Finite Group Databasebeta

← #13155671

Subgroups — #13155671

Order 1728.

662 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Dic3×#634C211yesmaximal, characteristic, complemented
2C3×Dic3×D12C211yesmaximal, characteristic
3C2×C3×#3716C211yesmaximal, characteristic
4#11722076C211yesmaximal, characteristic, complemented
5#11722055C211yesmaximal, characteristic, complemented
6#11722032C211yesmaximal, characteristic
7C3×#63554C211yesmaximal, characteristic, complemented
8#10591440—31nomaximal
9#10590766—31nomaximal
10#10590360—31nomaximal
11C2×C3×#634C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C3×D3×Dic3—21no—
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88807—21no—
20Dic3×C3⋊Dic3—21no—
21C3×(C2×D3)⋊Dic3—21no—
22C3×#3716C414yescomplemented
23C4×#634—31no—
24C3×D4×Dic3—31no—
25C2×C3×#214—31no—
26C2×C3×#204—31no—
27C3×C4×D12—31no—
28Dic3×D12D311yescharacteristic
29Dic3×Dic6—31no—
30Dic3×Dic6—31no—
31Dic3×Dic6—61no—
32C2×#3716D311yescharacteristic
33#63659—31no—
34#63645—31no—
35C3×#815—31no—
36C3×#803—31no—
37#63575—31no—
38#63575—31no—
39#63575—61no—
40#63567—31no—
41#63566—31no—
42#63565—31no—
43#63563—31no—
44#63563—31no—
45#63563—61no—
46#63554D311yescharacteristic, complemented
47C22×C32×D3—21no—
48C2×C33×C4C2311yescharacteristic
49C2×C3⋊(C3⋊Dic3)—21no—
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3—21no—
52C3×#634—21no—
53C3×#634C2×C412yescomplemented
54C2×C32×Dic3D411yescharacteristic
55C2×C32×Dic3D411yescharacteristic
56C32×D12—21no—
57C32×D12C2×C412yes—
58C3×D3×Dic3—42no—
59C3×C3⋊(C3⋊C8)—21no—
60C32×C3⋊C8C2×C412yescomplemented
61#5164—91no—
62#5018—91no—
63#4710—91no—
64C2×C32×D4—31no—
65C2×#634C2×D311yescharacteristic
66C22×C3×Dic3—61no—
67C2×C3×D12—21no—
68C2×C3×D12C2×D311yescharacteristic
69C2×C3×D12C2×D311yescharacteristic
70C2×C3×C4×D3—61no—
71C2×C3×Dic6—31no—
72C2×C3×Dic6—31no—
73C2×C3×Dic6—61no—
74C2×D3×Dic3—21no—
75C2×C32×C8—31no—
76C32×C42—31no—
77C4⋊(C3⋊Dic3)—31no—
78C4⋊(C3⋊Dic3)—31no—
79C4⋊(C3⋊Dic3)—31no—
80C4⋊(C3⋊Dic3)—61no—
81C4⋊(C3⋊Dic3)—61no—
82C4⋊(C3⋊Dic3)—61no—
83C4⋊(C3⋊Dic3)—121no—
84C4⋊(C3⋊Dic3)C2×D311yescharacteristic
85#3750—61no—
86C4×C3⋊Dic3—61no—
87C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
88C3×#217—61no—
89C3×#214—34no—
90C3×(C2×D3)⋊C4—61no—
91C3×C4⋊Dic3—31no—
92C3×C4⋊Dic3—31no—
93C3×C4⋊Dic3—61no—
94C3×C4×Dic3—21no—
95C3×C4×Dic3C2×D311yescharacteristic
96C3×C4×Dic3C2×D311yescharacteristic
97C2×C3×C3⋊C8—21no—
98C2×C3×C3⋊C8—31no—
99C2×C3×C3⋊C8—31no—
100C2×C3×C3⋊C8—61no—
101C2×C3×C3⋊C8C2×D311yescharacteristic
102C2×C3×C3⋊C8C2×D311yescharacteristic
103C3×#204—34no—
104Dic3⋊Dic3—62no—
105Dic3⋊Dic3—61no—
106Dic3⋊Dic3—61no—
107(C2×D3)⋊Dic3—21no—
108Dic32—62no—
109Dic32—61no—
110Dic32—61no—
111#3716Dic314yescomplemented
112C22×C33C2×D411yescharacteristic
113C2×C32×D3—22no—
114C2×C32×D3—41no—
115C2×C32×D3—41no—
116C33×C4C22×C411yescharacteristic
117C33×C4#4111yescharacteristic
118C3⋊(C3⋊Dic3)—41no—
119C3×C3⋊Dic3—22no—
120C3×C3⋊Dic3—41no—
121C32×Dic3—21no—
122C32×Dic3—21no—
123C2×C3×SD16—91no—
124C3×C4×D4—91no—
125Q8×Dic3—91no—
126D4×Dic3—31no—
127C2×#214—31no—
128C2×#204—31no—
129C4×D12—31no—
130C4×Dic6—91no—
131C4×Dic6—91no—
132C4×Dic6—181no—
133C3×#97—91no—
134#835—91no—
135#832—91no—
136#832—91no—
137#832—181no—
138#821—91no—
139#818—91no—
140#818—91no—
141#818—181no—
142#816—91no—
143#815—31no—
144#810—91no—
145#807—91no—
146#803—31no—
147C23×C32—61no—
148C22×C3×D3—21no—
149C22×C3×D3—21no—
150C22×C3×D3—41no—
151C32×D4—32no—
152C32×D4—61no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—31no—
157C2×C32×C4—31no—
158C2×C32×C4—41no—
159C2×C32×C4C22×D311yescharacteristic
160C2×C32×C4C22×D311yescharacteristic
161C2×C32×C4C22×D311yescharacteristic
162C2×C3⋊Dic3—21no—
163C2×C3⋊Dic3—21no—
164C2×C3⋊Dic3—61no—
165C2×C3⋊Dic3—62no—
166C2×C3⋊Dic3—61no—
167C2×C3⋊Dic3—61no—
168C2×C3⋊Dic3—121no—
169C2×C3⋊Dic3—121no—
170C2×C3⋊Dic3—122no—
171#634—21no—
172#634C2×Dic312yescomplemented
173C2×C3×Dic3—21no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—62no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—61no—
178C2×C3×Dic3—61no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—121no—
181C2×C3×Dic3—122no—
182C2×C3×Dic3D1211yescharacteristic
183C2×C3×Dic3D1211yescharacteristic
184C2×C3×Dic3#6711yescharacteristic
185C2×C3×Dic3#6711yescharacteristic
186C3×D12—21no—
187C3×D12—21no—
188C3×D12—22no—
189C3×D12—41no—
190C3×D12C4×D312yes—
191C3×D12C2×Dic312yes—
192C3×C4×D3—122no—
193C3×Dic6—32no—
194C3×Dic6—32no—
195C3×Dic6—62no—
196C3×Dic6—62no—
197C3×Dic6—61no—
198C3×Dic6—61no—
199D3×Dic3—42no—
200C32×C8—32no—
201C3⋊(C3⋊C8)—21no—
202C3×C3⋊C8—22no—
203C3×C3⋊C8—61no—
204C3×C3⋊C8—61no—
205C3×C3⋊C8—121no—
206C3×C3⋊C8C4×D312yescomplemented
207C3×C3⋊C8C2×Dic312yescomplemented
208#440—271noSylow
209C2×C33C4×D411yescharacteristic
210C2×C33AGL(1,ℤ/8)11yescharacteristic
211C2×C33#13711yescharacteristic
212C32×D3—42no—
213C2×C3×Q8—91no—
214C2×C3×D4—31no—
215C2×C3×D4—31no—
216C2×C3×D4—61no—
217C22×C3×C4—181no—
218C22×Dic3—61no—
219C2×D12D3211yescharacteristic
220C2×C4×D3—61no—
221C2×Dic6—31no—
222C2×Dic6—31no—
223C2×Dic6—61no—
224C3×SD16—94no—
225C2×C3×C8—31no—
226C2×C3×C8—31no—
227C2×C3×C8—61no—
228C2×C3×C8—91no—
229C3×C4⋊C4—91no—
230C3×C22⋊C4—181no—
231C3×C42—31no—
232C3×C42—31no—
233C3×C42—61no—
234#217—61no—
235#214—34no—
236(C2×D3)⋊C4—61no—
237C4⋊Dic3—31no—
238C4⋊Dic3—31no—
239C4⋊Dic3—61no—
240C4⋊Dic3—91no—
241C4⋊Dic3—91no—
242C4⋊Dic3—91no—
243C4⋊Dic3—181no—
244C4⋊Dic3—181no—
245C4⋊Dic3—181no—
246C4⋊Dic3—181no—
247C4⋊Dic3—361no—
248Dic3⋊C4—181no—
249Dic3⋊C4—181no—
250Dic3⋊C4—182no—
251C4×Dic3—181no—
252C4×Dic3—181no—
253C4×Dic3—182no—
254C4×Dic3—181no—
255C4×Dic3D3211yescharacteristic
256C2×C3⋊C8—31no—
257C2×C3⋊C8—31no—
258C2×C3⋊C8—61no—
259C2×C3⋊C8D3211yescharacteristic
260#204—34no—
261C22×C32—21no—
262C22×C32—21no—
263C22×C32—21no—
264C22×C32—41no—
265C22×C32—62no—
266C22×C32—121no—
267C22×C32—121no—
268C22×C32C2×D1211yescharacteristic
269C22×C32D3×D411yescharacteristic
270C22×C32C2×#6711yescharacteristic
271C2×C3×D3—22no—
272C2×C3×D3—22no—
273C2×C3×D3—42no—
274C2×C3×D3—41no—
275C2×C3×D3—41no—
276C2×C3×D3—41no—
277C2×C3×D3—41no—
278C2×C3×D3—81no—
279C2×C3×D3—81no—
280C32×C4—21no—
281C32×C4—21no—
282C32×C4—21no—
283C32×C4—21no—
284C32×C4—21no—
285C32×C4—21no—
286C32×C4—41no—
287C32×C4—41no—
288C32×C4—61no—
289C32×C4—61no—
290C32×C4C2×C4×D311yescharacteristic
291C32×C4C2×C4×D311yescharacteristic
292C32×C4#23511yescharacteristic
293C32×C4#23511yescharacteristic
294C32×C4#23711yescharacteristic
295C32×C4C22×Dic311yescharacteristic
296C3⋊Dic3—22no—
297C3⋊Dic3—41no—
298C3⋊Dic3—121no—
299C3⋊Dic3—122no—
300C3⋊Dic3—121no—
301C3⋊Dic3—121no—
302C3⋊Dic3—241no—
303C3⋊Dic3—241no—
304C3⋊Dic3—242no—
305C3×Dic3—22no—
306C3×Dic3—21no—
307C3×Dic3—21no—
308C3×Dic3—21no—
309C3×Dic3—21no—
310C3×Dic3—41no—
311C3×Dic3—64no—
312C3×Dic3—62no—
313C3×Dic3—62no—
314C3×Dic3—121no—
315C3×Dic3—121no—
316C3×Dic3—241no—
317C2×SD16—91no—
318C4×Q8—271no—
319C4×D4—91no—
320#107—271no—
321#103—271no—
322#102—271no—
323#97—91no—
324C33#44011yescharacteristic, complemented, Sylow
325C23×C3—61no—
326C23×C3—61no—
327C23×C3—121no—
328C22×D3—21no—
329C3×Q8—92no—
330C3×Q8—181no—
331C3×D4—32no—
332C3×D4—32no—
333C3×D4—62no—
334C3×D4—61no—
335C3×D4—61no—
336C3×D4—121no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—31no—
341C2×C3×C4—31no—
342C2×C3×C4—31no—
343C2×C3×C4—31no—
344C2×C3×C4—41no—
345C2×C3×C4—61no—
346C2×C3×C4—61no—
347C2×C3×C4—181no—
348C2×C3×C4—181no—
349C2×C3×C4—362no—
350C2×C3×C4C2×D3211yescharacteristic
351C2×C3×C4C2×D3211yescharacteristic
352C2×C3×C4C2×D3211yescharacteristic
353C2×Dic3—62no—
354C2×Dic3—61no—
355C2×Dic3—61no—
356C2×Dic3—61no—
357C2×Dic3—61no—
358C2×Dic3—121no—
359C2×Dic3—122no—
360C2×Dic3—182no—
361C2×Dic3—181no—
362C2×Dic3—181no—
363C2×Dic3—181no—
364C2×Dic3—361no—
365C2×Dic3—361no—
366C2×Dic3—362no—
367C2×Dic3#62611yescharacteristic
368C2×Dic3#62611yescharacteristic
369D12—21no—
370D12D3×Dic312yes—
371C4×D3—122no—
372Dic6—32no—
373Dic6—32no—
374Dic6—62no—
375Dic6—62no—
376Dic6—61no—
377Dic6—61no—
378C3×C8—32no—
379C3×C8—32no—
380C3×C8—62no—
381C3×C8—181no—
382C3⋊C8—61no—
383C3⋊C8—61no—
384C3⋊C8—121no—
385C3⋊C8D3×Dic312yescomplemented
386C2×C32—21no—
387C2×C32—21no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—21no—
391C2×C32—21no—
392C2×C32—21no—
393C2×C32—21no—
394C2×C32—21no—
395C2×C32—41no—
396C2×C32—41no—
397C2×C32—41no—
398C2×C32—122no—
399C2×C32C4×D1211yescharacteristic
400C2×C32#90811yescharacteristic
401C2×C32#90911yescharacteristic
402C2×C32#91411yescharacteristic
403C2×C32#91511yescharacteristic
404C2×C32C4×#6711yescharacteristic
405C2×C32#93211yescharacteristic
406C2×C32D4×Dic311yescharacteristic
407C2×C32#95111yescharacteristic
408C3×D3—42no—
409C3×D3—42no—
410C3×D3—82no—
411C2×Q8—91no—
412C2×D4—31no—
413C22×C4—181no—
414SD16—94no—
415C2×C8—31no—
416C2×C8—91no—
417C4⋊C4—91no—
418C4⋊C4—271no—
419C4⋊C4—541no—
420C22⋊C4—181no—
421C42—31no—
422C42—541no—
423C22×C3—21no—
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—41no—
427C22×C3—62no—
428C22×C3—62no—
429C22×C3—122no—
430C22×C3—121no—
431C22×C3—121no—
432C22×C3—121no—
433C22×C3—121no—
434C22×C3—241no—
435C22×C3—241no—
436C22×C3D3×D1211yescharacteristic
437C22×C3C2×#62611yescharacteristic
438C22×C3D3×#6711yescharacteristic
439C2×D3—22no—
440C2×D3—41no—
441C2×D3—41no—
442C3×C4—21no—
443C3×C4—21no—
444C3×C4—21no—
445C3×C4—21no—
446C3×C4—21no—
447C3×C4—21no—
448C3×C4—41no—
449C3×C4—41no—
450C3×C4—62no—
451C3×C4—61no—
452C3×C4—61no—
453C3×C4—61no—
454C3×C4—61no—
455C3×C4—121no—
456C3×C4—182no—
457C3×C4—361no—
458C3×C4#379511yescharacteristic
459C3×C4#379811yescharacteristic
460C3×C4C4×D3211yescharacteristic
461C3×C4C2×D3×Dic311yescharacteristic
462C3×C4C2×D3×Dic311yescharacteristic
463C3×C4#380411yescharacteristic
464Dic3—21no—
465Dic3—21no—
466Dic3—64no—
467Dic3—62no—
468Dic3—62no—
469Dic3—121no—
470Dic3—121no—
471Dic3—241no—
472Dic3—362no—
473Dic3—361no—
474Dic3—361no—
475Dic3—361no—
476Dic3—721no—
477Dic3—721no—
478Dic3—722no—
479C32—21no—
480C32—21no—
481C32—21no—
482C32—41no—
483C32#471011yescharacteristic, complemented
484C32#501811yescharacteristic, complemented
485C32#516411yescharacteristic, complemented
486C23—61no—
487Q8—92no—
488Q8—181no—
489D4—32no—
490D4—61no—
491C2×C4—31no—
492C2×C4—31no—
493C2×C4—181no—
494C2×C4—181no—
495C2×C4—362no—
496C2×C4—541no—
497C2×C4D3311yescharacteristic
498C8—32no—
499C8—181no—
500C2×C3—21no—
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—41no—
510C2×C3—41no—
511C2×C3—41no—
512C2×C3—122no—
513C2×C3—122no—
514C2×C3—242no—
515C2×C3#6379411yescharacteristic
516C2×C3#6380011yescharacteristic
517C2×C3#6383011yescharacteristic
518C2×C3#6383111yescharacteristic
519C2×C3Dic3×D1211yescharacteristic
520C2×C3C4×#62611yescharacteristic
521C2×C3#6392911yescharacteristic
522C2×C3#6393411yescharacteristic
523C2×C3Dic3×#6711yescharacteristic
524D3—42no—
525C22—62no—
526C22—121no—
527C22—121no—
528C22D3×#62611yescharacteristic
529C4—61no—
530C4—61no—
531C4—182no—
532C4—361no—
533C4—1081no—
534C4D32×Dic311yescharacteristic
535C4#8896011yescharacteristic
536C3—21no—
537C3—21no—
538C3—21no—
539C3—41no—
540C3#1059036011yescharacteristic, complemented
541C3#1059076611yescharacteristic, complemented
542C3#1059144011yescharacteristic, complemented
543C2—122no—
544C2#1172286811yescharacteristic
545C2#1172288611yescharacteristic
546C2Dic3×#62611yescharacteristic
subgroupG/H
centerC22D3×#626
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypocenterC33#440
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. Dic3×#634C2semidirect
4. #11722076C2semidirect
5. #11722055C2semidirect
7. C3×#63554C2semidirect
22. C3×#3716C4semidirect
46. #63554D3semidirect
53. C3×#634C2×C4semidirect
60. C32×C3⋊C8C2×C4semidirect
111. #3716Dic3semidirect
172. #634C2×Dic3semidirect
206. C3×C3⋊C8C4×D3semidirect
207. C3×C3⋊C8C2×Dic3semidirect
324. C33#440semidirect
385. C3⋊C8D3×Dic3semidirect
483. C32#4710semidirect
484. C32#5018semidirect
485. C32#5164semidirect
540. C3#10590360semidirect
541. C3#10590766semidirect
542. C3#10591440semidirect

Maximal subgroups

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

subgroupindex
Dic3×#6342
C3×Dic3×D122
C2×C3×#37162
#117220762
#117220552
#117220322
C3×#635542
#105914403
#105907663
#105903603