Finite Group Databasebeta

← #10900842

Subgroups — #10900842

Order 768.

1378 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5725C211yesmaximal, characteristic
2C2×#5187C212yesmaximal, complemented
3#76663C211yesmaximal, characteristic, complemented
4C3×#2016C211yesmaximal, characteristic, complemented
5#72072C212yesmaximal, complemented
6#71781C211yesmaximal, characteristic, complemented
7#70267C212yesmaximal, complemented
8#70245C212yesmaximal, complemented
9#69688C211yesmaximal, characteristic, complemented
10#68664C211yesmaximal, characteristic, complemented
11#68629C211yesmaximal, characteristic, complemented
12#29629—31nomaximal, Sylow
13C22×#239C2212yes—
14C22×C3×Q16C2212yes—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×#947C2212yes—
17C2×#946C2211yescharacteristic
18C22×#215C2212yes—
19#5725—28no—
20C2×#895—21no—
21C2×#895—21no—
22C2×#893—21no—
23C2×#893—21no—
24C2×#874C2211yescharacteristic
25C2×C42×D3C2211yescharacteristic
26C2×C3×#103C2212yes—
27C3×#391C2211yescharacteristic
28#5229C2211yescharacteristic
29C2×#835C2212yes—
30C2×#835C2212yes—
31#5187—28no—
32C2×#821C2212yes—
33C2×#821C2212yes—
34C2×(C2×D3)⋊C8C2212yes—
35#4987C2211yescharacteristic
36#4984C2211yescharacteristic
37C2×#809C2212yes—
38#4928C2211yescharacteristic
39#4668—21no—
40#4664—21no—
41#4654—21no—
42#4652—21no—
43C3×#336C2211yescharacteristic
44#4550C2212yes—
45#4524C2211yescharacteristic
46C2×#532—31no—
47C2×#452—32no—
48#2855—31no—
49#2043—31no—
50#2042—32no—
51#2041—31no—
52#2016D311yescharacteristic, complemented
53#2015—32no—
54#2014—31no—
55#1817—31no—
56#1788—32no—
57C22×C3×Q8C2311yescharacteristic
58C22×C3×Q8C2312yes—
59C2×#239—24no—
60C2×#239—42no—
61C2×#239—42no—
62C2×#239—42no—
63C2×#239—42no—
64C2×#239D418yes—
65C22×D12—22no—
66C22×D12—21no—
67C22×D12—21no—
68C22×D12C2312yes—
69C22×C4×D3—22no—
70C22×C4×D3D411yescharacteristic
71C22×C4×D3D411yescharacteristic
72C22×C4×D3C2311yescharacteristic
73C2×C3×Q16—24no—
74C2×C3×Q16—42no—
75C2×C3×Q16—42no—
76C22×C3×C8C2312yes—
77C3×C4⋊Q8—24no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C42C2311yescharacteristic
82#947—24no—
83#947—24no—
84#946—24no—
85C2×#215—24no—
86C2×#215—42no—
87C2×#215—42no—
88C2×(C2×D3)⋊C4—22no—
89C2×(C2×D3)⋊C4—22no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3C2311yescharacteristic
94C2×C4⋊Dic3C2311yescharacteristic
95C2×Dic3⋊C4—21no—
96C2×Dic3⋊C4—21no—
97C2×C4×Dic3D411yescharacteristic
98C2×C4×Dic3D411yescharacteristic
99C2×C4×Dic3C2311yescharacteristic
100C22×C3⋊C8C2312yes—
101#895—28no—
102#895—28no—
103#893—28no—
104#893—28no—
105#874—24no—
106C42×D3—24no—
107C3×#103—42no—
108C3×#95—21no—
109C3×#95—21no—
110#835—24no—
111#835—42no—
112C3⋊#95—21no—
113C3⋊#95—21no—
114C3⋊#95—21no—
115C3⋊#95—21no—
116C3⋊#95—21no—
117#821—24no—
118#821—42no—
119(C2×D3)⋊C8—24no—
120#809—42no—
121C22×#41—32no—
122C22×Q16C2×D312yes—
123C22×SD16—32no—
124#532—68no—
125C2×C4⋊Q8C2×D311yescharacteristic
126C2×#127—31no—
127C2×#124—32no—
128C2×C22⋊Q8—31no—
129C2×#121—61no—
130C2×#121—61no—
131C2×C4×D4—61no—
132C2×C4×D4—61no—
133C22×C42—31no—
134#452—68no—
135C2×#103—32no—
136C2×#103—32no—
137C2×#103C2×D312yes—
138C2×#102—32no—
139C2×#102—32no—
140C2×#102—32no—
141C2×C22⋊C8—32no—
142#391—31no—
143#391C2×D311yescharacteristic
144#389—31no—
145#389—31no—
146#387—61no—
147#387—61no—
148#385—61no—
149#381—61no—
150#380—31no—
151#336—32no—
152#336—31no—
153#336C2×D311yescharacteristic
154C23×D3—22no—
155C23×D3C2×D411yescharacteristic
156C2×C3×Q8—22no—
157C2×C3×Q8—22no—
158C2×C3×Q8—22no—
159C2×C3×Q8—22no—
160C2×C3×Q8—22no—
161C2×C3×Q8—22no—
162C2×C3×Q8—21no—
163C2×C3×Q8—22no—
164C2×C3×Q8—21no—
165C2×C3×Q8C2×D414yes—
166C22×C3×C4—22no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2411yescharacteristic
172C22×Dic3—21no—
173C22×Dic3—21no—
174C22×Dic3C2×D411yescharacteristic
175C22×Dic3C2×D411yescharacteristic
176C22×Dic3C2×D411yescharacteristic
177#239—48no—
178#239—48no—
179#239—48no—
180#239—48no—
181C2×D12—22no—
182C2×D12—22no—
183C2×D12—22no—
184C2×D12—22no—
185C2×D12—24no—
186C2×D12—24no—
187C2×D12—28no—
188C2×D12—22no—
189C2×D12—42no—
190C2×D12—42no—
191C2×D12—44no—
192C2×D12—44no—
193C2×D12—42no—
194C2×D12—42no—
195C2×D12C2×D414yes—
196C2×C4×D3—22no—
197C2×C4×D3—21no—
198C2×C4×D3—21no—
199C2×C4×D3—22no—
200C2×C4×D3—22no—
201C2×C4×D3—22no—
202C2×C4×D3—22no—
203C2×C4×D3—28no—
204C2×C4×D3—41no—
205C2×C4×D3—41no—
206C2×C4×D3—44no—
207C2×C4×D3—44no—
208C2×C4×D3C2×D414yes—
209C3×Q16—48no—
210C2×C3×C8—22no—
211C2×C3×C8—22no—
212C2×C3×C8—22no—
213C3×C4⋊C4—24no—
214C3×C4⋊C4—24no—
215C3×C4⋊C4—22no—
216C3×C42—22no—
217#215—48no—
218(C2×D3)⋊C4—28no—
219(C2×D3)⋊C4—28no—
220(C2×D3)⋊C4—42no—
221(C2×D3)⋊C4—42no—
222C4⋊Dic3—22no—
223C4⋊Dic3—21no—
224C4⋊Dic3—21no—
225C4⋊Dic3—42no—
226Dic3⋊C4—42no—
227Dic3⋊C4—42no—
228C4×Dic3—22no—
229C4×Dic3—22no—
230C4×Dic3—22no—
231C2×C3⋊C8—22no—
232C2×C3⋊C8—22no—
233C2×C3⋊C8—22no—
234C2×#41—38no—
235C2×#41—64no—
236C2×#41—122no—
237C2×#41—122no—
238C2×#41—122no—
239C2×#41—122no—
240C22×Q8C22×D311yescharacteristic
241C22×Q8C22×D312yes—
242C22×D4—32no—
243C22×D4—62no—
244C22×D4—61no—
245C22×D4—61no—
246C23×C4—31no—
247C23×C4—31no—
248C23×C4—31no—
249C23×C4—62no—
250C2×Q16—24no—
251C2×Q16—42no—
252C2×Q16—42no—
253C2×SD16—64no—
254C2×SD16—122no—
255C2×SD16—122no—
256C22×C8—32no—
257C22×C8C22×D312yes—
258C4⋊Q8—24no—
259#127—64no—
260#124—64no—
261#124—64no—
262C22⋊Q8—64no—
263#121—68no—
264#121—68no—
265C4×D4—68no—
266C4×D4—68no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—31no—
271C2×C4⋊C4—61no—
272C2×C4⋊C4—61no—
273C2×C4⋊C4—61no—
274C2×C4⋊C4C22×D311yescharacteristic
275C2×C22⋊C4—62no—
276C2×C22⋊C4—62no—
277C2×C22⋊C4—61no—
278C2×C22⋊C4—61no—
279C2×C42—31no—
280C2×C42—31no—
281C2×C42—31no—
282C2×C42—64no—
283C2×C42C22×D311yescharacteristic
284#103—42no—
285#103—64no—
286#103—122no—
287#102—64no—
288#102—122no—
289#102—122no—
290C22⋊C8—64no—
291#95—21no—
292#95—21no—
293#95—61no—
294#95—61no—
295#95—61no—
296#95—61no—
297#95—61no—
298C23×C3C22×D411yescharacteristic
299C22×D3—21no—
300C22×D3—21no—
301C22×D3—21no—
302C22×D3—21no—
303C22×D3—21no—
304C22×D3—21no—
305C22×D3—24no—
306C22×D3—42no—
307C22×D3—42no—
308C22×D3—42no—
309C22×D3—42no—
310C22×D3—42no—
311C22×D3—42no—
312C22×D3#12012yes—
313C3×Q8—24no—
314C3×Q8—24no—
315C3×Q8—42no—
316C3×Q8—42no—
317C3×Q8—42no—
318C3×Q8—42no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—24no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4#13512yes—
333C2×C3×C4#13712yes—
334C2×C3×C4C22×D412yes—
335C2×C3×C4H(5,𝔽2)12yes—
336C2×C3×C4H(5,𝔽2)11yescharacteristic
337C2×C3×C4H(5,𝔽2)11yescharacteristic
338C2×Dic3—21no—
339C2×Dic3—21no—
340C2×Dic3—21no—
341C2×Dic3—21no—
342C2×Dic3—21no—
343C2×Dic3—21no—
344C2×Dic3—21no—
345C2×Dic3—21no—
346C2×Dic3—41no—
347C2×Dic3—41no—
348C2×Dic3—41no—
349C2×Dic3—41no—
350C2×Dic3—41no—
351C2×Dic3—41no—
352C2×Dic3#12012yes—
353D12—24no—
354D12—24no—
355D12—48no—
356D12—48no—
357D12—416no—
358D12—42no—
359D12—42no—
360C4×D3—24no—
361C4×D3—24no—
362C4×D3—416no—
363C4×D3—44no—
364C4×D3—44no—
365C3×C8—42no—
366C3⋊C8—42no—
367C24—31no—
368C24—62no—
369#41—128no—
370#41—128no—
371#41—128no—
372#41—128no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—21no—
380C2×Q8—22no—
381C2×Q8—21no—
382C2×Q8D3×D414yes—
383C2×D4—34no—
384C2×D4—68no—
385C2×D4—64no—
386C2×D4—64no—
387C2×D4—62no—
388C2×D4—62no—
389C2×D4—62no—
390C2×D4—62no—
391C2×D4—62no—
392C2×D4—122no—
393C2×D4—122no—
394C2×D4—124no—
395C2×D4—124no—
396C2×D4—122no—
397C2×D4—122no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—31no—
402C22×C4—34no—
403C22×C4—31no—
404C22×C4—31no—
405C22×C4—68no—
406C22×C4—62no—
407C22×C4—62no—
408C22×C4—62no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—121no—
416C22×C4—121no—
417C22×C4—124no—
418C22×C4—124no—
419C22×C4D3×D411yescharacteristic
420C22×C4D3×D411yescharacteristic
421C22×C4C23×D311yescharacteristic
422Q16—48no—
423SD16—128no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—62no—
428C2×C8—62no—
429C2×C8—62no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—22no—
433C4⋊C4—62no—
434C4⋊C4—61no—
435C4⋊C4—61no—
436C4⋊C4—122no—
437C4⋊C4—122no—
438C4⋊C4—122no—
439C22⋊C4—68no—
440C22⋊C4—68no—
441C22⋊C4—122no—
442C22⋊C4—122no—
443C42—22no—
444C42—62no—
445C42—62no—
446C42—62no—
447C22×C3C2×#12011yescharacteristic
448C22×C3#53511yescharacteristic
449C22×C3#53511yescharacteristic
450C22×C3C2×#13511yescharacteristic
451C22×C3C2×#13711yescharacteristic
452C22×C3#57511yescharacteristic
453C22×C3#57511yescharacteristic
454C2×D3—22no—
455C2×D3—22no—
456C2×D3—22no—
457C2×D3—41no—
458C2×D3—41no—
459C2×D3—41no—
460C2×D3—41no—
461C2×D3—44no—
462C2×D3—44no—
463C2×D3—44no—
464C2×D3—82no—
465C2×D3—82no—
466C2×D3—82no—
467C2×D3—82no—
468C3×C4—22no—
469C3×C4—22no—
470C3×C4—44no—
471C3×C4—81no—
472C3×C4—81no—
473C3×C4#53412yes—
474C3×C4#53412yes—
475Dic3—22no—
476Dic3—41no—
477Dic3—41no—
478Dic3—81no—
479Dic3—81no—
480C23—32no—
481C23—61no—
482C23—61no—
483C23—61no—
484C23—61no—
485C23—61no—
486C23—61no—
487C23—64no—
488C23—122no—
489C23—122no—
490C23—122no—
491C23—122no—
492C23—122no—
493C23—122no—
494C23C2×D3×D411yescharacteristic
495Q8—24no—
496Q8—24no—
497Q8—42no—
498Q8—42no—
499Q8—42no—
500Q8—42no—
501D4—64no—
502D4—64no—
503D4—128no—
504D4—128no—
505D4—1216no—
506D4—122no—
507D4—122no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—24no—
513C2×C4—32no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—61no—
523C2×C4—61no—
524C2×C4—64no—
525C2×C4—64no—
526C2×C4—61no—
527C2×C4—61no—
528C2×C4—61no—
529C2×C4—61no—
530C2×C4—61no—
531C2×C4—61no—
532C2×C4—124no—
533C2×C4—124no—
534C2×C4—1216no—
535C2×C4—121no—
536C2×C4—121no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4#91812yes—
542C2×C4#91912yes—
543C2×C4C2×D3×D412yes—
544C2×C4#100411yescharacteristic
545C2×C4#100412yes—
546C2×C4#100911yescharacteristic
547C8—42no—
548C8—122no—
549C2×C3#261311yescharacteristic
550C2×C3#300211yescharacteristic
551C2×C3#300711yescharacteristic
552C2×C3#321011yescharacteristic
553C2×C3#321211yescharacteristic
554C2×C3#321611yescharacteristic
555C2×C3#321911yescharacteristic
556D3—42no—
557D3—84no—
558C22—62no—
559C22—62no—
560C22—62no—
561C22—121no—
562C22—121no—
563C22—121no—
564C22—121no—
565C22—124no—
566C22—124no—
567C22—124no—
568C22—242no—
569C22—242no—
570C22—242no—
571C22—242no—
572C22D3×#12011yescharacteristic
573C22#560711yescharacteristic
574C22#571511yescharacteristic
575C22C2×#91811yescharacteristic
576C22C2×#91911yescharacteristic
577C22#577711yescharacteristic
578C22#577711yescharacteristic
579C4—22no—
580C4—22no—
581C4—44no—
582C4—62no—
583C4—81no—
584C4—81no—
585C4—121no—
586C4—121no—
587C4—241no—
588C4—241no—
589C4#558912yes—
590C4#560512yes—
591C3#2962911yescharacteristic, complemented, Sylow
592C2—122no—
593C2—244no—
594C2#7676011yescharacteristic
595C2#8018211yescharacteristic
596C2#8018411yescharacteristic
597C2#8048111yescharacteristic
598C2#8048211yescharacteristic
599C2#8180311yescharacteristic
600C2#8187711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2016C2
socleC23×C3C22×D4
prime radicalC2×#5725C2
hypercenter#2016D3
hypocenterC3#29629
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2016

Complements

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

NH ≅ G/Ntype
2. C2×#5187C2semidirect
3. #76663C2semidirect
4. C3×#2016C2semidirect
5. #72072C2semidirect
6. #71781C2semidirect
7. #70267C2semidirect
8. #70245C2semidirect
9. #69688C2semidirect
10. #68664C2semidirect
11. #68629C2semidirect
52. #2016D3semidirect
591. C3#29629semidirect

Maximal subgroups

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

subgroupindex
C2×#57252
C2×#51872
#766632
C3×#20162
#720722
#717812
#702672
#702452
#696882
#686642
#686292
#296293