Finite Group Databasebeta

← #13178807

Subgroups — #13178807

Order 1728.

2014 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#638C211yesmaximal, characteristic, complemented
2C2×#89049C211yesmaximal, characteristic, complemented
3C3×#64162C212yesmaximal, complemented
4#11723729C212yesmaximal, complemented
5#11723725C214yesmaximal, complemented
6#11723724C212yesmaximal, complemented
7#11723723C211yesmaximal, characteristic, complemented
8#11723720C212yesmaximal, complemented
9#10595528—31nomaximal
10#10595524—32nomaximal
11C24×C33C2211yescharacteristic, complemented
12C23×C3×C3⋊D3C2211yescharacteristic
13C22×#1113C2211yescharacteristic, complemented
14C2×C3×#638—24no—
15C2×C3×#638C2212yes—
16C2×C3×#638C2212yes—
17C2×C3×#638C2212yescomplemented
18C2×C3×#638C2214yescomplemented
19C22×C3×C3⋊Dic3C2211yescharacteristic, complemented
20#89049—24no—
21C2×#6295C2212yescomplemented
22C2×#6294C2211yescharacteristic, complemented
23C2×#6294C2212yescomplemented
24C3×#3757C2212yescomplemented
25C3×#3757C2214yescomplemented
26#88815C2214yescomplemented
27#88814C2214yescomplemented
28#88814C2212yescomplemented
29C22×#638D311yescharacteristic, complemented
30C22×C3×#67—32no—
31C22×C3×#67—61no—
32C2×#3811—31no—
33C2×D3×#67—32no—
34#64162D312yescomplemented
35C3×#953—32no—
36C3×#953—32no—
37C3×#953—62no—
38#63983—32no—
39#63983—34no—
40#63975—32no—
41#63974—31no—
42#63971—34no—
43#63970—34no—
44#63969—34no—
45D3×#217—32no—
46#63966—32no—
47#63962—32no—
48#63961—32no—
49#63958—34no—
50C23×C33—22no—
51C23×C33—42no—
52C23×C33C2312yes—
53C23×C33C2311yescharacteristic
54C22×C3×C3⋊D3—42no—
55C22×C3×C3⋊D3D414yescomplemented
56C22×C3×C3⋊D3C2312yes—
57C22×C3×C3⋊D3C2311yescharacteristic
58C22×C3×C3⋊D3C2311yescharacteristic
59C2×#1113—22no—
60C2×#1113—42no—
61C3×#638—24no—
62C3×#638—24no—
63C3×#638—24no—
64C3×#638—44no—
65C2×C3×C3⋊Dic3—22no—
66C2×C3×C3⋊Dic3C2312yes—
67C2×C3×C3⋊Dic3C2314yes—
68C2×C3×C3⋊Dic3C2312yes—
69#6295—24no—
70#6294—22no—
71#6294—42no—
72#5804—91no—
73#5589—92no—
74C24×C32—21no—
75C24×C32—22no—
76C24×C32—41no—
77C24×C32C2×D311yescharacteristic, complemented
78C24×C32C2×D312yescomplemented
79C23×C3⋊D3C2×D311yescharacteristic
80C23×C3×D3—32no—
81C23×C3×D3—61no—
82C22×D32—32no—
83C22×D32—31no—
84C2×#638—24no—
85C2×#638C2×D312yes—
86C2×#638C2×D312yes—
87C2×#638C2×D312yescomplemented
88C2×#638C2×D314yescomplemented
89C22×C3⋊Dic3C2×D311yescharacteristic, complemented
90C2×C3×#67—32no—
91C2×C3×#67—32no—
92C2×C3×#67—34no—
93C2×C3×#67—34no—
94C2×C3×#67—34no—
95C2×C3×#67—34no—
96C2×C3×#67—68no—
97C2×C3×#67—62no—
98C2×C3×#67—62no—
99C2×C3×#67—68no—
100C2×C3×#67—62no—
101C2×C3×#67—64no—
102C22×C3×Dic3—32no—
103C22×C3×Dic3—61no—
104#3811—64no—
105D3×#67—68no—
106C2×#626—32no—
107C2×#626—32no—
108C2×#625—32no—
109C2×#624—31no—
110C2×#624—32no—
111C2×D3×Dic3—32no—
112C2×D3×Dic3—34no—
113#3757C2×D312yescomplemented
114#3757C2×D314yescomplemented
115C3×#217—34no—
116C3×#217—32no—
117C3×#217—32no—
118C3×#217—34no—
119C3×#217—62no—
120C3×#217—64no—
121#3724—34no—
122Dic3⋊Dic3—34no—
123Dic3⋊Dic3—34no—
124#3722—34no—
125#3722—32no—
126(C2×D3)⋊Dic3—34no—
127(C2×D3)⋊Dic3—34no—
128(C2×D3)⋊Dic3—34no—
129C22×C33—22no—
130C22×C33—22no—
131C22×C33—24no—
132C22×C33—44no—
133C22×C33C2×D412yes—
134C22×C33C2411yescharacteristic
135C2×C3×C3⋊D3—22no—
136C2×C3×C3⋊D3—22no—
137C2×C3×C3⋊D3—22no—
138C2×C3×C3⋊D3—24no—
139C2×C3×C3⋊D3—44no—
140C2×C3×C3⋊D3C2×D412yes—
141C2×C3×C3⋊D3C2×D412yes—
142#1113—44no—
143C3×C3⋊Dic3—22no—
144C3×C3⋊Dic3—22no—
145C3×C3⋊Dic3—24no—
146C22×C3×D4—91no—
147C22×#67—32no—
148C22×#67—61no—
149C22×#67—91no—
150C2×D3×D4—92no—
151C3×#120—92no—
152#953—32no—
153#953—32no—
154#953—62no—
155C2×#217—91no—
156#939—94no—
157#938—94no—
158#937—92no—
159#937—94no—
160#935—92no—
161#935—92no—
162#929—92no—
163#886—92no—
164#884—94no—
165#883—94no—
166D3×C22⋊C4—92no—
167C23×C32—22no—
168C23×C32—24no—
169C23×C32—24no—
170C23×C32—21no—
171C23×C32—24no—
172C23×C32—22no—
173C23×C32—22no—
174C23×C32—41no—
175C23×C32—42no—
176C23×C32—44no—
177C23×C32—44no—
178C23×C32—42no—
179C23×C32—44no—
180C23×C32—44no—
181C23×C32—48no—
182C23×C32—48no—
183C23×C32C22×D312yes—
184C23×C32C22×D314yes—
185C23×C32C22×D312yes—
186C23×C32C22×D311yescharacteristic
187C22×C3⋊D3—42no—
188C22×C3⋊D3#6714yescomplemented
189C22×C3⋊D3C22×D312yes—
190C22×C3⋊D3C22×D311yescharacteristic
191C22×C3⋊D3C22×D311yescharacteristic
192C22×C3×D3—32no—
193C22×C3×D3—32no—
194C22×C3×D3—38no—
195C22×C3×D3—32no—
196C22×C3×D3—32no—
197C22×C3×D3—64no—
198C22×C3×D3—61no—
199C22×C3×D3—61no—
200C22×C3×D3—62no—
201C22×C3×D3—124no—
202C22×C3×D3—124no—
203C2×D32—64no—
204C2×D32—62no—
205C2×D32—124no—
206C2×D32—122no—
207#638—24no—
208#638—24no—
209#638—24no—
210#638—44no—
211C2×C3⋊Dic3—22no—
212C2×C3⋊Dic3C22×D312yes—
213C2×C3⋊Dic3C22×D314yes—
214C2×C3⋊Dic3C22×D312yes—
215C3×#67—68no—
216C3×#67—68no—
217C3×#67—68no—
218C3×#67—68no—
219C3×#67—64no—
220C3×#67—64no—
221C3×#67—124no—
222C3×#67—124no—
223C3×#67—124no—
224C3×#67—128no—
225C2×C3×Dic3—34no—
226C2×C3×Dic3—34no—
227C2×C3×Dic3—34no—
228C2×C3×Dic3—32no—
229C2×C3×Dic3—32no—
230C2×C3×Dic3—62no—
231C2×C3×Dic3—62no—
232C2×C3×Dic3—64no—
233C2×C3×Dic3—64no—
234C2×C3×Dic3—64no—
235#626—64no—
236#626—64no—
237#625—64no—
238#624—62no—
239#624—122no—
240D3×Dic3—64no—
241D3×Dic3—124no—
242#534—271noSylow
243C2×C33—22no—
244C2×C33—42no—
245C2×C33C22×D411yescharacteristic
246C2×C33H(5,𝔽2)12yes—
247C3×C3⋊D3—22no—
248C3×C3⋊D3—22no—
249C3×C3⋊D3—42no—
250C24×C3—21no—
251C24×C3—22no—
252C24×C3—41no—
253C24×C3—91no—
254C24×C3D3211yescharacteristic, complemented
255C24×C3D3212yescomplemented
256C23×D3—32no—
257C23×D3—61no—
258C23×D3—92no—
259C23×D3—91no—
260C2×C3×D4—92no—
261C2×C3×D4—92no—
262C2×C3×D4—92no—
263C2×C3×D4—94no—
264C2×C3×D4—184no—
265C22×C3×C4—91no—
266C2×#67—34no—
267C2×#67—34no—
268C2×#67—32no—
269C2×#67—32no—
270C2×#67—34no—
271C2×#67—34no—
272C2×#67—62no—
273C2×#67—62no—
274C2×#67—62no—
275C2×#67—64no—
276C2×#67—68no—
277C2×#67—68no—
278C2×#67—92no—
279C2×#67—92no—
280C2×#67—184no—
281C22×Dic3—32no—
282C22×Dic3—61no—
283C22×Dic3—91no—
284C22×Dic3—92no—
285D3×D4—188no—
286C2×D12—92no—
287C2×C4×D3—92no—
288C2×C4×D3—94no—
289C3×C22⋊C4—92no—
290C3×C22⋊C4—94no—
291#217—32no—
292#217—34no—
293#217—32no—
294#217—34no—
295#217—62no—
296#217—64no—
297#217—94no—
298#217—92no—
299(C2×D3)⋊C4—94no—
300(C2×D3)⋊C4—94no—
301(C2×D3)⋊C4—94no—
302C4⋊Dic3—94no—
303Dic3⋊C4—94no—
304Dic3⋊C4—94no—
305C22×C32—22no—
306C22×C32—24no—
307C22×C32—22no—
308C22×C32—24no—
309C22×C32—24no—
310C22×C32—24no—
311C22×C32—28no—
312C22×C32—24no—
313C22×C32—24no—
314C22×C32—24no—
315C22×C32—22no—
316C22×C32—21no—
317C22×C32—24no—
318C22×C32—22no—
319C22×C32—42no—
320C22×C32—41no—
321C22×C32—44no—
322C22×C32—44no—
323C22×C32—44no—
324C22×C32—44no—
325C22×C32—44no—
326C22×C32—44no—
327C22×C32—48no—
328C22×C32—44no—
329C22×C32—48no—
330C22×C32—48no—
331C22×C32—416no—
332C22×C32—416no—
333C22×C32D3×D414yes—
334C22×C32C2×#6712yes—
335C22×C32C23×D312yes—
336C22×C32C23×D311yescharacteristic
337C2×C3⋊D3—22no—
338C2×C3⋊D3—22no—
339C2×C3⋊D3—22no—
340C2×C3⋊D3—24no—
341C2×C3⋊D3—44no—
342C2×C3⋊D3C2×#6712yes—
343C2×C3⋊D3C2×#6712yes—
344C2×C3×D3—34no—
345C2×C3×D3—34no—
346C2×C3×D3—62no—
347C2×C3×D3—62no—
348C2×C3×D3—64no—
349C2×C3×D3—64no—
350C2×C3×D3—62no—
351C2×C3×D3—64no—
352C2×C3×D3—64no—
353C2×C3×D3—62no—
354C2×C3×D3—122no—
355C2×C3×D3—122no—
356C2×C3×D3—122no—
357C2×C3×D3—124no—
358C2×C3×D3—128no—
359C2×C3×D3—128no—
360D32—128no—
361D32—124no—
362C3⋊Dic3—22no—
363C3⋊Dic3—22no—
364C3⋊Dic3—24no—
365C3×Dic3—64no—
366C3×Dic3—62no—
367C3×Dic3—64no—
368C3×Dic3—64no—
369C3×Dic3—64no—
370C3×Dic3—62no—
371C3×Dic3—122no—
372C3×Dic3—124no—
373C22×D4—91no—
374C22×D4—271no—
375#123—272no—
376#123—272no—
377#121—274no—
378#120—92no—
379#120—272no—
380C2×C22⋊C4—271no—
381C33#53411yescharacteristic, complemented, Sylow
382C23×C3—24no—
383C23×C3—24no—
384C23×C3—22no—
385C23×C3—21no—
386C23×C3—24no—
387C23×C3—22no—
388C23×C3—22no—
389C23×C3—44no—
390C23×C3—44no—
391C23×C3—44no—
392C23×C3—42no—
393C23×C3—44no—
394C23×C3—48no—
395C23×C3—48no—
396C23×C3—42no—
397C23×C3—41no—
398C23×C3—91no—
399C23×C3—91no—
400C23×C3—94no—
401C23×C3—92no—
402C23×C3—362no—
403C23×C3C2×D3212yes—
404C23×C3C2×D3214yes—
405C23×C3C2×D3212yes—
406C23×C3C2×D3211yescharacteristic
407C22×D3—32no—
408C22×D3—32no—
409C22×D3—38no—
410C22×D3—32no—
411C22×D3—32no—
412C22×D3—64no—
413C22×D3—61no—
414C22×D3—61no—
415C22×D3—62no—
416C22×D3—124no—
417C22×D3—124no—
418C22×D3—184no—
419C22×D3—182no—
420C22×D3—364no—
421C22×D3—362no—
422C3×D4—184no—
423C3×D4—184no—
424C3×D4—184no—
425C3×D4—364no—
426C2×C3×C4—92no—
427C2×C3×C4—92no—
428C2×C3×C4—94no—
429C2×C3×C4—182no—
430#67—68no—
431#67—68no—
432#67—68no—
433#67—68no—
434#67—64no—
435#67—64no—
436#67—124no—
437#67—124no—
438#67—124no—
439#67—128no—
440#67—184no—
441#67—184no—
442C2×Dic3—32no—
443C2×Dic3—34no—
444C2×Dic3—34no—
445C2×Dic3—34no—
446C2×Dic3—32no—
447C2×Dic3—62no—
448C2×Dic3—64no—
449C2×Dic3—62no—
450C2×Dic3—64no—
451C2×Dic3—64no—
452C2×Dic3—182no—
453C2×Dic3—362no—
454D12—184no—
455C4×D3—184no—
456C4×D3—364no—
457C2×C32—22no—
458C2×C32—24no—
459C2×C32—24no—
460C2×C32—22no—
461C2×C32—21no—
462C2×C32—24no—
463C2×C32—22no—
464C2×C32—41no—
465C2×C32—42no—
466C2×C32—44no—
467C2×C32—44no—
468C2×C32—48no—
469C2×C32—44no—
470C2×C32—48no—
471C2×C32—42no—
472C2×C32—44no—
473C2×C32C2×D3×D412yes—
474C2×C32#100412yes—
475C2×C32#100414yes—
476C2×C32C22×#6711yescharacteristic
477C3⋊D3—22no—
478C3⋊D3—22no—
479C3⋊D3—42no—
480C3×D3—64no—
481C3×D3—64no—
482C3×D3—122no—
483C3×D3—122no—
484C3×D3—122no—
485C3×D3—122no—
486C3×D3—242no—
487C24—91no—
488C24—271no—
489C24#111311yescharacteristic, complemented
490C2×D4—92no—
491C2×D4—92no—
492C2×D4—92no—
493C2×D4—94no—
494C2×D4—184no—
495C2×D4—272no—
496C2×D4—544no—
497C22×C4—91no—
498C22×C4—271no—
499C22×C4—272no—
500C4⋊C4—274no—
501C22⋊C4—92no—
502C22⋊C4—94no—
503C22⋊C4—272no—
504C22⋊C4—274no—
505C22×C3—24no—
506C22×C3—24no—
507C22×C3—22no—
508C22×C3—24no—
509C22×C3—28no—
510C22×C3—24no—
511C22×C3—22no—
512C22×C3—24no—
513C22×C3—24no—
514C22×C3—24no—
515C22×C3—22no—
516C22×C3—21no—
517C22×C3—22no—
518C22×C3—24no—
519C22×C3—44no—
520C22×C3—44no—
521C22×C3—44no—
522C22×C3—44no—
523C22×C3—44no—
524C22×C3—44no—
525C22×C3—48no—
526C22×C3—48no—
527C22×C3—48no—
528C22×C3—44no—
529C22×C3—416no—
530C22×C3—416no—
531C22×C3—41no—
532C22×C3—42no—
533C22×C3—92no—
534C22×C3—92no—
535C22×C3—182no—
536C22×C3—182no—
537C22×C3—182no—
538C22×C3—184no—
539C22×C3—364no—
540C22×C3D3×#6714yes—
541C22×C3#381112yes—
542C22×C3C22×D3212yes—
543C22×C3C22×D3211yescharacteristic
544C2×D3—34no—
545C2×D3—34no—
546C2×D3—62no—
547C2×D3—62no—
548C2×D3—64no—
549C2×D3—64no—
550C2×D3—62no—
551C2×D3—62no—
552C2×D3—64no—
553C2×D3—64no—
554C2×D3—122no—
555C2×D3—122no—
556C2×D3—122no—
557C2×D3—124no—
558C2×D3—128no—
559C2×D3—128no—
560C2×D3—368no—
561C2×D3—364no—
562C3×C4—182no—
563C3×C4—182no—
564C3×C4—184no—
565Dic3—64no—
566Dic3—62no—
567Dic3—64no—
568Dic3—62no—
569Dic3—64no—
570Dic3—64no—
571Dic3—122no—
572Dic3—124no—
573C32—22no—
574C32—21no—
575C32—41no—
576C32#558912yescomplemented
577C32#580411yescharacteristic, complemented
578C23—22no—
579C23—42no—
580C23—91no—
581C23—91no—
582C23—94no—
583C23—92no—
584C23—362no—
585C23—542no—
586C23—1082no—
587C23C2×#111312yes—
588C23C2×#111311yescharacteristic
589D4—184no—
590D4—184no—
591D4—184no—
592D4—364no—
593D4—544no—
594C2×C4—94no—
595C2×C4—92no—
596C2×C4—92no—
597C2×C4—182no—
598C2×C4—542no—
599C2×C4—1082no—
600C2×C3—24no—
601C2×C3—24no—
602C2×C3—22no—
603C2×C3—21no—
604C2×C3—22no—
605C2×C3—24no—
606C2×C3—22no—
607C2×C3—44no—
608C2×C3—44no—
609C2×C3—44no—
610C2×C3—48no—
611C2×C3—48no—
612C2×C3—44no—
613C2×C3—42no—
614C2×C3—41no—
615C2×C3—42no—
616C2×C3—182no—
617C2×C3—182no—
618C2×C3—362no—
619C2×C3C2×D3×#6712yes—
620C2×C3C2×#381111yescharacteristic
621C2×C3#6433014yes—
622C2×C3#6433012yes—
623D3—64no—
624D3—64no—
625D3—122no—
626D3—122no—
627D3—122no—
628D3—122no—
629D3—242no—
630C22—22no—
631C22—22no—
632C22—24no—
633C22—44no—
634C22—92no—
635C22—92no—
636C22—182no—
637C22—182no—
638C22—182no—
639C22—184no—
640C22—364no—
641C22—1084no—
642C22#8904912yes—
643C22C22×#111311yescharacteristic
644C4—182no—
645C4—182no—
646C4—184no—
647C3—21no—
648C3—22no—
649C3—41no—
650C3#1059552412yescomplemented
651C3#1059552811yescharacteristic, complemented
652C2—22no—
653C2—42no—
654C2—182no—
655C2—182no—
656C2—362no—
657C2C2×#8904911yescharacteristic
658C2#1172463312yes—
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC24×C33C22
hypercenterC24#1113
hypocenterC33#534
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C22×C3×#638C2semidirect
2. C2×#89049C2semidirect
3. C3×#64162C2semidirect
4. #11723729C2semidirect
5. #11723725C2semidirect
6. #11723724C2semidirect
7. #11723723C2semidirect
8. #11723720C2semidirect
11. C24×C33C22semidirect
13. C22×#1113C22semidirect
17. C2×C3×#638C22semidirect
18. C2×C3×#638C22semidirect
19. C22×C3×C3⋊Dic3C22semidirect
21. C2×#6295C22semidirect
22. C2×#6294C22semidirect
23. C2×#6294C22semidirect
24. C3×#3757C22semidirect
25. C3×#3757C22semidirect
26. #88815C22semidirect
27. #88814C22semidirect
28. #88814C22semidirect
29. C22×#638D3semidirect
34. #64162D3semidirect
55. C22×C3×C3⋊D3D4semidirect
77. C24×C32C2×D3semidirect
78. C24×C32C2×D3semidirect
87. C2×#638C2×D3semidirect
88. C2×#638C2×D3semidirect
89. C22×C3⋊Dic3C2×D3semidirect
113. #3757C2×D3semidirect
114. #3757C2×D3semidirect
188. C22×C3⋊D3#67semidirect
254. C24×C3D32semidirect
255. C24×C3D32semidirect
381. C33#534semidirect
489. C24#1113semidirect
576. C32#5589semidirect
577. C32#5804semidirect
650. C3#10595524semidirect
651. C3#10595528semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#6382
C2×#890492
C3×#641622
#117237292
#117237252
#117237242
#117237232
#117237202
#105955283
#105955243