Finite Group Databasebeta

← #10848768

Subgroups — #10848768

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×#5680C211yesmaximal, characteristic, complemented
2C2×#5058C211yesmaximal, characteristic, complemented
3C2×#4713C211yesmaximal, characteristic, complemented
4#74375C211yesmaximal, characteristic, complemented
5C3×#1959C211yesmaximal, characteristic, complemented
6#70272C212yesmaximal, complemented
7#70240C212yesmaximal, complemented
8#70204C212yesmaximal, complemented
9#70172C212yesmaximal, complemented
10#69807C211yesmaximal, characteristic, complemented
11#69583C211yesmaximal, characteristic, complemented
12#25605—31nomaximal, Sylow
13C2×C3×#124C2211yescharacteristic
14C2×#946C2211yescharacteristic
15C2×Q8×Dic3C2211yescharacteristic
16C22×#215C2212yes—
17C2×#938C2211yescharacteristic
18C2×D4×Dic3C2211yescharacteristic
19C22×#213C2212yes—
20#5680—24no—
21#5680—24no—
22C2×#895—21no—
23C2×#895—21no—
24C2×#895C2212yes—
25C2×#883—21no—
26C2×#67⋊C4—21no—
27C2×#879—21no—
28C2×#877—21no—
29C2×#874C2211yescharacteristic
30C2×C42×D3C2211yescharacteristic
31C2×#870C2211yescharacteristic
32C2×C3×#103C2212yes—
33C2×C3×#102C2212yes—
34C2×C3×#97C2211yescharacteristic
35C3×#384C2211yescharacteristic
36C2×#821C2212yes—
37C2×#821C2212yes—
38C2×(C2×D3)⋊C8C2212yes—
39#5058—24no—
40#4988—21no—
41#4988—21no—
42#4988C2212yes—
43C2×#803C2211yescharacteristic
44#4713—24no—
45#4550C2212yes—
46#4529C2212yes—
47#4529C2212yes—
48C2×#529—31no—
49C2×#496—31no—
50C2×#490—31no—
51#2856—31no—
52#2000—32no—
53#1999—32no—
54#1992—31no—
55#1959D311yescharacteristic, complemented
56#1880—32no—
57#1879—32no—
58#1749—31no—
59C22×C3×Q8C2311yescharacteristic
60C22×C3×D4C2311yescharacteristic
61C22×#67—21no—
62C23×Dic3—21no—
63C22×D12—22no—
64C22×D12—21no—
65C22×D12—21no—
66C22×D12C2312yes—
67C22×C4×D3D411yescharacteristic
68C22×C4×D3D411yescharacteristic
69C22×C4×D3C2311yescharacteristic
70C22×C3×C8C2312yes—
71C3×#124—22no—
72C3×#124D414yescomplemented
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C22⋊C4—21no—
77C2×C3×C22⋊C4—21no—
78C2×C3×C42C2311yescharacteristic
79C2×#217—21no—
80C2×#217—21no—
81#946—22no—
82#946—22no—
83Q8×Dic3—22no—
84Q8×Dic3—22no—
85C2×#215—24no—
86C2×#215—24no—
87C2×#215—42no—
88#938—22no—
89#938—22no—
90D4×Dic3—22no—
91D4×Dic3—22no—
92C2×#213—24no—
93C2×#213—24no—
94C2×#213—42no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—22no—
97C2×(C2×D3)⋊C4—22no—
98C2×(C2×D3)⋊C4—22no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3D411yescharacteristic
107C2×C4×Dic3D411yescharacteristic
108C2×C4×Dic3C2311yescharacteristic
109C22×C3⋊C8C2312yes—
110#895—42no—
111#895—42no—
112#895—44no—
113#895—44no—
114#883—28no—
115#67⋊C4—28no—
116#879—28no—
117#877—28no—
118#874—22no—
119#874—22no—
120C42×D3—22no—
121C42×D3—22no—
122#870—22no—
123#870D414yescomplemented
124C3×#103—42no—
125C3×#102—42no—
126C3×#97—22no—
127#821—24no—
128#821—24no—
129(C2×D3)⋊C8—24no—
130#803—22no—
131C22×SD16—32no—
132C22×D8—32no—
133C2×#127—31no—
134#529—64no—
135#529—64no—
136C2×#126—61no—
137C2×#125—31no—
138C2×#124C2×D311yescharacteristic
139C2×#123—61no—
140C2×C22⋊Q8—31no—
141C2×#121—31no—
142C2×#121—32no—
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—64no—
152C2×#103C2×D312yes—
153C2×#102—32no—
154C2×#102—32no—
155C2×#102C2×D312yes—
156C2×C22⋊C8—32no—
157C2×#97—31no—
158C2×#97C2×D311yescharacteristic
159#390—32no—
160#390—61no—
161#390—61no—
162#384C2×D311yescharacteristic
163#340—32no—
164#340—32no—
165#336—32no—
166C24×C3—21no—
167C23×D3—22no—
168C23×D3C2×D411yescharacteristic
169C2×C3×Q8—21no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8C2×D412yes—
173C2×C3×D4—21no—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4C2×D412yes—
177C22×C3×C4—21no—
178C22×C3×C4—22no—
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2411yescharacteristic
182C2×#67—24no—
183C2×#67—42no—
184C2×#67—42no—
185C22×Dic3—24no—
186C22×Dic3—21no—
187C22×Dic3—21no—
188C22×Dic3—42no—
189C22×Dic3—42no—
190C22×Dic3C2×D411yescharacteristic
191C22×Dic3C2×D411yescharacteristic
192C22×Dic3C2×D411yescharacteristic
193C2×D12—22no—
194C2×D12—22no—
195C2×D12—22no—
196C2×D12—22no—
197C2×D12—22no—
198C2×D12—22no—
199C2×D12—42no—
200C2×D12—42no—
201C2×D12—42no—
202C2×D12—42no—
203C2×D12—44no—
204C2×D12—44no—
205C2×D12—44no—
206C2×D12—42no—
207C2×D12—42no—
208C2×C4×D3—22no—
209C2×C4×D3—22no—
210C2×C4×D3—22no—
211C2×C4×D3—21no—
212C2×C4×D3—21no—
213C2×C4×D3—22no—
214C2×C4×D3—22no—
215C2×C4×D3—22no—
216C2×C4×D3—41no—
217C2×C4×D3—41no—
218C2×C3×C8—22no—
219C2×C3×C8—22no—
220C2×C3×C8—22no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—22no—
223C3×C4⋊C4—42no—
224C3×C4⋊C4—42no—
225C3×C22⋊C4—24no—
226C3×C22⋊C4—24no—
227C3×C42C2×D412yes—
228C3×C42C2×D412yes—
229#217—24no—
230#217—24no—
231#215—48no—
232#213—48no—
233(C2×D3)⋊C4—24no—
234(C2×D3)⋊C4—44no—
235(C2×D3)⋊C4—44no—
236(C2×D3)⋊C4—44no—
237C4⋊Dic3—24no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3C2×D412yes—
241C4⋊Dic3C2×D412yes—
242Dic3⋊C4—24no—
243Dic3⋊C4—24no—
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×Q8C22×D311yescharacteristic
254C22×D4—32no—
255C22×D4—62no—
256C22×D4—61no—
257C22×D4—61no—
258C22×D4—61no—
259C22×D4C22×D311yescharacteristic
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—62no—
273#127—62no—
274#126—68no—
275#125—34no—
276#125—62no—
277#124—22no—
278#124D1214yescomplemented
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—68no—
291C4×D4—62no—
292C4×D4—62no—
293#117—68no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4C22×D312yes—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
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—42no—
319#102—42no—
320#102—64no—
321#102—64no—
322C22⋊C8—64no—
323#97—22no—
324#97—62no—
325C23×C3—22no—
326C23×C3—41no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3—41no—
330C23×C3—41no—
331C23×C3—41no—
332C23×C3C22×D411yescharacteristic
333C22×D3—21no—
334C22×D3—21no—
335C22×D3—21no—
336C22×D3—21no—
337C22×D3—21no—
338C22×D3—21no—
339C22×D3—42no—
340C22×D3—42no—
341C22×D3—42no—
342C22×D3—42no—
343C22×D3—42no—
344C22×D3—42no—
345C22×D3—42no—
346C22×D3AGL(1,ℤ/8)12yes—
347C3×Q8—24no—
348C3×Q8—42no—
349C3×D4—24no—
350C3×D4—42no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—22no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4—42no—
361C2×C3×C4#12012yes—
362C2×C3×C4#12012yes—
363C2×C3×C4C22×D411yescharacteristic
364C2×C3×C4C22×D411yescharacteristic
365C2×C3×C4C2×#4112yes—
366C2×C3×C4H(5,𝔽2)12yes—
367#67—48no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—22no—
373C2×Dic3—22no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—48no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3AGL(1,ℤ/8)12yes—
384D12—42no—
385D12—42no—
386D12—42no—
387D12—42no—
388D12—84no—
389D12—84no—
390D12—88no—
391C4×D3—44no—
392C4×D3—44no—
393C4×D3—42no—
394C4×D3—42no—
395C3×C8—42no—
396C3⋊C8—24no—
397C24—21no—
398C24—31no—
399C24—62no—
400C2×Q8—21no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8C2×D1212yes—
404C2×D4—21no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—64no—
408C2×D4—62no—
409C2×D4—62no—
410C2×D4—62no—
411C2×D4—62no—
412C2×D4—62no—
413C2×D4—62no—
414C2×D4—122no—
415C2×D4—122no—
416C2×D4—122no—
417C2×D4—122no—
418C2×D4—124no—
419C2×D4—124no—
420C2×D4—124no—
421C2×D4—122no—
422C2×D4—122no—
423C2×D4—122no—
424C2×D4—122no—
425C2×D4C2×D1212yes—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—31no—
429C22×C4—31no—
430C22×C4—31no—
431C22×C4—62no—
432C22×C4—62no—
433C22×C4—64no—
434C22×C4—61no—
435C22×C4—61no—
436C22×C4—62no—
437C22×C4—62no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—62no—
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—22no—
458C4⋊C4—22no—
459C4⋊C4—32no—
460C4⋊C4—32no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—64no—
464C4⋊C4—64no—
465C4⋊C4—64no—
466C4⋊C4—64no—
467C4⋊C4—61no—
468C4⋊C4—61no—
469C22⋊C4—24no—
470C22⋊C4—24no—
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—
482C42C2×D1212yes—
483C42D3×D412yes—
484C22×C3—42no—
485C22×C3—42no—
486C22×C3—42no—
487C22×C3—81no—
488C22×C3—81no—
489C22×C3—81no—
490C22×C3—81no—
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—82no—
506C2×D3—82no—
507C2×D3—82no—
508C2×D3—82no—
509C2×D3—82no—
510C2×D3—82no—
511C2×D3—82no—
512C3×C4—22no—
513C3×C4—22no—
514C3×C4—42no—
515C3×C4—82no—
516C3×C4#54612yes—
517C3×C4#55012yes—
518Dic3—22no—
519Dic3—41no—
520Dic3—41no—
521Dic3—42no—
522Dic3—42no—
523C23—22no—
524C23—32no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—61no—
532C23—61no—
533C23—61no—
534C23—61no—
535C23—61no—
536C23—61no—
537C23—122no—
538C23—122no—
539C23—122no—
540C23—122no—
541C23—122no—
542C23—122no—
543C23—122no—
544C23C2×D3×D411yescharacteristic
545Q8—24no—
546Q8—42no—
547D4—24no—
548D4—42no—
549D4—128no—
550D4—122no—
551D4—122no—
552D4—122no—
553D4—122no—
554D4—244no—
555D4—244no—
556D4—248no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—32no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—62no—
575C2×C4—62no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—124no—
579C2×C4—124no—
580C2×C4—128no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—122no—
586C2×C4—122no—
587C2×C4#88212yes—
588C2×C4#88212yes—
589C2×C4C22×D1211yescharacteristic
590C2×C4C2×D3×D411yescharacteristic
591C2×C4#100412yes—
592C2×C4D3×#4112yes—
593C8—42no—
594C8—64no—
595C2×C3—82no—
596C2×C3#243111yescharacteristic
597C2×C3#300611yescharacteristic
598C2×C3#300611yescharacteristic
599C2×C3#317111yescharacteristic
600C2×C3#318011yescharacteristic
601C2×C3#321711yescharacteristic
602C2×C3#321911yescharacteristic
603D3—42no—
604D3—162no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—62no—
609C22—62no—
610C22—62no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—121no—
616C22—121no—
617C22—121no—
618C22—121no—
619C22—242no—
620C22—242no—
621C22—242no—
622C22—242no—
623C22—242no—
624C22—242no—
625C22—242no—
626C22C2×#88211yescharacteristic
627C22#566811yescharacteristic
628C22#571511yescharacteristic
629C22D3×AGL(1,ℤ/8)11yescharacteristic
630C22D3×AGL(1,ℤ/8)11yescharacteristic
631C22#577711yescharacteristic
632C22#577711yescharacteristic
633C4—22no—
634C4—22no—
635C4—42no—
636C4—62no—
637C4—82no—
638C4—121no—
639C4—121no—
640C4—122no—
641C4—122no—
642C4#555412yes—
643C4#559912yes—
644C3#2560511yescharacteristic, complemented, Sylow
645C2—82no—
646C2—122no—
647C2—482no—
648C2#7631311yescharacteristic
649C2#7888411yescharacteristic
650C2#7888411yescharacteristic
651C2#8166411yescharacteristic
652C2#8166811yescharacteristic
653C2#8185111yescharacteristic
654C2#8187711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1959C2
socleC23×C3C22×D4
hypercenter#1959D3
hypocenterC3#25605
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1959

Complements

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

NH ≅ G/Ntype
1. C2×#5680C2semidirect
2. C2×#5058C2semidirect
3. C2×#4713C2semidirect
4. #74375C2semidirect
5. C3×#1959C2semidirect
6. #70272C2semidirect
7. #70240C2semidirect
8. #70204C2semidirect
9. #70172C2semidirect
10. #69807C2semidirect
11. #69583C2semidirect
55. #1959D3semidirect
72. C3×#124D4semidirect
123. #870D4semidirect
278. #124D12semidirect
644. C3#25605semidirect

Maximal subgroups

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

subgroupindex
C2×#56802
C2×#50582
C2×#47132
#743752
C3×#19592
#702722
#702402
#702042
#701722
#698072
#695832
#256053