Finite Group Databasebeta

← #10848774

Subgroups — #10848774

Order 768.

1318 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5691C211yesmaximal, characteristic, complemented
2C2×C3×#490C211yesmaximal, characteristic, complemented
3#83192C211yesmaximal, characteristic, complemented
4C2×#5074C211yesmaximal, characteristic, complemented
5#71741C211yesmaximal, characteristic, complemented
6#70213C212yesmaximal, complemented
7#70208C212yesmaximal, complemented
8#70205C212yesmaximal, complemented
9#70204C212yesmaximal, complemented
10#69821C211yesmaximal, characteristic, complemented
11#69807C211yesmaximal, characteristic, complemented
12#25605—31nomaximal, Sylow
13C2×C3×#127C2211yescharacteristic
14C2×C3×#125C2211yescharacteristic
15C2×#938—21no—
16C2×#938—21no—
17C2×#938C2212yes—
18C22×#214C2212yes—
19C22×#213C2212yes—
20#5691—24no—
21#5691—24no—
22C2×#898—21no—
23C2×#897C2211yescharacteristic
24C2×#895C2211yescharacteristic
25C2×#894—21no—
26C2×#893—21no—
27C2×#893C2211yescharacteristic
28C2×#892—21no—
29C2×#887C2211yescharacteristic
30C2×#875C2211yescharacteristic
31C2×C42×D3C2211yescharacteristic
32C3×#490—24no—
33C2×C3×#102C2212yes—
34C2×C3×#102C2212yes—
35C2×C3×#97C2211yescharacteristic
36#5221—21no—
37#5221—21no—
38#5221C2212yes—
39C2×#821C2212yes—
40C2×(C2×D3)⋊C8C2212yes—
41C2×#816C2212yes—
42#5074—24no—
43#4935C2211yescharacteristic
44C2×#803C2211yescharacteristic
45#4550C2212yes—
46#4529C2212yes—
47#4529C2212yes—
48C2×#529—31no—
49C2×#496—31no—
50C2×#490D311yescharacteristic, complemented
51#2856—31no—
52#2000—32no—
53#1999—32no—
54#1992—31no—
55#1959—31no—
56#1880—32no—
57#1879—32no—
58#1749—31no—
59C22×C3×D4—21no—
60C22×C3×D4—21no—
61C22×C3×D4C2312yes—
62C22×#67—22no—
63C22×D12—21no—
64C22×D12C2311yescharacteristic
65C22×C4×D3—21no—
66C22×C4×D3D411yescharacteristic
67C22×C4×D3D411yescharacteristic
68C22×C4×D3C2311yescharacteristic
69C22×Dic6C2311yescharacteristic
70C22×C3×C8C2312yes—
71C3×#127—22no—
72C3×#127—22no—
73C3×#125—22no—
74C3×#125D414yescomplemented
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C42C2311yescharacteristic
80C2×#217—22no—
81C2×#217—22no—
82C2×#217—22no—
83#938—42no—
84#938—42no—
85#938—44no—
86#938—44no—
87C2×#214—24no—
88C2×#214—24no—
89C2×#214—42no—
90C2×#213—24no—
91C2×#213—24no—
92C2×#213—42no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3C2312yes—
102C2×Dic3⋊C4—21no—
103C2×C4×Dic3—21no—
104C2×C4×Dic3D411yescharacteristic
105C2×C4×Dic3D411yescharacteristic
106C2×C4×Dic3C2311yescharacteristic
107C22×C3⋊C8C2312yes—
108#898—28no—
109#897—22no—
110#897—22no—
111#895—22no—
112#895—22no—
113#894—28no—
114#893—28no—
115#893—22no—
116#893—22no—
117#892—28no—
118#887—22no—
119#887—22no—
120#875—22no—
121#875D414yescomplemented
122C42×D3—22no—
123C42×D3—22no—
124C3×#102—24no—
125C3×#102—24no—
126C3×#97—22no—
127#821—42no—
128(C2×D3)⋊C8—24no—
129#816—42no—
130#803—22no—
131C22×SD16—32no—
132C22×D8—32no—
133C2×#127C2×D311yescharacteristic
134#529—64no—
135#529—64no—
136C2×#126—61no—
137C2×#125C2×D311yescharacteristic
138C2×#124—31no—
139C2×#123—61no—
140C2×C22⋊Q8—31no—
141C2×#121—32no—
142C2×#121—31no—
143C2×#121—61no—
144C2×#121—61no—
145C2×C4×Q8—31no—
146C2×C4×D4—31no—
147C2×C4×D4—61no—
148C2×#117—61no—
149C22×C42—31no—
150#496—64no—
151#490—24no—
152C2×#103—32no—
153C2×#102—32no—
154C2×#102C2×D312yes—
155C2×#102C2×D312yes—
156C2×C22⋊C8—32no—
157C2×#97—31no—
158C2×#97C2×D311yescharacteristic
159#390—32no—
160#390—61no—
161#390—61no—
162#384—31no—
163#340—32no—
164#340—32no—
165#336—32no—
166C24×C3—22no—
167C23×D3—21no—
168C23×D3C2×D411yescharacteristic
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—42no—
176C2×C3×D4—42no—
177C2×C3×D4—42no—
178C2×C3×D4—42no—
179C2×C3×D4—42no—
180C2×C3×D4—42no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2411yescharacteristic
186C2×#67—44no—
187C2×#67—44no—
188C2×#67—44no—
189C22×Dic3—21no—
190C22×Dic3—22no—
191C22×Dic3C2×D411yescharacteristic
192C22×Dic3C2×D411yescharacteristic
193C22×Dic3C2×D411yescharacteristic
194C2×D12—22no—
195C2×D12—22no—
196C2×D12—24no—
197C2×D12—21no—
198C2×D12—42no—
199C2×D12—42no—
200C2×D12C2×D412yes—
201C2×C4×D3—24no—
202C2×C4×D3—22no—
203C2×C4×D3—22no—
204C2×C4×D3—22no—
205C2×C4×D3—21no—
206C2×C4×D3—21no—
207C2×C4×D3—22no—
208C2×C4×D3—22no—
209C2×C4×D3—22no—
210C2×C4×D3—41no—
211C2×C4×D3—41no—
212C2×C4×D3—42no—
213C2×C4×D3—42no—
214C2×Dic6—22no—
215C2×Dic6—22no—
216C2×Dic6—21no—
217C2×Dic6C2×D412yes—
218C2×C3×C8—22no—
219C2×C3×C8—22no—
220C2×C3×C8—22no—
221C3×C4⋊C4—24no—
222C3×C4⋊C4—24no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4C2×D412yes—
226C3×C4⋊C4C2×D412yes—
227C3×C42C2×D412yes—
228C3×C42C2×D412yes—
229#217—44no—
230#217—44no—
231#217—44no—
232#214—48no—
233#213—48no—
234(C2×D3)⋊C4—24no—
235(C2×D3)⋊C4—24no—
236(C2×D3)⋊C4—24no—
237(C2×D3)⋊C4—24no—
238(C2×D3)⋊C4—24no—
239C4⋊Dic3—24no—
240C4⋊Dic3—22no—
241C4⋊Dic3—22no—
242C4⋊Dic3—42no—
243C4⋊Dic3—42no—
244Dic3⋊C4—24no—
245C4×Dic3—24no—
246C4×Dic3—22no—
247C4×Dic3—22no—
248C4×Dic3—21no—
249C4×Dic3—21no—
250C2×C3⋊C8—22no—
251C2×C3⋊C8—22no—
252C2×C3⋊C8#4114yes—
253C22×Q8—31no—
254C22×D4—21no—
255C22×D4—21no—
256C22×D4—31no—
257C22×D4—61no—
258C22×D4—62no—
259C22×D4C22×D312yes—
260C23×C4—31no—
261C23×C4—31no—
262C23×C4—31no—
263C23×C4—61no—
264C2×SD16—64no—
265C2×SD16—64no—
266C2×SD16—122no—
267C2×D8—64no—
268C2×D8—64no—
269C2×D8—122no—
270C22×C8—32no—
271C22×C8C22×D312yes—
272#127—22no—
273#127—22no—
274#126—68no—
275#125—22no—
276#125#6714yescomplemented
277#124—34no—
278#124—62no—
279#123—68no—
280C22⋊Q8—62no—
281C22⋊Q8—62no—
282#121—62no—
283#121—62no—
284#121—122no—
285#121—122no—
286#121—124no—
287#121—124no—
288C4×Q8—62no—
289C4×Q8—62no—
290C4×D4—62no—
291C4×D4—68no—
292C4×D4—62no—
293#117—68no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—32no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4C22×D311yescharacteristic
302C2×C4⋊C4C22×D311yescharacteristic
303C2×C22⋊C4—61no—
304C2×C22⋊C4—61no—
305C2×C22⋊C4—61no—
306C2×C22⋊C4—61no—
307C2×C22⋊C4—61no—
308C2×C22⋊C4—62no—
309C2×C22⋊C4—62no—
310C2×C22⋊C4—62no—
311C2×C42—31no—
312C2×C42—31no—
313C2×C42—31no—
314C2×C42—61no—
315C2×C42—62no—
316C2×C42—62no—
317C2×C42C22×D311yescharacteristic
318#103—122no—
319#102—24no—
320#102—24no—
321#102—122no—
322C22⋊C8—64no—
323#97—22no—
324#97—62no—
325C23×C3—42no—
326C23×C3—42no—
327C23×C3—42no—
328C23×C3—42no—
329C23×C3—42no—
330C23×C3—42no—
331C23×C3—42no—
332C23×C3C22×D411yescharacteristic
333C22×D3—21no—
334C22×D3—21no—
335C22×D3—21no—
336C22×D3—21no—
337C22×D3—21no—
338C22×D3—21no—
339C22×D3—22no—
340C22×D3—41no—
341C22×D3—41no—
342C22×D3—41no—
343C22×D3—41no—
344C22×D3—41no—
345C22×D3—41no—
346C22×D3AGL(1,ℤ/8)12yes—
347C3×D4—42no—
348C3×D4—42no—
349C3×D4—42no—
350C3×D4—42no—
351C3×D4—84no—
352C3×D4—84no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4#12012yes—
364C2×C3×C4#12012yes—
365C2×C3×C4C22×D411yescharacteristic
366C2×C3×C4C22×D411yescharacteristic
367C2×C3×C4C2×#4112yes—
368C2×C3×C4H(5,𝔽2)12yes—
369#67—88no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—22no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—42no—
382C2×Dic3—42no—
383C2×Dic3—42no—
384C2×Dic3AGL(1,ℤ/8)12yes—
385D12—24no—
386D12—48no—
387D12—42no—
388C4×D3—44no—
389C4×D3—48no—
390C4×D3—44no—
391C4×D3—42no—
392C4×D3—42no—
393Dic6—24no—
394Dic6—42no—
395C3×C8—42no—
396C3⋊C8—24no—
397C24—22no—
398C24—31no—
399C24—61no—
400C2×Q8—32no—
401C2×Q8—61no—
402C2×Q8—62no—
403C2×Q8—62no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—32no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—64no—
418C2×D4—62no—
419C2×D4—62no—
420C2×D4—61no—
421C2×D4—122no—
422C2×D4—122no—
423C2×D4—124no—
424C2×D4—124no—
425C2×D4—124no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—31no—
429C22×C4—31no—
430C22×C4—31no—
431C22×C4—62no—
432C22×C4—62no—
433C22×C4—64no—
434C22×C4—61no—
435C22×C4—62no—
436C22×C4—62no—
437C22×C4—61no—
438C22×C4—62no—
439C22×C4—62no—
440C22×C4—61no—
441C22×C4—62no—
442C22×C4—121no—
443C22×C4—121no—
444C22×C4—122no—
445C22×C4—122no—
446C22×C4D3×D411yescharacteristic
447C22×C4D3×D411yescharacteristic
448C22×C4C23×D311yescharacteristic
449SD16—128no—
450D8—128no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—34no—
455C2×C8—62no—
456C2×C8—62no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—64no—
462C4⋊C4—64no—
463C4⋊C4—62no—
464C4⋊C4—62no—
465C4⋊C4—122no—
466C4⋊C4—122no—
467C4⋊C4C2×#6712yes—
468C4⋊C4C2×#6712yes—
469C22⋊C4—64no—
470C22⋊C4—64no—
471C22⋊C4—64no—
472C22⋊C4—64no—
473C22⋊C4—64no—
474C22⋊C4—124no—
475C22⋊C4—124no—
476C22⋊C4—124no—
477C42—64no—
478C42—62no—
479C42—62no—
480C42—61no—
481C42—61no—
482C42D3×D412yes—
483C42C2×#6712yes—
484C22×C3—82no—
485C22×C3—82no—
486C22×C3—82no—
487C22×C3—82no—
488C22×C3—82no—
489C22×C3—82no—
490C22×C3—82no—
491C22×C3C2×#12011yescharacteristic
492C22×C3#53211yescharacteristic
493C22×C3#53511yescharacteristic
494C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
495C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
496C22×C3#57511yescharacteristic
497C22×C3#57511yescharacteristic
498C2×D3—22no—
499C2×D3—22no—
500C2×D3—22no—
501C2×D3—41no—
502C2×D3—41no—
503C2×D3—41no—
504C2×D3—41no—
505C2×D3—42no—
506C2×D3—42no—
507C2×D3—42no—
508C2×D3—81no—
509C2×D3—81no—
510C2×D3—81no—
511C2×D3—81no—
512C3×C4—22no—
513C3×C4—22no—
514C3×C4—42no—
515C3×C4—42no—
516C3×C4#54612yes—
517C3×C4#55012yes—
518Dic3—22no—
519Dic3—42no—
520Dic3—41no—
521Dic3—41no—
522Dic3—82no—
523C23—32no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—61no—
532C23—61no—
533C23—61no—
534C23—61no—
535C23—61no—
536C23—61no—
537C23—62no—
538C23—121no—
539C23—121no—
540C23—121no—
541C23—121no—
542C23—121no—
543C23—121no—
544C23C2×D3×D411yescharacteristic
545Q8—64no—
546Q8—122no—
547D4—42no—
548D4—42no—
549D4—42no—
550D4—42no—
551D4—64no—
552D4—84no—
553D4—84no—
554D4—128no—
555D4—122no—
556D4—248no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—32no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—61no—
575C2×C4—61no—
576C2×C4—62no—
577C2×C4—124no—
578C2×C4—124no—
579C2×C4—128no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—122no—
583C2×C4—122no—
584C2×C4—122no—
585C2×C4—122no—
586C2×C4—122no—
587C2×C4#93712yes—
588C2×C4#93712yes—
589C2×C4C2×D3×D411yescharacteristic
590C2×C4D3×#4112yes—
591C2×C4#100912yes—
592C2×C4C22×#6711yescharacteristic
593C8—42no—
594C8—64no—
595C2×C3—162no—
596C2×C3#243111yescharacteristic
597C2×C3#300611yescharacteristic
598C2×C3#300611yescharacteristic
599C2×C3#317111yescharacteristic
600C2×C3#318011yescharacteristic
601C2×C3#321711yescharacteristic
602C2×C3#321911yescharacteristic
603D3—42no—
604D3—82no—
605C22—62no—
606C22—62no—
607C22—62no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—121no—
616C22—121no—
617C22—121no—
618C22—121no—
619C22—122no—
620C22—122no—
621C22—122no—
622C22—241no—
623C22—241no—
624C22—241no—
625C22—241no—
626C22#566811yescharacteristic
627C22#567411yescharacteristic
628C22D3×AGL(1,ℤ/8)11yescharacteristic
629C22D3×AGL(1,ℤ/8)11yescharacteristic
630C22#577311yescharacteristic
631C22#577311yescharacteristic
632C22C2×#93711yescharacteristic
633C4—22no—
634C4—22no—
635C4—42no—
636C4—42no—
637C4—62no—
638C4—121no—
639C4—122no—
640C4—121no—
641C4—242no—
642C4#567612yes—
643C4#583412yes—
644C3#2560511yescharacteristic, complemented, Sylow
645C2—122no—
646C2—162no—
647C2—242no—
648C2#8170411yescharacteristic
649C2#8170711yescharacteristic
650C2#8171711yescharacteristic
651C2#8171811yescharacteristic
652C2#8295811yescharacteristic
653C2#8394611yescharacteristic
654C2#8394611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#490C2
socleC23×C3C22×D4
hypercenterC2×#490D3
hypocenterC3#25605
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#490

Complements

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

NH ≅ G/Ntype
1. C2×#5691C2semidirect
2. C2×C3×#490C2semidirect
3. #83192C2semidirect
4. C2×#5074C2semidirect
5. #71741C2semidirect
6. #70213C2semidirect
7. #70208C2semidirect
8. #70205C2semidirect
9. #70204C2semidirect
10. #69821C2semidirect
11. #69807C2semidirect
50. C2×#490D3semidirect
74. C3×#125D4semidirect
121. #875D4semidirect
276. #125#67semidirect
644. C3#25605semidirect

Maximal subgroups

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

subgroupindex
C2×#56912
C2×C3×#4902
#831922
C2×#50742
#717412
#702132
#702082
#702052
#702042
#698212
#698072
#256053