Finite Group Databasebeta

← #10800215

Subgroups — #10800215

Order 768.

1256 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85680C211yesmaximal, characteristic
2C2×C3×C4×M16C211yesmaximal, characteristic, complemented
3C4×#926C211yesmaximal, characteristic, complemented
4#71984C214yesmaximal, complemented
5#69868C212yesmaximal, complemented
6#69867C212yesmaximal, complemented
7C4×#825C214yesmaximal, complemented
8#21382—31nomaximal, Sylow
9C22×C3×M16C2212yes—
10C22×C3×C42C2211yescharacteristic
11#5791—21no—
12#5791—21no—
13#5791C2212yes—
14C2×#926—21no—
15#5479—21no—
16#5479—21no—
17#5479—21no—
18#5479—21no—
19#5479C412yescomplemented
20#5479C412yescomplemented
21#5477C411yescharacteristic, complemented
22#5477C411yescharacteristic, complemented
23#5477C2211yescharacteristic
24C4×#235—21no—
25C4×#235—21no—
26C4×#235C2212yes—
27#5470C411yescharacteristic, complemented
28#5470C411yescharacteristic, complemented
29#5470C2211yescharacteristic
30C3×C4×M16—21no—
31C3×C4×M16—21no—
32C3×C4×M16C2212yes—
33C3×C4×M16C2212yes—
34C2×C3×#97—21no—
35C2×C3×C4×C8—21no—
36C4×#217—21no—
37C4×#217C2212yes—
38C2×#825C2218yes—
39C4×C4⋊Dic3—21no—
40C4×C4⋊Dic3C2212yes—
41#4932—21no—
42C42×Dic3C2212yes—
43#4555C2218yes—
44#3444—31no—
45C2×C4×M16D311yescharacteristic, complemented
46C4×#117—31no—
47#1986—32no—
48#1980—32no—
49#1871—34no—
50C4×#104—34no—
51C23×C3×C4D411yescharacteristic
52C23×C3×C4D411yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C2×#235—21no—
55C2×#235—21no—
56C2×#235C2312yes—
57C2×C3×M16—22no—
58C2×C3×M16—28no—
59C2×C3×M16C2314yes—
60C22×C3×C8—22no—
61C2×C3×C42C2×C411yescharacteristic
62C2×C3×C42C2×C411yescharacteristic
63C2×C3×C42C2×C411yescharacteristic
64C2×C3×C42C2×C411yescharacteristic
65C2×C3×C42D411yescharacteristic
66C2×C3×C42D411yescharacteristic
67C2×C3×C42D411yescharacteristic
68C2×C3×C42D411yescharacteristic
69C2×C3×C42D411yescharacteristic
70C2×C3×C42D411yescharacteristic
71C2×C3×C42C2311yescharacteristic
72C2×C3×C42C2311yescharacteristic
73#953—22no—
74C2×#217—22no—
75#930—21no—
76#930—21no—
77#930—21no—
78#930—21no—
79#930C2×C412yes—
80#930C2×C412yes—
81#929—22no—
82#929—22no—
83#929—22no—
84C4×#67—22no—
85C4×#67—22no—
86C4×#67—22no—
87C4×#67—22no—
88C4×#67—22no—
89C4×#67—22no—
90#926—28no—
91#926—21no—
92#926—21no—
93#926C2312yes—
94C2×C4⋊Dic3—22no—
95#924—21no—
96#924—21no—
97#924—21no—
98#924—21no—
99#924C2×C412yes—
100#924C2×C412yes—
101C2×C4×Dic3—22no—
102C2×C4×Dic3—22no—
103C2×C4×Dic3C2314yes—
104#876—22no—
105#876—22no—
106#875—21no—
107#875—21no—
108#875C2×C412yes—
109#874C2×C411yescharacteristic
110C4×D12—21no—
111C4×D12—21no—
112C4×D12—21no—
113C4×D12—21no—
114C4×D12C2×C412yes—
115C4×D12C2×C412yes—
116#872—22no—
117#872—22no—
118#872—22no—
119C42×D3—22no—
120#870—21no—
121#870—21no—
122#870C2×C412yes—
123#869C2×C411yescharacteristic
124#869C2×C411yescharacteristic
125#869C2×C411yescharacteristic
126C4×Dic6—21no—
127C4×Dic6—21no—
128C4×Dic6—21no—
129C4×Dic6—21no—
130C4×Dic6C2×C412yes—
131C4×Dic6C2×C412yes—
132C3×#97—22no—
133C3×#97—22no—
134C3×C4×C8—22no—
135C3×C4×C8—22no—
136C3⋊#95—22no—
137C3⋊#95—22no—
138#825—216no—
139C22×M16C2×D312yes—
140#533—31no—
141#533—31no—
142#533—31no—
143#532—31no—
144#532—31no—
145#532—31no—
146#529—32no—
147#529—32no—
148#529—61no—
149#529—61no—
150#529—61no—
151#529—61no—
152#525—32no—
153#525—61no—
154#525—61no—
155C4×#41—32no—
156C4×#41—61no—
157C4×#41—61no—
158C2×#117—61no—
159C22×C42C2×D311yescharacteristic
160C2×#104—38no—
161C4×M16—21no—
162C4×M16—21no—
163C4×M16C2×D312yes—
164C4×M16C2×D312yes—
165C2×#97—21no—
166C2×C4×C8—21no—
167C4×C4⋊C4—32no—
168C4×C4⋊C4—61no—
169C4×C22⋊C4—32no—
170C4×C22⋊C4—61no—
171#376—61no—
172C43—32no—
173#339—38no—
174C24×C3C2×D411yescharacteristic
175C22×C3×C4—24no—
176C22×C3×C4—24no—
177C22×C3×C4—42no—
178C22×C3×C4C22⋊C411yescharacteristic
179C22×C3×C4C22⋊C411yescharacteristic
180C22×C3×C4C22⋊C411yescharacteristic
181C22×C3×C4C22⋊C411yescharacteristic
182C22×C3×C4C22⋊C411yescharacteristic
183C22×C3×C4C22⋊C411yescharacteristic
184C22×C3×C4C22⋊C411yescharacteristic
185C22×C3×C4C22⋊C411yescharacteristic
186C22×C3×C4C22×C411yescharacteristic
187C22×C3×C4C22×C411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2411yescharacteristic
196C2×#67—22no—
197C2×#67—22no—
198C2×#67—22no—
199C22×Dic3—24no—
200#235—28no—
201#235—44no—
202#235—44no—
203C2×D12—21no—
204C2×D12—21no—
205C2×D12C22×C412yes—
206C2×C4×D3—22no—
207C2×C4×D3—22no—
208C2×C4×D3—22no—
209C2×Dic6—21no—
210C2×Dic6—21no—
211C2×Dic6C22×C412yes—
212C3×M16—28no—
213C3×M16—24no—
214C3×M16#4118yescomplemented
215C2×C3×C8—28no—
216C2×C3×C8—24no—
217C3×C42C22⋊C411yescharacteristic
218C3×C42C22⋊C411yescharacteristic
219C3×C42C22⋊C411yescharacteristic
220C3×C42C22⋊C411yescharacteristic
221C3×C42C22⋊C411yescharacteristic
222C3×C42C22⋊C411yescharacteristic
223C3×C42C22⋊C411yescharacteristic
224C3×C42C22⋊C411yescharacteristic
225C3×C42C22×C411yescharacteristic
226C3×C42C22×C411yescharacteristic
227C3×C42C22×C411yescharacteristic
228C3×C42C22×C411yescharacteristic
229C3×C42C2×D411yescharacteristic
230C3×C42C2×D411yescharacteristic
231C3×C42C2×D411yescharacteristic
232C3×C42C2×D411yescharacteristic
233#217—28no—
234#217—22no—
235#217—22no—
236#217—22no—
237(C2×D3)⋊C4—22no—
238(C2×D3)⋊C4—22no—
239(C2×D3)⋊C4—22no—
240(C2×D3)⋊C4—22no—
241(C2×D3)⋊C4—22no—
242(C2×D3)⋊C4—22no—
243C4⋊Dic3—28no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3C22×C412yes—
249C4⋊Dic3C22×C412yes—
250Dic3⋊C4—22no—
251Dic3⋊C4—22no—
252Dic3⋊C4—22no—
253Dic3⋊C4—22no—
254Dic3⋊C4—22no—
255Dic3⋊C4—22no—
256C4×Dic3—28no—
257C4×Dic3—22no—
258C4×Dic3—22no—
259C4×Dic3—22no—
260C4×Dic3—28no—
261C4×Dic3—44no—
262C2×#41—32no—
263C2×#41—61no—
264C2×#41—61no—
265C23×C4D1211yescharacteristic
266C23×C4D1211yescharacteristic
267C23×C4C22×D311yescharacteristic
268C2×M16—22no—
269C2×M16—28no—
270C2×M16C22×D314yes—
271C22×C8—22no—
272C4⋊Q8—31no—
273C4⋊Q8—31no—
274C4⋊Q8—31no—
275#127—31no—
276#126—62no—
277#126—62no—
278#125—32no—
279#125—61no—
280#125—61no—
281#124—32no—
282#124—61no—
283#124—61no—
284#123—62no—
285#123—62no—
286#123—62no—
287C22⋊Q8—32no—
288C22⋊Q8—32no—
289C22⋊Q8—61no—
290C22⋊Q8—61no—
291C22⋊Q8—61no—
292C22⋊Q8—61no—
293#121—32no—
294#121—32no—
295#121—61no—
296#121—61no—
297#121—61no—
298#121—61no—
299#120—62no—
300C4×Q8—32no—
301C4×Q8—32no—
302C4×Q8—61no—
303C4×Q8—61no—
304C4×Q8—61no—
305C4×Q8—61no—
306C4×D4—32no—
307C4×D4—32no—
308C4×D4—62no—
309C4×D4—62no—
310C4×D4—62no—
311C4×D4—62no—
312C4×D4—62no—
313C4×D4—62no—
314C4×D4—61no—
315C4×D4—61no—
316C4×D4—61no—
317C4×D4—61no—
318#117—32no—
319#117—68no—
320#117—61no—
321#117—61no—
322#117—62no—
323#117—62no—
324#117—62no—
325C2×C4⋊C4—62no—
326C2×C22⋊C4—62no—
327C2×C42—34no—
328C2×C42—62no—
329C2×C42—62no—
330C2×C42—62no—
331C2×C42C4×D311yescharacteristic
332C2×C42C4×D311yescharacteristic
333C2×C42C4×D311yescharacteristic
334C2×C42C4×D311yescharacteristic
335C2×C42D1211yescharacteristic
336C2×C42D1211yescharacteristic
337C2×C42#6711yescharacteristic
338C2×C42#6711yescharacteristic
339C2×C42#6711yescharacteristic
340C2×C42#6711yescharacteristic
341C2×C42C22×D311yescharacteristic
342C2×C42C22×D311yescharacteristic
343#104—616no—
344#97—22no—
345#97—22no—
346C4×C8—22no—
347C4×C8—22no—
348#95—62no—
349#95—62no—
350C23×C3—22no—
351C23×C3—22no—
352C23×C3—41no—
353C23×C3C2×C22⋊C411yescharacteristic
354C23×C3C2×C22⋊C411yescharacteristic
355C23×C3C22×D411yescharacteristic
356C22×D3—22no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4#10414yescomplemented
374C2×C3×C4C2×C22⋊C411yescharacteristic
375C2×C3×C4C2×C22⋊C411yescharacteristic
376C2×C3×C4C2×C22⋊C411yescharacteristic
377C2×C3×C4C2×C22⋊C411yescharacteristic
378C2×C3×C4C2×C22⋊C411yescharacteristic
379C2×C3×C4C2×C22⋊C411yescharacteristic
380C2×C3×C4C2×C22⋊C411yescharacteristic
381C2×C3×C4C2×C22⋊C411yescharacteristic
382C2×C3×C4C2×C22⋊C411yescharacteristic
383C2×C3×C4C2×C22⋊C411yescharacteristic
384C2×C3×C4C23×C411yescharacteristic
385C2×C3×C4C22×D411yescharacteristic
386C2×C3×C4C2×#4114yes—
387#67—44no—
388#67—44no—
389#67—44no—
390C2×Dic3—28no—
391C2×Dic3—22no—
392C2×Dic3—22no—
393C2×Dic3—22no—
394C2×Dic3—44no—
395C2×Dic3—44no—
396D12—24no—
397D12—42no—
398D12—42no—
399C4×D3—44no—
400C4×D3—44no—
401C4×D3—44no—
402Dic6—24no—
403Dic6—42no—
404Dic6—42no—
405C3×C8—28no—
406C24C2×D1211yescharacteristic
407#41—68no—
408#41—124no—
409#41—124no—
410C2×Q8—32no—
411C2×Q8—61no—
412C2×Q8—61no—
413C2×D4—32no—
414C2×D4—62no—
415C2×D4—62no—
416C2×D4—62no—
417C2×D4—61no—
418C2×D4—61no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—42no—
422C22×C4—64no—
423C22×C4—62no—
424C22×C4—62no—
425C22×C4—62no—
426C22×C4(C2×D3)⋊C411yescharacteristic
427C22×C4(C2×D3)⋊C411yescharacteristic
428C22×C4(C2×D3)⋊C411yescharacteristic
429C22×C4(C2×D3)⋊C411yescharacteristic
430C22×C4(C2×D3)⋊C411yescharacteristic
431C22×C4(C2×D3)⋊C411yescharacteristic
432C22×C4(C2×D3)⋊C411yescharacteristic
433C22×C4(C2×D3)⋊C411yescharacteristic
434C22×C4C2×C4×D311yescharacteristic
435C22×C4C2×C4×D311yescharacteristic
436C22×C4C2×D1211yescharacteristic
437C22×C4C2×D1211yescharacteristic
438C22×C4C2×D1211yescharacteristic
439C22×C4C2×D1211yescharacteristic
440C22×C4C2×D1211yescharacteristic
441C22×C4C2×#6711yescharacteristic
442C22×C4C2×#6711yescharacteristic
443C22×C4C23×D311yescharacteristic
444M16—28no—
445M16—24no—
446M16#23518yescomplemented
447C2×C8—28no—
448C2×C8—24no—
449C4⋊C4—32no—
450C4⋊C4—32no—
451C4⋊C4—68no—
452C4⋊C4—62no—
453C4⋊C4—62no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—62no—
459C4⋊C4—62no—
460C4⋊C4—62no—
461C4⋊C4—62no—
462C22⋊C4—68no—
463C22⋊C4—62no—
464C22⋊C4—62no—
465C22⋊C4—62no—
466C22⋊C4—62no—
467C22⋊C4—62no—
468C22⋊C4—62no—
469C22⋊C4—62no—
470C22⋊C4—62no—
471C22⋊C4—62no—
472C42—68no—
473C42—62no—
474C42—62no—
475C42—62no—
476C42—68no—
477C42—124no—
478C42(C2×D3)⋊C411yescharacteristic
479C42(C2×D3)⋊C411yescharacteristic
480C42(C2×D3)⋊C411yescharacteristic
481C42(C2×D3)⋊C411yescharacteristic
482C42(C2×D3)⋊C411yescharacteristic
483C42(C2×D3)⋊C411yescharacteristic
484C42(C2×D3)⋊C411yescharacteristic
485C42(C2×D3)⋊C411yescharacteristic
486C42C2×C4×D311yescharacteristic
487C42C2×C4×D311yescharacteristic
488C42C2×C4×D311yescharacteristic
489C42C2×C4×D311yescharacteristic
490C42C2×#6711yescharacteristic
491C42C2×#6711yescharacteristic
492C42C2×#6711yescharacteristic
493C42C2×#6711yescharacteristic
494C22×C3—21no—
495C22×C3—21no—
496C22×C3—21no—
497C22×C3—21no—
498C22×C3—21no—
499C22×C3—21no—
500C22×C3—21no—
501C22×C3—44no—
502C22×C3C2×#10412yes—
503C22×C3C22×C22⋊C411yescharacteristic
504C22×C3#53212yes—
505C2×D3—42no—
506C2×D3—42no—
507C2×D3—42no—
508C3×C4—21no—
509C3×C4—21no—
510C3×C4—21no—
511C3×C4—21no—
512C3×C4—21no—
513C3×C4—21no—
514C3×C4—21no—
515C3×C4C2×#10412yes—
516C3×C4C2×#10412yes—
517C3×C4#42112yes—
518C3×C4C4×#4112yes—
519C3×C4#52512yes—
520Dic3—48no—
521Dic3—42no—
522Dic3—42no—
523Dic3—42no—
524C23—22no—
525C23—22no—
526C23—41no—
527C23—62no—
528C23C2×(C2×D3)⋊C411yescharacteristic
529C23C2×(C2×D3)⋊C411yescharacteristic
530C23C22×D1211yescharacteristic
531Q8—64no—
532Q8—122no—
533Q8—122no—
534D4—64no—
535D4—122no—
536D4—124no—
537D4—124no—
538D4—124no—
539D4—122no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—68no—
557C2×C4—62no—
558C2×C4—62no—
559C2×C4—62no—
560C2×C4—124no—
561C2×C4—124no—
562C2×C4—124no—
563C2×C4—124no—
564C2×C4—124no—
565C2×C4#82514yescomplemented
566C2×C4C2×(C2×D3)⋊C411yescharacteristic
567C2×C4C2×(C2×D3)⋊C411yescharacteristic
568C2×C4C2×(C2×D3)⋊C411yescharacteristic
569C2×C4C2×(C2×D3)⋊C411yescharacteristic
570C2×C4C2×(C2×D3)⋊C411yescharacteristic
571C2×C4C2×(C2×D3)⋊C411yescharacteristic
572C2×C4C2×(C2×D3)⋊C411yescharacteristic
573C2×C4C2×(C2×D3)⋊C411yescharacteristic
574C2×C4C2×(C2×D3)⋊C411yescharacteristic
575C2×C4C2×(C2×D3)⋊C411yescharacteristic
576C2×C4C22×C4×D311yescharacteristic
577C2×C4C2×#23514yes—
578C2×C4C22×#6711yescharacteristic
579C8—28no—
580C2×C3—22no—
581C2×C3—22no—
582C2×C3—41no—
583C2×C3#229711yescharacteristic
584C2×C3C22×#10411yescharacteristic
585C2×C3C2×#42111yescharacteristic
586D3—82no—
587C22—21no—
588C22—21no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—21no—
594C22—44no—
595C22—122no—
596C22—122no—
597C22—122no—
598C22C2×#82512yes—
599C22#547712yes—
600C22C22×(C2×D3)⋊C411yescharacteristic
601C4—21no—
602C4—21no—
603C4—21no—
604C4—21no—
605C4—21no—
606C4—21no—
607C4—21no—
608C4—128no—
609C4—122no—
610C4—122no—
611C4—122no—
612C4C2×#82512yes—
613C4C2×#82512yes—
614C4#513912yes—
615C4C4×#23512yes—
616C4#579112yes—
617C3#2138211yescharacteristic, complemented, Sylow
618C2—22no—
619C2—22no—
620C2—41no—
621C2—242no—
622C2#8207011yescharacteristic
623C2C22×#82511yescharacteristic
624C2C2×#513911yescharacteristic
subgroupG/H
centerC42C2×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×C4×M16C2
socleC22×C3C22×C22⋊C4
prime radical#85680C2
cosocleC22×C3×C4C24
hypercenterC2×C4×M16D3
hypocenterC3#21382
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C42 ◁ C22×C42 ◁ C2×C4×M16

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4×M16C2semidirect
3. C4×#926C2semidirect
4. #71984C2semidirect
5. #69868C2semidirect
6. #69867C2semidirect
7. C4×#825C2semidirect
19. #5479C4semidirect
20. #5479C4semidirect
21. #5477C4semidirect
22. #5477C4semidirect
27. #5470C4semidirect
28. #5470C4semidirect
45. C2×C4×M16D3semidirect
214. C3×M16#41semidirect
373. C2×C3×C4#104semidirect
446. M16#235semidirect
565. C2×C4#825semidirect
617. C3#21382semidirect

Maximal subgroups

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

subgroupindex
#856802
C2×C3×C4×M162
C4×#9262
#719842
#698682
#698672
C4×#8252
#213823