Finite Group Databasebeta

← #13155643

Subgroups — #13155643

Order 1728.

686 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88811C211yesmaximal, characteristic
2Dic3×C4⋊(C3⋊D3)C211yesmaximal, characteristic
3C3×#63844C211yesmaximal, characteristic, complemented
4#11722054C211yesmaximal, characteristic, complemented
5#11722040C211yesmaximal, characteristic, complemented
6C3×#63564C211yesmaximal, characteristic
7C3×Dic3×C3⋊C8C211yesmaximal, characteristic, complemented
8#10590744—31nomaximal
9#10590453—31nomaximal
10C4×#3717—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×Dic3×C3⋊D3—21no—
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88811C414yescomplemented
20#88803—21no—
21C3×#3724—21no—
22C3×Dic32—21no—
23C4×C4⋊(C3⋊D3)—31no—
24C3×#887—31no—
25C3×C4×Dic6—31no—
26Dic3×D12—31no—
27Dic3×D12—31no—
28Dic3×D12—61no—
29#63844D311yescharacteristic, complemented
30C2×#3717—31no—
31C2×#3716—31no—
32C2×#3715—31no—
33#63643—31no—
34#63635—31no—
35C3×#821—31no—
36C3×C8×Dic3—31no—
37C3×#809—31no—
38C3×C4×C3⋊C8—31no—
39#63573—31no—
40#63573—31no—
41#63573—61no—
42#63566—31no—
43#63566—31no—
44#63566—61no—
45#63564D311yescharacteristic
46Dic3×C3⋊C8D311yescharacteristic, complemented
47C22×C3×C3⋊D3—21no—
48C2×C33×C4C2311yescharacteristic
49C2×C3⋊(C3⋊Dic3)—21no—
50C2×C3×C3⋊Dic3—21no—
51C3×C4⋊(C3⋊D3)—21no—
52C3×C4⋊(C3⋊D3)C2×C412yes—
53C3×#634—21no—
54C3×#634C2×C412yescomplemented
55C2×C32×Dic3—21no—
56C2×C32×Dic3D411yescharacteristic
57C2×C32×Dic3D411yescharacteristic
58Dic3×C3⋊D3—42no—
59C3×C3⋊(C3⋊C8)C2×C412yescomplemented
60C32×C3⋊C8—21no—
61C4×#215—91no—
62#4758—91no—
63C4×#204—91no—
64C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
65C2×C4×C3⋊D3—61no—
66C2×#634C2×D311yescharacteristic
67C2×C3×D12—31no—
68C2×C3×D12—31no—
69C2×C3×D12—61no—
70C2×C3×Dic6—31no—
71C2×C3×Dic6—31no—
72C2×C3×Dic6—61no—
73C2×D3×Dic3—62no—
74C2×D3×Dic3—61no—
75C2×D3×Dic3—61no—
76C2×C32×C8—31no—
77C32×C4⋊C4—31no—
78C32×C42—31no—
79#3752—61no—
80C4⋊(C3⋊Dic3)—31no—
81C4⋊(C3⋊Dic3)—31no—
82C4⋊(C3⋊Dic3)—31no—
83C4⋊(C3⋊Dic3)—61no—
84C4⋊(C3⋊Dic3)—61no—
85C4⋊(C3⋊Dic3)—61no—
86C4⋊(C3⋊Dic3)—121no—
87C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
88C3×C4⋊Dic3—21no—
89C3×C4⋊Dic3C2×D311yescharacteristic
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×Dic3⋊C4—61no—
92C3×Dic3⋊C4—61no—
93C3×C4×Dic3—21no—
94C3×C4×Dic3—61no—
95C3×C4×Dic3—61no—
96C3×C4×Dic3C2×D311yescharacteristic
97C3×C4×Dic3C2×D311yescharacteristic
98C2×C3×C3⋊C8—21no—
99C2×C3×C3⋊C8—31no—
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—61no—
102C2×C3×C3⋊C8C2×D311yescharacteristic
103C2×C3×C3⋊C8C2×D311yescharacteristic
104#3724—21no—
105(C2×D3)⋊Dic3—62no—
106(C2×D3)⋊Dic3—61no—
107(C2×D3)⋊Dic3—61no—
108Dic32—21no—
109#3717—34no—
110#3716—34no—
111#3715—34no—
112C22×C33C2×D411yescharacteristic
113C2×C3×C3⋊D3—22no—
114C2×C3×C3⋊D3—41no—
115C2×C3×C3⋊D3—41no—
116C33×C4C22×C411yescharacteristic
117C33×C4#4111yescharacteristic
118C3⋊(C3⋊Dic3)—41no—
119C3×C3⋊Dic3—22no—
120C32×Dic3—21no—
121C32×Dic3—41no—
122C32×Dic3SD1612yescomplemented
123C3×C4×Q8—91no—
124C2×#215—91no—
125D4×Dic3—91no—
126C2×#214—91no—
127C2×#204—91no—
128#887—31no—
129C4×D12—91no—
130C4×D12—91no—
131C4×D12—181no—
132C4×Dic6—31no—
133C3×#102—91no—
134C3×C4×C8—91no—
135#835—91no—
136#821—31no—
137#817—91no—
138#816—91no—
139#816—91no—
140#816—181no—
141C8×Dic3—31no—
142#809—31no—
143#808—91no—
144#808—91no—
145#808—181no—
146C4×C3⋊C8—31no—
147C22×C3⋊D3—21no—
148C22×C3×D3—62no—
149C22×C3×D3—61no—
150C22×C3×D3—61no—
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—31no—
155C2×C32×C4—31no—
156C2×C32×C4—41no—
157C2×C32×C4—61no—
158C2×C32×C4C22×D311yescharacteristic
159C2×C32×C4C22×D311yescharacteristic
160C2×C32×C4C22×D311yescharacteristic
161C2×C3⋊Dic3—21no—
162C2×C3⋊Dic3—61no—
163C2×C3⋊Dic3—62no—
164C2×C3⋊Dic3—61no—
165C2×C3⋊Dic3—61no—
166C2×C3⋊Dic3—121no—
167C2×C3⋊Dic3—121no—
168C2×C3⋊Dic3—122no—
169C4⋊(C3⋊D3)—21no—
170C4⋊(C3⋊D3)C4×D312yes—
171C4×C3⋊D3—122no—
172#634—21no—
173#634C4×D312yescomplemented
174C2×C3×Dic3—21no—
175C2×C3×Dic3—21no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—41no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—121no—
182C2×C3×Dic3D1211yescharacteristic
183C2×C3×Dic3D1211yescharacteristic
184C2×C3×Dic3#6711yescharacteristic
185C2×C3×Dic3#6711yescharacteristic
186C3×D12—32no—
187C3×D12—32no—
188C3×D12—62no—
189C3×D12—62no—
190C3×D12—61no—
191C3×D12—61no—
192C3×Dic6—32no—
193C3×Dic6—32no—
194C3×Dic6—62no—
195C3×Dic6—61no—
196C3×Dic6—61no—
197C3×Dic6—121no—
198D3×Dic3—124no—
199D3×Dic3—122no—
200D3×Dic3—122no—
201C32×C8—61no—
202C3⋊(C3⋊C8)C4×D312yescomplemented
203C3×C3⋊C8—21no—
204C3×C3⋊C8—21no—
205C3×C3⋊C8—32no—
206C3×C3⋊C8—32no—
207C3×C3⋊C8—41no—
208C3×C3⋊C8—62no—
209C4×SD16—271noSylow
210C2×C33C4×D411yescharacteristic
211C2×C33C2×SD1611yescharacteristic
212C2×C33#13511yescharacteristic
213C3×C3⋊D3—42no—
214C2×C3×Q8—91no—
215C2×C3×D4—91no—
216C22×Dic3—181no—
217C2×D12—31no—
218C2×D12—31no—
219C2×D12—61no—
220C2×C4×D3—181no—
221C2×C4×D3—181no—
222C2×C4×D3—182no—
223C2×Dic6—31no—
224C2×Dic6—31no—
225C2×Dic6—61no—
226C2×C3×C8—31no—
227C2×C3×C8—31no—
228C2×C3×C8—61no—
229C2×C3×C8—91no—
230C3×C4⋊C4—31no—
231C3×C4⋊C4—31no—
232C3×C4⋊C4—61no—
233C3×C4⋊C4—181no—
234C3×C42—31no—
235C3×C42—31no—
236C3×C42—61no—
237C3×C42—181no—
238#217—181no—
239#215—94no—
240#214—94no—
241(C2×D3)⋊C4—181no—
242(C2×D3)⋊C4—181no—
243(C2×D3)⋊C4—182no—
244C4⋊Dic3—91no—
245C4⋊Dic3—91no—
246C4⋊Dic3—91no—
247C4⋊Dic3—181no—
248C4⋊Dic3—181no—
249C4⋊Dic3—181no—
250C4⋊Dic3—361no—
251C4⋊Dic3D3211yescharacteristic
252Dic3⋊C4—61no—
253Dic3⋊C4—61no—
254C4×Dic3—61no—
255C4×Dic3—61no—
256C4×Dic3D3211yescharacteristic
257C2×C3⋊C8—31no—
258C2×C3⋊C8—31no—
259C2×C3⋊C8—61no—
260C2×C3⋊C8D3211yescharacteristic
261#204—94no—
262C22×C32—21no—
263C22×C32—21no—
264C22×C32—21no—
265C22×C32—41no—
266C22×C32C2×D1211yescharacteristic
267C22×C32D3×D411yescharacteristic
268C22×C32C2×#6711yescharacteristic
269C2×C3⋊D3—22no—
270C2×C3⋊D3—41no—
271C2×C3⋊D3—41no—
272C2×C3×D3—64no—
273C2×C3×D3—62no—
274C2×C3×D3—62no—
275C2×C3×D3—122no—
276C2×C3×D3—122no—
277C2×C3×D3—121no—
278C2×C3×D3—121no—
279C2×C3×D3—121no—
280C2×C3×D3—121no—
281C32×C4—21no—
282C32×C4—21no—
283C32×C4—21no—
284C32×C4—21no—
285C32×C4—21no—
286C32×C4—21no—
287C32×C4—32no—
288C32×C4—41no—
289C32×C4—41no—
290C32×C4—61no—
291C32×C4—121no—
292C32×C4C2×C4×D311yescharacteristic
293C32×C4C2×C4×D311yescharacteristic
294C32×C4C2×C4×D311yescharacteristic
295C32×C4#23511yescharacteristic
296C32×C4#23511yescharacteristic
297C32×C4#23711yescharacteristic
298C3⋊Dic3—22no—
299C3⋊Dic3—121no—
300C3⋊Dic3—122no—
301C3⋊Dic3—121no—
302C3⋊Dic3—121no—
303C3⋊Dic3—241no—
304C3⋊Dic3—241no—
305C3⋊Dic3—242no—
306C3×Dic3—21no—
307C3×Dic3—21no—
308C3×Dic3—22no—
309C3×Dic3—41no—
310C3×Dic3—41no—
311C3×Dic3—41no—
312C3×Dic3—62no—
313C3×Dic3—62no—
314C3×Dic3—81no—
315C3×Dic3—122no—
316C3×Dic3#20412yescomplemented
317C3×Dic3#21512yescomplemented
318C2×SD16—271no—
319C4×Q8—91no—
320C4×D4—271no—
321#106—271no—
322#103—271no—
323#102—91no—
324C4×C8—91no—
325C33C4×SD1611yescharacteristic, complemented, Sylow
326C23×C3—181no—
327C22×D3—62no—
328C22×D3—61no—
329C22×D3—61no—
330C3×Q8—92no—
331C3×Q8—181no—
332C3×D4—92no—
333C3×D4—181no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—31no—
338C2×C3×C4—31no—
339C2×C3×C4—31no—
340C2×C3×C4—31no—
341C2×C3×C4—41no—
342C2×C3×C4—61no—
343C2×C3×C4—61no—
344C2×C3×C4—61no—
345C2×C3×C4—61no—
346C2×C3×C4—121no—
347C2×C3×C4—181no—
348C2×C3×C4C2×D3211yescharacteristic
349C2×C3×C4C2×D3211yescharacteristic
350C2×C3×C4C2×D3211yescharacteristic
351C2×Dic3—21no—
352C2×Dic3—61no—
353C2×Dic3—61no—
354C2×Dic3—121no—
355C2×Dic3—182no—
356C2×Dic3—181no—
357C2×Dic3—181no—
358C2×Dic3—181no—
359C2×Dic3—361no—
360C2×Dic3—361no—
361C2×Dic3—362no—
362C2×Dic3—362no—
363C2×Dic3#62611yescharacteristic
364C2×Dic3#62611yescharacteristic
365D12—32no—
366D12—32no—
367D12—62no—
368D12—62no—
369D12—61no—
370D12—61no—
371C4×D3—362no—
372C4×D3—362no—
373C4×D3—364no—
374Dic6—32no—
375Dic6—32no—
376Dic6—62no—
377Dic6—61no—
378Dic6—61no—
379Dic6—121no—
380C3×C8—61no—
381C3×C8—61no—
382C3×C8—92no—
383C3×C8—121no—
384C3⋊C8—21no—
385C3⋊C8—32no—
386C3⋊C8—32no—
387C3⋊C8—62no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—21no—
391C2×C32—21no—
392C2×C32—21no—
393C2×C32—21no—
394C2×C32—21no—
395C2×C32—21no—
396C2×C32—21no—
397C2×C32—41no—
398C2×C32—41no—
399C2×C32—41no—
400C2×C32C4×D1211yescharacteristic
401C2×C32#67⋊C411yescharacteristic
402C2×C32C2×#20411yescharacteristic
403C2×C32#90411yescharacteristic
404C2×C32#91211yescharacteristic
405C2×C32D3×SD1611yescharacteristic
406C2×C32C4×#6711yescharacteristic
407C2×C32C2×#21511yescharacteristic
408C2×C32#95011yescharacteristic
409C3⋊D3—42no—
410C3×D3—124no—
411C3×D3—122no—
412C3×D3—122no—
413C2×Q8—91no—
414C2×D4—91no—
415C22×C4—541no—
416SD16—274no—
417C2×C8—31no—
418C2×C8—91no—
419C4⋊C4—31no—
420C4⋊C4—181no—
421C4⋊C4—271no—
422C22⋊C4—541no—
423C42—31no—
424C42—181no—
425C22×C3—21no—
426C22×C3—21no—
427C22×C3—21no—
428C22×C3—41no—
429C22×C3—182no—
430C22×C3—361no—
431C22×C3—361no—
432C22×C3D3×D1211yescharacteristic
433C22×C3C2×#62611yescharacteristic
434C22×C3D3×#6711yescharacteristic
435C2×D3—64no—
436C2×D3—62no—
437C2×D3—62no—
438C2×D3—122no—
439C2×D3—122no—
440C2×D3—121no—
441C2×D3—121no—
442C2×D3—121no—
443C2×D3—121no—
444C3×C4—21no—
445C3×C4—21no—
446C3×C4—21no—
447C3×C4—21no—
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—32no—
451C3×C4—32no—
452C3×C4—41no—
453C3×C4—41no—
454C3×C4—62no—
455C3×C4—61no—
456C3×C4—61no—
457C3×C4—121no—
458C3×C4—121no—
459C3×C4—121no—
460C3×C4—182no—
461C3×C4—241no—
462C3×C4#379511yescharacteristic
463C3×C4#379811yescharacteristic
464C3×C4C4×D3211yescharacteristic
465C3×C4C4×D3211yescharacteristic
466C3×C4C4×D3211yescharacteristic
467C3×C4#380411yescharacteristic
468Dic3—21no—
469Dic3—41no—
470Dic3—62no—
471Dic3—62no—
472Dic3—122no—
473Dic3—362no—
474Dic3—361no—
475Dic3—361no—
476Dic3—361no—
477Dic3—721no—
478Dic3—721no—
479Dic3—722no—
480Dic3#371712yescomplemented
481C32—21no—
482C32—21no—
483C32—21no—
484C32—41no—
485C32C4×#20411yescharacteristic, complemented
486C32#475811yescharacteristic, complemented
487C32C4×#21511yescharacteristic, complemented
488C23—181no—
489Q8—92no—
490Q8—181no—
491D4—92no—
492D4—181no—
493C2×C4—31no—
494C2×C4—31no—
495C2×C4—61no—
496C2×C4—181no—
497C2×C4—541no—
498C2×C4—1082no—
499C2×C4D3311yescharacteristic
500C8—61no—
501C8—92no—
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—21no—
511C2×C3—41no—
512C2×C3—41no—
513C2×C3—41no—
514C2×C3—362no—
515C2×C3D3×#20411yescharacteristic
516C2×C3#6380711yescharacteristic
517C2×C3#6382911yescharacteristic
518C2×C3C2×#371711yescharacteristic
519C2×C3#6387911yescharacteristic
520C2×C3#6388011yescharacteristic
521C2×C3C4×#62611yescharacteristic
522C2×C3#6393311yescharacteristic
523C2×C3D3×#21511yescharacteristic
524D3—124no—
525D3—122no—
526D3—122no—
527C22—182no—
528C22—361no—
529C22—361no—
530C22D3×#62611yescharacteristic
531C4—32no—
532C4—61no—
533C4—121no—
534C4—182no—
535C4—1081no—
536C4#8894911yescharacteristic
537C4#8896011yescharacteristic
538C3—21no—
539C3—21no—
540C3—21no—
541C3—41no—
542C3C4×#371711yescharacteristic, complemented
543C3#1059045311yescharacteristic, complemented
544C3#1059074411yescharacteristic, complemented
545C2—362no—
546C2D3×#371711yescharacteristic
547C2#1172285411yescharacteristic
548C2#1172293411yescharacteristic
subgroupG/H
centerC22D3×#626
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypocenterC33C4×SD16
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. C3×#63844C2semidirect
4. #11722054C2semidirect
5. #11722040C2semidirect
7. C3×Dic3×C3⋊C8C2semidirect
19. #88811C4semidirect
29. #63844D3semidirect
46. Dic3×C3⋊C8D3semidirect
54. C3×#634C2×C4semidirect
59. C3×C3⋊(C3⋊C8)C2×C4semidirect
122. C32×Dic3SD16semidirect
173. #634C4×D3semidirect
202. C3⋊(C3⋊C8)C4×D3semidirect
316. C3×Dic3#204semidirect
317. C3×Dic3#215semidirect
325. C33C4×SD16semidirect
480. Dic3#3717semidirect
485. C32C4×#204semidirect
486. C32#4758semidirect
487. C32C4×#215semidirect
542. C3C4×#3717semidirect
543. C3#10590453semidirect
544. C3#10590744semidirect

Maximal subgroups

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

subgroupindex
C2×#888112
Dic3×C4⋊(C3⋊D3)2
C3×#638442
#117220542
#117220402
C3×#635642
C3×Dic3×C3⋊C82
#105907443
#105904533
C4×#37173