Finite Group Databasebeta

← #10799105

Subgroups — #10799105

Order 768.

1782 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#908C211yesmaximal, characteristic
2C2×C3×#428C211yesmaximal, characteristic, complemented
3C2×Dic3×M16C211yesmaximal, characteristic, complemented
4#82228C211yesmaximal, characteristic, complemented
5C2×#5001C211yesmaximal, characteristic, complemented
6C2×#4988C211yesmaximal, characteristic, complemented
7C2×#4966C211yesmaximal, characteristic, complemented
8#70629C212yesmaximal, complemented
9#70628C212yesmaximal, complemented
10#70576C212yesmaximal, complemented
11#70574C212yesmaximal, complemented
12#21236—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C23×D12C2211yescharacteristic
15C22×C3×M16C2211yescharacteristic
16C2×C3×#117C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×(C2×D3)⋊C4—22no—
19C22×C4×Dic3C2211yescharacteristic
20C22×#208C2211yescharacteristic
21C2×#908—24no—
22C2×#908—24no—
23C2×#908C418yescomplemented
24C22×D24C2212yes—
25C22×#204C2212yes—
26C3×#428—24no—
27C2×C3×#107C2212yescomplemented
28C2×C3×#106C2212yescomplemented
29Dic3×M16—24no—
30C2×#815C2212yescomplemented
31C2×C8×Dic3C2212yescomplemented
32#5001—24no—
33#4988—22no—
34#4988C2212yes—
35#4988C2212yes—
36#4988C2211yescharacteristic, complemented
37#4988C2211yescharacteristic, complemented
38#4973C2211yescharacteristic, complemented
39#4973C2211yescharacteristic, complemented
40C2×#810C2212yescomplemented
41#4966—24no—
42C2×#809C2212yes—
43C2×#809C2212yes—
44C2×#809C2212yescomplemented
45#4958—22no—
46C4×(C2×D3)⋊C4—22no—
47C22×AGL(1,ℤ/8)—31no—
48C2×#428D311yescharacteristic, complemented
49C2×#417—31no—
50C2×#418—31no—
51C2×C4×M16—31no—
52#2297—31no—
53C2×#390—31no—
54#1973—32no—
55#1972—32no—
56#1970—32no—
57#1968—32no—
58C24×D3—21no—
59C23×C3×C4—21no—
60C23×C3×C4C2311yescharacteristic
61C22×#67—21no—
62C23×Dic3D412yes—
63C2×#235—22no—
64C2×#235—22no—
65C2×#235—22no—
66C2×#235—22no—
67C2×#235—22no—
68C2×#235C2×C414yes—
69C22×D12—22no—
70C22×D12—22no—
71C22×D12—22no—
72C22×D12—22no—
73C22×D12—21no—
74C22×D12—22no—
75C22×D12C2×C414yes—
76C22×D12C2312yes—
77C22×D12C2311yescharacteristic
78C22×C4×D3—21no—
79C22×Dic6C2311yescharacteristic
80C2×C3×M16C2×C414yes—
81C2×C3×M16D414yes—
82C2×C3×M16D414yes—
83C22×C3×C8C2312yes—
84C3×#117—22no—
85C3×#117—22no—
86C2×C3×C4⋊C4—22no—
87C2×C3×C4⋊C4—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2312yes—
92C2×C3×C22⋊C4—21no—
93C2×C3×C42—21no—
94C2×(C2×D3)⋊C4—22no—
95C2×(C2×D3)⋊C4—22no—
96C2×(C2×D3)⋊C4—24no—
97C2×(C2×D3)⋊C4—24no—
98C2×(C2×D3)⋊C4—48no—
99C2×C4×Dic3—22no—
100C2×C4×Dic3—22no—
101C2×C4×Dic3D412yes—
102C2×C4×Dic3C2311yescharacteristic
103C2×C4×Dic3C2311yescharacteristic
104C2×#208—22no—
105C2×#208—22no—
106C2×#208D414yes—
107C22×C3⋊C8C2312yes—
108#908—232no—
109C2×D24—24no—
110C2×D24—24no—
111C2×D24C2×C418yescomplemented
112C2×#204—24no—
113C2×#204—24no—
114C2×#204C2×C418yescomplemented
115C3×#107—24no—
116C3×#106—24no—
117C3⋊#95—22no—
118C3⋊#95—22no—
119#815—24no—
120C8×Dic3—24no—
121#810—24no—
122#809—24no—
123#809—24no—
124#809—24no—
125C22×#41—31no—
126C23×D4—31no—
127C2×AGL(1,ℤ/8)—38no—
128C2×AGL(1,ℤ/8)—64no—
129C2×AGL(1,ℤ/8)—64no—
130C22×SD16—32no—
131C22×D8—32no—
132C22×M16—31no—
133C22×M16C2×D311yescharacteristic
134C2×#117C2×D311yescharacteristic
135C22×C4⋊C4C2×D311yescharacteristic
136C22×C22⋊C4—62no—
137C22×C42—31no—
138#428—24no—
139C2×#107C2×D312yescomplemented
140C2×#106C2×D312yescomplemented
141#418—64no—
142#417—64no—
143C2×#103—32no—
144C2×#102—32no—
145C2×#102—32no—
146C2×#102—32no—
147C4×M16—64no—
148C2×#97—32no—
149C2×C4×C8—32no—
150#391—31no—
151#391—31no—
152#390—32no—
153#390—32no—
154#390—31no—
155#390—31no—
156#390—62no—
157#386—62no—
158C4×C22⋊C4—62no—
159C24×C3C2×D411yescharacteristic
160C23×D3—24no—
161C23×D3—21no—
162C23×D3—22no—
163C23×D3—44no—
164C23×D3—44no—
165C23×D3—42no—
166C23×D3—42no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—22no—
170C22×C3×C4—44no—
171C22×C3×C4—42no—
172C22×C3×C4C22×C412yes—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×D412yes—
177C22×C3×C4#4112yes—
178C22×C3×C4#4112yes—
179C22×C3×C4C2411yescharacteristic
180C2×#67—24no—
181C2×#67—42no—
182C2×#67—42no—
183C22×Dic3—24no—
184C22×Dic3—21no—
185C22×Dic3—24no—
186C22×Dic3—42no—
187C22×Dic3C2×D412yes—
188C22×Dic3C2×D412yes—
189#235—28no—
190#235—28no—
191#235—44no—
192#235—44no—
193C2×D12—28no—
194C2×D12—28no—
195C2×D12—22no—
196C2×D12—21no—
197C2×D12—22no—
198C2×D12—22no—
199C2×D12—21no—
200C2×D12—21no—
201C2×D12—22no—
202C2×D12—21no—
203C2×D12—24no—
204C2×D12—22no—
205C2×D12—21no—
206C2×D12—44no—
207C2×D12—42no—
208C2×D12—44no—
209C2×D12—42no—
210C2×D12C22×C414yes—
211C2×D12C22×C412yes—
212C2×C4×D3—24no—
213C2×C4×D3—42no—
214C2×C4×D3—42no—
215C2×Dic6—21no—
216C2×Dic6—21no—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6C22×C412yes—
221C3×M16C22⋊C4116yescomplemented
222C2×C3×C8C22×C414yes—
223C2×C3×C8C2×D414yes—
224C2×C3×C8C2×D414yes—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—22no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—42no—
232C3×C22⋊C4—42no—
233C3×C42—42no—
234(C2×D3)⋊C4—44no—
235(C2×D3)⋊C4—48no—
236(C2×D3)⋊C4—44no—
237(C2×D3)⋊C4—48no—
238C4×Dic3—24no—
239C4×Dic3—21no—
240C4×Dic3—21no—
241C4×Dic3—21no—
242C4×Dic3—21no—
243#208—28no—
244C2×C3⋊C8—22no—
245C2×C3⋊C8—22no—
246C2×C3⋊C8C2×D414yes—
247D24—216no—
248#204—216no—
249C25—61no—
250C2×#41—34no—
251C2×#41—62no—
252C2×#41—62no—
253C2×#41—62no—
254C2×#41—62no—
255C2×#41—62no—
256C22×Q8—31no—
257C22×D4—34no—
258C22×D4—31no—
259C22×D4—32no—
260C22×D4—61no—
261C22×D4—62no—
262C22×D4—62no—
263C22×D4—61no—
264C22×D4—62no—
265C22×D4—62no—
266C22×D4—62no—
267C23×C4—21no—
268C23×C4—32no—
269C23×C4—61no—
270C23×C4C22×D311yescharacteristic
271AGL(1,ℤ/8)—632no—
272C2×SD16—38no—
273C2×SD16—64no—
274C2×SD16—64no—
275C2×D8—38no—
276C2×D8—64no—
277C2×D8—64no—
278C2×M16—34no—
279C2×M16—62no—
280C2×M16—62no—
281C2×M16C4×D314yes—
282C2×M16D1214yes—
283C2×M16#6714yes—
284C22×C8—32no—
285C22×C8C22×D312yes—
286#117—22no—
287#117—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4C22×D312yes—
292C2×C4⋊C4C22×D311yescharacteristic
293C2×C4⋊C4C22×D311yescharacteristic
294C2×C22⋊C4—21no—
295C2×C22⋊C4—64no—
296C2×C22⋊C4—64no—
297C2×C22⋊C4—62no—
298C2×C22⋊C4—62no—
299C2×C22⋊C4—128no—
300C2×C42—21no—
301C2×C42—32no—
302C2×C42—31no—
303C2×C42—31no—
304C2×C42—62no—
305C2×C42—62no—
306#107—24no—
307#106—24no—
308#103—64no—
309#102—64no—
310#102—64no—
311#102—64no—
312#97—64no—
313C4×C8—64no—
314#95—62no—
315#95—62no—
316C23×C3—22no—
317C23×C3—22no—
318C23×C3C4×D412yes—
319C23×C3#12112yes—
320C23×C3#12712yes—
321C23×C3C22×D411yescharacteristic
322C22×D3—22no—
323C22×D3—24no—
324C22×D3—416no—
325C22×D3—41no—
326C22×D3—42no—
327C22×D3—41no—
328C22×D3—42no—
329C22×D3—44no—
330C22×D3—42no—
331C22×D3—42no—
332C22×D3—44no—
333C22×D3—48no—
334C22×D3—84no—
335C22×D3—84no—
336C22×D3—88no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—42no—
340C2×C3×C4—44no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—84no—
344C2×C3×C4C2×C22⋊C414yes—
345C2×C3×C4C4×D414yes—
346C2×C3×C4C4×D412yes—
347C2×C3×C4#12114yes—
348C2×C3×C4#12112yes—
349C2×C3×C4#12414yes—
350C2×C3×C4#12712yes—
351C2×C3×C4C23×C411yescharacteristic
352C2×C3×C4C22×D411yescharacteristic
353C2×C3×C4C22×D411yescharacteristic
354C2×C3×C4C22×D411yescharacteristic
355C2×C3×C4C2×#4111yescharacteristic
356C2×C3×C4C2×#4111yescharacteristic
357#67—48no—
358C2×Dic3—22no—
359C2×Dic3—22no—
360C2×Dic3—22no—
361C2×Dic3—22no—
362C2×Dic3—22no—
363C2×Dic3—22no—
364C2×Dic3—48no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3AGL(1,ℤ/8)14yescomplemented
368D12—24no—
369D12—22no—
370D12—24no—
371D12—22no—
372D12—42no—
373D12—41no—
374D12—42no—
375D12—41no—
376D12—48no—
377C4×D3—48no—
378Dic6—22no—
379Dic6—22no—
380Dic6—41no—
381Dic6—41no—
382C3×C8C2×C22⋊C418yescomplemented
383C3⋊C8—24no—
384C24—64no—
385C24—61no—
386C24—62no—
387C24—124no—
388C24—124no—
389C24—122no—
390C24—122no—
391C24D3×D411yescharacteristic
392#41—68no—
393#41—68no—
394#41—124no—
395#41—124no—
396C2×Q8—32no—
397C2×Q8—61no—
398C2×Q8—61no—
399C2×Q8—61no—
400C2×Q8—61no—
401C2×Q8—61no—
402C2×D4—32no—
403C2×D4—34no—
404C2×D4—64no—
405C2×D4—68no—
406C2×D4—64no—
407C2×D4—68no—
408C2×D4—61no—
409C2×D4—62no—
410C2×D4—62no—
411C2×D4—61no—
412C2×D4—62no—
413C2×D4—61no—
414C2×D4—62no—
415C2×D4—61no—
416C2×D4—61no—
417C2×D4—62no—
418C2×D4—122no—
419C2×D4—122no—
420C2×D4—122no—
421C2×D4—124no—
422C2×D4—122no—
423C2×D4—124no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—32no—
428C22×C4—32no—
429C22×C4—44no—
430C22×C4—42no—
431C22×C4—64no—
432C22×C4—64no—
433C22×C4—61no—
434C22×C4—64no—
435C22×C4—122no—
436C22×C4—122no—
437C22×C4—122no—
438C22×C4C2×C4×D312yes—
439C22×C4C2×D1212yes—
440C22×C4D3×D411yescharacteristic
441C22×C4D3×D412yes—
442C22×C4#23912yes—
443C22×C4#23912yes—
444C22×C4C2×#6712yes—
445C22×C4C23×D311yescharacteristic
446SD16—616no—
447D8—616no—
448M16—68no—
449M16(C2×D3)⋊C4116yescomplemented
450C2×C8—34no—
451C2×C8—62no—
452C2×C8—62no—
453C2×C8C2×C4×D314yes—
454C2×C8C2×D1214yes—
455C2×C8C2×#6714yes—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—22no—
461C4⋊C4—21no—
462C4⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—128no—
465C22⋊C4—124no—
466C22⋊C4—128no—
467C22⋊C4—124no—
468C42—42no—
469C42—64no—
470C42—61no—
471C42—61no—
472C42—61no—
473C42—61no—
474C22×C3—22no—
475C22×C3—22no—
476C22×C3—24no—
477C22×C3—24no—
478C22×C3#39014yes—
479C22×C3C2×C4×D411yescharacteristic
480C22×C3C2×#12111yescharacteristic
481C22×C3C2×#12711yescharacteristic
482C22×C3C2×AGL(1,ℤ/8)12yes—
483C22×C3#57512yes—
484C2×D3—44no—
485C2×D3—48no—
486C2×D3—42no—
487C2×D3—44no—
488C2×D3—816no—
489C2×D3—81no—
490C2×D3—82no—
491C2×D3—81no—
492C2×D3—82no—
493C2×D3—82no—
494C2×D3—84no—
495C3×C4—81no—
496C3×C4—82no—
497C3×C4—81no—
498C3×C4#39014yes—
499C3×C4C22×C22⋊C411yescharacteristic
500C3×C4C2×C4×D411yescharacteristic
501C3×C4C2×#12111yescharacteristic
502C3×C4C2×#12411yescharacteristic
503Dic3—24no—
504Dic3—42no—
505Dic3—42no—
506C23—22no—
507C23—22no—
508C23—62no—
509C23—64no—
510C23—1216no—
511C23—128no—
512C23—121no—
513C23—122no—
514C23—124no—
515C23—122no—
516C23—122no—
517C23—124no—
518C23—121no—
519C23—122no—
520C23—244no—
521C23—248no—
522C23—244no—
523C23#89312yes—
524C23#89512yes—
525C23#94012yes—
526C23C2×D3×D411yescharacteristic
527Q8—62no—
528Q8—62no—
529Q8—121no—
530Q8—121no—
531D4—64no—
532D4—62no—
533D4—62no—
534D4—64no—
535D4—128no—
536D4—128no—
537D4—122no—
538D4—121no—
539D4—122no—
540D4—121no—
541C2×C4—34no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—62no—
549C2×C4—62no—
550C2×C4—62no—
551C2×C4—62no—
552C2×C4—62no—
553C2×C4—62no—
554C2×C4—84no—
555C2×C4—128no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—128no—
559C2×C4#89312yes—
560C2×C4#89314yes—
561C2×C4#89512yes—
562C2×C4#89514yes—
563C2×C4C2×(C2×D3)⋊C414yes—
564C2×C4#94012yes—
565C2×C4#94714yes—
566C2×C4C22×C4×D311yescharacteristic
567C2×C4C22×D1211yescharacteristic
568C2×C4C2×D3×D411yescharacteristic
569C2×C4C2×#23911yescharacteristic
570C2×C4C2×#23911yescharacteristic
571C2×C4C22×#6711yescharacteristic
572C8—64no—
573C8C2×(C2×D3)⋊C418yescomplemented
574C2×C3—24no—
575C2×C3C2×#39011yescharacteristic
576C2×C3C4×AGL(1,ℤ/8)12yes—
577C2×C3#305512yes—
578C2×C3#315111yescharacteristic
579C2×C3#315311yescharacteristic
580D3—82no—
581D3—84no—
582C22—22no—
583C22—24no—
584C22—24no—
585C22—22no—
586C22—128no—
587C22—124no—
588C22—122no—
589C22—124no—
590C22—2416no—
591C22—242no—
592C22—241no—
593C22—244no—
594C22—242no—
595C22—242no—
596C22—241no—
597C22#498814yes—
598C22C2×#89311yescharacteristic
599C22C2×#89511yescharacteristic
600C22D3×AGL(1,ℤ/8)12yes—
601C22#577712yes—
602C22C2×#94011yescharacteristic
603C4—64no—
604C4—81no—
605C4—82no—
606C4—81no—
607C4—122no—
608C4—122no—
609C4#498814yes—
610C4C2×#89311yescharacteristic
611C4C2×#89511yescharacteristic
612C4C22×(C2×D3)⋊C411yescharacteristic
613C4C2×#94711yescharacteristic
614C3#2123611yescharacteristic, complemented, Sylow
615C2—24no—
616C2—244no—
617C2—242no—
618C2#7919912yes—
619C2#7921112yes—
620C2C2×#498811yescharacteristic
621C2#8393711yescharacteristic
622C2#8397311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×#428C2
socleC23×C3C22×D4
prime radicalC22×#908C2
cosocleC22×C3×C4C24
hypercenterC2×#428D3
hypocenterC3#21236
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
2. C2×C3×#428C2semidirect
3. C2×Dic3×M16C2semidirect
4. #82228C2semidirect
5. C2×#5001C2semidirect
6. C2×#4988C2semidirect
7. C2×#4966C2semidirect
8. #70629C2semidirect
9. #70628C2semidirect
10. #70576C2semidirect
11. #70574C2semidirect
23. C2×#908C4semidirect
27. C2×C3×#107C22semidirect
28. C2×C3×#106C22semidirect
30. C2×#815C22semidirect
31. C2×C8×Dic3C22semidirect
36. #4988C22semidirect
37. #4988C22semidirect
38. #4973C22semidirect
39. #4973C22semidirect
40. C2×#810C22semidirect
44. C2×#809C22semidirect
48. C2×#428D3semidirect
111. C2×D24C2×C4semidirect
114. C2×#204C2×C4semidirect
139. C2×#107C2×D3semidirect
140. C2×#106C2×D3semidirect
221. C3×M16C22⋊C4semidirect
367. C2×Dic3AGL(1,ℤ/8)semidirect
382. C3×C8C2×C22⋊C4semidirect
449. M16(C2×D3)⋊C4semidirect
573. C8C2×(C2×D3)⋊C4semidirect
614. C3#21236semidirect

Maximal subgroups

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

subgroupindex
C22×#9082
C2×C3×#4282
C2×Dic3×M162
#822282
C2×#50012
C2×#49882
C2×#49662
#706292
#706282
#705762
#705742
#212363