Finite Group Databasebeta

← #10943922

Subgroups — #10943922

Order 768.

3630 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5964C211yesmaximal, characteristic
2C3×#2432C211yesmaximal, characteristic, complemented
3C2×#5095C212yesmaximal, complemented
4C2×#5061C212yesmaximal, complemented
5#82735C212yesmaximal, complemented
6#72137C211yesmaximal, characteristic, complemented
7#69692C212yesmaximal, complemented
8#69662C212yesmaximal, complemented
9#69454C212yesmaximal, complemented
10#32893—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12#5964—28no—
13#5964—28no—
14C2×#1010—22no—
15C2×#1009—22no—
16C2×D3×#41—24no—
17C2×#1007C2212yes—
18C2×#1004—22no—
19C2×#1004C2212yes—
20C22×#235—22no—
21C22×#235C2212yes—
22C2×C3×C4⋊Q8C2211yescharacteristic, complemented
23C2×C3×#124C2211yescharacteristic, complemented
24C2×C3×C22⋊Q8C2211yescharacteristic, complemented
25C2×#942C2214yescomplemented
26C3×#394—22no—
27C3×#391C2211yescharacteristic, complemented
28C3×#390C2211yescharacteristic, complemented
29C3×#385—22no—
30C3×C4×C22⋊C4C2211yescharacteristic, complemented
31C2×#836C2212yescomplemented
32C2×#833C2212yescomplemented
33#5207C2212yescomplemented
34#5095—28no—
35#5061—28no—
36#5050—22no—
37#5047—24no—
38#5046—22no—
39#4966C2214yescomplemented
40C2×#805C2214yescomplemented
41C2×#805C2214yescomplemented
42C2×#585—31no—
43C2×#456—32no—
44C2×#455—32no—
45#3007—32no—
46#2432D311yescharacteristic, complemented
47#1793—32no—
48#1792—32no—
49#1789—32no—
50#1785—31no—
51C2×C3×#41—24no—
52C2×C3×#41—24no—
53C2×C3×#41—22no—
54C2×C3×#41—22no—
55C22×C3×Q8C2311yescharacteristic
56C22×C3×Q8C2311yescharacteristic
57C22×C3×D4D412yes—
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4D412yes—
60C23×C3×C4C2311yescharacteristic
61C22×#67—24no—
62C22×#67—22no—
63#1010—232no—
64#1009—232no—
65D3×#41—232no—
66D3×#41—232no—
67#1007—24no—
68#1007—28no—
69#1007D418yes—
70C2×#239—24no—
71C2×D3×Q8—24no—
72#1004—216no—
73#1004—216no—
74#1004—24no—
75#1004—28no—
76#1004D418yes—
77C2×#237—28no—
78C2×#237—24no—
79C2×D3×D4—28no—
80C2×D3×D4—24no—
81C2×#235—24no—
82C2×#235—24no—
83C2×#235—216no—
84C2×#235—216no—
85C2×#235—22no—
86C2×#235—24no—
87C2×#235—22no—
88C2×#235—28no—
89C2×#235—28no—
90C2×#235D414yes—
91C2×#235C2314yes—
92C22×D12—22no—
93C22×D12D412yes—
94C22×C4×D3—24no—
95C22×C4×D3—22no—
96C22×Dic6—22no—
97C22×Dic6D412yes—
98C3×C4⋊Q8—24no—
99C3×#124—24no—
100C3×C22⋊Q8—22no—
101C3×C22⋊Q8—22no—
102C2×C3×C4⋊C4—21no—
103C2×C3×C4⋊C4—22no—
104C2×C3×C4⋊C4C2311yescharacteristic
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—22no—
107C2×C3×C22⋊C4—22no—
108C2×C3×C22⋊C4—21no—
109C2×C3×C42C2311yescharacteristic
110C2×C3×C42C2311yescharacteristic
111C2×#216—24no—
112#942—28no—
113#942—28no—
114C2×#215—24no—
115C2×#208C2314yes—
116C3×#95—21no—
117C3×#95—22no—
118#836—24no—
119#833—24no—
120#830—22no—
121#830—22no—
122#810—24no—
123#809—24no—
124#805—28no—
125#805—28no—
126#585—68no—
127#585—68no—
128C2×2−1+4—32no—
129C2×2−1+4—62no—
130C2×H(5,𝔽2)—32no—
131C2×H(5,𝔽2)—62no—
132C2×H(5,𝔽2)—62no—
133C22×#41—32no—
134C22×#41—64no—
135C22×#41—62no—
136C22×#41C2×D311yescharacteristic
137C2×#137—34no—
138C2×C4⋊Q8C2×D311yescharacteristic, complemented
139C2×#124C2×D311yescharacteristic, complemented
140C2×C22⋊Q8C2×D311yescharacteristic, complemented
141#456—68no—
142#455—68no—
143#452—64no—
144#451—62no—
145#450—62no—
146C2×#104—34no—
147C2×#104—34no—
148#417—34no—
149C2×#101—32no—
150C2×#100—32no—
151#407—32no—
152#394—22no—
153#391C2×D311yescharacteristic, complemented
154#390C2×D311yescharacteristic, complemented
155#385—22no—
156C4×C22⋊C4C2×D311yescharacteristic, complemented
157C24×C3C2×D412yes—
158C24×C3C2×D411yescharacteristic
159C23×D3—24no—
160C23×D3—22no—
161C3×#41—216no—
162C3×#41—216no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8—22no—
168C2×C3×Q8C2×D414yes—
169C2×C3×D4—24no—
170C2×C3×D4—24no—
171C2×C3×D4—24no—
172C2×C3×D4—22no—
173C2×C3×D4—22no—
174C2×C3×D4C2×D414yes—
175C22×C3×C4—24no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—24no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2×D412yes—
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D412yes—
189C22×C3×C4C2411yescharacteristic
190C2×#67—28no—
191C2×#67—216no—
192C2×#67—28no—
193C2×#67—28no—
194C2×#67—216no—
195C2×#67—24no—
196C2×#67—24no—
197C22×Dic3—24no—
198C22×Dic3—22no—
199#239—216no—
200#239—216no—
201D3×Q8—216no—
202D3×Q8—216no—
203#237—264no—
204#237—216no—
205#237—216no—
206D3×D4—264no—
207D3×D4—216no—
208D3×D4—216no—
209#235—28no—
210#235—28no—
211#235—216no—
212#235—232no—
213#235—232no—
214#235—216no—
215#235—216no—
216#235—24no—
217#235—24no—
218#235—28no—
219#235—28no—
220#235C2×D418yes—
221#235C2×D418yes—
222C2×D12—24no—
223C2×D12—216no—
224C2×D12—22no—
225C2×D12—28no—
226C2×D12C2×D414yes—
227C2×C4×D3—28no—
228C2×C4×D3—232no—
229C2×C4×D3—216no—
230C2×C4×D3—24no—
231C2×C4×D3—24no—
232C2×Dic6—24no—
233C2×Dic6—216no—
234C2×Dic6—22no—
235C2×Dic6—28no—
236C2×Dic6C2×D414yes—
237C3×C4⋊C4—22no—
238C3×C4⋊C4—41no—
239C3×C4⋊C4—41no—
240C3×C4⋊C4—44no—
241C3×C22⋊C4—42no—
242C3×C22⋊C4—44no—
243C3×C22⋊C4—42no—
244C3×C22⋊C4—44no—
245C3×C42—22no—
246C3×C42—22no—
247#216—48no—
248#215—48no—
249#208—24no—
250#208—24no—
251C2×C3⋊C8—24no—
2522−1+4—38no—
2532−1+4—68no—
2542−1+4—632no—
2552−1+4—64no—
256H(5,𝔽2)—38no—
257H(5,𝔽2)—68no—
258H(5,𝔽2)—616no—
259H(5,𝔽2)—616no—
260H(5,𝔽2)—632no—
261H(5,𝔽2)—64no—
262C2×#41—24no—
263C2×#41—24no—
264C2×#41—22no—
265C2×#41—22no—
266C2×#41—34no—
267C2×#41—34no—
268C2×#41—68no—
269C2×#41—68no—
270C2×#41—64no—
271C2×#41—64no—
272C2×#41—616no—
273C2×#41—616no—
274C2×#41—64no—
275C2×#41—68no—
276C2×#41—64no—
277C2×#41—632no—
278C2×#41—632no—
279C2×#41—62no—
280C2×#41—62no—
281C2×#41—64no—
282C22×Q8—32no—
283C22×Q8—62no—
284C22×Q8—64no—
285C22×Q8C22×D311yescharacteristic
286C22×Q8C22×D311yescharacteristic
287C22×D4—32no—
288C22×D4—62no—
289C22×D4—64no—
290C22×D4—68no—
291C22×D4—62no—
292C22×D4—64no—
293C22×D4#6712yes—
294C22×D4C22×D311yescharacteristic
295C23×C4—64no—
296C23×C4—62no—
297C23×C4#6712yes—
298C23×C4C22×D311yescharacteristic
299#137—68no—
300#137—68no—
301C2×Q16—64no—
302C2×SD16—64no—
303C2×M16—34no—
304C4⋊Q8—24no—
305#124—24no—
306C22⋊Q8—22no—
307C22⋊Q8—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4C22×D311yescharacteristic
311C2×C22⋊C4—21no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—22no—
315C2×C42C22×D311yescharacteristic
316C2×C42C22×D311yescharacteristic
317#104—68no—
318#104—68no—
319#103—64no—
320#102—64no—
321#101—64no—
322#100—64no—
323C22⋊C8—62no—
324C22⋊C8—62no—
325#95—21no—
326#95—22no—
327C23×C3—22no—
328C23×C3—21no—
329C23×C3—22no—
330C23×C3—21no—
331C23×C3—41no—
332C23×C3—41no—
333C23×C3—42no—
334C23×C3—42no—
335C23×C3#12012yes—
336C23×C3#12014yes—
337C23×C3C22×D411yescharacteristic
338C22×D3—24no—
339C22×D3—216no—
340C22×D3—28no—
341C22×D3—82no—
342C22×D3—82no—
343C22×D3—84no—
344C22×D3—84no—
345C3×Q8—28no—
346C3×Q8—44no—
347C3×D4—216no—
348C3×D4—28no—
349C2×C3×C4—28no—
350C2×C3×C4—28no—
351C2×C3×C4—22no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—22no—
355C2×C3×C4—24no—
356C2×C3×C4—24no—
357C2×C3×C4—22no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—42no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4#12011yescharacteristic
369C2×C3×C4#12011yescharacteristic
370C2×C3×C4#12014yes—
371C2×C3×C4C22×D412yes—
372C2×C3×C4C22×D412yes—
373C2×C3×C4C22×D412yes—
374#67—232no—
375#67—216no—
376#67—28no—
377#67—28no—
378C2×Dic3—216no—
379C2×Dic3—28no—
380C2×Dic3—24no—
381D12—28no—
382D12—216no—
383D12—24no—
384D12#12018yes—
385C4×D3—216no—
386C4×D3—232no—
387C4×D3—28no—
388C4×D3—28no—
389Dic6—28no—
390Dic6—216no—
391Dic6—24no—
392Dic6#12018yes—
393C3⋊C8—44no—
394C24—64no—
395C24—62no—
396C24C2×#6712yes—
397C24C2×#6711yescharacteristic
398#41—216no—
399#41—216no—
400#41—38no—
401#41—38no—
402#41—68no—
403#41—68no—
404#41—632no—
405#41—68no—
406#41—68no—
407#41—632no—
408#41—616no—
409#41—616no—
410#41—616no—
411#41—664no—
412#41—616no—
413#41—616no—
414#41—616no—
415#41—616no—
416#41—64no—
417#41—64no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—34no—
424C2×Q8—68no—
425C2×Q8—64no—
426C2×Q8—616no—
427C2×Q8—616no—
428C2×Q8—616no—
429C2×Q8—62no—
430C2×Q8D3×D414yes—
431C2×D4—24no—
432C2×D4—24no—
433C2×D4—24no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—34no—
437C2×D4—68no—
438C2×D4—64no—
439C2×D4—616no—
440C2×D4—616no—
441C2×D4—664no—
442C2×D4—616no—
443C2×D4—68no—
444C2×D4—68no—
445C2×D4—68no—
446C2×D4—616no—
447C2×D4—64no—
448C2×D4—616no—
449C2×D4—64no—
450C2×D4—62no—
451C2×D4D3×D414yes—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—68no—
463C22×C4—632no—
464C22×C4—616no—
465C22×C4—64no—
466C22×C4—62no—
467C22×C4—64no—
468C22×C4—64no—
469C22×C4D3×D412yes—
470C22×C4D3×D412yes—
471C22×C4C2×#6711yescharacteristic
472C22×C4C2×#6712yes—
473C22×C4C23×D311yescharacteristic
474Q16—128no—
475SD16—128no—
476M16—64no—
477M16—64no—
478C2×C8—64no—
479C4⋊C4—22no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—44no—
483C22⋊C4—42no—
484C22⋊C4—44no—
485C22⋊C4—42no—
486C22⋊C4—44no—
487C42—22no—
488C42—22no—
489C22×C3—21no—
490C22×C3—21no—
491C22×C3—22no—
492C22×C3—24no—
493C22×C3—22no—
494C22×C3—22no—
495C22×C3—44no—
496C22×C3—42no—
497C22×C3—42no—
498C22×C3—81no—
499C22×C3—82no—
500C22×C3—81no—
501C22×C3C2×#12011yescharacteristic
502C22×C3C2×#12011yescharacteristic
503C22×C3C2×#12011yescharacteristic
504C2×D3—28no—
505C2×D3—44no—
506C2×D3—44no—
507C2×D3—88no—
508C2×D3—816no—
509C3×C4—21no—
510C3×C4—24no—
511C3×C4—21no—
512C3×C4—42no—
513C3×C4—81no—
514C3×C4—81no—
515C3×C4—81no—
516C3×C4C2×#12014yes—
517Dic3—28no—
518C23—22no—
519C23—21no—
520C23—22no—
521C23—21no—
522C23—41no—
523C23—41no—
524C23—42no—
525C23—42no—
526C23—68no—
527C23—616no—
528C23—64no—
529C23—242no—
530C23—242no—
531C23—244no—
532C23—244no—
533C23#93712yes—
534C23#93714yes—
535C23C22×#6711yescharacteristic
536Q8—28no—
537Q8—38no—
538Q8—44no—
539Q8—68no—
540Q8—616no—
541Q8—64no—
542D4—216no—
543D4—28no—
544D4—38no—
545D4—68no—
546D4—616no—
547D4—632no—
548D4—616no—
549D4—68no—
550D4—68no—
551D4—64no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—616no—
572C2×C4—632no—
573C2×C4—616no—
574C2×C4—68no—
575C2×C4—64no—
576C2×C4—68no—
577C2×C4—68no—
578C2×C4#93714yes—
579C2×C4#93711yescharacteristic
580C2×C4#93711yescharacteristic
581C2×C4C2×D3×D412yes—
582C2×C4C2×D3×D412yes—
583C2×C4C2×D3×D412yes—
584C8—124no—
585C2×C3—21no—
586C2×C3—21no—
587C2×C3—42no—
588C2×C3—44no—
589C2×C3#284611yescharacteristic
590C2×C3#301811yescharacteristic
591C2×C3#302211yescharacteristic
592D3—48no—
593C22—21no—
594C22—21no—
595C22—22no—
596C22—22no—
597C22—24no—
598C22—22no—
599C22—44no—
600C22—42no—
601C22—42no—
602C22—68no—
603C22—81no—
604C22—82no—
605C22—81no—
606C22—124no—
607C22—124no—
608C22—248no—
609C22—2416no—
610C22C2×#93711yescharacteristic
611C22C2×#93711yescharacteristic
612C22C2×#93711yescharacteristic
613C4—21no—
614C4—21no—
615C4—24no—
616C4—42no—
617C4—68no—
618C4—81no—
619C4—81no—
620C4—81no—
621C4D3×#12014yes—
622C3#3289311yescharacteristic, complemented, Sylow
623C2—21no—
624C2—21no—
625C2—42no—
626C2—44no—
627C2—128no—
628C2#8261211yescharacteristic
629C2#8332211yescharacteristic
630C2#8341611yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2432C2
socleC22×C3C2×#120
prime radicalC2×#5964C2
hypercenter#2432D3
hypocenterC3#32893
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2432

Complements

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

NH ≅ G/Ntype
2. C3×#2432C2semidirect
3. C2×#5095C2semidirect
4. C2×#5061C2semidirect
5. #82735C2semidirect
6. #72137C2semidirect
7. #69692C2semidirect
8. #69662C2semidirect
9. #69454C2semidirect
22. C2×C3×C4⋊Q8C22semidirect
23. C2×C3×#124C22semidirect
24. C2×C3×C22⋊Q8C22semidirect
25. C2×#942C22semidirect
27. C3×#391C22semidirect
28. C3×#390C22semidirect
30. C3×C4×C22⋊C4C22semidirect
31. C2×#836C22semidirect
32. C2×#833C22semidirect
33. #5207C22semidirect
39. #4966C22semidirect
40. C2×#805C22semidirect
41. C2×#805C22semidirect
46. #2432D3semidirect
138. C2×C4⋊Q8C2×D3semidirect
139. C2×#124C2×D3semidirect
140. C2×C22⋊Q8C2×D3semidirect
153. #391C2×D3semidirect
154. #390C2×D3semidirect
156. C4×C22⋊C4C2×D3semidirect
622. C3#32893semidirect

Maximal subgroups

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

subgroupindex
C2×#59642
C3×#24322
C2×#50952
C2×#50612
#827352
#721372
#696922
#696622
#694542
#328933