Finite Group Databasebeta

← #10823406

Subgroups — #10823406

Order 768.

1508 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#532C211yesmaximal, characteristic, complemented
2C2×#5014C212yesmaximal, complemented
3C4×#930C211yesmaximal, characteristic, complemented
4#71792C211yesmaximal, characteristic, complemented
5#71783C211yesmaximal, characteristic, complemented
6#71767C211yesmaximal, characteristic, complemented
7#71762C211yesmaximal, characteristic, complemented
8#71684C212yesmaximal, complemented
9#71680C212yesmaximal, complemented
10#71609C212yesmaximal, complemented
11#71529C211yesmaximal, characteristic, complemented
12#23573—31nomaximal, Sylow
13C22×C3×#41—21no—
14C3×#532—28no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#127C2211yescharacteristic
17C2×C3×#124—21no—
18C2×C3×#121—21no—
19C2×C3×#121C2212yes—
20C2×C3×C4×D4—21no—
21C2×C3×C4×D4C2212yes—
22C22×C3×C42C2211yescharacteristic
23C22×#213C2212yes—
24C2×#930C2211yescharacteristic
25C2×C4×#67—21no—
26C2×C4×D12C2211yescharacteristic
27C2×#832C2212yes—
28C2×#832C2212yes—
29C2×#832C2212yes—
30#5213—21no—
31#5212C2211yescharacteristic
32C2×#830C2212yes—
33#5014—28no—
34C2×#809C2212yes—
35C2×#808C2212yes—
36#4940—21no—
37#4939C2211yescharacteristic
38C4×(C2×D3)⋊C4—21no—
39C4×C4⋊Dic3C2211yescharacteristic
40#4928C2211yescharacteristic
41C2×#803C2212yes—
42#4529C2212yes—
43#4529C2212yes—
44#4524C2211yescharacteristic
45#4524C2211yescharacteristic
46C2×#532D311yescharacteristic, complemented
47C2×#442—32no—
48C4×#121—31no—
49#2058—31no—
50#2041—31no—
51#2034—32no—
52#2023—31no—
53#2011—31no—
54#1999—32no—
55#1762—32no—
56#1749—31no—
57C2×C3×#41—28no—
58C2×C3×#41—44no—
59C2×C3×#41—44no—
60C22×C3×Q8—21no—
61C22×C3×D4—22no—
62C22×C3×D4—21no—
63C22×C3×D4—21no—
64C22×C3×D4C2312yes—
65C23×C3×C4—22no—
66C23×C3×C4D411yescharacteristic
67C23×C3×C4D411yescharacteristic
68C23×C3×C4C2311yescharacteristic
69C22×#67—21no—
70C22×D12C2311yescharacteristic
71C22×C4×D3—21no—
72C3×C4⋊Q8—24no—
73C3×#127—24no—
74C3×#124—28no—
75C3×#121—28no—
76C3×#121—24no—
77C3×#121—24no—
78C3×C4×D4—28no—
79C3×C4×D4—24no—
80C3×C4×D4D418yescomplemented
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C42—24no—
86C2×C3×C42D411yescharacteristic
87C2×C3×C42D411yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C2×#217—21no—
91C2×#213—24no—
92C2×#213—24no—
93C2×#213—24no—
94#930—22no—
95#930—22no—
96C4×#67—44no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×Dic3⋊C4—21no—
105C2×C4×Dic3—21no—
106C22×C3⋊C8C2312yes—
107C22×C3⋊C8C2312yes—
108C4×D12—24no—
109#832—24no—
110#832—24no—
111#832—24no—
112C3⋊#95—21no—
113C3⋊#95—21no—
114C3⋊#95—21no—
115#830—24no—
116#809—24no—
117#808—24no—
118#803—24no—
119C22×#41—21no—
120C22×D8—32no—
121#532—28no—
122C2×C4⋊Q8C2×D311yescharacteristic
123C2×#127C2×D311yescharacteristic
124C2×#124—21no—
125C2×#121—21no—
126C2×#121—31no—
127C2×#121C2×D312yes—
128C2×C4×D4—21no—
129C2×C4×D4—31no—
130C2×C4×D4—61no—
131C2×C4×D4C2×D312yes—
132C22×C42C2×D311yescharacteristic
133#442—68no—
134C2×#106—32no—
135C2×#102—32no—
136C2×#102—32no—
137C2×#102—32no—
138C2×#102—32no—
139C2×C22⋊C8—32no—
140C2×#97—32no—
141#390—31no—
142#389—31no—
143#388—61no—
144#386—61no—
145#380—31no—
146C4×C4⋊C4—31no—
147C4×C22⋊C4—61no—
148#340—32no—
149#340—32no—
150#336—31no—
151#336—31no—
152C24×C3—22no—
153C24×C3C2×D411yescharacteristic
154C23×D3—21no—
155C3×#41—432no—
156C2×C3×Q8—24no—
157C2×C3×Q8—42no—
158C2×C3×Q8—42no—
159C2×C3×D4—24no—
160C2×C3×D4—28no—
161C2×C3×D4—24no—
162C2×C3×D4—22no—
163C2×C3×D4—22no—
164C2×C3×D4—22no—
165C2×C3×D4—22no—
166C2×C3×D4—22no—
167C2×C3×D4—42no—
168C2×C3×D4—42no—
169C2×C3×D4—44no—
170C2×C3×D4—44no—
171C2×C3×D4—42no—
172C2×C3×D4—42no—
173C2×C3×D4C2×D414yes—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—28no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—41no—
186C22×C3×C4—44no—
187C22×C3×C4—44no—
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2411yescharacteristic
194C2×#67—24no—
195C2×#67—42no—
196C2×#67—42no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C2×D12—22no—
201C2×D12—22no—
202C2×D12—21no—
203C2×D12#4112yescomplemented
204C2×C4×D3—42no—
205C2×C4×D3—42no—
206C2×C4×D3—42no—
207C3×C4⋊C4—24no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4C2×D414yes—
210C3×C22⋊C4—28no—
211C3×C22⋊C4—28no—
212C3×C42—22no—
213C3×C42—22no—
214C3×C42—22no—
215C3×C42C2×D414yes—
216#217—42no—
217#213—216no—
218(C2×D3)⋊C4—24no—
219(C2×D3)⋊C4—42no—
220(C2×D3)⋊C4—42no—
221(C2×D3)⋊C4—42no—
222C4⋊Dic3—21no—
223C4⋊Dic3—22no—
224C4⋊Dic3—22no—
225C4⋊Dic3#4112yescomplemented
226Dic3⋊C4—42no—
227C4×Dic3—42no—
228C2×C3⋊C8—22no—
229C2×C3⋊C8—22no—
230C2×C3⋊C8—22no—
231C2×C3⋊C8—22no—
232C2×C3⋊C8—22no—
233C2×C3⋊C8#4114yescomplemented
234C2×#41—28no—
235C2×#41—44no—
236C2×#41—44no—
237C22×Q8—21no—
238C22×D4—22no—
239C22×D4—21no—
240C22×D4—21no—
241C22×D4—31no—
242C22×D4—61no—
243C22×D4C22×D312yes—
244C23×C4—22no—
245C23×C4—61no—
246C23×C4#6711yescharacteristic
247C23×C4#6711yescharacteristic
248C23×C4C22×D311yescharacteristic
249C2×D8—64no—
250C2×D8—64no—
251C2×D8—64no—
252C22×C8—32no—
253C22×C8—32no—
254C4⋊Q8—24no—
255#127—24no—
256#124—28no—
257#121—28no—
258#121—24no—
259#121—24no—
260#121—62no—
261#121—62no—
262C4×D4—28no—
263C4×D4—24no—
264C4×D4—64no—
265C4×D4—124no—
266C4×D4#6718yescomplemented
267C2×C4⋊C4—21no—
268C2×C4⋊C4—31no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—31no—
271C2×C4⋊C4—61no—
272C2×C4⋊C4C22×D312yes—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—61no—
276C2×C22⋊C4—61no—
277C2×C22⋊C4—61no—
278C2×C22⋊C4—61no—
279C2×C22⋊C4—61no—
280C2×C42—24no—
281C2×C42—61no—
282C2×C42#6711yescharacteristic
283C2×C42#6711yescharacteristic
284C2×C42C22×D311yescharacteristic
285C2×C42C22×D311yescharacteristic
286#106—64no—
287#102—64no—
288#102—64no—
289#102—64no—
290#102—64no—
291C22⋊C8—64no—
292#97—64no—
293#95—61no—
294#95—61no—
295#95—61no—
296C23×C3—21no—
297C23×C3—21no—
298C23×C3—21no—
299C23×C3—21no—
300C23×C3—21no—
301C23×C3—21no—
302C23×C3—24no—
303C23×C3—42no—
304C23×C3—42no—
305C23×C3—42no—
306C23×C3—42no—
307C23×C3—42no—
308C23×C3—42no—
309C23×C3AGL(1,ℤ/8)12yes—
310C23×C3C22×D411yescharacteristic
311C22×D3—22no—
312C22×D3—41no—
313C22×D3—41no—
314C22×D3—41no—
315C22×D3—41no—
316C22×D3—41no—
317C22×D3—41no—
318C3×Q8—48no—
319C3×D4—24no—
320C3×D4—24no—
321C3×D4—48no—
322C3×D4—48no—
323C3×D4—416no—
324C3×D4—42no—
325C3×D4—42no—
326C2×C3×C4—28no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—24no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—44no—
340C2×C3×C4—416no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4#12012yes—
346C2×C3×C4#12012yes—
347C2×C3×C4AGL(1,ℤ/8)12yes—
348C2×C3×C4AGL(1,ℤ/8)12yes—
349C2×C3×C4C22×D412yes—
350C2×C3×C4C2×#4111yescharacteristic
351C2×C3×C4C2×#4112yes—
352C2×C3×C4C2×#4111yescharacteristic
353#67—48no—
354C2×Dic3—22no—
355C2×Dic3—41no—
356C2×Dic3—41no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363D12—24no—
364D12—42no—
365C4×D3—84no—
366C3⋊C8—24no—
367C3⋊C8—42no—
368C24—22no—
369C24—61no—
370C24C2×#6711yescharacteristic
371#41—432no—
372C2×Q8—24no—
373C2×Q8—42no—
374C2×Q8—42no—
375C2×D4—24no—
376C2×D4—28no—
377C2×D4—24no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—32no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—64no—
391C2×D4—62no—
392C2×D4—61no—
393C2×D4—62no—
394C2×D4—122no—
395C2×D4—122no—
396C2×D4C2×#6714yes—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—28no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—41no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—61no—
412C22×C4—61no—
413C22×C4—61no—
414C22×C4—122no—
415C22×C4—122no—
416C22×C4—122no—
417C22×C4C2×#6711yescharacteristic
418C22×C4C2×#6711yescharacteristic
419C22×C4C2×#6711yescharacteristic
420C22×C4C2×#6711yescharacteristic
421C22×C4C2×#6711yescharacteristic
422C22×C4C23×D311yescharacteristic
423D8—616no—
424C2×C8—34no—
425C2×C8—62no—
426C2×C8—62no—
427C2×C8—62no—
428C2×C8—62no—
429C2×C8—62no—
430C4⋊C4—24no—
431C4⋊C4—22no—
432C4⋊C4—32no—
433C4⋊C4—61no—
434C4⋊C4—62no—
435C4⋊C4—62no—
436C4⋊C4—122no—
437C4⋊C4C2×#6714yes—
438C22⋊C4—28no—
439C22⋊C4—28no—
440C22⋊C4—64no—
441C22⋊C4—122no—
442C22⋊C4—122no—
443C22⋊C4—122no—
444C22⋊C4—122no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—122no—
449C42C2×#6714yes—
450C22×C3—22no—
451C22×C3—22no—
452C22×C3—22no—
453C22×C3—41no—
454C22×C3—41no—
455C22×C3—41no—
456C22×C3—41no—
457C22×C3—44no—
458C22×C3—44no—
459C22×C3—44no—
460C22×C3—82no—
461C22×C3—82no—
462C22×C3—82no—
463C22×C3—82no—
464C22×C3C2×#12011yescharacteristic
465C22×C3#52511yescharacteristic
466C22×C3#53211yescharacteristic
467C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
468C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
469C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
470C22×C3#57511yescharacteristic
471C2×D3—42no—
472C2×D3—42no—
473C2×D3—42no—
474C2×D3—81no—
475C2×D3—81no—
476C2×D3—81no—
477C2×D3—81no—
478C3×C4—22no—
479C3×C4—22no—
480C3×C4—22no—
481C3×C4—22no—
482C3×C4—41no—
483C3×C4—44no—
484C3×C4#52512yes—
485C3×C4#53212yes—
486Dic3—42no—
487Dic3—81no—
488Dic3—81no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—24no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—62no—
503C23—121no—
504C23—121no—
505C23—121no—
506C23—121no—
507C23—121no—
508C23—121no—
509C23#93212yes—
510C23C22×#6711yescharacteristic
511Q8—48no—
512D4—24no—
513D4—24no—
514D4—48no—
515D4—48no—
516D4—416no—
517D4—42no—
518D4—42no—
519D4—64no—
520D4—128no—
521D4—122no—
522C2×C4—28no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—24no—
535C2×C4—44no—
536C2×C4—416no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—62no—
542C2×C4—121no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—244no—
551C2×C4#93212yes—
552C2×C4#94912yes—
553C2×C4#95312yes—
554C2×C4#95312yes—
555C2×C4D3×#4111yescharacteristic
556C2×C4D3×#4112yes—
557C2×C4D3×#4111yescharacteristic
558C2×C4C22×#6712yes—
559C8—64no—
560C8—122no—
561C2×C3—42no—
562C2×C3—84no—
563C2×C3#238811yescharacteristic
564C2×C3#300311yescharacteristic
565C2×C3#300611yescharacteristic
566C2×C3#310911yescharacteristic
567C2×C3#311511yescharacteristic
568C2×C3#318011yescharacteristic
569C2×C3#318411yescharacteristic
570D3—82no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—44no—
579C22—44no—
580C22—44no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—122no—
586C22—122no—
587C22—122no—
588C22—241no—
589C22—241no—
590C22—241no—
591C22—241no—
592C22C2×#93211yescharacteristic
593C22C2×#93211yescharacteristic
594C22C2×#94911yescharacteristic
595C22#582211yescharacteristic
596C22#582811yescharacteristic
597C22#583211yescharacteristic
598C22C2×#95311yescharacteristic
599C4—22no—
600C4—22no—
601C4—22no—
602C4—22no—
603C4—44no—
604C4—41no—
605C4—122no—
606C4—241no—
607C4—241no—
608C4#582812yes—
609C4#583212yes—
610C3#2357311yescharacteristic, complemented, Sylow
611C2—42no—
612C2—84no—
613C2—242no—
614C2#8329311yescharacteristic
615C2#8329511yescharacteristic
616C2#8330011yescharacteristic
617C2#8330111yescharacteristic
618C2#8403811yescharacteristic
619C2#8408211yescharacteristic
620C2#8409511yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#532C2
socleC23×C3C22×D4
hypercenterC2×#532D3
hypocenterC3#23573
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#532

Complements

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

NH ≅ G/Ntype
1. C2×C3×#532C2semidirect
2. C2×#5014C2semidirect
3. C4×#930C2semidirect
4. #71792C2semidirect
5. #71783C2semidirect
6. #71767C2semidirect
7. #71762C2semidirect
8. #71684C2semidirect
9. #71680C2semidirect
10. #71609C2semidirect
11. #71529C2semidirect
46. C2×#532D3semidirect
80. C3×C4×D4D4semidirect
203. C2×D12#41semidirect
225. C4⋊Dic3#41semidirect
233. C2×C3⋊C8#41semidirect
266. C4×D4#67semidirect
610. C3#23573semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5322
C2×#50142
C4×#9302
#717922
#717832
#717672
#717622
#716842
#716802
#716092
#715292
#235733