Finite Group Databasebeta

← #13178710

Subgroups — #13178710

Order 1728.

1358 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64149C211yesmaximal, characteristic, complemented
2C3×#64148C212yesmaximal, complemented
3#11723729C212yesmaximal, complemented
4#11723725C212yesmaximal, complemented
5#11723720C212yesmaximal, complemented
6#11723717C212yesmaximal, complemented
7#11723701C211yesmaximal, characteristic, complemented
8#11723686C211yesmaximal, characteristic, complemented
9#11723676C212yesmaximal, complemented
10#10595313—32nomaximal
11#10595312—31nomaximal
12C22×#1113C2211yescharacteristic
13C2×C33×D4C2211yescharacteristic, complemented
14C2×C3×#638C2212yes—
15C2×C3×#638C2212yes—
16C2×C3×#638C2212yes—
17C2×C3×#638C2212yescomplemented
18C22×C3×C3⋊Dic3C2212yescomplemented
19C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
20C2×#6295C2212yes—
21C2×#6294C2211yescharacteristic, complemented
22C3×#3757C2212yescomplemented
23C3×#3752C2212yes—
24C3×#3750C2212yescomplemented
25C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
26#88815C2212yescomplemented
27#88815C2212yescomplemented
28#88814C2212yes—
29#88814C2212yes—
30#88814C2212yes—
31#88814C2211yescharacteristic
32#88814C2211yescharacteristic, complemented
33#64149D311yescharacteristic, complemented
34#64148D312yescomplemented
35C3×#940—32no—
36C3×#940—61no—
37C3×#939—32no—
38C3×#939—32no—
39C3×#939—62no—
40#63983—32no—
41#63983—32no—
42#63983—32no—
43#63980—32no—
44#63978—32no—
45#63978—32no—
46#63971—32no—
47#63970—32no—
48#63966—32no—
49#63962—32no—
50#63961—32no—
51#63958—32no—
52#63923—31no—
53#63922—32no—
54#63897—31no—
55#63884—32no—
56#63868—32no—
57#63862—32no—
58#63859—32no—
59C23×C33C2312yes—
60C22×C3×C3⋊D3C2312yes—
61C22×C3×C3⋊D3C2311yescharacteristic
62C22×C3×C3⋊D3C2311yescharacteristic
63C2×#1113—42no—
64C2×#1113—41no—
65C33×D4—41no—
66C2×C33×C4C2311yescharacteristic
67C3×#638—42no—
68C3×#638—42no—
69C3×#638—42no—
70C3×#638—42no—
71C2×C3×C3⋊Dic3—42no—
72C2×C3×C3⋊Dic3C2312yes—
73C2×C3×C3⋊Dic3C2312yes—
74C2×C3×C3⋊Dic3C2312yes—
75C2×C3×C3⋊Dic3C2312yes—
76C3×C4⋊(C3⋊D3)—41no—
77#6295—42no—
78#6294—41no—
79#5720—91no—
80#5620—92no—
81C22×D32—32no—
82C22×D32—31no—
83C2×C32×D4—22no—
84C2×C32×D4—21no—
85C2×C32×D4—41no—
86C2×C32×D4C2×D311yescharacteristic, complemented
87C2×C32×D4C2×D312yescomplemented
88C2×#638C2×D312yes—
89C2×#638C2×D312yes—
90C2×#638C2×D312yes—
91C2×#638C2×D312yescomplemented
92C22×C3⋊Dic3C2×D312yescomplemented
93C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
94C2×C3×#67—32no—
95C2×C3×#67—32no—
96C2×C3×#67—32no—
97C2×C3×#67—32no—
98C2×C3×#67—32no—
99C2×C3×#67—32no—
100C2×C3×#67—32no—
101C2×C3×#67—32no—
102C2×C3×#67—62no—
103C2×C3×#67—62no—
104C2×C3×#67—62no—
105C2×C3×#67—62no—
106C22×C3×Dic3—32no—
107C22×C3×Dic3—32no—
108C22×C3×Dic3—62no—
109C2×C3×D12—32no—
110C2×C3×D12—61no—
111C2×#626—32no—
112C2×#626—32no—
113C2×#625—32no—
114C2×#624—31no—
115C2×D3×Dic3—32no—
116#3757C2×D312yescomplemented
117#3752C2×D312yes—
118#3750C2×D312yescomplemented
119C4×C3⋊Dic3C2×D311yescharacteristic, complemented
120C3×#217—32no—
121C3×#217—32no—
122C3×#217—62no—
123C3×(C2×D3)⋊C4—32no—
124C3×(C2×D3)⋊C4—32no—
125C3×(C2×D3)⋊C4—62no—
126C3×Dic3⋊C4—32no—
127C3×Dic3⋊C4—32no—
128C3×Dic3⋊C4—62no—
129C3×C4×Dic3—32no—
130C3×C4×Dic3—61no—
131#3724—32no—
132#3724—32no—
133Dic3⋊Dic3—32no—
134Dic3⋊Dic3—32no—
135Dic3⋊Dic3—32no—
136Dic3⋊Dic3—32no—
137#3722—32no—
138#3722—32no—
139#3722—32no—
140#3722—31no—
141#3722—31no—
142(C2×D3)⋊Dic3—32no—
143(C2×D3)⋊Dic3—32no—
144(C2×D3)⋊Dic3—32no—
145(C2×D3)⋊Dic3—32no—
146(C2×D3)⋊Dic3—32no—
147(C2×D3)⋊Dic3—32no—
148(C2×D3)⋊Dic3—32no—
149(C2×D3)⋊Dic3—32no—
150C22×C33—22no—
151C22×C33—22no—
152C22×C33—22no—
153C22×C33C2411yescharacteristic
154C2×C3×C3⋊D3—21no—
155C2×C3×C3⋊D3—22no—
156C2×C3×C3⋊D3—22no—
157C2×C3×C3⋊D3—21no—
158C2×C3×C3⋊D3—22no—
159C2×C3×C3⋊D3—22no—
160C2×C3×C3⋊D3—22no—
161#1113—44no—
162C33×C4—21no—
163C3×C3⋊Dic3—22no—
164C3×C3⋊Dic3—22no—
165C3×C3⋊Dic3—22no—
166C3×C3⋊Dic3—22no—
167C3×#127—91no—
168C3×#121—92no—
169#940—32no—
170#940—61no—
171#939—32no—
172#939—32no—
173#939—62no—
174#939—92no—
175#939—92no—
176#939—92no—
177#938—92no—
178#937—91no—
179#937—92no—
180#937—92no—
181#937—92no—
182#935—91no—
183#935—92no—
184#930—92no—
185#929—92no—
186#898—92no—
187#894—92no—
188#886—92no—
189#884—92no—
190#882—92no—
191#879—92no—
192#876—92no—
193C23×C32—22no—
194C23×C32—22no—
195C23×C32—22no—
196C23×C32—42no—
197C23×C32C22×D312yes—
198C23×C32C22×D312yes—
199C23×C32C22×D312yes—
200C22×C3⋊D3C22×D311yescharacteristic
201C22×C3⋊D3C22×D312yes—
202C22×C3⋊D3C22×D311yescharacteristic
203C22×C3×D3—32no—
204C22×C3×D3—32no—
205C22×C3×D3—32no—
206C22×C3×D3—32no—
207C22×C3×D3—62no—
208C22×C3×D3—61no—
209C22×C3×D3—61no—
210C2×D32—122no—
211C2×D32—122no—
212C2×D32—122no—
213C2×D32—121no—
214C2×D32—122no—
215C32×D4—41no—
216C32×D4—42no—
217C32×D4—42no—
218C32×D4—44no—
219C32×D4—44no—
220C2×C32×C4—21no—
221C2×C32×C4—22no—
222C2×C32×C4—41no—
223C2×C32×C4C22×D312yes—
224C2×C32×C4C22×D311yescharacteristic
225#638—42no—
226#638—42no—
227#638—42no—
228#638—42no—
229C2×C3⋊Dic3—42no—
230C2×C3⋊Dic3C22×D312yes—
231C2×C3⋊Dic3C22×D312yes—
232C2×C3⋊Dic3C22×D312yes—
233C2×C3⋊Dic3C22×D312yes—
234C4⋊(C3⋊D3)—41no—
235C3×#67—122no—
236C3×#67—122no—
237C3×#67—122no—
238C3×#67—122no—
239C3×#67—122no—
240C3×#67—122no—
241C3×#67—122no—
242C3×#67—122no—
243C3×#67—124no—
244C3×#67—124no—
245C3×#67—124no—
246C3×#67—124no—
247C2×C3×Dic3—32no—
248C2×C3×Dic3—32no—
249C2×C3×Dic3—32no—
250C2×C3×Dic3—32no—
251C2×C3×Dic3—32no—
252C2×C3×Dic3—32no—
253C2×C3×Dic3—32no—
254C2×C3×Dic3—32no—
255C2×C3×Dic3—62no—
256C2×C3×Dic3—62no—
257C2×C3×Dic3—62no—
258C2×C3×Dic3—62no—
259C2×C3×Dic3—122no—
260C2×C3×Dic3—122no—
261C2×C3×Dic3—124no—
262C3×D12—122no—
263C3×D12—122no—
264#626—122no—
265#626—122no—
266#625—122no—
267#624—121no—
268D3×Dic3—122no—
269#560—271noSylow
270C2×C33—42no—
271C2×C33H(5,𝔽2)11yescharacteristic
272C2×C33H(5,𝔽2)12yes—
273C3×C3⋊D3—42no—
274C3×C3⋊D3—41no—
275C3×C3⋊D3—41no—
276C23×D3—92no—
277C23×D3—91no—
278C2×C3×D4—22no—
279C2×C3×D4—21no—
280C2×C3×D4—41no—
281C2×C3×D4—92no—
282C2×C3×D4—92no—
283C2×C3×D4—92no—
284C2×C3×D4—92no—
285C2×C3×D4—91no—
286C2×C3×D4D3211yescharacteristic, complemented
287C2×C3×D4D3212yescomplemented
288C22×C3×C4—92no—
289C2×#67—32no—
290C2×#67—32no—
291C2×#67—32no—
292C2×#67—32no—
293C2×#67—32no—
294C2×#67—32no—
295C2×#67—32no—
296C2×#67—32no—
297C2×#67—62no—
298C2×#67—62no—
299C2×#67—62no—
300C2×#67—62no—
301C2×#67—92no—
302C2×#67—92no—
303C22×Dic3—32no—
304C22×Dic3—32no—
305C22×Dic3—62no—
306C22×Dic3—91no—
307C2×D12—32no—
308C2×D12—61no—
309C2×D12—92no—
310C2×C4×D3—92no—
311C3×C4⋊C4—92no—
312C3×C22⋊C4—92no—
313C3×C22⋊C4—92no—
314C3×C42—91no—
315#217—32no—
316#217—32no—
317#217—62no—
318#217—92no—
319#217—92no—
320#217—92no—
321#217—91no—
322#217—91no—
323(C2×D3)⋊C4—32no—
324(C2×D3)⋊C4—32no—
325(C2×D3)⋊C4—62no—
326(C2×D3)⋊C4—92no—
327(C2×D3)⋊C4—92no—
328(C2×D3)⋊C4—92no—
329(C2×D3)⋊C4—92no—
330(C2×D3)⋊C4—92no—
331(C2×D3)⋊C4—92no—
332(C2×D3)⋊C4—92no—
333(C2×D3)⋊C4—92no—
334C4⋊Dic3—92no—
335C4⋊Dic3—92no—
336Dic3⋊C4—32no—
337Dic3⋊C4—32no—
338Dic3⋊C4—62no—
339Dic3⋊C4—92no—
340Dic3⋊C4—92no—
341Dic3⋊C4—92no—
342Dic3⋊C4—92no—
343C4×Dic3—32no—
344C4×Dic3—61no—
345C22×C32—22no—
346C22×C32—24no—
347C22×C32—22no—
348C22×C32—22no—
349C22×C32—22no—
350C22×C32—22no—
351C22×C32—22no—
352C22×C32—22no—
353C22×C32—22no—
354C22×C32—22no—
355C22×C32—22no—
356C22×C32—21no—
357C22×C32—41no—
358C22×C32—42no—
359C22×C32—42no—
360C22×C32—44no—
361C22×C32—42no—
362C22×C32—42no—
363C22×C32—42no—
364C22×C32—42no—
365C22×C32—42no—
366C22×C32—42no—
367C22×C32—82no—
368C22×C32—82no—
369C22×C32C23×D311yescharacteristic
370C22×C32C23×D312yes—
371C2×C3⋊D3—21no—
372C2×C3⋊D3—22no—
373C2×C3⋊D3—21no—
374C2×C3⋊D3—22no—
375C2×C3⋊D3—22no—
376C2×C3⋊D3—22no—
377C2×C3⋊D3—22no—
378C2×C3×D3—62no—
379C2×C3×D3—62no—
380C2×C3×D3—62no—
381C2×C3×D3—62no—
382C2×C3×D3—62no—
383C2×C3×D3—62no—
384C2×C3×D3—62no—
385C2×C3×D3—62no—
386C2×C3×D3—62no—
387C2×C3×D3—62no—
388C2×C3×D3—62no—
389C2×C3×D3—62no—
390C2×C3×D3—121no—
391C2×C3×D3—121no—
392C2×C3×D3—122no—
393C2×C3×D3—122no—
394C2×C3×D3—122no—
395C2×C3×D3—122no—
396C2×C3×D3—122no—
397D32—124no—
398D32—128no—
399C32×C4—21no—
400C32×C4—22no—
401C32×C4—22no—
402C32×C4—42no—
403C32×C4—42no—
404C3⋊Dic3—22no—
405C3⋊Dic3—22no—
406C3⋊Dic3—22no—
407C3⋊Dic3—22no—
408C3×Dic3—64no—
409C3×Dic3—64no—
410C3×Dic3—62no—
411C3×Dic3—62no—
412C3×Dic3—62no—
413C3×Dic3—62no—
414C3×Dic3—62no—
415C3×Dic3—62no—
416C3×Dic3—62no—
417C3×Dic3—62no—
418C3×Dic3—122no—
419C3×Dic3—122no—
420#127—91no—
421#126—272no—
422#123—272no—
423#123—271no—
424#121—92no—
425#121—272no—
426#121—272no—
427#120—271no—
428#120—272no—
429C33#56011yescharacteristic, complemented, Sylow
430C23×C3—22no—
431C23×C3—22no—
432C23×C3—22no—
433C23×C3—42no—
434C23×C3—91no—
435C23×C3—92no—
436C23×C3—91no—
437C23×C3C2×D3212yes—
438C23×C3C2×D3212yes—
439C23×C3C2×D3212yes—
440C22×D3—32no—
441C22×D3—32no—
442C22×D3—32no—
443C22×D3—32no—
444C22×D3—62no—
445C22×D3—61no—
446C22×D3—61no—
447C22×D3—362no—
448C22×D3—362no—
449C22×D3—362no—
450C22×D3—362no—
451C22×D3—361no—
452C3×D4—42no—
453C3×D4—41no—
454C3×D4—42no—
455C3×D4—44no—
456C3×D4—44no—
457C3×D4—361no—
458C3×D4—362no—
459C3×D4—362no—
460C3×D4—362no—
461C3×D4—362no—
462C2×C3×C4—21no—
463C2×C3×C4—22no—
464C2×C3×C4—41no—
465C2×C3×C4—92no—
466C2×C3×C4—92no—
467C2×C3×C4—92no—
468C2×C3×C4—92no—
469C2×C3×C4—362no—
470C2×C3×C4C2×D3212yes—
471C2×C3×C4C2×D3211yescharacteristic
472#67—122no—
473#67—122no—
474#67—122no—
475#67—122no—
476#67—122no—
477#67—122no—
478#67—122no—
479#67—122no—
480#67—124no—
481#67—124no—
482#67—124no—
483#67—124no—
484#67—362no—
485#67—362no—
486C2×Dic3—32no—
487C2×Dic3—32no—
488C2×Dic3—32no—
489C2×Dic3—32no—
490C2×Dic3—32no—
491C2×Dic3—32no—
492C2×Dic3—32no—
493C2×Dic3—32no—
494C2×Dic3—62no—
495C2×Dic3—62no—
496C2×Dic3—62no—
497C2×Dic3—62no—
498C2×Dic3—122no—
499C2×Dic3—122no—
500C2×Dic3—124no—
501C2×Dic3—361no—
502D12—122no—
503D12—122no—
504D12—362no—
505C4×D3—362no—
506C2×C32—22no—
507C2×C32—22no—
508C2×C32—22no—
509C2×C32—22no—
510C2×C32—21no—
511C2×C32—41no—
512C2×C32—42no—
513C2×C32—42no—
514C2×C32—44no—
515C2×C32—42no—
516C2×C32—42no—
517C2×C32—82no—
518C2×C32—82no—
519C2×C32—84no—
520C2×C32#100412yes—
521C2×C32#100412yes—
522C2×C32#100411yescharacteristic
523C2×C32#100412yes—
524C2×C32#100912yes—
525C3⋊D3—41no—
526C3⋊D3—41no—
527C3⋊D3—42no—
528C3×D3—122no—
529C3×D3—122no—
530C3×D3—122no—
531C3×D3—122no—
532C3×D3—241no—
533C3×D3—241no—
534C3×D3—242no—
535C24—271no—
536C2×D4—92no—
537C2×D4—92no—
538C2×D4—92no—
539C2×D4—92no—
540C2×D4—91no—
541C2×D4—272no—
542C2×D4#111311yescharacteristic, complemented
543C22×C4—92no—
544C22×C4—271no—
545C4⋊C4—92no—
546C4⋊C4—272no—
547C4⋊C4—272no—
548C22⋊C4—92no—
549C22⋊C4—92no—
550C22⋊C4—272no—
551C22⋊C4—272no—
552C22⋊C4—271no—
553C22⋊C4—272no—
554C22⋊C4—271no—
555C42—91no—
556C22×C3—24no—
557C22×C3—22no—
558C22×C3—22no—
559C22×C3—22no—
560C22×C3—22no—
561C22×C3—22no—
562C22×C3—22no—
563C22×C3—22no—
564C22×C3—22no—
565C22×C3—22no—
566C22×C3—22no—
567C22×C3—21no—
568C22×C3—42no—
569C22×C3—42no—
570C22×C3—42no—
571C22×C3—42no—
572C22×C3—42no—
573C22×C3—42no—
574C22×C3—42no—
575C22×C3—42no—
576C22×C3—44no—
577C22×C3—41no—
578C22×C3—82no—
579C22×C3—82no—
580C22×C3—182no—
581C22×C3—182no—
582C22×C3—182no—
583C22×C3—182no—
584C22×C3—181no—
585C22×C3—181no—
586C22×C3—182no—
587C22×C3C22×D3212yes—
588C22×C3C22×D3211yescharacteristic
589C2×D3—62no—
590C2×D3—62no—
591C2×D3—62no—
592C2×D3—62no—
593C2×D3—62no—
594C2×D3—62no—
595C2×D3—62no—
596C2×D3—62no—
597C2×D3—62no—
598C2×D3—62no—
599C2×D3—62no—
600C2×D3—62no—
601C2×D3—121no—
602C2×D3—121no—
603C2×D3—122no—
604C2×D3—122no—
605C2×D3—122no—
606C2×D3—122no—
607C2×D3—122no—
608C2×D3—364no—
609C2×D3—368no—
610C3×C4—22no—
611C3×C4—22no—
612C3×C4—21no—
613C3×C4—42no—
614C3×C4—42no—
615C3×C4—182no—
616C3×C4—182no—
617C3×C4—182no—
618C3×C4—182no—
619Dic3—64no—
620Dic3—64no—
621Dic3—62no—
622Dic3—62no—
623Dic3—62no—
624Dic3—62no—
625Dic3—62no—
626Dic3—62no—
627Dic3—62no—
628Dic3—62no—
629Dic3—122no—
630Dic3—122no—
631C32—21no—
632C32—22no—
633C32—41no—
634C32#562012yescomplemented
635C32#572011yescharacteristic, complemented
636C23—92no—
637C23—91no—
638C23—91no—
639C23—1082no—
640C23—1081no—
641C23C2×#111312yes—
642D4—41no—
643D4—361no—
644D4—362no—
645D4—362no—
646D4—362no—
647D4—362no—
648D4—1082no—
649C2×C4—92no—
650C2×C4—92no—
651C2×C4—92no—
652C2×C4—92no—
653C2×C4—362no—
654C2×C4—1081no—
655C2×C4C2×#111311yescharacteristic
656C2×C3—22no—
657C2×C3—22no—
658C2×C3—22no—
659C2×C3—22no—
660C2×C3—21no—
661C2×C3—44no—
662C2×C3—42no—
663C2×C3—42no—
664C2×C3—42no—
665C2×C3—42no—
666C2×C3—41no—
667C2×C3—82no—
668C2×C3—82no—
669C2×C3—84no—
670C2×C3—361no—
671C2×C3—361no—
672C2×C3—362no—
673C2×C3#6431312yes—
674C2×C3#6431412yes—
675C2×C3#6431412yes—
676C2×C3#6433011yescharacteristic
677C2×C3#6433012yes—
678D3—122no—
679D3—122no—
680D3—122no—
681D3—122no—
682D3—241no—
683D3—242no—
684D3—241no—
685C22—22no—
686C22—22no—
687C22—22no—
688C22—182no—
689C22—182no—
690C22—182no—
691C22—182no—
692C22—181no—
693C22—181no—
694C22—182no—
695C22—1084no—
696C22C22×#111311yescharacteristic
697C4—21no—
698C4—182no—
699C4—182no—
700C4—182no—
701C4—182no—
702C3—21no—
703C3—22no—
704C3—41no—
705C3#1059531211yescharacteristic, complemented
706C3#1059531312yescomplemented
707C2—42no—
708C2—361no—
709C2—362no—
710C2—361no—
711C2#1172462312yes—
712C2#1172463311yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4#1113
hypocenterC33#560
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C3×#64149C2semidirect
2. C3×#64148C2semidirect
3. #11723729C2semidirect
4. #11723725C2semidirect
5. #11723720C2semidirect
6. #11723717C2semidirect
7. #11723701C2semidirect
8. #11723686C2semidirect
9. #11723676C2semidirect
13. C2×C33×D4C22semidirect
17. C2×C3×#638C22semidirect
18. C22×C3×C3⋊Dic3C22semidirect
21. C2×#6294C22semidirect
22. C3×#3757C22semidirect
24. C3×#3750C22semidirect
25. C3×C4×C3⋊Dic3C22semidirect
26. #88815C22semidirect
27. #88815C22semidirect
32. #88814C22semidirect
33. #64149D3semidirect
34. #64148D3semidirect
86. C2×C32×D4C2×D3semidirect
87. C2×C32×D4C2×D3semidirect
91. C2×#638C2×D3semidirect
92. C22×C3⋊Dic3C2×D3semidirect
116. #3757C2×D3semidirect
118. #3750C2×D3semidirect
119. C4×C3⋊Dic3C2×D3semidirect
286. C2×C3×D4D32semidirect
287. C2×C3×D4D32semidirect
429. C33#560semidirect
542. C2×D4#1113semidirect
634. C32#5620semidirect
635. C32#5720semidirect
705. C3#10595312semidirect
706. C3#10595313semidirect

Maximal subgroups

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

subgroupindex
C3×#641492
C3×#641482
#117237292
#117237252
#117237202
#117237172
#117237012
#117236862
#117236762
#105953133
#105953123