Finite Group Databasebeta

← #13169321

Subgroups — #13169321

Order 1728.

1008 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723679C211yesmaximal, characteristic
2#11723400C212yesmaximal
3Dic6×C3⋊Dic3C211yesmaximal, characteristic
4C3×#63839C212yesmaximal
5C3×#63838C211yesmaximal, characteristic
6#11723032C211yesmaximal, characteristic
7#11723018C211yesmaximal, characteristic
8#11723011C212yesmaximal
9#11722941C211yesmaximal, characteristic
10Dic3×#627C212yesmaximal
11Dic3×#627C211yesmaximal, characteristic
12#10595338—31nomaximal
13#10594317—32nomaximal
14C2×C32×Dic6C2211yescharacteristic
15C2×C3×#627C2212yes—
16C2×C3×#627C2211yescharacteristic
17C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
18C3×#3750C2212yes—
19C3×C4×C3⋊Dic3C2211yescharacteristic
20C32×Dic3⋊C4C2212yes—
21#88815C2212yes—
22#88815C2211yescharacteristic
23#88815C2211yescharacteristic
24#88806C2212yes—
25Dic3⋊(C3⋊Dic3)C2211yescharacteristic
26Dic3⋊(C3⋊Dic3)C2212yes—
27Dic3⋊(C3⋊Dic3)C2212yes—
28Dic3⋊(C3⋊Dic3)C2211yescharacteristic
29Dic3×C3⋊Dic3C2211yescharacteristic
30Dic3×C3⋊Dic3C2211yescharacteristic
31Dic3×C3⋊Dic3C2212yes—
32C3×#3724C2212yes—
33C3×#3724C2211yescharacteristic
34C3×Dic3⋊Dic3C2212yes—
35C3×Dic32C2212yes—
36C3×Dic32C2212yes—
37Q8×C3⋊Dic3—31no—
38#64099—32no—
39C3×#944—32no—
40C3×#888—32no—
41C3×#887—32no—
42C4×#627—31no—
43C4×#627—32no—
44#63873—31no—
45#63870—31no—
46#63870—31no—
47#63869—32no—
48#63867—32no—
49#63867—32no—
50#63845—32no—
51#63845—32no—
52#63845—32no—
53#63845—32no—
54#63845—62no—
55#63843—32no—
56Dic3×Dic6—32no—
57Dic3×Dic6—32no—
58Dic3×Dic6—32no—
59Dic3×Dic6—32no—
60Dic3×Dic6—61no—
61#63839—32no—
62#63839D312yes—
63#63838—31no—
64#63838D311yescharacteristic
65C2×C33×C4C2311yescharacteristic
66C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
67C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
68C2×C3×C3⋊Dic3C2311yescharacteristic
69C2×C3×C3⋊Dic3C2312yes—
70C2×C3×C3⋊Dic3C2312yes—
71C2×C3×C3⋊Dic3C2311yescharacteristic
72C2×C32×Dic3C2312yes—
73C2×C32×Dic3C2311yescharacteristic
74C2×C32×Dic3C2312yes—
75C2×C32×Dic3C2311yescharacteristic
76C32×Dic6—22no—
77C3×#627—24no—
78C3×#627Q814yescomplemented
79#5721—92no—
80#5569—91no—
81C2×C32×Q8—31no—
82C2×C3×Dic6—21no—
83C2×C3×Dic6—32no—
84C2×C3×Dic6—32no—
85C2×C3×Dic6—32no—
86C2×C3×Dic6C2×D312yes—
87C2×#627C2×D312yes—
88C2×#627C2×D311yescharacteristic
89C32×C4⋊C4—32no—
90C4⋊(C3⋊Dic3)—32no—
91C4⋊(C3⋊Dic3)—31no—
92C4⋊(C3⋊Dic3)—31no—
93C4⋊(C3⋊Dic3)—32no—
94C4⋊(C3⋊Dic3)—32no—
95C4⋊(C3⋊Dic3)—31no—
96C4⋊(C3⋊Dic3)—61no—
97C4⋊(C3⋊Dic3)—62no—
98C4⋊(C3⋊Dic3)—121no—
99#3750—32no—
100#3750C2×D312yes—
101C4×C3⋊Dic3—31no—
102C4×C3⋊Dic3—32no—
103C4×C3⋊Dic3—31no—
104C4×C3⋊Dic3C2×D311yescharacteristic
105C3×C4⋊Dic3—32no—
106C3×Dic3⋊C4—22no—
107C3×Dic3⋊C4—32no—
108C3×Dic3⋊C4—32no—
109C3×Dic3⋊C4—32no—
110C3×Dic3⋊C4—32no—
111C3×Dic3⋊C4—32no—
112C3×Dic3⋊C4—32no—
113C3×Dic3⋊C4—62no—
114C3×Dic3⋊C4C2×D312yes—
115C3×Dic3⋊C4C2×D312yes—
116C3×C4×Dic3—32no—
117C3×C4×Dic3—32no—
118C3×C4×Dic3—32no—
119C3×C4×Dic3—32no—
120C3×C4×Dic3—32no—
121C3×C4×Dic3—61no—
122#3724—31no—
123#3724—32no—
124#3724—31no—
125#3724—32no—
126#3724—32no—
127#3724—62no—
128#3724C2×D311yescharacteristic
129#3724C2×D312yes—
130Dic3⋊Dic3—32no—
131Dic3⋊Dic3—32no—
132Dic3⋊Dic3—32no—
133Dic3⋊Dic3—32no—
134Dic3⋊Dic3—32no—
135Dic3⋊Dic3—32no—
136Dic3⋊Dic3—32no—
137Dic3⋊Dic3—32no—
138Dic3⋊Dic3—32no—
139Dic3⋊Dic3—32no—
140Dic3⋊Dic3—61no—
141Dic3⋊Dic3—61no—
142Dic3⋊Dic3—62no—
143Dic3⋊Dic3—62no—
144Dic3⋊Dic3C2×D312yes—
145Dic32—32no—
146Dic32—32no—
147Dic32—32no—
148Dic32—32no—
149Dic32—61no—
150Dic32—61no—
151Dic32—62no—
152Dic32C2×D312yes—
153Dic32C2×D312yes—
154C22×C33C2411yescharacteristic
155C33×C4—21no—
156C3⋊(C3⋊Dic3)—21no—
157C3⋊(C3⋊Dic3)—21no—
158C3×C3⋊Dic3—22no—
159C3×C3⋊Dic3—21no—
160C3×C3⋊Dic3—22no—
161C3×C3⋊Dic3C2×Q812yes—
162C32×Dic3—21no—
163C32×Dic3—22no—
164C32×Dic3C2×Q814yes—
165C32×Dic3#4112yes—
166C3×C4⋊Q8—92no—
167C3×C4×Q8—91no—
168Q8×Dic3—92no—
169Q8×Dic3—91no—
170Q8×Dic3—92no—
171Q8×Dic3—91no—
172Q8×Dic3—181no—
173#944—32no—
174#944—92no—
175#890—92no—
176#890—91no—
177#890—92no—
178#890—92no—
179#890—182no—
180#889—92no—
181#889—92no—
182#889—92no—
183#889—92no—
184#888—32no—
185#888—91no—
186#887—32no—
187#887—92no—
188#870—91no—
189#870—92no—
190C4×Dic6—92no—
191C4×Dic6—92no—
192C4×Dic6—92no—
193C4×Dic6—91no—
194C32×Q8—62no—
195C2×C32×C4—21no—
196C2×C32×C4—22no—
197C2×C32×C4—32no—
198C2×C32×C4—31no—
199C2×C32×C4—32no—
200C2×C32×C4—31no—
201C2×C32×C4—41no—
202C2×C32×C4C22×D312yes—
203C2×C32×C4C22×D311yescharacteristic
204C2×C3⋊Dic3—31no—
205C2×C3⋊Dic3—32no—
206C2×C3⋊Dic3—32no—
207C2×C3⋊Dic3—31no—
208C2×C3⋊Dic3—61no—
209C2×C3⋊Dic3—62no—
210C2×C3⋊Dic3—61no—
211C2×C3⋊Dic3—62no—
212C2×C3⋊Dic3—121no—
213C2×C3⋊Dic3—121no—
214C2×C3⋊Dic3C22×D311yescharacteristic
215C2×C3⋊Dic3C22×D312yes—
216C2×C3⋊Dic3C22×D312yes—
217C2×C3⋊Dic3C22×D311yescharacteristic
218C2×C3×Dic3—22no—
219C2×C3×Dic3—21no—
220C2×C3×Dic3—21no—
221C2×C3×Dic3—22no—
222C2×C3×Dic3—32no—
223C2×C3×Dic3—32no—
224C2×C3×Dic3—32no—
225C2×C3×Dic3—32no—
226C2×C3×Dic3—32no—
227C2×C3×Dic3—32no—
228C2×C3×Dic3—61no—
229C2×C3×Dic3—62no—
230C2×C3×Dic3—61no—
231C2×C3×Dic3—62no—
232C2×C3×Dic3C22×D312yes—
233C2×C3×Dic3C22×D312yes—
234C2×C3×Dic3C22×D312yes—
235C2×C3×Dic3C22×D312yes—
236C2×C3×Dic3C22×D312yes—
237C2×C3×Dic3C22×D312yes—
238C3×Dic6—24no—
239C3×Dic6—24no—
240C3×Dic6—38no—
241C3×Dic6—64no—
242C3×Dic6—64no—
243#627—24no—
244#627Dic614yescomplemented
245#557—271noSylow
246C2×C33C22×Q811yescharacteristic
247C2×C33C2×#4111yescharacteristic
248C2×C332−1+411yescharacteristic
249C2×C3×Q8—32no—
250C2×C3×Q8—61no—
251C2×C3×Q8—91no—
252C2×C3×Q8—92no—
253C2×Dic6—32no—
254C2×Dic6—32no—
255C2×Dic6—32no—
256C2×Dic6D3211yescharacteristic
257C3×C4⋊C4—32no—
258C3×C4⋊C4—32no—
259C3×C4⋊C4—62no—
260C3×C4⋊C4—92no—
261C3×C4⋊C4—91no—
262C3×C4⋊C4—92no—
263C3×C4⋊C4—92no—
264C3×C42—92no—
265C3×C42—92no—
266C3×C42—91no—
267C4⋊Dic3—32no—
268C4⋊Dic3—91no—
269C4⋊Dic3—91no—
270C4⋊Dic3—92no—
271C4⋊Dic3—92no—
272C4⋊Dic3—92no—
273C4⋊Dic3—92no—
274C4⋊Dic3—92no—
275C4⋊Dic3—92no—
276C4⋊Dic3—92no—
277C4⋊Dic3—91no—
278C4⋊Dic3—92no—
279C4⋊Dic3—182no—
280C4⋊Dic3—182no—
281C4⋊Dic3—181no—
282C4⋊Dic3—181no—
283C4⋊Dic3—182no—
284C4⋊Dic3—181no—
285C4⋊Dic3—361no—
286Dic3⋊C4—32no—
287Dic3⋊C4—32no—
288Dic3⋊C4—32no—
289Dic3⋊C4—32no—
290Dic3⋊C4—32no—
291Dic3⋊C4—32no—
292Dic3⋊C4—62no—
293Dic3⋊C4—92no—
294Dic3⋊C4—92no—
295Dic3⋊C4—92no—
296Dic3⋊C4—92no—
297Dic3⋊C4—92no—
298Dic3⋊C4—92no—
299Dic3⋊C4—91no—
300Dic3⋊C4—91no—
301Dic3⋊C4—92no—
302Dic3⋊C4—92no—
303Dic3⋊C4—92no—
304Dic3⋊C4—182no—
305Dic3⋊C4D3212yes—
306C4×Dic3—32no—
307C4×Dic3—32no—
308C4×Dic3—32no—
309C4×Dic3—32no—
310C4×Dic3—32no—
311C4×Dic3—61no—
312C4×Dic3—92no—
313C4×Dic3—92no—
314C4×Dic3—92no—
315C4×Dic3—92no—
316C4×Dic3—91no—
317C4×Dic3—91no—
318C4×Dic3—92no—
319C4×Dic3—182no—
320C4×Dic3—181no—
321C4×Dic3—181no—
322C22×C32—22no—
323C22×C32—21no—
324C22×C32—41no—
325C22×C32C23×D311yescharacteristic
326C22×C32C23×D312yes—
327C32×C4—21no—
328C32×C4—22no—
329C32×C4—22no—
330C32×C4—34no—
331C32×C4—32no—
332C32×C4—42no—
333C32×C4—42no—
334C32×C4—61no—
335C32×C4—62no—
336C3⋊Dic3—22no—
337C3⋊Dic3—22no—
338C3⋊Dic3—21no—
339C3⋊Dic3—62no—
340C3⋊Dic3—62no—
341C3⋊Dic3—61no—
342C3⋊Dic3—61no—
343C3⋊Dic3—121no—
344C3⋊Dic3—121no—
345C3⋊Dic3—122no—
346C3⋊Dic3—122no—
347C3⋊Dic3—241no—
348C3⋊Dic3—241no—
349C3⋊Dic3C2×Dic612yes—
350C3×Dic3—22no—
351C3×Dic3—22no—
352C3×Dic3—22no—
353C3×Dic3—22no—
354C3×Dic3—24no—
355C3×Dic3—22no—
356C3×Dic3—34no—
357C3×Dic3—42no—
358C3×Dic3—62no—
359C3×Dic3—62no—
360C3×Dic3—62no—
361C3×Dic3—62no—
362C3×Dic3—62no—
363C3×Dic3—62no—
364C3×Dic3—122no—
365C3×Dic3—121no—
366C3×Dic3—122no—
367C3×Dic3C2×Dic614yes—
368C3×Dic3#23714yes—
369C3×Dic3D3×Q814yes—
370C4⋊Q8—92no—
371C4⋊Q8—271no—
372#125—271no—
373#125—272no—
374#125—272no—
375#125—271no—
376C4×Q8—91no—
377C4×Q8—272no—
378C4×Q8—271no—
379C4×Q8—271no—
380C4×Q8—271no—
381C33#55711yescharacteristic, complemented, Sylow
382C3×Q8—64no—
383C3×Q8—64no—
384C3×Q8—94no—
385C3×Q8—184no—
386C2×C3×C4—21no—
387C2×C3×C4—22no—
388C2×C3×C4—32no—
389C2×C3×C4—32no—
390C2×C3×C4—32no—
391C2×C3×C4—32no—
392C2×C3×C4—32no—
393C2×C3×C4—32no—
394C2×C3×C4—41no—
395C2×C3×C4—62no—
396C2×C3×C4—61no—
397C2×C3×C4—61no—
398C2×C3×C4—62no—
399C2×C3×C4—91no—
400C2×C3×C4—92no—
401C2×C3×C4—91no—
402C2×C3×C4—92no—
403C2×C3×C4C2×D3212yes—
404C2×C3×C4C2×D3211yescharacteristic
405C2×Dic3—32no—
406C2×Dic3—32no—
407C2×Dic3—32no—
408C2×Dic3—32no—
409C2×Dic3—32no—
410C2×Dic3—32no—
411C2×Dic3—61no—
412C2×Dic3—61no—
413C2×Dic3—62no—
414C2×Dic3—62no—
415C2×Dic3—92no—
416C2×Dic3—91no—
417C2×Dic3—92no—
418C2×Dic3—91no—
419C2×Dic3—182no—
420C2×Dic3—181no—
421C2×Dic3—181no—
422C2×Dic3—182no—
423C2×Dic3—361no—
424C2×Dic3—361no—
425C2×Dic3C2×D3212yes—
426C2×Dic3C2×D3211yescharacteristic
427C2×Dic3C2×D3212yes—
428C2×Dic3C2×D3211yescharacteristic
429Dic6—22no—
430Dic6—38no—
431Dic6—64no—
432Dic6—64no—
433C2×C32—22no—
434C2×C32—22no—
435C2×C32—21no—
436C2×C32—22no—
437C2×C32—21no—
438C2×C32—21no—
439C2×C32—41no—
440C2×C32—41no—
441C2×C32—41no—
442C2×C32C22×Dic611yescharacteristic
443C2×C32C2×#23712yes—
444C2×C32C2×D3×Q812yes—
445C2×C32#100712yes—
446C2×C32#100711yescharacteristic
447C2×C32D3×#4111yescharacteristic
448C2×Q8—31no—
449C2×Q8—91no—
450C2×Q8—92no—
451C4⋊C4—32no—
452C4⋊C4—91no—
453C4⋊C4—92no—
454C4⋊C4—92no—
455C4⋊C4—92no—
456C4⋊C4—271no—
457C4⋊C4—272no—
458C4⋊C4—272no—
459C4⋊C4—271no—
460C4⋊C4—272no—
461C4⋊C4—271no—
462C4⋊C4—272no—
463C4⋊C4—271no—
464C4⋊C4—271no—
465C42—92no—
466C42—92no—
467C42—91no—
468C42—272no—
469C42—271no—
470C42—271no—
471C22×C3—21no—
472C22×C3—22no—
473C22×C3—41no—
474C22×C3C22×D3212yes—
475C22×C3C22×D3211yescharacteristic
476C3×C4—22no—
477C3×C4—22no—
478C3×C4—21no—
479C3×C4—34no—
480C3×C4—34no—
481C3×C4—34no—
482C3×C4—42no—
483C3×C4—42no—
484C3×C4—62no—
485C3×C4—64no—
486C3×C4—62no—
487C3×C4—62no—
488C3×C4—62no—
489C3×C4—62no—
490C3×C4—92no—
491C3×C4—122no—
492C3×C4—182no—
493C3×C4—181no—
494C3×C4—182no—
495Dic3—22no—
496Dic3—21no—
497Dic3—34no—
498Dic3—62no—
499Dic3—62no—
500Dic3—62no—
501Dic3—62no—
502Dic3—62no—
503Dic3—62no—
504Dic3—121no—
505Dic3—122no—
506Dic3—122no—
507Dic3—181no—
508Dic3—181no—
509Dic3—182no—
510Dic3—182no—
511Dic3—362no—
512Dic3—361no—
513Dic3—361no—
514Dic3—362no—
515Dic3—721no—
516Dic3—721no—
517Dic3D3×Dic614yes—
518Dic3#380512yes—
519C32—21no—
520C32—22no—
521C32—41no—
522C32#556911yescharacteristic, complemented
523C32#572112yescomplemented
524Q8—62no—
525Q8—94no—
526Q8—184no—
527C2×C4—32no—
528C2×C4—31no—
529C2×C4—32no—
530C2×C4—31no—
531C2×C4—91no—
532C2×C4—91no—
533C2×C4—92no—
534C2×C4—92no—
535C2×C4—271no—
536C2×C4—271no—
537C2×C4D3311yescharacteristic
538C2×C3—21no—
539C2×C3—22no—
540C2×C3—22no—
541C2×C3—21no—
542C2×C3—22no—
543C2×C3—21no—
544C2×C3—41no—
545C2×C3—41no—
546C2×C3—41no—
547C2×C3C2×D3×Dic612yes—
548C2×C3D3×#23712yes—
549C2×C3#6431611yescharacteristic
550C2×C3#6431612yes—
551C2×C3D32×Q811yescharacteristic
552C2×C3C2×#380511yescharacteristic
553C22C2×D3311yescharacteristic
554C4—21no—
555C4—34no—
556C4—32no—
557C4—61no—
558C4—62no—
559C4—92no—
560C4—181no—
561C4—182no—
562C4—182no—
563C4—541no—
564C4—541no—
565C3—22no—
566C3—21no—
567C3—41no—
568C3#1059431712yescomplemented
569C3#1059533811yescharacteristic, complemented
570C2D32×Dic611yescharacteristic
571C2#1172439911yescharacteristic
572C2D3×#380511yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, prime radical, cosocleC22×C33C24
FittingC2×C33×C4C23
hypercenterC2×C4D33
hypocenterC33#557
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
78. C3×#627Q8semidirect
244. #627Dic6semidirect
381. C33#557semidirect
522. C32#5569semidirect
523. C32#5721semidirect
568. C3#10594317semidirect
569. C3#10595338semidirect

Maximal subgroups

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

subgroupindex
#117236792
#117234002
Dic6×C3⋊Dic32
C3×#638392
C3×#638382
#117230322
#117230182
#117230112
#117229412
Dic3×#6272
#105953383
#105943173