Finite Group Databasebeta

← #11235115

Subgroups — #11235115

Order 768.

1726 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2357C211yesmaximal, characteristic, complemented
2#82384C211yesmaximal, characteristic, complemented
3#82329C211yesmaximal, characteristic
4#76353C212yesmaximal
5#76337C212yesmaximal, complemented
6#75925C214yesmaximal, complemented
7#75603C214yesmaximal, complemented
8#74160C211yesmaximal, characteristic, complemented
9#74154C211yesmaximal, characteristic, complemented
10#74103C214yesmaximal, complemented
11#73800C211yesmaximal, characteristic, complemented
12#73695C212yesmaximal, complemented
13#73669C211yesmaximal, characteristic, complemented
14#73640C212yesmaximal, complemented
15#73527C214yesmaximal, complemented
16#45430—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×#940C2211yescharacteristic
19C2×#897C2214yes—
20C2×#885C2214yes—
21C2×#879C2214yes—
22C2×C4×D12C2211yescharacteristic
23C2×#869C2211yescharacteristic
24C2×C4×Dic6C2212yes—
25C3×#390C2212yes—
26C3×#388C2214yes—
27C3×#382C2211yescharacteristic
28C3×#381C2212yes—
29C3×C4×C4⋊C4C2211yescharacteristic
30C3×C4×C4⋊C4C2212yes—
31C3×C4×C22⋊C4C2212yes—
32#5221C2211yescharacteristic
33#5221C2212yes—
34#5213C2214yes—
35#5212C2214yes—
36#4988C2212yes—
37#4988C2211yescharacteristic
38#4982C2212yes—
39#4979C2212yes—
40#4977C2211yescharacteristic
41#4977C2212yes—
42Dic3×C4⋊C4C2212yes—
43Dic3×C4⋊C4C2212yes—
44#4972C2212yes—
45#4961C2214yes—
46#4959C2214yes—
47#4957C2214yes—
48#4956C2214yes—
49#4953C2214yes—
50#4951C2214yes—
51#4948C2214yes—
52#4947C2212yes—
53#4947C2212yes—
54#4940C2214yes—
55#4940C2214yes—
56#4939C2212yes—
57#4935C2212yes—
58C4×C4⋊Dic3C2214yes—
59C4×C4⋊Dic3C2211yescharacteristic
60#4932C2211yescharacteristic
61C4×Dic3⋊C4C2212yes—
62#4929C2214yes—
63#4677C2212yes—
64#4677C2212yes—
65#4675C2214yes—
66#4675C2214yes—
67#4670C2214yes—
68#4666C2212yes—
69#4664C2212yes—
70#4657C2212yes—
71#4656C2212yes—
72#4655C2214yes—
73#4655C2214yes—
74#4653C2214yes—
75#4646C2214yes—
76#2740—34no—
77#2700—34no—
78#2563—31no—
79#2560—34no—
80#2548—32no—
81#2542—31no—
82#2521—34no—
83#2513—32no—
84#2512—32no—
85#2399—31no—
86#2357—31no—
87#2357D311yescharacteristic, complemented
88#2355—32no—
89#2331—31no—
90#2328—31no—
91C22×C3×D4C2311yescharacteristic
92C23×C3×C4C2312yes—
93C22×#67C2314yes—
94C22×D12C2311yescharacteristic
95C22×C4×D3C2312yes—
96C22×Dic6C2312yes—
97C22×Dic6C2312yes—
98C3×C4×D4—24no—
99C2×C3×C4⋊C4C2312yes—
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C4⋊C4C2312yes—
102C2×C3×C22⋊C4C2312yes—
103C2×C3×C22⋊C4C2314yes—
104C2×C3×C22⋊C4C2314yes—
105C2×C3×C42C2312yes—
106C2×C3×C42C2314yes—
107C2×C3×C42C2311yescharacteristic
108C2×#217C2314yes—
109C2×#217C2314yes—
110C2×#217C2314yes—
111#940—24no—
112C2×(C2×D3)⋊C4C2312yes—
113C2×(C2×D3)⋊C4C2314yes—
114C2×(C2×D3)⋊C4C2314yes—
115C2×(C2×D3)⋊C4C2314yes—
116C2×(C2×D3)⋊C4C2314yes—
117C2×(C2×D3)⋊C4C2314yes—
118C2×C4⋊Dic3C2312yes—
119C2×C4⋊Dic3C2312yes—
120C2×C4⋊Dic3C2314yes—
121C2×C4⋊Dic3C2311yescharacteristic
122C2×C4⋊Dic3C2312yes—
123C2×C4⋊Dic3C2314yes—
124C2×C4⋊Dic3C2312yes—
125C2×Dic3⋊C4C2312yes—
126C2×Dic3⋊C4C2312yes—
127C2×Dic3⋊C4C2312yes—
128C2×Dic3⋊C4C2314yes—
129C2×Dic3⋊C4C2312yes—
130C2×C4×Dic3C2311yescharacteristic
131C2×C4×Dic3C2312yes—
132C2×C4×Dic3C2314yes—
133C2×C4×Dic3C2314yes—
134C2×C4×Dic3C2311yescharacteristic
135#897—48no—
136#885—48no—
137#879—48no—
138C4×D12—24no—
139#869—24no—
140C4×Dic6—28no—
141C3×#95C2312yes—
142C3×#95C2312yes—
143C3×#95C2314yes—
144C3×#95C2311yescharacteristic
145C3×#95C2311yescharacteristic
146C3⋊#95C2312yes—
147C3⋊#95C2314yes—
148C3⋊#95C2314yes—
149C3⋊#95C2312yes—
150C3⋊#95C2314yes—
151C3⋊#95C2314yes—
152C3⋊#95C2314yes—
153C3⋊#95C2314yes—
154C3⋊#95C2312yes—
155C3⋊#95C2312yes—
156C3⋊#95C2311yescharacteristic
157C3⋊#95C2311yescharacteristic
158C2×C4⋊Q8—31no—
159C2×#127—31no—
160C2×#126—34no—
161C2×#124—34no—
162C2×C22⋊Q8—34no—
163C2×C4×Q8—32no—
164C2×C4×D4—31no—
165C2×C4×D4C2×D311yescharacteristic
166#399—32no—
167#399—32no—
168#398—32no—
169#398—32no—
170#397—34no—
171#397—34no—
172#397—34no—
173#396—34no—
174#395—32no—
175#395—32no—
176#394—34no—
177#394—34no—
178#393—32no—
179#393—32no—
180#391—34no—
181#391—34no—
182#390—32no—
183#390—31no—
184#390—31no—
185#390—32no—
186#390—34no—
187#390—32no—
188#390C2×D312yes—
189#389—32no—
190#388—34no—
191#388—34no—
192#388—34no—
193#388—34no—
194#388C2×D314yes—
195#387—34no—
196#387—34no—
197#387—34no—
198#386—34no—
199#386—34no—
200#384—32no—
201#382—32no—
202#382—31no—
203#382C2×D311yescharacteristic
204#381—32no—
205#381C2×D312yes—
206#380—34no—
207C4×C4⋊C4—32no—
208C4×C4⋊C4—32no—
209C4×C4⋊C4—34no—
210C4×C4⋊C4—31no—
211C4×C4⋊C4—32no—
212C4×C4⋊C4C2×D312yes—
213C4×C4⋊C4C2×D311yescharacteristic
214C4×C22⋊C4—32no—
215C4×C22⋊C4C2×D312yes—
216#376—31no—
217C24×C3C2412yes—
218C23×D3C2412yes—
219C2×C3×D4—22no—
220C2×C3×D4—22no—
221C2×C3×D4—22no—
222C22×C3×C4—24no—
223C22×C3×C4—44no—
224C22×C3×C4C2412yes—
225C22×C3×C4C2414yes—
226C22×C3×C4C2412yes—
227C22×C3×C4C2411yescharacteristic
228C22×C3×C4C2411yescharacteristic
229C22×C3×C4C2412yes—
230C22×C3×C4C2411yescharacteristic
231C2×#67—28no—
232C2×#67—44no—
233C2×#67—44no—
234C22×Dic3C2412yes—
235C22×Dic3C2412yes—
236C22×Dic3C2412yes—
237C22×Dic3C2412yes—
238C22×Dic3C2414yes—
239C22×Dic3C2412yes—
240C2×D12—22no—
241C2×D12—22no—
242C2×D12—22no—
243C2×C4×D3—24no—
244C2×C4×D3—44no—
245C2×Dic6—24no—
246C2×Dic6—24no—
247C2×Dic6—24no—
248C2×Dic6—42no—
249C2×Dic6—44no—
250C3×C4⋊C4—22no—
251C3×C4⋊C4—24no—
252C3×C4⋊C4—24no—
253C3×C22⋊C4—24no—
254C3×C22⋊C4—28no—
255C3×C22⋊C4—44no—
256C3×C42—24no—
257C3×C42—44no—
258C3×C42#4114yes—
259#217—44no—
260#217—44no—
261#217—44no—
262(C2×D3)⋊C4—24no—
263(C2×D3)⋊C4—28no—
264(C2×D3)⋊C4—44no—
265(C2×D3)⋊C4—44no—
266(C2×D3)⋊C4—44no—
267(C2×D3)⋊C4—44no—
268C4⋊Dic3—24no—
269C4⋊Dic3—24no—
270C4⋊Dic3—22no—
271C4⋊Dic3—28no—
272C4⋊Dic3—24no—
273C4⋊Dic3—44no—
274C4⋊Dic3—42no—
275Dic3⋊C4—24no—
276Dic3⋊C4—24no—
277Dic3⋊C4—24no—
278Dic3⋊C4—44no—
279Dic3⋊C4—42no—
280C4×Dic3—24no—
281C4×Dic3—22no—
282C4×Dic3—44no—
283C4×Dic3—44no—
284C4×Dic3#4114yes—
285C22×Q8—32no—
286C22×Q8—32no—
287C22×D4—34no—
288C22×D4—31no—
289C22×D4C22×D311yescharacteristic
290C23×C4—32no—
291C23×C4C22×D312yes—
292C4⋊Q8—64no—
293#127—64no—
294#126—128no—
295#124—128no—
296C22⋊Q8—128no—
297C4×Q8—68no—
298C4×D4—24no—
299C4×D4—64no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—32no—
302C2×C4⋊C4—32no—
303C2×C4⋊C4—32no—
304C2×C4⋊C4—34no—
305C2×C4⋊C4—32no—
306C2×C4⋊C4—32no—
307C2×C4⋊C4—32no—
308C2×C4⋊C4—34no—
309C2×C4⋊C4—34no—
310C2×C4⋊C4—32no—
311C2×C4⋊C4—32no—
312C2×C4⋊C4C22×D311yescharacteristic
313C2×C4⋊C4C22×D312yes—
314C2×C4⋊C4C22×D312yes—
315C2×C22⋊C4—32no—
316C2×C22⋊C4—34no—
317C2×C22⋊C4—34no—
318C2×C22⋊C4—34no—
319C2×C22⋊C4—34no—
320C2×C22⋊C4—34no—
321C2×C22⋊C4—34no—
322C2×C22⋊C4—34no—
323C2×C22⋊C4—34no—
324C2×C22⋊C4C22×D312yes—
325C2×C22⋊C4C22×D314yes—
326C2×C22⋊C4C22×D314yes—
327C2×C42—31no—
328C2×C42—32no—
329C2×C42—31no—
330C2×C42—34no—
331C2×C42—34no—
332C2×C42C22×D312yes—
333C2×C42C22×D314yes—
334C2×C42C22×D311yescharacteristic
335#95—32no—
336#95—32no—
337#95—31no—
338#95—34no—
339#95—31no—
340#95—34no—
341#95—34no—
342#95—34no—
343#95—32no—
344#95—32no—
345#95—34no—
346#95—34no—
347#95C22×D312yes—
348#95C22×D311yescharacteristic
349#95C22×D312yes—
350#95C22×D311yescharacteristic
351#95C22×D314yes—
352C23×C3—22no—
353C23×C3—22no—
354C23×C3—24no—
355C23×C3—22no—
356C23×C3—24no—
357C23×C3C2511yescharacteristic
358C22×D3—22no—
359C22×D3—22no—
360C22×D3—24no—
361C22×D3—22no—
362C22×D3—24no—
363C3×D4—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—22no—
372C2×C3×C4—24no—
373C2×C3×C4—22no—
374C2×C3×C4—24no—
375C2×C3×C4—24no—
376C2×C3×C4—84no—
377C2×C3×C4C2×#4112yes—
378C2×C3×C4C2×#4112yes—
379C2×C3×C4C2×#4112yes—
380C2×C3×C4C2×#4112yes—
381C2×C3×C4H(5,𝔽2)14yes—
382C2×C3×C42−1+412yes—
383#67—88no—
384C2×Dic3—22no—
385C2×Dic3—22no—
386C2×Dic3—22no—
387C2×Dic3—22no—
388C2×Dic3—24no—
389C2×Dic3—22no—
390C2×Dic3—22no—
391C2×Dic3—24no—
392C2×Dic3—24no—
393C2×Dic3—22no—
394C2×Dic3—24no—
395C2×Dic3—22no—
396C2×Dic3—22no—
397C2×Dic3—22no—
398C2×Dic3—22no—
399C2×Dic3—22no—
400C2×Dic3C2×#4114yes—
401D12—44no—
402C4×D3—84no—
403Dic6—48no—
404Dic6—84no—
405C24—32no—
406C24C23×D312yes—
407C2×Q8—64no—
408C2×Q8—64no—
409C2×Q8—64no—
410C2×Q8—122no—
411C2×Q8—124no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—68no—
416C2×D4—62no—
417C2×D4—62no—
418C2×D4—62no—
419C2×D4—124no—
420C2×D4—124no—
421C22×C4—24no—
422C22×C4—32no—
423C22×C4—32no—
424C22×C4—32no—
425C22×C4—32no—
426C22×C4—32no—
427C22×C4—34no—
428C22×C4—44no—
429C22×C4—64no—
430C22×C4—124no—
431C22×C4C23×D312yes—
432C22×C4C23×D314yes—
433C22×C4C23×D312yes—
434C22×C4C23×D311yescharacteristic
435C22×C4C23×D311yescharacteristic
436C22×C4C23×D312yes—
437C22×C4C23×D311yescharacteristic
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—64no—
442C4⋊C4—64no—
443C4⋊C4—64no—
444C4⋊C4—62no—
445C4⋊C4—64no—
446C4⋊C4—64no—
447C4⋊C4—68no—
448C4⋊C4—64no—
449C4⋊C4—124no—
450C4⋊C4—124no—
451C4⋊C4—122no—
452C4⋊C4—122no—
453C22⋊C4—24no—
454C22⋊C4—28no—
455C22⋊C4—44no—
456C22⋊C4—64no—
457C22⋊C4—68no—
458C22⋊C4—124no—
459C22⋊C4—124no—
460C22⋊C4—124no—
461C22⋊C4—124no—
462C22⋊C4—124no—
463C22⋊C4—124no—
464C22⋊C4—124no—
465C42—24no—
466C42—34no—
467C42—44no—
468C42—64no—
469C42—62no—
470C42—124no—
471C42—124no—
472C42#23514yes—
473C22×C3—42no—
474C22×C3—44no—
475C22×C3—42no—
476C22×C3—42no—
477C22×C3—44no—
478C22×C3C22×#4111yescharacteristic
479C22×C3C22×#4111yescharacteristic
480C22×C3C2×H(5,𝔽2)12yes—
481C22×C3C2×2−1+411yescharacteristic
482C22×C3#58512yes—
483C2×D3—42no—
484C2×D3—44no—
485C2×D3—42no—
486C2×D3—42no—
487C2×D3—44no—
488C3×C4—24no—
489C3×C4—22no—
490C3×C4—22no—
491C3×C4—42no—
492C3×C4—42no—
493C3×C4—44no—
494C3×C4#55614yes—
495Dic3—24no—
496Dic3—42no—
497Dic3—42no—
498Dic3—44no—
499Dic3—42no—
500Dic3—42no—
501C23—22no—
502C23—22no—
503C23—24no—
504C23—22no—
505C23—24no—
506C23—62no—
507C23—62no—
508C23—64no—
509C23—62no—
510C23—64no—
511C23C24×D311yescharacteristic
512Q8—128no—
513Q8—244no—
514D4—44no—
515D4—124no—
516D4—248no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—34no—
530C2×C4—62no—
531C2×C4—62no—
532C2×C4—62no—
533C2×C4—64no—
534C2×C4—62no—
535C2×C4—62no—
536C2×C4—62no—
537C2×C4—64no—
538C2×C4—64no—
539C2×C4—62no—
540C2×C4—62no—
541C2×C4—64no—
542C2×C4—62no—
543C2×C4—62no—
544C2×C4—62no—
545C2×C4—62no—
546C2×C4—84no—
547C2×C4—244no—
548C2×C4C2×#23512yes—
549C2×C4C2×#23512yes—
550C2×C4C2×#23512yes—
551C2×C4D3×#4112yes—
552C2×C4#100914yes—
553C2×C4#101012yes—
554C2×C3—82no—
555C2×C3C2×#55611yescharacteristic
556C2×C3#349412yes—
557C2×C3#350811yescharacteristic
558C2×C3#351011yescharacteristic
559C2×C3#355412yes—
560D3—82no—
561C22—44no—
562C22—42no—
563C22—44no—
564C22—42no—
565C22—42no—
566C22—124no—
567C22—122no—
568C22—124no—
569C22—122no—
570C22—122no—
571C22C22×#23511yescharacteristic
572C22C2×D3×#4111yescharacteristic
573C22C2×#100912yes—
574C22C2×#101011yescharacteristic
575C22#596612yes—
576C4—22no—
577C4—24no—
578C4—22no—
579C4—44no—
580C4—42no—
581C4—42no—
582C4—64no—
583C4—122no—
584C4—122no—
585C4—124no—
586C4—122no—
587C4—122no—
588C4#553314yes—
589C3#4543011yescharacteristic, complemented, Sylow
590C2—82no—
591C2—242no—
592C2C2×#553311yescharacteristic
593C2#8584811yescharacteristic
594C2#8586311yescharacteristic
595C2#8687712yes—
596C2#8706712yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2357C2
prime radicalC2×#940C22
hypercenter#2357D3
hypocenterC3#45430
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2357

Complements

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

NH ≅ G/Ntype
1. C3×#2357C2semidirect
2. #82384C2semidirect
5. #76337C2semidirect
6. #75925C2semidirect
7. #75603C2semidirect
8. #74160C2semidirect
9. #74154C2semidirect
10. #74103C2semidirect
11. #73800C2semidirect
12. #73695C2semidirect
13. #73669C2semidirect
14. #73640C2semidirect
15. #73527C2semidirect
87. #2357D3semidirect
589. C3#45430semidirect

Maximal subgroups

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

subgroupindex
C3×#23572
#823842
#823292
#763532
#763372
#759252
#756032
#741602
#741542
#741032
#738002
#736952
#736692
#736402
#735272
#454303