Finite Group Databasebeta

← #11429951

Subgroups — #11429951

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×#5547C214yesmaximal, complemented
2C3×C4×#127C211yesmaximal, characteristic, complemented
3#83190C212yesmaximal, complemented
4#82401C214yesmaximal, complemented
5#82400C212yesmaximal, complemented
6C4×#937C212yesmaximal, complemented
7#76466C212yesmaximal, complemented
8#76465C214yesmaximal, complemented
9#76464C211yesmaximal, characteristic, complemented
10#76462C214yesmaximal, complemented
11#76380C211yesmaximal, characteristic, complemented
12#74371C212yesmaximal, complemented
13#73299C212yesmaximal, complemented
14#52103—31nomaximal, Sylow
15C22×#235C2212yes—
16C23×C4×D3C2212yescomplemented
17C2×C3×#127C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×C3×C4×D4C2214yescomplemented
20C2×C3×C4×D4C2211yescharacteristic, complemented
21C2×#939C2214yes—
22C2×#937C2212yes—
23C2×#937C2214yescomplemented
24C2×C4×#67C2214yes—
25C2×C4×#67C2214yescomplemented
26#5547—232no—
27C2×#896C2214yescomplemented
28C2×#894C2214yes—
29C2×#884C2214yes—
30C2×#883C2214yescomplemented
31C2×#882C2214yes—
32C2×#878C2214yescomplemented
33C2×C4×D12C2212yes—
34C2×#872C2214yescomplemented
35C3×#390C2211yescharacteristic
36C3×#390C2214yescomplemented
37C3×#390C2211yescharacteristic, complemented
38C3×#384C2211yescharacteristic, complemented
39C3×C43C2211yescharacteristic, complemented
40#5223C2214yes—
41#5223C2214yescomplemented
42#5222C2214yescomplemented
43#5220C2212yes—
44C4×#217C2212yescomplemented
45C4×#217C2212yescomplemented
46#4992C2214yescomplemented
47#4991C2212yes—
48#4989C2214yescomplemented
49#4983C2214yescomplemented
50#4979C2212yescomplemented
51#4961C2214yes—
52#4961C2214yescomplemented
53#4960C2212yes—
54#4959C2214yescomplemented
55#4956C2214yes—
56#4955C2212yescomplemented
57#4948C2214yescomplemented
58#4948C2214yescomplemented
59#4947C2212yescomplemented
60#4946C2212yescomplemented
61#4940C2214yescomplemented
62#4940C2214yescomplemented
63#4939C2212yes—
64#4939C2212yes—
65C4×(C2×D3)⋊C4C2214yescomplemented
66C4×C4⋊Dic3C2212yescomplemented
67#4928C2212yescomplemented
68#4928C2212yescomplemented
69C2×#525—34no—
70#2683—31no—
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×#127D311yescharacteristic, complemented
80C4×#120—32no—
81#2293—32no—
82C24×D3C2312yes—
83C22×C3×D4C2314yes—
84C22×C3×D4C2311yescharacteristic
85C22×C3×D4C2311yescharacteristic
86C23×C3×C4C2312yes—
87C23×C3×C4C2312yes—
88C22×#67C2314yes—
89C22×#67C2314yes—
90C22×#67C2314yes—
91C23×Dic3C2312yes—
92C2×#235—216no—
93C2×#235—44no—
94C22×D12C2312yes—
95C22×D12C2312yes—
96C22×C4×D3—216no—
97C22×C4×D3—44no—
98C22×C4×D3C2314yes—
99C22×C4×D3C2314yes—
100C22×Dic6C2312yes—
101C3×#127—24no—
102C3×C4×D4—216no—
103C3×C4×D4—42no—
104C3×C4×D4—42no—
105C2×C3×C4⋊C4C2314yes—
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C22⋊C4C2314yes—
109C2×C3×C22⋊C4C2314yes—
110C2×C3×C22⋊C4C2312yes—
111C2×C3×C22⋊C4C2312yes—
112C2×C3×C42C2311yescharacteristic
113C2×C3×C42C2311yescharacteristic
114C2×C3×C42C2311yescharacteristic
115C2×C3×C42C2312yes—
116C2×C3×C42C2312yes—
117C2×#217C2312yes—
118C2×#217C2314yes—
119C2×#217C2314yes—
120C2×#217C2312yes—
121C2×#217C2314yes—
122#939—216no—
123#937—28no—
124#937—216no—
125C4×#67—216no—
126C4×#67—216no—
127C2×(C2×D3)⋊C4C2314yes—
128C2×(C2×D3)⋊C4C2314yes—
129C2×(C2×D3)⋊C4C2314yes—
130C2×(C2×D3)⋊C4C2314yes—
131C2×(C2×D3)⋊C4C2314yes—
132C2×(C2×D3)⋊C4C2314yes—
133C2×(C2×D3)⋊C4C2314yes—
134C2×C4⋊Dic3C2312yes—
135C2×C4⋊Dic3C2312yes—
136C2×C4⋊Dic3C2312yes—
137C2×Dic3⋊C4C2314yes—
138C2×Dic3⋊C4C2314yes—
139C2×Dic3⋊C4C2314yes—
140C2×Dic3⋊C4C2314yes—
141C2×C4×Dic3C2314yes—
142#896—216no—
143#894—216no—
144#884—216no—
145#883—216no—
146#882—216no—
147#878—216no—
148C4×D12D4116yescomplemented
149#872—216no—
150C3⋊#95C2314yes—
151C3⋊#95C2314yes—
152C3⋊#95C2314yes—
153C3⋊#95C2312yes—
154C3⋊#95C2312yes—
155C3⋊#95C2314yes—
156C22×#41—32no—
157C24×C4—32no—
158C2×#127C2×D311yescharacteristic
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×#120—32no—
168C2×#120—34no—
169C2×C4×D4—34no—
170C2×C4×D4—34no—
171C2×C4×D4—32no—
172C2×C4×D4C2×D311yescharacteristic
173C2×C4×D4C2×D314yescomplemented
174C2×C4×D4C2×D311yescharacteristic, complemented
175C2×#117—34no—
176#398—34no—
177#397—34no—
178#396—34no—
179#396—34no—
180#395—32no—
181#394—34no—
182#394—32no—
183#394—34no—
184#394—34no—
185#394—34no—
186#393—32no—
187#392—34no—
188#392—34no—
189#392—32no—
190#390—32no—
191#390—32no—
192#390C2×D311yescharacteristic
193#390C2×D314yescomplemented
194#390C2×D311yescharacteristic, complemented
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#384C2×D311yescharacteristic, complemented
204#379—32no—
205C4×C4⋊C4—32no—
206C4×C22⋊C4—32no—
207C4×C22⋊C4—32no—
208C4×C22⋊C4—34no—
209C43C2×D311yescharacteristic, complemented
210C24×C3C2412yes—
211C24×C3C2412yes—
212C23×D3—24no—
213C23×D3—28no—
214C23×D3—42no—
215C23×D3—44no—
216C23×D3—42no—
217C23×D3C2414yes—
218C2×C3×D4—28no—
219C2×C3×D4—28no—
220C2×C3×D4—22no—
221C2×C3×D4—41no—
222C2×C3×D4—41no—
223C2×C3×D4—41no—
224C2×C3×D4—44no—
225C2×C3×D4—41no—
226C2×C3×D4#4114yescomplemented
227C22×C3×C4—28no—
228C22×C3×C4—28no—
229C22×C3×C4—24no—
230C22×C3×C4—24no—
231C22×C3×C4C2411yescharacteristic
232C22×C3×C4C2412yes—
233C22×C3×C4C2412yes—
234C22×C3×C4C2412yes—
235C22×C3×C4C2412yes—
236C22×C3×C4C2411yescharacteristic
237C22×C3×C4C2411yescharacteristic
238C2×#67—28no—
239C2×#67—28no—
240C2×#67—28no—
241C2×#67—28no—
242C2×#67—28no—
243C2×#67—44no—
244C2×#67—44no—
245C2×#67—44no—
246C2×#67—44no—
247C22×Dic3—28no—
248C22×Dic3—42no—
249C22×Dic3C2414yes—
250C22×Dic3C2414yes—
251C22×Dic3C2414yes—
252#235—432no—
253C2×D12—28no—
254C2×D12—42no—
255C2×D12—42no—
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—432no—
263C2×C4×D3—44no—
264C2×C4×D3C2×D4116yes—
265C2×Dic6—28no—
266C2×Dic6—42no—
267C3×C4⋊C4—28no—
268C3×C4⋊C4—41no—
269C3×C4⋊C4—41no—
270C3×C22⋊C4—28no—
271C3×C22⋊C4—28no—
272C3×C22⋊C4—42no—
273C3×C22⋊C4—42no—
274C3×C42—22no—
275C3×C42—22no—
276C3×C42—41no—
277C3×C42—42no—
278C3×C42C2×D418yes—
279#217—24no—
280#217—28no—
281#217—28no—
282#217—42no—
283#217—44no—
284(C2×D3)⋊C4—28no—
285(C2×D3)⋊C4—28no—
286(C2×D3)⋊C4—28no—
287(C2×D3)⋊C4—28no—
288(C2×D3)⋊C4—28no—
289(C2×D3)⋊C4—44no—
290(C2×D3)⋊C4C2×D4116yes—
291C4⋊Dic3—42no—
292C4⋊Dic3—42no—
293C4⋊Dic3C2×D418yes—
294Dic3⋊C4—28no—
295Dic3⋊C4—28no—
296Dic3⋊C4—28no—
297Dic3⋊C4—28no—
298C4×Dic3—28no—
299C25—32no—
300C2×#41—616no—
301C2×#41—124no—
302C22×Q8—32no—
303C22×D4—34no—
304C22×D4—34no—
305C22×D4—34no—
306C22×D4—32no—
307C22×D4—32no—
308C22×D4C22×D314yes—
309C22×D4C22×D311yescharacteristic
310C22×D4C22×D311yescharacteristic
311C23×C4—34no—
312C23×C4—32no—
313C23×C4—34no—
314C23×C4—616no—
315C23×C4—124no—
316C23×C4C22×D312yes—
317C23×C4C22×D312yes—
318#127—24no—
319#123—616no—
320#123—616no—
321C22⋊Q8—616no—
322C22⋊Q8—616no—
323#121—616no—
324#121—616no—
325#120—616no—
326#120—616no—
327#120—68no—
328C4×D4—216no—
329C4×D4—316no—
330C4×D4—42no—
331C4×D4—42no—
332C4×D4—616no—
333C4×D4—616no—
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—32no—
341C2×C4⋊C4—32no—
342C2×C4⋊C4C22×D314yes—
343C2×C4⋊C4C22×D311yescharacteristic
344C2×C4⋊C4C22×D311yescharacteristic
345C2×C22⋊C4—34no—
346C2×C22⋊C4—34no—
347C2×C22⋊C4—34no—
348C2×C22⋊C4—32no—
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—34no—
356C2×C22⋊C4—34no—
357C2×C22⋊C4C22×D314yes—
358C2×C22⋊C4C22×D314yes—
359C2×C22⋊C4C22×D312yes—
360C2×C22⋊C4C22×D312yes—
361C2×C42—34no—
362C2×C42C22×D311yescharacteristic
363C2×C42C22×D311yescharacteristic
364C2×C42C22×D312yes—
365C2×C42C22×D311yescharacteristic
366C2×C42C22×D312yes—
367#95—34no—
368#95—34no—
369#95—34no—
370#95—32no—
371#95—32no—
372#95—34no—
373C23×C3—24no—
374C23×C3—22no—
375C23×C3—22no—
376C23×C3—24no—
377C23×C3—24no—
378C23×C3—22no—
379C23×C3—22no—
380C23×C3—24no—
381C23×C3—22no—
382C23×C3C2×#4114yes—
383C23×C3C2511yescharacteristic
384C22×D3—24no—
385C22×D3—24no—
386C22×D3—24no—
387C22×D3—24no—
388C22×D3—24no—
389C22×D3—24no—
390C22×D3—48no—
391C22×D3—416no—
392C22×D3—416no—
393C22×D3—48no—
394C22×D3—48no—
395C22×D3C22×D418yes—
396C3×D4—44no—
397C3×D4—416no—
398C3×D4—44no—
399C2×C3×C4—22no—
400C2×C3×C4—21no—
401C2×C3×C4—22no—
402C2×C3×C4—22no—
403C2×C3×C4—21no—
404C2×C3×C4—22no—
405C2×C3×C4—24no—
406C2×C3×C4—24no—
407C2×C3×C4—21no—
408C2×C3×C4—22no—
409C2×C3×C4—22no—
410C2×C3×C4—22no—
411C2×C3×C4—48no—
412C2×C3×C4—48no—
413C2×C3×C4C22×D414yes—
414C2×C3×C4C22×D414yes—
415C2×C3×C4C22×D414yes—
416C2×C3×C4C2×#4112yes—
417C2×C3×C4H(5,𝔽2)12yes—
418#67—416no—
419#67—416no—
420#67—416no—
421C2×Dic3—24no—
422C2×Dic3—24no—
423C2×Dic3—24no—
424C2×Dic3—24no—
425C2×Dic3—24no—
426C2×Dic3—24no—
427C2×Dic3—24no—
428C2×Dic3—24no—
429C2×Dic3—48no—
430C2×Dic3C22×D418yes—
431D12—48no—
432D12—48no—
433C4×D3—232no—
434C4×D3—416no—
435Dic6—48no—
436C24—34no—
437C24—68no—
438C24—64no—
439C24—122no—
440C24—122no—
441C24—124no—
442C24C23×D312yes—
443C24C23×D312yes—
444#41—1232no—
445C2×Q8—68no—
446C2×Q8—122no—
447C2×D4—28no—
448C2×D4—22no—
449C2×D4—28no—
450C2×D4—38no—
451C2×D4—41no—
452C2×D4—41no—
453C2×D4—41no—
454C2×D4—44no—
455C2×D4—41no—
456C2×D4—68no—
457C2×D4—68no—
458C2×D4—68no—
459C2×D4—68no—
460C2×D4—68no—
461C2×D4—68no—
462C2×D4—122no—
463C2×D4—122no—
464C2×D4—122no—
465C2×D4—124no—
466C2×D4—124no—
467C2×D4—124no—
468C2×D4—124no—
469C2×D4#23514yescomplemented
470C22×C4—28no—
471C22×C4—28no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—34no—
475C22×C4—316no—
476C22×C4—34no—
477C22×C4—34no—
478C22×C4—68no—
479C22×C4—68no—
480C22×C4—68no—
481C22×C4—68no—
482C22×C4—68no—
483C22×C4—124no—
484C22×C4—1232no—
485C22×C4—122no—
486C22×C4C23×D311yescharacteristic
487C22×C4C23×D312yes—
488C22×C4C23×D312yes—
489C22×C4C23×D311yescharacteristic
490C22×C4C23×D312yes—
491C22×C4C23×D312yes—
492C22×C4C23×D311yescharacteristic
493C4⋊C4—28no—
494C4⋊C4—38no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—68no—
498C4⋊C4—68no—
499C4⋊C4—68no—
500C4⋊C4—68no—
501C4⋊C4—122no—
502C4⋊C4—122no—
503C22⋊C4—28no—
504C22⋊C4—28no—
505C22⋊C4—316no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—68no—
509C22⋊C4—68no—
510C22⋊C4—68no—
511C22⋊C4—68no—
512C22⋊C4—64no—
513C22⋊C4—68no—
514C22⋊C4—68no—
515C22⋊C4—68no—
516C22⋊C4—124no—
517C22⋊C4—124no—
518C22⋊C4—122no—
519C42—22no—
520C42—22no—
521C42—41no—
522C42—42no—
523C42—68no—
524C42D3×D418yes—
525C22×C3—24no—
526C22×C3—28no—
527C22×C3—44no—
528C22×C3—42no—
529C22×C3—42no—
530C22×C3—44no—
531C22×C3—42no—
532C22×C3—42no—
533C22×C3—44no—
534C22×C3—42no—
535C22×C3C23×D412yes—
536C22×C3C22×#4111yescharacteristic
537C22×C3C2×H(5,𝔽2)11yescharacteristic
538C22×C3#58511yescharacteristic
539C22×C3#58512yes—
540C2×D3—28no—
541C2×D3—28no—
542C2×D3—28no—
543C2×D3—432no—
544C2×D3—44no—
545C2×D3—44no—
546C2×D3—44no—
547C2×D3—44no—
548C3×C4—24no—
549C3×C4—22no—
550C3×C4—24no—
551C3×C4—42no—
552C3×C4—42no—
553C3×C4—41no—
554C3×C4D4214yes—
555Dic3—28no—
556Dic3—44no—
557Dic3—44no—
558C23—24no—
559C23—22no—
560C23—22no—
561C23—24no—
562C23—22no—
563C23—24no—
564C23—22no—
565C23—24no—
566C23—22no—
567C23—38no—
568C23—64no—
569C23—64no—
570C23—64no—
571C23—64no—
572C23—64no—
573C23—64no—
574C23—1216no—
575C23—128no—
576C23—128no—
577C23—1216no—
578C23—128no—
579C23C2×#23514yes—
580C23C24×D311yescharacteristic
581Q8—128no—
582D4—44no—
583D4—416no—
584D4—44no—
585D4—128no—
586D4—128no—
587D4—1216no—
588D4—1216no—
589D4—1216no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—38no—
603C2×C4—48no—
604C2×C4—48no—
605C2×C4—64no—
606C2×C4—64no—
607C2×C4—632no—
608C2×C4—64no—
609C2×C4—64no—
610C2×C4—64no—
611C2×C4—64no—
612C2×C4—64no—
613C2×C4—64no—
614C2×C4—1216no—
615C2×C4—128no—
616C2×C4C2×#23512yes—
617C2×C4C2×D3×D414yes—
618C2×C4C2×D3×D414yes—
619C2×C4C2×D3×D414yes—
620C2×C4#100412yes—
621C2×C3—44no—
622C2×C3—82no—
623C2×C3#344912yes—
624C2×C3#345911yescharacteristic
625C2×C3C2×D4211yescharacteristic
626C2×C3D4×#4112yes—
627C2×C3#352911yescharacteristic
628D3—48no—
629C22—24no—
630C22—28no—
631C22—42no—
632C22—44no—
633C22—42no—
634C22—42no—
635C22—44no—
636C22—42no—
637C22—44no—
638C22—42no—
639C22—68no—
640C22—68no—
641C22—68no—
642C22—1232no—
643C22—124no—
644C22—124no—
645C22—124no—
646C22—124no—
647C22C22×#23511yescharacteristic
648C22C22×D3×D412yes—
649C22C2×#100411yescharacteristic
650C22#596411yescharacteristic
651C22#596412yes—
652C4—22no—
653C4—24no—
654C4—24no—
655C4—42no—
656C4—42no—
657C4—41no—
658C4—68no—
659C4—124no—
660C4—124no—
661C4#571714yes—
662C3#5210311yescharacteristic, complemented, Sylow
663C2—44no—
664C2—82no—
665C2—128no—
666C2#8577611yescharacteristic
667C2D4×#23512yes—
668C2C2×#571711yescharacteristic
669C2#8623512yes—
670C2#8624811yescharacteristic
subgroupG/H
centerC22×C4C23×D3
commutator, socle, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC3×C4×#127C2
hypercenterC4×#127D3
hypocenterC3#52103
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C4×#127

Complements

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

NH ≅ G/Ntype
1. C2×#5547C2semidirect
2. C3×C4×#127C2semidirect
3. #83190C2semidirect
4. #82401C2semidirect
5. #82400C2semidirect
6. C4×#937C2semidirect
7. #76466C2semidirect
8. #76465C2semidirect
9. #76464C2semidirect
10. #76462C2semidirect
11. #76380C2semidirect
12. #74371C2semidirect
13. #73299C2semidirect
16. C23×C4×D3C22semidirect
19. C2×C3×C4×D4C22semidirect
20. C2×C3×C4×D4C22semidirect
23. C2×#937C22semidirect
25. C2×C4×#67C22semidirect
27. C2×#896C22semidirect
30. C2×#883C22semidirect
32. C2×#878C22semidirect
34. C2×#872C22semidirect
36. C3×#390C22semidirect
37. C3×#390C22semidirect
38. C3×#384C22semidirect
39. C3×C43C22semidirect
41. #5223C22semidirect
42. #5222C22semidirect
44. C4×#217C22semidirect
45. C4×#217C22semidirect
46. #4992C22semidirect
48. #4989C22semidirect
49. #4983C22semidirect
50. #4979C22semidirect
52. #4961C22semidirect
54. #4959C22semidirect
56. #4955C22semidirect
57. #4948C22semidirect
58. #4948C22semidirect
59. #4947C22semidirect
60. #4946C22semidirect
61. #4940C22semidirect
62. #4940C22semidirect
65. C4×(C2×D3)⋊C4C22semidirect
66. C4×C4⋊Dic3C22semidirect
67. #4928C22semidirect
68. #4928C22semidirect
79. C4×#127D3semidirect
148. C4×D12D4semidirect
173. C2×C4×D4C2×D3semidirect
174. C2×C4×D4C2×D3semidirect
193. #390C2×D3semidirect
194. #390C2×D3semidirect
203. #384C2×D3semidirect
209. C43C2×D3semidirect
226. C2×C3×D4#41semidirect
469. C2×D4#235semidirect
662. C3#52103semidirect

Maximal subgroups

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

subgroupindex
C2×#55472
C3×C4×#1272
#831902
#824012
#824002
C4×#9372
#764662
#764652
#764642
#764622
#763802
#743712
#732992
#521033