Finite Group Databasebeta

← #10992687

Subgroups — #10992687

Order 768.

1802 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2356C211yesmaximal, characteristic, complemented
2#82499C211yesmaximal, characteristic
3C4×#944C211yesmaximal, characteristic, complemented
4#76664C212yesmaximal, complemented
5#76630C212yesmaximal, complemented
6#75028C214yesmaximal, complemented
7#75021C214yesmaximal
8#74266C214yesmaximal, complemented
9#74240C211yesmaximal, characteristic
10#74220C211yesmaximal, characteristic, complemented
11#73786C212yesmaximal, complemented
12#73753C212yesmaximal, complemented
13#73643C211yesmaximal, characteristic, complemented
14C4×#893C211yesmaximal, characteristic
15#73392C214yesmaximal, complemented
16#36209—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×#946C2212yes—
19C2×#944C2211yescharacteristic
20C2×#897C2212yes—
21C2×#893C2212yes—
22C2×#890C2212yes—
23C2×C4×D12C2211yescharacteristic
24C2×#872C2214yes—
25C2×C42×D3C2212yes—
26C2×C4×Dic6C2211yescharacteristic
27C3×#391C2212yes—
28C3×#387C2214yes—
29C3×#383C2212yes—
30C3×#382C2211yescharacteristic
31C3×C4×C4⋊C4C2212yes—
32C3×C4×C4⋊C4C2211yescharacteristic
33C3×#376C2212yes—
34#5229C2211yescharacteristic
35#4991C2212yes—
36#4987C2212yes—
37#4987C2212yes—
38#4981C2212yes—
39#4978C2214yes—
40#4976C2214yes—
41#4975C2211yescharacteristic
42#4975C2212yes—
43Dic3×C4⋊C4C2212yes—
44#4973C2211yescharacteristic
45#4940C2214yes—
46#4940C2214yes—
47C4×(C2×D3)⋊C4C2212yes—
48#4936C2214yes—
49#4936C2211yescharacteristic
50C4×C4⋊Dic3C2212yes—
51C4×Dic3⋊C4C2214yes—
52C4×Dic3⋊C4C2214yes—
53C42×Dic3C2211yescharacteristic
54#4929C2212yes—
55#4675C2214yes—
56#4674C2214yes—
57#4672C2212yes—
58#4670C2214yes—
59#4670C2214yes—
60#4667C2214yes—
61#4667C2214yes—
62#4667C2214yes—
63#4666C2212yes—
64#4666C2212yes—
65#4661C2212yes—
66#4660C2214yes—
67#4659C2214yes—
68#4651C2212yes—
69#4650C2214yes—
70#4650C2214yes—
71#4648C2212yes—
72#4647C2214yes—
73#4647C2214yes—
74#4647C2214yes—
75#4647C2214yes—
76#2859—31no—
77#2857—32no—
78#2536—31no—
79#2536—31no—
80#2508—34no—
81#2495—34no—
82#2489—32no—
83#2489—32no—
84#2485—34no—
85#2356—31no—
86#2356D311yescharacteristic, complemented
87#2346—34no—
88C4×C4⋊Q8—31no—
89#2285—32no—
90C42×D4—31no—
91C22×C3×Q8C2311yescharacteristic
92C22×D12C2311yescharacteristic
93C22×C4×D3C2312yes—
94C22×C4×D3C2312yes—
95C22×C4×D3C2312yes—
96C22×Dic6C2311yescharacteristic
97C3×C4×Q8—24no—
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2312yes—
100C2×C3×C4⋊C4C2312yes—
101C2×C3×C4⋊C4C2312yes—
102C2×C3×C42C2312yes—
103C2×C3×C42C2312yes—
104C2×C3×C42C2314yes—
105C2×C3×C42C2311yescharacteristic
106#946—44no—
107#944—24no—
108C2×(C2×D3)⋊C4C2312yes—
109C2×(C2×D3)⋊C4C2312yes—
110C2×(C2×D3)⋊C4C2312yes—
111C2×(C2×D3)⋊C4C2314yes—
112C2×(C2×D3)⋊C4C2314yes—
113C2×(C2×D3)⋊C4C2314yes—
114C2×(C2×D3)⋊C4C2314yes—
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2312yes—
117C2×C4⋊Dic3C2311yescharacteristic
118C2×C4⋊Dic3C2314yes—
119C2×C4⋊Dic3C2314yes—
120C2×Dic3⋊C4C2314yes—
121C2×Dic3⋊C4C2314yes—
122C2×Dic3⋊C4C2314yes—
123C2×Dic3⋊C4C2314yes—
124C2×Dic3⋊C4C2312yes—
125C2×Dic3⋊C4C2314yes—
126C2×Dic3⋊C4C2314yes—
127C2×C4×Dic3C2312yes—
128C2×C4×Dic3C2312yes—
129C2×C4×Dic3C2312yes—
130C2×C4×Dic3C2312yes—
131C2×C4×Dic3C2314yes—
132C2×C4×Dic3C2312yes—
133C2×C4×Dic3C2311yescharacteristic
134C2×C4×Dic3C2311yescharacteristic
135#897—44no—
136#893—28no—
137#890—44no—
138C4×D12—24no—
139#872—216no—
140C42×D3—28no—
141C4×Dic6—24no—
142C3×#95C2312yes—
143C3×#95C2312yes—
144C3×#95C2314yes—
145C3×#95C2314yes—
146C3×#95C2311yescharacteristic
147C3×#95C2314yes—
148C3×#95C2311yescharacteristic
149C3⋊#95C2312yes—
150C3⋊#95C2312yes—
151C3⋊#95C2314yes—
152C3⋊#95C2314yes—
153C3⋊#95C2314yes—
154C3⋊#95C2312yes—
155C3⋊#95C2314yes—
156C3⋊#95C2314yes—
157C3⋊#95C2314yes—
158C3⋊#95C2312yes—
159C3⋊#95C2314yes—
160C2×C4⋊Q8—31no—
161C2×#125—32no—
162C2×C22⋊Q8—32no—
163C2×C22⋊Q8—32no—
164C2×C4×Q8—31no—
165C2×C4×Q8C2×D311yescharacteristic
166C2×C4×D4—32no—
167C2×C4×D4—31no—
168C2×#117—34no—
169C22×C42—32no—
170#400—34no—
171#400—34no—
172#398—32no—
173#398—32no—
174#397—34no—
175#397—34no—
176#394—32no—
177#394—32no—
178#391—31no—
179#391—31no—
180#391—32no—
181#391C2×D312yes—
182#389—34no—
183#389—34no—
184#389—34no—
185#389—34no—
186#389—34no—
187#388—34no—
188#388—34no—
189#388—34no—
190#388—34no—
191#387—34no—
192#387C2×D314yes—
193#384—31no—
194#384—32no—
195#383—32no—
196#383C2×D312yes—
197#382—34no—
198#382—31no—
199#382—34no—
200#382—34no—
201#382C2×D311yescharacteristic
202#380—32no—
203#380—32no—
204#380—34no—
205#380—34no—
206#380—34no—
207C4×C4⋊C4—32no—
208C4×C4⋊C4—34no—
209C4×C4⋊C4—34no—
210C4×C4⋊C4—32no—
211C4×C4⋊C4C2×D311yescharacteristic
212C4×C4⋊C4C2×D312yes—
213C4×C22⋊C4—32no—
214C4×C22⋊C4—32no—
215C4×C22⋊C4—32no—
216#376—32no—
217#376C2×D312yes—
218C43—31no—
219C23×D3C2412yes—
220C2×C3×Q8—22no—
221C2×C3×Q8—22no—
222C2×C3×Q8—22no—
223C22×C3×C4C2412yes—
224C22×C3×C4C2412yes—
225C22×C3×C4C2412yes—
226C22×C3×C4C2414yes—
227C22×C3×C4C2412yes—
228C22×C3×C4C2411yescharacteristic
229C22×C3×C4C2411yescharacteristic
230C22×C3×C4C2411yescharacteristic
231C22×Dic3C2412yes—
232C22×Dic3C2412yes—
233C22×Dic3C2412yes—
234C22×Dic3C2414yes—
235C22×Dic3C2412yes—
236C22×Dic3C2412yes—
237C2×D12—24no—
238C2×D12—22no—
239C2×C4×D3—24no—
240C2×C4×D3—24no—
241C2×C4×D3—24no—
242C2×C4×D3—24no—
243C2×C4×D3—24no—
244C2×C4×D3—24no—
245C2×C4×D3—24no—
246C2×C4×D3—24no—
247C2×C4×D3#4118yes—
248C2×Dic6—22no—
249C2×Dic6—22no—
250C2×Dic6—22no—
251C3×C4⋊C4—24no—
252C3×C4⋊C4—22no—
253C3×C4⋊C4—24no—
254C3×C4⋊C4—24no—
255C3×C42—24no—
256C3×C42—28no—
257C3×C42—24no—
258C3×C42#4114yes—
259(C2×D3)⋊C4—28no—
260(C2×D3)⋊C4—28no—
261(C2×D3)⋊C4—24no—
262(C2×D3)⋊C4—24no—
263(C2×D3)⋊C4—24no—
264(C2×D3)⋊C4—44no—
265(C2×D3)⋊C4—44no—
266C4⋊Dic3—22no—
267C4⋊Dic3—22no—
268C4⋊Dic3—24no—
269C4⋊Dic3—44no—
270C4⋊Dic3—44no—
271Dic3⋊C4—24no—
272Dic3⋊C4—28no—
273Dic3⋊C4—28no—
274Dic3⋊C4—28no—
275Dic3⋊C4—28no—
276Dic3⋊C4—44no—
277Dic3⋊C4—44no—
278C4×Dic3—24no—
279C4×Dic3—28no—
280C4×Dic3—22no—
281C4×Dic3—24no—
282C4×Dic3—24no—
283C4×Dic3—24no—
284C4×Dic3—24no—
285C4×Dic3#4114yes—
286C22×Q8—31no—
287C22×Q8C22×D311yescharacteristic
288C22×D4—31no—
289C23×C4—32no—
290C23×C4—32no—
291C23×C4—32no—
292C4⋊Q8—64no—
293#125—124no—
294C22⋊Q8—124no—
295C22⋊Q8—124no—
296C4×Q8—24no—
297C4×Q8—64no—
298C4×D4—68no—
299C4×D4—64no—
300#117—616no—
301C2×C4⋊C4—34no—
302C2×C4⋊C4—34no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—32no—
306C2×C4⋊C4—32no—
307C2×C4⋊C4—34no—
308C2×C4⋊C4—34no—
309C2×C4⋊C4—34no—
310C2×C4⋊C4—34no—
311C2×C4⋊C4—34no—
312C2×C4⋊C4—34no—
313C2×C4⋊C4C22×D312yes—
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C4⋊C4C22×D312yes—
316C2×C4⋊C4C22×D312yes—
317C2×C22⋊C4—34no—
318C2×C22⋊C4—34no—
319C2×C22⋊C4—32no—
320C2×C22⋊C4—32no—
321C2×C22⋊C4—32no—
322C2×C22⋊C4—34no—
323C2×C22⋊C4—34no—
324C2×C42—31no—
325C2×C42—34no—
326C2×C42—32no—
327C2×C42—32no—
328C2×C42—31no—
329C2×C42—32no—
330C2×C42—32no—
331C2×C42—32no—
332C2×C42—68no—
333C2×C42C22×D314yes—
334C2×C42C22×D312yes—
335C2×C42C22×D312yes—
336C2×C42C22×D311yescharacteristic
337#95—34no—
338#95—32no—
339#95—32no—
340#95—32no—
341#95—34no—
342#95—34no—
343#95—34no—
344#95—34no—
345#95—34no—
346#95—34no—
347#95—32no—
348#95C22×D311yescharacteristic
349#95C22×D314yes—
350#95C22×D312yes—
351#95C22×D311yescharacteristic
352#95C22×D314yes—
353#95C22×D314yes—
354#95C22×D312yes—
355C23×C3C2511yescharacteristic
356C22×D3—22no—
357C22×D3—22no—
358C22×D3—22no—
359C22×D3—22no—
360C22×D3—22no—
361C22×D3—22no—
362C22×D3C2×#4114yes—
363C3×Q8—44no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—24no—
371C2×C3×C4—24no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—24no—
377C2×C3×C4—24no—
378C2×C3×C4—22no—
379C2×C3×C4C2×#4112yes—
380C2×C3×C4C2×#4112yes—
381C2×C3×C4C2×#4112yes—
382C2×C3×C4C2×#4114yes—
383C2×C3×C4C2×#4112yes—
384C2×C3×C4H(5,𝔽2)12yes—
385C2×Dic3—22no—
386C2×Dic3—22no—
387C2×Dic3—24no—
388C2×Dic3—24no—
389C2×Dic3—22no—
390C2×Dic3—22no—
391C2×Dic3—22no—
392C2×Dic3—22no—
393C2×Dic3—22no—
394C2×Dic3—24no—
395C2×Dic3—22no—
396C2×Dic3—22no—
397C2×Dic3—22no—
398C2×Dic3—24no—
399C2×Dic3C2×#4114yes—
400C2×Dic3C2×#4114yes—
401C2×Dic32−1+414yes—
402D12—28no—
403C4×D3—216no—
404C4×D3—48no—
405C4×D3—48no—
406Dic6—44no—
407C24—32no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—62no—
412C2×Q8—62no—
413C2×Q8—62no—
414C2×D4—64no—
415C2×D4—62no—
416C22×C4—32no—
417C22×C4—38no—
418C22×C4—32no—
419C22×C4—32no—
420C22×C4—32no—
421C22×C4—34no—
422C22×C4—32no—
423C22×C4—64no—
424C22×C4—64no—
425C22×C4—64no—
426C22×C4—64no—
427C22×C4—64no—
428C22×C4—64no—
429C22×C4—64no—
430C22×C4—64no—
431C22×C4C23×D312yes—
432C22×C4C23×D312yes—
433C22×C4C23×D312yes—
434C22×C4C23×D314yes—
435C22×C4C23×D311yescharacteristic
436C22×C4C23×D311yescharacteristic
437C22×C4C23×D312yes—
438C22×C4C23×D311yescharacteristic
439C4⋊C4—22no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—68no—
444C4⋊C4—68no—
445C4⋊C4—68no—
446C4⋊C4—68no—
447C4⋊C4—62no—
448C4⋊C4—64no—
449C4⋊C4—62no—
450C4⋊C4—64no—
451C4⋊C4—124no—
452C4⋊C4—124no—
453C4⋊C4—124no—
454C4⋊C4—124no—
455C22⋊C4—68no—
456C22⋊C4—68no—
457C22⋊C4—64no—
458C22⋊C4—64no—
459C22⋊C4—64no—
460C22⋊C4—124no—
461C22⋊C4—124no—
462C42—24no—
463C42—24no—
464C42—28no—
465C42—34no—
466C42—64no—
467C42—64no—
468C42—64no—
469C42—64no—
470C42—64no—
471C42—68no—
472C42—62no—
473C42#23514yes—
474C22×C3C22×#4111yescharacteristic
475C22×C3C22×#4112yes—
476C22×C3C22×#4111yescharacteristic
477C22×C3C2×H(5,𝔽2)11yescharacteristic
478C22×C3C2×2−1+412yes—
479C2×D3—24no—
480C2×D3—24no—
481C2×D3—24no—
482C2×D3—42no—
483C2×D3—42no—
484C2×D3—42no—
485C2×D3—42no—
486C3×C4—22no—
487C3×C4—24no—
488C3×C4—22no—
489C3×C4—42no—
490C3×C4—42no—
491C3×C4—42no—
492C3×C4—44no—
493C3×C4#55514yes—
494Dic3—42no—
495Dic3—42no—
496Dic3—42no—
497Dic3—44no—
498Dic3—42no—
499Dic3#55218yes—
500C23—34no—
501C23—62no—
502C23—62no—
503C23—62no—
504C23—62no—
505C23—62no—
506C23—62no—
507C23C24×D311yescharacteristic
508Q8—44no—
509Q8—124no—
510D4—68no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—34no—
527C2×C4—34no—
528C2×C4—34no—
529C2×C4—616no—
530C2×C4—62no—
531C2×C4—62no—
532C2×C4—62no—
533C2×C4—62no—
534C2×C4—62no—
535C2×C4—64no—
536C2×C4—64no—
537C2×C4—62no—
538C2×C4—62no—
539C2×C4—62no—
540C2×C4—64no—
541C2×C4—62no—
542C2×C4—62no—
543C2×C4—64no—
544C2×C4—128no—
545C2×C4—128no—
546C2×C4C2×#23512yes—
547C2×C4C2×#23512yes—
548C2×C4C2×#23512yes—
549C2×C4D3×#4114yes—
550C2×C4D3×#4112yes—
551C2×C4#100912yes—
552C2×C3#344411yescharacteristic
553C2×C3C2×#55212yes—
554C2×C3C2×#55511yescharacteristic
555C2×C3#350911yescharacteristic
556C2×C3#351112yes—
557D3—44no—
558C22—64no—
559C22—64no—
560C22—64no—
561C22—122no—
562C22—122no—
563C22—122no—
564C22—122no—
565C22C22×#23511yescharacteristic
566C22C2×D3×#4111yescharacteristic
567C22C2×D3×#4112yes—
568C22C2×#100911yescharacteristic
569C22D3×2−1+412yes—
570C4—24no—
571C4—22no—
572C4—22no—
573C4—38no—
574C4—42no—
575C4—42no—
576C4—44no—
577C4—42no—
578C4—122no—
579C4—122no—
580C4—124no—
581C4—122no—
582C4—122no—
583C4#553214yes—
584C3#3620911yescharacteristic, complemented, Sylow
585C2—124no—
586C2C2×#553211yescharacteristic
587C2#8583611yescharacteristic
588C2#8587611yescharacteristic
589C2D3×#55212yes—
590C2#8698512yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2356C2
prime radicalC22×D12C23
hypercenter#2356D3
hypocenterC3#36209
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2356

Complements

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

NH ≅ G/Ntype
1. C3×#2356C2semidirect
3. C4×#944C2semidirect
4. #76664C2semidirect
5. #76630C2semidirect
6. #75028C2semidirect
8. #74266C2semidirect
10. #74220C2semidirect
11. #73786C2semidirect
12. #73753C2semidirect
13. #73643C2semidirect
15. #73392C2semidirect
86. #2356D3semidirect
584. C3#36209semidirect

Maximal subgroups

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

subgroupindex
C3×#23562
#824992
C4×#9442
#766642
#766302
#750282
#750212
#742662
#742402
#742202
#737862
#737532
#736432
C4×#8932
#733922
#362093