Finite Group Databasebeta

← #10828177

Subgroups — #10828177

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×#5552C211yesmaximal, characteristic
2C3×#2487C211yesmaximal, characteristic, complemented
3C22×#810C211yesmaximal, characteristic, complemented
4C2×#4968C211yesmaximal, characteristic, complemented
5#71610C211yesmaximal, characteristic, complemented
6#69632C212yesmaximal, complemented
7#69628C212yesmaximal, complemented
8#69526C212yesmaximal, complemented
9#69525C211yesmaximal, characteristic, complemented
10#69522C212yesmaximal, complemented
11#69515C211yesmaximal, characteristic, complemented
12#24036—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C23×Dic6C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17C2×#939—21no—
18C2×#930C2211yescharacteristic
19C2×#924C2211yescharacteristic
20C23×C3⋊C8C2211yescharacteristic
21#5552—24no—
22#5552—24no—
23C2×#896—21no—
24C2×#885—21no—
25C2×#67⋊C4—21no—
26C2×#875C2211yescharacteristic
27C2×C4×Dic6C2211yescharacteristic
28C3×#400—21no—
29C3×#399—21no—
30C3×#389C2212yes—
31C3×#386—21no—
32C3×#385—21no—
33C3×#380C2211yescharacteristic
34C3×#380C2212yes—
35C2×#830C2211yescharacteristic
36C2×#830C2212yes—
37C2×#810—24no—
38C2×#810C2212yes—
39C2×#810C2212yes—
40C2×#810C2212yes—
41C2×#810C2212yes—
42#4968—24no—
43C2×#809C2212yes—
44C2×#809C2212yes—
45C2×#808C2212yes—
46#4529C2212yes—
47#4524C2212yes—
48C2×#548—31no—
49C2×#481—31no—
50C22×#103—31no—
51#2487D311yescharacteristic, complemented
52#2042—32no—
53#2035—32no—
54#1893—31no—
55#1869—31no—
56#1865—32no—
57#1788—32no—
58#1787—31no—
59C22×C3×D4D411yescharacteristic
60C23×C3×C4—21no—
61C23×C3×C4C2311yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C22×#67—21no—
64C22×#67—21no—
65C23×Dic3—21no—
66C2×#235—22no—
67C2×#235—22no—
68C2×#235—24no—
69C2×#235—42no—
70C22×D12C2311yescharacteristic
71C22×C4×D3—21no—
72C22×Dic6—24no—
73C22×Dic6—21no—
74C22×Dic6—22no—
75C22×Dic6—42no—
76C22×Dic6D414yes—
77C22×Dic6C2311yescharacteristic
78C22×Dic6C2311yescharacteristic
79C3×C4×D4—22no—
80C3×C4×D4—22no—
81C2×C3×C4⋊C4—22no—
82C2×C3×C4⋊C4—22no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—22no—
85C2×C3×C4⋊C4D411yescharacteristic
86C2×C3×C4⋊C4C2312yes—
87C2×C3×C4⋊C4C2312yes—
88C2×C3×C22⋊C4—21no—
89C2×C3×C22⋊C4—22no—
90C2×C3×C22⋊C4D411yescharacteristic
91C2×C3×C22⋊C4D411yescharacteristic
92C2×C3×C42C2311yescharacteristic
93C2×#217—21no—
94#939—28no—
95#930—22no—
96#930D414yes—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×C4⋊Dic3C2311yescharacteristic
101#924—22no—
102#924—22no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×C4×Dic3—21no—
106C22×C3⋊C8—22no—
107C22×C3⋊C8—42no—
108C22×C3⋊C8C2312yes—
109C22×C3⋊C8C2312yes—
110#896—28no—
111#885—28no—
112#67⋊C4—28no—
113#875—22no—
114#875—22no—
115C4×Dic6—22no—
116C4×Dic6—22no—
117C3×#95—22no—
118C3×#95—21no—
119C3×#95—22no—
120#830—22no—
121#830—42no—
122#810—24no—
123#810—24no—
124#810—24no—
125#810—42no—
126#809—24no—
127#809—42no—
128#808—24no—
129C22×#41—31no—
130C23×Q8—31no—
131C23×C8—31no—
132#548—64no—
133#548—64no—
134C2×#124—31no—
135C2×#124—61no—
136C2×C22⋊Q8—31no—
137C2×C22⋊Q8—61no—
138C2×#121—31no—
139C2×#121—61no—
140C2×C4×Q8—31no—
141C2×C4×D4—61no—
142C2×C4×D4C2×D311yescharacteristic
143C22×C4⋊C4C2×D311yescharacteristic
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—21no—
156#399—21no—
157#389C2×D312yes—
158#386—21no—
159#385—21no—
160#380C2×D311yescharacteristic
161#380C2×D312yes—
162#340—32no—
163#336—32no—
164C24×C3C2×D411yescharacteristic
165C24×C3C2×D411yescharacteristic
166C23×D3—21no—
167C2×C3×D4—22no—
168C2×C3×D4—22no—
169C2×C3×D4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—42no—
179C22×C3×C4—82no—
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2411yescharacteristic
187C2×#67—24no—
188C2×#67—24no—
189C2×#67—44no—
190C2×#67—42no—
191C2×#67—42no—
192C22×Dic3—21no—
193C22×Dic3—24no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—44no—
197#235—48no—
198#235—44no—
199#235—44no—
200C2×D12—22no—
201C2×D12—22no—
202C2×D12—21no—
203C2×D12C2×D412yes—
204C2×C4×D3—24no—
205C2×C4×D3—44no—
206C2×Dic6—21no—
207C2×Dic6—22no—
208C2×Dic6—22no—
209C2×Dic6—22no—
210C2×Dic6—28no—
211C2×Dic6—24no—
212C2×Dic6—21no—
213C2×Dic6—44no—
214C2×Dic6—48no—
215C2×Dic6—44no—
216C2×Dic6C2×D412yes—
217C2×Dic6C2×D412yes—
218C2×Dic6#4114yes—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—22no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—42no—
225C3×C4⋊C4—82no—
226C3×C22⋊C4—22no—
227C3×C22⋊C4—22no—
228C3×C22⋊C4—42no—
229C3×C22⋊C4—82no—
230C3×C42—21no—
231C3×C42—21no—
232#217—24no—
233(C2×D3)⋊C4—24no—
234(C2×D3)⋊C4—24no—
235(C2×D3)⋊C4—24no—
236C4⋊Dic3—21no—
237C4⋊Dic3C2×D412yes—
238Dic3⋊C4—24no—
239Dic3⋊C4—24no—
240C4×Dic3—24no—
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—62no—
249C2×#41—62no—
250C2×#41—64no—
251C2×#41—122no—
252C22×Q8—34no—
253C22×Q8—31no—
254C22×Q8—31no—
255C22×Q8—64no—
256C22×Q8—62no—
257C22×Q8—61no—
258C22×Q8—122no—
259C22×D4—31no—
260C22×D4—61no—
261C22×D4—61no—
262C22×D4#6711yescharacteristic
263C23×C4—21no—
264C23×C4—61no—
265C23×C4—61no—
266C23×C4C22×D311yescharacteristic
267C23×C4C22×D311yescharacteristic
268C22×C8—32no—
269C22×C8—32no—
270C22×C8—62no—
271C22×C8—122no—
272#124—62no—
273#124—62no—
274#124—68no—
275C22⋊Q8—68no—
276C22⋊Q8—62no—
277C22⋊Q8—62no—
278#121—34no—
279#121—68no—
280#121—62no—
281C4×Q8—62no—
282C4×Q8—62no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—68no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4#6711yescharacteristic
294C2×C4⋊C4C22×D312yes—
295C2×C4⋊C4C22×D312yes—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—61no—
299C2×C22⋊C4—61no—
300C2×C22⋊C4—61no—
301C2×C22⋊C4—61no—
302C2×C22⋊C4#6711yescharacteristic
303C2×C22⋊C4#6711yescharacteristic
304C2×C42—61no—
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—22no—
316#95—21no—
317#95—22no—
318C23×C3—22no—
319C23×C3—22no—
320C23×C3—21no—
321C23×C3—22no—
322C23×C3—21no—
323C23×C3—21no—
324C23×C3—22no—
325C23×C3#12012yes—
326C23×C3#13512yes—
327C23×C3#13712yes—
328C23×C3C22×D411yescharacteristic
329C22×D3—22no—
330C22×D3—42no—
331C22×D3—41no—
332C22×D3—41no—
333C22×D3—42no—
334C3×D4—44no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—24no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—22no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—22no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4—42no—
355C2×C3×C4—84no—
356C2×C3×C4#12012yes—
357C2×C3×C4C22×D411yescharacteristic
358C2×C3×C4C22×D411yescharacteristic
359C2×C3×C4C2×#4112yes—
360C2×C3×C4H(5,𝔽2)12yes—
361#67—48no—
362#67—48no—
363C2×Dic3—22no—
364C2×Dic3—24no—
365C2×Dic3—22no—
366C2×Dic3—22no—
367C2×Dic3—48no—
368C2×Dic3—42no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371D12—24no—
372D12—41no—
373D12—41no—
374C4×D3—48no—
375Dic6—22no—
376Dic6—24no—
377Dic6—22no—
378Dic6—22no—
379Dic6—48no—
380Dic6—42no—
381Dic6#12314yes—
382C3⋊C8—42no—
383C3⋊C8—42no—
384C24—61no—
385C24C2×#6711yescharacteristic
386C24C2×#6711yescharacteristic
387#41—128no—
388#41—124no—
389#41—124no—
390C2×Q8—32no—
391C2×Q8—32no—
392C2×Q8—34no—
393C2×Q8—68no—
394C2×Q8—64no—
395C2×Q8—61no—
396C2×Q8—61no—
397C2×Q8—62no—
398C2×Q8—62no—
399C2×Q8—62no—
400C2×Q8—124no—
401C2×Q8—124no—
402C2×Q8—128no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—32no—
407C2×D4—64no—
408C2×D4—64no—
409C2×D4—61no—
410C2×D4—62no—
411C2×D4—62no—
412C2×D4—124no—
413C2×D4—122no—
414C2×D4—122no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—42no—
424C22×C4—64no—
425C22×C4—61no—
426C22×C4—64no—
427C22×C4—61no—
428C22×C4—61no—
429C22×C4—82no—
430C22×C4—124no—
431C22×C4—124no—
432C22×C4D3×D412yes—
433C22×C4D3×D412yes—
434C22×C4C2×#6711yescharacteristic
435C22×C4C2×#6711yescharacteristic
436C22×C4C2×#6711yescharacteristic
437C22×C4C2×#6711yescharacteristic
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—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—32no—
452C4⋊C4—42no—
453C4⋊C4—64no—
454C4⋊C4—64no—
455C4⋊C4—61no—
456C4⋊C4—82no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—42no—
460C22⋊C4—64no—
461C22⋊C4—64no—
462C22⋊C4—64no—
463C22⋊C4—64no—
464C22⋊C4—82no—
465C42—21no—
466C42—21no—
467C42—64no—
468C22×C3—22no—
469C22×C3—22no—
470C22×C3—24no—
471C22×C3—24no—
472C22×C3—41no—
473C22×C3—41no—
474C22×C3—41no—
475C22×C3—42no—
476C22×C3—42no—
477C22×C3#45214yes—
478C22×C3C2×#12011yescharacteristic
479C22×C3#54612yes—
480C22×C3C2×#13511yescharacteristic
481C22×C3C2×#13711yescharacteristic
482C22×C3#57712yes—
483C2×D3—42no—
484C2×D3—42no—
485C2×D3—42no—
486C2×D3—82no—
487C2×D3—82no—
488C3×C4—22no—
489C3×C4—41no—
490C3×C4—41no—
491C3×C4—41no—
492C3×C4—82no—
493C3×C4—82no—
494C3×C4C2×#12311yescharacteristic
495C3×C4#53411yescharacteristic
496C3×C4#54612yes—
497Dic3—24no—
498Dic3—42no—
499Dic3—42no—
500C23—22no—
501C23—22no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—22no—
506C23—22no—
507C23—62no—
508C23—121no—
509C23—121no—
510C23—122no—
511C23—122no—
512C23#93712yes—
513C23#94212yes—
514C23#95012yes—
515C23C22×#6711yescharacteristic
516Q8—34no—
517Q8—62no—
518Q8—64no—
519Q8—62no—
520Q8—62no—
521Q8—128no—
522Q8—122no—
523D4—44no—
524D4—64no—
525D4—128no—
526D4—128no—
527D4—121no—
528D4—121no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—62no—
550C2×C4—64no—
551C2×C4—62no—
552C2×C4—62no—
553C2×C4—84no—
554C2×C4—128no—
555C2×C4—128no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—122no—
559C2×C4#93712yes—
560C2×C4C2×D3×D411yescharacteristic
561C2×C4C2×D3×D411yescharacteristic
562C2×C4D3×#4112yes—
563C2×C4#100912yes—
564C8—122no—
565C8—122no—
566C2×C3—24no—
567C2×C3—81no—
568C2×C3#240511yescharacteristic
569C2×C3C2×#45211yescharacteristic
570C2×C3#301011yescharacteristic
571C2×C3#330112yes—
572C2×C3#331612yes—
573D3—82no—
574C22—24no—
575C22—24no—
576C22—22no—
577C22—22no—
578C22—41no—
579C22—42no—
580C22—42no—
581C22—41no—
582C22—41no—
583C22—122no—
584C22—122no—
585C22—122no—
586C22—242no—
587C22—242no—
588C22#504714yes—
589C22C2×#93711yescharacteristic
590C22C2×#94211yescharacteristic
591C22C2×#95011yescharacteristic
592C22#582912yes—
593C22#583612yes—
594C4—22no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—64no—
599C4—82no—
600C4—82no—
601C4—122no—
602C4—122no—
603C4D3×#12311yescharacteristic
604C4#565311yescharacteristic
605C4#565412yes—
606C3#2403611yescharacteristic, complemented, Sylow
607C2—24no—
608C2—81no—
609C2—242no—
610C2C2×#504711yescharacteristic
611C2#8289311yescharacteristic
612C2#8339011yescharacteristic
613C2#8361812yes—
614C2#8364112yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2487C2
socleC23×C3C22×D4
prime radicalC2×#5552C2
hypercenter#2487D3
hypocenterC3#24036
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2487

Complements

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

NH ≅ G/Ntype
2. C3×#2487C2semidirect
3. C22×#810C2semidirect
4. C2×#4968C2semidirect
5. #71610C2semidirect
6. #69632C2semidirect
7. #69628C2semidirect
8. #69526C2semidirect
9. #69525C2semidirect
10. #69522C2semidirect
11. #69515C2semidirect
51. #2487D3semidirect
606. C3#24036semidirect

Maximal subgroups

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

subgroupindex
C2×#55522
C3×#24872
C22×#8102
C2×#49682
#716102
#696322
#696282
#695262
#695252
#695222
#695152
#240363