Finite Group Databasebeta

← #10799112

Subgroups — #10799112

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×#949C211yesmaximal, characteristic
2C3×#2297C211yesmaximal, characteristic, complemented
3C2×#5232C211yesmaximal, characteristic, complemented
4C2×#5001C211yesmaximal, characteristic, complemented
5C2×#4995C211yesmaximal, characteristic, complemented
6C2×#4939C211yesmaximal, characteristic, complemented
7C2×C4×#208C211yesmaximal, characteristic, complemented
8#69679C212yesmaximal, complemented
9#69676C212yesmaximal, complemented
10#69648C212yesmaximal, complemented
11#69647C212yesmaximal, complemented
12#21236—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×D12C2211yescharacteristic
15C2×C3×#117C2211yescharacteristic
16C22×C3×C42C2211yescharacteristic
17C2×#949—24no—
18C2×#949—24no—
19C2×#949C418yescomplemented
20C22×#215C2212yes—
21C22×#213C2212yes—
22C22×(C2×D3)⋊C4—22no—
23C22×C4⋊Dic3C2211yescharacteristic
24C22×#208C2211yescharacteristic
25C22×#208C2211yescharacteristic
26C3×#391C2211yescharacteristic, complemented
27C3×#391C2211yescharacteristic, complemented
28C3×#390C2211yescharacteristic, complemented
29C3×#390C2211yescharacteristic, complemented
30C3×#386—22no—
31C3×C4×C22⋊C4—22no—
32#5232—24no—
33C2×#835C2212yescomplemented
34C2×#832C2212yescomplemented
35#5001—24no—
36#4995—24no—
37C2×#809C2212yes—
38C2×#809C2212yes—
39C2×#808C2212yescomplemented
40C2×#807C2212yescomplemented
41#4939—22no—
42#4939C2212yes—
43#4939C2212yes—
44C4×#208—24no—
45C2×#803C2212yescomplemented
46C2×C4×C3⋊C8C2212yescomplemented
47C22×AGL(1,ℤ/8)—31no—
48C2×#428—31no—
49C2×#417—31no—
50C2×#418—31no—
51C2×C4×M16—31no—
52#2297D311yescharacteristic, complemented
53C2×#390—31no—
54#1973—32no—
55#1972—32no—
56#1970—32no—
57#1968—32no—
58C24×D3—21no—
59C2×C3×#41—22no—
60C2×C3×#41—22no—
61C2×C3×#41—22no—
62C2×C3×#41—22no—
63C2×C3×#41—22no—
64C2×C3×#41C2×C414yes—
65C22×C3×Q8C2311yescharacteristic
66C22×C3×D4—21no—
67C22×C3×D4C2311yescharacteristic
68C23×C3×C4—21no—
69C23×C3×C4D412yes—
70C23×C3×C4C2311yescharacteristic
71C23×Dic3—21no—
72C22×D12—22no—
73C22×D12—22no—
74C22×D12—22no—
75C22×D12—22no—
76C22×D12—21no—
77C22×D12—22no—
78C22×D12C2×C414yes—
79C22×D12C2312yes—
80C3×#117—22no—
81C3×#117—22no—
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C22⋊C4—21no—
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4—22no—
87C2×C3×C42—22no—
88C2×C3×C42—22no—
89C2×C3×C42—21no—
90C2×C3×C42D412yes—
91C2×C3×C42C2311yescharacteristic
92C2×C3×C42C2311yescharacteristic
93#949—232no—
94C2×#215—24no—
95C2×#215—24no—
96C2×#215C2×C418yescomplemented
97C2×#213—24no—
98C2×#213—24no—
99C2×#213C2×C418yescomplemented
100C2×(C2×D3)⋊C4—24no—
101C2×(C2×D3)⋊C4—24no—
102C2×(C2×D3)⋊C4—48no—
103C2×C4⋊Dic3—22no—
104C2×C4⋊Dic3—22no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3C2312yes—
107C2×#208—22no—
108C2×#208—22no—
109C2×#208C2×C414yes—
110C2×#208D414yes—
111C2×#208D414yes—
112C2×#208D414yes—
113C22×C3⋊C8C2312yes—
114C22×C3⋊C8C2312yes—
115C3×#95—22no—
116C3×#95—22no—
117#835—24no—
118#832—24no—
119#809—24no—
120#809—24no—
121#808—24no—
122#807—24no—
123#803—24no—
124C4×C3⋊C8—24no—
125C22×#41C2×D311yescharacteristic
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×M16—31no—
134C2×#117C2×D311yescharacteristic
135C22×C4⋊C4—31no—
136C22×C22⋊C4—62no—
137C22×C42C2×D311yescharacteristic
138#428—64no—
139C2×#107—32no—
140C2×#106—32no—
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#391C2×D311yescharacteristic, complemented
151#391C2×D311yescharacteristic, complemented
152#390—32no—
153#390—32no—
154#390—62no—
155#390C2×D311yescharacteristic, complemented
156#390C2×D311yescharacteristic, complemented
157#386—22no—
158C4×C22⋊C4—22no—
159C24×C3—21no—
160C24×C3C2×D411yescharacteristic
161C23×D3—24no—
162C23×D3—22no—
163C23×D3—44no—
164C23×D3—44no—
165C23×D3—42no—
166C23×D3—42no—
167C3×#41—28no—
168C3×#41—28no—
169C3×#41—44no—
170C3×#41—44no—
171C2×C3×Q8—21no—
172C2×C3×Q8—21no—
173C2×C3×Q8—21no—
174C2×C3×Q8—21no—
175C2×C3×Q8—21no—
176C2×C3×Q8C22×C412yes—
177C2×C3×D4—24no—
178C2×C3×D4—21no—
179C2×C3×D4—21no—
180C2×C3×D4—21no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—42no—
184C2×C3×D4—42no—
185C2×C3×D4C22×C412yes—
186C22×C3×C4—24no—
187C22×C3×C4—21no—
188C22×C3×C4—24no—
189C22×C3×C4—24no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—42no—
193C22×C3×C4—42no—
194C22×C3×C4—42no—
195C22×C3×C4C22×C412yes—
196C22×C3×C4C2×D412yes—
197C22×C3×C4C2×D412yes—
198C22×C3×C4C2×D412yes—
199C22×C3×C4C2×D412yes—
200C22×C3×C4C2×D412yes—
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4#4112yes—
203C22×C3×C4#4112yes—
204C22×C3×C4C2411yescharacteristic
205C22×Dic3—22no—
206C22×Dic3—44no—
207C22×Dic3—42no—
208C2×D12—28no—
209C2×D12—28no—
210C2×D12—22no—
211C2×D12—22no—
212C2×D12—22no—
213C2×D12—22no—
214C2×D12—24no—
215C2×D12—22no—
216C2×D12—44no—
217C2×D12—42no—
218C2×D12—44no—
219C2×D12—42no—
220C2×D12C22×C414yes—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C22⋊C4—44no—
226C3×C22⋊C4—42no—
227C3×C22⋊C4—44no—
228C3×C42—24no—
229C3×C42—21no—
230C3×C42—21no—
231C3×C42—21no—
232C3×C42—21no—
233C3×C42—42no—
234#215—216no—
235#213—216no—
236(C2×D3)⋊C4—48no—
237(C2×D3)⋊C4—48no—
238C4⋊Dic3—22no—
239C4⋊Dic3—22no—
240C4⋊Dic3—42no—
241#208—28no—
242#208C22⋊C4116yescomplemented
243C2×C3⋊C8—22no—
244C2×C3⋊C8—22no—
245C2×C3⋊C8C22×C414yes—
246C2×C3⋊C8C2×D414yes—
247C2×C3⋊C8C2×D414yes—
248C2×C3⋊C8C2×D414yes—
249C25—61no—
250C2×#41—22no—
251C2×#41—22no—
252C2×#41—22no—
253C2×#41—22no—
254C2×#41—22no—
255C2×#41C4×D314yes—
256C22×Q8C22×D311yescharacteristic
257C22×D4—21no—
258C22×D4—34no—
259C22×D4—32no—
260C22×D4—62no—
261C22×D4—62no—
262C22×D4—62no—
263C22×D4—62no—
264C22×D4—62no—
265C22×D4—61no—
266C22×D4C22×D311yescharacteristic
267C23×C4—21no—
268C23×C4—61no—
269C23×C4#6712yes—
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—34no—
280C2×M16—34no—
281C2×M16—34no—
282C2×M16—62no—
283C2×M16—62no—
284C22×C8—32no—
285C22×C8—32no—
286#117—22no—
287#117—22no—
288C2×C4⋊C4—32no—
289C2×C4⋊C4—62no—
290C2×C4⋊C4—62no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4C22×D311yescharacteristic
293C2×C4⋊C4C22×D311yescharacteristic
294C2×C22⋊C4—21no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—64no—
298C2×C22⋊C4—64no—
299C2×C22⋊C4—128no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—21no—
303C2×C42#6712yes—
304C2×C42C22×D311yescharacteristic
305C2×C42C22×D311yescharacteristic
306#107—64no—
307#106—64no—
308#103—64no—
309#102—64no—
310#102—64no—
311#102—64no—
312#97—64no—
313C4×C8—64no—
314#95—22no—
315#95—22no—
316C23×C3—22no—
317C23×C3—22no—
318C23×C3—22no—
319C23×C3—41no—
320C23×C3—41no—
321C23×C3—42no—
322C23×C3—42no—
323C23×C3C4×D412yes—
324C23×C3#12112yes—
325C23×C3#12712yes—
326C23×C3C22×D411yescharacteristic
327C22×D3—24no—
328C22×D3—416no—
329C22×D3—42no—
330C22×D3—42no—
331C22×D3—44no—
332C22×D3—44no—
333C22×D3—48no—
334C22×D3—84no—
335C22×D3—84no—
336C22×D3—88no—
337C3×Q8—22no—
338C3×Q8—22no—
339C3×Q8—41no—
340C3×Q8—41no—
341C3×D4—22no—
342C3×D4—22no—
343C3×D4—48no—
344C3×D4—41no—
345C3×D4—41no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—48no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—48no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4C2×C22⋊C414yes—
361C2×C3×C4C4×D414yes—
362C2×C3×C4C4×D412yes—
363C2×C3×C4#12114yes—
364C2×C3×C4#12112yes—
365C2×C3×C4#12414yes—
366C2×C3×C4#12712yes—
367C2×C3×C4AGL(1,ℤ/8)14yescomplemented
368C2×C3×C4C23×C411yescharacteristic
369C2×C3×C4C22×D411yescharacteristic
370C2×C3×C4C22×D411yescharacteristic
371C2×C3×C4C22×D411yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374C2×Dic3—42no—
375C2×Dic3—44no—
376C2×Dic3—84no—
377D12—24no—
378D12—24no—
379D12—42no—
380D12—42no—
381D12—48no—
382C3⋊C8—24no—
383C3⋊C8C2×C22⋊C418yescomplemented
384C24—21no—
385C24—64no—
386C24—62no—
387C24—124no—
388C24—124no—
389C24—122no—
390C24—122no—
391C24C2×#6711yescharacteristic
392#41—28no—
393#41—28no—
394#41—44no—
395#41—44no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8C2×C4×D312yes—
402C2×D4—24no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—34no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—68no—
412C2×D4—68no—
413C2×D4—64no—
414C2×D4—62no—
415C2×D4—62no—
416C2×D4—62no—
417C2×D4—62no—
418C2×D4—62no—
419C2×D4—124no—
420C2×D4—124no—
421C2×D4—122no—
422C2×D4—122no—
423C2×D4C2×C4×D312yes—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—62no—
434C22×C4—124no—
435C22×C4—122no—
436C22×C4C2×C4×D312yes—
437C22×C4#23512yes—
438C22×C4D3×D412yes—
439C22×C4D3×D412yes—
440C22×C4D3×D412yes—
441C22×C4#23712yes—
442C22×C4C2×#6712yes—
443C22×C4C2×#6712yes—
444C22×C4C2×#6711yescharacteristic
445C22×C4C23×D311yescharacteristic
446SD16—616no—
447D8—616no—
448M16—316no—
449M16—68no—
450C2×C8—34no—
451C2×C8—34no—
452C2×C8—34no—
453C2×C8—34no—
454C2×C8—62no—
455C2×C8—62no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—62no—
461C4⋊C4—62no—
462C4⋊C4—122no—
463C22⋊C4—44no—
464C22⋊C4—42no—
465C22⋊C4—44no—
466C22⋊C4—128no—
467C22⋊C4—128no—
468C42—24no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—42no—
474C22×C3—22no—
475C22×C3—22no—
476C22×C3—24no—
477C22×C3—24no—
478C22×C3—44no—
479C22×C3—42no—
480C22×C3—81no—
481C22×C3—81no—
482C22×C3—82no—
483C22×C3#39014yes—
484C22×C3C2×C4×D411yescharacteristic
485C22×C3C2×#12111yescharacteristic
486C22×C3C2×#12711yescharacteristic
487C22×C3C2×AGL(1,ℤ/8)12yes—
488C22×C3#57512yes—
489C2×D3—48no—
490C2×D3—44no—
491C2×D3—816no—
492C2×D3—82no—
493C2×D3—82no—
494C2×D3—84no—
495C3×C4—24no—
496C3×C4—42no—
497C3×C4—42no—
498C3×C4—81no—
499C3×C4—81no—
500C3×C4#39014yes—
501C3×C4C22×C22⋊C411yescharacteristic
502C3×C4C2×C4×D411yescharacteristic
503C3×C4C2×#12111yescharacteristic
504C3×C4C2×#12411yescharacteristic
505Dic3—82no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—41no—
510C23—41no—
511C23—42no—
512C23—42no—
513C23—64no—
514C23—1216no—
515C23—128no—
516C23—122no—
517C23—122no—
518C23—124no—
519C23—124no—
520C23—244no—
521C23—244no—
522C23—248no—
523C23C4×#6712yes—
524C23#93912yes—
525C23#94012yes—
526C23C22×#6711yescharacteristic
527Q8—22no—
528Q8—22no—
529Q8—41no—
530Q8—41no—
531D4—22no—
532D4—22no—
533D4—48no—
534D4—41no—
535D4—41no—
536D4—64no—
537D4—64no—
538D4—128no—
539D4—122no—
540D4—122no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—48no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—48no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—122no—
556C2×C4—124no—
557C2×C4—244no—
558C2×C4D3×C22⋊C414yes—
559C2×C4#67⋊C414yes—
560C2×C4#88414yes—
561C2×C4#88514yes—
562C2×C4C4×#6712yes—
563C2×C4#93912yes—
564C2×C4#94012yes—
565C2×C4#94914yescomplemented
566C2×C4C22×C4×D311yescharacteristic
567C2×C4C2×#23511yescharacteristic
568C2×C4C2×D3×D411yescharacteristic
569C2×C4C2×D3×D411yescharacteristic
570C2×C4C2×D3×D411yescharacteristic
571C2×C4C2×#23711yescharacteristic
572C8—38no—
573C8—64no—
574C2×C3—24no—
575C2×C3—82no—
576C2×C3C2×#39011yescharacteristic
577C2×C3C4×AGL(1,ℤ/8)12yes—
578C2×C3#305512yes—
579C2×C3#315111yescharacteristic
580C2×C3#315311yescharacteristic
581D3—84no—
582C22—22no—
583C22—22no—
584C22—24no—
585C22—24no—
586C22—44no—
587C22—42no—
588C22—81no—
589C22—81no—
590C22—82no—
591C22—128no—
592C22—124no—
593C22—2416no—
594C22—242no—
595C22—242no—
596C22—244no—
597C22#495714yes—
598C22C2×C4×#6711yescharacteristic
599C22C2×#93911yescharacteristic
600C22C2×#94011yescharacteristic
601C22C2×#94912yes—
602C22#582212yes—
603C4—24no—
604C4—42no—
605C4—42no—
606C4—81no—
607C4—81no—
608C4—242no—
609C4#495714yes—
610C4C2×D3×C22⋊C411yescharacteristic
611C4C2×#67⋊C411yescharacteristic
612C4C2×#88411yescharacteristic
613C4C2×#88511yescharacteristic
614C3#2123611yescharacteristic, complemented, Sylow
615C2—24no—
616C2—82no—
617C2—244no—
618C2C2×#495711yescharacteristic
619C2C4×#94912yes—
620C2#8292412yes—
621C2#8328711yescharacteristic
622C2#8328911yescharacteristic
subgroupG/H
centerC23C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2297C2
socleC23×C3C22×D4
prime radicalC22×#949C2
cosocleC22×C3×C4C24
hypercenter#2297D3
hypocenterC3#21236
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2297

Complements

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

NH ≅ G/Ntype
2. C3×#2297C2semidirect
3. C2×#5232C2semidirect
4. C2×#5001C2semidirect
5. C2×#4995C2semidirect
6. C2×#4939C2semidirect
7. C2×C4×#208C2semidirect
8. #69679C2semidirect
9. #69676C2semidirect
10. #69648C2semidirect
11. #69647C2semidirect
19. C2×#949C4semidirect
26. C3×#391C22semidirect
27. C3×#391C22semidirect
28. C3×#390C22semidirect
29. C3×#390C22semidirect
33. C2×#835C22semidirect
34. C2×#832C22semidirect
39. C2×#808C22semidirect
40. C2×#807C22semidirect
45. C2×#803C22semidirect
46. C2×C4×C3⋊C8C22semidirect
52. #2297D3semidirect
96. C2×#215C2×C4semidirect
99. C2×#213C2×C4semidirect
150. #391C2×D3semidirect
151. #391C2×D3semidirect
155. #390C2×D3semidirect
156. #390C2×D3semidirect
242. #208C22⋊C4semidirect
367. C2×C3×C4AGL(1,ℤ/8)semidirect
383. C3⋊C8C2×C22⋊C4semidirect
565. C2×C4#949semidirect
614. C3#21236semidirect

Maximal subgroups

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

subgroupindex
C22×#9492
C3×#22972
C2×#52322
C2×#50012
C2×#49952
C2×#49392
C2×C4×#2082
#696792
#696762
#696482
#696472
#212363