Finite Group Databasebeta

← #10967479

Subgroups — #10967479

Order 768.

3114 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×D4C211yesmaximal, characteristic, complemented
2C2×#5479C211yesmaximal, characteristic, complemented
3#84050C211yesmaximal, characteristic, complemented
4#84049C211yesmaximal, characteristic
5C4×#953C214yesmaximal, complemented
6#82401C214yesmaximal, complemented
7#82390C214yesmaximal, complemented
8#82386C212yesmaximal, complemented
9#82381C212yesmaximal, complemented
10C4×#940C212yesmaximal, complemented
11C4×#936C212yesmaximal, complemented
12#82180C211yesmaximal, characteristic, complemented
13#82178C211yesmaximal, characteristic, complemented
14#82106C214yesmaximal, complemented
15C4×#926C211yesmaximal, characteristic, complemented
16#34965—31nomaximal, Sylow
17C23×C3×D4C2211yescharacteristic
18C24×C3×C4C2212yes—
19C2×C3×C4×D4—28no—
20C2×C3×C4×D4C2212yes—
21C2×C3×C4×D4C2212yes—
22C2×C3×C4×D4C2212yes—
23C2×C3×C4×D4C2212yes—
24C22×C3×C4⋊C4C2211yescharacteristic
25C22×C3×C22⋊C4C2212yes—
26C22×C3×C42C2211yescharacteristic
27C2×#953C2214yes—
28C2×#940C2212yes—
29C2×#936C2212yes—
30C2×#930C2211yescharacteristic
31C2×#929C2212yes—
32C2×C4×#67C2212yes—
33C2×C4×#67C2218yescomplemented
34C2×#926C2212yes—
35C2×#924C2211yescharacteristic
36#5479—28no—
37C2×#876C2212yescomplemented
38C2×#875C2211yescharacteristic, complemented
39C2×C4×D12C2211yescharacteristic, complemented
40C2×#872C2212yescomplemented
41C2×#870C2211yescharacteristic, complemented
42C2×C4×Dic6C2211yescharacteristic, complemented
43#5247C2214yes—
44#5223C2214yes—
45#5222C2214yes—
46#5221C2212yes—
47#5221C2212yes—
48#5213C2214yes—
49#5213C2218yescomplemented
50#5210C2214yes—
51#5210C2218yescomplemented
52C4×#217C2211yescharacteristic
53C4×#217C2211yescharacteristic
54C4×#217C2212yes—
55C4×#217C2218yescomplemented
56#4992C2214yescomplemented
57#4988C2212yescomplemented
58#4983C2214yescomplemented
59#4977C2212yescomplemented
60#4975C2212yescomplemented
61#4973C2212yescomplemented
62#4961C2218yescomplemented
63#4957C2218yescomplemented
64#4948C2218yescomplemented
65#4945C2218yescomplemented
66C4×C4⋊Dic3C2212yescomplemented
67C4×C4⋊Dic3C2212yescomplemented
68#4932C2212yescomplemented
69C42×Dic3C2212yescomplemented
70C2×#529—31no—
71C22×C4×D4D311yescharacteristic, complemented
72#2603—34no—
73#2566—32no—
74#2544—32no—
75#2504—31no—
76#2502—31no—
77#2435—31no—
78#2433—31no—
79#2411—34no—
80#2392—34no—
81C4×#127—32no—
82C4×#124—32no—
83C4×#120—34no—
84C4×#117—31no—
85C25×C3C2312yes—
86C22×C3×D4—28no—
87C22×C3×D4—24no—
88C22×C3×D4C2312yes—
89C22×C3×D4C2312yes—
90C23×C3×C4—28no—
91C23×C3×C4—48no—
92C23×C3×C4C2314yes—
93C23×C3×C4C2311yescharacteristic
94C23×C3×C4C2311yescharacteristic
95C23×C3×C4C2311yescharacteristic
96C23×C3×C4C2312yes—
97C23×C3×C4C2314yes—
98C22×#67C2312yes—
99C22×#67C2318yes—
100C22×D12C2311yescharacteristic
101C22×C4×D3C2312yes—
102C22×Dic6C2311yescharacteristic
103C3×C4×D4—28no—
104C3×C4×D4—28no—
105C3×C4×D4—28no—
106C3×C4×D4—28no—
107C2×C3×C4⋊C4—24no—
108C2×C3×C4⋊C4C2312yes—
109C2×C3×C4⋊C4C2312yes—
110C2×C3×C22⋊C4—28no—
111C2×C3×C22⋊C4C2314yes—
112C2×C3×C22⋊C4C2314yes—
113C2×C3×C42D418yescomplemented
114C2×C3×C42C2311yescharacteristic
115C2×C3×C42C2311yescharacteristic
116C2×C3×C42C2311yescharacteristic
117C2×C3×C42C2311yescharacteristic
118#953—216no—
119C2×#217C2318yes—
120C2×#217C2318yes—
121C2×#217C2312yes—
122C2×#217C2312yes—
123C2×#217C2318yes—
124C2×#217C2312yes—
125#940—28no—
126#936—28no—
127#930—24no—
128#929—28no—
129C4×#67—28no—
130C4×#67—416no—
131C2×(C2×D3)⋊C4C2312yes—
132C2×(C2×D3)⋊C4C2312yes—
133C2×(C2×D3)⋊C4C2312yes—
134C2×(C2×D3)⋊C4C2312yes—
135C2×(C2×D3)⋊C4C2318yes—
136#926—28no—
137C2×C4⋊Dic3C2314yes—
138C2×C4⋊Dic3C2311yescharacteristic
139C2×C4⋊Dic3C2311yescharacteristic
140#924—24no—
141C2×Dic3⋊C4C2312yes—
142C2×Dic3⋊C4C2312yes—
143C2×Dic3⋊C4C2312yes—
144C2×Dic3⋊C4C2312yes—
145C2×Dic3⋊C4C2318yes—
146C2×C4×Dic3C2314yes—
147C2×C4×Dic3C2314yes—
148C2×C4×Dic3C2312yes—
149C2×C4×Dic3C2312yes—
150C2×C4×Dic3C2312yes—
151#876—28no—
152#875—24no—
153C4×D12—24no—
154#872—28no—
155#870—24no—
156C4×Dic6—24no—
157C3⋊#95C2314yes—
158C3⋊#95C2312yes—
159C3⋊#95C2312yes—
160C3⋊#95C2318yes—
161C3⋊#95C2318yes—
162C23×D4C2×D311yescharacteristic
163C24×C4C2×D312yes—
164C2×#127—32no—
165#529—68no—
166C2×#126—32no—
167C2×#125—31no—
168C2×#124—32no—
169C2×#124—31no—
170C2×#123—32no—
171C2×C22⋊Q8—31no—
172C2×#121—31no—
173C2×#120—34no—
174C2×C4×Q8—31no—
175C2×C4×D4—28no—
176C2×C4×D4—32no—
177C2×C4×D4—38no—
178C2×C4×D4—31no—
179C2×C4×D4C2×D312yes—
180C2×C4×D4C2×D312yes—
181C2×C4×D4C2×D312yes—
182C2×C4×D4C2×D312yes—
183C2×#117—32no—
184C2×#117—32no—
185C22×C4⋊C4C2×D311yescharacteristic
186C22×C22⋊C4C2×D312yes—
187C22×C42C2×D311yescharacteristic
188#398—34no—
189#397—34no—
190#396—38no—
191#394—34no—
192#394—38no—
193#392—34no—
194#391—32no—
195#390—32no—
196#390—32no—
197#390—32no—
198#390—38no—
199#388—38no—
200#386—38no—
201#386—34no—
202#385—38no—
203#385—34no—
204#384—32no—
205#382—32no—
206#379—34no—
207C4×C4⋊C4—32no—
208C4×C4⋊C4—32no—
209C4×C22⋊C4—38no—
210C4×C22⋊C4—31no—
211C4×C22⋊C4—31no—
212C4×C22⋊C4—32no—
213#376—32no—
214C43—32no—
215C24×C3—24no—
216C24×C3—28no—
217C24×C3—28no—
218C24×C3—44no—
219C24×C3C2414yes—
220C24×C3C2411yescharacteristic
221C23×D3C2412yes—
222C2×C3×D4—232no—
223C2×C3×D4—28no—
224C2×C3×D4—24no—
225C2×C3×D4—24no—
226C2×C3×D4—24no—
227C2×C3×D4#4118yes—
228C22×C3×C4—24no—
229C22×C3×C4—24no—
230C22×C3×C4—24no—
231C22×C3×C4—24no—
232C22×C3×C4—28no—
233C22×C3×C4—28no—
234C22×C3×C4—216no—
235C22×C3×C4—48no—
236C22×C3×C4—44no—
237C22×C3×C4—432no—
238C22×C3×C4C2×D414yes—
239C22×C3×C4C2×D414yes—
240C22×C3×C4C2×D414yes—
241C22×C3×C4C2414yes—
242C22×C3×C4C2411yescharacteristic
243C22×C3×C4C2411yescharacteristic
244C22×C3×C4C2411yescharacteristic
245C22×C3×C4C2411yescharacteristic
246C22×C3×C4C2411yescharacteristic
247C22×C3×C4C2411yescharacteristic
248C2×#67—24no—
249C2×#67—216no—
250C2×#67—28no—
251C2×#67—216no—
252C2×#67—48no—
253C22×Dic3C2414yes—
254C22×Dic3C2412yes—
255C22×Dic3C2412yes—
256C22×Dic3C2412yes—
257C22×Dic3C2414yes—
258C2×D12—22no—
259C2×D12—24no—
260C2×C4×D3—24no—
261C2×C4×D3—28no—
262C2×Dic6—22no—
263C2×Dic6—24no—
264C3×C4⋊C4—24no—
265C3×C4⋊C4—24no—
266C3×C22⋊C4—28no—
267C3×C22⋊C4—28no—
268C3×C42C2×D414yes—
269C3×C42C2×D414yes—
270C3×C42C2×D414yes—
271C3×C42C2×D414yes—
272#217—24no—
273#217—24no—
274#217—216no—
275#217—216no—
276#217—48no—
277#217#4118yes—
278(C2×D3)⋊C4—24no—
279(C2×D3)⋊C4—24no—
280(C2×D3)⋊C4—24no—
281(C2×D3)⋊C4—24no—
282(C2×D3)⋊C4—48no—
283C4⋊Dic3—22no—
284C4⋊Dic3—22no—
285C4⋊Dic3—28no—
286Dic3⋊C4—24no—
287Dic3⋊C4—24no—
288Dic3⋊C4—24no—
289Dic3⋊C4—24no—
290Dic3⋊C4—48no—
291C4×Dic3—24no—
292C4×Dic3—24no—
293C4×Dic3—24no—
294C4×Dic3—28no—
295C4×Dic3—28no—
296C25C22×D312yes—
297C22×Q8—31no—
298C22×D4—28no—
299C22×D4—24no—
300C22×D4—32no—
301C22×D4—38no—
302C22×D4—31no—
303C22×D4C22×D312yes—
304C22×D4C22×D312yes—
305C23×C4—28no—
306C23×C4—32no—
307C23×C4—48no—
308C23×C4C22×D314yes—
309C23×C4C22×D311yescharacteristic
310C23×C4C22×D311yescharacteristic
311C23×C4C22×D311yescharacteristic
312C23×C4C22×D312yes—
313C23×C4C22×D314yes—
314#127—68no—
315#126—68no—
316#125—64no—
317#124—68no—
318#124—64no—
319#123—68no—
320C22⋊Q8—64no—
321#121—64no—
322#120—616no—
323C4×Q8—64no—
324C4×D4—28no—
325C4×D4—28no—
326C4×D4—28no—
327C4×D4—28no—
328C4×D4—68no—
329C4×D4—64no—
330C4×D4—1216no—
331#117—68no—
332#117—68no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—32no—
335C2×C4⋊C4—32no—
336C2×C4⋊C4—34no—
337C2×C4⋊C4—31no—
338C2×C4⋊C4—32no—
339C2×C4⋊C4—31no—
340C2×C4⋊C4—32no—
341C2×C4⋊C4—38no—
342C2×C4⋊C4C22×D312yes—
343C2×C4⋊C4C22×D312yes—
344C2×C22⋊C4—28no—
345C2×C22⋊C4—32no—
346C2×C22⋊C4—32no—
347C2×C22⋊C4—32no—
348C2×C22⋊C4—38no—
349C2×C22⋊C4—38no—
350C2×C22⋊C4—32no—
351C2×C22⋊C4—32no—
352C2×C22⋊C4—32no—
353C2×C22⋊C4—32no—
354C2×C22⋊C4—38no—
355C2×C22⋊C4—38no—
356C2×C22⋊C4C22×D314yes—
357C2×C22⋊C4C22×D314yes—
358C2×C42—34no—
359C2×C42—32no—
360C2×C42—32no—
361C2×C42—32no—
362C2×C42—34no—
363C2×C42#6718yescomplemented
364C2×C42C22×D311yescharacteristic
365C2×C42C22×D311yescharacteristic
366C2×C42C22×D311yescharacteristic
367C2×C42C22×D311yescharacteristic
368#95—34no—
369#95—32no—
370#95—32no—
371#95—38no—
372#95—38no—
373C23×C3—22no—
374C23×C3—22no—
375C23×C3—216no—
376C23×C3—24no—
377C23×C3—28no—
378C23×C3—24no—
379C23×C3—28no—
380C23×C3—432no—
381C23×C3—48no—
382C23×C3—48no—
383C23×C3C22×D412yes—
384C23×C3C2×#4118yes—
385C23×C3C2×#4114yes—
386C23×C3C2511yescharacteristic
387C22×D3—22no—
388C22×D3—24no—
389C22×D3—24no—
390C22×D3—24no—
391C3×D4—216no—
392C3×D4—216no—
393C2×C3×C4—28no—
394C2×C3×C4—22no—
395C2×C3×C4—22no—
396C2×C3×C4—22no—
397C2×C3×C4—22no—
398C2×C3×C4—28no—
399C2×C3×C4—28no—
400C2×C3×C4—24no—
401C2×C3×C4—28no—
402C2×C3×C4—416no—
403C2×C3×C4—48no—
404C2×C3×C4—416no—
405C2×C3×C4C22×D412yes—
406C2×C3×C4C22×D412yes—
407C2×C3×C4C22×D412yes—
408C2×C3×C4C22×D412yes—
409C2×C3×C4C22×D412yes—
410C2×C3×C4C22×D412yes—
411C2×C3×C4C2×#4114yes—
412C2×C3×C4H(5,𝔽2)14yes—
413#67—432no—
414#67—48no—
415C2×Dic3—24no—
416C2×Dic3—24no—
417C2×Dic3—22no—
418C2×Dic3—22no—
419C2×Dic3—22no—
420C2×Dic3—24no—
421C2×Dic3—24no—
422C2×Dic3—24no—
423C2×Dic3—24no—
424C2×Dic3—24no—
425C2×Dic3—24no—
426C2×Dic3C2×#4118yes—
427D12—44no—
428C4×D3—48no—
429Dic6—44no—
430C24—28no—
431C24—24no—
432C24—28no—
433C24—32no—
434C24—44no—
435C24C23×D314yes—
436C24C23×D311yescharacteristic
437C2×Q8—64no—
438C2×Q8—62no—
439C2×D4—232no—
440C2×D4—28no—
441C2×D4—24no—
442C2×D4—24no—
443C2×D4—24no—
444C2×D4—616no—
445C2×D4—616no—
446C2×D4—68no—
447C2×D4—64no—
448C2×D4—64no—
449C2×D4—62no—
450C2×D4—128no—
451C2×D4#23518yes—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—216no—
457C22×C4—28no—
458C22×C4—28no—
459C22×C4—34no—
460C22×C4—34no—
461C22×C4—32no—
462C22×C4—32no—
463C22×C4—32no—
464C22×C4—48no—
465C22×C4—44no—
466C22×C4—432no—
467C22×C4—68no—
468C22×C4—64no—
469C22×C4C2×#6714yes—
470C22×C4C2×#6714yes—
471C22×C4C2×#6714yes—
472C22×C4C23×D314yes—
473C22×C4C23×D311yescharacteristic
474C22×C4C23×D311yescharacteristic
475C22×C4C23×D311yescharacteristic
476C22×C4C23×D311yescharacteristic
477C22×C4C23×D311yescharacteristic
478C22×C4C23×D311yescharacteristic
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—68no—
482C4⋊C4—64no—
483C4⋊C4—64no—
484C4⋊C4—62no—
485C4⋊C4—64no—
486C4⋊C4—62no—
487C4⋊C4—64no—
488C4⋊C4—128no—
489C22⋊C4—28no—
490C22⋊C4—28no—
491C22⋊C4—38no—
492C22⋊C4—616no—
493C22⋊C4—616no—
494C22⋊C4—64no—
495C22⋊C4—64no—
496C22⋊C4—64no—
497C22⋊C4—64no—
498C22⋊C4—64no—
499C22⋊C4—64no—
500C22⋊C4—128no—
501C22⋊C4—128no—
502C42—68no—
503C42—64no—
504C42—64no—
505C42—68no—
506C42—64no—
507C42C2×#6714yes—
508C42C2×#6714yes—
509C42C2×#6714yes—
510C42C2×#6714yes—
511C22×C3—22no—
512C22×C3—22no—
513C22×C3—28no—
514C22×C3—28no—
515C22×C3—216no—
516C22×C3—432no—
517C22×C3—44no—
518C22×C3—44no—
519C22×C3—48no—
520C22×C3#53214yes—
521C22×C3C23×D411yescharacteristic
522C22×C3C22×#4112yes—
523C22×C3C22×#4112yes—
524C22×C3C2×H(5,𝔽2)12yes—
525C2×D3—42no—
526C2×D3—42no—
527C2×D3—42no—
528C2×D3—48no—
529C3×C4—22no—
530C3×C4—22no—
531C3×C4—22no—
532C3×C4—22no—
533C3×C4—44no—
534C3×C4#52514yes—
535C3×C4#53514yes—
536Dic3—28no—
537Dic3—42no—
538Dic3—42no—
539Dic3—42no—
540Dic3—44no—
541C23—22no—
542C23—22no—
543C23—216no—
544C23—24no—
545C23—28no—
546C23—28no—
547C23—24no—
548C23—432no—
549C23—48no—
550C23—48no—
551C23—64no—
552C23—64no—
553C23—64no—
554C23—62no—
555C23C2×#23518yes—
556C23D3×#4114yes—
557C23C22×#6712yes—
558C23C24×D311yescharacteristic
559Q8—124no—
560D4—216no—
561D4—216no—
562D4—1232no—
563D4—128no—
564D4—124no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—24no—
574C2×C4—38no—
575C2×C4—416no—
576C2×C4—48no—
577C2×C4—416no—
578C2×C4—64no—
579C2×C4—64no—
580C2×C4—64no—
581C2×C4—64no—
582C2×C4—64no—
583C2×C4—64no—
584C2×C4—64no—
585C2×C4—64no—
586C2×C4—62no—
587C2×C4—62no—
588C2×C4—62no—
589C2×C4—128no—
590C2×C4C2×#23514yes—
591C2×C4#100414yes—
592C2×C4C22×#6712yes—
593C2×C4C22×#6712yes—
594C2×C4C22×#6712yes—
595C2×C4C22×#6712yes—
596C2×C4C22×#6712yes—
597C2×C4C22×#6712yes—
598C2×C3—24no—
599C2×C3—48no—
600C2×C3C2×#52511yescharacteristic
601C2×C3C2×#53211yescharacteristic
602C2×C3C2×#53511yescharacteristic
603C2×C3#348114yes—
604D3—82no—
605C22—28no—
606C22—22no—
607C22—22no—
608C22—216no—
609C22—28no—
610C22—432no—
611C22—48no—
612C22—44no—
613C22—44no—
614C22—128no—
615C22—122no—
616C22—122no—
617C22—122no—
618C22#583214yes—
619C22C22×#23512yes—
620C22C2×#100412yes—
621C22C2×D3×#4112yes—
622C22C23×#6711yescharacteristic
623C4—22no—
624C4—22no—
625C4—22no—
626C4—22no—
627C4—44no—
628C4—68no—
629C4—124no—
630C4—122no—
631C4—122no—
632C4—122no—
633C4#579114yes—
634C4#580514yes—
635C3#3496511yescharacteristic, complemented, Sylow
636C2—24no—
637C2—48no—
638C2—242no—
639C2#8578514yes—
640C2C2×#579111yescharacteristic
641C2C2×#580511yescharacteristic
642C2C2×#583211yescharacteristic
subgroupG/H
centerC22×C4C23×D3
commutator, socle, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC22×C3×C4×D4C2
prime radical#84049C2
hypercenterC22×C4×D4D3
hypocenterC3#34965
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
1. C22×C3×C4×D4C2semidirect
2. C2×#5479C2semidirect
3. #84050C2semidirect
5. C4×#953C2semidirect
6. #82401C2semidirect
7. #82390C2semidirect
8. #82386C2semidirect
9. #82381C2semidirect
10. C4×#940C2semidirect
11. C4×#936C2semidirect
12. #82180C2semidirect
13. #82178C2semidirect
14. #82106C2semidirect
15. C4×#926C2semidirect
33. C2×C4×#67C22semidirect
37. C2×#876C22semidirect
38. C2×#875C22semidirect
39. C2×C4×D12C22semidirect
40. C2×#872C22semidirect
41. C2×#870C22semidirect
42. C2×C4×Dic6C22semidirect
49. #5213C22semidirect
51. #5210C22semidirect
55. C4×#217C22semidirect
56. #4992C22semidirect
57. #4988C22semidirect
58. #4983C22semidirect
59. #4977C22semidirect
60. #4975C22semidirect
61. #4973C22semidirect
62. #4961C22semidirect
63. #4957C22semidirect
64. #4948C22semidirect
65. #4945C22semidirect
66. C4×C4⋊Dic3C22semidirect
67. C4×C4⋊Dic3C22semidirect
68. #4932C22semidirect
69. C42×Dic3C22semidirect
71. C22×C4×D4D3semidirect
113. C2×C3×C42D4semidirect
363. C2×C42#67semidirect
635. C3#34965semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×D42
C2×#54792
#840502
#840492
C4×#9532
#824012
#823902
#823862
#823812
C4×#9402
C4×#9362
#821802
#821782
#821062
C4×#9262
#349653