no—
359C2×C3×C4—61no—
360C2×C3×C4—61no—
361C2×C3×C4—94no—
362C2×C3×C4—91no—
363C2×C3×C4—92no—
364C2×C3×C4—92no—
365C2×C3×C4—91no—
366C2×C3×C4—91no—
367C2×C3×C4—92no—
368C2×C3×C4—92no—
369C2×C3×C4C2×D3211yescharacteristic
370C2×C3×C4C2×D3211yescharacteristic
371C2×C3×C4C2×D3211yescharacteristic
372C2×Dic3—32no—
373C2×Dic3—32no—
374C2×Dic3—32no—
375C2×Dic3—31no—
376C2×Dic3—32no—
377C2×Dic3—32no—
378C2×Dic3—31no—
379C2×Dic3—32no—
380C2×Dic3—31no—
381C2×Dic3—62no—
382C2×Dic3—62no—
383C2×Dic3—91no—
384C2×Dic3—92no—
385C2×Dic3—91no—
386C2×Dic3—92no—
387C2×Dic3—91no—
388C2×Dic3—98no—
389C4×D3—34no—
390C4×D3—94no—
391C4×D3—98no—
392C4×D3—98no—
393Dic6—34no—
394Dic6—34no—
395Dic6—34no—
396Dic6—64no—
397Dic6—94no—
398Dic6—98no—
399Dic6—98no—
400Dic6—98no—
401C2×C32—21no—
402C2×C32—21no—
403C2×C32—21no—
404C2×C32C22×C4×D311yescharacteristic
405C2×C32C22×C4×D311yescharacteristic
406C2×C32#100711yescharacteristic
407C2×C32#101011yescharacteristic
408C2×C32#101011yescharacteristic
409C2×C32#101011yescharacteristic
410C3⋊D3—22no—
411C3×D3—62no—
412C2×Q8—91no—
413C2×Q8—91no—
414C2×Q8—272no—
415C2×Q8—278no—
416C22×C4—91no—
417C22×C4—272no—
418C4⋊C4—91no—
419C4⋊C4—91no—
420C4⋊C4—92no—
421C4⋊C4—92no—
422C4⋊C4—91no—
423C4⋊C4—91no—
424C4⋊C4—272no—
425C4⋊C4—272no—
426C22⋊C4—91no—
427C22⋊C4—91no—
428C22⋊C4—272no—
429C42—91no—
430C42—92no—
431C42—92no—
432C42—272no—
433C42—272no—
434C42—272no—
435C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
436C22×C3—31no—
437C22×C3—31no—
438C22×C3—61no—
439C22×C3—92no—
440C22×C3—181no—
441C22×C3—181no—
442C22×C3C22×D3211yescharacteristic
443C2×D3—32no—
444C2×D3—61no—
445C2×D3—61no—
446C2×D3—92no—
447C2×D3—181no—
448C2×D3—181no—
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—32no—
452C3×C4—32no—
453C3×C4—61no—
454C3×C4—61no—
455C3×C4—61no—
456C3×C4—61no—
457C3×C4—62no—
458C3×C4—62no—
459C3×C4—92no—
460C3×C4—94no—
461C3×C4—94no—
462C3×C4—121no—
463C3×C4—181no—
464C3×C4—181no—
465C3×C4—182no—
466C3×C4—182no—
467C3×C4C4×D3212yes—
468Dic3—34no—
469Dic3—34no—
470Dic3—34no—
471Dic3—32no—
472Dic3—64no—
473Dic3—62no—
474Dic3—62no—
475Dic3—61no—
476Dic3—62no—
477Dic3—61no—
478Dic3—92no—
479Dic3—94no—
480Dic3—122no—
481Dic3—181no—
482Dic3—182no—
483Dic3—181no—
484C32—21no—
485C32#551611yescharacteristic, complemented
486C32#557211yescharacteristic, complemented
487C23—91no—
488Q8—94no—
489Q8—94no—
490Q8—278no—
491C2×C4—92no—
492C2×C4—92no—
493C2×C4—91no—
494C2×C4—94no—
495C2×C4—91no—
496C2×C4—91no—
497C2×C4—92no—
498C2×C4—92no—
499C2×C4—278no—
500C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
501C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
502C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
503C2×C3—31no—
504C2×C3—31no—
505C2×C3—31no—
506C2×C3—31no—
507C2×C3—31no—
508C2×C3—31no—
509C2×C3—61no—
510C2×C3—61no—
511C2×C3—61no—
512C2×C3—182no—
513C2×C3#6429711yescharacteristic
514C2×C3#6429811yescharacteristic
515C2×C3C2×C4×D3211yescharacteristic
516D3—62no—
517D3—182no—
518C22—92no—
519C22—181no—
520C22—181no—
521C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
522C4—21no—
523C4—21no—
524C4—92no—
525C4—94no—
526C4—94no—
527C4—182no—
528C4—181no—
529C4—181no—
530C4—182no—
531C4C4×((C3⋊D3)⋊C3)⋊C212yes—
532C3—31no—
533C3—31no—
534C3—61no—
535C3#1059388511yescharacteristic
536C2—182no—
537C2#1172248211yescharacteristic
538C2#1172248311yescharacteristic
539C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radicalC22×(C3⋊D3)⋊C3C23
cosocleC22×H(3,𝔽3)C24
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#519
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C4×#6229C2semidirect
4. C4×#634⋊C3C2semidirect
7. #11721083C2semidirect
8. #11721082C2semidirect
10. #11721078C2semidirect
11. #11721077C2semidirect
19. ((C3⋊D3)⋊C3)⋊Q8C4semidirect
41. C4×#634D3semidirect
346. H(3,𝔽3)#519semidirect
485. C32#5516semidirect
486. C32#5572semidirect

Maximal subgroups

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

subgroupindex
C2×((C3⋊D3)⋊C3)⋊Q82
C4×#62292
#64081⋊C32
C4×#634⋊C32
#117211212
#117211202
#117210832
#117210822
#117210812
#117210782
#117210772
#105938853
#105938483