Finite Group Databasebeta

← #10848860

Subgroups — #10848860

Order 768.

1216 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#542C211yesmaximal, characteristic, complemented
2C2×#5088C211yesmaximal, characteristic, complemented
3C2×#5058C211yesmaximal, characteristic, complemented
4#82286C211yesmaximal, characteristic, complemented
5#71788C211yesmaximal, characteristic, complemented
6#71725C211yesmaximal, characteristic, complemented
7#71699C212yesmaximal, complemented
8#71684C212yesmaximal, complemented
9#71633C212yesmaximal, complemented
10#71613C212yesmaximal, complemented
11#71582C211yesmaximal, characteristic, complemented
12#25614—31nomaximal, Sylow
13C3×#542—24no—
14C3×#542—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#126—21no—
17C2×C3×#124—21no—
18C2×C3×#124C2211yescharacteristic
19C2×C3×#123—21no—
20C2×C3×C22⋊Q8—21no—
21C2×C3×C22⋊Q8C2211yescharacteristic
22C2×C3×#121C2211yescharacteristic
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×D4C2211yescharacteristic
25C2×C3×#117C2211yescharacteristic
26C22×#215C2212yes—
27C22×#213C2212yes—
28C2×#930C2212yes—
29C2×#874C2211yescharacteristic
30C2×#835C2212yes—
31C2×#832C2212yes—
32C2×#830C2212yes—
33#5088—24no—
34#5058—24no—
35#4991—21no—
36#4991—21no—
37C2×#809C2212yes—
38C2×#809C2212yes—
39#4960—21no—
40#4960—21no—
41#4939C2212yes—
42#4936C2211yescharacteristic
43C2×#803C2211yescharacteristic
44C2×C4×C3⋊C8C2211yescharacteristic
45#4529C2212yes—
46#4529C2212yes—
47#4524C2212yes—
48C2×#542D311yescharacteristic, complemented
49C2×#496—31no—
50C2×#486—31no—
51#2619—31no—
52#2000—32no—
53#1999—32no—
54#1993—31no—
55#1958—31no—
56#1880—32no—
57#1879—32no—
58#1750—31no—
59C22×C3×Q8—21no—
60C22×C3×Q8C2311yescharacteristic
61C22×C3×D4—21no—
62C22×C3×D4C2311yescharacteristic
63C23×C3×C4—21no—
64C23×C3×C4C2311yescharacteristic
65C22×#67—22no—
66C22×D12—21no—
67C22×D12—21no—
68C22×D12C2312yes—
69C3×C4⋊Q8—22no—
70C3×C4⋊Q8D414yescomplemented
71C3×#126—28no—
72C3×#124—28no—
73C3×#124—22no—
74C3×#124D414yescomplemented
75C3×#123—28no—
76C3×C22⋊Q8—28no—
77C3×C22⋊Q8—22no—
78C3×C22⋊Q8—22no—
79C3×#121—22no—
80C3×#121—22no—
81C3×C4×Q8—22no—
82C3×C4×Q8—22no—
83C3×C4×D4—22no—
84C3×C4×D4—22no—
85C3×#117—22no—
86C3×#117—22no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4D411yescharacteristic
91C2×C3×C4⋊C4D411yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4D411yescharacteristic
100C2×C3×C22⋊C4D411yescharacteristic
101C2×C3×C42—21no—
102C2×C3×C42C2311yescharacteristic
103C2×C3×C42C2311yescharacteristic
104C2×#215—24no—
105C2×#215—24no—
106C2×#215—42no—
107C2×#213—24no—
108C2×#213—24no—
109C2×#213—42no—
110#930—42no—
111#930—42no—
112C2×(C2×D3)⋊C4—22no—
113C2×(C2×D3)⋊C4—22no—
114C2×(C2×D3)⋊C4—22no—
115C2×C4⋊Dic3C2312yes—
116C22×C3⋊C8C2312yes—
117C22×C3⋊C8C2312yes—
118#874—22no—
119#874—22no—
120#835—42no—
121#832—42no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124#830—24no—
125#809—24no—
126#809—24no—
127#803—22no—
128C4×C3⋊C8—22no—
129C22×SD16—32no—
130C22×D8—32no—
131#542—24no—
132#542—24no—
133C2×C4⋊Q8C2×D311yescharacteristic
134C2×#127—31no—
135C2×#126—21no—
136C2×#124—21no—
137C2×#124C2×D311yescharacteristic
138C2×#123—21no—
139C2×C22⋊Q8—21no—
140C2×C22⋊Q8C2×D311yescharacteristic
141C2×#121—32no—
142C2×#121C2×D311yescharacteristic
143C2×C4×Q8C2×D311yescharacteristic
144C2×C4×D4C2×D311yescharacteristic
145C2×#117C2×D311yescharacteristic
146#496—64no—
147#486—64no—
148C2×#103—32no—
149C2×#102—32no—
150C2×#102—32no—
151C2×#102—32no—
152C2×C22⋊C8—32no—
153C2×#97—31no—
154C2×C4×C8—31no—
155#394—61no—
156#394—61no—
157#392—61no—
158#392—61no—
159#390—32no—
160#382—31no—
161#340—32no—
162#340—32no—
163#336—32no—
164C24×C3—21no—
165C24×C3C2×D411yescharacteristic
166C23×D3—22no—
167C2×C3×Q8—24no—
168C2×C3×Q8—21no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—82no—
172C2×C3×Q8C2×D412yes—
173C2×C3×D4—24no—
174C2×C3×D4—21no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—82no—
178C2×C3×D4C2×D412yes—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—24no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—82no—
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—44no—
195C2×#67—82no—
196C2×#67—82no—
197C22×Dic3—22no—
198C2×D12—22no—
199C2×D12—22no—
200C2×D12—22no—
201C2×D12—22no—
202C2×D12—22no—
203C2×D12—22no—
204C2×D12—42no—
205C2×D12—42no—
206C2×D12—82no—
207C2×D12—82no—
208C3×C4⋊C4—24no—
209C3×C4⋊C4—24no—
210C3×C4⋊C4—24no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4C2×D412yes—
216C3×C4⋊C4C2×D412yes—
217C3×C22⋊C4—24no—
218C3×C22⋊C4—24no—
219C3×C22⋊C4—24no—
220C3×C22⋊C4—24no—
221C3×C22⋊C4—24no—
222C3×C22⋊C4—22no—
223C3×C22⋊C4—22no—
224C3×C42—24no—
225C3×C42—21no—
226C3×C42—21no—
227C3×C42C2×D412yes—
228C3×C42C2×D412yes—
229#215—48no—
230#213—48no—
231(C2×D3)⋊C4—44no—
232(C2×D3)⋊C4—82no—
233(C2×D3)⋊C4—82no—
234C4⋊Dic3—22no—
235C4⋊Dic3—22no—
236C2×C3⋊C8—22no—
237C2×C3⋊C8—22no—
238C2×C3⋊C8—22no—
239C2×C3⋊C8—22no—
240C2×C3⋊C8—22no—
241C2×C3⋊C8#4114yes—
242C22×Q8—21no—
243C22×Q8C22×D311yescharacteristic
244C22×D4—21no—
245C22×D4—32no—
246C22×D4—62no—
247C22×D4—61no—
248C22×D4—61no—
249C22×D4C22×D311yescharacteristic
250C23×C4—21no—
251C23×C4C22×D311yescharacteristic
252C2×SD16—64no—
253C2×SD16—64no—
254C2×SD16—122no—
255C2×D8—64no—
256C2×D8—64no—
257C2×D8—122no—
258C22×C8—32no—
259C22×C8—32no—
260C4⋊Q8—22no—
261C4⋊Q8#6714yescomplemented
262#127—62no—
263#127—62no—
264#126—28no—
265#124—28no—
266#124—22no—
267#124#6714yescomplemented
268#123—28no—
269C22⋊Q8—28no—
270C22⋊Q8—22no—
271C22⋊Q8—22no—
272#121—22no—
273#121—22no—
274#121—122no—
275#121—122no—
276C4×Q8—22no—
277C4×Q8—22no—
278C4×D4—22no—
279C4×D4—22no—
280#117—22no—
281#117—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—32no—
286C2×C4⋊C4#6711yescharacteristic
287C2×C4⋊C4#6711yescharacteristic
288C2×C4⋊C4C22×D311yescharacteristic
289C2×C4⋊C4C22×D311yescharacteristic
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—62no—
296C2×C22⋊C4—62no—
297C2×C22⋊C4—62no—
298C2×C22⋊C4#6711yescharacteristic
299C2×C22⋊C4#6711yescharacteristic
300C2×C42—21no—
301C2×C42C22×D311yescharacteristic
302C2×C42C22×D311yescharacteristic
303#103—122no—
304#102—64no—
305#102—64no—
306#102—122no—
307C22⋊C8—64no—
308#97—62no—
309C4×C8—62no—
310#95—61no—
311#95—61no—
312C23×C3—21no—
313C23×C3—22no—
314C23×C3—21no—
315C23×C3—21no—
316C23×C3—22no—
317C23×C3—22no—
318C23×C3—41no—
319C23×C3—41no—
320C23×C3—42no—
321C23×C3—42no—
322C23×C3C22×D411yescharacteristic
323C22×D3—42no—
324C22×D3—42no—
325C22×D3—42no—
326C22×D3—42no—
327C22×D3—42no—
328C22×D3—42no—
329C22×D3—42no—
330C3×Q8—24no—
331C3×Q8—42no—
332C3×Q8—84no—
333C3×D4—24no—
334C3×D4—42no—
335C3×D4—84no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—21no—
339C2×C3×C4—22no—
340C2×C3×C4—21no—
341C2×C3×C4—22no—
342C2×C3×C4—21no—
343C2×C3×C4—22no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4—84no—
355C2×C3×C4#12012yes—
356C2×C3×C4#12012yes—
357C2×C3×C4C22×D411yescharacteristic
358C2×C3×C4C22×D411yescharacteristic
359C2×C3×C4C2×#4112yes—
360C2×C3×C4H(5,𝔽2)12yes—
361#67—88no—
362C2×Dic3—42no—
363C2×Dic3—42no—
364C2×Dic3—42no—
365D12—42no—
366D12—42no—
367D12—42no—
368D12—42no—
369D12—84no—
370D12—84no—
371C3⋊C8—24no—
372C3⋊C8—42no—
373C24—21no—
374C24—62no—
375C24C2×#6711yescharacteristic
376C2×Q8—24no—
377C2×Q8—21no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—82no—
381C2×Q8C2×#6712yes—
382C2×D4—24no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—21no—
386C2×D4—62no—
387C2×D4—62no—
388C2×D4—62no—
389C2×D4—62no—
390C2×D4—62no—
391C2×D4—62no—
392C2×D4—82no—
393C2×D4—124no—
394C2×D4—122no—
395C2×D4—122no—
396C2×D4—242no—
397C2×D4—242no—
398C2×D4—242no—
399C2×D4—242no—
400C2×D4C2×#6712yes—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—62no—
410C22×C4—82no—
411C22×C4C2×#6711yescharacteristic
412C22×C4C2×#6711yescharacteristic
413C22×C4C2×#6711yescharacteristic
414C22×C4C2×#6711yescharacteristic
415C22×C4C2×#6711yescharacteristic
416C22×C4C23×D311yescharacteristic
417SD16—128no—
418D8—128no—
419C2×C8—34no—
420C2×C8—62no—
421C2×C8—62no—
422C2×C8—62no—
423C2×C8—62no—
424C2×C8—62no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—62no—
433C4⋊C4—62no—
434C4⋊C4C2×#6712yes—
435C4⋊C4C2×#6712yes—
436C22⋊C4—24no—
437C22⋊C4—24no—
438C22⋊C4—24no—
439C22⋊C4—24no—
440C22⋊C4—24no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—124no—
444C22⋊C4—242no—
445C22⋊C4—242no—
446C42—24no—
447C42—21no—
448C42—21no—
449C42C2×#6712yes—
450C42C2×#6712yes—
451C22×C3—42no—
452C22×C3—41no—
453C22×C3—42no—
454C22×C3—42no—
455C22×C3—42no—
456C22×C3—41no—
457C22×C3—41no—
458C22×C3—42no—
459C22×C3—82no—
460C22×C3—82no—
461C22×C3C2×#12011yescharacteristic
462C22×C3#53211yescharacteristic
463C22×C3#53511yescharacteristic
464C22×C3#57612yes—
465C22×C3#57712yes—
466C2×D3—82no—
467C2×D3—82no—
468C2×D3—82no—
469C2×D3—82no—
470C2×D3—82no—
471C2×D3—82no—
472C2×D3—82no—
473C3×C4—22no—
474C3×C4—22no—
475C3×C4—41no—
476C3×C4—42no—
477C3×C4—41no—
478C3×C4—42no—
479C3×C4—42no—
480C3×C4—41no—
481C3×C4#54612yes—
482C3×C4#55012yes—
483Dic3—82no—
484C23—21no—
485C23—21no—
486C23—22no—
487C23—22no—
488C23—21no—
489C23—22no—
490C23—41no—
491C23—41no—
492C23—42no—
493C23—42no—
494C23—122no—
495C23—122no—
496C23—122no—
497C23—122no—
498C23—122no—
499C23—122no—
500C23—122no—
501C23C22×#6711yescharacteristic
502Q8—24no—
503Q8—42no—
504Q8—84no—
505D4—24no—
506D4—42no—
507D4—84no—
508D4—122no—
509D4—122no—
510D4—122no—
511D4—122no—
512D4—244no—
513D4—244no—
514D4—248no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—84no—
534C2×C4—122no—
535C2×C4—122no—
536C2×C4—122no—
537C2×C4#95312yes—
538C2×C4#95312yes—
539C2×C4D3×#4112yes—
540C2×C4#100912yes—
541C2×C4C22×#6711yescharacteristic
542C2×C4C22×#6711yescharacteristic
543C8—64no—
544C8—122no—
545C2×C3—81no—
546C2×C3—82no—
547C2×C3#243111yescharacteristic
548C2×C3#300811yescharacteristic
549C2×C3#301011yescharacteristic
550C2×C3#317912yes—
551C2×C3#322412yes—
552D3—162no—
553C22—42no—
554C22—42no—
555C22—41no—
556C22—42no—
557C22—42no—
558C22—41no—
559C22—41no—
560C22—42no—
561C22—82no—
562C22—82no—
563C22—242no—
564C22—242no—
565C22—242no—
566C22—242no—
567C22—242no—
568C22—242no—
569C22—242no—
570C22#583011yescharacteristic
571C22#583211yescharacteristic
572C22#583512yes—
573C22#583712yes—
574C22C2×#95311yescharacteristic
575C4—22no—
576C4—22no—
577C4—41no—
578C4—42no—
579C4—41no—
580C4—42no—
581C4—42no—
582C4—41no—
583C4—242no—
584C4#582912yes—
585C4#583412yes—
586C3#2561411yescharacteristic, complemented, Sylow
587C2—81no—
588C2—82no—
589C2—482no—
590C2#8345912yes—
591C2#8346312yes—
592C2#8409211yescharacteristic
593C2#8410111yescharacteristic
594C2#8411311yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#542C2
socleC23×C3C22×D4
hypercenterC2×#542D3
hypocenterC3#25614
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#542

Complements

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

NH ≅ G/Ntype
1. C2×C3×#542C2semidirect
2. C2×#5088C2semidirect
3. C2×#5058C2semidirect
4. #82286C2semidirect
5. #71788C2semidirect
6. #71725C2semidirect
7. #71699C2semidirect
8. #71684C2semidirect
9. #71633C2semidirect
10. #71613C2semidirect
11. #71582C2semidirect
48. C2×#542D3semidirect
70. C3×C4⋊Q8D4semidirect
74. C3×#124D4semidirect
261. C4⋊Q8#67semidirect
267. #124#67semidirect
586. C3#25614semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5422
C2×#50882
C2×#50582
#822862
#717882
#717252
#716992
#716842
#716332
#716132
#715822
#256143