Finite Group Databasebeta

← #13178709

Subgroups — #13178709

Order 1728.

1336 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64148C212yesmaximal, complemented
2C3×#64147C211yesmaximal, characteristic, complemented
3#11723729C212yesmaximal, complemented
4#11723726C212yesmaximal, complemented
5#11723720C212yesmaximal, complemented
6#11723718C212yesmaximal, complemented
7#11723700C211yesmaximal, characteristic, complemented
8#11723675C212yesmaximal, complemented
9#11723671C211yesmaximal, characteristic, complemented
10#10595311—31nomaximal
11#10595310—32nomaximal
12C22×#1113C2211yescharacteristic
13C2×C33×D4C2211yescharacteristic, complemented
14C2×C3×#638C2212yes—
15C2×C3×#638C2212yes—
16C2×C3×#638C2212yes—
17C22×C3×C3⋊Dic3C2212yescomplemented
18C2×C3×C4×C3⋊D3C2211yescharacteristic
19C2×#6296C2211yescharacteristic, complemented
20C2×#6294C2211yescharacteristic
21C3×#3757C2212yescomplemented
22C3×#3757C2212yescomplemented
23C3×#3752C2212yes—
24C3×C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
25C3×#3750C2212yescomplemented
26#88815C2211yescharacteristic, complemented
27#88815C2212yescomplemented
28#88814C2212yes—
29#88814C2212yes—
30#88814C2212yes—
31#88814C2212yes—
32#88813C2212yescomplemented
33#64148D312yescomplemented
34#64147D311yescharacteristic, complemented
35C3×#939—32no—
36C3×#939—32no—
37C3×#939—62no—
38C3×#938—32no—
39C3×#938—61no—
40#63983—32no—
41#63983—32no—
42#63983—32no—
43#67⋊Dic3—32no—
44Dic3×#67—32no—
45#63966—32no—
46#63962—32no—
47#63961—32no—
48#63959—32no—
49#63959—32no—
50#63953—32no—
51#63952—32no—
52#63921—31no—
53D3×(C2×D3)⋊C4—32no—
54#63865—31no—
55#63861—32no—
56#63858—32no—
57#63851—32no—
58#63848—32no—
59C23×C33C2312yes—
60C22×C3×C3⋊D3C2312yes—
61C22×C3×C3⋊D3C2311yescharacteristic
62C2×#1113—22no—
63C2×#1113—41no—
64C2×#1113—41no—
65C33×D4—41no—
66C2×C33×C4C2311yescharacteristic
67C3×#638—24no—
68C3×#638—42no—
69C3×#638—42no—
70C2×C3×C3⋊Dic3—42no—
71C2×C3×C3⋊Dic3C2312yes—
72C2×C3×C3⋊Dic3C2312yes—
73C2×C3×C3⋊Dic3C2312yes—
74C2×C3×C3⋊Dic3C2312yes—
75C2×C3×C3⋊Dic3C2311yescharacteristic
76C3×C4×C3⋊D3—41no—
77#6296—41no—
78#6294—41no—
79#5617—91no—
80#5613—92no—
81C22×D32—32no—
82C22×D32—31no—
83C2×C32×D4—21no—
84C2×C32×D4—22no—
85C2×C32×D4—41no—
86C2×C32×D4C2×D312yescomplemented
87C2×C32×D4C2×D311yescharacteristic, complemented
88C2×#638C2×D312yes—
89C2×#638C2×D312yes—
90C2×#638C2×D312yes—
91C22×C3⋊Dic3C2×D312yescomplemented
92C2×C4×C3⋊D3C2×D311yescharacteristic
93C2×C3×#67—32no—
94C2×C3×#67—32no—
95C2×C3×#67—32no—
96C2×C3×#67—32no—
97C2×C3×#67—32no—
98C2×C3×#67—32no—
99C2×C3×#67—62no—
100C2×C3×#67—62no—
101C2×C3×#67—62no—
102C22×C3×Dic3—32no—
103C22×C3×Dic3—32no—
104C22×C3×Dic3—62no—
105C2×C3×C4×D3—32no—
106C2×C3×C4×D3—61no—
107C2×#627—32no—
108C2×#627—31no—
109C2×#624—31no—
110C2×D3×Dic3—32no—
111#3757C2×D312yescomplemented
112#3757C2×D312yescomplemented
113#3752C2×D312yes—
114C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
115#3750C2×D312yescomplemented
116C3×#217—32no—
117C3×#217—32no—
118C3×#217—32no—
119C3×#217—32no—
120C3×#217—62no—
121C3×#217—62no—
122C3×(C2×D3)⋊C4—32no—
123C3×(C2×D3)⋊C4—32no—
124C3×(C2×D3)⋊C4—62no—
125C3×C4⋊Dic3—32no—
126C3×C4⋊Dic3—61no—
127C3×Dic3⋊C4—32no—
128C3×Dic3⋊C4—32no—
129C3×Dic3⋊C4—62no—
130#3724—32no—
131#3724—31no—
132Dic3⋊Dic3—32no—
133Dic3⋊Dic3—32no—
134Dic3⋊Dic3—32no—
135#3722—32no—
136#3722—32no—
137#3722—32no—
138#3722—32no—
139(C2×D3)⋊Dic3—32no—
140(C2×D3)⋊Dic3—32no—
141(C2×D3)⋊Dic3—32no—
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—
147Dic32—32no—
148Dic32—32no—
149Dic32—32no—
150C22×C33—22no—
151C22×C33—22no—
152C22×C33—22no—
153C22×C33C2411yescharacteristic
154C2×C3×C3⋊D3—21no—
155C2×C3×C3⋊D3—21no—
156C2×C3×C3⋊D3—22no—
157C2×C3×C3⋊D3—22no—
158C2×C3×C3⋊D3—22no—
159C2×C3×C3⋊D3#4112yes—
160#1113—44no—
161C33×C4—21no—
162C3×C3⋊Dic3—22no—
163C3×C3⋊Dic3—22no—
164C3×C3⋊Dic3—22no—
165C3×C3⋊Dic3—21no—
166C3×C3⋊Dic3—22no—
167C3×#121—91no—
168C3×#121—92no—
169#939—32no—
170#939—32no—
171#939—62no—
172#938—32no—
173#938—61no—
174#937—92no—
175#937—92no—
176#937—92no—
177#936—92no—
178#935—92no—
179D4×Dic3—92no—
180#929—92no—
181C4×#67—92no—
182C2×(C2×D3)⋊C4—91no—
183#924—91no—
184#898—92no—
185#898—92no—
186#896—92no—
187#886—92no—
188#885—92no—
189#885—92no—
190#67⋊C4—92no—
191D3×C22⋊C4—92no—
192#879—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×D312yes—
201C22×C3⋊D3C22×D311yescharacteristic
202C22×C3×D3—32no—
203C22×C3×D3—32no—
204C22×C3×D3—32no—
205C22×C3×D3—61no—
206C22×C3×D3—62no—
207C2×D32—64no—
208C2×D32—62no—
209C2×D32—122no—
210C2×D32—122no—
211C2×D32—121no—
212C2×D32—121no—
213C32×D4—41no—
214C32×D4—42no—
215C32×D4—42no—
216C32×D4—44no—
217C32×D4—44no—
218C2×C32×C4—21no—
219C2×C32×C4—22no—
220C2×C32×C4—41no—
221C2×C32×C4C22×D311yescharacteristic
222C2×C32×C4C22×D312yes—
223#638—24no—
224#638—42no—
225#638—42no—
226C2×C3⋊Dic3—42no—
227C2×C3⋊Dic3C22×D312yes—
228C2×C3⋊Dic3C22×D312yes—
229C2×C3⋊Dic3C22×D312yes—
230C2×C3⋊Dic3C22×D312yes—
231C2×C3⋊Dic3C22×D311yescharacteristic
232C4×C3⋊D3—41no—
233C3×#67—64no—
234C3×#67—64no—
235C3×#67—124no—
236C3×#67—122no—
237C3×#67—122no—
238C3×#67—122no—
239C3×#67—122no—
240C3×#67—124no—
241C3×#67—124no—
242C2×C3×Dic3—32no—
243C2×C3×Dic3—32no—
244C2×C3×Dic3—32no—
245C2×C3×Dic3—32no—
246C2×C3×Dic3—32no—
247C2×C3×Dic3—32no—
248C2×C3×Dic3—32no—
249C2×C3×Dic3—32no—
250C2×C3×Dic3—32no—
251C2×C3×Dic3—62no—
252C2×C3×Dic3—62no—
253C2×C3×Dic3—62no—
254C2×C3×Dic3—62no—
255C2×C3×Dic3—61no—
256C2×C3×Dic3—122no—
257C2×C3×Dic3—122no—
258C2×C3×Dic3—124no—
259C3×C4×D3—122no—
260C3×C4×D3—122no—
261#627—122no—
262#627—121no—
263#624—121no—
264D3×Dic3—122no—
265#538—271noSylow
266C2×C33—42no—
267C2×C33C2×#4111yescharacteristic
268C2×C33H(5,𝔽2)11yescharacteristic
269C2×C33H(5,𝔽2)11yescharacteristic
270C3×C3⋊D3—22no—
271C3×C3⋊D3—42no—
272C23×D3—92no—
273C23×D3—91no—
274C2×C3×D4—21no—
275C2×C3×D4—22no—
276C2×C3×D4—41no—
277C2×C3×D4—92no—
278C2×C3×D4—92no—
279C2×C3×D4—92no—
280C2×C3×D4D3212yescomplemented
281C2×C3×D4D3211yescharacteristic, complemented
282C22×C3×C4—92no—
283C22×C3×C4—91no—
284C2×#67—32no—
285C2×#67—32no—
286C2×#67—32no—
287C2×#67—32no—
288C2×#67—32no—
289C2×#67—32no—
290C2×#67—62no—
291C2×#67—62no—
292C2×#67—62no—
293C22×Dic3—32no—
294C22×Dic3—32no—
295C22×Dic3—62no—
296C22×Dic3—91no—
297C2×C4×D3—32no—
298C2×C4×D3—61no—
299C2×C4×D3—92no—
300C2×Dic6—91no—
301C2×Dic6—92no—
302C3×C4⋊C4—92no—
303C3×C4⋊C4—91no—
304C3×C22⋊C4—92no—
305C3×C22⋊C4—92no—
306C3×C22⋊C4—92no—
307#217—32no—
308#217—32no—
309#217—32no—
310#217—32no—
311#217—62no—
312#217—62no—
313#217—92no—
314#217—92no—
315#217—92no—
316#217—92no—
317(C2×D3)⋊C4—32no—
318(C2×D3)⋊C4—32no—
319(C2×D3)⋊C4—62no—
320(C2×D3)⋊C4—92no—
321(C2×D3)⋊C4—92no—
322(C2×D3)⋊C4—92no—
323(C2×D3)⋊C4—92no—
324(C2×D3)⋊C4—92no—
325(C2×D3)⋊C4—92no—
326(C2×D3)⋊C4—92no—
327(C2×D3)⋊C4—92no—
328C4⋊Dic3—32no—
329C4⋊Dic3—61no—
330C4⋊Dic3—91no—
331C4⋊Dic3—92no—
332Dic3⋊C4—32no—
333Dic3⋊C4—32no—
334Dic3⋊C4—62no—
335Dic3⋊C4—92no—
336Dic3⋊C4—92no—
337Dic3⋊C4—92no—
338C4×Dic3—92no—
339C4×Dic3—92no—
340C4×Dic3—92no—
341C22×C32—22no—
342C22×C32—24no—
343C22×C32—22no—
344C22×C32—22no—
345C22×C32—22no—
346C22×C32—22no—
347C22×C32—22no—
348C22×C32—22no—
349C22×C32—22no—
350C22×C32—22no—
351C22×C32—22no—
352C22×C32—21no—
353C22×C32—41no—
354C22×C32—42no—
355C22×C32—42no—
356C22×C32—44no—
357C22×C32—42no—
358C22×C32—42no—
359C22×C32—42no—
360C22×C32—42no—
361C22×C32—42no—
362C22×C32—42no—
363C22×C32—82no—
364C22×C32—82no—
365C22×C32C23×D311yescharacteristic
366C22×C32C23×D312yes—
367C2×C3⋊D3—21no—
368C2×C3⋊D3—21no—
369C2×C3⋊D3—22no—
370C2×C3⋊D3—22no—
371C2×C3⋊D3—22no—
372C2×C3⋊D3#23712yes—
373C2×C3×D3—34no—
374C2×C3×D3—62no—
375C2×C3×D3—62no—
376C2×C3×D3—62no—
377C2×C3×D3—62no—
378C2×C3×D3—62no—
379C2×C3×D3—62no—
380C2×C3×D3—62no—
381C2×C3×D3—62no—
382C2×C3×D3—62no—
383C2×C3×D3—121no—
384C2×C3×D3—121no—
385C2×C3×D3—122no—
386C2×C3×D3—122no—
387C2×C3×D3—122no—
388D32—128no—
389D32—124no—
390C32×C4—21no—
391C32×C4—22no—
392C32×C4—22no—
393C32×C4—42no—
394C32×C4—42no—
395C3⋊Dic3—22no—
396C3⋊Dic3—22no—
397C3⋊Dic3—22no—
398C3⋊Dic3—21no—
399C3⋊Dic3—22no—
400C3×Dic3—64no—
401C3×Dic3—62no—
402C3×Dic3—62no—
403C3×Dic3—62no—
404C3×Dic3—62no—
405C3×Dic3—62no—
406C3×Dic3—62no—
407C3×Dic3—62no—
408C3×Dic3—62no—
409C3×Dic3—62no—
410C3×Dic3—121no—
411C3×Dic3—122no—
412C3×Dic3—122no—
413C3×Dic3—122no—
414#126—272no—
415#124—272no—
416#123—272no—
417C22⋊Q8—271no—
418#121—91no—
419#121—92no—
420#120—272no—
421C4×D4—272no—
422C2×C22⋊C4—271no—
423C33#53811yescharacteristic, complemented, Sylow
424C23×C3—22no—
425C23×C3—22no—
426C23×C3—22no—
427C23×C3—42no—
428C23×C3—91no—
429C23×C3—92no—
430C23×C3C2×D3212yes—
431C23×C3C2×D3212yes—
432C23×C3C2×D3212yes—
433C22×D3—32no—
434C22×D3—32no—
435C22×D3—32no—
436C22×D3—61no—
437C22×D3—62no—
438C22×D3—184no—
439C22×D3—182no—
440C22×D3—362no—
441C22×D3—362no—
442C22×D3—361no—
443C22×D3—361no—
444C3×D4—42no—
445C3×D4—41no—
446C3×D4—42no—
447C3×D4—44no—
448C3×D4—44no—
449C3×D4—184no—
450C3×D4—362no—
451C3×D4—362no—
452C2×C3×C4—21no—
453C2×C3×C4—22no—
454C2×C3×C4—41no—
455C2×C3×C4—92no—
456C2×C3×C4—92no—
457C2×C3×C4—92no—
458C2×C3×C4—92no—
459C2×C3×C4—91no—
460C2×C3×C4—361no—
461C2×C3×C4—362no—
462C2×C3×C4C2×D3211yescharacteristic
463C2×C3×C4C2×D3212yes—
464#67—64no—
465#67—64no—
466#67—124no—
467#67—122no—
468#67—122no—
469#67—122no—
470#67—122no—
471#67—124no—
472#67—124no—
473C2×Dic3—32no—
474C2×Dic3—32no—
475C2×Dic3—32no—
476C2×Dic3—32no—
477C2×Dic3—32no—
478C2×Dic3—32no—
479C2×Dic3—32no—
480C2×Dic3—32no—
481C2×Dic3—32no—
482C2×Dic3—62no—
483C2×Dic3—62no—
484C2×Dic3—62no—
485C2×Dic3—62no—
486C2×Dic3—61no—
487C2×Dic3—122no—
488C2×Dic3—122no—
489C2×Dic3—124no—
490C2×Dic3—361no—
491C4×D3—122no—
492C4×D3—122no—
493C4×D3—362no—
494Dic6—362no—
495Dic6—361no—
496C2×C32—22no—
497C2×C32—21no—
498C2×C32—21no—
499C2×C32—22no—
500C2×C32—21no—
501C2×C32—22no—
502C2×C32—41no—
503C2×C32—41no—
504C2×C32—41no—
505C2×C32—42no—
506C2×C32—44no—
507C2×C32—42no—
508C2×C32—42no—
509C2×C32—82no—
510C2×C32—82no—
511C2×C32—84no—
512C2×C32C2×#23711yescharacteristic
513C2×C32#100412yes—
514C2×C32#100412yes—
515C2×C32#100411yescharacteristic
516C2×C32D3×#4112yes—
517C2×C32#100911yescharacteristic
518C3⋊D3—22no—
519C3⋊D3—42no—
520C3×D3—64no—
521C3×D3—122no—
522C3×D3—122no—
523C3×D3—122no—
524C3×D3—242no—
525C24—271no—
526C2×Q8—271no—
527C2×D4—92no—
528C2×D4—92no—
529C2×D4—92no—
530C2×D4#111311yescharacteristic, complemented
531C22×C4—92no—
532C22×C4—91no—
533C22×C4—271no—
534C4⋊C4—92no—
535C4⋊C4—91no—
536C4⋊C4—272no—
537C4⋊C4—271no—
538C22⋊C4—92no—
539C22⋊C4—92no—
540C22⋊C4—92no—
541C22⋊C4—272no—
542C22⋊C4—272no—
543C22⋊C4—272no—
544C22⋊C4—272no—
545C42—272no—
546C22×C3—24no—
547C22×C3—22no—
548C22×C3—22no—
549C22×C3—22no—
550C22×C3—22no—
551C22×C3—22no—
552C22×C3—22no—
553C22×C3—22no—
554C22×C3—22no—
555C22×C3—22no—
556C22×C3—21no—
557C22×C3—22no—
558C22×C3—42no—
559C22×C3—42no—
560C22×C3—42no—
561C22×C3—42no—
562C22×C3—42no—
563C22×C3—42no—
564C22×C3—42no—
565C22×C3—42no—
566C22×C3—44no—
567C22×C3—41no—
568C22×C3—82no—
569C22×C3—82no—
570C22×C3—92no—
571C22×C3—181no—
572C22×C3—181no—
573C22×C3—182no—
574C22×C3—182no—
575C22×C3—182no—
576C22×C3C22×D3212yes—
577C22×C3C22×D3211yescharacteristic
578C2×D3—34no—
579C2×D3—62no—
580C2×D3—62no—
581C2×D3—62no—
582C2×D3—62no—
583C2×D3—62no—
584C2×D3—62no—
585C2×D3—62no—
586C2×D3—62no—
587C2×D3—62no—
588C2×D3—121no—
589C2×D3—121no—
590C2×D3—122no—
591C2×D3—122no—
592C2×D3—122no—
593C2×D3—368no—
594C2×D3—364no—
595C3×C4—22no—
596C3×C4—22no—
597C3×C4—21no—
598C3×C4—42no—
599C3×C4—42no—
600C3×C4—182no—
601C3×C4—182no—
602C3×C4—182no—
603C3×C4—181no—
604C3×C4—182no—
605Dic3—64no—
606Dic3—62no—
607Dic3—62no—
608Dic3—62no—
609Dic3—62no—
610Dic3—62no—
611Dic3—62no—
612Dic3—62no—
613Dic3—62no—
614Dic3—62no—
615Dic3—121no—
616Dic3—122no—
617Dic3—122no—
618Dic3—122no—
619C32—22no—
620C32—21no—
621C32—41no—
622C32#561312yescomplemented
623C32#561711yescharacteristic, complemented
624C23—91no—
625C23—92no—
626C23—542no—
627C23—1081no—
628C23—1081no—
629C23C2×#111312yes—
630Q8—1081no—
631D4—41no—
632D4—184no—
633D4—362no—
634D4—362no—
635C2×C4—92no—
636C2×C4—92no—
637C2×C4—92no—
638C2×C4—91no—
639C2×C4—92no—
640C2×C4—361no—
641C2×C4—362no—
642C2×C4—1081no—
643C2×C4C2×#111311yescharacteristic
644C2×C3—21no—
645C2×C3—22no—
646C2×C3—22no—
647C2×C3—21no—
648C2×C3—21no—
649C2×C3—22no—
650C2×C3—44no—
651C2×C3—42no—
652C2×C3—42no—
653C2×C3—42no—
654C2×C3—41no—
655C2×C3—41no—
656C2×C3—41no—
657C2×C3—82no—
658C2×C3—82no—
659C2×C3—84no—
660C2×C3—182no—
661C2×C3—362no—
662C2×C3D3×#23712yes—
663C2×C3#6431211yescharacteristic
664C2×C3#6431311yescharacteristic
665C2×C3#6431412yes—
666C2×C3#6433012yes—
667C2×C3#6433011yescharacteristic
668D3—64no—
669D3—122no—
670D3—122no—
671D3—122no—
672D3—242no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—92no—
677C22—181no—
678C22—181no—
679C22—182no—
680C22—182no—
681C22—182no—
682C22—1084no—
683C22C22×#111311yescharacteristic
684C4—21no—
685C4—182no—
686C4—182no—
687C4—182no—
688C4—181no—
689C4—182no—
690C3—21no—
691C3—22no—
692C3—41no—
693C3#1059531012yescomplemented
694C3#1059531111yescharacteristic, complemented
695C2—42no—
696C2—182no—
697C2—362no—
698C2#1172462211yescharacteristic
699C2#1172462311yescharacteristic
700C2#1172463311yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4#1113
hypocenterC33#538
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C3×#64148C2semidirect
2. C3×#64147C2semidirect
3. #11723729C2semidirect
4. #11723726C2semidirect
5. #11723720C2semidirect
6. #11723718C2semidirect
7. #11723700C2semidirect
8. #11723675C2semidirect
9. #11723671C2semidirect
13. C2×C33×D4C22semidirect
17. C22×C3×C3⋊Dic3C22semidirect
19. C2×#6296C22semidirect
21. C3×#3757C22semidirect
22. C3×#3757C22semidirect
24. C3×C4⋊(C3⋊Dic3)C22semidirect
25. C3×#3750C22semidirect
26. #88815C22semidirect
27. #88815C22semidirect
32. #88813C22semidirect
33. #64148D3semidirect
34. #64147D3semidirect
86. C2×C32×D4C2×D3semidirect
87. C2×C32×D4C2×D3semidirect
91. C22×C3⋊Dic3C2×D3semidirect
111. #3757C2×D3semidirect
112. #3757C2×D3semidirect
114. C4⋊(C3⋊Dic3)C2×D3semidirect
115. #3750C2×D3semidirect
280. C2×C3×D4D32semidirect
281. C2×C3×D4D32semidirect
423. C33#538semidirect
530. C2×D4#1113semidirect
622. C32#5613semidirect
623. C32#5617semidirect
693. C3#10595310semidirect
694. C3#10595311semidirect

Maximal subgroups

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

subgroupindex
C3×#641482
C3×#641472
#117237292
#117237262
#117237202
#117237182
#117237002
#117236752
#117236712
#105953113
#105953103