Finite Group Databasebeta

← #11069652

Subgroups — #11069652

Order 768.

2030 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#541C211yesmaximal, characteristic, complemented
2C2×#5698C211yesmaximal, characteristic, complemented
3#83261C211yesmaximal, characteristic, complemented
4#83253C211yesmaximal, characteristic, complemented
5#83098C214yesmaximal, complemented
6#83084C214yesmaximal
7#83052C214yesmaximal, complemented
8#83037C214yesmaximal, complemented
9Dic3×#125C211yesmaximal, characteristic, complemented
10#82780C212yesmaximal
11#82757C212yesmaximal, complemented
12#82496C212yesmaximal, complemented
13#82465C212yesmaximal, complemented
14#82262C211yesmaximal, characteristic, complemented
15#82225C211yesmaximal, characteristic
16#39385—31nomaximal, Sylow
17C3×#541—28no—
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×#126C2214yes—
20C2×C3×#125C2214yes—
21C2×C3×#125C2211yescharacteristic
22C2×C3×C22⋊Q8C2212yes—
23C2×C3×C4×Q8C2212yes—
24C2×C3×#117C2211yescharacteristic
25C2×#946C2212yes—
26C2×#944C2212yes—
27C2×#944C2212yes—
28C2×C4×#67C2211yescharacteristic
29C2×#924C2212yes—
30#5698—28no—
31C2×#898C2214yes—
32C2×#897C2212yes—
33C2×#890C2214yes—
34C2×#887C2212yes—
35C2×#872C2211yescharacteristic
36C2×#869C2211yescharacteristic
37#5229C2212yes—
38C4×#217C2211yescharacteristic
39#4994C2214yes—
40#4993C2214yes—
41#4992C2214yes—
42#4990C2212yes—
43#4984C2212yes—
44#4983C2214yes—
45#4983C2214yes—
46#4981C2214yes—
47#4981C2214yes—
48#4980C2214yes—
49#4978C2214yes—
50#4976C2214yes—
51#4976C2212yes—
52#4975C2214yes—
53#4975C2214yes—
54#4975C2214yes—
55Dic3×C4⋊C4C2212yes—
56Dic3×C4⋊C4C2214yes—
57#4972C2214yes—
58#4957C2212yes—
59#4957C2214yes—
60#4953C2214yes—
61#4949C2214yes—
62#4948C2214yes—
63#4945C2212yes—
64#4944C2214yes—
65#4940C2214yes—
66C4×(C2×D3)⋊C4C2211yescharacteristic
67#4936C2211yescharacteristic
68#4933C2214yes—
69C4×Dic3⋊C4C2211yescharacteristic
70C4×Dic3⋊C4C2214yes—
71C42×Dic3C2211yescharacteristic
72#4929C2212yes—
73#4928C2214yes—
74C2×#541—31no—
75C2×#541D311yescharacteristic, complemented
76#2868—31no—
77#2721—34no—
78#2721—34no—
79#2621—31no—
80#2621—31no—
81#2552—34no—
82#2542—32no—
83#2512—34no—
84#2429—32no—
85#2429—32no—
86#2365—32no—
87C4×#125—31no—
88#2293—31no—
89C22×C3×Q8C2312yes—
90C23×C3×C4C2311yescharacteristic
91C22×#67C2311yescharacteristic
92C22×C4×D3C2311yescharacteristic
93C22×Dic6C2312yes—
94C3×C4⋊Q8—24no—
95C3×#126—216no—
96C3×#125—216no—
97C3×#125D418yescomplemented
98C3×C22⋊Q8—28no—
99C3×C4×Q8—28no—
100C3×#117—24no—
101C2×C3×C4⋊C4C2314yes—
102C2×C3×C4⋊C4C2314yes—
103C2×C3×C4⋊C4C2314yes—
104C2×C3×C4⋊C4C2312yes—
105C2×C3×C4⋊C4C2314yes—
106C2×C3×C4⋊C4C2312yes—
107C2×C3×C22⋊C4C2314yes—
108C2×C3×C22⋊C4C2312yes—
109C2×C3×C42C2314yes—
110C2×C3×C42C2311yescharacteristic
111C2×C3×C42C2311yescharacteristic
112C2×#217C2312yes—
113C2×#217C2311yescharacteristic
114C2×#217C2314yes—
115#946—44no—
116#944—28no—
117#944—44no—
118C4×#67—24no—
119C2×(C2×D3)⋊C4C2312yes—
120C2×(C2×D3)⋊C4C2314yes—
121C2×(C2×D3)⋊C4C2312yes—
122C2×(C2×D3)⋊C4C2311yescharacteristic
123C2×(C2×D3)⋊C4C2314yes—
124C2×C4⋊Dic3C2314yes—
125C2×C4⋊Dic3C2314yes—
126C2×C4⋊Dic3C2312yes—
127C2×C4⋊Dic3C2314yes—
128#924—44no—
129C2×Dic3⋊C4C2312yes—
130C2×Dic3⋊C4C2314yes—
131C2×Dic3⋊C4C2314yes—
132C2×Dic3⋊C4C2314yes—
133C2×Dic3⋊C4C2312yes—
134C2×Dic3⋊C4C2311yescharacteristic
135C2×Dic3⋊C4C2314yes—
136C2×Dic3⋊C4C2314yes—
137C2×Dic3⋊C4C2314yes—
138C2×Dic3⋊C4C2314yes—
139C2×C4×Dic3C2311yescharacteristic
140C2×C4×Dic3C2312yes—
141C2×C4×Dic3C2312yes—
142C2×C4×Dic3C2314yes—
143C2×C4×Dic3C2311yescharacteristic
144C2×C4×Dic3C2314yes—
145C2×C4×Dic3C2311yescharacteristic
146#898—216no—
147#897—28no—
148#890—216no—
149#887—28no—
150#872—24no—
151#869—24no—
152C3⋊#95C2312yes—
153C3⋊#95C2314yes—
154C3⋊#95C2314yes—
155C3⋊#95C2312yes—
156C3⋊#95C2314yes—
157C3⋊#95C2314yes—
158C3⋊#95C2314yes—
159C3⋊#95C2314yes—
160C3⋊#95C2314yes—
161C3⋊#95C2312yes—
162#541—28no—
163#541—68no—
164C2×C4⋊Q8—32no—
165C2×C4⋊Q8—32no—
166C2×C4⋊Q8—31no—
167C2×C4⋊Q8C2×D311yescharacteristic
168C2×#126—34no—
169C2×#126C2×D314yes—
170C2×#125—34no—
171C2×#125C2×D314yes—
172C2×#125C2×D311yescharacteristic
173C2×C22⋊Q8—32no—
174C2×C22⋊Q8—32no—
175C2×C22⋊Q8—32no—
176C2×C22⋊Q8C2×D312yes—
177C2×C4×Q8—32no—
178C2×C4×Q8C2×D312yes—
179C2×C4×D4—31no—
180C2×#117—31no—
181C2×#117C2×D311yescharacteristic
182#399—34no—
183#399—34no—
184#398—34no—
185#398—34no—
186#398—34no—
187#398—34no—
188#397—34no—
189#397—34no—
190#396—34no—
191#396—34no—
192#391—32no—
193#391—32no—
194#390—34no—
195#390—32no—
196#389—32no—
197#389—34no—
198#389—34no—
199#389—32no—
200#389—34no—
201#388—32no—
202#388—34no—
203#388—32no—
204#388—34no—
205#387—34no—
206#387—34no—
207#387—34no—
208#384—34no—
209#384—34no—
210#384—34no—
211#382—31no—
212C4×C4⋊C4—32no—
213C4×C4⋊C4—34no—
214C4×C4⋊C4—31no—
215C4×C4⋊C4—34no—
216C4×C22⋊C4—31no—
217C4×C22⋊C4—31no—
218C43—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×C4C2412yes—
228C22×C3×C4C2414yes—
229C22×C3×C4C2411yescharacteristic
230C22×C3×C4C2412yes—
231C22×C3×C4C2414yes—
232C22×C3×C4C2411yescharacteristic
233C2×#67—22no—
234C2×#67—24no—
235C22×Dic3C2414yes—
236C22×Dic3C2412yes—
237C22×Dic3C2414yes—
238C22×Dic3C2411yescharacteristic
239C22×Dic3C2411yescharacteristic
240C22×Dic3C2412yes—
241C22×Dic3C2411yescharacteristic
242C2×C4×D3—22no—
243C2×C4×D3—22no—
244C2×C4×D3—22no—
245C2×Dic6—24no—
246C2×Dic6—24no—
247C2×Dic6—24no—
248C3×C4⋊C4—24no—
249C3×C4⋊C4—28no—
250C3×C4⋊C4—28no—
251C3×C4⋊C4—28no—
252C3×C4⋊C4C2×D418yes—
253C3×C4⋊C4C2×D4116yes—
254C3×C22⋊C4—24no—
255C3×C22⋊C4—28no—
256C3×C42—22no—
257C3×C42—28no—
258C3×C42C2×D414yes—
259#217—22no—
260#217—24no—
261#217—44no—
262(C2×D3)⋊C4—24no—
263(C2×D3)⋊C4—28no—
264(C2×D3)⋊C4—22no—
265(C2×D3)⋊C4—24no—
266(C2×D3)⋊C4—44no—
267C4⋊Dic3—28no—
268C4⋊Dic3—28no—
269C4⋊Dic3—24no—
270C4⋊Dic3—28no—
271Dic3⋊C4—24no—
272Dic3⋊C4—28no—
273Dic3⋊C4—22no—
274Dic3⋊C4—28no—
275Dic3⋊C4—28no—
276Dic3⋊C4—24no—
277Dic3⋊C4—44no—
278Dic3⋊C4—44no—
279Dic3⋊C4—44no—
280Dic3⋊C4—44no—
281C4×Dic3—22no—
282C4×Dic3—28no—
283C4×Dic3—22no—
284C4×Dic3—24no—
285C4×Dic3—24no—
286C4×Dic3—28no—
287C4×Dic3#4114yes—
288C22×Q8—32no—
289C22×Q8C22×D312yes—
290C22×D4—31no—
291C23×C4—31no—
292C23×C4C22×D311yescharacteristic
293C4⋊Q8—24no—
294C4⋊Q8—68no—
295C4⋊Q8—64no—
296C4⋊Q8—124no—
297#126—216no—
298#126—616no—
299#125—216no—
300#125—616no—
301#125#6718yescomplemented
302C22⋊Q8—28no—
303C22⋊Q8—68no—
304C22⋊Q8—124no—
305C22⋊Q8—124no—
306C4×Q8—28no—
307C4×Q8—68no—
308C4×D4—64no—
309#117—24no—
310#117—64no—
311C2×C4⋊C4—32no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—34no—
314C2×C4⋊C4—32no—
315C2×C4⋊C4—34no—
316C2×C4⋊C4—34no—
317C2×C4⋊C4—34no—
318C2×C4⋊C4—34no—
319C2×C4⋊C4—32no—
320C2×C4⋊C4—34no—
321C2×C4⋊C4—34no—
322C2×C4⋊C4—34no—
323C2×C4⋊C4—34no—
324C2×C4⋊C4—34no—
325C2×C4⋊C4C22×D312yes—
326C2×C4⋊C4C22×D314yes—
327C2×C4⋊C4C22×D314yes—
328C2×C4⋊C4C22×D314yes—
329C2×C4⋊C4C22×D312yes—
330C2×C4⋊C4C22×D314yes—
331C2×C22⋊C4—31no—
332C2×C22⋊C4—32no—
333C2×C22⋊C4—32no—
334C2×C22⋊C4—31no—
335C2×C22⋊C4—34no—
336C2×C22⋊C4—32no—
337C2×C22⋊C4—34no—
338C2×C22⋊C4—34no—
339C2×C22⋊C4C22×D312yes—
340C2×C22⋊C4C22×D314yes—
341C2×C42—31no—
342C2×C42—31no—
343C2×C42—34no—
344C2×C42—31no—
345C2×C42—32no—
346C2×C42—32no—
347C2×C42—34no—
348C2×C42C22×D311yescharacteristic
349C2×C42C22×D314yes—
350C2×C42C22×D311yescharacteristic
351#95—32no—
352#95—32no—
353#95—34no—
354#95—32no—
355#95—34no—
356#95—34no—
357#95—34no—
358#95—34no—
359#95—34no—
360#95—34no—
361C23×C3—21no—
362C23×C3—21no—
363C23×C3—21no—
364C23×C3—22no—
365C23×C3—22no—
366C23×C3C2511yescharacteristic
367C22×D3—21no—
368C22×D3—21no—
369C22×D3—21no—
370C22×D3—22no—
371C22×D3—22no—
372C3×Q8—48no—
373C2×C3×C4—22no—
374C2×C3×C4—24no—
375C2×C3×C4—22no—
376C2×C3×C4—24no—
377C2×C3×C4—22no—
378C2×C3×C4—24no—
379C2×C3×C4—21no—
380C2×C3×C4—24no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—24no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—44no—
387C2×C3×C4C22×D418yes—
388C2×C3×C4C22×D414yes—
389C2×C3×C4C22×D412yes—
390C2×C3×C4C2×#4112yes—
391C2×C3×C4H(5,𝔽2)12yes—
392#67—44no—
393C2×Dic3—22no—
394C2×Dic3—22no—
395C2×Dic3—22no—
396C2×Dic3—22no—
397C2×Dic3—24no—
398C2×Dic3—21no—
399C2×Dic3—24no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—24no—
403C2×Dic3—24no—
404C2×Dic3—24no—
405C2×Dic3—22no—
406C2×Dic3—24no—
407C2×Dic3—22no—
408C2×Dic3C2×#4112yes—
409C2×Dic3C2×#4112yes—
410C2×Dic32−1+414yes—
411C2×Dic32−1+414yes—
412C4×D3—44no—
413Dic6—48no—
414C24—31no—
415C24C23×D311yescharacteristic
416C2×Q8—24no—
417C2×Q8—24no—
418C2×Q8—24no—
419C2×Q8—64no—
420C2×Q8—64no—
421C2×Q8—64no—
422C2×D4—62no—
423C2×D4—64no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—31no—
428C22×C4—31no—
429C22×C4—32no—
430C22×C4—34no—
431C22×C4—32no—
432C22×C4—34no—
433C22×C4—31no—
434C22×C4—62no—
435C22×C4—62no—
436C22×C4—62no—
437C22×C4C23×D312yes—
438C22×C4C23×D314yes—
439C22×C4C23×D311yescharacteristic
440C22×C4C23×D312yes—
441C22×C4C23×D314yes—
442C22×C4C23×D311yescharacteristic
443C4⋊C4—24no—
444C4⋊C4—28no—
445C4⋊C4—28no—
446C4⋊C4—28no—
447C4⋊C4—62no—
448C4⋊C4—68no—
449C4⋊C4—68no—
450C4⋊C4—64no—
451C4⋊C4—68no—
452C4⋊C4—64no—
453C4⋊C4—64no—
454C4⋊C4—68no—
455C4⋊C4—68no—
456C4⋊C4—68no—
457C4⋊C4—124no—
458C4⋊C4—124no—
459C4⋊C4—124no—
460C4⋊C4—124no—
461C4⋊C4C2×#6718yes—
462C4⋊C4C2×#67116yes—
463C22⋊C4—24no—
464C22⋊C4—28no—
465C22⋊C4—62no—
466C22⋊C4—64no—
467C22⋊C4—62no—
468C22⋊C4—64no—
469C22⋊C4—64no—
470C22⋊C4—68no—
471C22⋊C4—124no—
472C22⋊C4—124no—
473C42—22no—
474C42—28no—
475C42—34no—
476C42—64no—
477C42—68no—
478C42—64no—
479C42—62no—
480C42—62no—
481C42—68no—
482C42C2×#6714yes—
483C22×C3—41no—
484C22×C3—41no—
485C22×C3—42no—
486C22×C3—41no—
487C22×C3—42no—
488C22×C3C23×D411yescharacteristic
489C22×C3C22×#4111yescharacteristic
490C22×C3C2×H(5,𝔽2)11yescharacteristic
491C22×C3C2×2−1+412yes—
492C22×C3C2×2−1+412yes—
493C2×D3—41no—
494C2×D3—41no—
495C2×D3—42no—
496C2×D3—41no—
497C2×D3—42no—
498C3×C4—24no—
499C3×C4—28no—
500C3×C4—41no—
501C3×C4—44no—
502C3×C4—42no—
503C3×C4#55014yes—
504Dic3—22no—
505Dic3—24no—
506Dic3—44no—
507Dic3—41no—
508Dic3—42no—
509Dic3—44no—
510Dic3#54114yescomplemented
511C23—21no—
512C23—21no—
513C23—22no—
514C23—22no—
515C23—21no—
516C23—61no—
517C23—61no—
518C23—62no—
519C23—62no—
520C23—61no—
521C23C24×D311yescharacteristic
522Q8—48no—
523Q8—128no—
524D4—124no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—21no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—21no—
538C2×C4—32no—
539C2×C4—34no—
540C2×C4—32no—
541C2×C4—34no—
542C2×C4—44no—
543C2×C4—62no—
544C2×C4—62no—
545C2×C4—62no—
546C2×C4—61no—
547C2×C4—64no—
548C2×C4—64no—
549C2×C4—64no—
550C2×C4—61no—
551C2×C4—64no—
552C2×C4—62no—
553C2×C4—64no—
554C2×C4—62no—
555C2×C4—62no—
556C2×C4—64no—
557C2×C4—61no—
558C2×C4—124no—
559C2×C4D3×#4112yes—
560C2×C4#100912yes—
561C2×C4C22×#6718yes—
562C2×C4C22×#6714yes—
563C2×C4C22×#6712yes—
564C2×C3—81no—
565C2×C3C2×#54111yescharacteristic
566C2×C3C2×#55011yescharacteristic
567C2×C3#349912yes—
568C2×C3#350911yescharacteristic
569C2×C3#355212yes—
570D3—81no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—42no—
575C22—42no—
576C22—121no—
577C22—121no—
578C22—121no—
579C22—122no—
580C22—122no—
581C22C2×D3×#4111yescharacteristic
582C22C2×#100911yescharacteristic
583C22D3×2−1+412yes—
584C22D3×2−1+412yes—
585C22C23×#6711yescharacteristic
586C4—24no—
587C4—28no—
588C4—34no—
589C4—41no—
590C4—44no—
591C4—42no—
592C4—62no—
593C4—64no—
594C4—121no—
595C4—124no—
596C4—124no—
597C4—122no—
598C4#583414yes—
599C3#3938511yescharacteristic, complemented, Sylow
600C2—81no—
601C2—241no—
602C2D3×#54111yescharacteristic
603C2#8651011yescharacteristic
604C2#8651712yes—
605C2C2×#583411yescharacteristic
606C2#8729212yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC2×C3×#541C2
prime radicalC22×#67C23
hypercenterC2×#541D3
hypocenterC3#39385
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#541

Complements

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

NH ≅ G/Ntype
1. C2×C3×#541C2semidirect
2. C2×#5698C2semidirect
3. #83261C2semidirect
4. #83253C2semidirect
5. #83098C2semidirect
7. #83052C2semidirect
8. #83037C2semidirect
9. Dic3×#125C2semidirect
11. #82757C2semidirect
12. #82496C2semidirect
13. #82465C2semidirect
14. #82262C2semidirect
75. C2×#541D3semidirect
97. C3×#125D4semidirect
301. #125#67semidirect
510. Dic3#541semidirect
599. C3#39385semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5412
C2×#56982
#832612
#832532
#830982
#830842
#830522
#830372
Dic3×#1252
#827802
#827572
#824962
#824652
#822622
#822252
#393853