Finite Group Databasebeta

← #10762161

Subgroups — #10762161

Order 768.

1096 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85843C211yesmaximal, characteristic
2C3×C4×#131C211yesmaximal, characteristic, complemented
3#82522C211yesmaximal, characteristic, complemented
4#69855C212yesmaximal, complemented
5#69854C211yesmaximal, characteristic, complemented
6#69853C211yesmaximal, characteristic, complemented
7#69852C212yesmaximal, complemented
8#69736C212yesmaximal, complemented
9#69735C211yesmaximal, characteristic, complemented
10#69734C211yesmaximal, characteristic, complemented
11C2×#4459C212yesmaximal, complemented
12#17517—31nomaximal, Sylow
13C2×C3×#131—21no—
14C3×C4×#41C2211yescharacteristic
15#5829—21no—
16#5585C411yescharacteristic
17#5580—21no—
18#5579C411yescharacteristic
19#5565C412yes—
20#5550C412yes—
21D4×D12C411yescharacteristic
22#5547—22no—
23C4×D3×D4—21no—
24#5543C411yescharacteristic
25#5530—21no—
26#5527—22no—
27#5479C2212yes—
28#5477C2211yescharacteristic
29C4×#235C2211yescharacteristic
30C2×C4×D12C2212yes—
31C3×#405—22no—
32C3×#405—21no—
33C3×C4×M16C2212yescomplemented
34C3×C4×M16C2211yescharacteristic, complemented
35C2×C3×C4×C8C2212yescomplemented
36C2×C3×C4×C8C2211yescharacteristic, complemented
37#5024—21no—
38#5011—22no—
39D4×C3⋊C8—21no—
40#4926C2212yescomplemented
41#4925C2211yescharacteristic, complemented
42#4924C2211yescharacteristic, complemented
43C2×C4⋊(C3⋊C8)C2212yescomplemented
44#4459C2211yescharacteristic
45#4459C2212yes—
46#4459C2211yescharacteristic
47#4459C2212yescomplemented
48#4459C2212yescomplemented
49#4459C2212yescomplemented
50#4459C2212yescomplemented
51#4456C2211yescharacteristic, complemented
52#4456C2211yescharacteristic, complemented
53#4456C2212yescomplemented
54#3481—31no—
55#2990—31no—
56C4×#131D311yescharacteristic, complemented
57#1494—32no—
58#1492—31no—
59#1490—32no—
60#1489—31no—
61#1484—31no—
62#1483—32no—
63#1476—31no—
64C2×D4⋊C8—32no—
65C2×C3×#41D411yescharacteristic
66C2×D3×D4—21no—
67C2×#235—21no—
68C2×#235C2×C411yescharacteristic
69C22×D12C2×C412yes—
70C22×C4×D3—22no—
71C3×#131—28no—
72C2×C3×M16—22no—
73C2×C3×M16—21no—
74C22×C3×C8—22no—
75C22×C3×C8—21no—
76C3×C4×Q8C2×C411yescharacteristic
77C3×C4×Q8D411yescharacteristic
78C3×C4×D4C2×C411yescharacteristic
79C3×C4×D4C2×C412yes—
80C3×C4×D4D411yescharacteristic
81C3×C4×D4D412yes—
82C3×#117D412yes—
83C3×#117D411yescharacteristic
84C2×C3×C42C2312yes—
85C2×C3×C42C2311yescharacteristic
86#947—21no—
87#946—21no—
88#939—22no—
89#937—22no—
90D4×Dic3—21no—
91#930—21no—
92#930C2×C412yes—
93#930C2×C412yes—
94#929—22no—
95C4×#67—21no—
96C4×#67—21no—
97C4×#67—22no—
98C2×(C2×D3)⋊C4—22no—
99#926C2×C411yescharacteristic
100C2×C4⋊Dic3C2×C412yes—
101#924C2×C412yes—
102C2×#208—21no—
103C22×C3⋊C8—22no—
104#898—22no—
105#896—22no—
106#895—21no—
107#894—22no—
108#894—22no—
109#893—21no—
110D3×C4⋊C4—21no—
111#890—21no—
112#884—22no—
113#883—22no—
114#882—22no—
115#67⋊C4—22no—
116D3×C22⋊C4—22no—
117#878—22no—
118#876—22no—
119#875—21no—
120#875C2×C412yes—
121#874C2×C411yescharacteristic
122C4×D12—22no—
123C4×D12—22no—
124C4×D12—21no—
125C4×D12C2×C412yes—
126C4×D12C2×C412yes—
127C4×D12C2×C411yescharacteristic
128C4×D12C2311yescharacteristic
129C4×D12C2312yes—
130#872—21no—
131#872—22no—
132C42×D3—21no—
133#870C2×C412yes—
134#869C2×C411yescharacteristic
135C4×Dic6C2×C411yescharacteristic
136C4×Dic6C2311yescharacteristic
137C3×#97—21no—
138C3×#97—22no—
139C3×C4×C8—21no—
140C3×C4×C8—22no—
141C3×C4×C8C2311yescharacteristic
142C3×C4×C8C2311yescharacteristic
143C3×C4×C8C2312yes—
144#830—21no—
145#830—22no—
146C4⋊(C3⋊C8)—22no—
147C4⋊(C3⋊C8)—21no—
148C4⋊(C3⋊C8)C2311yescharacteristic
149C4⋊(C3⋊C8)C2312yes—
150C4⋊(C3⋊C8)C2311yescharacteristic
151#803—22no—
152C4×C3⋊C8—21no—
153C2×#131—21no—
154#559—31no—
155#554—31no—
156#554—31no—
157#553—32no—
158#553—61no—
159#551—62no—
160#546—32no—
161#546—61no—
162D42—31no—
163#532—31no—
164#529—32no—
165#529—61no—
166#525—62no—
167C4×#41—31no—
168C4×#41C2×D311yescharacteristic
169C2×C4×D4—32no—
170C2×C4×D4—61no—
171#446—61no—
172#435—62no—
173C8×D4—61no—
174#432—32no—
175#431—31no—
176#423—31no—
177C2×C4⋊C8—32no—
178#405—22no—
179#405—21no—
180C4×M16C2×D312yescomplemented
181C4×M16C2×D311yescharacteristic, complemented
182C2×C4×C8C2×D312yescomplemented
183C2×C4×C8C2×D311yescharacteristic, complemented
184Q8⋊C8—31no—
185Q8⋊C8—31no—
186Q8⋊C8—32no—
187D4⋊C8—31no—
188D4⋊C8—32no—
189D4⋊C8—31no—
190D4⋊C8—32no—
191D4⋊C8—32no—
192D4⋊C8—32no—
193D4⋊C8—32no—
194C23×D3—22no—
195C3×#41—24no—
196C2×C3×Q8C22⋊C411yescharacteristic
197C2×C3×D4C22⋊C412yes—
198C2×C3×D4C22⋊C411yescharacteristic
199C22×C3×C4C22×C412yes—
200C22×C3×C4C22×C411yescharacteristic
201C22×C3×C4C2×D412yes—
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D412yes—
204C22×C3×C4C2×D411yescharacteristic
205C2×#67—22no—
206C2×#67—21no—
207C2×#67—22no—
208C22×Dic3—22no—
209D3×D4—44no—
210#235—22no—
211#235—41no—
212#235—81no—
213#235—81no—
214C2×D12—24no—
215C2×D12—22no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—42no—
219C2×D12C22×C412yes—
220C2×D12C22×C411yescharacteristic
221C2×C4×D3—21no—
222C2×C4×D3—21no—
223C2×C4×D3—21no—
224C2×C4×D3—22no—
225C2×C4×D3—22no—
226C2×C4×D3—22no—
227C2×C4×D3—82no—
228C2×C4×D3—82no—
229C2×Dic6—21no—
230C2×Dic6C22×C411yescharacteristic
231C3×M16—28no—
232C3×M16—24no—
233C2×C3×C8—28no—
234C2×C3×C8—24no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—22no—
238C3×C4⋊C4C22⋊C411yescharacteristic
239C3×C4⋊C4C22⋊C411yescharacteristic
240C3×C4⋊C4C22⋊C412yes—
241C3×C4⋊C4C22⋊C412yes—
242C3×C22⋊C4C22⋊C412yes—
243C3×C22⋊C4C22⋊C412yes—
244C3×C22⋊C4C22⋊C412yes—
245C3×C42C22×C411yescharacteristic
246C3×C42C22×C412yes—
247C3×C42C2×D411yescharacteristic
248C3×C42C2×D412yes—
249C3×C42C2×D411yescharacteristic
250C3×C42C2×D412yes—
251C3×C42C2411yescharacteristic
252#217—22no—
253#217—21no—
254(C2×D3)⋊C4—21no—
255(C2×D3)⋊C4—22no—
256(C2×D3)⋊C4—22no—
257(C2×D3)⋊C4—22no—
258(C2×D3)⋊C4—22no—
259(C2×D3)⋊C4—22no—
260(C2×D3)⋊C4—22no—
261(C2×D3)⋊C4—21no—
262(C2×D3)⋊C4—22no—
263C4⋊Dic3—21no—
264C4⋊Dic3—22no—
265C4⋊Dic3C22×C411yescharacteristic
266C4⋊Dic3C22×C411yescharacteristic
267C4⋊Dic3C22×C412yes—
268Dic3⋊C4—21no—
269Dic3⋊C4—22no—
270Dic3⋊C4—22no—
271Dic3⋊C4—22no—
272Dic3⋊C4—21no—
273C4×Dic3—21no—
274C4×Dic3—21no—
275C4×Dic3—21no—
276#208—81no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—22no—
280C2×C3⋊C8—82no—
281C2×#41—31no—
282C2×#41—61no—
283C2×#41#6711yescharacteristic
284C22×D4—32no—
285C22×D4—61no—
286C23×C4—62no—
287#131—28no—
288C2×M16—22no—
289C2×M16—21no—
290C2×M16—61no—
291C22×C8—22no—
292C22×C8—21no—
293C22×C8—62no—
294C4⋊Q8—31no—
295#127—31no—
296#126—62no—
297#126—62no—
298#125—32no—
299#125—61no—
300#124—32no—
301#124—61no—
302#124—61no—
303#123—62no—
304#123—62no—
305#123—62no—
306#123—62no—
307C22⋊Q8—32no—
308C22⋊Q8—61no—
309C22⋊Q8—62no—
310C22⋊Q8—62no—
311#121—32no—
312#121—32no—
313#121—62no—
314#121—61no—
315#121—61no—
316#121—62no—
317#120—62no—
318#120—62no—
319C4×Q8—31no—
320C4×Q8—31no—
321C4×Q8C4×D311yescharacteristic
322C4×Q8D1211yescharacteristic
323C4×D4—32no—
324C4×D4—32no—
325C4×D4—31no—
326C4×D4—31no—
327C4×D4—32no—
328C4×D4—61no—
329C4×D4—62no—
330C4×D4—61no—
331C4×D4—61no—
332C4×D4—61no—
333C4×D4—62no—
334C4×D4—62no—
335C4×D4—61no—
336C4×D4—62no—
337C4×D4C4×D311yescharacteristic
338C4×D4C4×D312yes—
339C4×D4D1211yescharacteristic
340C4×D4D1212yes—
341#117—31no—
342#117—62no—
343#117—61no—
344#117#6712yes—
345#117#6711yescharacteristic
346C2×C4⋊C4—32no—
347C2×C4⋊C4—61no—
348C2×C22⋊C4—62no—
349C2×C22⋊C4—62no—
350C2×C42—61no—
351C2×C42C22×D312yes—
352C2×C42C22×D311yescharacteristic
353C4⋊C8—31no—
354C4⋊C8—32no—
355C4⋊C8—31no—
356C4⋊C8—62no—
357C4⋊C8—61no—
358C22⋊C8—61no—
359C22⋊C8—62no—
360#97—21no—
361#97—22no—
362#97—62no—
363C4×C8—21no—
364C4×C8—22no—
365C4×C8—61no—
366C4×C8C22×D311yescharacteristic
367C4×C8C22×D312yes—
368C4×C8C22×D311yescharacteristic
369C23×C3C2×C22⋊C411yescharacteristic
370C23×C3C2×C22⋊C412yes—
371C22×D3—21no—
372C22×D3—22no—
373C22×D3—44no—
374C22×D3—44no—
375C22×D3—82no—
376C3×Q8—22no—
377C3×D4—24no—
378C3×D4#10414yes—
379C2×C3×C4—24no—
380C2×C3×C4—22no—
381C2×C3×C4—21no—
382C2×C3×C4—22no—
383C2×C3×C4—21no—
384C2×C3×C4—22no—
385C2×C3×C4—21no—
386C2×C3×C4—22no—
387C2×C3×C4—21no—
388C2×C3×C4—22no—
389C2×C3×C4C2×C22⋊C411yescharacteristic
390C2×C3×C4C2×C22⋊C412yes—
391C2×C3×C4C2×C22⋊C412yes—
392C2×C3×C4C2×C22⋊C411yescharacteristic
393C2×C3×C4C2×C22⋊C412yes—
394C2×C3×C4C2×C22⋊C411yescharacteristic
395C2×C3×C4C23×C411yescharacteristic
396C2×C3×C4C22×D411yescharacteristic
397C2×C3×C4C22×D411yescharacteristic
398#67—42no—
399#67—44no—
400#67—82no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—22no—
405C2×Dic3—82no—
406D12—22no—
407D12—21no—
408D12—22no—
409D12—44no—
410D12—42no—
411D12—81no—
412D12#13112yescomplemented
413C4×D3—42no—
414C4×D3—41no—
415C4×D3—42no—
416C4×D3—41no—
417C4×D3—42no—
418C4×D3—42no—
419C4×D3—82no—
420C4×D3—82no—
421Dic6—21no—
422Dic6—21no—
423Dic6—81no—
424C3×C8—22no—
425C3×C8—24no—
426C3×C8—22no—
427C3⋊C8—41no—
428C3⋊C8—42no—
429C3⋊C8—41no—
430C24—62no—
431#41—24no—
432#41—62no—
433#41—121no—
434#41—241no—
435#41—241no—
436C2×Q8—31no—
437C2×Q8—61no—
438C2×Q8(C2×D3)⋊C411yescharacteristic
439C2×D4—31no—
440C2×D4—32no—
441C2×D4—64no—
442C2×D4—61no—
443C2×D4—62no—
444C2×D4—62no—
445C2×D4—62no—
446C2×D4—61no—
447C2×D4—61no—
448C2×D4—122no—
449C2×D4—124no—
450C2×D4(C2×D3)⋊C412yes—
451C2×D4(C2×D3)⋊C411yescharacteristic
452C22×C4—62no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—62no—
457C22×C4—62no—
458C22×C4—62no—
459C22×C4—242no—
460C22×C4—242no—
461C22×C4C2×C4×D312yes—
462C22×C4C2×C4×D311yescharacteristic
463C22×C4C2×D1212yes—
464C22×C4C2×D1211yescharacteristic
465C22×C4C2×#6712yes—
466C22×C4C2×#6711yescharacteristic
467M16—28no—
468M16—24no—
469M16—241no—
470C2×C8—28no—
471C2×C8—24no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—22no—
475C2×C8—61no—
476C2×C8—61no—
477C2×C8—62no—
478C2×C8—242no—
479C4⋊C4—31no—
480C4⋊C4—32no—
481C4⋊C4—31no—
482C4⋊C4—62no—
483C4⋊C4—61no—
484C4⋊C4—62no—
485C4⋊C4—61no—
486C4⋊C4—62no—
487C4⋊C4—62no—
488C4⋊C4—61no—
489C4⋊C4(C2×D3)⋊C411yescharacteristic
490C4⋊C4(C2×D3)⋊C411yescharacteristic
491C4⋊C4(C2×D3)⋊C412yes—
492C4⋊C4(C2×D3)⋊C412yes—
493C22⋊C4—62no—
494C22⋊C4—61no—
495C22⋊C4—62no—
496C22⋊C4—62no—
497C22⋊C4—62no—
498C22⋊C4—62no—
499C22⋊C4—61no—
500C22⋊C4—62no—
501C22⋊C4—62no—
502C22⋊C4—61no—
503C22⋊C4—62no—
504C22⋊C4(C2×D3)⋊C412yes—
505C22⋊C4(C2×D3)⋊C412yes—
506C22⋊C4(C2×D3)⋊C412yes—
507C42—61no—
508C42—61no—
509C42—61no—
510C42C2×C4×D312yes—
511C42C2×C4×D311yescharacteristic
512C42C2×D1211yescharacteristic
513C42C2×D1212yes—
514C42C2×#6711yescharacteristic
515C42C2×#6712yes—
516C42C23×D311yescharacteristic
517C22×C3—22no—
518C22×C3—22no—
519C22×C3—21no—
520C22×C3—21no—
521C22×C3—21no—
522C22×C3C2×#10414yes—
523C22×C3C22×C22⋊C411yescharacteristic
524C2×D3—22no—
525C2×D3—41no—
526C2×D3—41no—
527C2×D3—42no—
528C2×D3—42no—
529C2×D3—42no—
530C2×D3—88no—
531C3×C4—21no—
532C3×C4—22no—
533C3×C4—21no—
534C3×C4—22no—
535C3×C4—21no—
536C3×C4—22no—
537C3×C4C2×#10412yes—
538C3×C4C2×#13111yescharacteristic
539C3×C4#56811yescharacteristic
540C3×C4#57611yescharacteristic
541C3×C4#57711yescharacteristic
542Dic3—41no—
543Dic3—42no—
544Dic3—41no—
545Dic3—41no—
546C23—61no—
547C23—62no—
548C23—124no—
549C23—124no—
550C23—242no—
551C23C2×(C2×D3)⋊C411yescharacteristic
552C23C2×(C2×D3)⋊C412yes—
553Q8—22no—
554Q8—61no—
555Q8—61no—
556Q8—241no—
557D4—24no—
558D4—32no—
559D4—62no—
560D4—61no—
561D4—62no—
562D4—124no—
563D4—122no—
564D4—124no—
565D4—122no—
566D4—241no—
567D4—242no—
568D4#80514yes—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—61no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—62no—
583C2×C4—122no—
584C2×C4—122no—
585C2×C4—121no—
586C2×C4—122no—
587C2×C4—121no—
588C2×C4—122no—
589C2×C4—242no—
590C2×C4—242no—
591C2×C4—242no—
592C2×C4C2×(C2×D3)⋊C411yescharacteristic
593C2×C4C2×(C2×D3)⋊C412yes—
594C2×C4C2×(C2×D3)⋊C412yes—
595C2×C4C2×(C2×D3)⋊C411yescharacteristic
596C2×C4C2×(C2×D3)⋊C412yes—
597C2×C4C2×(C2×D3)⋊C411yescharacteristic
598C2×C4C22×C4×D311yescharacteristic
599C2×C4C22×D1211yescharacteristic
600C2×C4C22×#6711yescharacteristic
601C8—22no—
602C8—24no—
603C8—22no—
604C8—121no—
605C8—121no—
606C8—122no—
607C2×C3—24no—
608C2×C3—41no—
609C2×C3#288011yescharacteristic
610C2×C3#289711yescharacteristic
611C2×C3C22×#10411yescharacteristic
612D3—42no—
613D3—82no—
614C22—22no—
615C22—22no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—62no—
620C22—121no—
621C22—121no—
622C22—122no—
623C22—122no—
624C22—122no—
625C22—248no—
626C22C2×#80514yes—
627C22C22×(C2×D3)⋊C411yescharacteristic
628C4—21no—
629C4—22no—
630C4—22no—
631C4—21no—
632C4—21no—
633C4—22no—
634C4—121no—
635C4—121no—
636C4—122no—
637C4—121no—
638C4C2×#80512yes—
639C4D3×#13111yescharacteristic
640C4#575011yescharacteristic
641C4#575111yescharacteristic
642C4#575211yescharacteristic
643C3#1751711yescharacteristic, complemented, Sylow
644C2—24no—
645C2—41no—
646C2—122no—
647C2—242no—
648C2C22×#80511yescharacteristic
649C2#8379311yescharacteristic
650C2#8379711yescharacteristic
subgroupG/H
centerC2×C4C22×D12
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×C4×#131C2
socleC22×C3C22×C22⋊C4
prime radical#85843C2
cosocleC3×C42C24
hypercenterC4×#131D3
hypocenterC3#17517
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ C4×#131

Complements

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

NH ≅ G/Ntype
2. C3×C4×#131C2semidirect
3. #82522C2semidirect
4. #69855C2semidirect
5. #69854C2semidirect
6. #69853C2semidirect
7. #69852C2semidirect
8. #69736C2semidirect
9. #69735C2semidirect
10. #69734C2semidirect
11. C2×#4459C2semidirect
33. C3×C4×M16C22semidirect
34. C3×C4×M16C22semidirect
35. C2×C3×C4×C8C22semidirect
36. C2×C3×C4×C8C22semidirect
40. #4926C22semidirect
41. #4925C22semidirect
42. #4924C22semidirect
43. C2×C4⋊(C3⋊C8)C22semidirect
47. #4459C22semidirect
48. #4459C22semidirect
49. #4459C22semidirect
50. #4459C22semidirect
51. #4456C22semidirect
52. #4456C22semidirect
53. #4456C22semidirect
56. C4×#131D3semidirect
180. C4×M16C2×D3semidirect
181. C4×M16C2×D3semidirect
182. C2×C4×C8C2×D3semidirect
183. C2×C4×C8C2×D3semidirect
412. D12#131semidirect
643. C3#17517semidirect

Maximal subgroups

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

subgroupindex
#858432
C3×C4×#1312
#825222
#698552
#698542
#698532
#698522
#697362
#697352
#697342
C2×#44592
#175173