Finite Group Databasebeta

← #13155675

Subgroups — #13155675

Order 1728.

678 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
1#11723668C211yesmaximal, characteristic, complemented
2D12×C3⋊Dic3C211yesmaximal, characteristic
3C2×C3×#3716C211yesmaximal, characteristic
4#11722055C211yesmaximal, characteristic, complemented
5#11722040C211yesmaximal, characteristic, complemented
6C3×#63563C211yesmaximal, characteristic
7C3×#63555C211yesmaximal, characteristic, complemented
8#10591446—31nomaximal
9#10590766—31nomaximal
10#10590747—31nomaximal
11C2×C3×#634C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×D3×C3⋊Dic3—21no—
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88815—21no—
20#88813—21no—
21(C2×D3)⋊(C3⋊Dic3)—21no—
22C3×#3716C414yescomplemented
23D4×C3⋊Dic3—31no—
24C2×C3×#214—31no—
25C2×C3×#204—31no—
26Dic3×D12—31no—
27Dic3×D12—31no—
28Dic3×D12—61no—
29#63844—31no—
30#63843—31no—
31Dic3×Dic6—31no—
32C2×#3716D311yescharacteristic
33#63645—31no—
34#63643—31no—
35C3×#832—31no—
36C3×#821—31no—
37C3×#815—31no—
38C3×#815—31no—
39#63575—31no—
40#63575—31no—
41#63575—61no—
42#63566—31no—
43#63566—31no—
44#63566—61no—
45#63563D311yescharacteristic
46#63555D311yescharacteristic, complemented
47C22×C32×D3—21no—
48C2×C33×C4C2311yescharacteristic
49C2×C3⋊(C3⋊Dic3)—21no—
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3—21no—
52C2×C3×C3⋊Dic3D411yescharacteristic
53C2×C3×C3⋊Dic3D411yescharacteristic
54C3×#634—21no—
55C3×#634C2×C412yescomplemented
56C32×D12—21no—
57C32×D12C2×C412yes—
58D3×C3⋊Dic3—42no—
59C32×C3⋊C8—21no—
60C32×C3⋊C8C2×C412yescomplemented
61#5164—91no—
62#4757—91no—
63#4710—91no—
64C2×C32×D4—31no—
65C22×C3⋊Dic3—61no—
66C2×#634C2×D311yescharacteristic
67C2×C3×D12—21no—
68C2×C3×D12C2×D311yescharacteristic
69C2×C3×D12C2×D311yescharacteristic
70C2×C3×Dic6—31no—
71C2×C3×Dic6—31no—
72C2×C3×Dic6—61no—
73C2×D3×Dic3—61no—
74C2×D3×Dic3—61no—
75C2×D3×Dic3—62no—
76C2×C32×C8—31no—
77C2×C32×C8—31no—
78#3757—61no—
79C4⋊(C3⋊Dic3)—31no—
80C4⋊(C3⋊Dic3)—31no—
81C4⋊(C3⋊Dic3)—31no—
82C4⋊(C3⋊Dic3)—61no—
83C4⋊(C3⋊Dic3)—61no—
84C4⋊(C3⋊Dic3)—61no—
85C4⋊(C3⋊Dic3)—121no—
86C4⋊(C3⋊Dic3)C2×D311yescharacteristic
87C4×C3⋊Dic3C2×D311yescharacteristic
88C3×#214—34no—
89C3×C4⋊Dic3—31no—
90C3×C4⋊Dic3—31no—
91C3×C4⋊Dic3—61no—
92C3×C4×Dic3—31no—
93C3×C4×Dic3—31no—
94C3×C4×Dic3—61no—
95C2×C3×C3⋊C8—21no—
96C2×C3×C3⋊C8—21no—
97C2×C3×C3⋊C8C2×D311yescharacteristic
98C2×C3×C3⋊C8C2×D311yescharacteristic
99C2×C3×C3⋊C8C2×D311yescharacteristic
100C2×C3×C3⋊C8C2×D311yescharacteristic
101C3×#204—34no—
102#3724—61no—
103Dic3⋊Dic3—61no—
104Dic3⋊Dic3—61no—
105(C2×D3)⋊Dic3—61no—
106(C2×D3)⋊Dic3—61no—
107(C2×D3)⋊Dic3—62no—
108Dic32—61no—
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—21no—
121C3×C3⋊Dic3—21no—
122C3×C3⋊Dic3—41no—
123C2×C3×SD16—91no—
124Q8×Dic3—91no—
125D4×Dic3—91no—
126D4×Dic3—91no—
127D4×Dic3—181no—
128C2×#214—31no—
129C2×#204—31no—
130#887—91no—
131C4×D12—91no—
132C4×Dic6—91no—
133C3×#102—91no—
134C3×#97—91no—
135#835—91no—
136#832—31no—
137#821—31no—
138#818—91no—
139#818—91no—
140#818—181no—
141#816—91no—
142#816—91no—
143#816—181no—
144#815—31no—
145#815—31no—
146#807—91no—
147C23×C32—61no—
148C22×C3×D3—22no—
149C22×C3×D3—21no—
150C22×C3×D3—21no—
151C32×D4—32no—
152C32×D4—61no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—41no—
157C2×C32×C4C22×D311yescharacteristic
158C2×C32×C4C22×D311yescharacteristic
159C2×C32×C4C22×D311yescharacteristic
160C2×C3⋊Dic3—21no—
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—
169C2×C3⋊Dic3—122no—
170C2×C3⋊Dic3D1211yescharacteristic
171C2×C3⋊Dic3D1211yescharacteristic
172#634—21no—
173#634C2×Dic312yescomplemented
174C2×C3×Dic3—31no—
175C2×C3×Dic3—31no—
176C2×C3×Dic3—31no—
177C2×C3×Dic3—31no—
178C2×C3×Dic3—61no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—61no—
183C2×C3×Dic3—61no—
184C2×C3×Dic3—121no—
185C2×C3×Dic3—121no—
186C3×D12—22no—
187C3×D12—21no—
188C3×D12—21no—
189C3×D12—22no—
190C3×D12C4×D312yes—
191C3×D12C2×Dic312yes—
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—32no—
202C32×C8—61no—
203C3×C3⋊C8—21no—
204C3×C3⋊C8—21no—
205C3×C3⋊C8—22no—
206C3×C3⋊C8—41no—
207C3×C3⋊C8C4×D312yescomplemented
208C3×C3⋊C8C2×Dic312yescomplemented
209#440—271noSylow
210C2×C33C4×D411yescharacteristic
211C2×C33AGL(1,ℤ/8)11yescharacteristic
212C2×C33#13711yescharacteristic
213C32×D3—42no—
214C2×C3×Q8—91no—
215C2×C3×D4—31no—
216C2×C3×D4—31no—
217C2×C3×D4—61no—
218C22×Dic3—181no—
219C22×Dic3—181no—
220C22×Dic3—182no—
221C2×D12D3211yescharacteristic
222C2×C4×D3—181no—
223C2×Dic6—31no—
224C2×Dic6—31no—
225C2×Dic6—61no—
226C3×SD16—94no—
227C2×C3×C8—31no—
228C2×C3×C8—31no—
229C2×C3×C8—31no—
230C2×C3×C8—31no—
231C2×C3×C8—61no—
232C2×C3×C8—61no—
233C3×C4⋊C4—91no—
234C3×C42—91no—
235#217—181no—
236#217—181no—
237#217—182no—
238#214—34no—
239(C2×D3)⋊C4—181no—
240C4⋊Dic3—31no—
241C4⋊Dic3—31no—
242C4⋊Dic3—61no—
243C4⋊Dic3—91no—
244C4⋊Dic3—91no—
245C4⋊Dic3—91no—
246C4⋊Dic3—181no—
247C4⋊Dic3—181no—
248C4⋊Dic3—181no—
249C4⋊Dic3—181no—
250C4⋊Dic3—361no—
251Dic3⋊C4—181no—
252Dic3⋊C4—181no—
253C4×Dic3—31no—
254C4×Dic3—31no—
255C4×Dic3—61no—
256C4×Dic3—181no—
257C4×Dic3—181no—
258C4×Dic3—181no—
259C2×C3⋊C8D3211yescharacteristic
260C2×C3⋊C8D3211yescharacteristic
261#204—34no—
262C22×C32—21no—
263C22×C32—21no—
264C22×C32—21no—
265C22×C32—41no—
266C22×C32—62no—
267C22×C32—121no—
268C22×C32—121no—
269C22×C32C2×D1211yescharacteristic
270C22×C32D3×D411yescharacteristic
271C22×C32D3×D411yescharacteristic
272C2×C3×D3—24no—
273C2×C3×D3—22no—
274C2×C3×D3—22no—
275C2×C3×D3—42no—
276C2×C3×D3—42no—
277C2×C3×D3—41no—
278C2×C3×D3—41no—
279C2×C3×D3—41no—
280C2×C3×D3—41no—
281C32×C4—21no—
282C32×C4—21no—
283C32×C4—21no—
284C32×C4—21no—
285C32×C4—21no—
286C32×C4—21no—
287C32×C4—41no—
288C32×C4—41no—
289C32×C4C2×C4×D311yescharacteristic
290C32×C4C2×C4×D311yescharacteristic
291C32×C4#23511yescharacteristic
292C32×C4#23711yescharacteristic
293C32×C4#23711yescharacteristic
294C32×C4C22×Dic311yescharacteristic
295C3⋊Dic3—21no—
296C3⋊Dic3—21no—
297C3⋊Dic3—22no—
298C3⋊Dic3—41no—
299C3⋊Dic3—121no—
300C3⋊Dic3—122no—
301C3⋊Dic3—121no—
302C3⋊Dic3—121no—
303C3⋊Dic3—241no—
304C3⋊Dic3—241no—
305C3⋊Dic3—242no—
306C3×Dic3—62no—
307C3×Dic3—62no—
308C3×Dic3—62no—
309C3×Dic3—61no—
310C3×Dic3—61no—
311C3×Dic3—61no—
312C3×Dic3—61no—
313C3×Dic3—122no—
314C3×Dic3—121no—
315C3×Dic3—121no—
316C3×Dic3—121no—
317C3×Dic3—241no—
318C2×SD16—91no—
319C4×Q8—271no—
320C4×D4—271no—
321#107—271no—
322#103—271no—
323#102—91no—
324#97—91no—
325C33#44011yescharacteristic, complemented, Sylow
326C23×C3—62no—
327C23×C3—61no—
328C23×C3—61no—
329C22×D3—21no—
330C3×Q8—92no—
331C3×Q8—181no—
332C3×D4—32no—
333C3×D4—32no—
334C3×D4—62no—
335C3×D4—62no—
336C3×D4—61no—
337C3×D4—61no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—41no—
342C2×C3×C4—91no—
343C2×C3×C4—91no—
344C2×C3×C4—181no—
345C2×C3×C4—181no—
346C2×C3×C4C2×D3211yescharacteristic
347C2×C3×C4C2×D3211yescharacteristic
348C2×C3×C4C2×D3211yescharacteristic
349C2×Dic3—31no—
350C2×Dic3—31no—
351C2×Dic3—31no—
352C2×Dic3—31no—
353C2×Dic3—61no—
354C2×Dic3—61no—
355C2×Dic3—61no—
356C2×Dic3—61no—
357C2×Dic3—61no—
358C2×Dic3—61no—
359C2×Dic3—121no—
360C2×Dic3—121no—
361C2×Dic3—182no—
362C2×Dic3—181no—
363C2×Dic3—181no—
364C2×Dic3—181no—
365C2×Dic3—361no—
366C2×Dic3—361no—
367C2×Dic3—362no—
368C2×Dic3—364no—
369C2×Dic3—362no—
370C2×Dic3—362no—
371D12—21no—
372D12D3×Dic312yes—
373C4×D3—362no—
374Dic6—32no—
375Dic6—32no—
376Dic6—62no—
377Dic6—61no—
378Dic6—61no—
379Dic6—121no—
380C3×C8—32no—
381C3×C8—32no—
382C3×C8—62no—
383C3×C8—61no—
384C3×C8—61no—
385C3×C8—121no—
386C3⋊C8—21no—
387C3⋊C8D3×Dic312yescomplemented
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×C32—122no—
401C2×C32C4×D1211yescharacteristic
402C2×C32#67⋊C411yescharacteristic
403C2×C32#90811yescharacteristic
404C2×C32#90911yescharacteristic
405C2×C32#91111yescharacteristic
406C2×C32#91411yescharacteristic
407C2×C32#91511yescharacteristic
408C2×C32#91511yescharacteristic
409C2×C32D4×Dic311yescharacteristic
410C3×D3—44no—
411C3×D3—42no—
412C3×D3—42no—
413C2×Q8—91no—
414C2×D4—31no—
415C22×C4—541no—
416SD16—94no—
417C2×C8—31no—
418C2×C8—31no—
419C4⋊C4—91no—
420C4⋊C4—271no—
421C4⋊C4—541no—
422C22⋊C4—541no—
423C42—91no—
424C42—541no—
425C22×C3—21no—
426C22×C3—21no—
427C22×C3—21no—
428C22×C3—41no—
429C22×C3—64no—
430C22×C3—62no—
431C22×C3—62no—
432C22×C3—122no—
433C22×C3—122no—
434C22×C3—121no—
435C22×C3—121no—
436C22×C3—121no—
437C22×C3—121no—
438C22×C3D3×D1211yescharacteristic
439C22×C3D3×D1211yescharacteristic
440C22×C3#381111yescharacteristic
441C2×D3—22no—
442C2×D3—41no—
443C2×D3—41no—
444C3×C4—21no—
445C3×C4—21no—
446C3×C4—21no—
447C3×C4—21no—
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—41no—
451C3×C4—41no—
452C3×C4—182no—
453C3×C4—181no—
454C3×C4—181no—
455C3×C4—361no—
456C3×C4#379511yescharacteristic
457C3×C4C4×D3211yescharacteristic
458C3×C4C2×D3×Dic311yescharacteristic
459C3×C4C2×D3×Dic311yescharacteristic
460C3×C4#380411yescharacteristic
461C3×C4#380511yescharacteristic
462Dic3—62no—
463Dic3—61no—
464Dic3—61no—
465Dic3—61no—
466Dic3—61no—
467Dic3—62no—
468Dic3—62no—
469Dic3—122no—
470Dic3—121no—
471Dic3—121no—
472Dic3—121no—
473Dic3—241no—
474Dic3—362no—
475Dic3—361no—
476Dic3—361no—
477Dic3—361no—
478Dic3—721no—
479Dic3—721no—
480Dic3—722no—
481C32—21no—
482C32—21no—
483C32—21no—
484C32—41no—
485C32#471011yescharacteristic, complemented
486C32#475711yescharacteristic, complemented
487C32#516411yescharacteristic, complemented
488C23—61no—
489Q8—92no—
490Q8—181no—
491D4—32no—
492D4—61no—
493C2×C4—91no—
494C2×C4—91no—
495C2×C4—181no—
496C2×C4—181no—
497C2×C4—541no—
498C2×C4—1082no—
499C2×C4D3311yescharacteristic
500C8—32no—
501C8—61no—
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—124no—
515C2×C3—122no—
516C2×C3—122no—
517C2×C3#6379411yescharacteristic
518C2×C3#6379511yescharacteristic
519C2×C3#6380011yescharacteristic
520C2×C3#6380011yescharacteristic
521C2×C3#6388011yescharacteristic
522C2×C3Dic3×D1211yescharacteristic
523C2×C3#6393111yescharacteristic
524C2×C3#6393711yescharacteristic
525C2×C3#67⋊Dic311yescharacteristic
526D3—42no—
527C22—62no—
528C22—121no—
529C22—121no—
530C22#8895511yescharacteristic
531C4—181no—
532C4—181no—
533C4—182no—
534C4—361no—
535C4—1081no—
536C4D32×Dic311yescharacteristic
537C4#8896111yescharacteristic
538C3—21no—
539C3—21no—
540C3—21no—
541C3—41no—
542C3#1059074711yescharacteristic, complemented
543C3#1059076611yescharacteristic, complemented
544C3#1059144611yescharacteristic, complemented
545C2—122no—
546C2#1172287911yescharacteristic
547C2#1172289511yescharacteristic
548C2#1172292811yescharacteristic
subgroupG/H
centerC22#88955
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. #11723668C2semidirect
4. #11722055C2semidirect
5. #11722040C2semidirect
7. C3×#63555C2semidirect
22. C3×#3716C4semidirect
46. #63555D3semidirect
55. C3×#634C2×C4semidirect
60. C32×C3⋊C8C2×C4semidirect
111. #3716Dic3semidirect
173. #634C2×Dic3semidirect
207. C3×C3⋊C8C4×D3semidirect
208. C3×C3⋊C8C2×Dic3semidirect
325. C33#440semidirect
387. C3⋊C8D3×Dic3semidirect
485. C32#4710semidirect
486. C32#4757semidirect
487. C32#5164semidirect
542. C3#10590747semidirect
543. C3#10590766semidirect
544. C3#10591446semidirect

Maximal subgroups

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

subgroupindex
#117236682
D12×C3⋊Dic32
C2×C3×#37162
#117220552
#117220402
C3×#635632
C3×#635552
#105914463
#105907663
#105907473