Finite Group Databasebeta

← #10967713

Subgroups — #10967713

Order 768.

2474 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5728C211yesmaximal, characteristic, complemented
2C2×C4×#237C211yesmaximal, characteristic
3C3×#2547C211yesmaximal, characteristic, complemented
4#83239C211yesmaximal, characteristic, complemented
5#82959C212yesmaximal
6Dic3×#124C212yesmaximal, complemented
7#82379C211yesmaximal, characteristic, complemented
8#77537C214yesmaximal, complemented
9#77458C214yesmaximal, complemented
10#76518C212yesmaximal, complemented
11#76481C212yesmaximal, complemented
12#74020C214yesmaximal, complemented
13#73866C214yesmaximal, complemented
14#73722C211yesmaximal, characteristic, complemented
15C4×#892C211yesmaximal, characteristic, complemented
16#34979—31nomaximal, Sylow
17C22×#237C2211yescharacteristic
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×#124C2212yes—
20C2×C3×C4×D4C2211yescharacteristic
21C2×#946C2212yes—
22C2×Q8×Dic3C2212yes—
23C2×#944C2212yes—
24C2×D4×Dic3C2211yescharacteristic
25C2×C4×#67C2212yes—
26C2×#924C2214yes—
27C22×C4×Dic3C2212yes—
28#5728—28no—
29C4×#237—28no—
30C2×#897C2212yes—
31C2×#892C2212yes—
32C2×#892C2212yes—
33C2×#892C2211yescharacteristic
34C2×#890C2212yes—
35C2×#887C2212yes—
36C2×#887C2211yescharacteristic
37C2×#67⋊C4C2212yes—
38C2×#877C2212yes—
39C2×C42×D3C2211yescharacteristic
40C2×#869C2211yescharacteristic
41C2×C4×Dic6C2211yescharacteristic
42C3×#397C2214yes—
43C3×#391C2212yes—
44C3×#380C2214yes—
45C3×C4×C4⋊C4C2211yescharacteristic
46#5229C2212yes—
47#5221C2212yes—
48#4983C2214yes—
49#4975C2212yes—
50Dic3×C4⋊C4C2212yes—
51#4973C2214yes—
52#4972C2214yes—
53#4959C2218yes—
54#4951C2218yes—
55#4949C2214yes—
56#4946C2218yes—
57Dic3×C22⋊C4C2218yes—
58C4×(C2×D3)⋊C4C2212yes—
59#4936C2212yes—
60C4×C4⋊Dic3C2211yescharacteristic
61C42×Dic3C2212yes—
62#4929C2212yes—
63#4675C2218yes—
64#4666C2212yes—
65#4659C2214yes—
66#4652C2218yes—
67#4649C2218yes—
68#4648C2212yes—
69#4647C2218yes—
70C2×#533—31no—
71C2×C4×#41—31no—
72#2611—34no—
73#2568—31no—
74#2547D311yescharacteristic, complemented
75#2546—31no—
76#2542—31no—
77#2499—32no—
78#2432—32no—
79#2415—34no—
80#2395—34no—
81C4×C4⋊Q8—32no—
82C4×#124—32no—
83C4×C22⋊Q8—34no—
84C4×#117—31no—
85C22×C3×Q8C2312yes—
86C22×C3×D4C2311yescharacteristic
87C23×C3×C4C2312yes—
88C22×#67C2312yes—
89C23×Dic3C2312yes—
90C23×Dic3C2312yes—
91C2×#237—24no—
92C2×#237—24no—
93C2×#237—24no—
94C22×C4×D3C2311yescharacteristic
95C22×C4×D3C2311yescharacteristic
96C22×C4×D3C2311yescharacteristic
97C22×Dic6C2312yes—
98C22×Dic6C2311yescharacteristic
99C3×C4⋊Q8—24no—
100C3×#124—28no—
101C3×C4×D4—24no—
102C2×C3×C4⋊C4C2312yes—
103C2×C3×C4⋊C4C2312yes—
104C2×C3×C4⋊C4C2311yescharacteristic
105C2×C3×C4⋊C4C2312yes—
106C2×C3×C22⋊C4C2318yes—
107C2×C3×C22⋊C4C2312yes—
108C2×C3×C42C2312yes—
109C2×C3×C42C2311yescharacteristic
110C2×#217C2318yes—
111C2×#217C2312yes—
112#946—28no—
113Q8×Dic3—28no—
114#944—28no—
115D4×Dic3—24no—
116C4×#67—28no—
117C2×(C2×D3)⋊C4C2314yes—
118C2×(C2×D3)⋊C4C2312yes—
119C2×(C2×D3)⋊C4C2314yes—
120C2×C4⋊Dic3C2314yes—
121C2×C4⋊Dic3C2312yes—
122C2×C4⋊Dic3C2312yes—
123C2×C4⋊Dic3C2311yescharacteristic
124C2×C4⋊Dic3C2312yes—
125#924—216no—
126C2×Dic3⋊C4C2312yes—
127C2×Dic3⋊C4C2318yes—
128C2×Dic3⋊C4C2318yes—
129C2×Dic3⋊C4C2314yes—
130C2×Dic3⋊C4C2312yes—
131C2×C4×Dic3—28no—
132C2×C4×Dic3C2314yes—
133C2×C4×Dic3C2312yes—
134C2×C4×Dic3C2314yes—
135C2×C4×Dic3C2314yes—
136C2×C4×Dic3C2312yes—
137C2×C4×Dic3C2312yes—
138C2×C4×Dic3C2312yes—
139C2×C4×Dic3C2311yescharacteristic
140C2×C4×Dic3C2311yescharacteristic
141C2×C4×Dic3C2311yescharacteristic
142#897—28no—
143#892—28no—
144#892—24no—
145#892—28no—
146#890—28no—
147#887—28no—
148#887—24no—
149#67⋊C4—28no—
150#877—28no—
151C42×D3D418yes—
152#869—24no—
153C4×Dic6—24no—
154C3×#95C2318yes—
155C3×#95C2312yes—
156C3⋊#95C2318yes—
157C3⋊#95C2314yes—
158C3⋊#95C2312yes—
159C3⋊#95C2318yes—
160C3⋊#95C2318yes—
161C22×#41—31no—
162#533—68no—
163C2×C4⋊Q8—32no—
164C2×C4⋊Q8—31no—
165C2×C4⋊Q8C2×D311yescharacteristic
166C2×#125—32no—
167C2×#124C2×D312yes—
168C2×C22⋊Q8—32no—
169C2×C22⋊Q8—34no—
170C2×C22⋊Q8—32no—
171C4×#41—68no—
172C2×C4×Q8—32no—
173C2×C4×Q8—31no—
174C2×C4×Q8—32no—
175C2×C4×Q8—31no—
176C2×C4×D4—31no—
177C2×C4×D4—32no—
178C2×C4×D4—32no—
179C2×C4×D4C2×D311yescharacteristic
180C2×#117—32no—
181C2×#117—31no—
182C2×#117—32no—
183C2×#117—32no—
184C22×C42—31no—
185C22×C42—32no—
186#400—34no—
187#399—34no—
188#398—34no—
189#397—38no—
190#397C2×D314yes—
191#394—38no—
192#391—32no—
193#391—34no—
194#391—32no—
195#391C2×D312yes—
196#390—32no—
197#389—34no—
198#389—38no—
199#388—38no—
200#387—38no—
201#385—38no—
202#384—32no—
203#382—32no—
204#380C2×D314yes—
205C4×C4⋊C4—32no—
206C4×C4⋊C4—31no—
207C4×C4⋊C4—38no—
208C4×C4⋊C4C2×D311yescharacteristic
209C4×C22⋊C4—38no—
210C4×C22⋊C4—32no—
211C4×C22⋊C4—32no—
212#376—32no—
213C43—32no—
214C24×C3C2412yes—
215C23×D3C2411yescharacteristic
216C2×C3×Q8—24no—
217C2×C3×Q8—24no—
218C2×C3×Q8—24no—
219C2×C3×D4—22no—
220C2×C3×D4—22no—
221C2×C3×D4#4114yes—
222C22×C3×C4—24no—
223C22×C3×C4—28no—
224C22×C3×C4C2414yes—
225C22×C3×C4C2414yes—
226C22×C3×C4C2412yes—
227C22×C3×C4C2411yescharacteristic
228C22×C3×C4C2411yescharacteristic
229C22×C3×C4C2411yescharacteristic
230C2×#67—24no—
231C2×#67—24no—
232C2×#67—24no—
233C22×Dic3—24no—
234C22×Dic3—24no—
235C22×Dic3—24no—
236C22×Dic3—24no—
237C22×Dic3—28no—
238C22×Dic3C2412yes—
239C22×Dic3C2414yes—
240C22×Dic3C2411yescharacteristic
241C22×Dic3C2411yescharacteristic
242C22×Dic3C2414yes—
243C22×Dic3C2412yes—
244C22×Dic3C2411yescharacteristic
245#237—232no—
246C2×C4×D3—24no—
247C2×C4×D3—22no—
248C2×C4×D3—22no—
249C2×C4×D3—24no—
250C2×C4×D3C2×D414yes—
251C2×C4×D3C2×D414yes—
252C2×C4×D3C2×D414yes—
253C2×Dic6—22no—
254C2×Dic6—22no—
255C2×Dic6—24no—
256C2×Dic6—28no—
257C2×Dic6#4114yes—
258C3×C4⋊C4—22no—
259C3×C4⋊C4—24no—
260C3×C4⋊C4—24no—
261C3×C4⋊C4—24no—
262C3×C22⋊C4—216no—
263C3×C22⋊C4—24no—
264C3×C42—24no—
265C3×C42C2×D414yes—
266#217—24no—
267#217—216no—
268(C2×D3)⋊C4—24no—
269(C2×D3)⋊C4—28no—
270(C2×D3)⋊C4—28no—
271C4⋊Dic3—22no—
272C4⋊Dic3—24no—
273C4⋊Dic3—24no—
274C4⋊Dic3—28no—
275C4⋊Dic3#4118yescomplemented
276Dic3⋊C4—24no—
277Dic3⋊C4—24no—
278Dic3⋊C4—28no—
279Dic3⋊C4—216no—
280Dic3⋊C4—216no—
281C4×Dic3—24no—
282C4×Dic3—24no—
283C4×Dic3—24no—
284C4×Dic3—24no—
285C4×Dic3—28no—
286C4×Dic3—28no—
287C4×Dic3—28no—
288C4×Dic3C2×D414yes—
289C4×Dic3C2×D414yes—
290C4×Dic3C2×D414yes—
291C2×#41—64no—
292C2×#41—64no—
293C2×#41—64no—
294C22×Q8—31no—
295C22×Q8—32no—
296C22×Q8C22×D312yes—
297C22×D4—32no—
298C22×D4C22×D311yescharacteristic
299C23×C4—31no—
300C23×C4—31no—
301C23×C4—31no—
302C23×C4—32no—
303C23×C4—32no—
304C23×C4C22×D312yes—
305C4⋊Q8—24no—
306C4⋊Q8—68no—
307C4⋊Q8—64no—
308#125—68no—
309#124—28no—
310C22⋊Q8—68no—
311C22⋊Q8—616no—
312C22⋊Q8—68no—
313C4×Q8—64no—
314C4×Q8—68no—
315C4×Q8—68no—
316C4×Q8—64no—
317C4×D4—24no—
318C4×D4—64no—
319C4×D4—68no—
320C4×D4—68no—
321#117—68no—
322#117—68no—
323#117—64no—
324#117—68no—
325C2×C4⋊C4—32no—
326C2×C4⋊C4—31no—
327C2×C4⋊C4—32no—
328C2×C4⋊C4—32no—
329C2×C4⋊C4—32no—
330C2×C4⋊C4—34no—
331C2×C4⋊C4—34no—
332C2×C4⋊C4—32no—
333C2×C4⋊C4—38no—
334C2×C4⋊C4—38no—
335C2×C4⋊C4C22×D311yescharacteristic
336C2×C4⋊C4C22×D312yes—
337C2×C4⋊C4C22×D312yes—
338C2×C4⋊C4C22×D312yes—
339C2×C22⋊C4—32no—
340C2×C22⋊C4—38no—
341C2×C22⋊C4—32no—
342C2×C22⋊C4—34no—
343C2×C22⋊C4—34no—
344C2×C22⋊C4C22×D312yes—
345C2×C22⋊C4C22×D318yes—
346C2×C42—31no—
347C2×C42—31no—
348C2×C42—31no—
349C2×C42—38no—
350C2×C42—32no—
351C2×C42—34no—
352C2×C42—34no—
353C2×C42—32no—
354C2×C42—32no—
355C2×C42—32no—
356C2×C42—34no—
357C2×C42—68no—
358C2×C42C22×D312yes—
359C2×C42C22×D311yescharacteristic
360#95—34no—
361#95—32no—
362#95—38no—
363#95—38no—
364#95—38no—
365#95C22×D312yes—
366#95C22×D318yes—
367C23×C3—22no—
368C23×C3—24no—
369C23×C3—22no—
370C23×C3—24no—
371C23×C3C2×#4114yes—
372C23×C3C2511yescharacteristic
373C22×D3—22no—
374C22×D3—21no—
375C22×D3—21no—
376C22×D3—22no—
377C22×D3C22×D412yes—
378C3×Q8—48no—
379C3×D4—28no—
380C2×C3×C4—22no—
381C2×C3×C4—22no—
382C2×C3×C4—24no—
383C2×C3×C4—24no—
384C2×C3×C4—24no—
385C2×C3×C4—24no—
386C2×C3×C4—24no—
387C2×C3×C4—24no—
388C2×C3×C4—24no—
389C2×C3×C4—48no—
390C2×C3×C4C22×D412yes—
391C2×C3×C4C22×D412yes—
392C2×C3×C4C22×D412yes—
393C2×C3×C4C2×#4112yes—
394C2×C3×C4C2×#4112yes—
395C2×C3×C4C2×#4114yes—
396#67—216no—
397C2×Dic3—216no—
398C2×Dic3—22no—
399C2×Dic3—22no—
400C2×Dic3—24no—
401C2×Dic3—22no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—24no—
405C2×Dic3—22no—
406C2×Dic3—24no—
407C2×Dic3—28no—
408C2×Dic3—24no—
409C2×Dic3—48no—
410C2×Dic3C22×D412yes—
411C2×Dic3C22×D412yes—
412C2×Dic3C22×D412yes—
413C2×Dic3C2×#4118yes—
414C2×Dic3C2×#4114yes—
415C2×Dic32−1+414yes—
416C4×D3—28no—
417C4×D3—28no—
418C4×D3—28no—
419Dic6—28no—
420Dic6—48no—
421C24—31no—
422C24C23×D312yes—
423#41—632no—
424C2×Q8—24no—
425C2×Q8—24no—
426C2×Q8—24no—
427C2×Q8—34no—
428C2×Q8—62no—
429C2×Q8—64no—
430C2×Q8—62no—
431C2×Q8—68no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—64no—
435C2×D4—64no—
436C2×D4—64no—
437C2×D4#23714yes—
438C22×C4—24no—
439C22×C4—28no—
440C22×C4—31no—
441C22×C4—31no—
442C22×C4—34no—
443C22×C4—32no—
444C22×C4—31no—
445C22×C4—34no—
446C22×C4—34no—
447C22×C4—34no—
448C22×C4—32no—
449C22×C4—34no—
450C22×C4—64no—
451C22×C4—64no—
452C22×C4—64no—
453C22×C4—64no—
454C22×C4—68no—
455C22×C4—64no—
456C22×C4—62no—
457C22×C4—64no—
458C22×C4—62no—
459C22×C4C23×D314yes—
460C22×C4C23×D314yes—
461C22×C4C23×D312yes—
462C22×C4C23×D311yescharacteristic
463C22×C4C23×D311yescharacteristic
464C22×C4C23×D311yescharacteristic
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—38no—
470C4⋊C4—64no—
471C4⋊C4—68no—
472C4⋊C4—68no—
473C4⋊C4—616no—
474C4⋊C4—616no—
475C4⋊C4—64no—
476C4⋊C4—62no—
477C4⋊C4—64no—
478C4⋊C4—64no—
479C22⋊C4—216no—
480C22⋊C4—24no—
481C22⋊C4—68no—
482C22⋊C4—64no—
483C22⋊C4—68no—
484C22⋊C4—64no—
485C22⋊C4—616no—
486C42—24no—
487C42—34no—
488C42—34no—
489C42—34no—
490C42—64no—
491C42—64no—
492C42—68no—
493C42—68no—
494C42—68no—
495C42—64no—
496C42—64no—
497C42D3×D414yes—
498C22×C3—24no—
499C22×C3—28no—
500C22×C3—42no—
501C22×C3—42no—
502C22×C3—44no—
503C22×C3C23×D411yescharacteristic
504C22×C3C22×#4111yescharacteristic
505C22×C3C22×#4111yescharacteristic
506C22×C3C22×#4112yes—
507C22×C3C2×2−1+412yes—
508C2×D3—22no—
509C2×D3—22no—
510C2×D3—22no—
511C2×D3—42no—
512C2×D3—42no—
513C3×C4—22no—
514C3×C4—44no—
515C3×C4—42no—
516C3×C4—44no—
517C3×C4#52514yes—
518C3×C4#53214yes—
519Dic3—24no—
520Dic3—22no—
521Dic3—22no—
522Dic3—28no—
523Dic3—44no—
524Dic3—42no—
525Dic3#53614yes—
526C23—22no—
527C23—24no—
528C23—22no—
529C23—24no—
530C23—32no—
531C23—61no—
532C23—62no—
533C23—61no—
534C23—62no—
535C23C2×#23714yes—
536C23C24×D311yescharacteristic
537Q8—48no—
538Q8—68no—
539Q8—128no—
540D4—28no—
541D4—616no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—32no—
552C2×C4—34no—
553C2×C4—32no—
554C2×C4—32no—
555C2×C4—38no—
556C2×C4—34no—
557C2×C4—48no—
558C2×C4—68no—
559C2×C4—616no—
560C2×C4—62no—
561C2×C4—61no—
562C2×C4—62no—
563C2×C4—62no—
564C2×C4—64no—
565C2×C4—61no—
566C2×C4—64no—
567C2×C4—68no—
568C2×C4—68no—
569C2×C4—62no—
570C2×C4—64no—
571C2×C4—68no—
572C2×C4—64no—
573C2×C4—128no—
574C2×C4C2×D3×D412yes—
575C2×C4C2×D3×D412yes—
576C2×C4C2×D3×D412yes—
577C2×C4C2×#23712yes—
578C2×C4D3×#4114yes—
579C2×C4D3×#4112yes—
580C2×C3—44no—
581C2×C3C2×#52511yescharacteristic
582C2×C3C2×#53211yescharacteristic
583C2×C3C2×#53611yescharacteristic
584C2×C3#348212yes—
585C2×C3#348212yes—
586D3—42no—
587C22—28no—
588C22—24no—
589C22—42no—
590C22—42no—
591C22—44no—
592C22—62no—
593C22—62no—
594C22—62no—
595C22—122no—
596C22—122no—
597C22C22×D3×D411yescharacteristic
598C22C22×#23711yescharacteristic
599C22C2×D3×#4112yes—
600C22C2×D3×#4111yescharacteristic
601C22D3×2−1+412yes—
602C4—22no—
603C4—34no—
604C4—42no—
605C4—44no—
606C4—44no—
607C4—62no—
608C4—62no—
609C4—64no—
610C4—68no—
611C4—124no—
612C4—122no—
613C4#560614yes—
614C4#566814yes—
615C3#3497911yescharacteristic, complemented, Sylow
616C2—44no—
617C2—122no—
618C2C2×#560611yescharacteristic
619C2C2×#566811yescharacteristic
620C2D3×#53611yescharacteristic
621C2#8667612yes—
622C2#8698112yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2547C2
prime radicalC22×#237C22
hypercenter#2547D3
hypocenterC3#34979
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2547

Complements

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

NH ≅ G/Ntype
1. C2×#5728C2semidirect
3. C3×#2547C2semidirect
4. #83239C2semidirect
6. Dic3×#124C2semidirect
7. #82379C2semidirect
8. #77537C2semidirect
9. #77458C2semidirect
10. #76518C2semidirect
11. #76481C2semidirect
12. #74020C2semidirect
13. #73866C2semidirect
14. #73722C2semidirect
15. C4×#892C2semidirect
74. #2547D3semidirect
275. C4⋊Dic3#41semidirect
615. C3#34979semidirect

Maximal subgroups

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

subgroupindex
C2×#57282
C2×C4×#2372
C3×#25472
#832392
#829592
Dic3×#1242
#823792
#775372
#774582
#765182
#764812
#740202
#738662
#737222
C4×#8922
#349793