Finite Group Databasebeta

← #11429957

Subgroups — #11429957

Order 768.

3356 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5627C214yesmaximal, complemented
2C3×#2683C211yesmaximal, characteristic, complemented
3#82829C214yesmaximal, complemented
4#77430C214yesmaximal, complemented
5#77281C214yesmaximal, complemented
6#77262C212yesmaximal, complemented
7#77169C212yesmaximal, complemented
8#76464C211yesmaximal, characteristic, complemented
9#76293C211yesmaximal, characteristic, complemented
10#76260C212yesmaximal, complemented
11#75961C212yesmaximal, complemented
12#74912C212yesmaximal, complemented
13#74659C212yesmaximal, complemented
14#52103—31nomaximal, Sylow
15C22×#239C2212yes—
16C23×C4×D3C2212yescomplemented
17C2×C3×C22⋊Q8C2214yescomplemented
18C2×#946C2214yescomplemented
19C2×#940C2211yescharacteristic
20C2×#937C2212yes—
21C2×D4×Dic3C2211yescharacteristic
22C2×#930C2214yes—
23#5627—232no—
24C2×#895C2214yes—
25C2×#894C2214yes—
26C2×#894C2214yescomplemented
27C2×#893C2212yes—
28C2×#893C2214yescomplemented
29C2×#893C2211yescharacteristic, complemented
30C2×#892C2214yescomplemented
31C2×#882C2214yes—
32C2×#882C2214yescomplemented
33C2×#67⋊C4C2214yes—
34C2×#67⋊C4C2214yescomplemented
35C3×#396C2214yescomplemented
36C3×#395C2212yescomplemented
37C3×#390C2211yescharacteristic
38C3×#389C2212yescomplemented
39C3×#386C2212yes—
40#5231C2212yescomplemented
41#5210C2212yescomplemented
42#4994C2214yescomplemented
43#4991C2212yes—
44#4991C2214yescomplemented
45#4991C2214yescomplemented
46#4990C2214yescomplemented
47#4988C2212yes—
48#4984C2212yescomplemented
49#4982C2214yescomplemented
50#4975C2211yescharacteristic, complemented
51Dic3×C4⋊C4C2212yescomplemented
52#4961C2214yes—
53#4961C2214yescomplemented
54#4960C2214yes—
55#4957C2214yescomplemented
56#4956C2214yes—
57#4955C2212yescomplemented
58Dic3×C22⋊C4C2212yescomplemented
59#4939C2212yes—
60#4939C2211yescharacteristic, complemented
61C42×Dic3C2211yescharacteristic, complemented
62#4675C2214yescomplemented
63#4671C2212yes—
64#4671C2214yescomplemented
65#4670C2214yescomplemented
66#4668C2214yescomplemented
67#4666C2214yescomplemented
68#4666C2212yescomplemented
69C2×#525—34no—
70#2683D311yescharacteristic, complemented
71#2682—32no—
72#2681—34no—
73#2680—32no—
74#2679—31no—
75#2678—34no—
76#2669—32no—
77#2392—34no—
78#2388—32no—
79C4×#127—31no—
80C4×#120—32no—
81#2293—32no—
82C24×D3C2312yes—
83C22×C3×Q8C2312yes—
84C22×C3×D4C2311yescharacteristic
85C23×C3×C4C2312yes—
86C22×#67C2314yes—
87C22×#67C2314yes—
88C23×Dic3C2312yes—
89C2×#239—216no—
90C2×#239—44no—
91C22×D12C2312yes—
92C22×D12C2314yes—
93C22×D12C2314yes—
94C22×D12C2311yescharacteristic
95C22×D12C2312yes—
96C22×C4×D3—216no—
97C22×C4×D3—44no—
98C22×C4×D3C2312yes—
99C22×C4×D3C2314yes—
100C22×C4×D3C2314yes—
101C3×C22⋊Q8—216no—
102C2×C3×C4⋊C4C2314yes—
103C2×C3×C4⋊C4C2314yes—
104C2×C3×C4⋊C4C2312yes—
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C4⋊C4C2312yes—
107C2×C3×C22⋊C4C2314yes—
108C2×C3×C22⋊C4C2314yes—
109C2×C3×C22⋊C4C2312yes—
110C2×C3×C42C2311yescharacteristic
111C2×#217C2312yes—
112C2×#217C2312yes—
113#946—216no—
114#940—24no—
115#937—28no—
116D4×Dic3—42no—
117#930—216no—
118C2×(C2×D3)⋊C4C2314yes—
119C2×(C2×D3)⋊C4C2314yes—
120C2×(C2×D3)⋊C4C2314yes—
121C2×(C2×D3)⋊C4C2314yes—
122C2×(C2×D3)⋊C4C2314yes—
123C2×(C2×D3)⋊C4C2314yes—
124C2×(C2×D3)⋊C4C2314yes—
125C2×(C2×D3)⋊C4C2314yes—
126C2×(C2×D3)⋊C4C2314yes—
127C2×(C2×D3)⋊C4C2312yes—
128C2×(C2×D3)⋊C4C2314yes—
129C2×C4⋊Dic3C2314yes—
130C2×C4⋊Dic3C2311yescharacteristic
131C2×C4⋊Dic3C2312yes—
132C2×Dic3⋊C4C2314yes—
133C2×Dic3⋊C4C2314yes—
134C2×C4×Dic3C2311yescharacteristic
135C2×C4×Dic3C2314yes—
136C2×C4×Dic3C2311yescharacteristic
137C2×C4×Dic3C2312yes—
138C2×C4×Dic3C2312yes—
139#895—216no—
140#894—216no—
141#894—216no—
142#893—216no—
143#893—42no—
144#893D4116yescomplemented
145#892—216no—
146#882—216no—
147#882—216no—
148#67⋊C4—216no—
149#67⋊C4—216no—
150C3×#95C2314yes—
151C3×#95C2312yes—
152C3⋊#95C2314yes—
153C3⋊#95C2312yes—
154C3⋊#95C2314yes—
155C3⋊#95C2314yes—
156C22×#41—32no—
157C24×C4—32no—
158C2×#127—31no—
159#525—632no—
160C2×#123—34no—
161C2×#123—34no—
162C2×C22⋊Q8—34no—
163C2×C22⋊Q8C2×D314yescomplemented
164C2×#121—34no—
165C2×#121—34no—
166C2×#120—32no—
167C2×#120—34no—
168C2×#120—34no—
169C2×C4×D4—31no—
170C2×C4×D4—32no—
171C2×C4×D4—34no—
172C2×C4×D4—31no—
173C2×C4×D4—34no—
174C2×C4×D4—34no—
175C2×#117—34no—
176#398—34no—
177#397—34no—
178#396—34no—
179#396C2×D314yescomplemented
180#395C2×D312yescomplemented
181#394—34no—
182#394—32no—
183#394—34no—
184#394—34no—
185#394—34no—
186#393—32no—
187#392—34no—
188#392—32no—
189#392—34no—
190#390—32no—
191#390—34no—
192#390—31no—
193#390—32no—
194#390C2×D311yescharacteristic
195#389—32no—
196#389C2×D312yescomplemented
197#388—34no—
198#388—34no—
199#386—34no—
200#386C2×D312yes—
201#385—32no—
202#385—34no—
203#384—31no—
204#379—32no—
205C4×C4⋊C4—32no—
206C4×C22⋊C4—32no—
207C4×C22⋊C4—32no—
208C4×C22⋊C4—34no—
209C43—31no—
210C24×C3C2412yes—
211C23×D3—24no—
212C23×D3—28no—
213C23×D3—42no—
214C23×D3—44no—
215C23×D3—42no—
216C23×D3C2412yes—
217C23×D3C2414yes—
218C2×C3×Q8—28no—
219C2×C3×Q8—42no—
220C2×C3×D4—41no—
221C2×C3×D4—41no—
222C2×C3×D4#4114yescomplemented
223C22×C3×C4—28no—
224C22×C3×C4—42no—
225C22×C3×C4C2414yes—
226C22×C3×C4C2412yes—
227C22×C3×C4C2414yes—
228C22×C3×C4C2412yes—
229C22×C3×C4C2411yescharacteristic
230C2×#67—28no—
231C2×#67—28no—
232C2×#67—28no—
233C2×#67—28no—
234C2×#67—44no—
235C2×#67—44no—
236C22×Dic3—28no—
237C22×Dic3—24no—
238C22×Dic3C2411yescharacteristic
239C22×Dic3C2414yes—
240C22×Dic3C2412yes—
241C22×Dic3C2412yes—
242C22×Dic3C2411yescharacteristic
243#239—432no—
244C2×D12—28no—
245C2×D12—28no—
246C2×D12—28no—
247C2×D12—28no—
248C2×D12—22no—
249C2×D12—41no—
250C2×D12—42no—
251C2×D12—42no—
252C2×D12—41no—
253C2×D12—44no—
254C2×D12—44no—
255C2×D12—44no—
256C2×D12—42no—
257C2×D12C2×D418yes—
258C2×C4×D3—28no—
259C2×C4×D3—28no—
260C2×C4×D3—28no—
261C2×C4×D3—28no—
262C2×C4×D3—28no—
263C2×C4×D3—24no—
264C2×C4×D3—432no—
265C2×C4×D3—44no—
266C2×C4×D3C2×D4116yes—
267C3×C4⋊C4—28no—
268C3×C4⋊C4—28no—
269C3×C4⋊C4—41no—
270C3×C4⋊C4—42no—
271C3×C4⋊C4C2×D418yes—
272C3×C22⋊C4—28no—
273C3×C22⋊C4—28no—
274C3×C22⋊C4—42no—
275C3×C42—41no—
276#217—24no—
277#217—42no—
278(C2×D3)⋊C4—28no—
279(C2×D3)⋊C4—28no—
280(C2×D3)⋊C4—28no—
281(C2×D3)⋊C4—28no—
282(C2×D3)⋊C4—28no—
283(C2×D3)⋊C4—28no—
284(C2×D3)⋊C4—28no—
285(C2×D3)⋊C4—42no—
286(C2×D3)⋊C4—44no—
287(C2×D3)⋊C4—44no—
288(C2×D3)⋊C4C2×D4116yes—
289C4⋊Dic3—28no—
290C4⋊Dic3—42no—
291C4⋊Dic3—41no—
292Dic3⋊C4—28no—
293Dic3⋊C4—28no—
294C4×Dic3—22no—
295C4×Dic3—28no—
296C4×Dic3—22no—
297C4×Dic3—42no—
298C4×Dic3C2×D418yes—
299C25—32no—
300C2×#41—616no—
301C2×#41—124no—
302C22×Q8C22×D312yes—
303C22×D4—34no—
304C22×D4—34no—
305C22×D4—32no—
306C22×D4—32no—
307C22×D4—34no—
308C22×D4—34no—
309C22×D4—31no—
310C22×D4C22×D311yescharacteristic
311C23×C4—34no—
312C23×C4—32no—
313C23×C4—32no—
314C23×C4—34no—
315C23×C4—616no—
316C23×C4—124no—
317C23×C4C22×D312yes—
318#127—64no—
319#123—616no—
320#123—616no—
321C22⋊Q8—216no—
322C22⋊Q8—616no—
323#121—616no—
324#121—616no—
325#120—616no—
326#120—616no—
327#120—68no—
328C4×D4—316no—
329C4×D4—616no—
330C4×D4—616no—
331C4×D4—616no—
332C4×D4—122no—
333C4×D4—122no—
334#117—616no—
335C2×C4⋊C4—34no—
336C2×C4⋊C4—34no—
337C2×C4⋊C4—34no—
338C2×C4⋊C4—31no—
339C2×C4⋊C4—32no—
340C2×C4⋊C4C22×D314yes—
341C2×C4⋊C4C22×D314yes—
342C2×C4⋊C4C22×D311yescharacteristic
343C2×C4⋊C4C22×D312yes—
344C2×C4⋊C4C22×D312yes—
345C2×C22⋊C4—34no—
346C2×C22⋊C4—32no—
347C2×C22⋊C4—34no—
348C2×C22⋊C4—34no—
349C2×C22⋊C4—34no—
350C2×C22⋊C4—34no—
351C2×C22⋊C4—34no—
352C2×C22⋊C4—34no—
353C2×C22⋊C4—34no—
354C2×C22⋊C4—32no—
355C2×C22⋊C4—32no—
356C2×C22⋊C4—34no—
357C2×C22⋊C4—34no—
358C2×C22⋊C4C22×D314yes—
359C2×C22⋊C4C22×D314yes—
360C2×C22⋊C4C22×D312yes—
361C2×C42—32no—
362C2×C42—34no—
363C2×C42—31no—
364C2×C42—31no—
365C2×C42—32no—
366C2×C42C22×D311yescharacteristic
367#95—34no—
368#95—34no—
369#95—32no—
370#95—34no—
371#95C22×D314yes—
372#95C22×D312yes—
373C23×C3—24no—
374C23×C3—22no—
375C23×C3—22no—
376C23×C3—24no—
377C23×C3C2×#4114yes—
378C23×C3C2511yescharacteristic
379C22×D3—24no—
380C22×D3—24no—
381C22×D3—24no—
382C22×D3—24no—
383C22×D3—24no—
384C22×D3—24no—
385C22×D3—24no—
386C22×D3—22no—
387C22×D3—22no—
388C22×D3—24no—
389C22×D3—22no—
390C22×D3—48no—
391C22×D3—416no—
392C22×D3—416no—
393C22×D3—48no—
394C22×D3—48no—
395C22×D3C22×D418yes—
396C3×Q8—48no—
397C3×D4—44no—
398C2×C3×C4—24no—
399C2×C3×C4—21no—
400C2×C3×C4—24no—
401C2×C3×C4—22no—
402C2×C3×C4—22no—
403C2×C3×C4—21no—
404C2×C3×C4—24no—
405C2×C3×C4—24no—
406C2×C3×C4—22no—
407C2×C3×C4—24no—
408C2×C3×C4—24no—
409C2×C3×C4—48no—
410C2×C3×C4C22×D414yes—
411C2×C3×C4C22×D418yes—
412C2×C3×C4C2×#4112yes—
413#67—416no—
414#67—416no—
415C2×Dic3—22no—
416C2×Dic3—22no—
417C2×Dic3—24no—
418C2×Dic3—24no—
419C2×Dic3—24no—
420C2×Dic3—21no—
421C2×Dic3—22no—
422C2×Dic3—22no—
423C2×Dic3—24no—
424C2×Dic3—48no—
425C2×Dic3C22×D414yes—
426C2×Dic3C22×D414yes—
427C2×Dic3H(5,𝔽2)12yes—
428D12—48no—
429D12—416no—
430D12—44no—
431D12—48no—
432D12—416no—
433C4×D3—232no—
434C4×D3—416no—
435C4×D3—48no—
436C24—34no—
437C24—32no—
438C24—68no—
439C24—64no—
440C24—122no—
441C24—122no—
442C24—124no—
443C24C23×D312yes—
444#41—1232no—
445C2×Q8—28no—
446C2×Q8—42no—
447C2×D4—38no—
448C2×D4—41no—
449C2×D4—41no—
450C2×D4—68no—
451C2×D4—68no—
452C2×D4—68no—
453C2×D4—68no—
454C2×D4—68no—
455C2×D4—68no—
456C2×D4—68no—
457C2×D4—68no—
458C2×D4—62no—
459C2×D4—121no—
460C2×D4—124no—
461C2×D4—122no—
462C2×D4—122no—
463C2×D4—121no—
464C2×D4—122no—
465C2×D4—124no—
466C2×D4—124no—
467C2×D4—124no—
468C2×D4—124no—
469C2×D4#23914yescomplemented
470C22×C4—28no—
471C22×C4—31no—
472C22×C4—32no—
473C22×C4—316no—
474C22×C4—31no—
475C22×C4—34no—
476C22×C4—32no—
477C22×C4—42no—
478C22×C4—68no—
479C22×C4—68no—
480C22×C4—68no—
481C22×C4—64no—
482C22×C4—68no—
483C22×C4—68no—
484C22×C4—64no—
485C22×C4—68no—
486C22×C4—124no—
487C22×C4—1232no—
488C22×C4C23×D314yes—
489C22×C4C23×D312yes—
490C22×C4C23×D314yes—
491C22×C4C23×D312yes—
492C22×C4C23×D311yescharacteristic
493C4⋊C4—28no—
494C4⋊C4—28no—
495C4⋊C4—41no—
496C4⋊C4—42no—
497C4⋊C4—68no—
498C4⋊C4—68no—
499C4⋊C4—68no—
500C4⋊C4—121no—
501C4⋊C4—122no—
502C4⋊C4D3×D418yes—
503C22⋊C4—28no—
504C22⋊C4—28no—
505C22⋊C4—316no—
506C22⋊C4—42no—
507C22⋊C4—68no—
508C22⋊C4—68no—
509C22⋊C4—68no—
510C22⋊C4—64no—
511C22⋊C4—68no—
512C22⋊C4—68no—
513C22⋊C4—68no—
514C22⋊C4—68no—
515C22⋊C4—122no—
516C22⋊C4—124no—
517C22⋊C4—124no—
518C22⋊C4—122no—
519C42—38no—
520C42—41no—
521C42—68no—
522C42—62no—
523C42—62no—
524C42—122no—
525C22×C3—24no—
526C22×C3—28no—
527C22×C3—44no—
528C22×C3—42no—
529C22×C3—42no—
530C22×C3C23×D412yes—
531C22×C3C22×#4111yescharacteristic
532C22×C3C2×H(5,𝔽2)11yescharacteristic
533C22×C3#58511yescharacteristic
534C22×C3#58512yes—
535C2×D3—28no—
536C2×D3—28no—
537C2×D3—28no—
538C2×D3—432no—
539C2×D3—44no—
540C2×D3—44no—
541C2×D3—44no—
542C2×D3—44no—
543C2×D3—44no—
544C2×D3—42no—
545C2×D3—42no—
546C2×D3—44no—
547C2×D3—42no—
548C3×C4—24no—
549C3×C4—22no—
550C3×C4—28no—
551C3×C4—44no—
552C3×C4—42no—
553Dic3—24no—
554Dic3—42no—
555Dic3—44no—
556Dic3—41no—
557Dic3D4214yes—
558C23—24no—
559C23—22no—
560C23—22no—
561C23—24no—
562C23—38no—
563C23—64no—
564C23—64no—
565C23—64no—
566C23—64no—
567C23—64no—
568C23—64no—
569C23—64no—
570C23—62no—
571C23—62no—
572C23—62no—
573C23—64no—
574C23—1216no—
575C23—128no—
576C23—128no—
577C23—1216no—
578C23—128no—
579C23C2×#23914yes—
580C23C24×D311yescharacteristic
581Q8—48no—
582D4—44no—
583D4—128no—
584D4—1216no—
585D4—124no—
586D4—128no—
587D4—1216no—
588D4—1216no—
589D4—1216no—
590C2×C4—24no—
591C2×C4—21no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—34no—
602C2×C4—32no—
603C2×C4—34no—
604C2×C4—48no—
605C2×C4—632no—
606C2×C4—62no—
607C2×C4—62no—
608C2×C4—64no—
609C2×C4—64no—
610C2×C4—64no—
611C2×C4—61no—
612C2×C4—62no—
613C2×C4—62no—
614C2×C4—64no—
615C2×C4—1216no—
616C2×C4—128no—
617C2×C4—128no—
618C2×C4C2×D3×D414yes—
619C2×C4C2×D3×D418yes—
620C2×C4C2×#23912yes—
621C2×C3—44no—
622C2×C3#344912yes—
623C2×C3#345911yescharacteristic
624C2×C3C2×D4211yescharacteristic
625C2×C3D4×#4112yes—
626C2×C3#352911yescharacteristic
627D3—48no—
628D3—82no—
629C22—24no—
630C22—28no—
631C22—42no—
632C22—44no—
633C22—42no—
634C22—68no—
635C22—68no—
636C22—68no—
637C22—1232no—
638C22—124no—
639C22—124no—
640C22—124no—
641C22—124no—
642C22—122no—
643C22—122no—
644C22—124no—
645C22—122no—
646C22—124no—
647C22C22×D3×D412yes—
648C22C22×#23911yescharacteristic
649C22D3×H(5,𝔽2)11yescharacteristic
650C22#596611yescharacteristic
651C22#596812yes—
652C4—22no—
653C4—24no—
654C4—28no—
655C4—34no—
656C4—44no—
657C4—42no—
658C4—64no—
659C4—121no—
660C4—122no—
661C4—124no—
662C3#5210311yescharacteristic, complemented, Sylow
663C2—44no—
664C2—128no—
665C2—242no—
666C2#8636312yes—
667C2#8657211yescharacteristic
668C2D3×D4211yescharacteristic
669C2D4×#23912yes—
670C2#8711511yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2683C2
hypercenter#2683D3
hypocenterC3#52103
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2683

Complements

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

NH ≅ G/Ntype
1. C2×#5627C2semidirect
2. C3×#2683C2semidirect
3. #82829C2semidirect
4. #77430C2semidirect
5. #77281C2semidirect
6. #77262C2semidirect
7. #77169C2semidirect
8. #76464C2semidirect
9. #76293C2semidirect
10. #76260C2semidirect
11. #75961C2semidirect
12. #74912C2semidirect
13. #74659C2semidirect
16. C23×C4×D3C22semidirect
17. C2×C3×C22⋊Q8C22semidirect
18. C2×#946C22semidirect
26. C2×#894C22semidirect
28. C2×#893C22semidirect
29. C2×#893C22semidirect
30. C2×#892C22semidirect
32. C2×#882C22semidirect
34. C2×#67⋊C4C22semidirect
35. C3×#396C22semidirect
36. C3×#395C22semidirect
38. C3×#389C22semidirect
40. #5231C22semidirect
41. #5210C22semidirect
42. #4994C22semidirect
44. #4991C22semidirect
45. #4991C22semidirect
46. #4990C22semidirect
48. #4984C22semidirect
49. #4982C22semidirect
50. #4975C22semidirect
51. Dic3×C4⋊C4C22semidirect
53. #4961C22semidirect
55. #4957C22semidirect
57. #4955C22semidirect
58. Dic3×C22⋊C4C22semidirect
60. #4939C22semidirect
61. C42×Dic3C22semidirect
62. #4675C22semidirect
64. #4671C22semidirect
65. #4670C22semidirect
66. #4668C22semidirect
67. #4666C22semidirect
68. #4666C22semidirect
70. #2683D3semidirect
144. #893D4semidirect
163. C2×C22⋊Q8C2×D3semidirect
179. #396C2×D3semidirect
180. #395C2×D3semidirect
196. #389C2×D3semidirect
222. C2×C3×D4#41semidirect
469. C2×D4#239semidirect
662. C3#52103semidirect

Maximal subgroups

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

subgroupindex
C2×#56272
C3×#26832
#828292
#774302
#772812
#772622
#771692
#764642
#762932
#762602
#759612
#749122
#746592
#521033