Finite Group Databasebeta

← #13156507

Subgroups — #13156507

Order 1728.

666 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723534C211yesmaximal, characteristic, complemented
2C3×#63911C211yesmaximal, characteristic
3#11722075C211yesmaximal, characteristic, complemented
4#11722034C211yesmaximal, characteristic, complemented
5#11722032C211yesmaximal, characteristic, complemented
6C3×#63563C211yesmaximal, characteristic, complemented
7C3×Dic3×C3⋊C8C211yesmaximal, characteristic, complemented
8#10591487—31nomaximal
9#10590878—31nomaximal
10#10590273—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×#6293—21no—
14C2×C3×#626—21no—
15C2×C3×#626—21no—
16C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C32×C4×Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21Dic3⋊(C3⋊Dic3)—21no—
22#64078—31no—
23C3×#940—31no—
24C3×#874—31no—
25#63918—31no—
26#63918—31no—
27#63918—61no—
28#63911D311yescharacteristic
29#63659—31no—
30#63648—31no—
31C3×#832—31no—
32C3×#821—31no—
33C3×C8×Dic3—31no—
34C3×C4×C3⋊C8—31no—
35#63564—31no—
36#63563—31no—
37#63563—31no—
38#63563—31no—
39#63563—31no—
40#63563—31no—
41#63563—61no—
42#63563—61no—
43#63563D311yescharacteristic, complemented
44#63562—31no—
45Dic3×C3⋊C8D311yescharacteristic, complemented
46C22×C3×C3⋊D3—21no—
47C22×C32×D3—21no—
48C2×C33×C4C2311yescharacteristic
49C2×C3⋊(C3⋊Dic3)—21no—
50C2×C3×C3⋊Dic3—21no—
51C3×C4⋊(C3⋊D3)—21no—
52C3×C4⋊(C3⋊D3)—21no—
53C2×C32×Dic3D411yescharacteristic
54C2×C32×Dic3D411yescharacteristic
55C32×D12—21no—
56C32×D12—21no—
57#6293—42no—
58C3×#626—42no—
59C3×#626—42no—
60C3×C3⋊(C3⋊C8)—21no—
61C32×C3⋊C8—21no—
62#5070—91no—
63#4760—91no—
64#4694—91no—
65C2×C32×D4—31no—
66C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
67C2×#634—61no—
68C2×C3×#67—61no—
69C2×C3×#67—61no—
70C2×C3×D12—21no—
71C2×C3×D12—31no—
72C2×C3×D12—31no—
73C2×C3×D12—61no—
74C2×C3×D12—61no—
75C2×C3×D12—61no—
76C2×C3×D12C2×D311yescharacteristic
77C2×C3×D12C2×D311yescharacteristic
78C2×#627—62no—
79C2×#627—61no—
80C2×#627—61no—
81C2×#626—21no—
82C2×#626—21no—
83C2×C32×C8—31no—
84C32×C42—31no—
85C4⋊(C3⋊Dic3)—31no—
86C4⋊(C3⋊Dic3)—31no—
87C4⋊(C3⋊Dic3)—31no—
88C4⋊(C3⋊Dic3)—61no—
89C4⋊(C3⋊Dic3)—61no—
90C4⋊(C3⋊Dic3)—61no—
91C4⋊(C3⋊Dic3)—61no—
92C4⋊(C3⋊Dic3)—121no—
93C4⋊(C3⋊Dic3)C2×D311yescharacteristic
94C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
95C3×C4⋊Dic3—31no—
96C3×C4⋊Dic3—31no—
97C3×C4⋊Dic3—61no—
98C3×C4×Dic3—21no—
99C3×C4×Dic3C2×D311yescharacteristic
100C3×C4×Dic3C2×D311yescharacteristic
101C2×C3×C3⋊C8—21no—
102C2×C3×C3⋊C8—31no—
103C2×C3×C3⋊C8—31no—
104C2×C3×C3⋊C8—61no—
105C2×C3×C3⋊C8C2×D311yescharacteristic
106C2×C3×C3⋊C8C2×D311yescharacteristic
107Dic3⋊Dic3—62no—
108Dic3⋊Dic3—61no—
109Dic3⋊Dic3—61no—
110C22×C33C2×D411yescharacteristic
111C2×C3×C3⋊D3—41no—
112C2×C3×C3⋊D3—41no—
113C2×C3×C3⋊D3—41no—
114C2×C32×D3—41no—
115C2×C32×D3—41no—
116C2×C32×D3—41no—
117C33×C4#4111yescharacteristic
118C33×C4#4111yescharacteristic
119C3⋊(C3⋊Dic3)—41no—
120C3×C3⋊Dic3—41no—
121C32×Dic3D812yes—
122C32×Dic3SD1612yes—
123C3×#127—91no—
124#940—31no—
125#888—91no—
126#874—31no—
127#869—91no—
128#869—91no—
129#869—181no—
130C3×#102—91no—
131C3×C4×C8—91no—
132#832—31no—
133#832—91no—
134#832—91no—
135#832—91no—
136#832—91no—
137#832—181no—
138#821—31no—
139#821—91no—
140#821—91no—
141#821—91no—
142#821—91no—
143#821—181no—
144C8×Dic3—31no—
145#809—91no—
146C4×C3⋊C8—31no—
147C23×C32—61no—
148C22×C3⋊D3—21no—
149C22×C3×D3—21no—
150C22×C3×D3—21no—
151C22×C3×D3—41no—
152C22×C3×D3—61no—
153C22×C3×D3—61no—
154C22×C3×D3—121no—
155C32×D4—61no—
156C32×D4—61no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—31no—
161C2×C32×C4—31no—
162C2×C32×C4—41no—
163C2×C32×C4C22×D311yescharacteristic
164C2×C32×C4C22×D311yescharacteristic
165C2×C32×C4C22×D311yescharacteristic
166C2×C3⋊Dic3—21no—
167C2×C3⋊Dic3—61no—
168C2×C3⋊Dic3—62no—
169C2×C3⋊Dic3—61no—
170C2×C3⋊Dic3—61no—
171C2×C3⋊Dic3—121no—
172C2×C3⋊Dic3—121no—
173C2×C3⋊Dic3—122no—
174C4⋊(C3⋊D3)—21no—
175C4⋊(C3⋊D3)—21no—
176#634—122no—
177C3×#67—122no—
178C3×#67—122no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—62no—
182C2×C3×Dic3—61no—
183C2×C3×Dic3—61no—
184C2×C3×Dic3D1211yescharacteristic
185C2×C3×Dic3D1211yescharacteristic
186C2×C3×Dic3#6711yescharacteristic
187C2×C3×Dic3#6711yescharacteristic
188C3×D12—22no—
189C3×D12—21no—
190C3×D12—21no—
191C3×D12—21no—
192C3×D12—21no—
193C3×D12—41no—
194C3×D12—62no—
195C3×D12—61no—
196C3×D12—61no—
197C3×D12—61no—
198C3×D12—61no—
199C3×D12—121no—
200C3×D12—122no—
201C3×D12—122no—
202#627—124no—
203#627—122no—
204#627—122no—
205#626—42no—
206#626—42no—
207C32×C8—61no—
208C3⋊(C3⋊C8)—21no—
209C3×C3⋊C8—21no—
210C3×C3⋊C8—21no—
211C3×C3⋊C8—41no—
212C3×C3⋊C8—61no—
213C3×C3⋊C8—61no—
214C3×C3⋊C8—121no—
215#486—271noSylow
216C2×C33#12411yescharacteristic
217C2×C33C2×D811yescharacteristic
218C2×C33C2×SD1611yescharacteristic
219C3×C3⋊D3—81no—
220C32×D3—81no—
221C2×C3×D4—31no—
222C2×C3×D4—31no—
223C2×C3×D4—61no—
224C2×C3×D4—91no—
225C2×C3×D4—181no—
226C2×C3×D4—181no—
227C2×#67—61no—
228C2×#67—61no—
229C2×D12—31no—
230C2×D12—31no—
231C2×D12—61no—
232C2×D12—61no—
233C2×D12—61no—
234C2×D12D3211yescharacteristic
235C2×Dic6—181no—
236C2×Dic6—181no—
237C2×Dic6—182no—
238C2×Dic6—181no—
239C2×C3×C8—31no—
240C2×C3×C8—31no—
241C2×C3×C8—61no—
242C2×C3×C8—91no—
243C3×C4⋊C4—91no—
244C3×C42—31no—
245C3×C42—31no—
246C3×C42—61no—
247C4⋊Dic3—31no—
248C4⋊Dic3—31no—
249C4⋊Dic3—61no—
250C4⋊Dic3—91no—
251C4⋊Dic3—91no—
252C4⋊Dic3—91no—
253C4⋊Dic3—181no—
254C4⋊Dic3—181no—
255C4⋊Dic3—181no—
256C4⋊Dic3—181no—
257C4⋊Dic3—181no—
258C4⋊Dic3—182no—
259C4⋊Dic3—361no—
260Dic3⋊C4—181no—
261C4×Dic3D3211yescharacteristic
262C2×C3⋊C8—31no—
263C2×C3⋊C8—31no—
264C2×C3⋊C8—61no—
265C2×C3⋊C8D3211yescharacteristic
266C22×C32—21no—
267C22×C32—21no—
268C22×C32—21no—
269C22×C32—41no—
270C22×C32—121no—
271C22×C32—121no—
272C22×C32—121no—
273C22×C32C2×D1211yescharacteristic
274C22×C32D3×D411yescharacteristic
275C22×C32C2×#6711yescharacteristic
276C2×C3⋊D3—41no—
277C2×C3⋊D3—41no—
278C2×C3⋊D3—41no—
279C2×C3×D3—42no—
280C2×C3×D3—41no—
281C2×C3×D3—41no—
282C2×C3×D3—41no—
283C2×C3×D3—41no—
284C2×C3×D3—41no—
285C2×C3×D3—41no—
286C2×C3×D3—81no—
287C2×C3×D3—81no—
288C2×C3×D3—122no—
289C2×C3×D3—121no—
290C2×C3×D3—121no—
291C2×C3×D3—121no—
292C2×C3×D3—121no—
293C2×C3×D3—121no—
294C2×C3×D3—121no—
295C2×C3×D3—241no—
296C2×C3×D3—241no—
297C32×C4—21no—
298C32×C4—21no—
299C32×C4—21no—
300C32×C4—21no—
301C32×C4—21no—
302C32×C4—21no—
303C32×C4—32no—
304C32×C4—32no—
305C32×C4—41no—
306C32×C4—41no—
307C32×C4#23511yescharacteristic
308C32×C4#23511yescharacteristic
309C32×C4#23511yescharacteristic
310C32×C4#23711yescharacteristic
311C32×C4#23711yescharacteristic
312C32×C4#23711yescharacteristic
313C3⋊Dic3—41no—
314C3⋊Dic3—121no—
315C3⋊Dic3—122no—
316C3⋊Dic3—121no—
317C3⋊Dic3—121no—
318C3⋊Dic3—241no—
319C3⋊Dic3—241no—
320C3⋊Dic3—242no—
321C3×Dic3—22no—
322C3×Dic3—22no—
323C3×Dic3—122no—
324C3×Dic3—121no—
325C3×Dic3—121no—
326C3×Dic3#20412yes—
327C3×Dic3D2412yes—
328C3×Dic3#21312yes—
329C3×Dic3#21412yes—
330C4⋊Q8—271no—
331#127—91no—
332#102—91no—
333#102—271no—
334#102—271no—
335#102—271no—
336C4×C8—91no—
337C33#48611yescharacteristic, complemented, Sylow
338C23×C3—61no—
339C23×C3—61no—
340C23×C3—121no—
341C23×C3—181no—
342C22×D3—21no—
343C22×D3—61no—
344C22×D3—61no—
345C22×D3—121no—
346C3×D4—62no—
347C3×D4—61no—
348C3×D4—61no—
349C3×D4—61no—
350C3×D4—61no—
351C3×D4—121no—
352C3×D4—181no—
353C3×D4—181no—
354C3×D4—362no—
355C3×D4—362no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—31no—
360C2×C3×C4—31no—
361C2×C3×C4—31no—
362C2×C3×C4—31no—
363C2×C3×C4—41no—
364C2×C3×C4—61no—
365C2×C3×C4—61no—
366C2×C3×C4—181no—
367C2×C3×C4C2×D3211yescharacteristic
368C2×C3×C4C2×D3211yescharacteristic
369C2×C3×C4C2×D3211yescharacteristic
370#67—122no—
371#67—122no—
372C2×Dic3—62no—
373C2×Dic3—61no—
374C2×Dic3—61no—
375C2×Dic3—182no—
376C2×Dic3—181no—
377C2×Dic3—181no—
378C2×Dic3—181no—
379C2×Dic3—361no—
380C2×Dic3—361no—
381C2×Dic3—362no—
382C2×Dic3#62611yescharacteristic
383C2×Dic3#62611yescharacteristic
384D12—21no—
385D12—21no—
386D12—62no—
387D12—61no—
388D12—61no—
389D12—61no—
390D12—61no—
391D12—121no—
392D12—122no—
393D12—122no—
394Dic6—362no—
395Dic6—362no—
396Dic6—364no—
397Dic6—362no—
398C3×C8—61no—
399C3×C8—61no—
400C3×C8—121no—
401C3×C8—181no—
402C3⋊C8—21no—
403C3⋊C8—61no—
404C3⋊C8—61no—
405C3⋊C8—121no—
406C2×C32—21no—
407C2×C32—21no—
408C2×C32—21no—
409C2×C32—21no—
410C2×C32—21no—
411C2×C32—21no—
412C2×C32—21no—
413C2×C32—21no—
414C2×C32—21no—
415C2×C32—41no—
416C2×C32—41no—
417C2×C32—41no—
418C2×C32—241no—
419C2×C32#87511yescharacteristic
420C2×C32#88511yescharacteristic
421C2×C32C2×#20411yescharacteristic
422C2×C32C2×D2411yescharacteristic
423C2×C32D3×D811yescharacteristic
424C2×C32D3×SD1611yescharacteristic
425C2×C32C2×#21311yescharacteristic
426C2×C32C2×#21411yescharacteristic
427C2×C32#93611yescharacteristic
428C3⋊D3—81no—
429C3×D3—82no—
430C3×D3—81no—
431C3×D3—81no—
432C3×D3—242no—
433C3×D3—241no—
434C3×D3—241no—
435C2×Q8—541no—
436C2×D4—31no—
437C2×D4—91no—
438C2×D4—181no—
439C2×D4—181no—
440C2×C8—31no—
441C2×C8—91no—
442C4⋊C4—91no—
443C4⋊C4—271no—
444C4⋊C4—541no—
445C42—31no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—41no—
450C22×C3—122no—
451C22×C3—121no—
452C22×C3—121no—
453C22×C3—121no—
454C22×C3—121no—
455C22×C3—121no—
456C22×C3—121no—
457C22×C3—241no—
458C22×C3—241no—
459C22×C3—361no—
460C22×C3—361no—
461C22×C3—361no—
462C22×C3D3×D1211yescharacteristic
463C22×C3C2×#62611yescharacteristic
464C22×C3D3×#6711yescharacteristic
465C2×D3—41no—
466C2×D3—41no—
467C2×D3—41no—
468C2×D3—122no—
469C2×D3—121no—
470C2×D3—121no—
471C2×D3—121no—
472C2×D3—121no—
473C2×D3—121no—
474C2×D3—121no—
475C2×D3—241no—
476C2×D3—241no—
477C3×C4—21no—
478C3×C4—21no—
479C3×C4—21no—
480C3×C4—21no—
481C3×C4—21no—
482C3×C4—21no—
483C3×C4—32no—
484C3×C4—32no—
485C3×C4—32no—
486C3×C4—32no—
487C3×C4—41no—
488C3×C4—41no—
489C3×C4—62no—
490C3×C4—62no—
491C3×C4—361no—
492C3×C4#379511yescharacteristic
493C3×C4#379511yescharacteristic
494C3×C4#379811yescharacteristic
495C3×C4#380411yescharacteristic
496C3×C4#380411yescharacteristic
497C3×C4#380511yescharacteristic
498Dic3—122no—
499Dic3—121no—
500Dic3—121no—
501Dic3—362no—
502Dic3—361no—
503Dic3—361no—
504Dic3—361no—
505Dic3—721no—
506Dic3—721no—
507Dic3—722no—
508Dic3#371412yes—
509Dic3#371612yes—
510C32—21no—
511C32—21no—
512C32—21no—
513C32—41no—
514C32#469411yescharacteristic, complemented
515C32#476011yescharacteristic, complemented
516C32#507011yescharacteristic, complemented
517C23—61no—
518C23—181no—
519Q8—1082no—
520D4—61no—
521D4—61no—
522D4—181no—
523D4—181no—
524D4—362no—
525D4—362no—
526C2×C4—31no—
527C2×C4—31no—
528C2×C4—181no—
529C2×C4—541no—
530C2×C4D3311yescharacteristic
531C8—61no—
532C8—181no—
533C2×C3—21no—
534C2×C3—21no—
535C2×C3—21no—
536C2×C3—21no—
537C2×C3—21no—
538C2×C3—21no—
539C2×C3—21no—
540C2×C3—21no—
541C2×C3—21no—
542C2×C3—41no—
543C2×C3—41no—
544C2×C3—41no—
545C2×C3—242no—
546C2×C3—241no—
547C2×C3—241no—
548C2×C3—721no—
549C2×C3D3×#20411yescharacteristic
550C2×C3D3×D2411yescharacteristic
551C2×C3C2×#371411yescharacteristic
552C2×C3C2×#371611yescharacteristic
553C2×C3#6385211yescharacteristic
554C2×C3#6386011yescharacteristic
555C2×C3D3×#21311yescharacteristic
556C2×C3D3×#21411yescharacteristic
557C2×C3#6395911yescharacteristic
558D3—81no—
559D3—242no—
560D3—241no—
561D3—241no—
562C22—121no—
563C22—121no—
564C22—121no—
565C22—361no—
566C22—361no—
567C22—361no—
568C22D3×#62611yescharacteristic
569C4—32no—
570C4—32no—
571C4—361no—
572C4—1081no—
573C4#8896011yescharacteristic
574C4#8896111yescharacteristic
575C3—21no—
576C3—21no—
577C3—21no—
578C3—41no—
579C3#1059027311yescharacteristic, complemented
580C3#1059087811yescharacteristic, complemented
581C3#1059148711yescharacteristic, complemented
582C2—241no—
583C2—721no—
584C2D3×#371411yescharacteristic
585C2D3×#371611yescharacteristic
586C2#1172300211yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC3×#63911C2
hypocenterC33#486
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
1. #11723534C2semidirect
3. #11722075C2semidirect
4. #11722034C2semidirect
5. #11722032C2semidirect
6. C3×#63563C2semidirect
7. C3×Dic3×C3⋊C8C2semidirect
43. #63563D3semidirect
45. Dic3×C3⋊C8D3semidirect
337. C33#486semidirect
514. C32#4694semidirect
515. C32#4760semidirect
516. C32#5070semidirect
579. C3#10590273semidirect
580. C3#10590878semidirect
581. C3#10591487semidirect

Maximal subgroups

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

subgroupindex
#117235342
C3×#639112
#117220752
#117220342
#117220322
C3×#635632
C3×Dic3×C3⋊C82
#105914873
#105908783
#105902733