Finite Group Databasebeta

← #13178691

Subgroups — #13178691

Order 1728.

1454 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3830C211yesmaximal, characteristic
2C22×#6296C211yesmaximal, characteristic, complemented
3C3×#64144C212yesmaximal, complemented
4#11723721C212yesmaximal, complemented
5#11723720C212yesmaximal, complemented
6#11723718C212yesmaximal, complemented
7#11723715C211yesmaximal, characteristic, complemented
8#11723671C212yesmaximal, complemented
9#11723666C212yesmaximal, complemented
10#10595271—31nomaximal
11#10595270—32nomaximal
12C2×C33×D4C2211yescharacteristic
13C2×C3×#638C2211yescharacteristic
14C2×C3×#638C2211yescharacteristic
15C22×C3×C3⋊Dic3C2212yes—
16C22×C3×C3⋊Dic3C2211yescharacteristic
17C22×C3×C3⋊Dic3C2211yescharacteristic
18C3×#3830—24no—
19C2×C3×C4×C3⋊D3C2211yescharacteristic
20C2×C3×#634C2211yescharacteristic
21C2×#6296—24no—
22C2×#6296C2212yes—
23C2×#6296C2212yes—
24C3×#3757C2212yes—
25C3×#3757C2214yes—
26C3×#3750C2214yes—
27#88815C2214yes—
28#88815C2212yes—
29#88814C2214yes—
30#88813C2212yes—
31C2×#3830D311yescharacteristic
32C2×C3×#237—32no—
33C2×C3×#237—61no—
34C22×#627—32no—
35C22×#627—31no—
36#64144D312yescomplemented
37C3×#935—32no—
38C3×#935—32no—
39C3×#935—62no—
40#63982—32no—
41#63966—32no—
42#63964—34no—
43#63962—32no—
44#63961—32no—
45#63959—34no—
46#63957—31no—
47#63955—32no—
48#63953—32no—
49#63865—32no—
50#63851—34no—
51#63840—32no—
52#63839—34no—
53C23×C33C2311yescharacteristic
54C23×C33C2311yescharacteristic
55C22×C3×C3⋊D3C2311yescharacteristic
56C33×D4—22no—
57C2×C33×C4C2311yescharacteristic
58C3×#638—22no—
59C3×#638—22no—
60C2×C3×C3⋊Dic3—24no—
61C2×C3×C3⋊Dic3—22no—
62C2×C3×C3⋊Dic3D414yes—
63C2×C3×C3⋊Dic3C2314yes—
64C2×C3×C3⋊Dic3C2314yes—
65C2×C3×C3⋊Dic3C2311yescharacteristic
66C2×C3×C3⋊Dic3C2311yescharacteristic
67C2×C3×C3⋊Dic3C2311yescharacteristic
68C3×C4×C3⋊D3—22no—
69C3×#634—22no—
70#6296—24no—
71#6296—24no—
72#5825—91no—
73#5648—92no—
74C2×C32×D4—21no—
75C2×C32×D4—22no—
76C2×C32×D4—41no—
77C2×C32×D4C2×D311yescharacteristic
78C2×C32×D4C2×D312yes—
79C2×#638C2×D311yescharacteristic
80C2×#638C2×D311yescharacteristic
81C22×C3⋊Dic3C2×D312yes—
82C22×C3⋊Dic3C2×D311yescharacteristic
83C22×C3⋊Dic3C2×D311yescharacteristic
84#3830—24no—
85C2×C4×C3⋊D3C2×D311yescharacteristic
86C2×#634C2×D311yescharacteristic
87C2×C3×#67—32no—
88C2×C3×#67—32no—
89C2×C3×#67—61no—
90C2×C3×#67—61no—
91C22×C3×Dic3—32no—
92C22×C3×Dic3—32no—
93C22×C3×Dic3—32no—
94C22×C3×Dic3—32no—
95C22×C3×Dic3—61no—
96C22×C3×Dic3—61no—
97C22×C3×Dic3—62no—
98C3×#237—68no—
99C3×#237—68no—
100C2×C3×C4×D3—32no—
101C2×C3×C4×D3—61no—
102C2×C3×Dic6—32no—
103C2×C3×Dic6—61no—
104C2×#627—34no—
105C2×#627—32no—
106C2×#627—34no—
107C2×#627—32no—
108C2×#627—68no—
109C2×#627—64no—
110#3757C2×D312yes—
111#3757C2×D314yes—
112#3750C2×D314yes—
113C3×#217—34no—
114C3×#217—32no—
115C3×#217—32no—
116C3×#217—34no—
117C3×#217—62no—
118C3×#217—64no—
119C3×Dic3⋊C4—34no—
120C3×Dic3⋊C4—34no—
121C3×Dic3⋊C4—64no—
122#3724—34no—
123#3724—32no—
124Dic3⋊Dic3—34no—
125Dic3⋊Dic3—34no—
126Dic3⋊Dic3—34no—
127#3722—34no—
128(C2×D3)⋊Dic3—34no—
129(C2×D3)⋊Dic3—34no—
130Dic32—34no—
131Dic32—32no—
132C22×C33—22no—
133C22×C33—21no—
134C22×C33—22no—
135C22×C33C2×D412yes—
136C22×C33C2411yescharacteristic
137C2×C3×C3⋊D3—21no—
138C2×C3×C3⋊D3—22no—
139C33×C4—21no—
140C3×C3⋊Dic3—21no—
141C3×C3⋊Dic3—24no—
142C3×C3⋊Dic3—24no—
143C3×C3⋊Dic3C2×D412yes—
144C3×C3⋊Dic3C2×D412yes—
145C2×C3×#41—91no—
146C2×#237—32no—
147C2×#237—61no—
148C22×Dic6—92no—
149C22×Dic6—91no—
150C3×#123—92no—
151#944—92no—
152#936—92no—
153#935—32no—
154#935—32no—
155#935—62no—
156#935—92no—
157#929—92no—
158#926—91no—
159#924—92no—
160#924—92no—
161#896—94no—
162#888—94no—
163#886—92no—
164#885—94no—
165#878—94no—
166#877—92no—
167C23×C32—22no—
168C23×C32—21no—
169C23×C32—21no—
170C23×C32—22no—
171C23×C32—41no—
172C23×C32—41no—
173C23×C32C22×D311yescharacteristic
174C23×C32C22×D312yes—
175C23×C32C22×D312yes—
176C23×C32C22×D311yescharacteristic
177C22×C3⋊D3C22×D311yescharacteristic
178C22×C3×D3—32no—
179C22×C3×D3—61no—
180C32×D4—22no—
181C32×D4—24no—
182C32×D4—24no—
183C32×D4—44no—
184C32×D4—44no—
185C2×C32×C4—21no—
186C2×C32×C4—22no—
187C2×C32×C4—41no—
188C2×C32×C4C22×D311yescharacteristic
189C2×C32×C4C22×D312yes—
190#638—22no—
191#638—22no—
192C2×C3⋊Dic3—22no—
193C2×C3⋊Dic3—24no—
194C2×C3⋊Dic3#6714yes—
195C2×C3⋊Dic3C22×D314yes—
196C2×C3⋊Dic3C22×D314yes—
197C2×C3⋊Dic3C22×D311yescharacteristic
198C2×C3⋊Dic3C22×D311yescharacteristic
199C2×C3⋊Dic3C22×D311yescharacteristic
200C4×C3⋊D3—22no—
201#634—22no—
202C3×#67—64no—
203C3×#67—64no—
204C3×#67—64no—
205C3×#67—64no—
206C2×C3×Dic3—38no—
207C2×C3×Dic3—32no—
208C2×C3×Dic3—32no—
209C2×C3×Dic3—34no—
210C2×C3×Dic3—34no—
211C2×C3×Dic3—34no—
212C2×C3×Dic3—34no—
213C2×C3×Dic3—32no—
214C2×C3×Dic3—64no—
215C2×C3×Dic3—61no—
216C2×C3×Dic3—61no—
217C2×C3×Dic3—61no—
218C2×C3×Dic3—64no—
219C2×C3×Dic3—64no—
220C2×C3×Dic3—64no—
221C2×C3×Dic3—64no—
222C2×C3×Dic3—64no—
223C2×C3×Dic3—64no—
224C2×C3×Dic3—124no—
225C3×C4×D3—64no—
226C3×C4×D3—64no—
227C3×Dic6—64no—
228C3×Dic6—64no—
229#627—68no—
230#627—68no—
231#627—64no—
232#627—64no—
233#536—271noSylow
234C2×C33—22no—
235C2×C33—41no—
236C2×C33C22×D411yescharacteristic
237C2×C332−1+412yes—
238C3×C3⋊D3—41no—
239C3×#41—184no—
240C2×C3×Q8—91no—
241C2×C3×D4—21no—
242C2×C3×D4—22no—
243C2×C3×D4—41no—
244C2×C3×D4—91no—
245C2×C3×D4—91no—
246C2×C3×D4D3211yescharacteristic
247C2×C3×D4D3212yes—
248C22×C3×C4—91no—
249C22×C3×C4—91no—
250C22×C3×C4—92no—
251C22×C3×C4—91no—
252C2×#67—32no—
253C2×#67—32no—
254C2×#67—61no—
255C2×#67—61no—
256C22×Dic3—32no—
257C22×Dic3—32no—
258C22×Dic3—32no—
259C22×Dic3—32no—
260C22×Dic3—61no—
261C22×Dic3—61no—
262C22×Dic3—62no—
263#237—68no—
264#237—68no—
265C2×C4×D3—32no—
266C2×C4×D3—61no—
267C2×Dic6—32no—
268C2×Dic6—61no—
269C2×Dic6—94no—
270C2×Dic6—94no—
271C2×Dic6—92no—
272C2×Dic6—92no—
273C2×Dic6—188no—
274C2×Dic6—184no—
275C3×C4⋊C4—94no—
276C3×C22⋊C4—92no—
277C3×C22⋊C4—94no—
278#217—32no—
279#217—34no—
280#217—34no—
281#217—32no—
282#217—62no—
283#217—64no—
284#217—94no—
285(C2×D3)⋊C4—94no—
286(C2×D3)⋊C4—94no—
287C4⋊Dic3—94no—
288C4⋊Dic3—92no—
289Dic3⋊C4—34no—
290Dic3⋊C4—34no—
291Dic3⋊C4—64no—
292Dic3⋊C4—94no—
293Dic3⋊C4—94no—
294Dic3⋊C4—94no—
295C4×Dic3—94no—
296C4×Dic3—92no—
297C22×C32—21no—
298C22×C32—24no—
299C22×C32—22no—
300C22×C32—22no—
301C22×C32—22no—
302C22×C32—22no—
303C22×C32—24no—
304C22×C32—22no—
305C22×C32—21no—
306C22×C32—24no—
307C22×C32—22no—
308C22×C32—41no—
309C22×C32—42no—
310C22×C32—42no—
311C22×C32—42no—
312C22×C32—44no—
313C22×C32—42no—
314C22×C32—42no—
315C22×C32—44no—
316C22×C32—82no—
317C22×C32—82no—
318C22×C32D3×D414yes—
319C22×C32C2×#6712yes—
320C22×C32C23×D312yes—
321C22×C32C23×D311yescharacteristic
322C2×C3⋊D3—21no—
323C2×C3⋊D3—22no—
324C2×C3×D3—62no—
325C2×C3×D3—62no—
326C2×C3×D3—64no—
327C2×C3×D3—122no—
328C32×C4—21no—
329C32×C4—22no—
330C32×C4—22no—
331C32×C4—42no—
332C32×C4—42no—
333C3⋊Dic3—21no—
334C3⋊Dic3—24no—
335C3⋊Dic3—24no—
336C3⋊Dic3C2×#6712yes—
337C3⋊Dic3C2×#6712yes—
338C3×Dic3—34no—
339C3×Dic3—34no—
340C3×Dic3—62no—
341C3×Dic3—62no—
342C3×Dic3—62no—
343C3×Dic3—64no—
344C3×Dic3—62no—
345C3×Dic3—64no—
346C3×Dic3—64no—
347C3×Dic3—64no—
348C3×Dic3—124no—
349C3×Dic3—124no—
350C2×#41—91no—
351C22×Q8—271no—
352C4⋊Q8—272no—
353#124—272no—
354#123—92no—
355#123—272no—
356C22⋊Q8—272no—
357C22⋊Q8—272no—
358#117—271no—
359C33#53611yescharacteristic, complemented, Sylow
360C23×C3—22no—
361C23×C3—21no—
362C23×C3—21no—
363C23×C3—22no—
364C23×C3—41no—
365C23×C3—41no—
366C23×C3—91no—
367C23×C3C2×D3211yescharacteristic
368C23×C3C2×D3212yes—
369C23×C3C2×D3212yes—
370C23×C3C2×D3211yescharacteristic
371C22×D3—32no—
372C22×D3—61no—
373C3×Q8—182no—
374C3×D4—24no—
375C3×D4—24no—
376C3×D4—22no—
377C3×D4—44no—
378C3×D4—44no—
379C3×D4—182no—
380C3×D4—182no—
381C2×C3×C4—21no—
382C2×C3×C4—22no—
383C2×C3×C4—41no—
384C2×C3×C4—91no—
385C2×C3×C4—91no—
386C2×C3×C4—94no—
387C2×C3×C4—94no—
388C2×C3×C4—91no—
389C2×C3×C4—94no—
390C2×C3×C4—182no—
391C2×C3×C4—182no—
392C2×C3×C4—184no—
393C2×C3×C4C2×D3211yescharacteristic
394C2×C3×C4C2×D3212yes—
395#67—64no—
396#67—64no—
397#67—64no—
398#67—64no—
399C2×Dic3—32no—
400C2×Dic3—32no—
401C2×Dic3—38no—
402C2×Dic3—34no—
403C2×Dic3—34no—
404C2×Dic3—34no—
405C2×Dic3—34no—
406C2×Dic3—32no—
407C2×Dic3—64no—
408C2×Dic3—61no—
409C2×Dic3—61no—
410C2×Dic3—61no—
411C2×Dic3—64no—
412C2×Dic3—64no—
413C2×Dic3—64no—
414C2×Dic3—64no—
415C2×Dic3—64no—
416C2×Dic3—64no—
417C2×Dic3—124no—
418C4×D3—64no—
419C4×D3—64no—
420Dic6—64no—
421Dic6—64no—
422Dic6—188no—
423Dic6—188no—
424Dic6—184no—
425Dic6—184no—
426C2×C32—24no—
427C2×C32—22no—
428C2×C32—24no—
429C2×C32—22no—
430C2×C32—22no—
431C2×C32—21no—
432C2×C32—41no—
433C2×C32—42no—
434C2×C32—44no—
435C2×C32—42no—
436C2×C32—41no—
437C2×C32—42no—
438C2×C32—42no—
439C2×C32—82no—
440C2×C32—82no—
441C2×C32—82no—
442C2×C32C2×D3×D412yes—
443C2×C32#101014yes—
444C2×C32#101012yes—
445C2×C32C22×#6711yescharacteristic
446C3⋊D3—41no—
447C3×D3—122no—
448C3×D3—122no—
449#41—184no—
450C2×Q8—91no—
451C2×Q8—272no—
452C2×Q8—272no—
453C2×Q8—544no—
454C2×D4—91no—
455C2×D4—91no—
456C2×D4#111311yescharacteristic
457C22×C4—91no—
458C22×C4—91no—
459C22×C4—92no—
460C22×C4—91no—
461C4⋊C4—94no—
462C4⋊C4—272no—
463C4⋊C4—274no—
464C22⋊C4—92no—
465C22⋊C4—94no—
466C22⋊C4—274no—
467C42—272no—
468C22×C3—24no—
469C22×C3—22no—
470C22×C3—22no—
471C22×C3—24no—
472C22×C3—22no—
473C22×C3—22no—
474C22×C3—21no—
475C22×C3—22no—
476C22×C3—21no—
477C22×C3—22no—
478C22×C3—24no—
479C22×C3—44no—
480C22×C3—42no—
481C22×C3—42no—
482C22×C3—42no—
483C22×C3—44no—
484C22×C3—42no—
485C22×C3—41no—
486C22×C3—42no—
487C22×C3—82no—
488C22×C3—82no—
489C22×C3—182no—
490C22×C3—181no—
491C22×C3D3×#6714yes—
492C22×C3#381112yes—
493C22×C3C22×D3212yes—
494C22×C3C22×D3211yescharacteristic
495C2×D3—62no—
496C2×D3—64no—
497C2×D3—62no—
498C2×D3—122no—
499C3×C4—22no—
500C3×C4—22no—
501C3×C4—21no—
502C3×C4—42no—
503C3×C4—42no—
504C3×C4—92no—
505C3×C4—92no—
506C3×C4—184no—
507C3×C4—184no—
508C3×C4—181no—
509Dic3—34no—
510Dic3—34no—
511Dic3—62no—
512Dic3—62no—
513Dic3—62no—
514Dic3—64no—
515Dic3—64no—
516Dic3—62no—
517Dic3—64no—
518Dic3—64no—
519Dic3—124no—
520Dic3—124no—
521C32—22no—
522C32—21no—
523C32—41no—
524C32#564812yescomplemented
525C32#582511yescharacteristic, complemented
526C23—91no—
527C23C2×#111311yescharacteristic
528C23C2×#111311yescharacteristic
529Q8—182no—
530Q8—544no—
531Q8—544no—
532D4—22no—
533D4—182no—
534D4—182no—
535C2×C4—91no—
536C2×C4—91no—
537C2×C4—94no—
538C2×C4—94no—
539C2×C4—94no—
540C2×C4—91no—
541C2×C4—182no—
542C2×C4—182no—
543C2×C4—184no—
544C2×C4C2×#111311yescharacteristic
545C2×C3—24no—
546C2×C3—22no—
547C2×C3—24no—
548C2×C3—22no—
549C2×C3—21no—
550C2×C3—22no—
551C2×C3—42no—
552C2×C3—44no—
553C2×C3—42no—
554C2×C3—42no—
555C2×C3—41no—
556C2×C3—42no—
557C2×C3—41no—
558C2×C3—82no—
559C2×C3—82no—
560C2×C3—82no—
561C2×C3—361no—
562C2×C3#6430914yes—
563C2×C3#6430912yes—
564C2×C3C2×D3×#6712yes—
565C2×C3C2×#381111yescharacteristic
566D3—122no—
567D3—122no—
568C22—22no—
569C22—22no—
570C22—21no—
571C22—182no—
572C22—181no—
573C22#8904912yes—
574C22C22×#111311yescharacteristic
575C4—21no—
576C4—92no—
577C4—92no—
578C4—184no—
579C4—181no—
580C4—184no—
581C3—21no—
582C3—22no—
583C3—41no—
584C3#1059527012yescomplemented
585C3#1059527111yescharacteristic, complemented
586C2—22no—
587C2—41no—
588C2—361no—
589C2#1172462012yes—
590C2C2×#8904911yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
prime radicalC2×C3×#3830C2
hypercenterC2×D4#1113
hypocenterC33#536
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C22×#6296C2semidirect
3. C3×#64144C2semidirect
4. #11723721C2semidirect
5. #11723720C2semidirect
6. #11723718C2semidirect
7. #11723715C2semidirect
8. #11723671C2semidirect
9. #11723666C2semidirect
36. #64144D3semidirect
359. C33#536semidirect
524. C32#5648semidirect
525. C32#5825semidirect
584. C3#10595270semidirect
585. C3#10595271semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38302
C22×#62962
C3×#641442
#117237212
#117237202
#117237182
#117237152
#117236712
#117236662
#105952713
#105952703