Finite Group Databasebeta

← #11024926

Subgroups — #11024926

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×#5731C211yesmaximal, characteristic, complemented
2C2×D3×C4⋊Q8C211yesmaximal, characteristic, complemented
3C3×#2327C211yesmaximal, characteristic, complemented
4#82496C212yesmaximal, complemented
5Q8×(C2×D3)⋊C4C212yesmaximal, complemented
6#76619C212yesmaximal
7#76503C212yesmaximal, complemented
8#75163C214yesmaximal, complemented
9#75157C214yesmaximal, complemented
10#75020C214yesmaximal
11#75009C214yesmaximal, complemented
12#74257C211yesmaximal, characteristic, complemented
13#74193C211yesmaximal, characteristic, complemented
14#73414C211yesmaximal, characteristic
15#73407C211yesmaximal, characteristic, complemented
16#37695—31nomaximal, Sylow
17C22×D3×Q8C2212yes—
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×C4×Q8C2212yes—
20C2×#946C2212yes—
21C2×Q8×Dic3C2212yes—
22C2×#944C2212yes—
23#5731—28no—
24D3×C4⋊Q8—28no—
25C2×#898C2214yes—
26C2×#897C2212yes—
27C2×D3×C4⋊C4C2214yes—
28C2×#889C2214yes—
29C2×#888C2214yes—
30C2×C4×D12C2211yescharacteristic
31C2×#872C2211yescharacteristic
32C2×C42×D3C2211yescharacteristic
33C2×#870C2211yescharacteristic
34C2×#869C2211yescharacteristic
35C3×#391C2212yes—
36C3×#387C2214yes—
37C3×#387C2214yes—
38C3×#382C2211yescharacteristic
39C3×#376C2211yescharacteristic
40#4994C2214yes—
41#4987C2214yes—
42#4987C2212yes—
43#4984C2212yes—
44#4981C2214yes—
45#4973C2214yes—
46#4973C2212yes—
47#4940C2214yes—
48C4×(C2×D3)⋊C4C2214yes—
49C4×(C2×D3)⋊C4C2211yescharacteristic
50C4×(C2×D3)⋊C4C2211yescharacteristic
51C4×C4⋊Dic3C2211yescharacteristic
52#4929C2212yes—
53#4929C2214yes—
54#4929C2212yes—
55#4928C2214yes—
56#4674C2214yes—
57#4673C2214yes—
58#4670C2212yes—
59#4668C2214yes—
60#4668C2212yes—
61#4668C2214yes—
62#4665C2214yes—
63#4665C2212yes—
64#4661C2214yes—
65#4661C2214yes—
66#4659C2214yes—
67#4659C2214yes—
68#4659C2214yes—
69#4656C2214yes—
70#4653C2214yes—
71#4653C2214yes—
72#4653C2214yes—
73#4653C2212yes—
74C2×#541—31no—
75C22×C4⋊Q8—31no—
76#2851—31no—
77#2716—34no—
78#2713—34no—
79#2642—31no—
80#2621—31no—
81#2481—34no—
82#2473—34no—
83#2446—32no—
84#2429—32no—
85#2409—32no—
86Q8×C22⋊C4—32no—
87#2327D311yescharacteristic, complemented
88#2290—31no—
89C22×C3×Q8C2312yes—
90C2×D3×Q8—28no—
91C2×D3×Q8—28no—
92C2×D3×Q8—44no—
93C22×D12C2311yescharacteristic
94C22×C4×D3C2312yes—
95C22×C4×D3C2311yescharacteristic
96C22×C4×D3C2314yes—
97C22×C4×D3C2311yescharacteristic
98C22×Dic6C2312yes—
99C22×Dic6C2314yes—
100C3×C4⋊Q8D418yescomplemented
101C3×C4×Q8—44no—
102C2×C3×C4⋊C4C2312yes—
103C2×C3×C4⋊C4C2314yes—
104C2×C3×C4⋊C4C2314yes—
105C2×C3×C42C2311yescharacteristic
106C2×C3×C42C2311yescharacteristic
107C2×C3×C42C2314yes—
108C2×C3×C42C2311yescharacteristic
109#946—28no—
110Q8×Dic3—28no—
111#944—28no—
112C2×(C2×D3)⋊C4C2314yes—
113C2×(C2×D3)⋊C4C2312yes—
114C2×(C2×D3)⋊C4C2312yes—
115C2×(C2×D3)⋊C4C2312yes—
116C2×(C2×D3)⋊C4C2314yes—
117C2×(C2×D3)⋊C4C2311yescharacteristic
118C2×(C2×D3)⋊C4C2311yescharacteristic
119C2×(C2×D3)⋊C4C2314yes—
120C2×C4⋊Dic3C2311yescharacteristic
121C2×C4⋊Dic3C2312yes—
122C2×C4⋊Dic3C2314yes—
123C2×C4⋊Dic3C2314yes—
124C2×C4⋊Dic3C2314yes—
125C2×C4⋊Dic3C2312yes—
126C2×C4⋊Dic3C2314yes—
127C2×Dic3⋊C4C2314yes—
128C2×Dic3⋊C4C2314yes—
129C2×Dic3⋊C4C2312yes—
130C2×Dic3⋊C4C2314yes—
131C2×Dic3⋊C4C2314yes—
132C2×C4×Dic3C2312yes—
133C2×C4×Dic3C2314yes—
134C2×C4×Dic3C2311yescharacteristic
135C2×C4×Dic3C2311yescharacteristic
136#898—216no—
137#897—44no—
138D3×C4⋊C4—216no—
139#889—216no—
140#888—216no—
141C4×D12—24no—
142#872—24no—
143C42×D3—24no—
144#870—24no—
145#869—24no—
146C3×#95C2314yes—
147C3×#95C2312yes—
148C3×#95C2314yes—
149C3×#95C2314yes—
150C3×#95C2312yes—
151C3⋊#95C2312yes—
152C3⋊#95C2314yes—
153C3⋊#95C2314yes—
154C3⋊#95C2314yes—
155C3⋊#95C2314yes—
156C3⋊#95C2314yes—
157C3⋊#95C2312yes—
158C3⋊#95C2312yes—
159C3⋊#95C2314yes—
160C3⋊#95C2314yes—
161C3⋊#95C2314yes—
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⋊Q8—32no—
174C2×C4×Q8—32no—
175C2×C4×Q8C2×D312yes—
176C2×C4×D4—31no—
177C2×#117—31no—
178C22×C4⋊C4—34no—
179C22×C42—31no—
180#400—34no—
181#400—34no—
182#400—34no—
183#398—34no—
184#398—34no—
185#398—34no—
186#397—34no—
187#396—34no—
188#395—34no—
189#393—34no—
190#391—32no—
191#391—34no—
192#391—32no—
193#391—34no—
194#391—32no—
195#391C2×D312yes—
196#389—32no—
197#389—34no—
198#388—34no—
199#388—32no—
200#387—34no—
201#387—34no—
202#387—34no—
203#387—32no—
204#387C2×D314yes—
205#387C2×D314yes—
206#386—34no—
207#386—32no—
208#385—34no—
209#385—34no—
210#385—32no—
211#382C2×D311yescharacteristic
212#380—32no—
213#380—34no—
214C4×C4⋊C4—31no—
215C4×C22⋊C4—31no—
216C4×C22⋊C4—34no—
217C4×C22⋊C4—31no—
218#376C2×D311yescharacteristic
219C23×D3C2411yescharacteristic
220C23×D3C2411yescharacteristic
221C2×C3×Q8—24no—
222C2×C3×Q8—24no—
223C2×C3×Q8C2×D418yes—
224C22×C3×C4C2411yescharacteristic
225C22×C3×C4C2412yes—
226C22×C3×C4C2411yescharacteristic
227C22×C3×C4C2414yes—
228C22×C3×C4C2412yes—
229C22×C3×C4C2414yes—
230C22×C3×C4C2411yescharacteristic
231C22×Dic3C2412yes—
232C22×Dic3C2414yes—
233C22×Dic3C2414yes—
234C22×Dic3C2411yescharacteristic
235C22×Dic3C2411yescharacteristic
236C22×Dic3C2412yes—
237D3×Q8—432no—
238C2×D12—22no—
239C2×D12—24no—
240C2×C4×D3—24no—
241C2×C4×D3—24no—
242C2×C4×D3—22no—
243C2×C4×D3—22no—
244C2×C4×D3—24no—
245C2×C4×D3—28no—
246C2×C4×D3—22no—
247C2×C4×D3—28no—
248C2×C4×D3—22no—
249C2×C4×D3—22no—
250C2×C4×D3—44no—
251C2×C4×D3#4114yes—
252C2×Dic6—24no—
253C2×Dic6—28no—
254C2×Dic6—24no—
255C2×Dic6—24no—
256C2×Dic6—28no—
257C2×Dic6—44no—
258C3×C4⋊C4—24no—
259C3×C4⋊C4—44no—
260C3×C4⋊C4C2×D4116yes—
261C3×C42—22no—
262C3×C42—22no—
263C3×C42—44no—
264C3×C42C2×D414yes—
265(C2×D3)⋊C4—22no—
266(C2×D3)⋊C4—28no—
267(C2×D3)⋊C4—24no—
268(C2×D3)⋊C4—24no—
269(C2×D3)⋊C4—28no—
270(C2×D3)⋊C4—22no—
271(C2×D3)⋊C4—24no—
272(C2×D3)⋊C4—44no—
273C4⋊Dic3—28no—
274C4⋊Dic3—24no—
275C4⋊Dic3—28no—
276C4⋊Dic3—22no—
277C4⋊Dic3—24no—
278C4⋊Dic3—28no—
279C4⋊Dic3—44no—
280Dic3⋊C4—28no—
281Dic3⋊C4—24no—
282Dic3⋊C4—28no—
283Dic3⋊C4—28no—
284Dic3⋊C4—28no—
285C4×Dic3—24no—
286C4×Dic3—22no—
287C4×Dic3—22no—
288C4×Dic3—28no—
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—31no—
299C23×C4—34no—
300C4⋊Q8—68no—
301C4⋊Q8—616no—
302C4⋊Q8—64no—
303C4⋊Q8D1218yescomplemented
304#126—616no—
305#125—616no—
306#125—64no—
307C22⋊Q8—68no—
308C22⋊Q8—124no—
309C4×Q8—44no—
310C4×Q8—68no—
311C4×D4—64no—
312#117—64no—
313C2×C4⋊C4—34no—
314C2×C4⋊C4—34no—
315C2×C4⋊C4—34no—
316C2×C4⋊C4—32no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—32no—
319C2×C4⋊C4—34no—
320C2×C4⋊C4—34no—
321C2×C4⋊C4—34no—
322C2×C4⋊C4—32no—
323C2×C4⋊C4—34no—
324C2×C4⋊C4—34no—
325C2×C4⋊C4—616no—
326C2×C4⋊C4C22×D312yes—
327C2×C4⋊C4C22×D314yes—
328C2×C4⋊C4C22×D314yes—
329C2×C22⋊C4—32no—
330C2×C22⋊C4—31no—
331C2×C22⋊C4—31no—
332C2×C22⋊C4—34no—
333C2×C22⋊C4—32no—
334C2×C22⋊C4—34no—
335C2×C22⋊C4—32no—
336C2×C22⋊C4—34no—
337C2×C42—31no—
338C2×C42—34no—
339C2×C42—31no—
340C2×C42—32no—
341C2×C42—64no—
342C2×C42C22×D311yescharacteristic
343C2×C42C22×D311yescharacteristic
344C2×C42C22×D314yes—
345C2×C42C22×D311yescharacteristic
346#95—32no—
347#95—32no—
348#95—34no—
349#95—34no—
350#95—32no—
351#95—34no—
352#95—34no—
353#95—34no—
354#95—34no—
355#95—34no—
356#95—34no—
357#95C22×D312yes—
358#95C22×D312yes—
359#95C22×D314yes—
360#95C22×D314yes—
361#95C22×D314yes—
362C23×C3C2511yescharacteristic
363C22×D3—21no—
364C22×D3—21no—
365C22×D3—22no—
366C22×D3—21no—
367C22×D3—21no—
368C22×D3—21no—
369C22×D3—22no—
370C22×D3—22no—
371C22×D3C2×#4112yes—
372C22×D32−1+414yes—
373C3×Q8—216no—
374C2×C3×C4—24no—
375C2×C3×C4—24no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—24no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—24no—
384C2×C3×C4—22no—
385C2×C3×C4—24no—
386C2×C3×C4—22no—
387C2×C3×C4—22no—
388C2×C3×C4C22×D414yes—
389C2×C3×C4C22×D418yes—
390C2×C3×C4C22×D412yes—
391C2×C3×C4C2×#4112yes—
392C2×C3×C4H(5,𝔽2)12yes—
393C2×Dic3—22no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—22no—
397C2×Dic3—22no—
398C2×Dic3—24no—
399C2×Dic3—21no—
400C2×Dic3—24no—
401C2×Dic3—21no—
402C2×Dic3—24no—
403C2×Dic3—21no—
404C2×Dic3—24no—
405C2×Dic3—22no—
406C2×Dic3—24no—
407C2×Dic3—22no—
408C2×Dic3—24no—
409C2×Dic3C2×#4112yes—
410C2×Dic32−1+414yes—
411D12—44no—
412C4×D3—216no—
413C4×D3—28no—
414C4×D3—416no—
415C4×D3—44no—
416Dic6—416no—
417Dic6—48no—
418C24—31no—
419C24—31no—
420C2×Q8—24no—
421C2×Q8—24no—
422C2×Q8—68no—
423C2×Q8—68no—
424C2×Q8—64no—
425C2×Q8—64no—
426C2×Q8—64no—
427C2×Q8—124no—
428C2×Q8—1232no—
429C2×Q8C2×D1218yes—
430C2×D4—62no—
431C2×D4—64no—
432C22×C4—31no—
433C22×C4—32no—
434C22×C4—34no—
435C22×C4—32no—
436C22×C4—34no—
437C22×C4—34no—
438C22×C4—31no—
439C22×C4—64no—
440C22×C4—64no—
441C22×C4—68no—
442C22×C4—64no—
443C22×C4—62no—
444C22×C4—62no—
445C22×C4—68no—
446C22×C4—62no—
447C22×C4—62no—
448C22×C4—62no—
449C22×C4—124no—
450C22×C4C23×D311yescharacteristic
451C22×C4C23×D312yes—
452C22×C4C23×D311yescharacteristic
453C22×C4C23×D314yes—
454C22×C4C23×D312yes—
455C22×C4C23×D314yes—
456C22×C4C23×D311yescharacteristic
457C4⋊C4—24no—
458C4⋊C4—44no—
459C4⋊C4—62no—
460C4⋊C4—64no—
461C4⋊C4—68no—
462C4⋊C4—68no—
463C4⋊C4—68no—
464C4⋊C4—68no—
465C4⋊C4—64no—
466C4⋊C4—64no—
467C4⋊C4—68no—
468C4⋊C4—68no—
469C4⋊C4—68no—
470C4⋊C4—124no—
471C4⋊C4C2×D12116yes—
472C22⋊C4—62no—
473C22⋊C4—64no—
474C22⋊C4—62no—
475C22⋊C4—64no—
476C22⋊C4—68no—
477C22⋊C4—64no—
478C22⋊C4—68no—
479C22⋊C4—124no—
480C42—22no—
481C42—22no—
482C42—44no—
483C42—62no—
484C42—64no—
485C42—62no—
486C42—68no—
487C42C2×D1214yes—
488C22×C3C23×D411yescharacteristic
489C22×C3C22×#4111yescharacteristic
490C22×C3C2×H(5,𝔽2)11yescharacteristic
491C22×C3C2×2−1+412yes—
492C22×C3C2×2−1+412yes—
493C2×D3—22no—
494C2×D3—24no—
495C2×D3—22no—
496C2×D3—24no—
497C2×D3—41no—
498C2×D3—41no—
499C2×D3—42no—
500C2×D3—41no—
501C2×D3—42no—
502C2×D3#54114yescomplemented
503C3×C4—28no—
504C3×C4—24no—
505C3×C4—41no—
506C3×C4—41no—
507C3×C4—44no—
508C3×C4—42no—
509C3×C4#54614yes—
510Dic3—22no—
511Dic3—24no—
512Dic3—44no—
513Dic3—41no—
514Dic3—42no—
515Dic3—44no—
516C23—32no—
517C23—34no—
518C23—61no—
519C23—61no—
520C23—62no—
521C23—61no—
522C23—61no—
523C23—61no—
524C23—62no—
525C23—62no—
526C23C24×D311yescharacteristic
527Q8—216no—
528Q8—1216no—
529Q8—128no—
530D4—124no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—32no—
546C2×C4—34no—
547C2×C4—616no—
548C2×C4—62no—
549C2×C4—61no—
550C2×C4—61no—
551C2×C4—62no—
552C2×C4—68no—
553C2×C4—62no—
554C2×C4—64no—
555C2×C4—61no—
556C2×C4—64no—
557C2×C4—64no—
558C2×C4—61no—
559C2×C4—61no—
560C2×C4—64no—
561C2×C4—62no—
562C2×C4—64no—
563C2×C4—62no—
564C2×C4—64no—
565C2×C4—1216no—
566C2×C4—124no—
567C2×C4C22×D1214yes—
568C2×C4C22×D1218yes—
569C2×C4C22×D1212yes—
570C2×C4#100412yes—
571C2×C4D3×#4112yes—
572C2×C3C2×#54111yescharacteristic
573C2×C3C2×#54611yescharacteristic
574C2×C3#348612yes—
575C2×C3#350911yescharacteristic
576C2×C3#355212yes—
577D3—24no—
578D3—81no—
579C22—34no—
580C22—62no—
581C22—64no—
582C22—64no—
583C22—62no—
584C22—121no—
585C22—122no—
586C22—122no—
587C22—121no—
588C22—121no—
589C22C23×D1211yescharacteristic
590C22C2×#100411yescharacteristic
591C22C2×D3×#4111yescharacteristic
592C22D3×2−1+412yes—
593C22D3×2−1+412yes—
594C4—28no—
595C4—24no—
596C4—42no—
597C4—44no—
598C4—41no—
599C4—41no—
600C4—62no—
601C4—64no—
602C4—124no—
603C4—121no—
604C4—122no—
605C4—124no—
606C4#555414yes—
607C3#3769511yescharacteristic, complemented, Sylow
608C2—64no—
609C2—241no—
610C2C2×#555411yescharacteristic
611C2#8588112yes—
612C2#8636112yes—
613C2D3×#54111yescharacteristic
614C2#8651011yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2327C2
prime radicalC22×D12C23
hypercenter#2327D3
hypocenterC3#37695
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2327

Complements

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

NH ≅ G/Ntype
1. C2×#5731C2semidirect
2. C2×D3×C4⋊Q8C2semidirect
3. C3×#2327C2semidirect
4. #82496C2semidirect
5. Q8×(C2×D3)⋊C4C2semidirect
7. #76503C2semidirect
8. #75163C2semidirect
9. #75157C2semidirect
11. #75009C2semidirect
12. #74257C2semidirect
13. #74193C2semidirect
15. #73407C2semidirect
87. #2327D3semidirect
100. C3×C4⋊Q8D4semidirect
303. C4⋊Q8D12semidirect
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×#57312
C2×D3×C4⋊Q82
C3×#23272
#824962
Q8×(C2×D3)⋊C42
#766192
#765032
#751632
#751572
#750202
#750092
#742572
#741932
#734142
#734072
#376953