Finite Group Databasebeta

← #10828170

Subgroups — #10828170

Order 768.

1344 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×#548C211yesmaximal, characteristic
2C22×#835C211yesmaximal, characteristic, complemented
3C2×#5034C211yesmaximal, characteristic, complemented
4#82374C211yesmaximal, characteristic, complemented
5#72135C212yesmaximal, complemented
6#72132C211yesmaximal, characteristic, complemented
7#71722C212yesmaximal, complemented
8#71694C212yesmaximal, complemented
9#71681C211yesmaximal, characteristic, complemented
10#71629C212yesmaximal, complemented
11#71610C211yesmaximal, characteristic, complemented
12#24036—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×C3×Q8C2211yescharacteristic
15C3×#548—24no—
16C3×#548—24no—
17C2×C3×#124—21no—
18C2×C3×#124C2211yescharacteristic
19C2×C3×C22⋊Q8—21no—
20C2×C3×C22⋊Q8C2211yescharacteristic
21C2×C3×#121—21no—
22C2×C3×#121C2211yescharacteristic
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×D4—21no—
25C2×C3×C4×D4C2211yescharacteristic
26C22×C4⋊Dic3C2211yescharacteristic
27C23×C3⋊C8C2211yescharacteristic
28C2×#835—24no—
29C2×#835C2212yes—
30C2×#835C2212yes—
31C2×#835C2212yes—
32C2×#835C2212yes—
33#5220—21no—
34C2×#832C2212yes—
35C2×#832C2212yes—
36#5212C2211yescharacteristic
37#5212C2212yes—
38C2×#830C2211yescharacteristic
39C2×#830C2212yes—
40#5034—24no—
41#4986—21no—
42C2×#808C2212yes—
43#4949—21no—
44#4946—21no—
45#4928C2212yes—
46#4529C2212yes—
47#4524C2212yes—
48C2×#548D311yescharacteristic
49C2×#481—31no—
50C22×#103—31no—
51#2487—31no—
52#2042—32no—
53#2035—32no—
54#1893—31no—
55#1869—31no—
56#1865—32no—
57#1788—32no—
58#1787—31no—
59C2×C3×#41—22no—
60C2×C3×#41—22no—
61C2×C3×#41—24no—
62C2×C3×#41—42no—
63C22×C3×Q8—21no—
64C22×C3×Q8—22no—
65C22×C3×Q8—24no—
66C22×C3×Q8—42no—
67C22×C3×Q8D414yes—
68C22×C3×Q8C2311yescharacteristic
69C22×C3×Q8C2311yescharacteristic
70C22×C3×D4—21no—
71C22×C3×D4—21no—
72C22×C3×D4D411yescharacteristic
73C22×C3×D4C2311yescharacteristic
74C23×C3×C4—21no—
75C23×C3×C4—21no—
76C23×C3×C4C2311yescharacteristic
77C23×C3×C4C2311yescharacteristic
78C23×Dic3—21no—
79C3×#124—28no—
80C3×#124—22no—
81C3×#124—22no—
82C3×C22⋊Q8—28no—
83C3×C22⋊Q8—22no—
84C3×C22⋊Q8—22no—
85C3×#121—28no—
86C3×#121—22no—
87C3×#121D414yes—
88C3×C4×Q8—22no—
89C3×C4×Q8—22no—
90C3×C4×D4—28no—
91C3×C4×D4—22no—
92C3×C4×D4—22no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4D411yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4D411yescharacteristic
102C2×C3×C22⋊C4D411yescharacteristic
103C2×C3×C42—21no—
104C2×C3×C42C2311yescharacteristic
105C2×#217—21no—
106C2×#217—22no—
107C2×C4⋊Dic3—22no—
108C2×C4⋊Dic3—22no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3C2312yes—
111C2×C4⋊Dic3C2312yes—
112C2×Dic3⋊C4—22no—
113C22×C3⋊C8—22no—
114C22×C3⋊C8—42no—
115C22×C3⋊C8C2312yes—
116C22×C3⋊C8C2312yes—
117#835—24no—
118#835—24no—
119#835—24no—
120#835—42no—
121#832—24no—
122#832—42no—
123C3⋊#95—22no—
124C3⋊#95—22no—
125C3⋊#95—21no—
126#830—22no—
127#830—42no—
128#808—24no—
129C22×#41C2×D311yescharacteristic
130C23×Q8C2×D311yescharacteristic
131C23×C8—31no—
132#548—24no—
133#548—24no—
134C2×#124—21no—
135C2×#124C2×D311yescharacteristic
136C2×C22⋊Q8—21no—
137C2×C22⋊Q8C2×D311yescharacteristic
138C2×#121—21no—
139C2×#121C2×D311yescharacteristic
140C2×C4×Q8C2×D311yescharacteristic
141C2×C4×D4—21no—
142C2×C4×D4C2×D311yescharacteristic
143C22×C4⋊C4—31no—
144#481—64no—
145C2×#106—32no—
146C2×#103—32no—
147C2×#103—32no—
148C2×#103—32no—
149C2×#103—32no—
150C2×#103—64no—
151C2×#102—32no—
152C2×#102—32no—
153C2×C22⋊C8—31no—
154C2×C22⋊C8—32no—
155#400—61no—
156#399—61no—
157#389—32no—
158#386—61no—
159#385—61no—
160#380—31no—
161#380—32no—
162#340—32no—
163#336—32no—
164C24×C3—21no—
165C24×C3C2×D411yescharacteristic
166C24×C3C2×D411yescharacteristic
167C3×#41—48no—
168C3×#41—44no—
169C3×#41—44no—
170C2×C3×Q8—24no—
171C2×C3×Q8—21no—
172C2×C3×Q8—21no—
173C2×C3×Q8—22no—
174C2×C3×Q8—28no—
175C2×C3×Q8—22no—
176C2×C3×Q8—22no—
177C2×C3×Q8—44no—
178C2×C3×Q8—44no—
179C2×C3×Q8—48no—
180C2×C3×Q8C2×D412yes—
181C2×C3×Q8C2×D412yes—
182C2×C3×Q8#4114yes—
183C2×C3×D4—24no—
184C2×C3×D4—22no—
185C2×C3×D4—24no—
186C2×C3×D4—22no—
187C2×C3×D4—22no—
188C2×C3×D4—21no—
189C2×C3×D4—22no—
190C2×C3×D4—22no—
191C2×C3×D4—44no—
192C2×C3×D4—42no—
193C2×C3×D4—42no—
194C2×C3×D4C2×D412yes—
195C22×C3×C4—21no—
196C22×C3×C4—22no—
197C22×C3×C4—22no—
198C22×C3×C4—24no—
199C22×C3×C4—24no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—22no—
205C22×C3×C4—22no—
206C22×C3×C4—44no—
207C22×C3×C4—44no—
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D412yes—
213C22×C3×C4C2×D412yes—
214C22×C3×C4C2411yescharacteristic
215C22×Dic3—22no—
216C22×Dic3—22no—
217C22×Dic3—42no—
218C22×Dic3—82no—
219C3×C4⋊C4—24no—
220C3×C4⋊C4—24no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4C2×D412yes—
224C3×C22⋊C4—24no—
225C3×C22⋊C4—24no—
226C3×C22⋊C4—24no—
227C3×C22⋊C4—22no—
228C3×C22⋊C4—24no—
229C3×C22⋊C4—22no—
230C3×C42—24no—
231C3×C42—21no—
232C3×C42—21no—
233#217—42no—
234#217—82no—
235C4⋊Dic3—22no—
236C4⋊Dic3—22no—
237C4⋊Dic3—22no—
238C4⋊Dic3—22no—
239C4⋊Dic3—42no—
240Dic3⋊C4—82no—
241C2×C3⋊C8—22no—
242C2×C3⋊C8—22no—
243C2×C3⋊C8—22no—
244C2×C3⋊C8—22no—
245C2×C3⋊C8—22no—
246C2×C3⋊C8—22no—
247C2×C3⋊C8—44no—
248C2×#41—22no—
249C2×#41—22no—
250C2×#41—24no—
251C2×#41—42no—
252C22×Q8—21no—
253C22×Q8—22no—
254C22×Q8—24no—
255C22×Q8—42no—
256C22×Q8#6714yes—
257C22×Q8C22×D311yescharacteristic
258C22×Q8C22×D311yescharacteristic
259C22×D4—21no—
260C22×D4—21no—
261C22×D4#6711yescharacteristic
262C22×D4C22×D311yescharacteristic
263C23×C4—21no—
264C23×C4—21no—
265C23×C4—61no—
266C23×C4C22×D311yescharacteristic
267C23×C4C22×D311yescharacteristic
268C22×C8—32no—
269C22×C8—32no—
270C22×C8—62no—
271C22×C8—122no—
272#124—28no—
273#124—22no—
274#124—22no—
275C22⋊Q8—28no—
276C22⋊Q8—22no—
277C22⋊Q8—22no—
278#121—28no—
279#121—22no—
280#121#6714yes—
281C4×Q8—22no—
282C4×Q8—22no—
283C4×D4—28no—
284C4×D4—22no—
285C4×D4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—32no—
289C2×C4⋊C4—32no—
290C2×C4⋊C4—62no—
291C2×C4⋊C4—62no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4—62no—
294C2×C4⋊C4#6711yescharacteristic
295C2×C4⋊C4C22×D311yescharacteristic
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—61no—
301C2×C22⋊C4—62no—
302C2×C22⋊C4#6711yescharacteristic
303C2×C22⋊C4#6711yescharacteristic
304C2×C42—21no—
305C2×C42C22×D311yescharacteristic
306#106—64no—
307#103—64no—
308#103—64no—
309#103—64no—
310#103—122no—
311#102—64no—
312#102—122no—
313C22⋊C8—62no—
314C22⋊C8—122no—
315#95—62no—
316#95—61no—
317#95—62no—
318C23×C3—22no—
319C23×C3—22no—
320C23×C3—21no—
321C23×C3—22no—
322C23×C3—22no—
323C23×C3—21no—
324C23×C3—21no—
325C23×C3—22no—
326C23×C3—42no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3—42no—
330C23×C3#12012yes—
331C23×C3#13512yes—
332C23×C3#13712yes—
333C23×C3C22×D411yescharacteristic
334C3×Q8—22no—
335C3×Q8—24no—
336C3×Q8—22no—
337C3×Q8—22no—
338C3×Q8—48no—
339C3×Q8—42no—
340C3×Q8#12314yes—
341C3×D4—24no—
342C3×D4—48no—
343C3×D4—44no—
344C3×D4—48no—
345C3×D4—41no—
346C3×D4—41no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—24no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—24no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—21no—
358C2×C3×C4—22no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—22no—
362C2×C3×C4—48no—
363C2×C3×C4—48no—
364C2×C3×C4—42no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—42no—
370C2×C3×C4#12012yes—
371C2×C3×C4C22×D411yescharacteristic
372C2×C3×C4C22×D411yescharacteristic
373C2×C3×C4C2×#4112yes—
374C2×C3×C4H(5,𝔽2)12yes—
375C2×Dic3—42no—
376C2×Dic3—42no—
377C2×Dic3—42no—
378C2×Dic3—42no—
379C2×Dic3—42no—
380C2×Dic3—42no—
381C2×Dic3—84no—
382C3⋊C8—42no—
383C3⋊C8—42no—
384C24—21no—
385C24C2×#6711yescharacteristic
386C24C2×#6711yescharacteristic
387#41—48no—
388#41—44no—
389#41—44no—
390C2×Q8—24no—
391C2×Q8—22no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—28no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—48no—
400C2×Q8#23714yes—
401C2×Q8C2×#6712yes—
402C2×Q8C2×#6712yes—
403C2×D4—24no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—24no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—21no—
411C2×D4—44no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4C2×#6712yes—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—62no—
429C22×C4—62no—
430C22×C4—122no—
431C22×C4—242no—
432C22×C4C2×#6711yescharacteristic
433C22×C4C2×#6711yescharacteristic
434C22×C4C2×#6711yescharacteristic
435C22×C4C2×#6711yescharacteristic
436C22×C4C2×#6712yes—
437C22×C4C2×#6712yes—
438C22×C4C23×D311yescharacteristic
439C2×C8—62no—
440C2×C8—62no—
441C2×C8—62no—
442C2×C8—62no—
443C2×C8—62no—
444C2×C8—62no—
445C2×C8—124no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—62no—
451C4⋊C4—62no—
452C4⋊C4—62no—
453C4⋊C4—62no—
454C4⋊C4—122no—
455C4⋊C4—242no—
456C4⋊C4C2×#6712yes—
457C22⋊C4—24no—
458C22⋊C4—24no—
459C22⋊C4—24no—
460C22⋊C4—22no—
461C22⋊C4—24no—
462C22⋊C4—22no—
463C22⋊C4—122no—
464C22⋊C4—242no—
465C42—24no—
466C42—21no—
467C42—21no—
468C22×C3—22no—
469C22×C3—22no—
470C22×C3—24no—
471C22×C3—24no—
472C22×C3—42no—
473C22×C3—41no—
474C22×C3—41no—
475C22×C3—42no—
476C22×C3—41no—
477C22×C3—42no—
478C22×C3—42no—
479C22×C3—42no—
480C22×C3—82no—
481C22×C3—82no—
482C22×C3#45214yes—
483C22×C3C2×#12011yescharacteristic
484C22×C3#54612yes—
485C22×C3C2×#13511yescharacteristic
486C22×C3C2×#13711yescharacteristic
487C22×C3#57712yes—
488C3×C4—22no—
489C3×C4—24no—
490C3×C4—41no—
491C3×C4—42no—
492C3×C4—41no—
493C3×C4—42no—
494C3×C4—41no—
495C3×C4C2×#12311yescharacteristic
496C3×C4#53411yescharacteristic
497C3×C4#54612yes—
498Dic3—82no—
499Dic3—82no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—21no—
504C23—21no—
505C23—22no—
506C23—21no—
507C23—22no—
508C23—42no—
509C23—41no—
510C23—41no—
511C23—42no—
512C23#95012yes—
513C23#95112yes—
514C23#95312yes—
515C23C22×#6711yescharacteristic
516Q8—22no—
517Q8—24no—
518Q8—22no—
519Q8—22no—
520Q8—48no—
521Q8—42no—
522Q8#93514yes—
523D4—24no—
524D4—48no—
525D4—44no—
526D4—48no—
527D4—41no—
528D4—41no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—48no—
545C2×C4—48no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—122no—
553C2×C4—122no—
554C2×C4—122no—
555C2×C4—122no—
556C2×C4—122no—
557C2×C4—122no—
558C2×C4—244no—
559C2×C4#95312yes—
560C2×C4C2×#23712yes—
561C2×C4#100412yes—
562C2×C4C22×#6711yescharacteristic
563C2×C4C22×#6711yescharacteristic
564C8—122no—
565C8—122no—
566C2×C3—24no—
567C2×C3—81no—
568C2×C3—82no—
569C2×C3#240511yescharacteristic
570C2×C3C2×#45211yescharacteristic
571C2×C3#301011yescharacteristic
572C2×C3#330112yes—
573C2×C3#331612yes—
574C22—24no—
575C22—22no—
576C22—24no—
577C22—22no—
578C22—42no—
579C22—42no—
580C22—41no—
581C22—42no—
582C22—41no—
583C22—42no—
584C22—41no—
585C22—42no—
586C22—82no—
587C22—82no—
588C22#523914yes—
589C22#580612yes—
590C22C2×#95011yescharacteristic
591C22C2×#95111yescharacteristic
592C22#583712yes—
593C22C2×#95311yescharacteristic
594C4—22no—
595C4—24no—
596C4—41no—
597C4—42no—
598C4—41no—
599C4—42no—
600C4—41no—
601C4—242no—
602C4—242no—
603C4C2×#93511yescharacteristic
604C4#580411yescharacteristic
605C4#580612yes—
606C3#2403611yescharacteristic, complemented, Sylow
607C2—24no—
608C2—81no—
609C2—82no—
610C2#8353512yes—
611C2#8353812yes—
612C2#8404711yescharacteristic
613C2C2×#523911yescharacteristic
614C2#8411311yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#548C2
socleC23×C3C22×D4
hypercenterC2×#548D3
hypocenterC3#24036
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#548

Complements

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

NH ≅ G/Ntype
2. C22×#835C2semidirect
3. C2×#5034C2semidirect
4. #82374C2semidirect
5. #72135C2semidirect
6. #72132C2semidirect
7. #71722C2semidirect
8. #71694C2semidirect
9. #71681C2semidirect
10. #71629C2semidirect
11. #71610C2semidirect
606. C3#24036semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5482
C22×#8352
C2×#50342
#823742
#721352
#721322
#717222
#716942
#716812
#716292
#716102
#240363