Finite Group Databasebeta

← #10847378

Subgroups — #10847378

Order 768.

1474 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#532C211yesmaximal, characteristic, complemented
2C2×#5038C212yesmaximal, complemented
3#82080C211yesmaximal, characteristic, complemented
4#71792C211yesmaximal, characteristic, complemented
5#71782C211yesmaximal, characteristic, complemented
6#71767C211yesmaximal, characteristic, complemented
7#71762C211yesmaximal, characteristic, complemented
8#71681C212yesmaximal, complemented
9#71613C212yesmaximal, complemented
10#71606C212yesmaximal, complemented
11C2×#4524C211yesmaximal, characteristic, complemented
12#25496—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×#929—21no—
26C23×C3⋊C8C2212yes—
27C2×#832C2212yes—
28C2×#832C2212yes—
29C2×#832C2212yes—
30#5213—21no—
31#5212C2211yescharacteristic
32#5038—28no—
33C2×#809C2212yes—
34C2×#807C2212yes—
35#4940—21no—
36#4940—21no—
37#4939C2211yescharacteristic
38#4939C2211yescharacteristic
39#4928C2211yescharacteristic
40#4928C2211yescharacteristic
41#4529C2212yes—
42#4529C2212yes—
43#4524C2211yescharacteristic
44#4524C2211yescharacteristic
45#4524C2212yes—
46C2×#532D311yescharacteristic, complemented
47C2×#466—32no—
48#2854—31no—
49#2057—31no—
50#2041—31no—
51#2023—31no—
52#2011—31no—
53#1893—32no—
54#1879—32no—
55#1862—32no—
56C2×#336—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
71C3×C4⋊Q8—24no—
72C3×#127—24no—
73C3×#124—28no—
74C3×#121—28no—
75C3×#121—24no—
76C3×#121D418yes—
77C3×C4×D4—28no—
78C3×C4×D4—24no—
79C3×C4×D4—24no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C42—24no—
85C2×C3×C42D411yescharacteristic
86C2×C3×C42D411yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C2×#217—21no—
90C2×#213—24no—
91C2×#213—24no—
92C2×#213—42no—
93#930—24no—
94#929—44no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C22×C3⋊C8—24no—
106C22×C3⋊C8—24no—
107C22×C3⋊C8—42no—
108C22×C3⋊C8C2312yes—
109C22×C3⋊C8C2312yes—
110#832—24no—
111#832—24no—
112#832—42no—
113C3⋊#95—21no—
114C3⋊#95—21no—
115C3⋊#95—21no—
116#809—24no—
117#807—24no—
118C22×#41—21no—
119C22×D8—32no—
120C23×C8—32no—
121#532—28no—
122C2×C4⋊Q8C2×D311yescharacteristic
123C2×#127C2×D311yescharacteristic
124C2×#124—21no—
125C2×#123—61no—
126C2×#121—21no—
127C2×#121—31no—
128C2×#121C2×D312yes—
129C2×C4×D4—21no—
130C2×C4×D4C2×D312yes—
131C22×C42C2×D311yescharacteristic
132#466—68no—
133C2×#107—32no—
134C2×#102—32no—
135C2×#102—32no—
136C2×#102—32no—
137C2×#102—32no—
138#390—31no—
139#390—31no—
140#389—31no—
141#389—31no—
142#388—61no—
143#388—61no—
144#386—61no—
145#380—31no—
146#340—32no—
147#340—32no—
148#336—31no—
149#336—32no—
150#336—31no—
151C24×C3—22no—
152C24×C3C2×D411yescharacteristic
153C23×D3—21no—
154C3×#41—432no—
155C2×C3×Q8—24no—
156C2×C3×Q8—42no—
157C2×C3×Q8—42no—
158C2×C3×D4—24no—
159C2×C3×D4—24no—
160C2×C3×D4—28no—
161C2×C3×D4—22no—
162C2×C3×D4—22no—
163C2×C3×D4—22no—
164C2×C3×D4—22no—
165C2×C3×D4—22no—
166C2×C3×D4—42no—
167C2×C3×D4—42no—
168C2×C3×D4—44no—
169C2×C3×D4—44no—
170C2×C3×D4—42no—
171C2×C3×D4—42no—
172C2×C3×D4C2×D414yes—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—28no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—41no—
183C22×C3×C4—44no—
184C22×C3×C4—44no—
185C22×C3×C4D814yescomplemented
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D414yes—
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2411yescharacteristic
193C2×#67—42no—
194C2×#67—42no—
195C2×#67—42no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C2×D12—22no—
200C2×D12—22no—
201C2×D12—21no—
202C2×D12—21no—
203C3×C4⋊C4—24no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4C2×D414yes—
206C3×C22⋊C4—28no—
207C3×C22⋊C4—28no—
208C3×C42—22no—
209C3×C42—22no—
210C3×C42—22no—
211C3×C42—22no—
212#217—42no—
213#213—48no—
214(C2×D3)⋊C4—42no—
215(C2×D3)⋊C4—42no—
216(C2×D3)⋊C4—42no—
217(C2×D3)⋊C4—42no—
218(C2×D3)⋊C4—42no—
219C4⋊Dic3—22no—
220C4⋊Dic3—22no—
221C4⋊Dic3—21no—
222C4⋊Dic3—21no—
223Dic3⋊C4—42no—
224Dic3⋊C4—42no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8—22no—
228C2×C3⋊C8—22no—
229C2×C3⋊C8—22no—
230C2×C3⋊C8—48no—
231C2×C3⋊C8#4114yescomplemented
232C2×#41—28no—
233C2×#41—44no—
234C2×#41—44no—
235C22×Q8—21no—
236C22×D4—22no—
237C22×D4—21no—
238C22×D4—21no—
239C22×D4—31no—
240C22×D4—61no—
241C22×D4C22×D312yes—
242C23×C4—22no—
243C23×C4#6711yescharacteristic
244C23×C4#6711yescharacteristic
245C23×C4C22×D311yescharacteristic
246C2×D8—64no—
247C2×D8—64no—
248C2×D8—122no—
249C22×C8—32no—
250C22×C8—32no—
251C22×C8—64no—
252C22×C8—64no—
253C22×C8—122no—
254C4⋊Q8—24no—
255#127—24no—
256#124—28no—
257#123—124no—
258#121—28no—
259#121—24no—
260#121—64no—
261#121#6718yes—
262C4×D4—28no—
263C4×D4—24no—
264C4×D4—24no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—31no—
267C2×C4⋊C4—31no—
268C2×C4⋊C4—31no—
269C2×C4⋊C4—61no—
270C2×C4⋊C4—61no—
271C2×C4⋊C4C22×D312yes—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—61no—
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#6711yescharacteristic
282C2×C42#6711yescharacteristic
283C2×C42C22×D311yescharacteristic
284C2×C42C22×D311yescharacteristic
285#107—64no—
286#102—64no—
287#102—64no—
288#102—64no—
289#102—122no—
290#95—61no—
291#95—61no—
292#95—61no—
293C23×C3—21no—
294C23×C3—21no—
295C23×C3—21no—
296C23×C3—21no—
297C23×C3—24no—
298C23×C3—42no—
299C23×C3—42no—
300C23×C3—42no—
301C23×C3—42no—
302C23×C3—42no—
303C23×C3—42no—
304C23×C3#12012yes—
305C23×C3C2×D812yes—
306C23×C3AGL(1,ℤ/8)12yes—
307C23×C3C22×D411yescharacteristic
308C22×D3—41no—
309C22×D3—41no—
310C22×D3—41no—
311C22×D3—41no—
312C22×D3—41no—
313C22×D3—41no—
314C22×D3—41no—
315C3×Q8—48no—
316C3×D4—24no—
317C3×D4—24no—
318C3×D4—48no—
319C3×D4—48no—
320C3×D4—416no—
321C3×D4—42no—
322C3×D4—42no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—28no—
332C2×C3×C4—24no—
333C2×C3×C4—24no—
334C2×C3×C4—21no—
335C2×C3×C4—24no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—44no—
339C2×C3×C4—416no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4#12012yes—
343C2×C3×C4C2×D812yes—
344C2×C3×C4C2×D812yes—
345C2×C3×C4#13512yes—
346C2×C3×C4C22×D412yes—
347C2×C3×C4C2×#4112yes—
348C2×C3×C4H(5,𝔽2)11yescharacteristic
349C2×C3×C42−1+411yescharacteristic
350#67—84no—
351C2×Dic3—41no—
352C2×Dic3—41no—
353C2×Dic3—41no—
354C2×Dic3—41no—
355C2×Dic3—41no—
356C2×Dic3—41no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360D12—42no—
361D12—42no—
362C3⋊C8—24no—
363C3⋊C8—42no—
364C24—22no—
365C24—61no—
366C24C2×#6711yescharacteristic
367#41—432no—
368C2×Q8—24no—
369C2×Q8—42no—
370C2×Q8—42no—
371C2×D4—24no—
372C2×D4—24no—
373C2×D4—28no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—42no—
380C2×D4—42no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—62no—
386C2×D4—61no—
387C2×D4—61no—
388C2×D4—62no—
389C2×D4—122no—
390C2×D4—122no—
391C2×D4—122no—
392C2×D4C2×#6714yes—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—28no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—41no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—61no—
406C22×C4—61no—
407C22×C4—61no—
408C22×C4#21314yescomplemented
409C22×C4C2×#6711yescharacteristic
410C22×C4C2×#6711yescharacteristic
411C22×C4C2×#6711yescharacteristic
412C22×C4C2×#6711yescharacteristic
413C22×C4C2×#6711yescharacteristic
414C22×C4C2×#6714yes—
415C22×C4C23×D311yescharacteristic
416D8—128no—
417C2×C8—34no—
418C2×C8—62no—
419C2×C8—62no—
420C2×C8—62no—
421C2×C8—62no—
422C2×C8—62no—
423C2×C8—128no—
424C4⋊C4—24no—
425C4⋊C4—22no—
426C4⋊C4—62no—
427C4⋊C4—61no—
428C4⋊C4—61no—
429C4⋊C4—62no—
430C4⋊C4—122no—
431C4⋊C4—122no—
432C4⋊C4C2×#6714yes—
433C22⋊C4—28no—
434C22⋊C4—28no—
435C22⋊C4—122no—
436C22⋊C4—122no—
437C22⋊C4—122no—
438C22⋊C4—122no—
439C22⋊C4—122no—
440C22⋊C4—122no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—22no—
445C22×C3—22no—
446C22×C3—22no—
447C22×C3—22no—
448C22×C3—22no—
449C22×C3—22no—
450C22×C3—22no—
451C22×C3—44no—
452C22×C3—44no—
453C22×C3—44no—
454C22×C3—82no—
455C22×C3—82no—
456C22×C3—82no—
457C22×C3—82no—
458C22×C3#44714yes—
459C22×C3C2×#12011yescharacteristic
460C22×C3#53211yescharacteristic
461C22×C3#53711yescharacteristic
462C22×C3C22×D811yescharacteristic
463C22×C3C2×#13511yescharacteristic
464C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
465C22×C3#57511yescharacteristic
466C2×D3—81no—
467C2×D3—81no—
468C2×D3—81no—
469C2×D3—81no—
470C2×D3—81no—
471C2×D3—81no—
472C2×D3—81no—
473C3×C4—22no—
474C3×C4—22no—
475C3×C4—22no—
476C3×C4—22no—
477C3×C4—44no—
478C3×C4—41no—
479C3×C4#54612yes—
480C3×C4#54712yes—
481Dic3—81no—
482Dic3—81no—
483Dic3—81no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—24no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—121no—
496C23—121no—
497C23—121no—
498C23—121no—
499C23—121no—
500C23—121no—
501C23—121no—
502C23C2×#21312yes—
503C23#93212yes—
504C23#95312yes—
505C23C22×#6711yescharacteristic
506Q8—48no—
507D4—24no—
508D4—24no—
509D4—48no—
510D4—48no—
511D4—416no—
512D4—42no—
513D4—42no—
514D4—122no—
515D4—122no—
516D4—244no—
517C2×C4—28no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—24no—
532C2×C4—44no—
533C2×C4—416no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—121no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4C2×#21312yes—
546C2×C4C2×#21312yes—
547C2×C4#95012yes—
548C2×C4#95312yes—
549C2×C4D3×#4112yes—
550C2×C4#100911yescharacteristic
551C2×C4#101011yescharacteristic
552C2×C4C22×#6712yes—
553C8—64no—
554C8—122no—
555C2×C3—24no—
556C2×C3—84no—
557C2×C3#243011yescharacteristic
558C2×C3C2×#44711yescharacteristic
559C2×C3#300211yescharacteristic
560C2×C3#316911yescharacteristic
561C2×C3#318011yescharacteristic
562C2×C3#324511yescharacteristic
563C2×C3#324711yescharacteristic
564D3—161no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—44no—
572C22—44no—
573C22—44no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22—241no—
579C22—241no—
580C22—241no—
581C22—241no—
582C22—241no—
583C22—241no—
584C22—241no—
585C22#521714yes—
586C22C22×#21311yescharacteristic
587C22C2×#93211yescharacteristic
588C22C2×#95011yescharacteristic
589C22#582211yescharacteristic
590C22#583211yescharacteristic
591C22#583311yescharacteristic
592C22C2×#95311yescharacteristic
593C4—22no—
594C4—22no—
595C4—22no—
596C4—22no—
597C4—44no—
598C4—41no—
599C4—241no—
600C4—241no—
601C4—241no—
602C4#582412yes—
603C4#582912yes—
604C3#2549611yescharacteristic, complemented, Sylow
605C2—24no—
606C2—84no—
607C2—481no—
608C2#8329111yescharacteristic
609C2#8329311yescharacteristic
610C2#8331011yescharacteristic
611C2#8331111yescharacteristic
612C2C2×#521711yescharacteristic
613C2#8408011yescharacteristic
614C2#8409111yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#532C2
socleC23×C3C22×D4
hypercenterC2×#532D3
hypocenterC3#25496
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×#5038C2semidirect
3. #82080C2semidirect
4. #71792C2semidirect
5. #71782C2semidirect
6. #71767C2semidirect
7. #71762C2semidirect
8. #71681C2semidirect
9. #71613C2semidirect
10. #71606C2semidirect
11. C2×#4524C2semidirect
46. C2×#532D3semidirect
185. C22×C3×C4D8semidirect
231. C2×C3⋊C8#41semidirect
408. C22×C4#213semidirect
604. C3#25496semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5322
C2×#50382
#820802
#717922
#717822
#717672
#717622
#716812
#716132
#716062
C2×#45242
#254963