Finite Group Databasebeta

← #11429953

Subgroups — #11429953

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×#5586C214yesmaximal, complemented
2C3×#2679C211yesmaximal, characteristic, complemented
3#82596C214yesmaximal, complemented
4#82594C212yesmaximal, complemented
5Dic3×#120C212yesmaximal, complemented
6#77006C214yesmaximal, complemented
7#77004C212yesmaximal, complemented
8#76890C214yesmaximal, complemented
9#76889C212yesmaximal, complemented
10#76380C211yesmaximal, characteristic, complemented
11#76379C212yesmaximal, complemented
12#76293C211yesmaximal, characteristic, complemented
13#76289C212yesmaximal, complemented
14#52103—31nomaximal, Sylow
15C24×Dic3C2212yescomplemented
16C22×#237C2212yes—
17C2×C3×#120C2212yes—
18C2×C3×#120C2214yes—
19C2×#953C2214yescomplemented
20C2×#940C2211yescharacteristic
21C2×#939C2214yes—
22C2×#939C2214yes—
23C2×#935C2214yes—
24C2×D4×Dic3C2214yes—
25C2×D4×Dic3C2211yescharacteristic
26C2×D4×Dic3C2212yes—
27#5586—232no—
28C2×#893C2211yescharacteristic, complemented
29C2×#886C2214yescomplemented
30C2×#67⋊C4C2214yescomplemented
31C2×#67⋊C4C2214yescomplemented
32C2×#878C2214yescomplemented
33C2×#878C2214yescomplemented
34C2×#877C2214yescomplemented
35C3×#394C2212yes—
36C3×#394C2214yes—
37C3×#390C2212yes—
38C3×#390C2211yescharacteristic
39#5247C2212yescomplemented
40#5223C2212yes—
41#5223C2214yes—
42#5221C2212yes—
43#5220C2212yes—
44#5220C2214yes—
45#4994C2214yescomplemented
46#4979C2212yescomplemented
47#4975C2211yescharacteristic, complemented
48Dic3×C4⋊C4C2212yescomplemented
49#4972C2212yescomplemented
50#4961C2214yescomplemented
51#4960C2214yescomplemented
52#4959C2214yescomplemented
53#4959C2214yescomplemented
54#4957C2214yescomplemented
55#4952C2214yescomplemented
56#4950C2214yescomplemented
57#4950C2212yescomplemented
58#4948C2214yescomplemented
59#4945C2214yescomplemented
60#4944C2214yescomplemented
61Dic3×C22⋊C4C2214yescomplemented
62Dic3×C22⋊C4C2212yescomplemented
63#4939C2211yescharacteristic, complemented
64C42×Dic3C2211yescharacteristic, complemented
65#4928C2212yescomplemented
66#4673C2212yescomplemented
67#4666C2212yescomplemented
68#4657C2214yescomplemented
69C2×#525—34no—
70#2683—31no—
71#2682—32no—
72#2681—34no—
73#2680—32no—
74#2679D311yescharacteristic, complemented
75#2678—34no—
76#2669—32no—
77#2392—34no—
78#2388—32no—
79C4×#127—31no—
80C4×#120—32no—
81#2293—32no—
82C25×C3C2312yes—
83C22×C3×D4C2312yes—
84C22×C3×D4C2314yes—
85C22×C3×D4C2312yes—
86C22×C3×D4C2311yescharacteristic
87C22×#67C2314yes—
88C22×#67C2314yes—
89C22×#67C2314yes—
90C23×Dic3—216no—
91C23×Dic3—44no—
92C23×Dic3C2312yes—
93C23×Dic3C2312yes—
94C23×Dic3C2314yes—
95C23×Dic3C2314yes—
96C2×#237—216no—
97C2×#237—44no—
98C22×D12C2311yescharacteristic
99C22×C4×D3C2312yes—
100C22×Dic6C2312yes—
101C3×#120—216no—
102C3×#120—28no—
103C2×C3×C4⋊C4C2312yes—
104C2×C3×C4⋊C4C2311yescharacteristic
105C2×C3×C22⋊C4C2314yes—
106C2×C3×C22⋊C4C2312yes—
107C2×C3×C22⋊C4C2314yes—
108C2×C3×C22⋊C4C2314yes—
109C2×C3×C22⋊C4C2314yes—
110C2×C3×C42C2311yescharacteristic
111#953—216no—
112C2×#217C2314yes—
113C2×#217C2314yes—
114C2×#217C2314yes—
115C2×#217C2314yes—
116C2×#217C2312yes—
117C2×#217C2312yes—
118C2×#217C2314yes—
119#940—24no—
120#939—216no—
121#939—216no—
122#935—216no—
123D4×Dic3—216no—
124D4×Dic3—42no—
125D4×Dic3D4116yescomplemented
126C2×(C2×D3)⋊C4C2314yes—
127C2×(C2×D3)⋊C4C2314yes—
128C2×(C2×D3)⋊C4C2314yes—
129C2×(C2×D3)⋊C4C2312yes—
130C2×C4⋊Dic3C2314yes—
131C2×C4⋊Dic3C2312yes—
132C2×C4⋊Dic3C2311yescharacteristic
133C2×C4⋊Dic3C2312yes—
134C2×Dic3⋊C4C2314yes—
135C2×Dic3⋊C4C2314yes—
136C2×Dic3⋊C4C2314yes—
137C2×Dic3⋊C4C2314yes—
138C2×C4×Dic3C2314yes—
139C2×C4×Dic3C2311yescharacteristic
140C2×C4×Dic3C2311yescharacteristic
141C2×C4×Dic3C2312yes—
142C2×C4×Dic3C2312yes—
143#893—42no—
144#886—216no—
145#67⋊C4—216no—
146#67⋊C4—216no—
147#878—216no—
148#878—216no—
149#877—216no—
150C3×#95C2312yes—
151C3⋊#95C2314yes—
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⋊Q8—34no—
164C2×#121—34no—
165C2×#121—34no—
166C2×#120—34no—
167C2×#120C2×D314yes—
168C2×#120C2×D312yes—
169C2×C4×D4—32no—
170C2×C4×D4—34no—
171C2×C4×D4—31no—
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#396—34no—
180#395—32no—
181#394—34no—
182#394—34no—
183#394—34no—
184#394C2×D312yes—
185#394C2×D314yes—
186#393—32no—
187#392—32no—
188#392—34no—
189#392—34no—
190#390—32no—
191#390—34no—
192#390—31no—
193#390C2×D311yescharacteristic
194#390C2×D312yes—
195#389—32no—
196#389—32no—
197#388—34no—
198#388—34no—
199#386—32no—
200#386—34no—
201#385—34no—
202#385—32no—
203#384—31no—
204#379—32no—
205C4×C4⋊C4—32no—
206C4×C22⋊C4—34no—
207C4×C22⋊C4—32no—
208C4×C22⋊C4—32no—
209C43—31no—
210C24×C3—28no—
211C24×C3—24no—
212C24×C3—44no—
213C24×C3—42no—
214C24×C3—42no—
215C24×C3C2414yes—
216C24×C3C2412yes—
217C23×D3C2412yes—
218C2×C3×D4—28no—
219C2×C3×D4—28no—
220C2×C3×D4—28no—
221C2×C3×D4—41no—
222C2×C3×D4—42no—
223C2×C3×D4—42no—
224C2×C3×D4—42no—
225C2×C3×D4—41no—
226C2×C3×D4—44no—
227C2×C3×D4C2×D418yes—
228C2×C3×D4#4114yescomplemented
229C22×C3×C4C2414yes—
230C22×C3×C4C2412yes—
231C22×C3×C4C2412yes—
232C22×C3×C4C2411yescharacteristic
233C2×#67—28no—
234C2×#67—28no—
235C2×#67—28no—
236C2×#67—28no—
237C2×#67—28no—
238C2×#67—44no—
239C2×#67—44no—
240C2×#67—44no—
241C2×#67—44no—
242C22×Dic3—28no—
243C22×Dic3—28no—
244C22×Dic3—28no—
245C22×Dic3—28no—
246C22×Dic3—28no—
247C22×Dic3—28no—
248C22×Dic3—24no—
249C22×Dic3—42no—
250C22×Dic3—44no—
251C22×Dic3—432no—
252C22×Dic3C2×D4116yes—
253C22×Dic3C2411yescharacteristic
254C22×Dic3C2414yes—
255C22×Dic3C2412yes—
256C22×Dic3C2412yes—
257C22×Dic3C2414yes—
258C22×Dic3C2411yescharacteristic
259#237—432no—
260C2×D12—22no—
261C2×D12—41no—
262C2×D12—41no—
263C2×C4×D3—28no—
264C2×C4×D3—24no—
265C2×Dic6—28no—
266C2×Dic6—42no—
267C3×C4⋊C4—41no—
268C3×C4⋊C4—42no—
269C3×C22⋊C4—28no—
270C3×C22⋊C4—24no—
271C3×C22⋊C4—28no—
272C3×C22⋊C4—28no—
273C3×C22⋊C4—28no—
274C3×C42—41no—
275#217—28no—
276#217—28no—
277#217—28no—
278#217—42no—
279#217—44no—
280#217—42no—
281#217C2×D4116yes—
282(C2×D3)⋊C4—28no—
283(C2×D3)⋊C4—28no—
284(C2×D3)⋊C4—42no—
285(C2×D3)⋊C4—44no—
286C4⋊Dic3—28no—
287C4⋊Dic3—42no—
288C4⋊Dic3—41no—
289C4⋊Dic3C2×D418yes—
290Dic3⋊C4—28no—
291Dic3⋊C4—28no—
292Dic3⋊C4—28no—
293Dic3⋊C4—28no—
294C4×Dic3—22no—
295C4×Dic3—28no—
296C4×Dic3—22no—
297C4×Dic3—42no—
298C4×Dic3C2×D418yes—
299C25C22×D312yes—
300C2×#41—616no—
301C2×#41—124no—
302C22×Q8—32no—
303C22×D4—34no—
304C22×D4—34no—
305C22×D4—34no—
306C22×D4—31no—
307C22×D4C22×D314yes—
308C22×D4C22×D312yes—
309C22×D4C22×D312yes—
310C22×D4C22×D311yescharacteristic
311C23×C4—34no—
312C23×C4—32no—
313C23×C4—32no—
314C23×C4—34no—
315C23×C4—32no—
316C23×C4—616no—
317C23×C4—124no—
318#127—64no—
319#123—616no—
320#123—616no—
321C22⋊Q8—616no—
322C22⋊Q8—616no—
323#121—616no—
324#121—616no—
325#120—216no—
326#120—28no—
327#120—616no—
328C4×D4—316no—
329C4×D4—616no—
330C4×D4—616no—
331C4×D4—616no—
332C4×D4—122no—
333C4×D4—122no—
334#117—616no—
335C2×C4⋊C4—32no—
336C2×C4⋊C4—34no—
337C2×C4⋊C4—34no—
338C2×C4⋊C4—34no—
339C2×C4⋊C4—34no—
340C2×C4⋊C4—34no—
341C2×C4⋊C4—31no—
342C2×C4⋊C4—32no—
343C2×C4⋊C4C22×D311yescharacteristic
344C2×C4⋊C4C22×D312yes—
345C2×C22⋊C4—34no—
346C2×C22⋊C4—34no—
347C2×C22⋊C4—34no—
348C2×C22⋊C4—34no—
349C2×C22⋊C4—34no—
350C2×C22⋊C4—34no—
351C2×C22⋊C4—32no—
352C2×C22⋊C4—34no—
353C2×C22⋊C4—32no—
354C2×C22⋊C4—32no—
355C2×C22⋊C4—34no—
356C2×C22⋊C4C22×D314yes—
357C2×C22⋊C4C22×D314yes—
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—34no—
370#95—32no—
371#95—34no—
372#95C22×D312yes—
373C23×C3—24no—
374C23×C3—24no—
375C23×C3—24no—
376C23×C3—22no—
377C23×C3—24no—
378C23×C3—22no—
379C23×C3—24no—
380C23×C3—24no—
381C23×C3—24no—
382C23×C3—24no—
383C23×C3—48no—
384C23×C3—416no—
385C23×C3—416no—
386C23×C3—48no—
387C23×C3—48no—
388C23×C3C22×D418yes—
389C23×C3C2×#4114yes—
390C23×C3C2511yescharacteristic
391C22×D3—24no—
392C22×D3—22no—
393C22×D3—22no—
394C22×D3—24no—
395C22×D3—22no—
396C3×D4—48no—
397C3×D4—44no—
398C3×D4—48no—
399C3×D4—416no—
400C2×C3×C4—21no—
401C2×C3×C4—22no—
402C2×C3×C4—22no—
403C2×C3×C4—21no—
404C2×C3×C4—24no—
405C2×C3×C4—24no—
406C2×C3×C4—24no—
407C2×C3×C4—22no—
408C2×C3×C4—24no—
409C2×C3×C4C22×D414yes—
410C2×C3×C4C2×#4112yes—
411#67—416no—
412#67—416no—
413#67—416no—
414C2×Dic3—24no—
415C2×Dic3—22no—
416C2×Dic3—24no—
417C2×Dic3—22no—
418C2×Dic3—232no—
419C2×Dic3—24no—
420C2×Dic3—24no—
421C2×Dic3—21no—
422C2×Dic3—22no—
423C2×Dic3—22no—
424C2×Dic3—24no—
425C2×Dic3—24no—
426C2×Dic3—48no—
427C2×Dic3—416no—
428C2×Dic3—48no—
429C2×Dic3C22×D414yes—
430C2×Dic3C22×D418yes—
431C2×Dic3C22×D414yes—
432C2×Dic3H(5,𝔽2)12yes—
433D12—44no—
434C4×D3—48no—
435Dic6—48no—
436C24—28no—
437C24—24no—
438C24—32no—
439C24—42no—
440C24—44no—
441C24—42no—
442C24C23×D314yes—
443C24C23×D312yes—
444#41—1232no—
445C2×Q8—68no—
446C2×Q8—122no—
447C2×D4—28no—
448C2×D4—28no—
449C2×D4—28no—
450C2×D4—41no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4—41no—
454C2×D4—42no—
455C2×D4—44no—
456C2×D4—68no—
457C2×D4—68no—
458C2×D4—68no—
459C2×D4—68no—
460C2×D4—68no—
461C2×D4—62no—
462C2×D4—121no—
463C2×D4—121no—
464C2×D4—124no—
465C2×D4—124no—
466C2×D4—124no—
467C2×D4—124no—
468C2×D4D3×D418yes—
469C2×D4#23714yescomplemented
470C22×C4—31no—
471C22×C4—316no—
472C22×C4—32no—
473C22×C4—34no—
474C22×C4—31no—
475C22×C4—34no—
476C22×C4—32no—
477C22×C4—68no—
478C22×C4—64no—
479C22×C4—68no—
480C22×C4—68no—
481C22×C4—68no—
482C22×C4—64no—
483C22×C4—68no—
484C22×C4—68no—
485C22×C4—68no—
486C22×C4—122no—
487C22×C4—124no—
488C22×C4—1232no—
489C22×C4C23×D314yes—
490C22×C4C23×D312yes—
491C22×C4C23×D312yes—
492C22×C4C23×D311yescharacteristic
493C4⋊C4—38no—
494C4⋊C4—41no—
495C4⋊C4—42no—
496C4⋊C4—68no—
497C4⋊C4—68no—
498C4⋊C4—68no—
499C4⋊C4—68no—
500C4⋊C4—68no—
501C4⋊C4—121no—
502C4⋊C4—122no—
503C22⋊C4—28no—
504C22⋊C4—24no—
505C22⋊C4—28no—
506C22⋊C4—28no—
507C22⋊C4—28no—
508C22⋊C4—316no—
509C22⋊C4—68no—
510C22⋊C4—68no—
511C22⋊C4—68no—
512C22⋊C4—68no—
513C22⋊C4—68no—
514C22⋊C4—122no—
515C22⋊C4—124no—
516C22⋊C4—122no—
517C22⋊C4—124no—
518C22⋊C4—122no—
519C42—38no—
520C42—41no—
521C42—68no—
522C42—62no—
523C42—62no—
524C42—122no—
525C22×C3—28no—
526C22×C3—24no—
527C22×C3—28no—
528C22×C3—28no—
529C22×C3—28no—
530C22×C3—432no—
531C22×C3—44no—
532C22×C3—44no—
533C22×C3—44no—
534C22×C3—42no—
535C22×C3—44no—
536C22×C3—42no—
537C22×C3—44no—
538C22×C3C23×D412yes—
539C22×C3C22×#4111yescharacteristic
540C22×C3C2×H(5,𝔽2)11yescharacteristic
541C22×C3#58511yescharacteristic
542C22×C3#58512yes—
543C2×D3—44no—
544C2×D3—42no—
545C2×D3—42no—
546C2×D3—44no—
547C2×D3—42no—
548C3×C4—24no—
549C3×C4—22no—
550C3×C4—42no—
551C3×C4—44no—
552Dic3—28no—
553Dic3—24no—
554Dic3—44no—
555Dic3—42no—
556Dic3—41no—
557Dic3D4214yes—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—22no—
562C23—24no—
563C23—22no—
564C23—24no—
565C23—24no—
566C23—24no—
567C23—24no—
568C23—48no—
569C23—416no—
570C23—416no—
571C23—48no—
572C23—48no—
573C23—64no—
574C23—62no—
575C23—62no—
576C23—64no—
577C23—62no—
578C23C2×D3×D418yes—
579C23C2×#23714yes—
580C23C24×D311yescharacteristic
581Q8—128no—
582D4—48no—
583D4—44no—
584D4—48no—
585D4—416no—
586D4—124no—
587D4—1216no—
588D4—1216no—
589D4—1216no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—34no—
600C2×C4—32no—
601C2×C4—34no—
602C2×C4—38no—
603C2×C4—632no—
604C2×C4—64no—
605C2×C4—62no—
606C2×C4—64no—
607C2×C4—62no—
608C2×C4—64no—
609C2×C4—64no—
610C2×C4—61no—
611C2×C4—62no—
612C2×C4—62no—
613C2×C4—64no—
614C2×C4—64no—
615C2×C4—128no—
616C2×C4—128no—
617C2×C4—128no—
618C2×C4—1216no—
619C2×C4C2×D3×D414yes—
620C2×C4C2×#23712yes—
621C2×C3—48no—
622C2×C3—44no—
623C2×C3#344912yes—
624C2×C3#345911yescharacteristic
625C2×C3C2×D4211yescharacteristic
626C2×C3D4×#4112yes—
627C2×C3#352911yescharacteristic
628D3—82no—
629C22—28no—
630C22—24no—
631C22—28no—
632C22—28no—
633C22—28no—
634C22—432no—
635C22—44no—
636C22—44no—
637C22—44no—
638C22—42no—
639C22—44no—
640C22—44no—
641C22—42no—
642C22—124no—
643C22—122no—
644C22—122no—
645C22—124no—
646C22—122no—
647C22C22×D3×D412yes—
648C22C22×#23711yescharacteristic
649C22D3×H(5,𝔽2)11yescharacteristic
650C22#596611yescharacteristic
651C22#596612yes—
652C4—24no—
653C4—22no—
654C4—34no—
655C4—44no—
656C4—42no—
657C4—64no—
658C4—68no—
659C4—124no—
660C4—121no—
661C4—122no—
662C3#5210311yescharacteristic, complemented, Sylow
663C2—48no—
664C2—44no—
665C2—242no—
666C2#8629412yes—
667C2#8629511yescharacteristic
668C2D4×#23712yes—
669C2D3×D4211yescharacteristic
670C2#8708511yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2679C2
hypercenter#2679D3
hypocenterC3#52103
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2679

Complements

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

NH ≅ G/Ntype
1. C2×#5586C2semidirect
2. C3×#2679C2semidirect
3. #82596C2semidirect
4. #82594C2semidirect
5. Dic3×#120C2semidirect
6. #77006C2semidirect
7. #77004C2semidirect
8. #76890C2semidirect
9. #76889C2semidirect
10. #76380C2semidirect
11. #76379C2semidirect
12. #76293C2semidirect
13. #76289C2semidirect
15. C24×Dic3C22semidirect
19. C2×#953C22semidirect
28. C2×#893C22semidirect
29. C2×#886C22semidirect
30. C2×#67⋊C4C22semidirect
31. C2×#67⋊C4C22semidirect
32. C2×#878C22semidirect
33. C2×#878C22semidirect
34. C2×#877C22semidirect
39. #5247C22semidirect
45. #4994C22semidirect
46. #4979C22semidirect
47. #4975C22semidirect
48. Dic3×C4⋊C4C22semidirect
49. #4972C22semidirect
50. #4961C22semidirect
51. #4960C22semidirect
52. #4959C22semidirect
53. #4959C22semidirect
54. #4957C22semidirect
55. #4952C22semidirect
56. #4950C22semidirect
57. #4950C22semidirect
58. #4948C22semidirect
59. #4945C22semidirect
60. #4944C22semidirect
61. Dic3×C22⋊C4C22semidirect
62. Dic3×C22⋊C4C22semidirect
63. #4939C22semidirect
64. C42×Dic3C22semidirect
65. #4928C22semidirect
66. #4673C22semidirect
67. #4666C22semidirect
68. #4657C22semidirect
74. #2679D3semidirect
125. D4×Dic3D4semidirect
228. C2×C3×D4#41semidirect
469. C2×D4#237semidirect
662. C3#52103semidirect

Maximal subgroups

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

subgroupindex
C2×#55862
C3×#26792
#825962
#825942
Dic3×#1202
#770062
#770042
#768902
#768892
#763802
#763792
#762932
#762892
#521033