Finite Group Databasebeta

← #13162062

Subgroups — #13162062

Order 1728.

2026 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#638C212yesmaximal, complemented
2C22×#6295C211yesmaximal, characteristic, complemented
3C2×C3×#3757C211yesmaximal, characteristic, complemented
4C2×#88814C212yesmaximal, complemented
5#11722082C211yesmaximal, characteristic, complemented
6#10591724—32nomaximal
7#10591723—31nomaximal
8C24×C33C2211yescharacteristic
9C23×C3×C3⋊D3C2212yes—
10C2×C3×#638—24no—
11C2×C3×#638—24no—
12C2×C3×#638—24no—
13C22×C3×C3⋊Dic3C2211yescharacteristic
14C22×C3×C3⋊Dic3C2211yescharacteristic
15C22×C3×C3⋊Dic3C2212yes—
16C2×#6295—24no—
17C2×#6295—22no—
18C3×#3757—22no—
19#88814—24no—
20C22×#638D312yescomplemented
21C22×C3×#67—32no—
22C22×C3×#67—32no—
23C22×C3×#67—62no—
24C22×#626—32no—
25C22×#625—31no—
26C2×#3757D311yescharacteristic, complemented
27C2×C3×#217—32no—
28C2×C3×#217—61no—
29C2×#3722—32no—
30C2×(C2×D3)⋊Dic3—32no—
31C2×(C2×D3)⋊Dic3—32no—
32#63579—31no—
33#63579—32no—
34C23×C33—21no—
35C23×C33—22no—
36C23×C33—22no—
37C23×C33—21no—
38C23×C33D412yescomplemented
39C23×C33C2311yescharacteristic
40C22×C3×C3⋊D3—22no—
41C22×C3×C3⋊D3—22no—
42C22×C3×C3⋊D3—22no—
43C22×C3×C3⋊D3—22no—
44C22×C3×C3⋊D3—22no—
45C22×C3×C3⋊D3—22no—
46C22×C3×C3⋊D3D414yescomplemented
47C3×#638—216no—
48C2×C3×C3⋊Dic3—22no—
49C2×C3×C3⋊Dic3—22no—
50C2×C3×C3⋊Dic3—22no—
51C2×C3×C3⋊Dic3—21no—
52C2×C3×C3⋊Dic3—22no—
53C2×C3×C3⋊Dic3D412yescomplemented
54C2×C3×C3⋊Dic3D414yescomplemented
55#6295—28no—
56#5223—92no—
57#4960—91no—
58C24×C32—21no—
59C24×C32—22no—
60C24×C32—41no—
61C24×C32C2×D311yescharacteristic
62C24×C32C2×D312yes—
63C23×C3⋊D3C2×D312yes—
64C23×C3×D3—32no—
65C23×C3×D3—32no—
66C23×C3×D3—62no—
67C2×#638—24no—
68C2×#638—24no—
69C2×#638—24no—
70C22×C3⋊Dic3C2×D311yescharacteristic
71C22×C3⋊Dic3C2×D311yescharacteristic
72C22×C3⋊Dic3C2×D312yes—
73C2×C3×#67—64no—
74C2×C3×#67—64no—
75C2×C3×#67—64no—
76C2×C3×#67—64no—
77C2×C3×#67—68no—
78C2×C3×#67—68no—
79C2×C3×#67—68no—
80C2×C3×#67—64no—
81C2×C3×#67—64no—
82C22×C3×Dic3—32no—
83C22×C3×Dic3—32no—
84C22×C3×Dic3—32no—
85C22×C3×Dic3—32no—
86C22×C3×Dic3—62no—
87C22×C3×Dic3—61no—
88C22×C3×Dic3—61no—
89C2×#626—64no—
90C2×#626—64no—
91C2×#626—64no—
92C2×#625—64no—
93C2×#625—62no—
94#3757—22no—
95C3×#217—64no—
96C3×#217—64no—
97#3722—64no—
98(C2×D3)⋊Dic3—64no—
99(C2×D3)⋊Dic3—64no—
100C22×C33—24no—
101C22×C33—22no—
102C22×C33—41no—
103C22×C33—42no—
104C22×C33—41no—
105C22×C33C2×D411yescharacteristic
106C22×C33C2×D412yes—
107C22×C33C2×D412yes—
108C22×C33C2×D411yescharacteristic
109C22×C33#4111yescharacteristic
110C2×C3×C3⋊D3—24no—
111C2×C3×C3⋊D3—24no—
112C2×C3×C3⋊D3—24no—
113C2×C3×C3⋊D3—42no—
114C2×C3×C3⋊D3—42no—
115C2×C3×C3⋊D3—42no—
116C2×C3×C3⋊D3—42no—
117C3×C3⋊Dic3—24no—
118C3×C3⋊Dic3—22no—
119C3×C3⋊Dic3—41no—
120C22×C3×D4—92no—
121C22×#67—32no—
122C22×#67—32no—
123C22×#67—62no—
124C22×#67—92no—
125C22×D12—91no—
126C2×C3×C22⋊C4—91no—
127C2×#217—32no—
128C2×#217—61no—
129C2×#217—92no—
130C2×(C2×D3)⋊C4—92no—
131C2×(C2×D3)⋊C4—92no—
132C3⋊#95—91no—
133C3⋊#95—92no—
134C23×C32—21no—
135C23×C32—21no—
136C23×C32—22no—
137C23×C32—22no—
138C23×C32—22no—
139C23×C32—22no—
140C23×C32—24no—
141C23×C32—24no—
142C23×C32—22no—
143C23×C32—22no—
144C23×C32—22no—
145C23×C32—22no—
146C23×C32—22no—
147C23×C32—22no—
148C23×C32—22no—
149C23×C32—24no—
150C23×C32—22no—
151C23×C32—21no—
152C23×C32—42no—
153C23×C32—41no—
154C23×C32—42no—
155C23×C32—42no—
156C23×C32—42no—
157C23×C32—42no—
158C23×C32—42no—
159C23×C32—42no—
160C23×C32—42no—
161C23×C32—44no—
162C23×C32—44no—
163C23×C32—42no—
164C23×C32D1212yescomplemented
165C23×C32#6714yescomplemented
166C23×C32C22×D311yescharacteristic
167C23×C32C22×D312yes—
168C22×C3⋊D3—22no—
169C22×C3⋊D3—22no—
170C22×C3⋊D3—22no—
171C22×C3⋊D3—22no—
172C22×C3⋊D3—22no—
173C22×C3⋊D3—22no—
174C22×C3⋊D3#6714yescomplemented
175C22×C3×D3—34no—
176C22×C3×D3—34no—
177C22×C3×D3—64no—
178C22×C3×D3—64no—
179C22×C3×D3—64no—
180C22×C3×D3—62no—
181C22×C3×D3—62no—
182C22×C3×D3—62no—
183C22×C3×D3—62no—
184C22×C3×D3—62no—
185C22×C3×D3—62no—
186C22×C3×D3—62no—
187C22×C3×D3—62no—
188C22×C3×D3—62no—
189C22×C3×D3—62no—
190C22×C3×D3—62no—
191C22×C3×D3—62no—
192C22×C3×D3—122no—
193C22×C3×D3—122no—
194C22×C3×D3—122no—
195C22×C3×D3—122no—
196#638—216no—
197C2×C3⋊Dic3—22no—
198C2×C3⋊Dic3—22no—
199C2×C3⋊Dic3—21no—
200C2×C3⋊Dic3—22no—
201C2×C3⋊Dic3—22no—
202C2×C3⋊Dic3#6714yescomplemented
203C2×C3⋊Dic3#6712yescomplemented
204C3×#67—632no—
205C3×#67—616no—
206C3×#67—616no—
207C2×C3×Dic3—34no—
208C2×C3×Dic3—34no—
209C2×C3×Dic3—34no—
210C2×C3×Dic3—64no—
211C2×C3×Dic3—62no—
212C2×C3×Dic3—64no—
213C2×C3×Dic3—64no—
214C2×C3×Dic3—62no—
215C2×C3×Dic3—62no—
216C2×C3×Dic3—62no—
217C2×C3×Dic3—62no—
218C2×C3×Dic3—62no—
219C2×C3×Dic3—62no—
220C2×C3×Dic3—62no—
221C2×C3×Dic3—62no—
222C2×C3×Dic3—62no—
223C2×C3×Dic3—62no—
224C2×C3×Dic3—122no—
225C2×C3×Dic3—122no—
226#626—616no—
227#625—68no—
228#392—271noSylow
229C2×C33—42no—
230C2×C33#12012yes—
231C2×C33#12011yescharacteristic
232C2×C33#12112yes—
233C2×C33#12111yescharacteristic
234C2×C33#12711yescharacteristic
235C3×C3⋊D3—44no—
236C24×C3—21no—
237C24×C3—22no—
238C24×C3—41no—
239C24×C3—92no—
240C24×C3D3211yescharacteristic
241C24×C3D3212yes—
242C23×D3—32no—
243C23×D3—32no—
244C23×D3—62no—
245C2×C3×D4—184no—
246C2×C3×D4—184no—
247C2×C3×D4—184no—
248C22×C3×C4—92no—
249C22×C3×C4—91no—
250C22×C3×C4—91no—
251C2×#67—64no—
252C2×#67—64no—
253C2×#67—64no—
254C2×#67—64no—
255C2×#67—68no—
256C2×#67—68no—
257C2×#67—68no—
258C2×#67—64no—
259C2×#67—64no—
260C2×#67—184no—
261C2×#67—184no—
262C2×#67—184no—
263C22×Dic3—32no—
264C22×Dic3—32no—
265C22×Dic3—32no—
266C22×Dic3—32no—
267C22×Dic3—62no—
268C22×Dic3—61no—
269C22×Dic3—61no—
270C2×D12—182no—
271C2×D12—184no—
272C3×C22⋊C4—182no—
273#217—64no—
274#217—64no—
275#217—184no—
276(C2×D3)⋊C4—184no—
277(C2×D3)⋊C4—184no—
278C22×C32—22no—
279C22×C32—24no—
280C22×C32—28no—
281C22×C32—24no—
282C22×C32—24no—
283C22×C32—24no—
284C22×C32—24no—
285C22×C32—22no—
286C22×C32—21no—
287C22×C32—21no—
288C22×C32—22no—
289C22×C32—22no—
290C22×C32—22no—
291C22×C32—22no—
292C22×C32—22no—
293C22×C32—22no—
294C22×C32—21no—
295C22×C32—22no—
296C22×C32—22no—
297C22×C32—41no—
298C22×C32—41no—
299C22×C32—42no—
300C22×C32—42no—
301C22×C32—41no—
302C22×C32—44no—
303C22×C32—44no—
304C22×C32—44no—
305C22×C32—48no—
306C22×C32—44no—
307C22×C32—44no—
308C22×C32—44no—
309C22×C32—44no—
310C22×C32—42no—
311C22×C32—44no—
312C22×C32—42no—
313C22×C32—44no—
314C22×C32—44no—
315C22×C32—48no—
316C22×C32—44no—
317C22×C32—42no—
318C22×C32—41no—
319C22×C32—42no—
320C22×C32—42no—
321C22×C32—42no—
322C22×C32—41no—
323C22×C32—42no—
324C22×C32C2×D1211yescharacteristic
325C22×C32#23511yescharacteristic
326C22×C32D3×D412yes—
327C22×C32D3×D412yes—
328C22×C32D3×D412yes—
329C22×C32D3×D412yes—
330C22×C32D3×D412yes—
331C22×C32#23712yes—
332C22×C32C2×#6712yes—
333C22×C32C2×#6711yescharacteristic
334C22×C32C2×#6712yes—
335C22×C32C2×#6712yes—
336C2×C3⋊D3—24no—
337C2×C3⋊D3—24no—
338C2×C3⋊D3—24no—
339C2×C3⋊D3—42no—
340C2×C3⋊D3—42no—
341C2×C3⋊D3—42no—
342C2×C3⋊D3—42no—
343C2×C3×D3—68no—
344C2×C3×D3—64no—
345C2×C3×D3—64no—
346C2×C3×D3—64no—
347C2×C3×D3—64no—
348C2×C3×D3—64no—
349C2×C3×D3—64no—
350C2×C3×D3—124no—
351C2×C3×D3—124no—
352C2×C3×D3—124no—
353C2×C3×D3—124no—
354C2×C3×D3—124no—
355C2×C3×D3—124no—
356C2×C3×D3—122no—
357C2×C3×D3—122no—
358C2×C3×D3—122no—
359C2×C3×D3—122no—
360C2×C3×D3—122no—
361C2×C3×D3—122no—
362C2×C3×D3—122no—
363C2×C3×D3—122no—
364C3⋊Dic3—24no—
365C3⋊Dic3—22no—
366C3⋊Dic3—41no—
367C3×Dic3—68no—
368C3×Dic3—64no—
369C3×Dic3—64no—
370C3×Dic3—64no—
371C3×Dic3—122no—
372C3×Dic3—122no—
373C3×Dic3—122no—
374C22×D4—92no—
375C22×D4—271no—
376C2×C22⋊C4—91no—
377C2×C22⋊C4—272no—
378#95—271no—
379C33#39211yescharacteristic, complemented, Sylow
380C23×C3—22no—
381C23×C3—24no—
382C23×C3—24no—
383C23×C3—22no—
384C23×C3—22no—
385C23×C3—21no—
386C23×C3—22no—
387C23×C3—22no—
388C23×C3—22no—
389C23×C3—22no—
390C23×C3—22no—
391C23×C3—21no—
392C23×C3—22no—
393C23×C3—22no—
394C23×C3—22no—
395C23×C3—24no—
396C23×C3—21no—
397C23×C3—22no—
398C23×C3—42no—
399C23×C3—42no—
400C23×C3—42no—
401C23×C3—42no—
402C23×C3—42no—
403C23×C3—42no—
404C23×C3—42no—
405C23×C3—42no—
406C23×C3—44no—
407C23×C3—44no—
408C23×C3—42no—
409C23×C3—41no—
410C23×C3—94no—
411C23×C3—182no—
412C23×C3—182no—
413C23×C3—182no—
414C23×C3—182no—
415C23×C3—182no—
416C23×C3—182no—
417C23×C3#62512yescomplemented
418C23×C3#62614yescomplemented
419C23×C3C2×D3212yes—
420C23×C3C2×D3211yescharacteristic
421C22×D3—34no—
422C22×D3—34no—
423C22×D3—64no—
424C22×D3—64no—
425C22×D3—64no—
426C22×D3—62no—
427C22×D3—62no—
428C22×D3—62no—
429C22×D3—62no—
430C22×D3—62no—
431C22×D3—62no—
432C22×D3—62no—
433C22×D3—62no—
434C22×D3—62no—
435C22×D3—62no—
436C22×D3—62no—
437C22×D3—62no—
438C22×D3—122no—
439C22×D3—122no—
440C22×D3—122no—
441C22×D3—122no—
442C3×D4—1816no—
443C2×C3×C4—94no—
444C2×C3×C4—92no—
445C2×C3×C4—182no—
446C2×C3×C4—182no—
447C2×C3×C4—182no—
448C2×C3×C4—182no—
449C2×C3×C4—181no—
450#67—632no—
451#67—616no—
452#67—616no—
453#67—1816no—
454C2×Dic3—34no—
455C2×Dic3—34no—
456C2×Dic3—34no—
457C2×Dic3—64no—
458C2×Dic3—62no—
459C2×Dic3—64no—
460C2×Dic3—64no—
461C2×Dic3—62no—
462C2×Dic3—62no—
463C2×Dic3—62no—
464C2×Dic3—62no—
465C2×Dic3—62no—
466C2×Dic3—62no—
467C2×Dic3—62no—
468C2×Dic3—62no—
469C2×Dic3—62no—
470C2×Dic3—62no—
471C2×Dic3—122no—
472C2×Dic3—122no—
473D12—188no—
474C2×C32—22no—
475C2×C32—22no—
476C2×C32—22no—
477C2×C32—22no—
478C2×C32—22no—
479C2×C32—22no—
480C2×C32—21no—
481C2×C32—22no—
482C2×C32—21no—
483C2×C32—22no—
484C2×C32—21no—
485C2×C32—22no—
486C2×C32—42no—
487C2×C32—41no—
488C2×C32—41no—
489C2×C32—42no—
490C2×C32—41no—
491C2×C32—44no—
492C2×C32—48no—
493C2×C32—48no—
494C2×C32—42no—
495C2×C32—44no—
496C2×C32#88212yes—
497C2×C32#88412yes—
498C2×C32#93011yescharacteristic
499C2×C32#93712yes—
500C2×C32#93711yescharacteristic
501C2×C32#93712yes—
502C2×C32#93812yes—
503C2×C32#93912yes—
504C2×C32#93912yes—
505C2×C32#94011yescharacteristic
506C2×C32#94012yes—
507C2×C32#95312yes—
508C3⋊D3—44no—
509C3×D3—128no—
510C3×D3—124no—
511C3×D3—124no—
512C24—92no—
513C24#111311yescharacteristic
514C2×D4—184no—
515C2×D4—184no—
516C2×D4—184no—
517C2×D4—542no—
518C2×D4—544no—
519C22×C4—91no—
520C22×C4—92no—
521C22×C4—91no—
522C22⋊C4—182no—
523C22⋊C4—544no—
524C22×C3—28no—
525C22×C3—24no—
526C22×C3—24no—
527C22×C3—24no—
528C22×C3—24no—
529C22×C3—22no—
530C22×C3—24no—
531C22×C3—22no—
532C22×C3—21no—
533C22×C3—22no—
534C22×C3—21no—
535C22×C3—21no—
536C22×C3—22no—
537C22×C3—22no—
538C22×C3—22no—
539C22×C3—22no—
540C22×C3—22no—
541C22×C3—22no—
542C22×C3—22no—
543C22×C3—44no—
544C22×C3—44no—
545C22×C3—44no—
546C22×C3—44no—
547C22×C3—48no—
548C22×C3—44no—
549C22×C3—44no—
550C22×C3—44no—
551C22×C3—42no—
552C22×C3—44no—
553C22×C3—42no—
554C22×C3—44no—
555C22×C3—48no—
556C22×C3—44no—
557C22×C3—44no—
558C22×C3—42no—
559C22×C3—42no—
560C22×C3—42no—
561C22×C3—41no—
562C22×C3—42no—
563C22×C3—41no—
564C22×C3—42no—
565C22×C3—41no—
566C22×C3—41no—
567C22×C3—41no—
568C22×C3—42no—
569C22×C3—42no—
570C22×C3—184no—
571C22×C3—184no—
572C22×C3—184no—
573C22×C3—362no—
574C22×C3—362no—
575C22×C3—362no—
576C22×C3—362no—
577C22×C3#380412yes—
578C22×C3#380511yescharacteristic
579C22×C3C2×#62511yescharacteristic
580C22×C3C2×#62612yes—
581C22×C3D3×#6712yes—
582C22×C3D3×#6712yes—
583C22×C3D3×#6712yes—
584C22×C3D3×#6712yes—
585C22×C3D3×#6712yes—
586C22×C3#381112yes—
587C22×C3#381112yes—
588C22×C3#381111yescharacteristic
589C2×D3—68no—
590C2×D3—64no—
591C2×D3—64no—
592C2×D3—64no—
593C2×D3—64no—
594C2×D3—64no—
595C2×D3—64no—
596C2×D3—124no—
597C2×D3—124no—
598C2×D3—124no—
599C2×D3—124no—
600C2×D3—124no—
601C2×D3—124no—
602C2×D3—122no—
603C2×D3—122no—
604C2×D3—122no—
605C2×D3—122no—
606C2×D3—122no—
607C2×D3—122no—
608C2×D3—122no—
609C2×D3—122no—
610C3×C4—184no—
611C3×C4—182no—
612C3×C4—361no—
613Dic3—68no—
614Dic3—64no—
615Dic3—64no—
616Dic3—64no—
617Dic3—122no—
618Dic3—122no—
619Dic3—122no—
620C32—21no—
621C32—22no—
622C32—41no—
623C32#496011yescharacteristic, complemented
624C32#522312yescomplemented
625C23—21no—
626C23—21no—
627C23—22no—
628C23—22no—
629C23—94no—
630C23—182no—
631C23—182no—
632C23—182no—
633C23—182no—
634C23—182no—
635C23—182no—
636C23#629512yescomplemented
637C23C2×#111311yescharacteristic
638D4—1816no—
639D4—548no—
640C2×C4—92no—
641C2×C4—94no—
642C2×C4—182no—
643C2×C4—182no—
644C2×C4—182no—
645C2×C4—181no—
646C2×C4—182no—
647C2×C3—22no—
648C2×C3—21no—
649C2×C3—22no—
650C2×C3—22no—
651C2×C3—22no—
652C2×C3—22no—
653C2×C3—22no—
654C2×C3—22no—
655C2×C3—21no—
656C2×C3—22no—
657C2×C3—21no—
658C2×C3—22no—
659C2×C3—48no—
660C2×C3—44no—
661C2×C3—48no—
662C2×C3—42no—
663C2×C3—44no—
664C2×C3—41no—
665C2×C3—42no—
666C2×C3—42no—
667C2×C3—41no—
668C2×C3—41no—
669C2×C3—364no—
670C2×C3#6395812yes—
671C2×C3#6397012yes—
672C2×C3#6397112yes—
673C2×C3#6397612yes—
674C2×C3#6397712yes—
675C2×C3#6397812yes—
676C2×C3#6397911yescharacteristic
677C2×C3#6397912yes—
678C2×C3#6398011yescharacteristic
679C2×C3#6398112yes—
680C2×C3#6398311yescharacteristic
681C2×C3#6398312yes—
682D3—128no—
683D3—124no—
684D3—124no—
685C22—22no—
686C22—24no—
687C22—42no—
688C22—41no—
689C22—41no—
690C22—184no—
691C22—184no—
692C22—184no—
693C22—362no—
694C22—362no—
695C22—362no—
696C22—362no—
697C22#8904611yescharacteristic
698C22C2×#629511yescharacteristic
699C22#8904911yescharacteristic
700C22#8904912yes—
701C22#8904912yes—
702C4—184no—
703C4—182no—
704C4—361no—
705C3—22no—
706C3—21no—
707C3—41no—
708C3#1059172311yescharacteristic, complemented
709C3#1059172412yescomplemented
710C2—42no—
711C2—364no—
712C2#1172371711yescharacteristic
713C2#1172372512yes—
714C2#1172372712yes—
715C2#1172372811yescharacteristic
716C2#1172372911yescharacteristic
subgroupG/H
center, FrattiniC23C2×#1113
commutator, socle, cosocleC23×C33C23
FittingC24×C33C22
hypercenterC24#1113
hypocenterC33#392
LCSG ▷ C23×C33 ▷ C33
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C22×C3×#638C2semidirect
2. C22×#6295C2semidirect
3. C2×C3×#3757C2semidirect
4. C2×#88814C2semidirect
5. #11722082C2semidirect
20. C22×#638D3semidirect
26. C2×#3757D3semidirect
38. C23×C33D4semidirect
46. C22×C3×C3⋊D3D4semidirect
53. C2×C3×C3⋊Dic3D4semidirect
54. C2×C3×C3⋊Dic3D4semidirect
164. C23×C32D12semidirect
165. C23×C32#67semidirect
174. C22×C3⋊D3#67semidirect
202. C2×C3⋊Dic3#67semidirect
203. C2×C3⋊Dic3#67semidirect
379. C33#392semidirect
417. C23×C3#625semidirect
418. C23×C3#626semidirect
623. C32#4960semidirect
624. C32#5223semidirect
636. C23#6295semidirect
708. C3#10591723semidirect
709. C3#10591724semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#6382
C22×#62952
C2×C3×#37572
C2×#888142
#117220822
#105917243
#105917233