Finite Group Databasebeta

← #11024928

Subgroups — #11024928

Order 768.

2232 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×D3×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×#5518C211yesmaximal, characteristic, complemented
3C3×#2621C211yesmaximal, characteristic, complemented
4#83245C211yesmaximal, characteristic, complemented
5#82757C212yesmaximal, complemented
6#82751C212yesmaximal
7#82268C211yesmaximal, characteristic
8#77856C214yesmaximal, complemented
9#77756C214yesmaximal, complemented
10#77588C214yesmaximal, complemented
11#77483C214yesmaximal
12#76614C212yesmaximal, complemented
13#76502C212yesmaximal, complemented
14#74257C211yesmaximal, characteristic, complemented
15#74152C211yesmaximal, characteristic, complemented
16#37695—31nomaximal, Sylow
17C22×D3×Q8C2212yes—
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×C22⋊Q8C2212yes—
20C2×C3×#117C2211yescharacteristic
21C2×#944C2212yes—
22C2×C4×#67C2211yescharacteristic
23C2×#924C2212yes—
24D3×C4⋊Q8—28no—
25#5518—28no—
26C2×D3×C4⋊C4C2214yes—
27C2×#889C2214yes—
28C2×#888C2214yes—
29C2×#887C2212yes—
30C2×#879C2214yes—
31C2×C42×D3C2211yescharacteristic
32C2×#870C2211yescharacteristic
33C2×#869C2211yescharacteristic
34C2×C4×Dic6C2212yes—
35C3×#398C2214yes—
36C3×#397C2214yes—
37C3×#391C2212yes—
38C3×#382C2211yescharacteristic
39#5229C2212yes—
40#5229C2212yes—
41C4×#217C2211yescharacteristic
42#4989C2214yes—
43#4989C2212yes—
44#4987C2212yes—
45#4985C2214yes—
46#4985C2214yes—
47#4984C2212yes—
48#4983C2214yes—
49#4979C2214yes—
50#4978C2212yes—
51#4973C2212yes—
52#4972C2214yes—
53#4956C2214yes—
54#4956C2212yes—
55#4948C2214yes—
56#4947C2214yes—
57#4944C2212yes—
58#4944C2214yes—
59C4×(C2×D3)⋊C4C2211yescharacteristic
60#4932C2211yescharacteristic
61C4×Dic3⋊C4C2211yescharacteristic
62#4668C2214yes—
63#4667C2214yes—
64#4666C2214yes—
65#4658C2214yes—
66#4657C2214yes—
67#4654C2214yes—
68#4654C2214yes—
69#4654C2214yes—
70#4653C2214yes—
71#4652C2214yes—
72#4646C2214yes—
73#4646C2214yes—
74C2×#541—31no—
75C22×C4⋊Q8—31no—
76#2851—31no—
77#2716—34no—
78#2713—34no—
79#2642—31no—
80#2621D311yescharacteristic, complemented
81#2481—34no—
82#2473—34no—
83#2446—32no—
84#2429—32no—
85#2409—32no—
86Q8×C22⋊C4—32no—
87#2327—31no—
88#2290—31no—
89C22×C3×Q8C2312yes—
90C23×C3×C4C2311yescharacteristic
91C22×#67C2311yescharacteristic
92C2×D3×Q8—28no—
93C2×D3×Q8—28no—
94C2×D3×Q8—44no—
95C22×C4×D3C2312yes—
96C22×C4×D3C2314yes—
97C22×C4×D3C2311yescharacteristic
98C22×Dic6C2314yes—
99C22×Dic6C2312yes—
100C3×C4⋊Q8—24no—
101C3×C22⋊Q8—44no—
102C3×#117—24no—
103C2×C3×C4⋊C4C2312yes—
104C2×C3×C4⋊C4C2312yes—
105C2×C3×C4⋊C4C2314yes—
106C2×C3×C4⋊C4C2314yes—
107C2×C3×C22⋊C4C2312yes—
108C2×C3×C22⋊C4C2314yes—
109C2×C3×C42C2311yescharacteristic
110C2×C3×C42C2311yescharacteristic
111C2×#217C2312yes—
112C2×#217C2311yescharacteristic
113C2×#217C2314yes—
114#944—28no—
115C4×#67—24no—
116C2×(C2×D3)⋊C4C2312yes—
117C2×(C2×D3)⋊C4C2314yes—
118C2×(C2×D3)⋊C4C2311yescharacteristic
119C2×C4⋊Dic3C2314yes—
120C2×C4⋊Dic3C2312yes—
121C2×C4⋊Dic3C2314yes—
122#924—28no—
123C2×Dic3⋊C4C2314yes—
124C2×Dic3⋊C4C2312yes—
125C2×Dic3⋊C4C2314yes—
126C2×Dic3⋊C4C2314yes—
127C2×Dic3⋊C4C2314yes—
128C2×Dic3⋊C4C2314yes—
129C2×Dic3⋊C4C2311yescharacteristic
130C2×Dic3⋊C4C2314yes—
131C2×C4×Dic3C2314yes—
132C2×C4×Dic3C2312yes—
133C2×C4×Dic3C2311yescharacteristic
134C2×C4×Dic3C2311yescharacteristic
135C2×C4×Dic3C2311yescharacteristic
136C2×C4×Dic3C2314yes—
137D3×C4⋊C4—216no—
138#889—216no—
139#888—216no—
140#887—44no—
141#879—216no—
142C42×D3—24no—
143#870—24no—
144#869D418yescomplemented
145C4×Dic6—28no—
146C3×#95C2314yes—
147C3×#95C2314yes—
148C3×#95C2314yes—
149C3⋊#95C2312yes—
150C3⋊#95C2314yes—
151C3⋊#95C2314yes—
152C3⋊#95C2314yes—
153C3⋊#95C2314yes—
154C3⋊#95C2314yes—
155C3⋊#95C2314yes—
156C3⋊#95C2314yes—
157C3⋊#95C2312yes—
158C3⋊#95C2312yes—
159C3⋊#95C2312yes—
160C3⋊#95C2314yes—
161C3⋊#95C2312yes—
162C23×Q8—32no—
163#541—68no—
164C2×C4⋊Q8—32no—
165C2×C4⋊Q8—34no—
166C2×C4⋊Q8—31no—
167C2×C4⋊Q8—68no—
168C2×C4⋊Q8C2×D311yescharacteristic
169C2×#126—34no—
170C2×#125—34no—
171C2×#125—31no—
172C2×C22⋊Q8—32no—
173C2×C22⋊Q8C2×D312yes—
174C2×C4×Q8—32no—
175C2×C4×Q8—32no—
176C2×C4×D4—31no—
177C2×#117C2×D311yescharacteristic
178C22×C4⋊C4—34no—
179C22×C42—31no—
180#400—34no—
181#400—34no—
182#400—34no—
183#398—34no—
184#398—34no—
185#398C2×D314yes—
186#397C2×D314yes—
187#396—34no—
188#395—34no—
189#393—34no—
190#391—32no—
191#391—32no—
192#391—32no—
193#391—34no—
194#391—34no—
195#391C2×D312yes—
196#389—32no—
197#389—34no—
198#388—34no—
199#388—32no—
200#387—32no—
201#387—34no—
202#387—34no—
203#387—34no—
204#387—34no—
205#387—34no—
206#386—34no—
207#386—32no—
208#385—34no—
209#385—32no—
210#385—34no—
211#382C2×D311yescharacteristic
212#380—32no—
213#380—34no—
214C4×C4⋊C4—31no—
215C4×C22⋊C4—31no—
216C4×C22⋊C4—31no—
217C4×C22⋊C4—34no—
218#376—31no—
219C24×C3C2411yescharacteristic
220C23×D3C2411yescharacteristic
221C2×C3×Q8—24no—
222C2×C3×Q8—24no—
223C2×C3×Q8—24no—
224C22×C3×C4—22no—
225C22×C3×C4—22no—
226C22×C3×C4—22no—
227C22×C3×C4C2414yes—
228C22×C3×C4C2414yes—
229C22×C3×C4C2412yes—
230C22×C3×C4C2411yescharacteristic
231C22×C3×C4C2412yes—
232C22×C3×C4C2411yescharacteristic
233C2×#67—22no—
234C2×#67—24no—
235C22×Dic3C2411yescharacteristic
236C22×Dic3C2411yescharacteristic
237C22×Dic3C2412yes—
238C22×Dic3C2414yes—
239C22×Dic3C2414yes—
240C22×Dic3C2411yescharacteristic
241C22×Dic3C2412yes—
242D3×Q8—432no—
243C2×C4×D3—24no—
244C2×C4×D3—24no—
245C2×C4×D3—22no—
246C2×C4×D3—22no—
247C2×C4×D3—24no—
248C2×C4×D3—28no—
249C2×C4×D3—28no—
250C2×C4×D3—44no—
251C2×C4×D3#4114yes—
252C2×Dic6—24no—
253C2×Dic6—24no—
254C2×Dic6—28no—
255C2×Dic6—28no—
256C2×Dic6—44no—
257C2×Dic6C2×D418yes—
258C3×C4⋊C4—28no—
259C3×C4⋊C4—24no—
260C3×C4⋊C4—24no—
261C3×C4⋊C4—44no—
262C3×C22⋊C4—24no—
263C3×C22⋊C4—44no—
264C3×C42—22no—
265C3×C42C2×D414yes—
266#217—28no—
267#217—22no—
268#217—24no—
269(C2×D3)⋊C4—22no—
270(C2×D3)⋊C4—24no—
271(C2×D3)⋊C4—28no—
272C4⋊Dic3—24no—
273C4⋊Dic3—28no—
274C4⋊Dic3C2×D4116yes—
275Dic3⋊C4—28no—
276Dic3⋊C4—28no—
277Dic3⋊C4—28no—
278Dic3⋊C4—28no—
279Dic3⋊C4—28no—
280Dic3⋊C4—22no—
281Dic3⋊C4—24no—
282Dic3⋊C4—44no—
283C4×Dic3—24no—
284C4×Dic3—22no—
285C4×Dic3—22no—
286C4×Dic3—22no—
287C4×Dic3—28no—
288C4×Dic3—44no—
289C22×Q8—32no—
290C22×Q8—34no—
291C22×Q8—68no—
292C22×Q8—68no—
293C22×Q8—124no—
294C22×Q8C22×D312yes—
295C22×D4—31no—
296C23×C4—31no—
297C23×C4—32no—
298C23×C4—34no—
299C23×C4C22×D311yescharacteristic
300C4⋊Q8—24no—
301C4⋊Q8—38no—
302C4⋊Q8—616no—
303C4⋊Q8—68no—
304#126—616no—
305#125—616no—
306#125—64no—
307C22⋊Q8—44no—
308C22⋊Q8—68no—
309C4×Q8—68no—
310C4×Q8—124no—
311C4×D4—64no—
312#117—24no—
313C2×C4⋊C4—34no—
314C2×C4⋊C4—34no—
315C2×C4⋊C4—34no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—32no—
318C2×C4⋊C4—34no—
319C2×C4⋊C4—34no—
320C2×C4⋊C4—34no—
321C2×C4⋊C4—32no—
322C2×C4⋊C4—34no—
323C2×C4⋊C4—34no—
324C2×C4⋊C4—616no—
325C2×C4⋊C4C22×D314yes—
326C2×C4⋊C4C22×D312yes—
327C2×C4⋊C4C22×D312yes—
328C2×C4⋊C4C22×D314yes—
329C2×C22⋊C4—31no—
330C2×C22⋊C4—34no—
331C2×C22⋊C4—32no—
332C2×C22⋊C4—31no—
333C2×C22⋊C4—32no—
334C2×C22⋊C4—34no—
335C2×C22⋊C4C22×D312yes—
336C2×C22⋊C4C22×D314yes—
337C2×C42—31no—
338C2×C42—31no—
339C2×C42—34no—
340C2×C42—31no—
341C2×C42—32no—
342C2×C42—34no—
343C2×C42—64no—
344C2×C42C22×D311yescharacteristic
345C2×C42C22×D311yescharacteristic
346#95—32no—
347#95—32no—
348#95—32no—
349#95—32no—
350#95—34no—
351#95—32no—
352#95—34no—
353#95—34no—
354#95—34no—
355#95—34no—
356#95—34no—
357#95—34no—
358#95—34no—
359#95C22×D314yes—
360#95C22×D314yes—
361#95C22×D314yes—
362C23×C3—21no—
363C23×C3—21no—
364C23×C3—21no—
365C23×C3—22no—
366C23×C3—22no—
367C23×C3C2511yescharacteristic
368C22×D3—21no—
369C22×D3—21no—
370C22×D3—22no—
371C22×D3C2×#4112yes—
372C22×D32−1+414yes—
373C3×Q8—48no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—21no—
378C2×C3×C4—24no—
379C2×C3×C4—21no—
380C2×C3×C4—24no—
381C2×C3×C4—24no—
382C2×C3×C4—24no—
383C2×C3×C4—21no—
384C2×C3×C4—22no—
385C2×C3×C4—24no—
386C2×C3×C4—22no—
387C2×C3×C4—24no—
388C2×C3×C4—44no—
389C2×C3×C4C22×D414yes—
390C2×C3×C4C22×D412yes—
391C2×C3×C4C2×#4112yes—
392C2×C3×C4H(5,𝔽2)12yes—
393#67—44no—
394C2×Dic3—22no—
395C2×Dic3—21no—
396C2×Dic3—24no—
397C2×Dic3—24no—
398C2×Dic3—21no—
399C2×Dic3—22no—
400C2×Dic3—24no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—22no—
404C2×Dic3—22no—
405C2×Dic3—24no—
406C2×Dic3—24no—
407C2×Dic3—22no—
408C2×Dic3—22no—
409C2×Dic3—22no—
410C2×Dic3C22×D418yes—
411C2×Dic3C2×#4112yes—
412C2×Dic32−1+414yes—
413C4×D3—216no—
414C4×D3—28no—
415C4×D3—416no—
416Dic6—216no—
417Dic6—416no—
418C24—31no—
419C24C23×D311yescharacteristic
420C2×Q8—24no—
421C2×Q8—24no—
422C2×Q8—24no—
423C2×Q8—38no—
424C2×Q8—68no—
425C2×Q8—68no—
426C2×Q8—64no—
427C2×Q8—64no—
428C2×Q8—124no—
429C2×Q8—1232no—
430C2×D4—62no—
431C2×D4—64no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—31no—
436C22×C4—32no—
437C22×C4—34no—
438C22×C4—34no—
439C22×C4—31no—
440C22×C4—31no—
441C22×C4—32no—
442C22×C4—34no—
443C22×C4—64no—
444C22×C4—68no—
445C22×C4—64no—
446C22×C4—68no—
447C22×C4—64no—
448C22×C4—62no—
449C22×C4—62no—
450C22×C4—124no—
451C22×C4C23×D314yes—
452C22×C4C23×D314yes—
453C22×C4C23×D312yes—
454C22×C4C23×D311yescharacteristic
455C22×C4C23×D312yes—
456C22×C4C23×D311yescharacteristic
457C4⋊C4—28no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—316no—
461C4⋊C4—44no—
462C4⋊C4—68no—
463C4⋊C4—68no—
464C4⋊C4—62no—
465C4⋊C4—68no—
466C4⋊C4—68no—
467C4⋊C4—64no—
468C4⋊C4—68no—
469C4⋊C4—64no—
470C4⋊C4—68no—
471C4⋊C4—124no—
472C22⋊C4—24no—
473C22⋊C4—44no—
474C22⋊C4—62no—
475C22⋊C4—64no—
476C22⋊C4—68no—
477C22⋊C4—62no—
478C22⋊C4—68no—
479C22⋊C4—64no—
480C42—22no—
481C42—68no—
482C42—62no—
483C42—64no—
484C42—62no—
485C42—62no—
486C42—124no—
487C42D3×D414yes—
488C22×C3—41no—
489C22×C3—41no—
490C22×C3—42no—
491C22×C3—41no—
492C22×C3—42no—
493C22×C3C23×D411yescharacteristic
494C22×C3C22×#4111yescharacteristic
495C22×C3C2×H(5,𝔽2)11yescharacteristic
496C22×C3C2×2−1+412yes—
497C22×C3C2×2−1+412yes—
498C2×D3—22no—
499C2×D3—24no—
500C2×D3—22no—
501C2×D3—24no—
502C2×D3#54114yescomplemented
503C3×C4—24no—
504C3×C4—44no—
505C3×C4—41no—
506C3×C4—42no—
507C3×C4—44no—
508C3×C4#54614yes—
509Dic3—28no—
510Dic3—22no—
511Dic3—24no—
512Dic3—41no—
513Dic3—44no—
514Dic3—41no—
515Dic3—42no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—22no—
520C23—22no—
521C23—32no—
522C23—34no—
523C23—61no—
524C23—61no—
525C23—62no—
526C23C24×D311yescharacteristic
527Q8—48no—
528Q8—616no—
529Q8—1216no—
530D4—124no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—38no—
546C2×C4—32no—
547C2×C4—34no—
548C2×C4—44no—
549C2×C4—616no—
550C2×C4—62no—
551C2×C4—61no—
552C2×C4—64no—
553C2×C4—61no—
554C2×C4—64no—
555C2×C4—62no—
556C2×C4—68no—
557C2×C4—64no—
558C2×C4—62no—
559C2×C4—61no—
560C2×C4—61no—
561C2×C4—64no—
562C2×C4—62no—
563C2×C4—62no—
564C2×C4—64no—
565C2×C4—62no—
566C2×C4—62no—
567C2×C4—1216no—
568C2×C4C2×D3×D414yes—
569C2×C4C2×D3×D412yes—
570C2×C4#100412yes—
571C2×C4D3×#4112yes—
572C2×C3—81no—
573C2×C3C2×#54111yescharacteristic
574C2×C3C2×#54611yescharacteristic
575C2×C3#348612yes—
576C2×C3#350911yescharacteristic
577C2×C3#355212yes—
578D3—24no—
579C22—34no—
580C22—42no—
581C22—42no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—64no—
586C22—64no—
587C22—62no—
588C22—62no—
589C22C22×D3×D411yescharacteristic
590C22C2×#100411yescharacteristic
591C22C2×D3×#4111yescharacteristic
592C22D3×2−1+412yes—
593C22D3×2−1+412yes—
594C4—24no—
595C4—42no—
596C4—44no—
597C4—41no—
598C4—44no—
599C4—64no—
600C4—62no—
601C4—68no—
602C4—124no—
603C4—122no—
604C4—121no—
605C4—121no—
606C4#561214yes—
607C3#3769511yescharacteristic, complemented, Sylow
608C2—64no—
609C2—81no—
610C2C2×#561211yescharacteristic
611C2#8635512yes—
612C2#8636112yes—
613C2D3×#54111yescharacteristic
614C2#8651011yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2621C2
prime radicalC22×#67C23
hypercenter#2621D3
hypocenterC3#37695
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2621

Complements

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

NH ≅ G/Ntype
1. C2×D3×C4⋊Q8C2semidirect
2. C2×#5518C2semidirect
3. C3×#2621C2semidirect
4. #83245C2semidirect
5. #82757C2semidirect
8. #77856C2semidirect
9. #77756C2semidirect
10. #77588C2semidirect
12. #76614C2semidirect
13. #76502C2semidirect
14. #74257C2semidirect
15. #74152C2semidirect
80. #2621D3semidirect
144. #869D4semidirect
502. C2×D3#541semidirect
607. C3#37695semidirect

Maximal subgroups

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

subgroupindex
C2×D3×C4⋊Q82
C2×#55182
C3×#26212
#832452
#827572
#827512
#822682
#778562
#777562
#775882
#774832
#766142
#765022
#742572
#741522
#376953