Finite Group Databasebeta

← #10828179

Subgroups — #10828179

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
2C22×#816C211yesmaximal, characteristic, complemented
3#82374C211yesmaximal, characteristic, complemented
4C2×#4735C211yesmaximal, characteristic, complemented
5C3×#1869C211yesmaximal, characteristic, complemented
6#71973C211yesmaximal, characteristic, complemented
7#69963C211yesmaximal, characteristic, complemented
8#69960C212yesmaximal, complemented
9#69936C212yesmaximal, complemented
10#68684C212yesmaximal, complemented
11#68630C212yesmaximal, complemented
12#24036—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C23×Dic6C2211yescharacteristic
15C23×C3×C8C2211yescharacteristic
16C2×C3×C4×D4C2211yescharacteristic
17C2×#939—21no—
18C2×#930C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C2×#924C2211yescharacteristic
21#5552—24no—
22#5552—24no—
23C2×#896—21no—
24C2×#885—21no—
25C2×#67⋊C4—21no—
26C2×#875C2211yescharacteristic
27C2×C4×Dic6C2211yescharacteristic
28C2×C3×C22⋊C8C2211yescharacteristic
29C2×C3×C22⋊C8C2212yes—
30#5220—21no—
31#5212C2211yescharacteristic
32#5212C2212yes—
33C2×#821C2212yes—
34C2×#821C2212yes—
35C2×#817C2212yes—
36C2×#816—24no—
37C2×#816C2212yes—
38C2×#816C2212yes—
39C2×#816C2212yes—
40C2×#816C2212yes—
41#4986—21no—
42#4949—21no—
43#4946—21no—
44#4928C2212yes—
45#4735—24no—
46C3×#336C2212yes—
47#4551C2212yes—
48C2×#548—31no—
49C2×#481—31no—
50C22×#103—31no—
51#2487—31no—
52#2042—32no—
53#2035—32no—
54#1893—31no—
55#1869D311yescharacteristic, complemented
56#1865—32no—
57#1788—32no—
58#1787—31no—
59C22×C3×D4D411yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C22×#67—21no—
63C22×#67—21no—
64C23×Dic3—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
79C22×C3×C8—22no—
80C22×C3×C8—42no—
81C22×C3×C8C2312yes—
82C22×C3×C8C2312yes—
83C3×C4×D4—22no—
84C3×C4×D4—22no—
85C2×C3×C4⋊C4D411yescharacteristic
86C2×C3×C22⋊C4D411yescharacteristic
87C2×C3×C22⋊C4D411yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C2×#217—21no—
90C2×#217—21no—
91C2×#217—22no—
92#939—28no—
93#930—22no—
94#930D414yes—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×C4⋊Dic3—22no—
99C2×C4⋊Dic3—22no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3C2312yes—
102C2×C4⋊Dic3C2312yes—
103C2×C4⋊Dic3C2311yescharacteristic
104#924—22no—
105#924—22no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—22no—
109C2×C4×Dic3—21no—
110#896—28no—
111#885—28no—
112#67⋊C4—28no—
113#875—22no—
114#875—22no—
115C4×Dic6—22no—
116C4×Dic6—22no—
117C3×C22⋊C8—22no—
118C3×C22⋊C8—42no—
119C3⋊#95—22no—
120C3⋊#95—22no—
121C3⋊#95—21no—
122#821—24no—
123#821—42no—
124#817—24no—
125#816—24no—
126#816—24no—
127#816—24no—
128#816—42no—
129C22×#41—31no—
130C23×Q8—31no—
131C23×C8C2×D311yescharacteristic
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⋊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⋊C8C2×D311yescharacteristic
154C2×C22⋊C8C2×D312yes—
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#336C2×D312yes—
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×C4C2×D411yescharacteristic
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2411yescharacteristic
183C2×#67—24no—
184C2×#67—24no—
185C2×#67—44no—
186C2×#67—42no—
187C2×#67—42no—
188C22×Dic3—21no—
189C22×Dic3—24no—
190C22×Dic3—21no—
191C22×Dic3—21no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—44no—
195C22×Dic3—42no—
196C22×Dic3—82no—
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—
219C2×C3×C8—22no—
220C2×C3×C8—22no—
221C2×C3×C8—22no—
222C2×C3×C8—22no—
223C2×C3×C8—22no—
224C2×C3×C8—22no—
225C2×C3×C8—44no—
226C3×C4⋊C4—22no—
227C3×C22⋊C4—22no—
228C3×C22⋊C4—22no—
229C3×C42—21no—
230C3×C42—21no—
231#217—24no—
232#217—42no—
233#217—82no—
234(C2×D3)⋊C4—24no—
235(C2×D3)⋊C4—24no—
236(C2×D3)⋊C4—24no—
237C4⋊Dic3—21no—
238C4⋊Dic3—22no—
239C4⋊Dic3—22no—
240C4⋊Dic3—22no—
241C4⋊Dic3—22no—
242C4⋊Dic3—42no—
243C4⋊Dic3C2×D412yes—
244Dic3⋊C4—24no—
245Dic3⋊C4—24no—
246Dic3⋊C4—82no—
247C4×Dic3—24no—
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—61no—
257C22×Q8—62no—
258C22×Q8—122no—
259C22×D4—31no—
260C22×D4—61no—
261C22×D4—61no—
262C22×D4D1211yescharacteristic
263C23×C4—61no—
264C23×C4—61no—
265C23×C4—61no—
266C23×C4C22×D311yescharacteristic
267C23×C4C22×D311yescharacteristic
268C22×C8—22no—
269C22×C8—42no—
270C22×C8C22×D312yes—
271C22×C8C22×D312yes—
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—31no—
287C2×C4⋊C4—32no—
288C2×C4⋊C4—32no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4—62no—
292C2×C4⋊C4—62no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—62no—
295C2×C4⋊C4D1211yescharacteristic
296C2×C22⋊C4—61no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4—61no—
299C2×C22⋊C4—61no—
300C2×C22⋊C4—61no—
301C2×C22⋊C4—62no—
302C2×C22⋊C4D1211yescharacteristic
303C2×C22⋊C4D1211yescharacteristic
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—22no—
314C22⋊C8—42no—
315#95—62no—
316#95—61no—
317#95—62no—
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#12012yes—
350C2×C3×C4C22×D411yescharacteristic
351C2×C3×C4C22×D411yescharacteristic
352C2×C3×C4C2×#4112yes—
353C2×C3×C4H(5,𝔽2)12yes—
354#67—48no—
355#67—48no—
356C2×Dic3—22no—
357C2×Dic3—24no—
358C2×Dic3—22no—
359C2×Dic3—22no—
360C2×Dic3—48no—
361C2×Dic3—42no—
362C2×Dic3—41no—
363C2×Dic3—41no—
364C2×Dic3—42no—
365C2×Dic3—42no—
366C2×Dic3—42no—
367C2×Dic3—42no—
368C2×Dic3—42no—
369C2×Dic3—42no—
370C2×Dic3—84no—
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×D1211yescharacteristic
386C24C2×D1211yescharacteristic
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—64no—
422C22×C4—61no—
423C22×C4—64no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—62no—
427C22×C4—62no—
428C22×C4—124no—
429C22×C4—124no—
430C22×C4—122no—
431C22×C4—242no—
432C22×C4C2×D1211yescharacteristic
433C22×C4C2×D1211yescharacteristic
434C22×C4C2×D1211yescharacteristic
435C22×C4C2×D1211yescharacteristic
436C22×C4D3×D412yes—
437C22×C4D3×D412yes—
438C22×C4C23×D311yescharacteristic
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—44no—
446C4⋊C4—22no—
447C4⋊C4—32no—
448C4⋊C4—64no—
449C4⋊C4—64no—
450C4⋊C4—61no—
451C4⋊C4—62no—
452C4⋊C4—62no—
453C4⋊C4—62no—
454C4⋊C4—62no—
455C4⋊C4—122no—
456C4⋊C4—242no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—64no—
460C22⋊C4—64no—
461C22⋊C4—64no—
462C22⋊C4—64no—
463C22⋊C4—122no—
464C22⋊C4—242no—
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×C4C2×#12311yescharacteristic
493C3×C4#53411yescharacteristic
494C3×C4#54612yes—
495Dic3—24no—
496Dic3—42no—
497Dic3—42no—
498Dic3—82no—
499Dic3—82no—
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#88212yes—
513C23#90412yes—
514C23#90912yes—
515C23C22×D1211yescharacteristic
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—62no—
544C2×C4—64no—
545C2×C4—62no—
546C2×C4—62no—
547C2×C4—128no—
548C2×C4—128no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—122no—
552C2×C4—122no—
553C2×C4—122no—
554C2×C4—122no—
555C2×C4—122no—
556C2×C4—122no—
557C2×C4—122no—
558C2×C4—244no—
559C2×C4#88212yes—
560C2×C4C2×D3×D411yescharacteristic
561C2×C4C2×D3×D411yescharacteristic
562C2×C4#100412yes—
563C2×C4D3×#4112yes—
564C8—42no—
565C8—42no—
566C2×C3—24no—
567C2×C3—81no—
568C2×C3#240511yescharacteristic
569C2×C3C2×#45211yescharacteristic
570C2×C3#301011yescharacteristic
571C2×C3#330112yes—
572C2×C3#331612yes—
573D3—82no—
574C22—22no—
575C22—24no—
576C22—24no—
577C22—22no—
578C22—41no—
579C22—41no—
580C22—42no—
581C22—42no—
582C22—41no—
583C22—122no—
584C22—122no—
585C22—122no—
586C22—242no—
587C22—242no—
588C22#473314yes—
589C22C2×#88211yescharacteristic
590C22#555412yes—
591C22C2×#90411yescharacteristic
592C22C2×#90911yescharacteristic
593C22#575112yes—
594C4—22no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—64no—
599C4—122no—
600C4—122no—
601C4—242no—
602C4—242no—
603C4#558911yescharacteristic
604C4#561012yes—
605C4D3×#12311yescharacteristic
606C3#2403611yescharacteristic, complemented, Sylow
607C2—24no—
608C2—81no—
609C2—242no—
610C2#7468611yescharacteristic
611C2C2×#473311yescharacteristic
612C2#7867811yescharacteristic
613C2#7945012yes—
614C2#7951312yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1869C2
socleC23×C3C22×D4
prime radicalC2×#5552C2
hypercenter#1869D3
hypocenterC3#24036
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #1869

Complements

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

NH ≅ G/Ntype
2. C22×#816C2semidirect
3. #82374C2semidirect
4. C2×#4735C2semidirect
5. C3×#1869C2semidirect
6. #71973C2semidirect
7. #69963C2semidirect
8. #69960C2semidirect
9. #69936C2semidirect
10. #68684C2semidirect
11. #68630C2semidirect
55. #1869D3semidirect
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
C22×#8162
#823742
C2×#47352
C3×#18692
#719732
#699632
#699602
#699362
#686842
#686302
#240363