Finite Group Databasebeta

← #11024923

Subgroups — #11024923

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
1C22×C3×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×#5731C211yesmaximal, characteristic, complemented
3#84068C212yesmaximal
4Q8×#217C212yesmaximal, complemented
5#83261C211yesmaximal, characteristic, complemented
6#83254C212yesmaximal, complemented
7#83248C214yesmaximal, complemented
8#83247C214yesmaximal, complemented
9#83245C211yesmaximal, characteristic, complemented
10#83234C212yesmaximal, complemented
11#83227C211yesmaximal, characteristic, complemented
12#82175C214yesmaximal
13#82174C214yesmaximal, complemented
14#82074C211yesmaximal, characteristic, complemented
15#82065C211yesmaximal, characteristic
16#37695—31nomaximal, Sylow
17C23×C3×Q8C2212yes—
18C2×C3×C4⋊Q8—28no—
19C2×C3×C4⋊Q8C2212yes—
20C2×C3×C4⋊Q8C2211yescharacteristic
21C2×C3×C4⋊Q8C2214yes—
22C2×C3×C4⋊Q8C2211yescharacteristic
23C22×C3×C4⋊C4C2214yes—
24C22×C3×C42C2211yescharacteristic
25C2×#946C2212yes—
26C2×#946C2212yes—
27C2×Q8×Dic3C2212yes—
28C2×Q8×Dic3C2212yes—
29C2×C4×#67C2211yescharacteristic
30#5731—28no—
31C2×#898C2214yes—
32C2×#889C2214yes—
33C2×#872C2211yescharacteristic
34C2×#870C2211yescharacteristic
35#5231C2214yes—
36#5230C2214yes—
37#5229C2212yes—
38#5229C2212yes—
39#5229C2212yes—
40#5229C2212yes—
41#5229C2214yes—
42#5229C2214yes—
43#5213C2214yes—
44#5213C2212yes—
45#5212C2212yes—
46#5212C2214yes—
47#5210C2214yes—
48#5210C2212yes—
49#5210C2214yes—
50C4×#217C2211yescharacteristic
51C4×#217C2214yes—
52#4994C2214yes—
53#4992C2214yes—
54#4990C2214yes—
55#4986C2214yes—
56#4985C2214yes—
57#4985C2214yes—
58#4983C2214yes—
59#4982C2214yes—
60#4981C2214yes—
61#4980C2214yes—
62#4980C2214yes—
63#4978C2214yes—
64#4978C2214yes—
65#4976C2214yes—
66#4940C2212yes—
67C4×(C2×D3)⋊C4C2211yescharacteristic
68#4936C2211yescharacteristic
69#4933C2212yes—
70#4933C2214yes—
71#4932C2211yescharacteristic
72C4×Dic3⋊C4C2211yescharacteristic
73#4928C2212yes—
74C2×#541—31no—
75C22×C4⋊Q8D311yescharacteristic, complemented
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#2327—31no—
88#2290—31no—
89C22×C3×Q8—28no—
90C22×C3×Q8—28no—
91C22×C3×Q8—44no—
92C22×C3×Q8C2312yes—
93C22×C3×Q8C2314yes—
94C22×C3×Q8C2312yes—
95C23×C3×C4C2312yes—
96C23×C3×C4C2314yes—
97C23×C3×C4C2311yescharacteristic
98C22×#67C2311yescharacteristic
99C22×C4×D3C2311yescharacteristic
100C3×C4⋊Q8—28no—
101C3×C4⋊Q8—216no—
102C3×C4⋊Q8—24no—
103C3×C4⋊Q8D418yescomplemented
104C2×C3×C4⋊C4—216no—
105C2×C3×C4⋊C4C2314yes—
106C2×C3×C4⋊C4C2314yes—
107C2×C3×C4⋊C4C2314yes—
108C2×C3×C4⋊C4C2314yes—
109C2×C3×C42—24no—
110C2×C3×C42C2311yescharacteristic
111C2×C3×C42C2312yes—
112C2×C3×C42C2311yescharacteristic
113C2×#217C2312yes—
114C2×#217C2314yes—
115C2×#217C2311yescharacteristic
116#946—28no—
117#946—44no—
118Q8×Dic3—28no—
119Q8×Dic3—44no—
120C4×#67—24no—
121C2×(C2×D3)⋊C4C2312yes—
122C2×(C2×D3)⋊C4C2311yescharacteristic
123C2×(C2×D3)⋊C4C2312yes—
124C2×(C2×D3)⋊C4C2314yes—
125C2×(C2×D3)⋊C4C2314yes—
126C2×C4⋊Dic3C2312yes—
127C2×C4⋊Dic3C2312yes—
128C2×C4⋊Dic3C2314yes—
129C2×C4⋊Dic3C2314yes—
130C2×Dic3⋊C4C2314yes—
131C2×Dic3⋊C4C2312yes—
132C2×Dic3⋊C4C2311yescharacteristic
133C2×Dic3⋊C4C2314yes—
134C2×Dic3⋊C4C2312yes—
135C2×Dic3⋊C4C2314yes—
136C2×Dic3⋊C4C2314yes—
137C2×C4×Dic3C2311yescharacteristic
138C2×C4×Dic3C2311yescharacteristic
139C2×C4×Dic3C2311yescharacteristic
140C2×C4×Dic3C2314yes—
141C2×C4×Dic3C2314yes—
142#898—216no—
143#889—216no—
144#872—24no—
145#870—24no—
146C3⋊#95C2314yes—
147C3⋊#95C2312yes—
148C3⋊#95C2312yes—
149C3⋊#95C2312yes—
150C3⋊#95C2314yes—
151C3⋊#95C2314yes—
152C3⋊#95C2312yes—
153C3⋊#95C2312yes—
154C3⋊#95C2314yes—
155C3⋊#95C2314yes—
156C3⋊#95C2314yes—
157C3⋊#95C2314yes—
158C3⋊#95C2314yes—
159C3⋊#95C2314yes—
160C3⋊#95C2314yes—
161C3⋊#95C2314yes—
162C23×Q8C2×D312yes—
163#541—68no—
164C2×C4⋊Q8—28no—
165C2×C4⋊Q8C2×D311yescharacteristic
166C2×C4⋊Q8C2×D312yes—
167C2×C4⋊Q8C2×D314yes—
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×Q8—32no—
176C2×C4×D4—31no—
177C2×#117—31no—
178C22×C4⋊C4C2×D314yes—
179C22×C42C2×D311yescharacteristic
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—32no—
192#391—32no—
193#391—34no—
194#391—34no—
195#391—32no—
196#389—34no—
197#389—32no—
198#388—34no—
199#388—32no—
200#387—34no—
201#387—34no—
202#387—34no—
203#387—34no—
204#387—34no—
205#387—32no—
206#386—34no—
207#386—32no—
208#385—34no—
209#385—34no—
210#385—32no—
211#382—31no—
212#380—32no—
213#380—34no—
214C4×C4⋊C4—31no—
215C4×C22⋊C4—31no—
216C4×C22⋊C4—34no—
217C4×C22⋊C4—31no—
218#376—31no—
219C24×C3C2411yescharacteristic
220C23×D3C2411yescharacteristic
221C2×C3×Q8—28no—
222C2×C3×Q8—24no—
223C2×C3×Q8—24no—
224C2×C3×Q8—28no—
225C2×C3×Q8—24no—
226C2×C3×Q8—24no—
227C2×C3×Q8—24no—
228C2×C3×Q8—44no—
229C2×C3×Q8—432no—
230C2×C3×Q8C2×D418yes—
231C22×C3×C4—24no—
232C22×C3×C4—24no—
233C22×C3×C4—22no—
234C22×C3×C4—22no—
235C22×C3×C4—24no—
236C22×C3×C4—28no—
237C22×C3×C4—28no—
238C22×C3×C4—44no—
239C22×C3×C4#4114yes—
240C22×C3×C4C2414yes—
241C22×C3×C4C2412yes—
242C22×C3×C4C2414yes—
243C22×C3×C4C2411yescharacteristic
244C22×C3×C4C2412yes—
245C22×C3×C4C2411yescharacteristic
246C2×#67—22no—
247C2×#67—24no—
248C22×Dic3C2411yescharacteristic
249C22×Dic3C2411yescharacteristic
250C22×Dic3C2412yes—
251C22×Dic3C2414yes—
252C22×Dic3C2412yes—
253C22×Dic3C2414yes—
254C22×Dic3C2411yescharacteristic
255C2×C4×D3—22no—
256C2×C4×D3—22no—
257C2×C4×D3—22no—
258C3×C4⋊C4—28no—
259C3×C4⋊C4—28no—
260C3×C4⋊C4—28no—
261C3×C4⋊C4C2×D4116yes—
262C3×C42—22no—
263C3×C42—24no—
264C3×C42C2×D414yes—
265#217—22no—
266#217—24no—
267#217—28no—
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—24no—
274C4⋊Dic3—28no—
275C4⋊Dic3—24no—
276C4⋊Dic3—44no—
277Dic3⋊C4—28no—
278Dic3⋊C4—24no—
279Dic3⋊C4—28no—
280Dic3⋊C4—22no—
281Dic3⋊C4—28no—
282Dic3⋊C4—24no—
283Dic3⋊C4—44no—
284C4×Dic3—22no—
285C4×Dic3—28no—
286C4×Dic3—22no—
287C4×Dic3—22no—
288C4×Dic3—44no—
289C22×Q8—28no—
290C22×Q8—28no—
291C22×Q8—44no—
292C22×Q8C22×D312yes—
293C22×Q8C22×D314yes—
294C22×Q8C22×D312yes—
295C22×D4—31no—
296C23×C4—31no—
297C23×C4C22×D312yes—
298C23×C4C22×D314yes—
299C23×C4C22×D311yescharacteristic
300C4⋊Q8—28no—
301C4⋊Q8—216no—
302C4⋊Q8—24no—
303C4⋊Q8#6718yescomplemented
304#126—616no—
305#125—616no—
306#125—64no—
307C22⋊Q8—68no—
308C22⋊Q8—124no—
309C4×Q8—68no—
310C4×Q8—124no—
311C4×D4—64no—
312#117—64no—
313C2×C4⋊C4—216no—
314C2×C4⋊C4—32no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—32no—
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—34no—
325C2×C4⋊C4C22×D314yes—
326C2×C4⋊C4C22×D314yes—
327C2×C4⋊C4C22×D314yes—
328C2×C4⋊C4C22×D314yes—
329C2×C22⋊C4—31no—
330C2×C22⋊C4—32no—
331C2×C22⋊C4—34no—
332C2×C22⋊C4—32no—
333C2×C22⋊C4—31no—
334C2×C22⋊C4—34no—
335C2×C22⋊C4—32no—
336C2×C22⋊C4—34no—
337C2×C42—24no—
338C2×C42—31no—
339C2×C42—31no—
340C2×C42—31no—
341C2×C42—34no—
342C2×C42—34no—
343C2×C42C22×D311yescharacteristic
344C2×C42C22×D312yes—
345C2×C42C22×D311yescharacteristic
346#95—32no—
347#95—32no—
348#95—32no—
349#95—32no—
350#95—34no—
351#95—34no—
352#95—32no—
353#95—34no—
354#95—34no—
355#95—34no—
356#95—34no—
357#95—34no—
358#95—34no—
359#95—34no—
360#95—34no—
361#95—34no—
362C23×C3—21no—
363C23×C3—21no—
364C23×C3—22no—
365C23×C3C2×#4112yes—
366C23×C32−1+414yes—
367C23×C3C2511yescharacteristic
368C22×D3—21no—
369C22×D3—21no—
370C22×D3—21no—
371C22×D3—22no—
372C22×D3—22no—
373C3×Q8—216no—
374C3×Q8—416no—
375C3×Q8—48no—
376C2×C3×C4—216no—
377C2×C3×C4—22no—
378C2×C3×C4—21no—
379C2×C3×C4—24no—
380C2×C3×C4—24no—
381C2×C3×C4—21no—
382C2×C3×C4—22no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—28no—
386C2×C3×C4—24no—
387C2×C3×C4—24no—
388C2×C3×C4—24no—
389C2×C3×C4—416no—
390C2×C3×C4C22×D418yes—
391C2×C3×C4C22×D414yes—
392C2×C3×C4C22×D412yes—
393C2×C3×C4C2×#4112yes—
394C2×C3×C4C2×#4112yes—
395C2×C3×C4H(5,𝔽2)12yes—
396C2×C3×C42−1+414yes—
397#67—44no—
398C2×Dic3—22no—
399C2×Dic3—24no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—22no—
405C2×Dic3—24no—
406C2×Dic3—21no—
407C2×Dic3—22no—
408C2×Dic3—24no—
409C2×Dic3—22no—
410C2×Dic3—24no—
411C2×Dic3—22no—
412C2×Dic3—22no—
413C2×Dic3—24no—
414C2×Dic3—22no—
415C2×Dic3—24no—
416C2×Dic3—22no—
417C4×D3—44no—
418C24—31no—
419C24C23×D311yescharacteristic
420C2×Q8—24no—
421C2×Q8—28no—
422C2×Q8—24no—
423C2×Q8—24no—
424C2×Q8—24no—
425C2×Q8—28no—
426C2×Q8—24no—
427C2×Q8—44no—
428C2×Q8—432no—
429C2×Q8C2×#6718yes—
430C2×D4—62no—
431C2×D4—64no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—28no—
438C22×C4—28no—
439C22×C4—31no—
440C22×C4—31no—
441C22×C4—32no—
442C22×C4—34no—
443C22×C4—32no—
444C22×C4—34no—
445C22×C4—31no—
446C22×C4—44no—
447C22×C4—62no—
448C22×C4—62no—
449C22×C4—62no—
450C22×C4#23714yes—
451C22×C4C23×D314yes—
452C22×C4C23×D311yescharacteristic
453C22×C4C23×D312yes—
454C22×C4C23×D312yes—
455C22×C4C23×D314yes—
456C22×C4C23×D311yescharacteristic
457C4⋊C4—28no—
458C4⋊C4—28no—
459C4⋊C4—28no—
460C4⋊C4—62no—
461C4⋊C4—64no—
462C4⋊C4—64no—
463C4⋊C4—68no—
464C4⋊C4—64no—
465C4⋊C4—64no—
466C4⋊C4—68no—
467C4⋊C4—68no—
468C4⋊C4—68no—
469C4⋊C4—124no—
470C4⋊C4—124no—
471C4⋊C4C2×#67116yes—
472C22⋊C4—62no—
473C22⋊C4—64no—
474C22⋊C4—68no—
475C22⋊C4—62no—
476C22⋊C4—64no—
477C22⋊C4—64no—
478C22⋊C4—68no—
479C22⋊C4—124no—
480C42—22no—
481C42—24no—
482C42—62no—
483C42—62no—
484C42—62no—
485C42—68no—
486C42—124no—
487C42C2×#6714yes—
488C22×C3—22no—
489C22×C3—24no—
490C22×C3—22no—
491C22×C3—24no—
492C22×C3#54114yescomplemented
493C22×C3C23×D411yescharacteristic
494C22×C3C22×#4111yescharacteristic
495C22×C3C2×H(5,𝔽2)11yescharacteristic
496C22×C3C2×2−1+412yes—
497C22×C3C2×2−1+412yes—
498C2×D3—41no—
499C2×D3—41no—
500C2×D3—42no—
501C2×D3—41no—
502C2×D3—42no—
503C3×C4—22no—
504C3×C4—28no—
505C3×C4—24no—
506C3×C4—24no—
507C3×C4—44no—
508C3×C4#54614yes—
509Dic3—41no—
510Dic3—41no—
511Dic3—42no—
512Dic3—44no—
513Dic3—41no—
514Dic3—42no—
515Dic3—44no—
516C23—21no—
517C23—22no—
518C23—21no—
519C23—61no—
520C23—61no—
521C23—62no—
522C23—62no—
523C23—61no—
524C23C2×#23712yes—
525C23#100714yes—
526C23C24×D311yescharacteristic
527Q8—216no—
528Q8—416no—
529Q8—48no—
530D4—124no—
531C2×C4—216no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—28no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—416no—
545C2×C4—61no—
546C2×C4—61no—
547C2×C4—61no—
548C2×C4—62no—
549C2×C4—64no—
550C2×C4—61no—
551C2×C4—62no—
552C2×C4—62no—
553C2×C4—64no—
554C2×C4—62no—
555C2×C4—64no—
556C2×C4—62no—
557C2×C4—64no—
558C2×C4—62no—
559C2×C4—64no—
560C2×C4—62no—
561C2×C4—62no—
562C2×C4—64no—
563C2×C4—61no—
564C2×C4—124no—
565C2×C4C2×#23712yes—
566C2×C4C2×#23712yes—
567C2×C4#100412yes—
568C2×C4#100714yes—
569C2×C4C22×#6714yes—
570C2×C4C22×#6718yes—
571C2×C4C22×#6712yes—
572C2×C3—24no—
573C2×C3C2×#54111yescharacteristic
574C2×C3C2×#54611yescharacteristic
575C2×C3#348612yes—
576C2×C3#350911yescharacteristic
577C2×C3#355212yes—
578D3—81no—
579C22—22no—
580C22—22no—
581C22—24no—
582C22—24no—
583C22—121no—
584C22—121no—
585C22—121no—
586C22—122no—
587C22—122no—
588C22#573114yescomplemented
589C22C22×#23711yescharacteristic
590C22C2×#100411yescharacteristic
591C22C2×#100712yes—
592C22C2×#100712yes—
593C22C23×#6711yescharacteristic
594C4—22no—
595C4—28no—
596C4—24no—
597C4—24no—
598C4—44no—
599C4—121no—
600C4—121no—
601C4—121no—
602C4—122no—
603C4—124no—
604C4—122no—
605C4—124no—
606C4#580614yes—
607C3#3769511yescharacteristic, complemented, Sylow
608C2—24no—
609C2—241no—
610C2#8620411yescharacteristic
611C2C2×#573111yescharacteristic
612C2#8621812yes—
613C2C2×#580611yescharacteristic
614C2#8725112yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC22×C3×C4⋊Q8C2
prime radicalC22×#67C23
hypercenterC22×C4⋊Q8D3
hypocenterC3#37695
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×C3×C4⋊Q8C2semidirect
2. C2×#5731C2semidirect
4. Q8×#217C2semidirect
5. #83261C2semidirect
6. #83254C2semidirect
7. #83248C2semidirect
8. #83247C2semidirect
9. #83245C2semidirect
10. #83234C2semidirect
11. #83227C2semidirect
13. #82174C2semidirect
14. #82074C2semidirect
75. C22×C4⋊Q8D3semidirect
103. C3×C4⋊Q8D4semidirect
303. C4⋊Q8#67semidirect
492. C22×C3#541semidirect
588. C22#5731semidirect
607. C3#37695semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊Q82
C2×#57312
#840682
Q8×#2172
#832612
#832542
#832482
#832472
#832452
#832342
#832272
#821752
#821742
#820742
#820652
#376953