Finite Group Databasebeta

← #11025181

Subgroups — #11025181

Order 768.

2000 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
1C2×C3×#533C211yesmaximal, characteristic, complemented
2C2×#5698C211yesmaximal, characteristic, complemented
3#83258C212yesmaximal, complemented
4#83235C212yesmaximal, complemented
5#83065C211yesmaximal, characteristic, complemented
6#83043C211yesmaximal, characteristic, complemented
7#83023C211yesmaximal, characteristic, complemented
8#82801C214yesmaximal
9#82796C214yesmaximal, complemented
10#82473C214yesmaximal, complemented
11#82471C214yesmaximal, complemented
12#82277C212yesmaximal
13#82243C212yesmaximal, complemented
14#82076C211yesmaximal, characteristic, complemented
15#82065C211yesmaximal, characteristic
16#37709—31nomaximal, Sylow
17C3×#533—28no—
18C2×C3×C4⋊Q8C2212yes—
19C2×C3×#125C2211yescharacteristic
20C2×C3×#125C2211yescharacteristic
21C2×C3×C22⋊Q8C2214yes—
22C2×C3×C4×Q8C2214yes—
23C2×C3×#117C2212yes—
24C22×C3×C42C2211yescharacteristic
25C2×C4×#67C2211yescharacteristic
26C2×#924C2212yes—
27#5698—28no—
28C2×#898C2214yes—
29C2×#897C2212yes—
30C2×#890C2214yes—
31C2×#887C2212yes—
32C2×#872C2211yescharacteristic
33C2×#869C2211yescharacteristic
34#5231C2214yes—
35#5230C2214yes—
36#5229C2214yes—
37#5229C2214yes—
38#5213C2212yes—
39#5212C2214yes—
40#5210C2212yes—
41C4×#217C2211yescharacteristic
42#4994C2214yes—
43#4990C2214yes—
44#4987C2212yes—
45#4986C2214yes—
46#4986C2214yes—
47#4986C2214yes—
48#4984C2212yes—
49#4984C2214yes—
50#4982C2214yes—
51#4982C2214yes—
52#4980C2214yes—
53#4978C2214yes—
54Dic3×C4⋊C4C2212yes—
55#4973C2212yes—
56#4973C2212yes—
57#4972C2212yes—
58#4961C2214yes—
59#4953C2214yes—
60#4945C2214yes—
61#4944C2214yes—
62#4940C2214yes—
63#4940C2212yes—
64C4×(C2×D3)⋊C4C2211yescharacteristic
65#4935C2211yescharacteristic
66#4933C2212yes—
67#4933C2214yes—
68#4932C2211yescharacteristic
69C4×Dic3⋊C4C2211yescharacteristic
70C4×Dic3⋊C4C2214yes—
71#4929C2212yes—
72#4928C2214yes—
73#4928C2214yes—
74C2×#541—31no—
75C2×#533D311yescharacteristic, complemented
76#2857—31no—
77#2702—34no—
78#2690—34no—
79#2642—31no—
80#2629—31no—
81#2492—32no—
82#2469—34no—
83#2453—34no—
84#2451—32no—
85#2415—32no—
86#2362—32no—
87#2330—31no—
88#2290—31no—
89C22×C3×Q8C2314yes—
90C23×C3×C4C2312yes—
91C23×C3×C4C2311yescharacteristic
92C22×#67C2311yescharacteristic
93C22×C4×D3C2311yescharacteristic
94C22×Dic6C2312yes—
95C3×C4⋊Q8—28no—
96C3×#125—24no—
97C3×#125D418yescomplemented
98C3×C22⋊Q8—216no—
99C3×C4×Q8—216no—
100C3×#117—28no—
101C2×C3×C4⋊C4C2312yes—
102C2×C3×C4⋊C4C2314yes—
103C2×C3×C4⋊C4C2314yes—
104C2×C3×C4⋊C4C2312yes—
105C2×C3×C4⋊C4C2314yes—
106C2×C3×C22⋊C4C2314yes—
107C2×C3×C42—24no—
108C2×C3×C42C2312yes—
109C2×C3×C42C2314yes—
110C2×C3×C42C2311yescharacteristic
111C2×C3×C42C2311yescharacteristic
112C2×#217C2314yes—
113C2×#217C2311yescharacteristic
114C2×#217C2312yes—
115C4×#67—24no—
116C2×(C2×D3)⋊C4C2312yes—
117C2×(C2×D3)⋊C4C2311yescharacteristic
118C2×(C2×D3)⋊C4C2312yes—
119C2×(C2×D3)⋊C4C2314yes—
120C2×(C2×D3)⋊C4C2314yes—
121C2×C4⋊Dic3C2314yes—
122C2×C4⋊Dic3C2314yes—
123C2×C4⋊Dic3C2312yes—
124C2×C4⋊Dic3C2314yes—
125C2×C4⋊Dic3C2314yes—
126#924—28no—
127C2×Dic3⋊C4C2311yescharacteristic
128C2×Dic3⋊C4C2314yes—
129C2×Dic3⋊C4C2312yes—
130C2×Dic3⋊C4C2314yes—
131C2×Dic3⋊C4C2312yes—
132C2×Dic3⋊C4C2314yes—
133C2×C4×Dic3C2311yescharacteristic
134C2×C4×Dic3C2314yes—
135C2×C4×Dic3C2311yescharacteristic
136C2×C4×Dic3C2311yescharacteristic
137C2×C4×Dic3C2314yes—
138#898—216no—
139#897—28no—
140#890—216no—
141#887—28no—
142#872—24no—
143#869—24no—
144C3⋊#95C2314yes—
145C3⋊#95C2314yes—
146C3⋊#95C2314yes—
147C3⋊#95C2314yes—
148C3⋊#95C2312yes—
149C3⋊#95C2312yes—
150C3⋊#95C2314yes—
151C3⋊#95C2312yes—
152C3⋊#95C2312yes—
153C3⋊#95C2312yes—
154C3⋊#95C2314yes—
155C3⋊#95C2314yes—
156C3⋊#95C2314yes—
157C3⋊#95C2314yes—
158C3⋊#95C2314yes—
159#541—68no—
160#533—28no—
161C2×C4⋊Q8—31no—
162C2×C4⋊Q8C2×D312yes—
163C2×#126—34no—
164C2×#125—34no—
165C2×#125C2×D311yescharacteristic
166C2×#125C2×D311yescharacteristic
167C2×C22⋊Q8—32no—
168C2×C22⋊Q8—32no—
169C2×C22⋊Q8C2×D314yes—
170C2×C4×Q8—32no—
171C2×C4×Q8C2×D314yes—
172C2×C4×D4—31no—
173C2×#117—31no—
174C2×#117C2×D312yes—
175C22×C42C2×D311yescharacteristic
176#400—34no—
177#400—34no—
178#400—34no—
179#398—34no—
180#398—34no—
181#397—34no—
182#396—34no—
183#395—34no—
184#394—34no—
185#393—34no—
186#391—34no—
187#391—34no—
188#391—32no—
189#391—32no—
190#391—32no—
191#389—32no—
192#389—34no—
193#389—32no—
194#389—34no—
195#389—34no—
196#388—34no—
197#388—34no—
198#388—34no—
199#388—34no—
200#388—32no—
201#387—34no—
202#387—34no—
203#387—34no—
204#387—32no—
205#386—32no—
206#385—32no—
207#384—31no—
208#380—34no—
209#380—32no—
210C4×C4⋊C4—32no—
211C4×C4⋊C4—34no—
212C4×C4⋊C4—31no—
213C4×C22⋊C4—31no—
214C4×C22⋊C4—31no—
215#376—31no—
216C24×C3C2411yescharacteristic
217C23×D3C2411yescharacteristic
218C2×C3×Q8—28no—
219C2×C3×Q8—28no—
220C2×C3×Q8—44no—
221C22×C3×C4—22no—
222C22×C3×C4—22no—
223C22×C3×C4—24no—
224C22×C3×C4—24no—
225C22×C3×C4—42no—
226C22×C3×C4#4114yes—
227C22×C3×C4C2414yes—
228C22×C3×C4C2414yes—
229C22×C3×C4C2412yes—
230C22×C3×C4C2411yescharacteristic
231C22×C3×C4C2412yes—
232C22×C3×C4C2411yescharacteristic
233C2×#67—22no—
234C2×#67—42no—
235C22×Dic3C2412yes—
236C22×Dic3C2411yescharacteristic
237C22×Dic3C2411yescharacteristic
238C22×Dic3C2414yes—
239C22×Dic3C2414yes—
240C22×Dic3C2411yescharacteristic
241C22×Dic3C2412yes—
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⋊C4C2×D418yes—
252C3×C4⋊C4C2×D4116yes—
253C3×C22⋊C4—28no—
254C3×C42—24no—
255C3×C42—22no—
256C3×C42—28no—
257C3×C42C2×D414yes—
258#217—28no—
259#217—22no—
260#217—42no—
261(C2×D3)⋊C4—24no—
262(C2×D3)⋊C4—28no—
263(C2×D3)⋊C4—22no—
264(C2×D3)⋊C4—24no—
265(C2×D3)⋊C4—44no—
266C4⋊Dic3—28no—
267C4⋊Dic3—28no—
268C4⋊Dic3—24no—
269C4⋊Dic3—44no—
270C4⋊Dic3—44no—
271Dic3⋊C4—24no—
272Dic3⋊C4—28no—
273Dic3⋊C4—28no—
274Dic3⋊C4—22no—
275Dic3⋊C4—24no—
276Dic3⋊C4—44no—
277C4×Dic3—22no—
278C4×Dic3—28no—
279C4×Dic3—22no—
280C4×Dic3—22no—
281C4×Dic3—44no—
282C22×Q8—32no—
283C22×Q8C22×D314yes—
284C22×D4—31no—
285C23×C4—31no—
286C23×C4C22×D312yes—
287C23×C4C22×D311yescharacteristic
288C4⋊Q8—28no—
289C4⋊Q8—64no—
290#126—616no—
291#125—24no—
292#125—616no—
293#125#6718yescomplemented
294C22⋊Q8—216no—
295C22⋊Q8—68no—
296C22⋊Q8—68no—
297C4×Q8—216no—
298C4×Q8—68no—
299C4×D4—64no—
300#117—28no—
301#117—64no—
302C2×C4⋊C4—32no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—34no—
305C2×C4⋊C4—34no—
306C2×C4⋊C4—32no—
307C2×C4⋊C4—34no—
308C2×C4⋊C4—34no—
309C2×C4⋊C4—32no—
310C2×C4⋊C4—34no—
311C2×C4⋊C4—34no—
312C2×C4⋊C4—34no—
313C2×C4⋊C4C22×D312yes—
314C2×C4⋊C4C22×D314yes—
315C2×C4⋊C4C22×D314yes—
316C2×C4⋊C4C22×D312yes—
317C2×C4⋊C4C22×D314yes—
318C2×C22⋊C4—31no—
319C2×C22⋊C4—34no—
320C2×C22⋊C4—32no—
321C2×C22⋊C4—31no—
322C2×C22⋊C4—34no—
323C2×C22⋊C4—32no—
324C2×C22⋊C4—32no—
325C2×C22⋊C4—34no—
326C2×C22⋊C4C22×D314yes—
327C2×C42—24no—
328C2×C42—31no—
329C2×C42—31no—
330C2×C42—34no—
331C2×C42—31no—
332C2×C42—34no—
333C2×C42C22×D314yes—
334C2×C42C22×D312yes—
335C2×C42C22×D311yescharacteristic
336C2×C42C22×D311yescharacteristic
337#95—32no—
338#95—32no—
339#95—32no—
340#95—32no—
341#95—32no—
342#95—34no—
343#95—34no—
344#95—34no—
345#95—34no—
346#95—34no—
347#95—34no—
348#95—34no—
349#95—34no—
350#95—34no—
351#95—34no—
352C23×C3—21no—
353C23×C3—21no—
354C23×C3—22no—
355C23×C3—22no—
356C23×C3C2×#4112yes—
357C23×C3C2511yescharacteristic
358C22×D3—21no—
359C22×D3—21no—
360C22×D3—22no—
361C22×D3—21no—
362C22×D3—22no—
363C3×Q8—416no—
364C2×C3×C4—21no—
365C2×C3×C4—22no—
366C2×C3×C4—21no—
367C2×C3×C4—22no—
368C2×C3×C4—24no—
369C2×C3×C4—24no—
370C2×C3×C4—22no—
371C2×C3×C4—22no—
372C2×C3×C4—28no—
373C2×C3×C4—24no—
374C2×C3×C4—24no—
375C2×C3×C4—24no—
376C2×C3×C4—48no—
377C2×C3×C4C22×D418yes—
378C2×C3×C4C22×D414yes—
379C2×C3×C4C22×D412yes—
380C2×C3×C4C2×#4112yes—
381C2×C3×C4C2×#4112yes—
382C2×C3×C42−1+412yes—
383C2×C3×C42−1+414yes—
384#67—44no—
385C2×Dic3—22no—
386C2×Dic3—24no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—24no—
391C2×Dic3—24no—
392C2×Dic3—21no—
393C2×Dic3—22no—
394C2×Dic3—22no—
395C2×Dic3—22no—
396C2×Dic3—22no—
397C2×Dic3—24no—
398C2×Dic3—21no—
399C2×Dic3—22no—
400C2×Dic3—24no—
401C2×Dic3—22no—
402C2×Dic3—24no—
403C2×Dic3—22no—
404C4×D3—44no—
405Dic6—48no—
406C24—31no—
407C24C23×D311yescharacteristic
408C2×Q8—28no—
409C2×Q8—28no—
410C2×Q8—44no—
411C2×Q8—64no—
412C2×Q8—64no—
413C2×Q8—64no—
414C2×D4—62no—
415C2×D4—122no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—32no—
421C22×C4—31no—
422C22×C4—31no—
423C22×C4—34no—
424C22×C4—34no—
425C22×C4—31no—
426C22×C4—32no—
427C22×C4—42no—
428C22×C4—62no—
429C22×C4—62no—
430C22×C4—62no—
431C22×C4#23914yes—
432C22×C4C23×D314yes—
433C22×C4C23×D311yescharacteristic
434C22×C4C23×D312yes—
435C22×C4C23×D314yes—
436C22×C4C23×D312yes—
437C22×C4C23×D311yescharacteristic
438C4⋊C4—24no—
439C4⋊C4—28no—
440C4⋊C4—28no—
441C4⋊C4—62no—
442C4⋊C4—68no—
443C4⋊C4—64no—
444C4⋊C4—64no—
445C4⋊C4—64no—
446C4⋊C4—68no—
447C4⋊C4—68no—
448C4⋊C4—68no—
449C4⋊C4—124no—
450C4⋊C4—124no—
451C4⋊C4—124no—
452C4⋊C4C2×#6718yes—
453C4⋊C4C2×#67116yes—
454C22⋊C4—28no—
455C22⋊C4—62no—
456C22⋊C4—68no—
457C22⋊C4—62no—
458C22⋊C4—64no—
459C22⋊C4—64no—
460C22⋊C4—68no—
461C22⋊C4—124no—
462C22⋊C4—122no—
463C42—24no—
464C42—22no—
465C42—28no—
466C42—62no—
467C42—62no—
468C42—62no—
469C42—68no—
470C42—124no—
471C42C2×#6714yes—
472C22×C3—22no—
473C22×C3—22no—
474C22×C3—22no—
475C22×C3—42no—
476C22×C3—42no—
477C22×C3C23×D411yescharacteristic
478C22×C3C22×#4111yescharacteristic
479C22×C3C2×2−1+411yescharacteristic
480C22×C3C2×2−1+412yes—
481C22×C3#58512yes—
482C2×D3—41no—
483C2×D3—41no—
484C2×D3—42no—
485C2×D3—41no—
486C2×D3—42no—
487C3×C4—22no—
488C3×C4—24no—
489C3×C4—24no—
490C3×C4—28no—
491C3×C4—44no—
492C3×C4#54814yes—
493Dic3—44no—
494Dic3—41no—
495Dic3—44no—
496Dic3—41no—
497Dic3—42no—
498Dic3—41no—
499Dic3—42no—
500C23—21no—
501C23—22no—
502C23—22no—
503C23—21no—
504C23—61no—
505C23—61no—
506C23—62no—
507C23—62no—
508C23—61no—
509C23C2×#23912yes—
510C23C24×D311yescharacteristic
511Q8—416no—
512Q8—128no—
513D4—124no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—28no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—48no—
527C2×C4—61no—
528C2×C4—61no—
529C2×C4—64no—
530C2×C4—61no—
531C2×C4—64no—
532C2×C4—62no—
533C2×C4—61no—
534C2×C4—64no—
535C2×C4—62no—
536C2×C4—62no—
537C2×C4—62no—
538C2×C4—62no—
539C2×C4—64no—
540C2×C4—62no—
541C2×C4—64no—
542C2×C4—62no—
543C2×C4—62no—
544C2×C4—64no—
545C2×C4—61no—
546C2×C4—124no—
547C2×C4C2×#23912yes—
548C2×C4C2×#23912yes—
549C2×C4#100712yes—
550C2×C4#101014yes—
551C2×C4C22×#6718yes—
552C2×C4C22×#6714yes—
553C2×C4C22×#6712yes—
554C2×C3—42no—
555C2×C3#345511yescharacteristic
556C2×C3C2×#54811yescharacteristic
557C2×C3#348912yes—
558C2×C3#350911yescharacteristic
559C2×C3#355512yes—
560D3—81no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—42no—
565C22—42no—
566C22—121no—
567C22—121no—
568C22—122no—
569C22—121no—
570C22—122no—
571C22C22×#23911yescharacteristic
572C22C2×#100711yescharacteristic
573C22C2×#101012yes—
574C22#596412yes—
575C22C23×#6711yescharacteristic
576C4—22no—
577C4—24no—
578C4—24no—
579C4—28no—
580C4—44no—
581C4—121no—
582C4—124no—
583C4—121no—
584C4—124no—
585C4—122no—
586C4—121no—
587C4—122no—
588C4#581714yes—
589C3#3770911yescharacteristic, complemented, Sylow
590C2—42no—
591C2—241no—
592C2#8626211yescharacteristic
593C2#8626912yes—
594C2#8627311yescharacteristic
595C2C2×#581711yescharacteristic
596C2#8726412yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC2×C3×#533C2
prime radicalC22×#67C23
hypercenterC2×#533D3
hypocenterC3#37709
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
1. C2×C3×#533C2semidirect
2. C2×#5698C2semidirect
3. #83258C2semidirect
4. #83235C2semidirect
5. #83065C2semidirect
6. #83043C2semidirect
7. #83023C2semidirect
9. #82796C2semidirect
10. #82473C2semidirect
11. #82471C2semidirect
13. #82243C2semidirect
14. #82076C2semidirect
75. C2×#533D3semidirect
97. C3×#125D4semidirect
293. #125#67semidirect
589. C3#37709semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5332
C2×#56982
#832582
#832352
#830652
#830432
#830232
#828012
#827962
#824732
#824712
#822772
#822432
#820762
#820652
#377093