Finite Group Databasebeta

← #10845937

Subgroups — #10845937

Order 768.

1154 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#460C211yesmaximal, characteristic, complemented
2C2×#5171C212yesmaximal, complemented
3#83188C211yesmaximal, characteristic
4#82500C211yesmaximal, characteristic, complemented
5C2×#4834C211yesmaximal, characteristic, complemented
6#72054C212yesmaximal, complemented
7#71764C211yesmaximal, characteristic, complemented
8C4×#835C211yesmaximal, characteristic, complemented
9#70454C211yesmaximal, characteristic, complemented
10#70202C212yesmaximal, complemented
11#70176C212yesmaximal, complemented
12#25387—31nomaximal, Sylow
13C22×C3×SD16C2212yes—
14C2×C3×#127C2211yescharacteristic
15C2×C3×C4×Q8C2211yescharacteristic
16C2×#946C2212yes—
17C2×#940—21no—
18C2×#938C2212yes—
19C2×#938C2212yes—
20C22×D3×C8C2212yes—
21C2×C4×D12C2211yescharacteristic
22C3×#460—24no—
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#102C2212yes—
25C2×#835C2211yescharacteristic
26C2×#835C2211yescharacteristic
27#5223—22no—
28#5221—21no—
29C2×#832C2212yes—
30C2×#832C2212yes—
31#5171—28no—
32C2×(C2×D3)⋊C8C2212yes—
33C2×#817C2212yes—
34#4994—22no—
35#4987C2212yes—
36#4939—21no—
37#4935C2211yescharacteristic
38C4×C4⋊Dic3C2211yescharacteristic
39#4928—21no—
40C2×C4×C3⋊C8C2211yescharacteristic
41#4834—24no—
42#4551C2212yes—
43#4529C2211yescharacteristic
44#4529C2211yescharacteristic
45C2×#465—32no—
46C2×#460D311yescharacteristic, complemented
47C2×C8×D4—31no—
48#2541—31no—
49#2538—31no—
50#2034—32no—
51#2000—32no—
52#1957—31no—
53#1948—31no—
54#1893—32no—
55C4×#103—31no—
56C22×C3×Q8C2311yescharacteristic
57C22×C3×D4—21no—
58C22×C3×D4—21no—
59C22×C3×D4C2312yes—
60C22×#67—22no—
61C22×#67—22no—
62C22×D12D411yescharacteristic
63C22×C4×D3C2312yes—
64C2×C3×SD16—24no—
65C2×C3×SD16—24no—
66C2×C3×SD16—42no—
67C22×C3×C8C2312yes—
68C3×#127—22no—
69C3×#127—22no—
70C3×C4×Q8—22no—
71C3×C4×Q8—22no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C42—21no—
75C2×C3×C42C2311yescharacteristic
76C2×#217—22no—
77C2×#217—22no—
78#946—24no—
79#946—42no—
80#940—44no—
81#938—24no—
82#938—24no—
83#938—42no—
84#938—42no—
85C2×(C2×D3)⋊C4—22no—
86C2×(C2×D3)⋊C4—22no—
87C2×(C2×D3)⋊C4D412yes—
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3—21no—
90C2×C4⋊Dic3D411yescharacteristic
91C2×C4⋊Dic3C2312yes—
92C2×C4⋊Dic3C2311yescharacteristic
93C2×C4⋊Dic3C2311yescharacteristic
94C2×Dic3⋊C4—22no—
95C2×C4×Dic3—21no—
96C22×C3⋊C8C2311yescharacteristic
97C22×C3⋊C8C2311yescharacteristic
98C2×D3×C8—24no—
99C2×D3×C8—24no—
100C2×D3×C8—42no—
101C4×D12—22no—
102C4×D12—22no—
103C3×C4⋊C8D414yescomplemented
104C3×#102—24no—
105#835—22no—
106#835D414yescomplemented
107#832—24no—
108#832—42no—
109C3⋊#95—21no—
110C3⋊#95—22no—
111(C2×D3)⋊C8—24no—
112#817—24no—
113C4×C3⋊C8—22no—
114C22×SD16C2×D312yes—
115C23×C8—32no—
116C2×#127—61no—
117C2×#127C2×D311yescharacteristic
118C2×C22⋊Q8—32no—
119C2×#121—32no—
120C2×#121—32no—
121C2×C4×Q8C2×D311yescharacteristic
122C2×C4×D4—31no—
123#465—68no—
124#460—24no—
125C8×D4—64no—
126C2×#106—32no—
127C2×C4⋊C8C2×D311yescharacteristic
128C2×#103—31no—
129C2×#103—31no—
130C2×#102—32no—
131C2×#102—32no—
132C2×#102C2×D312yes—
133C2×C22⋊C8—32no—
134C2×C4×C8—31no—
135#396—62no—
136#392—62no—
137#390—61no—
138#390—61no—
139#389—61no—
140#384—31no—
141#380—32no—
142C4×C4⋊C4—31no—
143#340—32no—
144#340—31no—
145#340—31no—
146C24×C3—22no—
147C23×D3C2×D412yes—
148C2×C3×Q8—21no—
149C2×C3×Q8—21no—
150C2×C3×Q8—21no—
151C2×C3×Q8—21no—
152C2×C3×Q8—21no—
153C2×C3×Q8C2×D412yes—
154C2×C3×D4—22no—
155C2×C3×D4—22no—
156C2×C3×D4—22no—
157C2×C3×D4—22no—
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4—42no—
161C2×C3×D4—42no—
162C2×C3×D4—42no—
163C2×C3×D4—42no—
164C2×C3×D4—81no—
165C2×C3×D4—81no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2411yescharacteristic
171C2×#67—44no—
172C2×#67—44no—
173C2×#67—44no—
174C2×#67—44no—
175C2×#67—82no—
176C2×#67—82no—
177C22×Dic3—21no—
178C22×Dic3—21no—
179C22×Dic3—22no—
180C22×Dic3C2×D412yes—
181C2×D12—22no—
182C2×D12—22no—
183C2×D12SD1614yescomplemented
184C2×C4×D3—22no—
185C2×C4×D3—22no—
186C2×C4×D3—22no—
187C2×C4×D3—22no—
188C2×C4×D3—22no—
189C2×C4×D3—22no—
190C3×SD16—48no—
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C2×C3×C8C2×D414yes—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—42no—
197C3×C42—42no—
198C3×C42C2×D412yes—
199C3×C42#4112yes—
200#217—44no—
201#217—44no—
202(C2×D3)⋊C4—24no—
203(C2×D3)⋊C4—44no—
204(C2×D3)⋊C4—82no—
205C4⋊Dic3—22no—
206C4⋊Dic3—21no—
207C4⋊Dic3—21no—
208C4⋊Dic3—22no—
209C4⋊Dic3—22no—
210C4⋊Dic3—42no—
211C4⋊Dic3—81no—
212C4⋊Dic3C2×D412yes—
213C4⋊Dic3#4112yes—
214Dic3⋊C4—44no—
215C4×Dic3—42no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8—21no—
218C2×C3⋊C8—21no—
219C2×C3⋊C8—21no—
220C2×C3⋊C8—21no—
221C2×C3⋊C8C2×D412yes—
222D3×C8—48no—
223C22×Q8C22×D311yescharacteristic
224C22×D4—21no—
225C22×D4—21no—
226C22×D4—31no—
227C22×D4—62no—
228C22×D4—62no—
229C22×D4C22×D312yes—
230C23×C4—32no—
231C2×SD16—24no—
232C2×SD16—24no—
233C2×SD16—42no—
234C22×C8—31no—
235C22×C8—31no—
236C22×C8—64no—
237C22×C8—64no—
238C22×C8—122no—
239C22×C8C22×D312yes—
240#127—22no—
241#127—22no—
242#127—124no—
243C22⋊Q8—64no—
244C22⋊Q8—122no—
245#121—64no—
246#121—64no—
247#121—122no—
248#121—122no—
249C4×Q8—22no—
250C4×Q8—22no—
251C4×D4—62no—
252C4×D4—62no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—31no—
255C2×C4⋊C4—31no—
256C2×C4⋊C4—31no—
257C2×C4⋊C4—32no—
258C2×C4⋊C4—62no—
259C2×C4⋊C4—61no—
260C2×C4⋊C4—61no—
261C2×C4⋊C4C22×D311yescharacteristic
262C2×C22⋊C4—32no—
263C2×C22⋊C4—62no—
264C2×C22⋊C4—62no—
265C2×C22⋊C4—62no—
266C2×C22⋊C4—62no—
267C2×C42—21no—
268C2×C42—61no—
269C2×C42C22×D311yescharacteristic
270#106—64no—
271C4⋊C8#6714yescomplemented
272#103—34no—
273#103—62no—
274#102—24no—
275#102—64no—
276#102—122no—
277C22⋊C8—64no—
278C4×C8—62no—
279#95—62no—
280#95—61no—
281C23×C3—42no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3—42no—
285C23×C3—42no—
286C23×C3—42no—
287C23×C3—42no—
288C23×C3C22×D411yescharacteristic
289C22×D3—22no—
290C22×D3—22no—
291C22×D3—22no—
292C22×D3—22no—
293C22×D3—22no—
294C22×D3—22no—
295C22×D3C2×SD1614yes—
296C3×Q8—22no—
297C3×Q8—22no—
298C3×Q8—41no—
299C3×Q8—41no—
300C3×D4—42no—
301C3×D4—42no—
302C3×D4—42no—
303C3×D4—42no—
304C3×D4—84no—
305C3×D4—84no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—22no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4#12112yes—
315C2×C3×C4#12112yes—
316C2×C3×C4C2×SD1612yes—
317C2×C3×C4#13512yes—
318C2×C3×C4C22×D411yescharacteristic
319C2×C3×C4C22×D411yescharacteristic
320C2×C3×C4C2×#4111yescharacteristic
321C2×C3×C4C2×#4111yescharacteristic
322C2×C3×C4H(5,𝔽2)11yescharacteristic
323C2×C3×C4H(5,𝔽2)11yescharacteristic
324#67—88no—
325#67—88no—
326C2×Dic3—22no—
327C2×Dic3—22no—
328C2×Dic3—22no—
329C2×Dic3—22no—
330C2×Dic3—22no—
331C2×Dic3—41no—
332C2×Dic3—41no—
333C2×Dic3—41no—
334C2×Dic3—41no—
335C2×Dic3—42no—
336C2×Dic3—42no—
337C2×Dic3—42no—
338D12—28no—
339C4×D3—42no—
340C4×D3—42no—
341C4×D3—42no—
342C4×D3—42no—
343C3×C8—24no—
344C3⋊C8—22no—
345C3⋊C8—41no—
346C24—22no—
347C24—32no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8D3×D412yes—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—34no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—62no—
366C2×D4—62no—
367C2×D4—81no—
368C2×D4—81no—
369C2×D4—124no—
370C2×D4—124no—
371C2×D4—124no—
372C2×D4—124no—
373C2×D4—242no—
374C2×D4—242no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—32no—
378C22×C4—61no—
379C22×C4—61no—
380C22×C4—62no—
381C22×C4—62no—
382C22×C4—62no—
383C22×C4—62no—
384C22×C4—62no—
385C22×C4—62no—
386C22×C4—62no—
387C22×C4D3×D411yescharacteristic
388C22×C4D3×D411yescharacteristic
389C22×C4C23×D311yescharacteristic
390SD16—48no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—32no—
394C2×C8—61no—
395C2×C8—61no—
396C2×C8—61no—
397C2×C8—61no—
398C2×C8—61no—
399C2×C8—128no—
400C2×C8C2×#6714yes—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—32no—
404C4⋊C4—32no—
405C4⋊C4—42no—
406C4⋊C4—62no—
407C4⋊C4—61no—
408C4⋊C4—61no—
409C4⋊C4—62no—
410C4⋊C4—62no—
411C4⋊C4—124no—
412C4⋊C4—122no—
413C4⋊C4—241no—
414C22⋊C4—64no—
415C22⋊C4—124no—
416C22⋊C4—124no—
417C22⋊C4—124no—
418C22⋊C4—242no—
419C42—42no—
420C42—122no—
421C42#23712yes—
422C42C2×#6712yes—
423C22×C3—82no—
424C22×C3—82no—
425C22×C3—82no—
426C22×C3—82no—
427C22×C3—82no—
428C22×C3—82no—
429C22×C3—82no—
430C22×C3C2×#12111yescharacteristic
431C22×C3D4211yescharacteristic
432C22×C3#54611yescharacteristic
433C22×C3C22×SD1611yescharacteristic
434C22×C3C2×#13511yescharacteristic
435C22×C3#57611yescharacteristic
436C22×C3#57711yescharacteristic
437C2×D3—24no—
438C2×D3—24no—
439C2×D3—24no—
440C2×D3—42no—
441C2×D3—42no—
442C2×D3—42no—
443C2×D3—42no—
444C3×C4—22no—
445C3×C4—42no—
446C3×C4—81no—
447C3×C4#46514yescomplemented
448C3×C4#53211yescharacteristic
449C3×C4#54011yescharacteristic
450C3×C4D4211yescharacteristic
451C3×C4#55011yescharacteristic
452Dic3—42no—
453Dic3—42no—
454Dic3—42no—
455Dic3—82no—
456C23—34no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—62no—
465C23—62no—
466C23—62no—
467C23—62no—
468C23—62no—
469C23—62no—
470C23C2×D3×D411yescharacteristic
471Q8—22no—
472Q8—22no—
473Q8—41no—
474Q8—41no—
475D4—42no—
476D4—42no—
477D4—42no—
478D4—42no—
479D4—68no—
480D4—84no—
481D4—84no—
482D4—248no—
483D4—248no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—62no—
493C2×C4—62no—
494C2×C4—62no—
495C2×C4—62no—
496C2×C4—62no—
497C2×C4—121no—
498C2×C4—121no—
499C2×C4—121no—
500C2×C4—121no—
501C2×C4—122no—
502C2×C4—122no—
503C2×C4—122no—
504C2×C4—122no—
505C2×C4—122no—
506C2×C4—122no—
507C2×C4—122no—
508C2×C4D3×SD1612yes—
509C2×C4#91612yes—
510C2×C4#93812yes—
511C2×C4#93812yes—
512C2×C4C2×D3×D411yescharacteristic
513C2×C4C2×#23711yescharacteristic
514C2×C4#100411yescharacteristic
515C2×C4D3×#4111yescharacteristic
516C2×C4#100911yescharacteristic
517C2×C4C22×#6711yescharacteristic
518C8—24no—
519C8—62no—
520C8—121no—
521C2×C3—162no—
522C2×C3#242811yescharacteristic
523C2×C3C2×#46511yescharacteristic
524C2×C3#306211yescharacteristic
525C2×C3D4×SD1611yescharacteristic
526C2×C3#328711yescharacteristic
527C2×C3#329511yescharacteristic
528C2×C3#330411yescharacteristic
529D3—44no—
530C22—64no—
531C22—64no—
532C22—64no—
533C22—82no—
534C22—82no—
535C22—82no—
536C22—82no—
537C22—82no—
538C22—82no—
539C22—82no—
540C22—122no—
541C22—122no—
542C22—122no—
543C22—122no—
544C22#560911yescharacteristic
545C22#561211yescharacteristic
546C22C2×D3×SD1611yescharacteristic
547C22C2×#91611yescharacteristic
548C22#577411yescharacteristic
549C22#577511yescharacteristic
550C22C2×#93811yescharacteristic
551C4—22no—
552C4—42no—
553C4—81no—
554C4—122no—
555C4—122no—
556C4—122no—
557C4—242no—
558C4#517114yescomplemented
559C4#559611yescharacteristic
560C4#561911yescharacteristic
561C4D4×#6711yescharacteristic
562C4#583411yescharacteristic
563C3#2538711yescharacteristic, complemented, Sylow
564C2—124no—
565C2—162no—
566C2#8039411yescharacteristic
567C2#8039511yescharacteristic
568C2#8039611yescharacteristic
569C2#8040611yescharacteristic
570C2#8264811yescharacteristic
571C2C2×#517111yescharacteristic
572C2#8395111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#460C2
socleC23×C3C22×D4
prime radical#83188C2
hypercenterC2×#460D3
hypocenterC3#25387
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#460

Complements

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

NH ≅ G/Ntype
1. C2×C3×#460C2semidirect
2. C2×#5171C2semidirect
4. #82500C2semidirect
5. C2×#4834C2semidirect
6. #72054C2semidirect
7. #71764C2semidirect
8. C4×#835C2semidirect
9. #70454C2semidirect
10. #70202C2semidirect
11. #70176C2semidirect
46. C2×#460D3semidirect
103. C3×C4⋊C8D4semidirect
106. #835D4semidirect
183. C2×D12SD16semidirect
271. C4⋊C8#67semidirect
447. C3×C4#465semidirect
558. C4#5171semidirect
563. C3#25387semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4602
C2×#51712
#831882
#825002
C2×#48342
#720542
#717642
C4×#8352
#704542
#702022
#701762
#253873