Finite Group Databasebeta

← #13149474

Subgroups — #13149474

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
2#11721755C211yesmaximal, characteristic, complemented
3#64101⋊C3C211yesmaximal, characteristic
4#64096⋊C3C211yesmaximal, characteristic, complemented
5#11721121C212yesmaximal
6#11721120C212yesmaximal
7C4×#6196C212yesmaximal, complemented
8#11721105C211yesmaximal, characteristic
9#11721084C211yesmaximal, characteristic, complemented
10#11721083C211yesmaximal, characteristic, complemented
11#11721077C212yesmaximal, complemented
12#10594357—31nomaximal
13#10594347—31nomaximal
14C2×#6229C2211yescharacteristic
15C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
16C2×#634⋊C3C2211yescharacteristic
17C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
18C2×#6196C2212yes—
19((C3⋊D3)⋊C3)⋊Q8C418yescomplemented
20C4⋊C4×H(3,𝔽3)C2211yescharacteristic
21#88540C2212yes—
22C4×#1084C2211yescharacteristic
23C4×#1084C2212yes—
24#3752⋊C3C2212yes—
25C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
26#3750⋊C3C2212yes—
27C4×(C3⋊Dic3)⋊C3C2212yes—
28C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
29C4×(C3⋊Dic3)⋊C3C2212yes—
30#88450C2211yescharacteristic
31#88450C2211yescharacteristic
32#88449C2212yes—
33#88448C2211yescharacteristic
34#88448C2212yes—
35#88448C2211yescharacteristic
36#88447C2212yes—
37#88447C2212yes—
38C2×#3797—31no—
39C2×D3×Dic6—31no—
40#64101D311yescharacteristic
41#64096D311yescharacteristic, complemented
42C3×#892—31no—
43C3×#887—31no—
44C3×#887—31no—
45C3×#887—31no—
46C3×#887—61no—
47C4×#627—32no—
48C4×#627—32no—
49#63878—32no—
50#63877—32no—
51#63874—31no—
52#63855—31no—
53#63844—31no—
54#63843—31no—
55Dic3×Dic6—31no—
56Dic3×Dic6—31no—
57#63838—32no—
58#63837—32no—
59#63836—32no—
60#63836—32no—
61C22×(C3⋊D3)⋊C3C2311yescharacteristic
62C2×C4×H(3,𝔽3)C2312yes—
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×#1084C2311yescharacteristic
65C2×#1084C2311yescharacteristic
66C2×#1084C2312yes—
67#6229C2×C414yes—
68C2×(C3⋊Dic3)⋊C3C2312yes—
69C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
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×C4⋊C4—21no—
94C32×C4⋊C4C2×D311yescharacteristic
95C32×C4⋊C4C2×D311yescharacteristic
96#3752C2×D312yes—
97C4⋊(C3⋊Dic3)C2×D311yescharacteristic
98#3750C2×D312yes—
99C4×C3⋊Dic3C2×D312yes—
100C4×C3⋊Dic3C2×D311yescharacteristic
101C4×C3⋊Dic3C2×D312yes—
102C3×(C2×D3)⋊C4—32no—
103C3×C4⋊Dic3—31no—
104C3×Dic3⋊C4—32no—
105C3×Dic3⋊C4—32no—
106C3×Dic3⋊C4—32no—
107C3×Dic3⋊C4—62no—
108C3×C4×Dic3—32no—
109C3×C4×Dic3—32no—
110C3×C4×Dic3—31no—
111C3×C4×Dic3—31no—
112C3×C4×Dic3—31no—
113C3×C4×Dic3—32no—
114C3×C4×Dic3—32no—
115C3×C4×Dic3—62no—
116C3×C4×Dic3—61no—
117#3724—31no—
118#3724—32no—
119#3724—31no—
120Dic3⋊Dic3—32no—
121Dic3⋊Dic3—32no—
122Dic3⋊Dic3—31no—
123Dic3⋊Dic3—32no—
124Dic3⋊Dic3—31no—
125#3722—31no—
126#3722—31no—
127(C2×D3)⋊Dic3—31no—
128(C2×D3)⋊Dic3—31no—
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)—22no—
138C4×H(3,𝔽3)C22×C412yes—
139#1084—21no—
140#1084—21no—
141#1084C22×C414yes—
142(C3⋊Dic3)⋊C3—22no—
143(C3⋊Dic3)⋊C3—21no—
144(C3⋊Dic3)⋊C3—21no—
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—31no—
157#892—91no—
158#892—92no—
159#892—92no—
160#887—31no—
161#887—31no—
162#887—31no—
163#887—61no—
164#887—91no—
165#887—92no—
166#887—91no—
167#887—92no—
168#887—91no—
169#877—92no—
170#877—92no—
171#872—91no—
172#872—91no—
173#872—92no—
174#872—92no—
175C4×Dic6—92no—
176C4×Dic6—92no—
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—22no—
185C2×C32×C4C22×D312yes—
186C2×C32×C4C22×D312yes—
187C2×C32×C4C22×D311yescharacteristic
188C2×C32×C4C22×D311yescharacteristic
189C2×C3⋊Dic3C22×D311yescharacteristic
190C2×C3⋊Dic3C22×D312yes—
191C2×C3⋊Dic3C22×D311yescharacteristic
192C2×C3⋊Dic3C22×D312yes—
193C2×C3⋊Dic3C22×D311yescharacteristic
194C4×C3⋊D3C4×D314yes—
195#634C4×D314yes—
196C2×C3×Dic3—32no—
197C2×C3×Dic3—31no—
198C2×C3×Dic3—32no—
199C2×C3×Dic3—32no—
200C2×C3×Dic3—31no—
201C2×C3×Dic3—32no—
202C2×C3×Dic3—31no—
203C2×C3×Dic3—31no—
204C2×C3×Dic3—31no—
205C2×C3×Dic3—31no—
206C2×C3×Dic3—31no—
207C2×C3×Dic3—61no—
208C2×C3×Dic3—62no—
209C2×C3×Dic3—61no—
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—31no—
244C3×C4⋊C4—31no—
245C3×C4⋊C4—61no—
246C3×C4⋊C4—91no—
247C3×C4⋊C4—92no—
248C3×C4⋊C4—92no—
249C3×C4⋊C4D3211yescharacteristic
250C3×C22⋊C4—92no—
251C3×C42—92no—
252C3×C42—91no—
253C3×C42—92no—
254C3×C42—92no—
255C3×C42—91no—
256#217—91no—
257#217—91no—
258(C2×D3)⋊C4—32no—
259(C2×D3)⋊C4—92no—
260(C2×D3)⋊C4—91no—
261(C2×D3)⋊C4—91no—
262(C2×D3)⋊C4—91no—
263(C2×D3)⋊C4—91no—
264C4⋊Dic3—31no—
265C4⋊Dic3—91no—
266C4⋊Dic3—92no—
267C4⋊Dic3—91no—
268C4⋊Dic3—91no—
269C4⋊Dic3—92no—
270Dic3⋊C4—32no—
271Dic3⋊C4—32no—
272Dic3⋊C4—32no—
273Dic3⋊C4—62no—
274Dic3⋊C4—92no—
275Dic3⋊C4—91no—
276Dic3⋊C4—92no—
277Dic3⋊C4—91no—
278Dic3⋊C4—92no—
279Dic3⋊C4—91no—
280Dic3⋊C4—92no—
281Dic3⋊C4—92no—
282Dic3⋊C4—91no—
283C4×Dic3—32no—
284C4×Dic3—32no—
285C4×Dic3—31no—
286C4×Dic3—31no—
287C4×Dic3—32no—
288C4×Dic3—32no—
289C4×Dic3—31no—
290C4×Dic3—62no—
291C4×Dic3—61no—
292C4×Dic3—92no—
293C4×Dic3—92no—
294C4×Dic3—91no—
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—22no—
312C32×C4—22no—
313C32×C4—42no—
314C32×C4C2×C4×D312yes—
315C32×C4C2×C4×D312yes—
316C3⋊Dic3—21no—
317C3⋊Dic3—21no—
318C3⋊Dic3—22no—
319C3⋊Dic3C2×C4×D314yes—
320C3⋊Dic3C2×C4×D312yes—
321C3×Dic3—34no—
322C3×Dic3—32no—
323C3×Dic3—34no—
324C3×Dic3—34no—
325C3×Dic3—64no—
326C3×Dic3—62no—
327C3×Dic3—62no—
328C3×Dic3—61no—
329C3×Dic3—61no—
330C3×Dic3—61no—
331C3×Dic3—61no—
332C3×Dic3—61no—
333C3×Dic3—61no—
334C3×Dic3—121no—
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—271no—
344#117—272no—
345#117—272no—
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—32no—
355C2×C3×C4—32no—
356C2×C3×C4—61no—
357C2×C3×C4—62no—
358C2×C3×C4—94no—
359C2×C3×C4—92no—
360C2×C3×C4—91no—
361C2×C3×C4—92no—
362C2×C3×C4—92no—
363C2×C3×C4—91no—
364C2×C3×C4—91no—
365C2×C3×C4—91no—
366C2×C3×C4—91no—
367C2×C3×C4C2×D3212yes—
368C2×C3×C4C2×D3211yescharacteristic
369C2×Dic3—32no—
370C2×Dic3—32no—
371C2×Dic3—31no—
372C2×Dic3—32no—
373C2×Dic3—31no—
374C2×Dic3—31no—
375C2×Dic3—31no—
376C2×Dic3—31no—
377C2×Dic3—31no—
378C2×Dic3—31no—
379C2×Dic3—32no—
380C2×Dic3—61no—
381C2×Dic3—62no—
382C2×Dic3—61no—
383C2×Dic3—91no—
384C2×Dic3—92no—
385C2×Dic3—91no—
386C2×Dic3—91no—
387C2×Dic3—92no—
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—92no—
420C4⋊C4—92no—
421C4⋊C4—271no—
422C4⋊C4—271no—
423C4⋊C4—272no—
424C4⋊C4—272no—
425C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
426C22⋊C4—92no—
427C22⋊C4—271no—
428C22⋊C4—271no—
429C42—92no—
430C42—92no—
431C42—91no—
432C42—92no—
433C42—91no—
434C42—272no—
435C42—272no—
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—22no—
450C3×C4—32no—
451C3×C4—32no—
452C3×C4—62no—
453C3×C4—62no—
454C3×C4—62no—
455C3×C4—92no—
456C3×C4—94no—
457C3×C4—94no—
458C3×C4—122no—
459C3×C4—181no—
460C3×C4—181no—
461C3×C4—182no—
462C3×C4—181no—
463C3×C4—181no—
464C3×C4C4×D3212yes—
465Dic3—34no—
466Dic3—34no—
467Dic3—34no—
468Dic3—32no—
469Dic3—64no—
470Dic3—61no—
471Dic3—61no—
472Dic3—61no—
473Dic3—61no—
474Dic3—62no—
475Dic3—61no—
476Dic3—61no—
477Dic3—62no—
478Dic3—92no—
479Dic3—94no—
480Dic3—121no—
481Dic3—181no—
482Dic3—182no—
483Dic3—181no—
484C32—21no—
485C32#551611yescharacteristic, complemented
486C32#557211yescharacteristic, complemented
487C23—91no—
488Q8—94no—
489Q8—94no—
490Q8—278no—
491C2×C4—91no—
492C2×C4—92no—
493C2×C4—92no—
494C2×C4—94no—
495C2×C4—91no—
496C2×C4—91no—
497C2×C4—92no—
498C2×C4—91no—
499C2×C4—91no—
500C2×C4—278no—
501C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
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×C3C2×C4×D3211yescharacteristic
514C2×C3#6430911yescharacteristic
515C2×C3#6431511yescharacteristic
516D3—62no—
517D3—182no—
518C22—92no—
519C22—181no—
520C22—181no—
521C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
522C4—22no—
523C4—92no—
524C4—94no—
525C4—94no—
526C4—181no—
527C4—181no—
528C4—181no—
529C4—181no—
530C4—182no—
531C4C4×((C3⋊D3)⋊C3)⋊C212yes—
532C3—31no—
533C3—31no—
534C3—61no—
535C3#1059435711yescharacteristic
536C2—182no—
537C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
538C2#1172250211yescharacteristic
539C2#1172251611yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
Frattini, socleC22×C3C22×D32
FittingC4⋊C4×H(3,𝔽3)C22
prime radicalC22×(C3⋊D3)⋊C3C23
cosocleC22×H(3,𝔽3)C24
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#519
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #11721755C2semidirect
4. #64096⋊C3C2semidirect
7. C4×#6196C2semidirect
9. #11721084C2semidirect
10. #11721083C2semidirect
11. #11721077C2semidirect
19. ((C3⋊D3)⋊C3)⋊Q8C4semidirect
41. #64096D3semidirect
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
#117217552
#64101⋊C32
#64096⋊C32
#117211212
#117211202
C4×#61962
#117211052
#117210842
#117210832
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