Finite Group Databasebeta

← #13178627

Subgroups — #13178627

Order 1728.

1814 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×C3⋊D3C211yesmaximal, characteristic
2C22×#6294C211yesmaximal, characteristic, complemented
3C3×#64144C212yesmaximal, complemented
4#11723724C212yesmaximal, complemented
5#11723723C211yesmaximal, characteristic, complemented
6#11723723C212yesmaximal, complemented
7#11723720C212yesmaximal, complemented
8#11723686C212yesmaximal, complemented
9#11723682C212yesmaximal, complemented
10#10595169—31nomaximal
11D3×#935—32nomaximal
12C23×C3×C3⋊D3C2211yescharacteristic
13C23×C3×C3⋊D3C2211yescharacteristic
14C2×C33×D4C2211yescharacteristic
15C2×C3×#638C2211yescharacteristic
16C2×C3×#638C2211yescharacteristic
17C22×C3×C3⋊Dic3C2212yescomplemented
18C3×D4×C3⋊D3—24no—
19C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
20C2×C3×C4×C3⋊D3C2211yescharacteristic
21C2×#6294—24no—
22C2×#6294C2212yes—
23C2×#6294C2212yes—
24C2×#6294C2212yescomplemented
25C3×#3757C2212yescomplemented
26C3×#3757C2214yescomplemented
27C3×#3750C2214yescomplemented
28#88815C2214yescomplemented
29#88814C2214yescomplemented
30#88814C2212yescomplemented
31C2×D4×C3⋊D3D311yescharacteristic
32C2×C3×D3×D4—32no—
33C2×C3×D3×D4—61no—
34C22×#624—31no—
35C22×D3×Dic3—32no—
36#64144D312yescomplemented
37C3×#935—32no—
38C3×#935—32no—
39C3×#935—62no—
40#63975—32no—
41#63974—32no—
42#63974—31no—
43#63969—34no—
44D3×#217—32no—
45D3×#217—34no—
46#63966—32no—
47#63962—32no—
48#63961—32no—
49#63897—32no—
50#63884—34no—
51#63883—32no—
52D3×Dic3⋊C4—34no—
53C23×C33C2311yescharacteristic
54C23×C33C2311yescharacteristic
55C22×C3×C3⋊D3—24no—
56C22×C3×C3⋊D3—22no—
57C22×C3×C3⋊D3—24no—
58C22×C3×C3⋊D3D414yes—
59C22×C3×C3⋊D3C2311yescharacteristic
60C22×C3×C3⋊D3C2311yescharacteristic
61C22×C3×C3⋊D3C2311yescharacteristic
62C33×D4—22no—
63C2×C33×C4C2311yescharacteristic
64C3×#638—22no—
65C3×#638—22no—
66C2×C3×C3⋊Dic3—24no—
67C2×C3×C3⋊Dic3C2314yes—
68C2×C3×C3⋊Dic3C2314yes—
69C2×C3×C3⋊Dic3C2311yescharacteristic
70C3×C4⋊(C3⋊D3)—22no—
71C3×C4×C3⋊D3—22no—
72#6294—24no—
73#6294—24no—
74#6294—24no—
75C2×#935—91no—
76D3×#123—92no—
77C23×C3⋊D3C2×D311yescharacteristic
78C23×C3⋊D3C2×D311yescharacteristic
79C23×C3×D3—32no—
80C23×C3×D3—32no—
81C23×C3×D3—61no—
82C23×C3×D3—61no—
83C2×C32×D4—21no—
84C2×C32×D4—22no—
85C2×C32×D4—41no—
86C2×C32×D4C2×D311yescharacteristic
87C2×C32×D4C2×D312yes—
88C2×#638C2×D311yescharacteristic
89C2×#638C2×D311yescharacteristic
90C22×C3⋊Dic3C2×D312yescomplemented
91D4×C3⋊D3—24no—
92C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
93C2×C4×C3⋊D3C2×D311yescharacteristic
94C2×C3×#67—32no—
95C2×C3×#67—32no—
96C2×C3×#67—61no—
97C2×C3×#67—61no—
98C22×C3×Dic3—32no—
99C22×C3×Dic3—32no—
100C22×C3×Dic3—62no—
101C3×D3×D4—68no—
102C3×D3×D4—68no—
103C2×C3×D12—32no—
104C2×C3×D12—61no—
105C2×C3×C4×D3—32no—
106C2×C3×C4×D3—61no—
107C2×#624—32no—
108C2×#624—32no—
109C2×#624—32no—
110C2×#624—64no—
111C2×D3×Dic3—34no—
112C2×D3×Dic3—34no—
113C2×D3×Dic3—34no—
114C2×D3×Dic3—68no—
115#3757C2×D312yescomplemented
116#3757C2×D314yescomplemented
117#3750C2×D314yescomplemented
118C3×#217—32no—
119C3×#217—32no—
120C3×#217—34no—
121C3×#217—34no—
122C3×#217—62no—
123C3×#217—64no—
124C3×Dic3⋊C4—34no—
125C3×Dic3⋊C4—34no—
126C3×Dic3⋊C4—64no—
127#3724—34no—
128Dic3⋊Dic3—34no—
129Dic3⋊Dic3—34no—
130#3722—34no—
131#3722—32no—
132(C2×D3)⋊Dic3—34no—
133(C2×D3)⋊Dic3—34no—
134(C2×D3)⋊Dic3—34no—
135C22×C33—22no—
136C22×C33—21no—
137C22×C33—22no—
138C22×C33C2×D412yes—
139C22×C33C2411yescharacteristic
140C2×C3×C3⋊D3—22no—
141C2×C3×C3⋊D3—28no—
142C2×C3×C3⋊D3—21no—
143C2×C3×C3⋊D3—22no—
144C2×C3×C3⋊D3—44no—
145C2×C3×C3⋊D3C2×D412yes—
146C2×C3×C3⋊D3C2×D412yes—
147C2×C3×C3⋊D3#4114yes—
148C33×C4—21no—
149C3×C3⋊Dic3—21no—
150C3×C3⋊Dic3—24no—
151C3×C3⋊Dic3—24no—
152C22×C3×D4—91no—
153C23×Dic3—91no—
154C2×D3×D4—32no—
155C2×D3×D4—61no—
156C22×C4×D3—92no—
157C3×#123—92no—
158C2×#217—91no—
159C2×#217—92no—
160#935—32no—
161#935—32no—
162#935—62no—
163#935—92no—
164#935—92no—
165#935—92no—
166#929—92no—
167C2×Dic3⋊C4—92no—
168#894—94no—
169D3×C4⋊C4—94no—
170#886—92no—
171#883—94no—
172D3×C22⋊C4—92no—
173D3×C22⋊C4—94no—
174C23×C32—21no—
175C23×C32—21no—
176C23×C32—22no—
177C23×C32—22no—
178C23×C32—41no—
179C23×C32—41no—
180C23×C32C22×D312yes—
181C23×C32C22×D311yescharacteristic
182C23×C32C22×D312yes—
183C23×C32C22×D311yescharacteristic
184C22×C3⋊D3—22no—
185C22×C3⋊D3—24no—
186C22×C3⋊D3—24no—
187C22×C3⋊D3#6714yes—
188C22×C3⋊D3C22×D311yescharacteristic
189C22×C3⋊D3C22×D311yescharacteristic
190C22×C3⋊D3C22×D311yescharacteristic
191C22×C3×D3—32no—
192C22×C3×D3—38no—
193C22×C3×D3—32no—
194C22×C3×D3—32no—
195C22×C3×D3—61no—
196C22×C3×D3—64no—
197C22×C3×D3—61no—
198C22×C3×D3—61no—
199C22×C3×D3—68no—
200C22×C3×D3—68no—
201C22×C3×D3—64no—
202C22×C3×D3—64no—
203C22×C3×D3—124no—
204C22×C3×D3—124no—
205C32×D4—22no—
206C32×D4—24no—
207C32×D4—24no—
208C32×D4—44no—
209C32×D4—44no—
210C2×C32×C4—21no—
211C2×C32×C4—22no—
212C2×C32×C4—41no—
213C2×C32×C4C22×D311yescharacteristic
214C2×C32×C4C22×D312yes—
215#638—22no—
216#638—22no—
217C2×C3⋊Dic3—24no—
218C2×C3⋊Dic3C22×D314yes—
219C2×C3⋊Dic3C22×D314yes—
220C2×C3⋊Dic3C22×D311yescharacteristic
221C4⋊(C3⋊D3)—22no—
222C4×C3⋊D3—22no—
223C3×#67—64no—
224C3×#67—64no—
225C3×#67—64no—
226C3×#67—64no—
227C2×C3×Dic3—34no—
228C2×C3×Dic3—34no—
229C2×C3×Dic3—34no—
230C2×C3×Dic3—34no—
231C2×C3×Dic3—32no—
232C2×C3×Dic3—61no—
233C2×C3×Dic3—64no—
234C2×C3×Dic3—64no—
235C2×C3×Dic3—64no—
236C2×C3×Dic3—64no—
237C2×C3×Dic3—124no—
238C3×D12—64no—
239C3×D12—64no—
240C3×C4×D3—64no—
241C3×C4×D3—64no—
242#624—64no—
243#624—64no—
244#624—64no—
245D3×Dic3—68no—
246D3×Dic3—68no—
247D3×Dic3—68no—
248C2×#123—271noSylow
249C2×C33—22no—
250C2×C33—41no—
251C2×C33C22×D411yescharacteristic
252C2×C33C2×#4112yes—
253C3×C3⋊D3—22no—
254C3×C3⋊D3—41no—
255C3×C3⋊D3#12314yescomplemented
256C24×C3—91no—
257C24×C3—91no—
258C23×D3—32no—
259C23×D3—32no—
260C23×D3—61no—
261C23×D3—61no—
262C2×C3×D4—21no—
263C2×C3×D4—22no—
264C2×C3×D4—41no—
265C2×C3×D4—91no—
266C2×C3×D4—91no—
267C2×C3×D4—91no—
268C2×C3×D4—184no—
269C2×C3×D4D3211yescharacteristic
270C2×C3×D4D3212yes—
271C22×C3×C4—92no—
272C22×C3×C4—91no—
273C2×#67—32no—
274C2×#67—32no—
275C2×#67—61no—
276C2×#67—61no—
277C22×Dic3—32no—
278C22×Dic3—32no—
279C22×Dic3—62no—
280C22×Dic3—92no—
281C22×Dic3—92no—
282C22×Dic3—92no—
283C22×Dic3—184no—
284D3×D4—68no—
285D3×D4—68no—
286C2×D12—32no—
287C2×D12—61no—
288C2×C4×D3—32no—
289C2×C4×D3—61no—
290C2×C4×D3—94no—
291C2×C4×D3—94no—
292C2×C4×D3—94no—
293C2×C4×D3—188no—
294C3×C4⋊C4—94no—
295C3×C22⋊C4—94no—
296C3×C22⋊C4—92no—
297#217—34no—
298#217—32no—
299#217—32no—
300#217—34no—
301#217—62no—
302#217—64no—
303#217—92no—
304#217—94no—
305(C2×D3)⋊C4—94no—
306(C2×D3)⋊C4—94no—
307(C2×D3)⋊C4—94no—
308C4⋊Dic3—94no—
309Dic3⋊C4—34no—
310Dic3⋊C4—34no—
311Dic3⋊C4—64no—
312Dic3⋊C4—94no—
313Dic3⋊C4—94no—
314C22×C32—21no—
315C22×C32—22no—
316C22×C32—24no—
317C22×C32—22no—
318C22×C32—22no—
319C22×C32—22no—
320C22×C32—24no—
321C22×C32—22no—
322C22×C32—21no—
323C22×C32—24no—
324C22×C32—22no—
325C22×C32—41no—
326C22×C32—42no—
327C22×C32—42no—
328C22×C32—42no—
329C22×C32—42no—
330C22×C32—42no—
331C22×C32—44no—
332C22×C32—44no—
333C22×C32—82no—
334C22×C32—82no—
335C22×C32D3×D414yes—
336C22×C32C2×#6712yes—
337C22×C32C23×D312yes—
338C22×C32C23×D311yescharacteristic
339C2×C3⋊D3—21no—
340C2×C3⋊D3—22no—
341C2×C3⋊D3—28no—
342C2×C3⋊D3—22no—
343C2×C3⋊D3—44no—
344C2×C3⋊D3#23714yes—
345C2×C3⋊D3C2×#6712yes—
346C2×C3⋊D3C2×#6712yes—
347C2×C3×D3—34no—
348C2×C3×D3—38no—
349C2×C3×D3—34no—
350C2×C3×D3—62no—
351C2×C3×D3—64no—
352C2×C3×D3—62no—
353C2×C3×D3—616no—
354C2×C3×D3—64no—
355C2×C3×D3—62no—
356C2×C3×D3—62no—
357C2×C3×D3—64no—
358C2×C3×D3—128no—
359C2×C3×D3—122no—
360C2×C3×D3—122no—
361C2×C3×D3—128no—
362C2×C3×D3—128no—
363C32×C4—21no—
364C32×C4—22no—
365C32×C4—22no—
366C32×C4—42no—
367C32×C4—42no—
368C3⋊Dic3—21no—
369C3⋊Dic3—24no—
370C3⋊Dic3—24no—
371C3×Dic3—62no—
372C3×Dic3—64no—
373C3×Dic3—62no—
374C3×Dic3—64no—
375C3×Dic3—64no—
376C3×Dic3—64no—
377C3×Dic3—124no—
378C3×Dic3—124no—
379C22×D4—91no—
380C23×C4—271no—
381#123—92no—
382#123—272no—
383#123—272no—
384#123—272no—
385C2×C4⋊C4—272no—
386C2×C22⋊C4—271no—
387C2×C22⋊C4—272no—
388C33C2×#12311yescharacteristic, complemented, Sylow
389C23×C3—22no—
390C23×C3—21no—
391C23×C3—21no—
392C23×C3—22no—
393C23×C3—41no—
394C23×C3—41no—
395C23×C3—91no—
396C23×C3—91no—
397C23×C3—94no—
398C23×C3—91no—
399C23×C3—184no—
400C23×C3—182no—
401C23×C3—184no—
402C23×C3C2×D3211yescharacteristic
403C23×C3C2×D3212yes—
404C23×C3C2×D3212yes—
405C23×C3C2×D3211yescharacteristic
406C22×D3—32no—
407C22×D3—32no—
408C22×D3—38no—
409C22×D3—32no—
410C22×D3—61no—
411C22×D3—64no—
412C22×D3—61no—
413C22×D3—61no—
414C22×D3—68no—
415C22×D3—64no—
416C22×D3—68no—
417C22×D3—64no—
418C22×D3—124no—
419C22×D3—124no—
420C3×D4—24no—
421C3×D4—24no—
422C3×D4—22no—
423C3×D4—44no—
424C3×D4—44no—
425C3×D4—182no—
426C3×D4—182no—
427C3×D4—182no—
428C2×C3×C4—21no—
429C2×C3×C4—22no—
430C2×C3×C4—41no—
431C2×C3×C4—91no—
432C2×C3×C4—94no—
433C2×C3×C4—94no—
434C2×C3×C4—182no—
435C2×C3×C4—184no—
436C2×C3×C4C2×D3211yescharacteristic
437C2×C3×C4C2×D3212yes—
438#67—64no—
439#67—64no—
440#67—64no—
441#67—64no—
442C2×Dic3—34no—
443C2×Dic3—34no—
444C2×Dic3—34no—
445C2×Dic3—34no—
446C2×Dic3—32no—
447C2×Dic3—61no—
448C2×Dic3—64no—
449C2×Dic3—64no—
450C2×Dic3—64no—
451C2×Dic3—64no—
452C2×Dic3—124no—
453C2×Dic3—184no—
454C2×Dic3—184no—
455C2×Dic3—184no—
456D12—64no—
457D12—64no—
458C4×D3—64no—
459C4×D3—64no—
460C4×D3—188no—
461C4×D3—188no—
462C4×D3—188no—
463C2×C32—24no—
464C2×C32—22no—
465C2×C32—24no—
466C2×C32—22no—
467C2×C32—22no—
468C2×C32—21no—
469C2×C32—41no—
470C2×C32—42no—
471C2×C32—42no—
472C2×C32—44no—
473C2×C32—41no—
474C2×C32—42no—
475C2×C32—42no—
476C2×C32—82no—
477C2×C32—82no—
478C2×C32—82no—
479C2×C32C2×D3×D412yes—
480C2×C32C2×#23712yes—
481C2×C32D3×#4114yes—
482C2×C32C22×#6711yescharacteristic
483C3⋊D3—22no—
484C3⋊D3—41no—
485C3⋊D3#93514yescomplemented
486C3×D3—38no—
487C3×D3—64no—
488C3×D3—64no—
489C3×D3—122no—
490C3×D3—122no—
491C3×D3—122no—
492C24—91no—
493C24—91no—
494C2×D4—91no—
495C2×D4—91no—
496C2×D4—91no—
497C2×D4—184no—
498C2×D4#111311yescharacteristic
499C22×C4—92no—
500C22×C4—91no—
501C22×C4—272no—
502C22×C4—272no—
503C22×C4—272no—
504C22×C4—544no—
505C4⋊C4—94no—
506C4⋊C4—274no—
507C22⋊C4—92no—
508C22⋊C4—94no—
509C22⋊C4—272no—
510C22⋊C4—274no—
511C22×C3—24no—
512C22×C3—22no—
513C22×C3—22no—
514C22×C3—24no—
515C22×C3—22no—
516C22×C3—22no—
517C22×C3—21no—
518C22×C3—22no—
519C22×C3—21no—
520C22×C3—24no—
521C22×C3—22no—
522C22×C3—42no—
523C22×C3—42no—
524C22×C3—44no—
525C22×C3—42no—
526C22×C3—44no—
527C22×C3—42no—
528C22×C3—41no—
529C22×C3—42no—
530C22×C3—82no—
531C22×C3—82no—
532C22×C3—94no—
533C22×C3—92no—
534C22×C3—92no—
535C22×C3—188no—
536C22×C3—182no—
537C22×C3—182no—
538C22×C3—181no—
539C22×C3—364no—
540C22×C3D3×#6714yes—
541C22×C3#381112yes—
542C22×C3C22×D3212yes—
543C22×C3C22×D3211yescharacteristic
544C2×D3—38no—
545C2×D3—34no—
546C2×D3—34no—
547C2×D3—62no—
548C2×D3—64no—
549C2×D3—62no—
550C2×D3—616no—
551C2×D3—64no—
552C2×D3—62no—
553C2×D3—64no—
554C2×D3—62no—
555C2×D3—128no—
556C2×D3—122no—
557C2×D3—122no—
558C2×D3—128no—
559C2×D3—128no—
560C3×C4—22no—
561C3×C4—22no—
562C3×C4—21no—
563C3×C4—42no—
564C3×C4—42no—
565C3×C4—184no—
566C3×C4—184no—
567C3×C4—181no—
568Dic3—62no—
569Dic3—64no—
570Dic3—64no—
571Dic3—62no—
572Dic3—64no—
573Dic3—64no—
574Dic3—124no—
575Dic3—124no—
576C32—22no—
577C32—21no—
578C32—41no—
579C32D3×#12312yescomplemented
580C32C2×#93511yescharacteristic, complemented
581C23—91no—
582C23—91no—
583C23—94no—
584C23—91no—
585C23—184no—
586C23—182no—
587C23—184no—
588C23C2×#111311yescharacteristic
589C23C2×#111311yescharacteristic
590D4—22no—
591D4—182no—
592D4—182no—
593D4—182no—
594C2×C4—94no—
595C2×C4—94no—
596C2×C4—91no—
597C2×C4—182no—
598C2×C4—184no—
599C2×C4—544no—
600C2×C4—544no—
601C2×C4—544no—
602C2×C4C2×#111311yescharacteristic
603C2×C3—24no—
604C2×C3—22no—
605C2×C3—24no—
606C2×C3—22no—
607C2×C3—21no—
608C2×C3—22no—
609C2×C3—42no—
610C2×C3—44no—
611C2×C3—42no—
612C2×C3—42no—
613C2×C3—41no—
614C2×C3—42no—
615C2×C3—41no—
616C2×C3—82no—
617C2×C3—82no—
618C2×C3—82no—
619C2×C3—94no—
620C2×C3—182no—
621C2×C3—361no—
622C2×C3D3×#23714yes—
623C2×C3#6431212yes—
624C2×C3C2×D3×#6712yes—
625C2×C3C2×#381111yescharacteristic
626D3—38no—
627D3—64no—
628D3—64no—
629D3—122no—
630D3—122no—
631D3—122no—
632C22—22no—
633C22—22no—
634C22—21no—
635C22—94no—
636C22—92no—
637C22—92no—
638C22—188no—
639C22—182no—
640C22—182no—
641C22—181no—
642C22—364no—
643C22#8904912yes—
644C22C22×#111311yescharacteristic
645C4—21no—
646C4—184no—
647C4—184no—
648C4—181no—
649C3—21no—
650C3—22no—
651C3—41no—
652C3D3×#93512yescomplemented
653C3#1059516911yescharacteristic, complemented
654C2—22no—
655C2—41no—
656C2—94no—
657C2—182no—
658C2—361no—
659C2#1172462212yes—
660C2C2×#8904911yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
prime radicalC2×C3×D4×C3⋊D3C2
hypercenterC2×D4#1113
hypocenterC33C2×#123
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C22×#6294C2semidirect
3. C3×#64144C2semidirect
4. #11723724C2semidirect
5. #11723723C2semidirect
6. #11723723C2semidirect
7. #11723720C2semidirect
8. #11723686C2semidirect
9. #11723682C2semidirect
17. C22×C3×C3⋊Dic3C22semidirect
24. C2×#6294C22semidirect
25. C3×#3757C22semidirect
26. C3×#3757C22semidirect
27. C3×#3750C22semidirect
28. #88815C22semidirect
29. #88814C22semidirect
30. #88814C22semidirect
36. #64144D3semidirect
90. C22×C3⋊Dic3C2×D3semidirect
115. #3757C2×D3semidirect
116. #3757C2×D3semidirect
117. #3750C2×D3semidirect
255. C3×C3⋊D3#123semidirect
388. C33C2×#123semidirect
485. C3⋊D3#935semidirect
579. C32D3×#123semidirect
580. C32C2×#935semidirect
652. C3D3×#935semidirect
653. C3#10595169semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×C3⋊D32
C22×#62942
C3×#641442
#117237242
#117237232
#117237202
#117236862
#117236822
#105951693
D3×#9353