Finite Group Databasebeta

← #11235110

Subgroups — #11235110

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×#2328C211yesmaximal, characteristic
2#83179C212yesmaximal
3#82384C211yesmaximal, characteristic, complemented
4#82379C211yesmaximal, characteristic, complemented
5#82338C211yesmaximal, characteristic, complemented
6#82338C211yesmaximal, characteristic, complemented
7#76342C212yesmaximal, complemented
8#76339C212yesmaximal, complemented
9#74832C214yesmaximal, complemented
10#74811C214yesmaximal, complemented
11#74762C214yesmaximal, complemented
12#74748C214yesmaximal, complemented
13#74158C212yesmaximal, complemented
14#73397C211yesmaximal, characteristic, complemented
15#73391C211yesmaximal, characteristic, complemented
16#45430—31nomaximal, Sylow
17C2×C3×#127C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C2×#936C2214yes—
21C2×#924C2214yes—
22C2×#879C2214yes—
23C2×#869C2211yescharacteristic
24C2×C4×Dic6C2212yes—
25C3×#390C2211yescharacteristic
26C3×#390C2211yescharacteristic
27C3×#386C2214yes—
28C3×#386C2214yes—
29C3×#382C2211yescharacteristic
30C3×#376C2211yescharacteristic
31#5222C2214yes—
32#5222C2214yes—
33#5221C2214yes—
34#5221C2212yes—
35#5221C2212yes—
36#5221C2212yes—
37#5221C2212yes—
38#5212C2214yes—
39#5211C2212yes—
40#5211C2212yes—
41C4×#217C2212yes—
42C4×#217C2212yes—
43#4977C2212yes—
44#4975C2212yes—
45Dic3×C4⋊C4C2212yes—
46Dic3×C4⋊C4C2212yes—
47#4953C2214yes—
48#4952C2214yes—
49#4952C2214yes—
50#4951C2214yes—
51#4951C2214yes—
52#4951C2214yes—
53#4949C2212yes—
54#4949C2212yes—
55#4948C2214yes—
56#4947C2212yes—
57#4947C2212yes—
58#4945C2214yes—
59#4944C2214yes—
60#4936C2211yescharacteristic
61C4×C4⋊Dic3C2212yes—
62C4×C4⋊Dic3C2211yescharacteristic
63C4×C4⋊Dic3C2211yescharacteristic
64C4×C4⋊Dic3C2212yes—
65#4928C2212yes—
66#4661C2212yes—
67#4661C2212yes—
68#4656C2212yes—
69#4656C2212yes—
70#4654C2214yes—
71#4653C2214yes—
72#4652C2214yes—
73#4649C2214yes—
74#4646C2214yes—
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#2357—31no—
88#2355—32no—
89#2331—31no—
90#2328D311yescharacteristic
91C22×C3×D4C2311yescharacteristic
92C22×C3×D4C2311yescharacteristic
93C22×C3×D4C2314yes—
94C23×C3×C4C2312yes—
95C23×C3×C4C2312yes—
96C22×Dic6C2312yes—
97C22×Dic6C2312yes—
98C3×#127—24no—
99C3×C4×D4—24no—
100C3×C4×D4—24no—
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C22⋊C4C2312yes—
104C2×C3×C22⋊C4C2312yes—
105C2×C3×C22⋊C4C2314yes—
106C2×C3×C22⋊C4C2314yes—
107C2×C3×C42C2311yescharacteristic
108C2×C3×C42C2311yescharacteristic
109C2×C3×C42C2311yescharacteristic
110C2×#217C2314yes—
111C2×#217C2314yes—
112C2×#217C2314yes—
113C2×#217C2314yes—
114C2×#217C2314yes—
115C2×#217C2314yes—
116C2×#217C2314yes—
117C2×#217C2314yes—
118#936—48no—
119C2×C4⋊Dic3C2312yes—
120C2×C4⋊Dic3C2312yes—
121C2×C4⋊Dic3C2312yes—
122C2×C4⋊Dic3C2312yes—
123C2×C4⋊Dic3C2312yes—
124C2×C4⋊Dic3C2312yes—
125C2×C4⋊Dic3C2312yes—
126C2×C4⋊Dic3C2312yes—
127#924—48no—
128C2×Dic3⋊C4C2314yes—
129C2×Dic3⋊C4C2312yes—
130C2×Dic3⋊C4C2312yes—
131C2×Dic3⋊C4C2314yes—
132C2×Dic3⋊C4C2314yes—
133C2×C4×Dic3C2312yes—
134C2×C4×Dic3C2312yes—
135C2×C4×Dic3C2314yes—
136C2×C4×Dic3C2314yes—
137C2×C4×Dic3C2314yes—
138#879—48no—
139#869—24no—
140C4×Dic6—28no—
141C3×#95C2312yes—
142C3×#95C2314yes—
143C3×#95C2312yes—
144C3⋊#95C2314yes—
145C3⋊#95C2312yes—
146C3⋊#95C2314yes—
147C3⋊#95C2314yes—
148C3⋊#95C2314yes—
149C3⋊#95C2314yes—
150C3⋊#95C2314yes—
151C3⋊#95C2312yes—
152C3⋊#95C2312yes—
153C3⋊#95C2311yescharacteristic
154C3⋊#95C2311yescharacteristic
155C3⋊#95C2312yes—
156C3⋊#95C2311yescharacteristic
157C3⋊#95C2311yescharacteristic
158C2×C4⋊Q8—31no—
159C2×#127C2×D311yescharacteristic
160C2×#126—34no—
161C2×#124—34no—
162C2×C22⋊Q8—34no—
163C2×C4×Q8—32no—
164C2×C4×D4C2×D311yescharacteristic
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—34no—
183#390—32no—
184#390—32no—
185#390—32no—
186#390—32no—
187#390C2×D311yescharacteristic
188#390C2×D311yescharacteristic
189#389—32no—
190#388—34no—
191#388—34no—
192#388—34no—
193#388—34no—
194#388—34no—
195#387—34no—
196#387—34no—
197#387—34no—
198#386C2×D314yes—
199#386C2×D314yes—
200#384—32no—
201#382—32no—
202#382—31no—
203#382C2×D311yescharacteristic
204#381—32no—
205#381—32no—
206#380—34no—
207C4×C4⋊C4—32no—
208C4×C4⋊C4—32no—
209C4×C4⋊C4—31no—
210C4×C4⋊C4—32no—
211C4×C4⋊C4—31no—
212C4×C4⋊C4—32no—
213C4×C4⋊C4—34no—
214C4×C22⋊C4—32no—
215C4×C22⋊C4—32no—
216#376C2×D311yescharacteristic
217C24×C3C2412yes—
218C24×C3C2412yes—
219C2×C3×D4—22no—
220C2×C3×D4—22no—
221C2×C3×D4—22no—
222C2×C3×D4—22no—
223C2×C3×D4—28no—
224C2×C3×D4—22no—
225C2×C3×D4—22no—
226C2×C3×D4—44no—
227C2×C3×D4—44no—
228C22×C3×C4—24no—
229C22×C3×C4—24no—
230C22×C3×C4—44no—
231C22×C3×C4—44no—
232C22×C3×C4C2412yes—
233C22×C3×C4C2412yes—
234C22×C3×C4C2412yes—
235C22×C3×C4C2411yescharacteristic
236C22×C3×C4C2411yescharacteristic
237C22×C3×C4C2412yes—
238C22×C3×C4C2411yescharacteristic
239C22×Dic3C2414yes—
240C22×Dic3C2412yes—
241C22×Dic3C2412yes—
242C22×Dic3C2412yes—
243C22×Dic3C2414yes—
244C22×Dic3C2412yes—
245C2×Dic6—24no—
246C2×Dic6—24no—
247C2×Dic6—24no—
248C2×Dic6—42no—
249C2×Dic6—44no—
250C3×C4⋊C4—22no—
251C3×C4⋊C4—22no—
252C3×C22⋊C4—24no—
253C3×C22⋊C4—24no—
254C3×C22⋊C4—44no—
255C3×C22⋊C4—44no—
256C3×C42—22no—
257C3×C42#4114yes—
258C3×C42#4114yes—
259#217—28no—
260#217—28no—
261#217—44no—
262#217—44no—
263#217—44no—
264#217—44no—
265#217—44no—
266#217—44no—
267C4⋊Dic3—24no—
268C4⋊Dic3—24no—
269C4⋊Dic3—24no—
270C4⋊Dic3—24no—
271C4⋊Dic3—24no—
272C4⋊Dic3—24no—
273C4⋊Dic3—42no—
274C4⋊Dic3—42no—
275Dic3⋊C4—24no—
276Dic3⋊C4—24no—
277Dic3⋊C4—28no—
278Dic3⋊C4—44no—
279Dic3⋊C4—44no—
280C4×Dic3—24no—
281C4×Dic3—24no—
282C4×Dic3—44no—
283C4×Dic3—44no—
284C4×Dic3—44no—
285C22×Q8—32no—
286C22×Q8—32no—
287C22×D4C22×D311yescharacteristic
288C22×D4C22×D311yescharacteristic
289C22×D4C22×D314yes—
290C23×C4C22×D312yes—
291C23×C4C22×D312yes—
292C4⋊Q8—64no—
293#127—24no—
294#126—128no—
295#124—128no—
296C22⋊Q8—128no—
297C4×Q8—68no—
298C4×D4—24no—
299C4×D4—24no—
300C2×C4⋊C4—32no—
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—32no—
309C2×C4⋊C4—34no—
310C2×C4⋊C4—34no—
311C2×C4⋊C4—32no—
312C2×C4⋊C4—32no—
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C22⋊C4—34no—
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⋊C4C22×D312yes—
324C2×C22⋊C4C22×D312yes—
325C2×C22⋊C4C22×D314yes—
326C2×C22⋊C4C22×D314yes—
327C2×C42—32no—
328C2×C42—32no—
329C2×C42—34no—
330C2×C42—34no—
331C2×C42—34no—
332C2×C42C22×D311yescharacteristic
333C2×C42C22×D311yescharacteristic
334C2×C42C22×D311yescharacteristic
335#95—31no—
336#95—31no—
337#95—32no—
338#95—32no—
339#95—34no—
340#95—31no—
341#95—31no—
342#95—34no—
343#95—34no—
344#95—34no—
345#95—32no—
346#95—32no—
347#95—34no—
348#95—34no—
349#95C22×D312yes—
350#95C22×D312yes—
351#95C22×D314yes—
352C23×C3—22no—
353C23×C3—22no—
354C23×C3—22no—
355C23×C3—22no—
356C23×C3—24no—
357C23×C3—24no—
358C23×C3—22no—
359C23×C3—22no—
360C23×C3—24no—
361C23×C3—24no—
362C23×C3C2511yescharacteristic
363C3×D4—44no—
364C3×D4—44no—
365C3×D4—88no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—22no—
372C2×C3×C4—24no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—24no—
376C2×C3×C4—22no—
377C2×C3×C4—84no—
378C2×C3×C4—84no—
379C2×C3×C4C2×#4114yes—
380C2×C3×C4C2×#4112yes—
381C2×C3×C4C2×#4112yes—
382C2×C3×C4C2×#4112yes—
383C2×C3×C4C2×#4112yes—
384C2×C3×C42−1+412yes—
385C2×Dic3—22no—
386C2×Dic3—22no—
387C2×Dic3—22no—
388C2×Dic3—22no—
389C2×Dic3—22no—
390C2×Dic3—24no—
391C2×Dic3—24no—
392C2×Dic3—22no—
393C2×Dic3—24no—
394C2×Dic3—24no—
395C2×Dic3—22no—
396C2×Dic3—22no—
397C2×Dic3—24no—
398C2×Dic3—24no—
399C2×Dic3—22no—
400C2×Dic3—22no—
401C2×Dic3—22no—
402C2×Dic3H(5,𝔽2)14yes—
403Dic6—48no—
404Dic6—84no—
405C24C23×D312yes—
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—28no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—44no—
420C2×D4—44no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—32no—
424C22×C4—34no—
425C22×C4—32no—
426C22×C4—32no—
427C22×C4—32no—
428C22×C4—34no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4C23×D312yes—
432C22×C4C23×D312yes—
433C22×C4C23×D312yes—
434C22×C4C23×D312yes—
435C22×C4C23×D311yescharacteristic
436C22×C4C23×D311yescharacteristic
437C22×C4C23×D311yescharacteristic
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—64no—
441C4⋊C4—68no—
442C4⋊C4—64no—
443C4⋊C4—64no—
444C4⋊C4—64no—
445C4⋊C4—64no—
446C4⋊C4—64no—
447C4⋊C4—64no—
448C4⋊C4—64no—
449C4⋊C4—124no—
450C4⋊C4—124no—
451C4⋊C4—122no—
452C4⋊C4—122no—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C22⋊C4—44no—
456C22⋊C4—44no—
457C22⋊C4—68no—
458C22⋊C4—68no—
459C22⋊C4—124no—
460C22⋊C4—124no—
461C22⋊C4—124no—
462C22⋊C4—124no—
463C22⋊C4—124no—
464C22⋊C4—124no—
465C42—22no—
466C42—64no—
467C42—64no—
468C42—124no—
469C42—124no—
470C42—124no—
471C42#23514yes—
472C42#23714yes—
473C22×C3—42no—
474C22×C3—42no—
475C22×C3—44no—
476C22×C3—44no—
477C22×C3—42no—
478C22×C3—42no—
479C22×C3—42no—
480C22×C3—42no—
481C22×C3—44no—
482C22×C3—44no—
483C22×C3C22×#4111yescharacteristic
484C22×C3C22×#4111yescharacteristic
485C22×C3C2×H(5,𝔽2)12yes—
486C22×C3C2×2−1+411yescharacteristic
487C22×C3#58512yes—
488C3×C4—24no—
489C3×C4—22no—
490C3×C4—22no—
491C3×C4—42no—
492C3×C4—42no—
493C3×C4—42no—
494C3×C4#55614yes—
495Dic3—24no—
496Dic3—42no—
497Dic3—42no—
498Dic3—42no—
499Dic3—44no—
500Dic3—44no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23C24×D311yescharacteristic
512Q8—128no—
513Q8—244no—
514D4—44no—
515D4—44no—
516D4—88no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—34no—
529C2×C4—62no—
530C2×C4—62no—
531C2×C4—62no—
532C2×C4—62no—
533C2×C4—62no—
534C2×C4—64no—
535C2×C4—64no—
536C2×C4—64no—
537C2×C4—62no—
538C2×C4—64no—
539C2×C4—62no—
540C2×C4—62no—
541C2×C4—64no—
542C2×C4—64no—
543C2×C4—62no—
544C2×C4—62no—
545C2×C4—62no—
546C2×C4—84no—
547C2×C4—84no—
548C2×C4C2×#23514yes—
549C2×C4C2×#23512yes—
550C2×C4C2×#23712yes—
551C2×C4C2×#23712yes—
552C2×C4C2×#23712yes—
553C2×C4#101012yes—
554C2×C3—82no—
555C2×C3—82no—
556C2×C3C2×#55611yescharacteristic
557C2×C3#349412yes—
558C2×C3#350811yescharacteristic
559C2×C3#351011yescharacteristic
560C2×C3#355412yes—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—44no—
568C22—44no—
569C22—44no—
570C22—44no—
571C22C22×#23511yescharacteristic
572C22C22×#23711yescharacteristic
573C22C2×#101011yescharacteristic
574C22D3×H(5,𝔽2)12yes—
575C22#596612yes—
576C4—22no—
577C4—22no—
578C4—24no—
579C4—42no—
580C4—42no—
581C4—42no—
582C4—64no—
583C4—124no—
584C4—122no—
585C4—122no—
586C4—122no—
587C4—124no—
588C4#554214yes—
589C3#4543011yescharacteristic, complemented, Sylow
590C2—82no—
591C2—82no—
592C2C2×#554211yescharacteristic
593C2#8586912yes—
594C2#8631911yescharacteristic
595C2#8632011yescharacteristic
596C2#8647812yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2328C2
prime radicalC2×C3×#127C22
hypercenter#2328D3
hypocenterC3#45430
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2328

Complements

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

NH ≅ G/Ntype
3. #82384C2semidirect
4. #82379C2semidirect
5. #82338C2semidirect
6. #82338C2semidirect
7. #76342C2semidirect
8. #76339C2semidirect
9. #74832C2semidirect
10. #74811C2semidirect
11. #74762C2semidirect
12. #74748C2semidirect
13. #74158C2semidirect
14. #73397C2semidirect
15. #73391C2semidirect
589. C3#45430semidirect

Maximal subgroups

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

subgroupindex
C3×#23282
#831792
#823842
#823792
#823382
#763422
#763392
#748322
#748112
#747622
#747482
#741582
#733972
#733912
#454303