Finite Group Databasebeta

← #10944111

Subgroups — #10944111

Order 768.

1406 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5523C211yesmaximal, characteristic
2C3×#2621C211yesmaximal, characteristic, complemented
3C2×#5090C211yesmaximal, characteristic, complemented
4C2×#5083C211yesmaximal, characteristic, complemented
5C2×#5047C212yesmaximal
6#71692C212yesmaximal, complemented
7#71583C211yesmaximal, characteristic, complemented
8#69690C212yesmaximal, complemented
9#69687C212yesmaximal, complemented
10#69559C211yesmaximal, characteristic, complemented
11#69556C211yesmaximal, characteristic, complemented
12#32916—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C22⋊Q8C2212yes—
16C2×C3×#117C2211yescharacteristic
17C22×#216C2212yes—
18C22×#215C2212yes—
19C22×C4⋊Dic3C2211yescharacteristic
20#5523—24no—
21#5523—24no—
22C2×#892—21no—
23C2×D3×C4⋊C4—21no—
24C2×#67⋊C4—21no—
25C2×#67⋊C4—21no—
26C2×C4×D12C2211yescharacteristic
27C2×C4×D12C2211yescharacteristic
28C2×C4×Dic6C2211yescharacteristic
29C2×C4×Dic6C2211yescharacteristic
30C3×#398—21no—
31C3×#398—21no—
32C3×#397—21no—
33C3×#397—21no—
34C3×#391C2212yes—
35C3×#382C2211yescharacteristic
36C2×#835C2212yes—
37C2×#835C2212yes—
38C2×#830C2212yes—
39#5090—24no—
40#5083—24no—
41#5047—28no—
42C2×#810C2212yes—
43C2×#809C2212yes—
44C2×C4⋊(C3⋊C8)C2211yescharacteristic
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46#4529C2212yes—
47#4529C2212yes—
48#4524C2212yes—
49C2×#520—31no—
50C2×#462—31no—
51C2×#461—31no—
52C2×#452—32no—
53#2621D311yescharacteristic, complemented
54#1878—32no—
55#1877—32no—
56#1872—32no—
57#1805—31no—
58#1804—31no—
59#1798—31no—
60C22×C3×Q8C2312yes—
61C23×C3×C4C2311yescharacteristic
62C22×#67—21no—
63C23×Dic3—21no—
64C2×#235—22no—
65C2×#235—22no—
66C2×#235—22no—
67C2×#235—22no—
68C2×#235—22no—
69C2×#235D414yes—
70C22×D12C2311yescharacteristic
71C22×C4×D3—21no—
72C22×C4×D3—21no—
73C22×C4×D3—21no—
74C22×Dic6C2311yescharacteristic
75C3×C4⋊Q8—22no—
76C3×C4⋊Q8—22no—
77C3×C22⋊Q8—24no—
78C3×C22⋊Q8—24no—
79C3×#117—22no—
80C3×#117—22no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—22no—
84C2×C3×C4⋊C4D411yescharacteristic
85C2×C3×C4⋊C4D411yescharacteristic
86C2×C3×C4⋊C4C2312yes—
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4D411yescharacteristic
89C2×C3×C22⋊C4D411yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C2×#216—24no—
93C2×#216—24no—
94C2×#216—24no—
95C2×#215—24no—
96C2×#215—24no—
97C2×#215—24no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3—22no—
102C2×C4⋊Dic3—22no—
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3—21no—
109C22×C3⋊C8C2312yes—
110C22×C3⋊C8C2312yes—
111#892—28no—
112D3×C4⋊C4—28no—
113#67⋊C4—28no—
114#67⋊C4—28no—
115C4×D12—22no—
116C4×D12—22no—
117C4×D12—22no—
118C4×D12D414yes—
119C4×Dic6—22no—
120C4×Dic6—22no—
121C4×Dic6—22no—
122C4×Dic6D414yes—
123C3×#95—21no—
124C3×#95—21no—
125C3×#95—22no—
126C3×#95—22no—
127#835—24no—
128#835—24no—
129#830—24no—
130#810—24no—
131#809—24no—
132C4⋊(C3⋊C8)—22no—
133C4⋊(C3⋊C8)—22no—
134C22×#41—31no—
135C22×Q16—32no—
136C22×SD16—32no—
137C2×C4⋊Q8C2×D311yescharacteristic
138C2×C22⋊Q8C2×D312yes—
139#520—64no—
140#520—64no—
141C2×C4×Q8—31no—
142C2×C4×Q8—31no—
143C2×C4×D4—31no—
144C2×C4×D4—31no—
145C2×C4×D4—61no—
146C2×C4×D4—61no—
147C2×#117—61no—
148C2×#117C2×D311yescharacteristic
149C22×C4⋊C4—31no—
150C22×C4⋊C4—61no—
151#462—64no—
152#461—64no—
153#452—68no—
154C2×C4⋊C8—31no—
155C2×C4⋊C8—31no—
156C2×#103—32no—
157C2×#103—32no—
158C2×#103—32no—
159C2×#102—32no—
160C2×C22⋊C8—32no—
161#398—21no—
162#398—21no—
163#397—21no—
164#397—21no—
165#391C2×D312yes—
166#382C2×D311yescharacteristic
167#340—32no—
168#340—32no—
169#336—32no—
170C24×C3C2×D411yescharacteristic
171C23×D3—21no—
172C2×C3×Q8—22no—
173C2×C3×Q8—22no—
174C2×C3×Q8—22no—
175C2×C3×Q8—22no—
176C2×C3×Q8—22no—
177C2×C3×Q8—22no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4#4112yes—
191C22×C3×C4C2411yescharacteristic
192C2×#67—24no—
193C2×#67—24no—
194C2×#67—24no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—24no—
199C22×Dic3—82no—
200#235—28no—
201#235—28no—
202#235—28no—
203#235—28no—
204C2×D12—21no—
205C2×D12—21no—
206C2×D12—21no—
207C2×D12C2×D412yes—
208C2×D12C2×D412yes—
209C2×D12#4112yes—
210C2×C4×D3—24no—
211C2×C4×D3—24no—
212C2×C4×D3—24no—
213C2×C4×D3—24no—
214C2×C4×D3—24no—
215C2×C4×D3—82no—
216C2×C4×D3—82no—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6C2×D412yes—
221C2×Dic6C2×D412yes—
222C2×Dic6#4112yes—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—22no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4—22no—
227C3×C4⋊C4—42no—
228C3×C4⋊C4—42no—
229C3×C4⋊C4—44no—
230C3×C22⋊C4—22no—
231C3×C22⋊C4—22no—
232C3×C22⋊C4—44no—
233C3×C42—21no—
234C3×C42—21no—
235C3×C42C2×D412yes—
236C3×C42C2×D412yes—
237#216—48no—
238#215—48no—
239(C2×D3)⋊C4—24no—
240(C2×D3)⋊C4—24no—
241C4⋊Dic3—24no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3C2×D412yes—
245C4⋊Dic3C2×D412yes—
246Dic3⋊C4—24no—
247Dic3⋊C4—24no—
248C4×Dic3—24no—
249C4×Dic3—24no—
250C2×C3⋊C8—22no—
251C2×C3⋊C8—22no—
252C2×C3⋊C8—22no—
253C2×C3⋊C8—22no—
254C2×C3⋊C8—22no—
255C2×C3⋊C8—22no—
256C2×#41—34no—
257C2×#41—62no—
258C2×#41—62no—
259C2×#41—62no—
260C2×#41—62no—
261C2×#41—62no—
262C22×Q8—31no—
263C22×Q8C22×D312yes—
264C22×D4—31no—
265C22×D4—61no—
266C23×C4—61no—
267C23×C4—61no—
268C23×C4—61no—
269C23×C4—61no—
270C23×C4C22×D311yescharacteristic
271C2×Q16—64no—
272C2×Q16—64no—
273C2×Q16—64no—
274C2×SD16—64no—
275C2×SD16—64no—
276C2×SD16—64no—
277C22×C8—32no—
278C22×C8—32no—
279C4⋊Q8—22no—
280C4⋊Q8—22no—
281C22⋊Q8—24no—
282C22⋊Q8—24no—
283C4×Q8—34no—
284C4×Q8—62no—
285C4×Q8—62no—
286C4×Q8—62no—
287C4×D4—34no—
288C4×D4—62no—
289C4×D4—62no—
290C4×D4—68no—
291C4×D4—68no—
292C4×D4—62no—
293#117—22no—
294#117—22no—
295#117—68no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—68no—
302C2×C4⋊C4—62no—
303C2×C4⋊C4—62no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4#6711yescharacteristic
308C2×C4⋊C4#6711yescharacteristic
309C2×C4⋊C4C22×D312yes—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C22⋊C4#6711yescharacteristic
314C2×C22⋊C4#6711yescharacteristic
315C2×C42—61no—
316C2×C42—61no—
317C2×C42C22×D311yescharacteristic
318C2×C42C22×D311yescharacteristic
319C4⋊C8—62no—
320C4⋊C8—62no—
321#103—64no—
322#103—64no—
323#103—64no—
324#102—64no—
325C22⋊C8—64no—
326#95—21no—
327#95—21no—
328#95—22no—
329#95—22no—
330C23×C3—21no—
331C23×C3—21no—
332C23×C3—22no—
333C23×C3—22no—
334C23×C3#12112yes—
335C23×C3C22×D411yescharacteristic
336C22×D3—22no—
337C22×D3—22no—
338C22×D3—22no—
339C22×D3—42no—
340C22×D3—42no—
341C3×Q8—42no—
342C3×Q8—42no—
343C3×Q8—42no—
344C3×Q8—42no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4#12012yes—
364C2×C3×C4#12012yes—
365C2×C3×C4#12112yes—
366C2×C3×C4C22×D411yescharacteristic
367C2×C3×C4C22×D411yescharacteristic
368C2×C3×C4C22×D411yescharacteristic
369C2×C3×C4C2×#4111yescharacteristic
370C2×C3×C4C2×#4111yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372#67—216no—
373C2×Dic3—22no—
374C2×Dic3—22no—
375C2×Dic3—22no—
376C2×Dic3—22no—
377C2×Dic3—22no—
378C2×Dic3—42no—
379C2×Dic3—42no—
380C2×Dic3—84no—
381D12—22no—
382D12—22no—
383D12—22no—
384D12#12114yes—
385C4×D3—216no—
386C4×D3—84no—
387C4×D3—84no—
388Dic6—22no—
389Dic6—22no—
390Dic6—22no—
391Dic6#12114yes—
392C3⋊C8—42no—
393C3⋊C8—42no—
394C24—61no—
395C24C2×#6711yescharacteristic
396#41—68no—
397#41—68no—
398#41—68no—
399#41—68no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—32no—
407C2×Q8—32no—
408C2×Q8—32no—
409C2×Q8—61no—
410C2×Q8—61no—
411C2×Q8—61no—
412C2×D4—32no—
413C2×D4—32no—
414C2×D4—32no—
415C2×D4—64no—
416C2×D4—64no—
417C2×D4—64no—
418C2×D4—61no—
419C2×D4—61no—
420C2×D4—61no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—64no—
428C22×C4—64no—
429C22×C4—64no—
430C22×C4—61no—
431C22×C4—64no—
432C22×C4—61no—
433C22×C4—61no—
434C22×C4—64no—
435C22×C4—64no—
436C22×C4—242no—
437C22×C4—242no—
438C22×C4—242no—
439C22×C4D3×D412yes—
440C22×C4#23712yes—
441C22×C4C2×#6711yescharacteristic
442C22×C4C2×#6711yescharacteristic
443C22×C4C2×#6711yescharacteristic
444C22×C4C2×#6711yescharacteristic
445C22×C4C2×#6711yescharacteristic
446C22×C4C23×D311yescharacteristic
447Q16—128no—
448SD16—128no—
449C2×C8—62no—
450C2×C8—62no—
451C2×C8—62no—
452C2×C8—62no—
453C2×C8—62no—
454C2×C8—62no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—32no—
460C4⋊C4—32no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—44no—
464C4⋊C4—64no—
465C4⋊C4—64no—
466C4⋊C4—64no—
467C4⋊C4—61no—
468C4⋊C4—61no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—44no—
472C22⋊C4—64no—
473C22⋊C4—64no—
474C42—21no—
475C42—21no—
476C42—64no—
477C42—64no—
478C42D3×D412yes—
479C42D3×D412yes—
480C22×C3—22no—
481C22×C3—24no—
482C22×C3—42no—
483C22×C3—41no—
484C22×C3—41no—
485C22×C3C2×#12011yescharacteristic
486C22×C3C2×#12111yescharacteristic
487C22×C3#52511yescharacteristic
488C22×C3#57712yes—
489C22×C3#57812yes—
490C2×D3—24no—
491C2×D3—44no—
492C2×D3—42no—
493C2×D3—42no—
494C2×D3—44no—
495C3×C4—22no—
496C3×C4—22no—
497C3×C4—22no—
498C3×C4—41no—
499C3×C4—41no—
500C3×C4—82no—
501C3×C4—82no—
502C3×C4C2×#12111yescharacteristic
503C3×C4C2×#12111yescharacteristic
504C3×C4#52511yescharacteristic
505C3×C4#52511yescharacteristic
506Dic3—24no—
507Dic3—42no—
508Dic3—42no—
509C23—21no—
510C23—21no—
511C23—22no—
512C23—22no—
513C23—62no—
514C23—62no—
515C23—62no—
516C23—122no—
517C23—122no—
518C23#93812yes—
519C23C22×#6711yescharacteristic
520Q8—34no—
521Q8—42no—
522Q8—42no—
523Q8—42no—
524Q8—42no—
525Q8—62no—
526Q8—62no—
527Q8—62no—
528D4—34no—
529D4—616no—
530D4—62no—
531D4—62no—
532D4—62no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—616no—
552C2×C4—62no—
553C2×C4—62no—
554C2×C4—62no—
555C2×C4—62no—
556C2×C4—62no—
557C2×C4—122no—
558C2×C4—122no—
559C2×C4—244no—
560C2×C4—244no—
561C2×C4—244no—
562C2×C4#93712yes—
563C2×C4#93712yes—
564C2×C4#93812yes—
565C2×C4C2×D3×D411yescharacteristic
566C2×C4C2×D3×D411yescharacteristic
567C2×C4C2×D3×D411yescharacteristic
568C2×C4C2×#23711yescharacteristic
569C2×C4D3×#4111yescharacteristic
570C2×C4D3×#4111yescharacteristic
571C8—122no—
572C8—122no—
573C2×C3—42no—
574C2×C3#284711yescharacteristic
575C2×C3#300911yescharacteristic
576C2×C3#301111yescharacteristic
577C2×C3#304612yes—
578C2×C3#312012yes—
579D3—44no—
580C22—22no—
581C22—24no—
582C22—41no—
583C22—41no—
584C22—42no—
585C22—64no—
586C22—122no—
587C22—124no—
588C22—122no—
589C22—124no—
590C22C2×#93711yescharacteristic
591C22C2×#93811yescharacteristic
592C22#582811yescharacteristic
593C22#583712yes—
594C22#583812yes—
595C4—22no—
596C4—22no—
597C4—22no—
598C4—41no—
599C4—41no—
600C4—64no—
601C4—82no—
602C4—82no—
603C4—122no—
604C4—122no—
605C4D3×#12111yescharacteristic
606C4D3×#12111yescharacteristic
607C4#560611yescharacteristic
608C4#560611yescharacteristic
609C3#3291611yescharacteristic, complemented, Sylow
610C2—42no—
611C2—124no—
612C2#8272311yescharacteristic
613C2#8340411yescharacteristic
614C2#8340612yes—
615C2#8341111yescharacteristic
616C2#8344612yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2621C2
socleC23×C3C22×D4
prime radicalC22×#235C22
hypercenter#2621D3
hypocenterC3#32916
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2621

Complements

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

NH ≅ G/Ntype
2. C3×#2621C2semidirect
3. C2×#5090C2semidirect
4. C2×#5083C2semidirect
6. #71692C2semidirect
7. #71583C2semidirect
8. #69690C2semidirect
9. #69687C2semidirect
10. #69559C2semidirect
11. #69556C2semidirect
53. #2621D3semidirect
609. C3#32916semidirect

Maximal subgroups

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

subgroupindex
C2×#55232
C3×#26212
C2×#50902
C2×#50832
C2×#50472
#716922
#715832
#696902
#696872
#695592
#695562
#329163