Finite Group Databasebeta

← #10795229

Subgroups — #10795229

Order 768.

1782 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D3×D4C211yesmaximal, characteristic
2C2×#4705C211yesmaximal, characteristic, complemented
3#73753C211yesmaximal, characteristic, complemented
4C3×#1762C211yesmaximal, characteristic, complemented
5#71609C211yesmaximal, characteristic, complemented
6#70198C212yesmaximal
7#70153C212yesmaximal, complemented
8#69812C211yesmaximal, characteristic, complemented
9#69474C211yesmaximal, characteristic, complemented
10#68892C212yesmaximal, complemented
11#68872C212yesmaximal, complemented
12#20738—31nomaximal, Sylow
13C22×D3×D4C2211yescharacteristic
14C23×C4×D3—21no—
15C2×C3×C4×D4C2211yescharacteristic
16C2×D4×Dic3C2211yescharacteristic
17C2×C4×#67—21no—
18C4×D3×D4—24no—
19C4×D3×D4C418yescomplemented
20C2×#893C2211yescharacteristic
21C2×D3×C4⋊C4C2211yescharacteristic
22C2×#67⋊C4—21no—
23C2×D3×C22⋊C4—21no—
24C2×C4×D12C2211yescharacteristic
25C2×#872—21no—
26C2×C42×D3C2211yescharacteristic
27C2×C3×#102C2212yes—
28C2×C3×#97C2211yescharacteristic, complemented
29C3×C4×C4⋊C4C2211yescharacteristic, complemented
30C2×#832C2212yes—
31C2×#821C2212yes—
32C2×(C2×D3)⋊C8C2212yes—
33C2×Dic3⋊C8C2212yescomplemented
34#4987C2212yes—
35#4975C2212yescomplemented
36C2×#809C2212yes—
37C4×C4⋊Dic3C2211yescharacteristic, complemented
38C4×Dic3⋊C4—21no—
39C2×#803C2211yescharacteristic, complemented
40#4705—24no—
41#4667—21no—
42#4650—21no—
43C3×#340C2212yescomplemented
44#4551C2212yescomplemented
45#4529C2212yescomplemented
46#4529C2212yescomplemented
47C22×C4×D4—31no—
48C2×#432—31no—
49#2285—31no—
50#1824—32no—
51#1802—32no—
52#1802—32no—
53#1786—32no—
54#1762—31no—
55#1762—31no—
56#1762—31no—
57#1762D311yescharacteristic, complemented
58C24×D3—21no—
59C22×C3×D4C2311yescharacteristic
60C23×C3×C4—21no—
61C22×#67—21no—
62C23×Dic3—21no—
63C2×D3×D4—24no—
64C2×D3×D4—24no—
65C2×D3×D4—22no—
66C2×D3×D4C2×C414yes—
67C22×D12C2311yescharacteristic
68C22×C4×D3—21no—
69C22×C4×D3—21no—
70C22×C4×D3—21no—
71C22×C4×D3—28no—
72C22×C4×D3—48no—
73C22×C4×D3D411yescharacteristic
74C22×C4×D3D411yescharacteristic
75C22×C4×D3C2311yescharacteristic
76C22×C3×C8C2312yes—
77C3×C4×D4—22no—
78C3×C4×D4C2×C414yes—
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C22⋊C4—21no—
82C2×C3×C42—21no—
83C2×C3×C42C2311yescharacteristic
84C2×#217—21no—
85D4×Dic3—22no—
86D4×Dic3C2×C414yes—
87C4×#67—28no—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—22no—
91C2×C4⋊Dic3C2312yes—
92C2×C4⋊Dic3C2311yescharacteristic
93C2×Dic3⋊C4—21no—
94C2×Dic3⋊C4—22no—
95C2×Dic3⋊C4—22no—
96C2×C4×Dic3—21no—
97C2×C4×Dic3D411yescharacteristic
98C2×C4×Dic3D411yescharacteristic
99C2×C4×Dic3C2311yescharacteristic
100C22×C3⋊C8C2312yes—
101#893—22no—
102#893C2×C414yes—
103D3×C4⋊C4—22no—
104D3×C4⋊C4C2×C414yes—
105#67⋊C4—28no—
106D3×C22⋊C4—28no—
107C4×D12—22no—
108C4×D12C2×C414yes—
109#872—44no—
110C42×D3C2×C414yes—
111C42×D3D414yes—
112C3×#102—24no—
113C3×#97—22no—
114C3×#95—21no—
115#832—24no—
116C3⋊#95—21no—
117C3⋊#95—22no—
118#821—24no—
119(C2×D3)⋊C8—24no—
120Dic3⋊C8—24no—
121#809—24no—
122#803—22no—
123C23×D4—31no—
124C24×C4—61no—
125C2×C4×D4—31no—
126C2×C4×D4—38no—
127C2×C4×D4—31no—
128C2×C4×D4—31no—
129C2×C4×D4—61no—
130C2×C4×D4—64no—
131C2×C4×D4—61no—
132C2×C4×D4C2×D311yescharacteristic
133C2×#117—61no—
134C22×C4⋊C4—31no—
135C22×C22⋊C4—61no—
136C22×C42—31no—
137#432—64no—
138C2×C4⋊C8—32no—
139C2×#102—32no—
140C2×#102—32no—
141C2×#102—32no—
142C2×#102C2×D312yes—
143C2×C22⋊C8—32no—
144C2×#97—31no—
145C2×#97C2×D311yescharacteristic, complemented
146#384—32no—
147#382—61no—
148#380—32no—
149#380—61no—
150C4×C4⋊C4—31no—
151C4×C4⋊C4—61no—
152C4×C4⋊C4C2×D311yescharacteristic, complemented
153#340—32no—
154#340—32no—
155#340—32no—
156#340C2×D312yescomplemented
157C24×C3—21no—
158C23×D3—28no—
159C23×D3—21no—
160C23×D3—24no—
161C23×D3—44no—
162C23×D3—42no—
163C23×D3—42no—
164C23×D3C2×D411yescharacteristic
165C2×C3×D4—21no—
166C2×C3×D4—22no—
167C2×C3×D4C22×C412yes—
168C2×C3×D4#4114yes—
169C22×C3×C4—21no—
170C22×C3×C4—24no—
171C22×C3×C4—22no—
172C22×C3×C4—44no—
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2411yescharacteristic
176C2×#67—28no—
177C2×#67—24no—
178C22×Dic3—21no—
179C22×Dic3—24no—
180C22×Dic3—22no—
181C22×Dic3—44no—
182C22×Dic3C2×D411yescharacteristic
183C22×Dic3C2×D411yescharacteristic
184C22×Dic3C2×D411yescharacteristic
185D3×D4—216no—
186D3×D4—28no—
187D3×D4—28no—
188C2×D12—21no—
189C2×D12—22no—
190C2×D12C22×C412yes—
191C2×D12#4114yes—
192C2×C4×D3—22no—
193C2×C4×D3—22no—
194C2×C4×D3—22no—
195C2×C4×D3—22no—
196C2×C4×D3—24no—
197C2×C4×D3—24no—
198C2×C4×D3—24no—
199C2×C4×D3—24no—
200C2×C4×D3—432no—
201C2×C4×D3—44no—
202C2×C4×D3—44no—
203C2×C4×D3—44no—
204C2×C4×D3C22⋊C414yes—
205C2×C4×D3C22⋊C414yes—
206C2×C4×D3C22×C412yes—
207C2×C4×D3C2×D412yes—
208C2×C3×C8—22no—
209C2×C3×C8—22no—
210C2×C3×C8—22no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4C22×C412yes—
215C3×C22⋊C4—24no—
216C3×C42—42no—
217C3×C42C22×C412yes—
218C3×C42C2×D412yes—
219#217—24no—
220(C2×D3)⋊C4—24no—
221(C2×D3)⋊C4—24no—
222(C2×D3)⋊C4—44no—
223C4⋊Dic3—21no—
224C4⋊Dic3—22no—
225C4⋊Dic3—22no—
226C4⋊Dic3C22×C412yes—
227Dic3⋊C4—24no—
228Dic3⋊C4—44no—
229Dic3⋊C4—44no—
230C4×Dic3—42no—
231C4×Dic3C22⋊C414yes—
232C4×Dic3C22⋊C414yes—
233C4×Dic3C22×C412yes—
234C4×Dic3C2×D412yes—
235C2×C3⋊C8—22no—
236C2×C3⋊C8—22no—
237C2×C3⋊C8—22no—
238C25—61no—
239C22×D4—34no—
240C22×D4—31no—
241C22×D4—62no—
242C22×D4—61no—
243C22×D4—64no—
244C22×D4—64no—
245C22×D4C22×D311yescharacteristic
246C23×C4—21no—
247C23×C4—31no—
248C23×C4—31no—
249C23×C4—31no—
250C23×C4—61no—
251C23×C4—68no—
252C23×C4—61no—
253C23×C4—61no—
254C23×C4—61no—
255C23×C4—128no—
256C22×C8—32no—
257C22×C8C22×D312yes—
258C4×D4—22no—
259C4×D4—34no—
260C4×D4—34no—
261C4×D4—34no—
262C4×D4—68no—
263C4×D4—62no—
264C4×D4—68no—
265C4×D4—62no—
266C4×D4—62no—
267C4×D4C4×D314yes—
268#117—124no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—34no—
271C2×C4⋊C4—32no—
272C2×C4⋊C4—62no—
273C2×C4⋊C4—61no—
274C2×C4⋊C4—62no—
275C2×C4⋊C4—62no—
276C2×C4⋊C4C22×D312yes—
277C2×C4⋊C4C22×D311yescharacteristic
278C2×C22⋊C4—21no—
279C2×C22⋊C4—68no—
280C2×C22⋊C4—61no—
281C2×C22⋊C4—61no—
282C2×C22⋊C4—61no—
283C2×C22⋊C4—62no—
284C2×C42—21no—
285C2×C42—31no—
286C2×C42—31no—
287C2×C42—31no—
288C2×C42—34no—
289C2×C42—34no—
290C2×C42—61no—
291C2×C42C22×D311yescharacteristic
292C4⋊C8—64no—
293#102—24no—
294#102—64no—
295#102—64no—
296#102—64no—
297C22⋊C8—64no—
298#97—22no—
299#97—62no—
300#95—21no—
301#95—62no—
302#95—61no—
303C23×C3—22no—
304C23×C3—24no—
305C23×C3—42no—
306C23×C3—41no—
307C23×C3—41no—
308C23×C3C22×D411yescharacteristic
309C22×D3—22no—
310C22×D3—22no—
311C22×D3—216no—
312C22×D3—22no—
313C22×D3—24no—
314C22×D3—48no—
315C22×D3—416no—
316C22×D3—416no—
317C22×D3—42no—
318C22×D3—41no—
319C22×D3—41no—
320C22×D3—48no—
321C22×D3C2×C22⋊C412yes—
322C22×D3AGL(1,ℤ/8)12yes—
323C22×D3#13712yes—
324C3×D4—22no—
325C3×D4—24no—
326C3×D4#11714yescomplemented
327C2×C3×C4—21no—
328C2×C3×C4—22no—
329C2×C3×C4—21no—
330C2×C3×C4—22no—
331C2×C3×C4—48no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4C2×C22⋊C412yes—
337C2×C3×C4C2×C22⋊C412yes—
338C2×C3×C4C23×C411yescharacteristic
339C2×C3×C4C22×D411yescharacteristic
340C2×C3×C4C2×#4112yes—
341C2×C3×C4C2×#4112yes—
342#67—216no—
343C2×Dic3—22no—
344C2×Dic3—21no—
345C2×Dic3—21no—
346C2×Dic3—22no—
347C2×Dic3—42no—
348C2×Dic3—48no—
349C2×Dic3—42no—
350C2×Dic3—42no—
351C2×Dic3—42no—
352C2×Dic3C2×C22⋊C412yes—
353C2×Dic3C2×C22⋊C412yes—
354C2×Dic3C2×C22⋊C412yes—
355C2×Dic3AGL(1,ℤ/8)14yes—
356D12—22no—
357D12—24no—
358D12#11714yescomplemented
359C4×D3—28no—
360C4×D3—22no—
361C4×D3—22no—
362C4×D3—24no—
363C4×D3—28no—
364C4×D3—48no—
365C4×D3—48no—
366C4×D3—48no—
367C3×C8—42no—
368C3⋊C8—42no—
369C24—21no—
370C24—31no—
371C24—64no—
372C24—68no—
373C24—61no—
374C24—122no—
375C24—122no—
376C24—124no—
377C2×D4—21no—
378C2×D4—22no—
379C2×D4—32no—
380C2×D4—34no—
381C2×D4—64no—
382C2×D4—68no—
383C2×D4—616no—
384C2×D4—68no—
385C2×D4—68no—
386C2×D4—62no—
387C2×D4—61no—
388C2×D4C2×C4×D312yes—
389C2×D4#23514yes—
390C22×C4—21no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—31no—
394C22×C4—31no—
395C22×C4—31no—
396C22×C4—34no—
397C22×C4—34no—
398C22×C4—32no—
399C22×C4—32no—
400C22×C4—44no—
401C22×C4—64no—
402C22×C4—64no—
403C22×C4—62no—
404C22×C4—62no—
405C22×C4—64no—
406C22×C4—64no—
407C22×C4—62no—
408C22×C4—62no—
409C22×C4—61no—
410C22×C4—64no—
411C22×C4—62no—
412C22×C4—124no—
413C22×C4—124no—
414C22×C4—124no—
415C22×C4—1232no—
416C22×C4—124no—
417C22×C4D3×D411yescharacteristic
418C22×C4D3×D411yescharacteristic
419C22×C4C23×D311yescharacteristic
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—62no—
424C2×C8—62no—
425C2×C8—62no—
426C4⋊C4—21no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—32no—
430C4⋊C4—64no—
431C4⋊C4—61no—
432C4⋊C4—62no—
433C4⋊C4—62no—
434C4⋊C4—124no—
435C4⋊C4—124no—
436C4⋊C4C2×C4×D312yes—
437C22⋊C4—24no—
438C22⋊C4—64no—
439C22⋊C4—64no—
440C22⋊C4—64no—
441C22⋊C4—124no—
442C42—34no—
443C42—34no—
444C42—32no—
445C42—32no—
446C42—42no—
447C42—122no—
448C42C2×C4×D312yes—
449C42D3×D412yes—
450C22×C3—24no—
451C22×C3—44no—
452C22×C3—44no—
453C22×C3—42no—
454C22×C3—42no—
455C22×C3C22×C22⋊C411yescharacteristic
456C22×C3#52512yes—
457C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
458C22×C3C2×AGL(1,ℤ/8)12yes—
459C22×C3C2×#13711yescharacteristic
460C2×D3—24no—
461C2×D3—24no—
462C2×D3—22no—
463C2×D3—22no—
464C2×D3—24no—
465C2×D3—432no—
466C2×D3—44no—
467C2×D3—44no—
468C2×D3—42no—
469C2×D3—42no—
470C2×D3#41714yes—
471C3×C4—22no—
472C3×C4—22no—
473C3×C4—42no—
474C3×C4—82no—
475C3×C4C2×#11711yescharacteristic
476C3×C4C2×#11711yescharacteristic
477C3×C4#52512yes—
478Dic3—22no—
479Dic3—22no—
480Dic3—42no—
481Dic3—82no—
482Dic3#41814yes—
483C23—22no—
484C23—24no—
485C23—32no—
486C23—32no—
487C23—32no—
488C23—42no—
489C23—41no—
490C23—41no—
491C23—616no—
492C23—62no—
493C23—62no—
494C23—62no—
495C23—64no—
496C23—128no—
497C23—128no—
498C23—1216no—
499C23—1216no—
500C23—122no—
501C23—121no—
502C23—121no—
503C23C2×D3×D411yescharacteristic
504D4—22no—
505D4—24no—
506D4—34no—
507D4—616no—
508D4—62no—
509D4—64no—
510D4#87214yescomplemented
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—32no—
516C2×C4—34no—
517C2×C4—32no—
518C2×C4—32no—
519C2×C4—48no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—68no—
525C2×C4—68no—
526C2×C4—61no—
527C2×C4—61no—
528C2×C4—62no—
529C2×C4—62no—
530C2×C4—62no—
531C2×C4—64no—
532C2×C4—62no—
533C2×C4—128no—
534C2×C4—128no—
535C2×C4—128no—
536C2×C4—128no—
537C2×C4—122no—
538C2×C4—122no—
539C2×C4—122no—
540C2×C4—122no—
541C2×C4D3×C22⋊C412yes—
542C2×C4D3×C22⋊C412yes—
543C2×C4C22×C4×D311yescharacteristic
544C2×C4C2×#23512yes—
545C2×C4C2×D3×D411yescharacteristic
546C2×C4D3×#4112yes—
547C8—42no—
548C8—122no—
549C2×C3—44no—
550C2×C3#228011yescharacteristic
551C2×C3C2×#41811yescharacteristic
552C2×C3C2×#41711yescharacteristic
553C2×C3#310912yes—
554C2×C3#311012yes—
555D3—24no—
556D3—44no—
557C22—24no—
558C22—34no—
559C22—42no—
560C22—42no—
561C22—44no—
562C22—44no—
563C22—62no—
564C22—62no—
565C22—64no—
566C22—64no—
567C22—64no—
568C22—1232no—
569C22—122no—
570C22—122no—
571C22—124no—
572C22—124no—
573C22C2×D3×C22⋊C411yescharacteristic
574C22#554712yes—
575C22D3×AGL(1,ℤ/8)11yescharacteristic
576C22D3×AGL(1,ℤ/8)12yes—
577C22D3×#13711yescharacteristic
578C4—22no—
579C4—22no—
580C4—34no—
581C4—42no—
582C4—62no—
583C4—62no—
584C4—82no—
585C4—122no—
586C4—242no—
587C4C2×#87211yescharacteristic
588C4D3×#11711yescharacteristic
589C4#554712yes—
590C3#2073811yescharacteristic, complemented, Sylow
591C2—44no—
592C2—64no—
593C2—124no—
594C2#7356111yescharacteristic
595C2D3×#41711yescharacteristic
596C2D3×#41811yescharacteristic
597C2#7977912yes—
598C2#7986912yes—
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1762C2
socleC23×C3C22×D4
prime radicalC22×D3×D4C22
cosocleC22×C3×C4C24
hypercenter#1762D3
hypocenterC3#20738
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
2. C2×#4705C2semidirect
3. #73753C2semidirect
4. C3×#1762C2semidirect
5. #71609C2semidirect
7. #70153C2semidirect
8. #69812C2semidirect
9. #69474C2semidirect
10. #68892C2semidirect
11. #68872C2semidirect
19. C4×D3×D4C4semidirect
28. C2×C3×#97C22semidirect
29. C3×C4×C4⋊C4C22semidirect
33. C2×Dic3⋊C8C22semidirect
35. #4975C22semidirect
37. C4×C4⋊Dic3C22semidirect
39. C2×#803C22semidirect
43. C3×#340C22semidirect
44. #4551C22semidirect
45. #4529C22semidirect
46. #4529C22semidirect
57. #1762D3semidirect
145. C2×#97C2×D3semidirect
152. C4×C4⋊C4C2×D3semidirect
156. #340C2×D3semidirect
326. C3×D4#117semidirect
358. D12#117semidirect
510. D4#872semidirect
590. C3#20738semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D3×D42
C2×#47052
#737532
C3×#17622
#716092
#701982
#701532
#698122
#694742
#688922
#688722
#207383