Finite Group Databasebeta

← #10972886

Subgroups — #10972886

Order 768.

2230 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5728C211yesmaximal, characteristic
2C2×C4×D3×Q8C211yesmaximal, characteristic
3C3×#2546C211yesmaximal, characteristic, complemented
4#83239C211yesmaximal, characteristic, complemented
5#83232C212yesmaximal
6#83227C212yesmaximal, complemented
7#82479C211yesmaximal, characteristic, complemented
8#77954C214yesmaximal
9#77823C214yesmaximal, complemented
10#76520C212yesmaximal
11#76485C212yesmaximal, complemented
12#76287C214yesmaximal
13#76076C214yesmaximal, complemented
14#73752C211yesmaximal, characteristic
15#73722C211yesmaximal, characteristic, complemented
16#35248—31nomaximal, Sylow
17C22×D3×Q8C2211yescharacteristic
18C2×C3×C4⋊Q8C2212yes—
19C2×C3×C4⋊Q8C2211yescharacteristic
20C2×C3×C4×Q8C2211yescharacteristic
21C2×#946C2212yes—
22C2×Q8×Dic3C2211yescharacteristic
23C2×Q8×Dic3C2212yes—
24C2×#944C2212yes—
25#5728—28no—
26C4×D3×Q8—28no—
27C2×#897C2212yes—
28C2×#892C2212yes—
29C2×D3×C4⋊C4C2212yes—
30C2×D3×C4⋊C4C2211yescharacteristic
31C2×#890C2212yes—
32C2×#887C2212yes—
33C2×#887C2211yescharacteristic
34C2×#887C2212yes—
35C2×C42×D3C2212yes—
36C2×C42×D3C2211yescharacteristic
37C2×#869C2211yescharacteristic
38C2×C4×Dic6C2211yescharacteristic
39C2×C4×Dic6C2212yes—
40C3×#398C2214yes—
41C3×#391C2212yes—
42C3×#389C2214yes—
43C3×C4×C4⋊C4C2211yescharacteristic
44#5229C2212yes—
45#5229C2212yes—
46#4989C2218yes—
47#4987C2212yes—
48#4986C2214yes—
49#4983C2214yes—
50#4980C2218yes—
51#4980C2218yes—
52#4979C2218yes—
53#4977C2212yes—
54#4977C2212yes—
55#4973C2214yes—
56#4972C2214yes—
57C4×(C2×D3)⋊C4C2212yes—
58#4936C2212yes—
59C4×C4⋊Dic3C2211yescharacteristic
60#4932C2212yes—
61#4929C2212yes—
62#4929C2214yes—
63#4664C2212yes—
64#4664C2218yes—
65#4659C2214yes—
66#4658C2218yes—
67#4655C2218yes—
68#4652C2218yes—
69#4651C2212yes—
70C2×#533—31no—
71C22×C4×Q8—31no—
72#2855—34no—
73#2634—34no—
74#2581—34no—
75#2568—31no—
76#2568—31no—
77#2546—31no—
78#2546D311yescharacteristic, complemented
79#2503—32no—
80#2484—34no—
81#2434—32no—
82#2327—32no—
83#2327—32no—
84#2283—31no—
85C22×C3×Q8C2312yes—
86C22×C3×Q8C2311yescharacteristic
87C2×D3×Q8—24no—
88C2×D3×Q8—24no—
89C2×D3×Q8—24no—
90C22×C4×D3C2312yes—
91C22×C4×D3C2312yes—
92C22×C4×D3C2311yescharacteristic
93C22×C4×D3C2311yescharacteristic
94C22×C4×D3C2311yescharacteristic
95C22×Dic6C2312yes—
96C22×Dic6C2312yes—
97C22×Dic6C2311yescharacteristic
98C3×C4⋊Q8—28no—
99C3×C4⋊Q8—24no—
100C3×C4×Q8—24no—
101C2×C3×C4⋊C4C2312yes—
102C2×C3×C4⋊C4C2312yes—
103C2×C3×C4⋊C4C2311yescharacteristic
104C2×C3×C4⋊C4C2318yes—
105C2×C3×C4⋊C4C2312yes—
106C2×C3×C4⋊C4C2312yes—
107C2×C3×C42C2312yes—
108C2×C3×C42C2312yes—
109C2×C3×C42C2311yescharacteristic
110#946—28no—
111Q8×Dic3—28no—
112Q8×Dic3—24no—
113#944—28no—
114C2×(C2×D3)⋊C4C2314yes—
115C2×(C2×D3)⋊C4C2314yes—
116C2×C4⋊Dic3C2314yes—
117C2×C4⋊Dic3C2312yes—
118C2×C4⋊Dic3C2312yes—
119C2×C4⋊Dic3C2312yes—
120C2×C4⋊Dic3C2311yescharacteristic
121C2×C4⋊Dic3C2312yes—
122C2×Dic3⋊C4C2312yes—
123C2×Dic3⋊C4C2318yes—
124C2×Dic3⋊C4C2312yes—
125C2×Dic3⋊C4C2314yes—
126C2×Dic3⋊C4C2312yes—
127C2×C4×Dic3C2314yes—
128C2×C4×Dic3C2312yes—
129C2×C4×Dic3C2312yes—
130C2×C4×Dic3C2312yes—
131C2×C4×Dic3C2312yes—
132C2×C4×Dic3C2311yescharacteristic
133C2×C4×Dic3C2311yescharacteristic
134C2×C4×Dic3C2311yescharacteristic
135#897—28no—
136#892—28no—
137D3×C4⋊C4—28no—
138D3×C4⋊C4—24no—
139#890—28no—
140#887—28no—
141#887—24no—
142#887—28no—
143C42×D3—28no—
144C42×D3D418yes—
145#869—24no—
146C4×Dic6—24no—
147C4×Dic6—28no—
148C3×#95C2312yes—
149C3×#95C2318yes—
150C3⋊#95C2314yes—
151C3⋊#95C2314yes—
152C3⋊#95C2312yes—
153C3⋊#95C2314yes—
154C3⋊#95C2318yes—
155C3⋊#95C2318yes—
156C3⋊#95C2318yes—
157C3⋊#95C2318yes—
158C3⋊#95C2318yes—
159C23×Q8—31no—
160#533—68no—
161C2×C4⋊Q8—32no—
162C2×C4⋊Q8—31no—
163C2×C4⋊Q8C2×D312yes—
164C2×C4⋊Q8C2×D311yescharacteristic
165C2×#125—32no—
166C2×C22⋊Q8—32no—
167C2×C22⋊Q8—32no—
168C2×C4×Q8—31no—
169C2×C4×Q8—32no—
170C2×C4×Q8—32no—
171C2×C4×Q8—31no—
172C2×C4×Q8—32no—
173C2×C4×Q8—31no—
174C2×C4×Q8—32no—
175C2×C4×Q8—68no—
176C2×C4×Q8C2×D311yescharacteristic
177C2×#117—32no—
178C22×C4⋊C4—32no—
179C22×C4⋊C4—31no—
180C22×C42—31no—
181C22×C42—32no—
182#400—34no—
183#400—34no—
184#400—38no—
185#398—34no—
186#398C2×D314yes—
187#395—38no—
188#391—32no—
189#391—32no—
190#391—34no—
191#391—34no—
192#391—32no—
193#391C2×D312yes—
194#389—34no—
195#389C2×D314yes—
196#387—38no—
197#387—38no—
198#387—38no—
199#387—38no—
200#385—38no—
201#383—32no—
202#382—32no—
203#382—32no—
204#382—32no—
205#381—38no—
206#381—32no—
207#380—32no—
208C4×C4⋊C4—31no—
209C4×C4⋊C4C2×D311yescharacteristic
210C4×C22⋊C4—32no—
211#376—32no—
212C23×D3C2411yescharacteristic
213C2×C3×Q8—22no—
214C2×C3×Q8—24no—
215C2×C3×Q8—22no—
216C2×C3×Q8—24no—
217C2×C3×Q8—24no—
218C2×C3×Q8#4114yes—
219C22×C3×C4C2414yes—
220C22×C3×C4C2412yes—
221C22×C3×C4C2414yes—
222C22×C3×C4C2411yescharacteristic
223C22×C3×C4C2412yes—
224C22×C3×C4C2411yescharacteristic
225C22×C3×C4C2411yescharacteristic
226C22×Dic3C2414yes—
227C22×Dic3C2412yes—
228C22×Dic3C2414yes—
229C22×Dic3C2411yescharacteristic
230C22×Dic3C2411yescharacteristic
231C22×Dic3C2411yescharacteristic
232C22×Dic3C2412yes—
233D3×Q8—232no—
234C2×C4×D3—24no—
235C2×C4×D3—24no—
236C2×C4×D3—24no—
237C2×C4×D3—22no—
238C2×C4×D3—22no—
239C2×C4×D3—24no—
240C2×C4×D3—24no—
241C2×C4×D3—24no—
242C2×C4×D3—28no—
243C2×C4×D3C2×D414yes—
244C2×C4×D3C2×D414yes—
245C2×C4×D3C2×D414yes—
246C2×Dic6—22no—
247C2×Dic6—24no—
248C2×Dic6—22no—
249C2×Dic6—24no—
250C2×Dic6—24no—
251C2×Dic6—24no—
252C2×Dic6—28no—
253C2×Dic6#4114yes—
254C3×C4⋊C4—24no—
255C3×C4⋊C4—22no—
256C3×C4⋊C4—24no—
257C3×C4⋊C4—24no—
258C3×C4⋊C4—216no—
259C3×C4⋊C4—24no—
260C3×C42—24no—
261C3×C42—24no—
262C3×C42C2×D414yes—
263(C2×D3)⋊C4—28no—
264(C2×D3)⋊C4—28no—
265C4⋊Dic3—22no—
266C4⋊Dic3—24no—
267C4⋊Dic3—24no—
268C4⋊Dic3—24no—
269C4⋊Dic3—24no—
270C4⋊Dic3—28no—
271Dic3⋊C4—24no—
272Dic3⋊C4—24no—
273Dic3⋊C4—24no—
274Dic3⋊C4—28no—
275Dic3⋊C4—216no—
276C4×Dic3—24no—
277C4×Dic3—24no—
278C4×Dic3—24no—
279C4×Dic3—24no—
280C4×Dic3—28no—
281C4×Dic3C2×D414yes—
282C4×Dic3C2×D414yes—
283C4×Dic3C2×D414yes—
284C22×Q8—31no—
285C22×Q8—32no—
286C22×Q8—32no—
287C22×Q8—64no—
288C22×Q8—64no—
289C22×Q8—64no—
290C22×Q8C22×D312yes—
291C22×Q8C22×D311yescharacteristic
292C23×C4—31no—
293C23×C4—31no—
294C23×C4—31no—
295C23×C4—32no—
296C23×C4—32no—
297C4⋊Q8—28no—
298C4⋊Q8—24no—
299C4⋊Q8—68no—
300C4⋊Q8—64no—
301#125—68no—
302C22⋊Q8—68no—
303C22⋊Q8—68no—
304C4×Q8—24no—
305C4×Q8—68no—
306C4×Q8—64no—
307C4×Q8—68no—
308C4×Q8—68no—
309C4×Q8—64no—
310C4×Q8—68no—
311C4×Q8—64no—
312#117—68no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—32no—
315C2×C4⋊C4—32no—
316C2×C4⋊C4—32no—
317C2×C4⋊C4—32no—
318C2×C4⋊C4—32no—
319C2×C4⋊C4—34no—
320C2×C4⋊C4—34no—
321C2×C4⋊C4—32no—
322C2×C4⋊C4—38no—
323C2×C4⋊C4—32no—
324C2×C4⋊C4—64no—
325C2×C4⋊C4—68no—
326C2×C4⋊C4C22×D311yescharacteristic
327C2×C4⋊C4C22×D312yes—
328C2×C4⋊C4C22×D312yes—
329C2×C4⋊C4C22×D312yes—
330C2×C4⋊C4C22×D318yes—
331C2×C4⋊C4C22×D312yes—
332C2×C22⋊C4—34no—
333C2×C22⋊C4—34no—
334C2×C42—31no—
335C2×C42—31no—
336C2×C42—31no—
337C2×C42—38no—
338C2×C42—34no—
339C2×C42—32no—
340C2×C42—32no—
341C2×C42—32no—
342C2×C42—32no—
343C2×C42—68no—
344C2×C42C22×D312yes—
345C2×C42C22×D312yes—
346C2×C42C22×D311yescharacteristic
347#95—34no—
348#95—34no—
349#95—34no—
350#95—32no—
351#95—38no—
352#95—38no—
353#95—38no—
354#95—38no—
355#95—38no—
356#95C22×D312yes—
357#95C22×D318yes—
358C23×C3C2511yescharacteristic
359C22×D3—21no—
360C22×D3—21no—
361C22×D3—22no—
362C22×D3C22×D412yes—
363C22×D32−1+414yes—
364C3×Q8—28no—
365C3×Q8—48no—
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—24no—
373C2×C3×C4—24no—
374C2×C3×C4—24no—
375C2×C3×C4—24no—
376C2×C3×C4—24no—
377C2×C3×C4C22×D412yes—
378C2×C3×C4C22×D412yes—
379C2×C3×C4C22×D412yes—
380C2×C3×C4C2×#4114yes—
381C2×C3×C4C2×#4112yes—
382C2×C3×C4C2×#4112yes—
383C2×C3×C42−1+414yes—
384C2×Dic3—22no—
385C2×Dic3—21no—
386C2×Dic3—22no—
387C2×Dic3—21no—
388C2×Dic3—22no—
389C2×Dic3—22no—
390C2×Dic3—22no—
391C2×Dic3—24no—
392C2×Dic3—24no—
393C2×Dic3—24no—
394C2×Dic3—24no—
395C2×Dic3—28no—
396C2×Dic3—24no—
397C2×Dic3C22×D412yes—
398C2×Dic3C22×D412yes—
399C2×Dic3C22×D412yes—
400C2×Dic3C2×#4114yes—
401C4×D3—216no—
402C4×D3—28no—
403C4×D3—28no—
404C4×D3—28no—
405C4×D3—48no—
406Dic6—216no—
407Dic6—28no—
408Dic6—48no—
409C24—31no—
410C2×Q8—22no—
411C2×Q8—24no—
412C2×Q8—24no—
413C2×Q8—22no—
414C2×Q8—24no—
415C2×Q8—34no—
416C2×Q8—64no—
417C2×Q8—64no—
418C2×Q8—632no—
419C2×Q8—62no—
420C2×Q8—64no—
421C2×Q8—64no—
422C2×Q8—62no—
423C2×Q8—68no—
424C2×Q8#23714yes—
425C22×C4—31no—
426C22×C4—31no—
427C22×C4—31no—
428C22×C4—32no—
429C22×C4—34no—
430C22×C4—34no—
431C22×C4—34no—
432C22×C4—34no—
433C22×C4—32no—
434C22×C4—34no—
435C22×C4—64no—
436C22×C4—64no—
437C22×C4—64no—
438C22×C4—64no—
439C22×C4—68no—
440C22×C4—64no—
441C22×C4—62no—
442C22×C4—64no—
443C22×C4—62no—
444C22×C4C23×D314yes—
445C22×C4C23×D312yes—
446C22×C4C23×D314yes—
447C22×C4C23×D311yescharacteristic
448C22×C4C23×D312yes—
449C22×C4C23×D311yescharacteristic
450C22×C4C23×D311yescharacteristic
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—216no—
456C4⋊C4—24no—
457C4⋊C4—64no—
458C4⋊C4—64no—
459C4⋊C4—64no—
460C4⋊C4—68no—
461C4⋊C4—68no—
462C4⋊C4—616no—
463C4⋊C4—64no—
464C4⋊C4—62no—
465C4⋊C4—64no—
466C4⋊C4—64no—
467C4⋊C4—64no—
468C22⋊C4—68no—
469C22⋊C4—68no—
470C42—24no—
471C42—24no—
472C42—34no—
473C42—34no—
474C42—34no—
475C42—64no—
476C42—64no—
477C42—68no—
478C42—64no—
479C42—64no—
480C42D3×D414yes—
481C22×C3C23×D411yescharacteristic
482C22×C3C22×#4111yescharacteristic
483C22×C3C22×#4111yescharacteristic
484C22×C3C2×2−1+412yes—
485C22×C3C2×2−1+412yes—
486C2×D3—22no—
487C2×D3—22no—
488C2×D3—24no—
489C2×D3—24no—
490C2×D3#53614yes—
491C3×C4—24no—
492C3×C4—22no—
493C3×C4—44no—
494C3×C4—42no—
495C3×C4—44no—
496C3×C4#52514yes—
497C3×C4#53614yes—
498Dic3—22no—
499Dic3—24no—
500Dic3—22no—
501Dic3—22no—
502Dic3—44no—
503Dic3—42no—
504Dic3—44no—
505C23—32no—
506C23—34no—
507C23—61no—
508C23—62no—
509C23—61no—
510C23C24×D311yescharacteristic
511Q8—28no—
512Q8—48no—
513Q8—616no—
514Q8—68no—
515Q8—128no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—32no—
528C2×C4—32no—
529C2×C4—32no—
530C2×C4—34no—
531C2×C4—68no—
532C2×C4—616no—
533C2×C4—62no—
534C2×C4—61no—
535C2×C4—62no—
536C2×C4—62no—
537C2×C4—61no—
538C2×C4—62no—
539C2×C4—68no—
540C2×C4—68no—
541C2×C4—62no—
542C2×C4—64no—
543C2×C4—64no—
544C2×C4—68no—
545C2×C4—64no—
546C2×C4—64no—
547C2×C4—64no—
548C2×C4—128no—
549C2×C4C2×D3×D412yes—
550C2×C4C2×D3×D412yes—
551C2×C4C2×D3×D412yes—
552C2×C4C2×#23714yes—
553C2×C4C2×#23712yes—
554C2×C4#100714yes—
555C2×C4D3×#4112yes—
556C2×C3C2×#52511yescharacteristic
557C2×C3C2×#53611yescharacteristic
558C2×C3C2×#53611yescharacteristic
559C2×C3#352312yes—
560C2×C3#352312yes—
561D3—24no—
562C22—34no—
563C22—62no—
564C22—62no—
565C22—64no—
566C22—64no—
567C22C22×D3×D411yescharacteristic
568C22C22×#23711yescharacteristic
569C22C2×#100712yes—
570C22C2×D3×#4111yescharacteristic
571C22D3×2−1+412yes—
572C4—22no—
573C4—24no—
574C4—42no—
575C4—44no—
576C4—44no—
577C4—62no—
578C4—62no—
579C4—64no—
580C4—62no—
581C4—122no—
582C4—124no—
583C4—124no—
584C4#560614yes—
585C4#572414yes—
586C3#3524811yescharacteristic, complemented, Sylow
587C2—64no—
588C2C2×#560611yescharacteristic
589C2C2×#572411yescharacteristic
590C2D3×#53611yescharacteristic
591C2#8675312yes—
592C2#8702512yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2546C2
prime radicalC23×D3C24
hypercenter#2546D3
hypocenterC3#35248
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2546

Complements

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

NH ≅ G/Ntype
3. C3×#2546C2semidirect
4. #83239C2semidirect
6. #83227C2semidirect
7. #82479C2semidirect
9. #77823C2semidirect
11. #76485C2semidirect
13. #76076C2semidirect
15. #73722C2semidirect
78. #2546D3semidirect
586. C3#35248semidirect

Maximal subgroups

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

subgroupindex
C2×#57282
C2×C4×D3×Q82
C3×#25462
#832392
#832322
#832272
#824792
#779542
#778232
#765202
#764852
#762872
#760762
#737522
#737222
#352483