Finite Group Databasebeta

← #10827383

Subgroups — #10827383

Order 768.

1304 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5618C211yesmaximal, characteristic
2C2×C3×#448C211yesmaximal, characteristic, complemented
3#84068C211yesmaximal, characteristic, complemented
4C22×#835C211yesmaximal, characteristic, complemented
5#72054C212yesmaximal, complemented
6#72047C212yesmaximal, complemented
7#71681C211yesmaximal, characteristic, complemented
8#70269C212yesmaximal, complemented
9#70232C212yesmaximal, complemented
10#69952C211yesmaximal, characteristic, complemented
11#69925C211yesmaximal, characteristic, complemented
12#23964—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×C3×SD16C2212yes—
15C2×C3×#121C2211yescharacteristic
16C2×#946C2212yes—
17C2×#939—21no—
18C2×#938C2211yescharacteristic
19C2×D4×Dic3—21no—
20C2×D4×Dic3C2211yescharacteristic
21C2×C4×#67C2211yescharacteristic
22C22×C4⋊Dic3C2211yescharacteristic
23C23×C3⋊C8C2211yescharacteristic
24#5618—24no—
25#5618—24no—
26C2×#892C2211yescharacteristic
27C2×#890C2211yescharacteristic
28C2×#886—21no—
29C2×#879—21no—
30C3×#448—24no—
31C2×C3×#103C2212yes—
32C2×C3×C22⋊C8C2211yescharacteristic
33#5231—21no—
34#5230—21no—
35#5229C2212yes—
36C2×#835—24no—
37C2×#835C2212yes—
38C2×#835C2212yes—
39C2×#832C2212yes—
40#5213—21no—
41#5212C2211yescharacteristic
42#5210—21no—
43C2×(C2×D3)⋊C8C2212yes—
44#4551C2212yes—
45#4551C2212yes—
46#4550C2212yes—
47#4529C2212yes—
48C2×#553—31no—
49C2×#448D311yescharacteristic, complemented
50C22×#103—31no—
51#2409—31no—
52#2026—32no—
53#2000—32no—
54#1893—31no—
55#1877—32no—
56#1873—32no—
57#1869—31no—
58#1824—31no—
59C22×C3×Q8—21no—
60C22×C3×Q8—24no—
61C22×C3×Q8—22no—
62C22×C3×Q8—22no—
63C22×C3×Q8—42no—
64C22×C3×Q8C2312yes—
65C22×C3×D4—21no—
66C22×C3×D4C2311yescharacteristic
67C23×C3×C4—21no—
68C23×C3×C4C2311yescharacteristic
69C22×#67—21no—
70C22×#67D411yescharacteristic
71C23×Dic3—21no—
72C23×Dic3—21no—
73C22×C4×D3C2311yescharacteristic
74C2×C3×SD16—24no—
75C2×C3×SD16—24no—
76C2×C3×SD16—42no—
77C22×C3×C8C2312yes—
78C3×#121—22no—
79C3×#121D414yes—
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C22⋊C4—21no—
82C2×#217—21no—
83C2×#217—21no—
84C2×#217—21no—
85C2×#217D411yescharacteristic
86#946—24no—
87#946—42no—
88#939—28no—
89#938—22no—
90#938—22no—
91D4×Dic3—28no—
92D4×Dic3—22no—
93D4×Dic3—22no—
94C4×#67—22no—
95C4×#67—22no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—22no—
98C2×(C2×D3)⋊C4D411yescharacteristic
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3—22no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3D414yes—
103C2×C4⋊Dic3C2312yes—
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—22no—
108C2×Dic3⋊C4D411yescharacteristic
109C2×C4×Dic3—21no—
110C2×C4×Dic3C2311yescharacteristic
111C22×C3⋊C8—22no—
112C22×C3⋊C8—42no—
113C22×C3⋊C8C2312yes—
114#892—22no—
115#892—22no—
116#890—22no—
117#890—22no—
118#886—28no—
119#879—28no—
120C3×#103—24no—
121C3×C22⋊C8—22no—
122#835—24no—
123#835—24no—
124#832—42no—
125C3⋊#95—21no—
126C3⋊#95—22no—
127C3⋊#95—22no—
128(C2×D3)⋊C8—42no—
129C23×Q8C2×D311yescharacteristic
130C22×SD16C2×D312yes—
131C23×C8—31no—
132#553—64no—
133#553—64no—
134C2×#126—61no—
135C2×#125—31no—
136C2×#123—61no—
137C2×C22⋊Q8—32no—
138C2×#121—31no—
139C2×#121—61no—
140C2×#121C2×D311yescharacteristic
141C2×C4×D4—31no—
142C2×C4×D4—31no—
143C2×C4×D4—61no—
144C2×#117—31no—
145C22×C4⋊C4—31no—
146#448—24no—
147C2×#103—32no—
148C2×#103—32no—
149C2×#103—64no—
150C2×#103C2×D312yes—
151C2×#102—32no—
152C2×C22⋊C8—32no—
153C2×C22⋊C8C2×D311yescharacteristic
154#395—61no—
155#393—61no—
156#391—32no—
157#386—61no—
158#385—61no—
159#380—31no—
160#340—32no—
161#340—32no—
162#340—32no—
163#336—32no—
164C24×C3—21no—
165C24×C3C2×D411yescharacteristic
166C23×D3C2×D411yescharacteristic
167C2×C3×Q8—22no—
168C2×C3×Q8—28no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8—44no—
174C2×C3×Q8—48no—
175C2×C3×Q8—44no—
176C2×C3×Q8—42no—
177C2×C3×Q8#4114yes—
178C2×C3×D4—24no—
179C2×C3×D4—21no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—42no—
183C2×C3×D4—42no—
184C2×C3×D4C2×D412yes—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—21no—
189C22×C3×C4—44no—
190C22×C3×C4—42no—
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D412yes—
193C22×C3×C4C2×D412yes—
194C22×C3×C4C2411yescharacteristic
195C2×#67—22no—
196C2×#67—22no—
197C2×#67—22no—
198C2×#67—24no—
199C2×#67—82no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—24no—
203C22×Dic3—24no—
204C22×Dic3—22no—
205C22×Dic3—82no—
206C22×Dic3—82no—
207C22×Dic3C2×D411yescharacteristic
208C22×Dic3C2×D411yescharacteristic
209C22×Dic3C2×D411yescharacteristic
210C2×C4×D3—21no—
211C2×C4×D3—21no—
212C2×C4×D3—22no—
213C2×C4×D3—22no—
214C3×SD16—48no—
215C2×C3×C8—22no—
216C2×C3×C8—22no—
217C2×C3×C8—22no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4C2×D412yes—
220C3×C22⋊C4—24no—
221#217—24no—
222#217—24no—
223#217—22no—
224#217—42no—
225(C2×D3)⋊C4—24no—
226(C2×D3)⋊C4—22no—
227(C2×D3)⋊C4—44no—
228C4⋊Dic3—24no—
229C4⋊Dic3—24no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—22no—
233C4⋊Dic3—22no—
234C4⋊Dic3C2×D412yes—
235C4⋊Dic3C2×D412yes—
236Dic3⋊C4—24no—
237Dic3⋊C4—22no—
238Dic3⋊C4—44no—
239C4×Dic3—24no—
240C4×Dic3—21no—
241C4×Dic3—21no—
242C2×C3⋊C8—22no—
243C2×C3⋊C8—22no—
244C2×C3⋊C8—22no—
245C2×C3⋊C8—44no—
246C22×Q8—21no—
247C22×Q8—24no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—42no—
251C22×Q8C22×D312yes—
252C22×D4—21no—
253C22×D4—31no—
254C22×D4—61no—
255C22×D4C22×D311yescharacteristic
256C23×C4—21no—
257C23×C4—31no—
258C23×C4—61no—
259C23×C4—61no—
260C23×C4C22×D311yescharacteristic
261C2×SD16—24no—
262C2×SD16—24no—
263C2×SD16—42no—
264C22×C8—32no—
265C22×C8—62no—
266C22×C8—122no—
267C22×C8C22×D312yes—
268#126—68no—
269#125—62no—
270#125—62no—
271#123—68no—
272C22⋊Q8—64no—
273C22⋊Q8—122no—
274#121—22no—
275#121—62no—
276#121—68no—
277#121—62no—
278#121#6714yes—
279C4×D4—62no—
280C4×D4—68no—
281C4×D4—62no—
282C4×D4—62no—
283C4×D4—62no—
284#117—62no—
285#117—62no—
286C2×C4⋊C4—34no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—32no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—62no—
295C2×C4⋊C4—62no—
296C2×C4⋊C4C22×D311yescharacteristic
297C2×C22⋊C4—21no—
298C2×C22⋊C4—31no—
299C2×C22⋊C4—31no—
300C2×C22⋊C4—61no—
301C2×C22⋊C4—61no—
302C2×C22⋊C4—61no—
303C2×C22⋊C4—62no—
304C2×C22⋊C4—61no—
305C2×C42—31no—
306C2×C42—61no—
307#103—24no—
308#103—64no—
309#103—64no—
310#102—122no—
311C22⋊C8—22no—
312C22⋊C8—122no—
313#95—62no—
314#95—62no—
315#95—61no—
316C23×C3—22no—
317C23×C3—22no—
318C23×C3—22no—
319C23×C3—42no—
320C23×C3—41no—
321C23×C3—41no—
322C23×C3—42no—
323C23×C3#12012yes—
324C23×C3#13512yes—
325C23×C3#13712yes—
326C23×C3C22×D411yescharacteristic
327C22×D3—21no—
328C22×D3—22no—
329C22×D3—21no—
330C22×D3—21no—
331C22×D3—22no—
332C3×Q8—24no—
333C3×Q8—24no—
334C3×Q8—48no—
335C3×Q8—42no—
336C3×Q8—42no—
337C3×D4—24no—
338C3×D4—48no—
339C3×D4—41no—
340C3×D4—41no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—24no—
345C2×C3×C4—22no—
346C2×C3×C4—24no—
347C2×C3×C4—48no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4#12012yes—
353C2×C3×C4C22×D411yescharacteristic
354C2×C3×C4C22×D411yescharacteristic
355C2×C3×C4C2×#4112yes—
356C2×C3×C4H(5,𝔽2)12yes—
357#67—44no—
358#67—84no—
359C2×Dic3—21no—
360C2×Dic3—21no—
361C2×Dic3—21no—
362C2×Dic3—22no—
363C2×Dic3—22no—
364C2×Dic3—22no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—22no—
368C2×Dic3—42no—
369C2×Dic3—42no—
370C2×Dic3—42no—
371C2×Dic3—42no—
372C2×Dic3—42no—
373C2×Dic3—84no—
374C2×Dic3—84no—
375C4×D3—41no—
376C4×D3—41no—
377C4×D3—42no—
378C3×C8—42no—
379C3⋊C8—42no—
380C24—21no—
381C24—31no—
382C24D3×D411yescharacteristic
383C2×Q8—22no—
384C2×Q8—28no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—44no—
390C2×Q8—48no—
391C2×Q8—44no—
392C2×Q8—42no—
393C2×Q8#23714yes—
394C2×D4—24no—
395C2×D4—21no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—62no—
401C2×D4—62no—
402C2×D4—62no—
403C2×D4—64no—
404C2×D4—242no—
405C2×D4C2×#6712yes—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—31no—
411C22×C4—31no—
412C22×C4—31no—
413C22×C4—44no—
414C22×C4—42no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4—64no—
418C22×C4—64no—
419C22×C4—61no—
420C22×C4—62no—
421C22×C4—62no—
422C22×C4—61no—
423C22×C4—62no—
424C22×C4—242no—
425C22×C4—242no—
426C22×C4D3×D411yescharacteristic
427C22×C4D3×D412yes—
428C22×C4C2×#6712yes—
429C22×C4C23×D311yescharacteristic
430SD16—48no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—62no—
435C2×C8—62no—
436C2×C8—62no—
437C2×C8—124no—
438C4⋊C4—21no—
439C4⋊C4—32no—
440C4⋊C4—32no—
441C4⋊C4—64no—
442C4⋊C4—64no—
443C4⋊C4—64no—
444C4⋊C4—62no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—62no—
448C4⋊C4—62no—
449C4⋊C4—124no—
450C4⋊C4C2×#6712yes—
451C22⋊C4—24no—
452C22⋊C4—64no—
453C22⋊C4—64no—
454C22⋊C4—64no—
455C22⋊C4—62no—
456C22⋊C4—62no—
457C22⋊C4—124no—
458C22⋊C4—122no—
459C42—64no—
460C42—61no—
461C42—61no—
462C22×C3—22no—
463C22×C3—22no—
464C22×C3—24no—
465C22×C3—24no—
466C22×C3—42no—
467C22×C3—42no—
468C22×C3—42no—
469C22×C3—82no—
470C22×C3—82no—
471C22×C3#45214yes—
472C22×C3C2×#12011yescharacteristic
473C22×C3#54612yes—
474C22×C3C2×#13511yescharacteristic
475C22×C3C2×#13711yescharacteristic
476C22×C3#57712yes—
477C2×D3—41no—
478C2×D3—41no—
479C2×D3—41no—
480C2×D3—42no—
481C2×D3—42no—
482C3×C4—22no—
483C3×C4—44no—
484C3×C4—42no—
485C3×C4#52511yescharacteristic
486C3×C4#53511yescharacteristic
487C3×C4#54612yes—
488Dic3—41no—
489Dic3—41no—
490Dic3—41no—
491Dic3—42no—
492Dic3—42no—
493Dic3—82no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—41no—
498C23—41no—
499C23—42no—
500C23—42no—
501C23—61no—
502C23—61no—
503C23—62no—
504C23—61no—
505C23—62no—
506C23#91512yes—
507C23#91612yes—
508C23#93712yes—
509C23C2×D3×D411yescharacteristic
510Q8—24no—
511Q8—24no—
512Q8—48no—
513Q8—42no—
514Q8—42no—
515D4—24no—
516D4—48no—
517D4—41no—
518D4—41no—
519D4—124no—
520D4—244no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—48no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—61no—
533C2×C4—61no—
534C2×C4—62no—
535C2×C4—61no—
536C2×C4—62no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—62no—
540C2×C4—62no—
541C2×C4—122no—
542C2×C4—122no—
543C2×C4—122no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—122no—
547C2×C4—122no—
548C2×C4—122no—
549C2×C4—244no—
550C2×C4—244no—
551C2×C4#93712yes—
552C2×C4C2×D3×D411yescharacteristic
553C2×C4C2×#23712yes—
554C2×C4#100412yes—
555C2×C4C22×#6711yescharacteristic
556C8—42no—
557C8—122no—
558C2×C3—24no—
559C2×C3—82no—
560C2×C3#240411yescharacteristic
561C2×C3C2×#45211yescharacteristic
562C2×C3#300911yescharacteristic
563C2×C3#330112yes—
564C2×C3#332012yes—
565D3—81no—
566C22—22no—
567C22—22no—
568C22—24no—
569C22—24no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—82no—
574C22—82no—
575C22—121no—
576C22—121no—
577C22—121no—
578C22—122no—
579C22—122no—
580C22#517314yes—
581C22#559112yes—
582C22C2×#91511yescharacteristic
583C22C2×#91611yescharacteristic
584C22#577812yes—
585C22C2×#93711yescharacteristic
586C4—22no—
587C4—42no—
588C4—44no—
589C4—121no—
590C4—122no—
591C4—122no—
592C4—121no—
593C4—121no—
594C4—242no—
595C4#558611yescharacteristic
596C4#559211yescharacteristic
597C4#580612yes—
598C3#2396411yescharacteristic, complemented, Sylow
599C2—24no—
600C2—82no—
601C2—241no—
602C2#8005012yes—
603C2#8005212yes—
604C2#8258111yescharacteristic
605C2C2×#517311yescharacteristic
606C2#8398211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#448C2
socleC23×C3C22×D4
prime radicalC2×#5618C2
hypercenterC2×#448D3
hypocenterC3#23964
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#448

Complements

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

NH ≅ G/Ntype
2. C2×C3×#448C2semidirect
3. #84068C2semidirect
4. C22×#835C2semidirect
5. #72054C2semidirect
6. #72047C2semidirect
7. #71681C2semidirect
8. #70269C2semidirect
9. #70232C2semidirect
10. #69952C2semidirect
11. #69925C2semidirect
49. C2×#448D3semidirect
598. C3#23964semidirect

Maximal subgroups

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

subgroupindex
C2×#56182
C2×C3×#4482
#840682
C22×#8352
#720542
#720472
#716812
#702692
#702322
#699522
#699252
#239643