Finite Group Databasebeta

← #11025120

Subgroups — #11025120

Order 768.

2012 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×#5731C211yesmaximal, characteristic, complemented
2C2×D3×#125C211yesmaximal, characteristic, complemented
3C3×#2642C211yesmaximal, characteristic, complemented
4#83269C211yesmaximal, characteristic, complemented
5#83065C211yesmaximal, characteristic, complemented
6#78249C214yesmaximal, complemented
7#78245C214yesmaximal, complemented
8#78145C214yesmaximal
9#78135C214yesmaximal, complemented
10#76630C212yesmaximal, complemented
11#76621C212yesmaximal
12#76615C212yesmaximal, complemented
13#76512C212yesmaximal, complemented
14#73414C211yesmaximal, characteristic
15#73390C211yesmaximal, characteristic, complemented
16#37705—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#125C2211yescharacteristic
19C2×#946C2212yes—
20C2×#946C2212yes—
21C2×Q8×Dic3C2212yes—
22#5731—28no—
23D3×#125—28no—
24C2×#898C2214yes—
25C2×D3×C4⋊C4C2214yes—
26C2×D3×C4⋊C4C2212yes—
27C2×#890C2212yes—
28C2×#889C2214yes—
29C2×#889C2214yes—
30C2×C4×D12C2211yescharacteristic
31C2×#872C2211yescharacteristic
32C2×C42×D3C2211yescharacteristic
33C2×#870C2211yescharacteristic
34C3×#400C2214yes—
35C3×#400C2214yes—
36C3×#391C2212yes—
37C3×#391C2212yes—
38C3×#376C2211yescharacteristic
39#5229C2212yes—
40#4994C2214yes—
41#4987C2214yes—
42#4987C2212yes—
43#4982C2214yes—
44#4972C2214yes—
45#4972C2212yes—
46C4×(C2×D3)⋊C4C2211yescharacteristic
47C4×(C2×D3)⋊C4C2211yescharacteristic
48#4935C2211yescharacteristic
49C4×C4⋊Dic3C2212yes—
50C4×C4⋊Dic3C2211yescharacteristic
51#4928C2212yes—
52#4676C2214yes—
53#4675C2214yes—
54#4670C2214yes—
55#4670C2212yes—
56#4668C2212yes—
57#4668C2214yes—
58#4667C2214yes—
59#4666C2214yes—
60#4665C2214yes—
61#4665C2212yes—
62#4661C2214yes—
63#4660C2214yes—
64#4660C2214yes—
65#4657C2214yes—
66#4655C2214yes—
67#4655C2214yes—
68#4653C2212yes—
69#4652C2214yes—
70#4650C2214yes—
71#4647C2214yes—
72#4647C2214yes—
73#4647C2214yes—
74C2×#541—31no—
75C22×#125—31no—
76#2857—31no—
77#2725—34no—
78#2719—34no—
79#2642D311yescharacteristic, complemented
80#2629—31no—
81#2489—32no—
82#2481—34no—
83#2453—32no—
84#2442—34no—
85#2417—32no—
86#2360—32no—
87#2330—31no—
88#2290—31no—
89C22×C3×Q8C2312yes—
90C22×D12C2311yescharacteristic
91C22×C4×D3C2311yescharacteristic
92C22×C4×D3C2314yes—
93C22×C4×D3C2312yes—
94C22×C4×D3C2311yescharacteristic
95C3×C4⋊Q8—24no—
96C3×#125—24no—
97C2×C3×C4⋊C4C2314yes—
98C2×C3×C4⋊C4C2312yes—
99C2×C3×C4⋊C4C2314yes—
100C2×C3×C42C2311yescharacteristic
101C2×C3×C42C2311yescharacteristic
102C2×C3×C42C2311yescharacteristic
103#946—28no—
104#946—28no—
105Q8×Dic3—28no—
106C2×(C2×D3)⋊C4C2312yes—
107C2×(C2×D3)⋊C4C2314yes—
108C2×(C2×D3)⋊C4C2314yes—
109C2×(C2×D3)⋊C4C2312yes—
110C2×(C2×D3)⋊C4C2311yescharacteristic
111C2×(C2×D3)⋊C4C2311yescharacteristic
112C2×(C2×D3)⋊C4C2312yes—
113C2×(C2×D3)⋊C4C2314yes—
114C2×C4⋊Dic3C2314yes—
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2314yes—
117C2×C4⋊Dic3C2312yes—
118C2×C4⋊Dic3C2312yes—
119C2×C4⋊Dic3C2314yes—
120C2×C4⋊Dic3C2312yes—
121C2×Dic3⋊C4C2314yes—
122C2×Dic3⋊C4C2314yes—
123C2×Dic3⋊C4C2314yes—
124C2×Dic3⋊C4C2314yes—
125C2×Dic3⋊C4C2312yes—
126C2×Dic3⋊C4C2314yes—
127C2×Dic3⋊C4C2314yes—
128C2×Dic3⋊C4C2314yes—
129C2×C4×Dic3C2314yes—
130C2×C4×Dic3C2312yes—
131C2×C4×Dic3C2311yescharacteristic
132C2×C4×Dic3C2311yescharacteristic
133C2×C4×Dic3C2314yes—
134#898—216no—
135D3×C4⋊C4—28no—
136D3×C4⋊C4—216no—
137#890—28no—
138#889—216no—
139#889—216no—
140C4×D12—24no—
141#872—24no—
142C42×D3—24no—
143#870D418yescomplemented
144C3×#95C2312yes—
145C3×#95C2314yes—
146C3×#95C2314yes—
147C3×#95C2314yes—
148C3×#95C2312yes—
149C3×#95C2314yes—
150C3⋊#95C2312yes—
151C3⋊#95C2314yes—
152C3⋊#95C2314yes—
153C3⋊#95C2312yes—
154C3⋊#95C2314yes—
155C3⋊#95C2314yes—
156C3⋊#95C2314yes—
157C3⋊#95C2312yes—
158C3⋊#95C2314yes—
159#541—68no—
160C2×C4⋊Q8C2×D311yescharacteristic
161C2×#126—34no—
162C2×#125—34no—
163C2×#125—32no—
164C2×#125—34no—
165C2×#125—31no—
166C2×#125—68no—
167C2×#125C2×D311yescharacteristic
168C2×C22⋊Q8—32no—
169C2×C22⋊Q8—32no—
170C2×C4×Q8—32no—
171C2×C4×D4—31no—
172C2×#117—31no—
173C22×C4⋊C4—32no—
174C22×C4⋊C4—34no—
175C22×C42—31no—
176#400—34no—
177#400—34no—
178#400C2×D314yes—
179#400C2×D314yes—
180#398—34no—
181#398—34no—
182#398—34no—
183#397—34no—
184#396—34no—
185#396—34no—
186#391—32no—
187#391C2×D312yes—
188#391C2×D312yes—
189#389—34no—
190#389—32no—
191#389—32no—
192#389—34no—
193#389—34no—
194#389—34no—
195#388—32no—
196#388—34no—
197#387—34no—
198#387—34no—
199#387—32no—
200#387—34no—
201#386—34no—
202#386—32no—
203#385—34no—
204#385—32no—
205#384—31no—
206#382—34no—
207#380—34no—
208#380—32no—
209#380—34no—
210C4×C4⋊C4—32no—
211C4×C4⋊C4—31no—
212C4×C22⋊C4—31no—
213C4×C22⋊C4—31no—
214C4×C22⋊C4—34no—
215#376C2×D311yescharacteristic
216C23×D3C2411yescharacteristic
217C23×D3C2411yescharacteristic
218C2×C3×Q8—24no—
219C2×C3×Q8—24no—
220C2×C3×Q8—24no—
221C22×C3×C4C2411yescharacteristic
222C22×C3×C4C2412yes—
223C22×C3×C4C2414yes—
224C22×C3×C4C2414yes—
225C22×C3×C4C2411yescharacteristic
226C22×C3×C4C2412yes—
227C22×C3×C4C2411yescharacteristic
228C22×Dic3C2412yes—
229C22×Dic3C2414yes—
230C22×Dic3C2411yescharacteristic
231C22×Dic3C2412yes—
232C22×Dic3C2414yes—
233C22×Dic3C2411yescharacteristic
234C2×D12—22no—
235C2×D12—24no—
236C2×C4×D3—22no—
237C2×C4×D3—22no—
238C2×C4×D3—28no—
239C2×C4×D3—24no—
240C2×C4×D3—24no—
241C2×C4×D3—22no—
242C2×C4×D3—22no—
243C2×C4×D3—22no—
244C2×C4×D3—28no—
245C2×C4×D3—44no—
246C2×C4×D3—42no—
247C2×C4×D3#4114yes—
248C3×C4⋊C4—28no—
249C3×C4⋊C4—28no—
250C3×C4⋊C4—24no—
251C3×C42—22no—
252C3×C42—22no—
253C3×C42C2×D414yes—
254(C2×D3)⋊C4—22no—
255(C2×D3)⋊C4—28no—
256(C2×D3)⋊C4—24no—
257(C2×D3)⋊C4—24no—
258(C2×D3)⋊C4—28no—
259(C2×D3)⋊C4—22no—
260(C2×D3)⋊C4—44no—
261(C2×D3)⋊C4—42no—
262C4⋊Dic3—24no—
263C4⋊Dic3—28no—
264C4⋊Dic3—28no—
265C4⋊Dic3—22no—
266C4⋊Dic3—44no—
267C4⋊Dic3—42no—
268C4⋊Dic3C2×D418yes—
269Dic3⋊C4—24no—
270Dic3⋊C4—28no—
271Dic3⋊C4—28no—
272Dic3⋊C4—28no—
273Dic3⋊C4—28no—
274Dic3⋊C4—44no—
275Dic3⋊C4—44no—
276Dic3⋊C4C2×D4116yes—
277C4×Dic3—24no—
278C4×Dic3—22no—
279C4×Dic3—22no—
280C4×Dic3—28no—
281C4×Dic3—44no—
282C22×Q8C22×D312yes—
283C22×D4—31no—
284C23×C4—31no—
285C23×C4—31no—
286C23×C4—34no—
287C23×C4—32no—
288C4⋊Q8—24no—
289#126—616no—
290#125—24no—
291#125—38no—
292#125—616no—
293#125—68no—
294#125—616no—
295C22⋊Q8—68no—
296C22⋊Q8—68no—
297C4×Q8—68no—
298C4×D4—64no—
299#117—64no—
300C2×C4⋊C4—32no—
301C2×C4⋊C4—34no—
302C2×C4⋊C4—34no—
303C2×C4⋊C4—34no—
304C2×C4⋊C4—34no—
305C2×C4⋊C4—32no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—34no—
308C2×C4⋊C4—34no—
309C2×C4⋊C4—32no—
310C2×C4⋊C4—34no—
311C2×C4⋊C4—32no—
312C2×C4⋊C4—34no—
313C2×C4⋊C4—34no—
314C2×C4⋊C4—34no—
315C2×C4⋊C4—68no—
316C2×C4⋊C4—616no—
317C2×C4⋊C4C22×D312yes—
318C2×C4⋊C4C22×D314yes—
319C2×C4⋊C4C22×D314yes—
320C2×C22⋊C4—32no—
321C2×C22⋊C4—31no—
322C2×C22⋊C4—31no—
323C2×C22⋊C4—34no—
324C2×C22⋊C4—34no—
325C2×C22⋊C4—32no—
326C2×C22⋊C4—32no—
327C2×C22⋊C4—34no—
328C2×C42—31no—
329C2×C42—34no—
330C2×C42—31no—
331C2×C42—32no—
332C2×C42—34no—
333C2×C42—64no—
334C2×C42C22×D311yescharacteristic
335C2×C42C22×D311yescharacteristic
336C2×C42C22×D311yescharacteristic
337#95—32no—
338#95—32no—
339#95—34no—
340#95—34no—
341#95—32no—
342#95—34no—
343#95—34no—
344#95—34no—
345#95—34no—
346#95C22×D312yes—
347#95C22×D312yes—
348#95C22×D314yes—
349#95C22×D314yes—
350#95C22×D314yes—
351#95C22×D314yes—
352C23×C3C2511yescharacteristic
353C22×D3—21no—
354C22×D3—21no—
355C22×D3—22no—
356C22×D3—21no—
357C22×D3—21no—
358C22×D3—21no—
359C22×D3—22no—
360C22×D3—22no—
361C22×D3C2×#4112yes—
362C22×D32−1+414yes—
363C3×Q8—48no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—24no—
367C2×C3×C4—24no—
368C2×C3×C4—24no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—24no—
374C2×C3×C4—22no—
375C2×C3×C4—24no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—24no—
379C2×C3×C4C22×D414yes—
380C2×C3×C4C22×D412yes—
381C2×C3×C4C2×#4112yes—
382C2×C3×C42−1+412yes—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—24no—
386C2×Dic3—24no—
387C2×Dic3—22no—
388C2×Dic3—22no—
389C2×Dic3—21no—
390C2×Dic3—24no—
391C2×Dic3—21no—
392C2×Dic3—22no—
393C2×Dic3—21no—
394C2×Dic3—22no—
395C2×Dic3—24no—
396C2×Dic3—22no—
397C2×Dic3—24no—
398C2×Dic3—22no—
399C2×Dic3C22×D418yes—
400C2×Dic3C2×#4112yes—
401D12—44no—
402C4×D3—28no—
403C4×D3—416no—
404C4×D3—48no—
405C4×D3—44no—
406C24—31no—
407C24—31no—
408C2×Q8—24no—
409C2×Q8—24no—
410C2×Q8—24no—
411C2×D4—62no—
412C2×D4—64no—
413C22×C4—34no—
414C22×C4—31no—
415C22×C4—34no—
416C22×C4—32no—
417C22×C4—34no—
418C22×C4—31no—
419C22×C4—32no—
420C22×C4—68no—
421C22×C4—64no—
422C22×C4—68no—
423C22×C4—62no—
424C22×C4—62no—
425C22×C4—64no—
426C22×C4—62no—
427C22×C4—62no—
428C22×C4—62no—
429C22×C4—122no—
430C22×C4—124no—
431C22×C4C23×D311yescharacteristic
432C22×C4C23×D312yes—
433C22×C4C23×D314yes—
434C22×C4C23×D314yes—
435C22×C4C23×D311yescharacteristic
436C22×C4C23×D312yes—
437C22×C4C23×D311yescharacteristic
438C4⋊C4—28no—
439C4⋊C4—24no—
440C4⋊C4—28no—
441C4⋊C4—38no—
442C4⋊C4—316no—
443C4⋊C4—68no—
444C4⋊C4—68no—
445C4⋊C4—68no—
446C4⋊C4—62no—
447C4⋊C4—68no—
448C4⋊C4—68no—
449C4⋊C4—68no—
450C4⋊C4—64no—
451C4⋊C4—64no—
452C4⋊C4—124no—
453C4⋊C4—124no—
454C4⋊C4—124no—
455C4⋊C4—122no—
456C22⋊C4—62no—
457C22⋊C4—68no—
458C22⋊C4—62no—
459C22⋊C4—68no—
460C22⋊C4—64no—
461C22⋊C4—64no—
462C22⋊C4—122no—
463C22⋊C4—124no—
464C42—22no—
465C42—22no—
466C42—68no—
467C42—64no—
468C42—62no—
469C42—62no—
470C42—124no—
471C42D3×D414yes—
472C22×C3C23×D411yescharacteristic
473C22×C3C22×#4111yescharacteristic
474C22×C3C2×2−1+411yescharacteristic
475C22×C3C2×2−1+412yes—
476C22×C3#58512yes—
477C2×D3—22no—
478C2×D3—24no—
479C2×D3—22no—
480C2×D3—24no—
481C2×D3—41no—
482C2×D3—41no—
483C2×D3—42no—
484C2×D3—41no—
485C2×D3—42no—
486C2×D3#54114yescomplemented
487C3×C4—24no—
488C3×C4—41no—
489C3×C4—44no—
490C3×C4—44no—
491C3×C4—41no—
492C3×C4—42no—
493C3×C4#54814yes—
494Dic3—22no—
495Dic3—28no—
496Dic3—41no—
497Dic3—42no—
498Dic3—42no—
499Dic3—44no—
500C23—32no—
501C23—34no—
502C23—61no—
503C23—61no—
504C23—62no—
505C23—61no—
506C23—61no—
507C23—61no—
508C23—62no—
509C23—62no—
510C23C24×D311yescharacteristic
511Q8—48no—
512D4—124no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—38no—
529C2×C4—32no—
530C2×C4—61no—
531C2×C4—61no—
532C2×C4—64no—
533C2×C4—64no—
534C2×C4—68no—
535C2×C4—62no—
536C2×C4—62no—
537C2×C4—61no—
538C2×C4—64no—
539C2×C4—61no—
540C2×C4—61no—
541C2×C4—62no—
542C2×C4—62no—
543C2×C4—64no—
544C2×C4—64no—
545C2×C4—62no—
546C2×C4—62no—
547C2×C4—128no—
548C2×C4—1216no—
549C2×C4—124no—
550C2×C4C2×D3×D414yes—
551C2×C4C2×D3×D412yes—
552C2×C4D3×#4112yes—
553C2×C4#101012yes—
554C2×C3C2×#54111yescharacteristic
555C2×C3C2×#54811yescharacteristic
556C2×C3#348912yes—
557C2×C3#351211yescharacteristic
558C2×C3#356312yes—
559D3—24no—
560D3—81no—
561C22—34no—
562C22—62no—
563C22—64no—
564C22—62no—
565C22—64no—
566C22—121no—
567C22—122no—
568C22—121no—
569C22—121no—
570C22—122no—
571C22C22×D3×D411yescharacteristic
572C22C2×D3×#4111yescharacteristic
573C22C2×#101011yescharacteristic
574C22D3×2−1+412yes—
575C22#596812yes—
576C4—24no—
577C4—44no—
578C4—44no—
579C4—41no—
580C4—41no—
581C4—42no—
582C4—62no—
583C4—68no—
584C4—121no—
585C4—122no—
586C4—122no—
587C4—124no—
588C4#567714yes—
589C3#3770511yescharacteristic, complemented, Sylow
590C2—64no—
591C2—241no—
592C2C2×#567711yescharacteristic
593C2#8635612yes—
594C2#8649711yescharacteristic
595C2D3×#54111yescharacteristic
596C2#8652012yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2642C2
prime radicalC22×D12C23
hypercenter#2642D3
hypocenterC3#37705
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2642

Complements

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

NH ≅ G/Ntype
1. C2×#5731C2semidirect
2. C2×D3×#125C2semidirect
3. C3×#2642C2semidirect
4. #83269C2semidirect
5. #83065C2semidirect
6. #78249C2semidirect
7. #78245C2semidirect
9. #78135C2semidirect
10. #76630C2semidirect
12. #76615C2semidirect
13. #76512C2semidirect
15. #73390C2semidirect
79. #2642D3semidirect
143. #870D4semidirect
486. C2×D3#541semidirect
589. C3#37705semidirect

Maximal subgroups

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

subgroupindex
C2×#57312
C2×D3×#1252
C3×#26422
#832692
#830652
#782492
#782452
#781452
#781352
#766302
#766212
#766152
#765122
#734142
#733902
#377053