Finite Group Databasebeta

← #10847377

Subgroups — #10847377

Order 768.

1474 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5716C211yesmaximal, characteristic, complemented
2C2×#5157C212yesmaximal, complemented
3#76475C211yesmaximal, characteristic, complemented
4C3×#2011C211yesmaximal, characteristic, complemented
5#72037C212yesmaximal, complemented
6#71762C211yesmaximal, characteristic, complemented
7#70202C212yesmaximal, complemented
8#70168C212yesmaximal, complemented
9#69638C211yesmaximal, characteristic, complemented
10D3×#336C211yesmaximal, characteristic, complemented
11#68632C211yesmaximal, characteristic, complemented
12#25496—31nomaximal, Sylow
13C22×#237—21no—
14C22×C3×D8C2212yes—
15C2×C3×#127C2211yescharacteristic
16C2×#938—21no—
17C2×#938C2211yescharacteristic
18C2×#938C2212yes—
19C2×#936—21no—
20C2×D4×Dic3—21no—
21C2×D4×Dic3C2212yes—
22C22×D3×C8C2212yes—
23#5716—28no—
24C2×#883—21no—
25C2×C42×D3C2211yescharacteristic
26C2×#869C2211yescharacteristic
27C2×C3×#102C2212yes—
28C3×#390C2211yescharacteristic
29#5221C2211yescharacteristic
30C2×#832C2212yes—
31C2×#832C2212yes—
32C2×#832C2212yes—
33#5157—28no—
34C2×#818C2212yes—
35#4987C2211yescharacteristic
36#4984C2211yescharacteristic
37#4956—21no—
38#4951—21no—
39#4945—21no—
40#4928C2211yescharacteristic
41C3×#336C2211yescharacteristic
42#4551C2212yes—
43#4550C2212yes—
44#4529C2212yes—
45#4524C2211yescharacteristic
46C2×#532—31no—
47C2×#466—32no—
48#2854—31no—
49#2057—31no—
50#2041—31no—
51#2023—31no—
52#2011D311yescharacteristic, complemented
53#1893—32no—
54#1879—32no—
55#1862—32no—
56C2×#336—31no—
57C22×C3×D4—21no—
58C22×C3×D4—21no—
59C22×C3×D4C2311yescharacteristic
60C22×C3×D4C2312yes—
61C22×#67—22no—
62C22×#67—21no—
63C23×Dic3—22no—
64C2×#237—28no—
65C2×#237—44no—
66C2×#237—44no—
67C22×C4×D3D411yescharacteristic
68C22×C4×D3D411yescharacteristic
69C22×C4×D3C2311yescharacteristic
70C22×Dic6—21no—
71C2×C3×D8—24no—
72C2×C3×D8—24no—
73C2×C3×D8—42no—
74C22×C3×C8C2312yes—
75C3×#127—24no—
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C22⋊C4—21no—
78C2×C3×C22⋊C4—21no—
79C2×C3×C42C2311yescharacteristic
80C2×#217—22no—
81C2×#217—22no—
82C2×#217—21no—
83C2×#217—21no—
84C2×#217—21no—
85#938—28no—
86#938—24no—
87#938—24no—
88#938D418yes—
89#936—28no—
90D4×Dic3—28no—
91D4×Dic3—24no—
92D4×Dic3—24no—
93C2×(C2×D3)⋊C4—21no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2312yes—
99C2×Dic3⋊C4—21no—
100C2×C4×Dic3D411yescharacteristic
101C2×C4×Dic3D411yescharacteristic
102C2×C4×Dic3C2311yescharacteristic
103C22×C3⋊C8C2312yes—
104C2×D3×C8—24no—
105C2×D3×C8—24no—
106C2×D3×C8—42no—
107#883—44no—
108C42×D3—24no—
109#869—24no—
110C3×#102—24no—
111#832—24no—
112#832—24no—
113#832—42no—
114C3⋊#95—21no—
115C3⋊#95—21no—
116C3⋊#95—21no—
117#818—24no—
118C22×#41—61no—
119C22×D8C2×D312yes—
120C23×C8—32no—
121#532—68no—
122C2×C4⋊Q8—31no—
123C2×#127C2×D311yescharacteristic
124C2×#124—61no—
125C2×#123—61no—
126C2×#121—32no—
127C2×#121—31no—
128C2×#121—61no—
129C2×C4×D4—32no—
130C2×C4×D4—61no—
131C22×C42—31no—
132#466—68no—
133C2×#107—32no—
134C2×#102—32no—
135C2×#102—32no—
136C2×#102—32no—
137C2×#102C2×D312yes—
138#390—31no—
139#390C2×D311yescharacteristic
140#389—31no—
141#389—31no—
142#388—61no—
143#388—61no—
144#386—61no—
145#380—31no—
146#340—32no—
147#340—32no—
148#336—32no—
149#336—31no—
150#336C2×D311yescharacteristic
151C24×C3—22no—
152C24×C3—21no—
153C23×D3C2×D411yescharacteristic
154C2×C3×D4—24no—
155C2×C3×D4—24no—
156C2×C3×D4—22no—
157C2×C3×D4—22no—
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4—22no—
161C2×C3×D4—22no—
162C2×C3×D4—21no—
163C2×C3×D4—22no—
164C2×C3×D4—21no—
165C2×C3×D4—42no—
166C2×C3×D4—42no—
167C2×C3×D4—42no—
168C2×C3×D4—42no—
169C2×C3×D4C2×D414yes—
170C22×C3×C4—21no—
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2411yescharacteristic
174C2×#67—28no—
175C2×#67—44no—
176C2×#67—44no—
177C2×#67—42no—
178C2×#67—42no—
179C2×#67—42no—
180C22×Dic3—28no—
181C22×Dic3—22no—
182C22×Dic3—21no—
183C22×Dic3—21no—
184C22×Dic3—44no—
185C22×Dic3—44no—
186C22×Dic3C2×D411yescharacteristic
187C22×Dic3C2×D411yescharacteristic
188C22×Dic3C2×D411yescharacteristic
189#237—432no—
190C2×C4×D3—22no—
191C2×C4×D3—22no—
192C2×C4×D3—22no—
193C2×C4×D3—21no—
194C2×C4×D3—21no—
195C2×C4×D3—22no—
196C2×C4×D3—22no—
197C2×C4×D3—41no—
198C2×C4×D3D814yescomplemented
199C2×C4×D3C2×D414yes—
200C2×Dic6—24no—
201C2×Dic6—42no—
202C2×Dic6—42no—
203C3×D8—48no—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C2×C3×C8#4114yescomplemented
207C3×C4⋊C4—22no—
208C3×C22⋊C4—42no—
209C3×C22⋊C4—42no—
210C3×C42—22no—
211#217—28no—
212#217—28no—
213#217—42no—
214#217—42no—
215#217—42no—
216(C2×D3)⋊C4—42no—
217C4⋊Dic3—24no—
218C4⋊Dic3—22no—
219C4⋊Dic3—22no—
220C4⋊Dic3—21no—
221C4⋊Dic3—21no—
222C4⋊Dic3—42no—
223C4⋊Dic3C2×D414yes—
224Dic3⋊C4—42no—
225C4×Dic3—22no—
226C4×Dic3—22no—
227C4×Dic3—22no—
228C2×C3⋊C8—22no—
229C2×C3⋊C8—22no—
230C2×C3⋊C8—22no—
231D3×C8—48no—
232C2×#41—68no—
233C2×#41—124no—
234C2×#41—124no—
235C22×Q8—61no—
236C22×D4—21no—
237C22×D4—21no—
238C22×D4—62no—
239C22×D4—61no—
240C22×D4C22×D311yescharacteristic
241C22×D4C22×D312yes—
242C23×C4—31no—
243C23×C4—31no—
244C23×C4—31no—
245C23×C4—62no—
246C2×D8—24no—
247C2×D8—24no—
248C2×D8—42no—
249C22×C8—32no—
250C22×C8—64no—
251C22×C8—64no—
252C22×C8—122no—
253C22×C8C22×D312yes—
254C4⋊Q8—64no—
255#127—24no—
256#124—68no—
257#123—124no—
258#121—38no—
259#121—64no—
260#121—68no—
261#121—64no—
262C4×D4—64no—
263C4×D4—68no—
264C4×D4—64no—
265C2×C4⋊C4—32no—
266C2×C4⋊C4—31no—
267C2×C4⋊C4—31no—
268C2×C4⋊C4—61no—
269C2×C4⋊C4—61no—
270C2×C4⋊C4—61no—
271C2×C4⋊C4C22×D311yescharacteristic
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—62no—
275C2×C22⋊C4—62no—
276C2×C22⋊C4—61no—
277C2×C22⋊C4—61no—
278C2×C22⋊C4—61no—
279C2×C22⋊C4—61no—
280C2×C42—31no—
281C2×C42—31no—
282C2×C42—31no—
283C2×C42—64no—
284C2×C42C22×D311yescharacteristic
285#107—64no—
286#102—24no—
287#102—64no—
288#102—64no—
289#102—122no—
290#95—61no—
291#95—61no—
292#95—61no—
293C23×C3—24no—
294C23×C3—42no—
295C23×C3—42no—
296C23×C3—42no—
297C23×C3—42no—
298C23×C3—42no—
299C23×C3—42no—
300C23×C3—41no—
301C23×C3—41no—
302C23×C3—41no—
303C23×C3—41no—
304C23×C3—41no—
305C23×C3—41no—
306C23×C3—41no—
307C23×C3C22×D411yescharacteristic
308C22×D3—21no—
309C22×D3—21no—
310C22×D3—21no—
311C22×D3—21no—
312C22×D3#12012yes—
313C22×D3C2×D812yes—
314C22×D3AGL(1,ℤ/8)12yes—
315C3×D4—24no—
316C3×D4—24no—
317C3×D4—48no—
318C3×D4—48no—
319C3×D4—42no—
320C3×D4—42no—
321C3×D4—42no—
322C3×D4—42no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4C2×D812yes—
331C2×C3×C4#13512yes—
332C2×C3×C4C22×D412yes—
333C2×C3×C4C2×#4112yes—
334C2×C3×C4H(5,𝔽2)11yescharacteristic
335C2×C3×C42−1+411yescharacteristic
336#67—416no—
337#67—84no—
338C2×Dic3—21no—
339C2×Dic3—21no—
340C2×Dic3—21no—
341C2×Dic3—21no—
342C2×Dic3—21no—
343C2×Dic3—24no—
344C2×Dic3—21no—
345C2×Dic3—21no—
346C2×Dic3—416no—
347C2×Dic3—42no—
348C2×Dic3—42no—
349C2×Dic3—41no—
350C2×Dic3—41no—
351C2×Dic3—41no—
352C2×Dic3—41no—
353C2×Dic3—41no—
354C2×Dic3—41no—
355C2×Dic3#12012yes—
356C2×Dic3C2×D812yes—
357C4×D3—28no—
358C4×D3—24no—
359C4×D3—24no—
360C4×D3—44no—
361Dic6—48no—
362C3×C8—24no—
363C3⋊C8—42no—
364C24—22no—
365C24—21no—
366C24—31no—
367#41—1232no—
368C2×Q8—64no—
369C2×Q8—122no—
370C2×Q8—122no—
371C2×D4—24no—
372C2×D4—24no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—21no—
380C2×D4—22no—
381C2×D4—21no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—68no—
387C2×D4—124no—
388C2×D4—124no—
389C2×D4—122no—
390C2×D4—122no—
391C2×D4—122no—
392C2×D4D3×D414yes—
393C22×C4—21no—
394C22×C4—31no—
395C22×C4—31no—
396C22×C4—34no—
397C22×C4—34no—
398C22×C4—31no—
399C22×C4—62no—
400C22×C4—62no—
401C22×C4—62no—
402C22×C4—62no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—68no—
406C22×C4—62no—
407C22×C4—62no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—121no—
411C22×C4—124no—
412C22×C4—124no—
413C22×C4D3×D411yescharacteristic
414C22×C4D3×D411yescharacteristic
415C22×C4C23×D311yescharacteristic
416D8—48no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—62no—
420C2×C8—62no—
421C2×C8—62no—
422C2×C8—128no—
423C2×C8#23714yescomplemented
424C4⋊C4—22no—
425C4⋊C4—34no—
426C4⋊C4—64no—
427C4⋊C4—62no—
428C4⋊C4—62no—
429C4⋊C4—61no—
430C4⋊C4—61no—
431C4⋊C4—122no—
432C4⋊C4—122no—
433C22⋊C4—42no—
434C22⋊C4—42no—
435C22⋊C4—68no—
436C22⋊C4—68no—
437C22⋊C4—122no—
438C22⋊C4—122no—
439C22⋊C4—122no—
440C22⋊C4—122no—
441C42—22no—
442C42—62no—
443C42—62no—
444C42—62no—
445C22×C3—44no—
446C22×C3—44no—
447C22×C3—44no—
448C22×C3—82no—
449C22×C3—82no—
450C22×C3—82no—
451C22×C3—82no—
452C22×C3—81no—
453C22×C3—81no—
454C22×C3—81no—
455C22×C3—81no—
456C22×C3—81no—
457C22×C3—81no—
458C22×C3—81no—
459C22×C3C2×#12011yescharacteristic
460C22×C3#53211yescharacteristic
461C22×C3#53711yescharacteristic
462C22×C3C22×D811yescharacteristic
463C22×C3C2×#13511yescharacteristic
464C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
465C22×C3#57511yescharacteristic
466C2×D3—22no—
467C2×D3—22no—
468C2×D3—22no—
469C2×D3—22no—
470C2×D3—22no—
471C2×D3—22no—
472C2×D3#44714yes—
473C3×C4—22no—
474C3×C4—22no—
475C3×C4—81no—
476C3×C4#54612yes—
477C3×C4#54712yes—
478Dic3—22no—
479Dic3—22no—
480Dic3—44no—
481Dic3—41no—
482Dic3—81no—
483Dic3—81no—
484C23—24no—
485C23—32no—
486C23—32no—
487C23—32no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—61no—
502C23—61no—
503C23—61no—
504C23—61no—
505C23C2×D3×D411yescharacteristic
506Q8—128no—
507D4—24no—
508D4—24no—
509D4—48no—
510D4—48no—
511D4—42no—
512D4—42no—
513D4—42no—
514D4—42no—
515D4—1216no—
516D4—244no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—32no—
522C2×C4—32no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—68no—
527C2×C4—61no—
528C2×C4—61no—
529C2×C4—61no—
530C2×C4—61no—
531C2×C4—64no—
532C2×C4—64no—
533C2×C4—61no—
534C2×C4—61no—
535C2×C4—61no—
536C2×C4—64no—
537C2×C4—124no—
538C2×C4—1216no—
539C2×C4—122no—
540C2×C4—122no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4D3×D812yes—
548C2×C4#91212yes—
549C2×C4C2×D3×D412yes—
550C2×C4C2×#23712yes—
551C2×C4#100411yescharacteristic
552C2×C4#101011yescharacteristic
553C8—24no—
554C8—122no—
555C2×C3—84no—
556C2×C3—161no—
557C2×C3#243011yescharacteristic
558C2×C3C2×#44711yescharacteristic
559C2×C3#300211yescharacteristic
560C2×C3#316911yescharacteristic
561C2×C3#318011yescharacteristic
562C2×C3#324511yescharacteristic
563C2×C3#324711yescharacteristic
564D3—24no—
565C22—34no—
566C22—44no—
567C22—44no—
568C22—44no—
569C22—62no—
570C22—62no—
571C22—62no—
572C22—62no—
573C22—62no—
574C22—62no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22—82no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22D3×#12011yescharacteristic
587C22#560811yescharacteristic
588C22#571611yescharacteristic
589C22C2×D3×D811yescharacteristic
590C22C2×#91211yescharacteristic
591C22D3×AGL(1,ℤ/8)11yescharacteristic
592C22#577311yescharacteristic
593C4—22no—
594C4—22no—
595C4—62no—
596C4—62no—
597C4—81no—
598C4—124no—
599C4—121no—
600C4—241no—
601C4—241no—
602C4#559112yes—
603C4#561112yes—
604C3#2549611yescharacteristic, complemented, Sylow
605C2—64no—
606C2—84no—
607C2—161no—
608C2#7675711yescharacteristic
609C2D3×#44711yescharacteristic
610C2#8000111yescharacteristic
611C2#8034211yescharacteristic
612C2#8034311yescharacteristic
613C2#8177511yescharacteristic
614C2#8183811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2011C2
socleC23×C3C22×D4
hypercenter#2011D3
hypocenterC3#25496
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2011

Complements

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

NH ≅ G/Ntype
1. C2×#5716C2semidirect
2. C2×#5157C2semidirect
3. #76475C2semidirect
4. C3×#2011C2semidirect
5. #72037C2semidirect
6. #71762C2semidirect
7. #70202C2semidirect
8. #70168C2semidirect
9. #69638C2semidirect
10. D3×#336C2semidirect
11. #68632C2semidirect
52. #2011D3semidirect
198. C2×C4×D3D8semidirect
206. C2×C3×C8#41semidirect
423. C2×C8#237semidirect
604. C3#25496semidirect

Maximal subgroups

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

subgroupindex
C2×#57162
C2×#51572
#764752
C3×#20112
#720372
#717622
#702022
#701682
#696382
D3×#3362
#686322
#254963