Finite Group Databasebeta

← #10763264

Subgroups — #10763264

Order 768.

984 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5691C211yesmaximal, characteristic
2D3×#432C211yesmaximal, characteristic, complemented
3#78517C211yesmaximal, characteristic, complemented
4C3×#1528C211yesmaximal, characteristic, complemented
5#71735C211yesmaximal, characteristic, complemented
6#70804C211yesmaximal, characteristic, complemented
7#69860C211yesmaximal, characteristic, complemented
8#68368C214yesmaximal, complemented
9#68366C214yesmaximal, complemented
10#17668—31nomaximal, Sylow
11C2×C3×#125C2211yescharacteristic
12#5691—21no—
13#5691—21no—
14#5691—21no—
15#5691—21no—
16#5691C412yes—
17#5691C412yes—
18#5691C2212yes—
19#5691C2212yes—
20C2×#898—21no—
21C2×#897C411yescharacteristic
22C2×#895C411yescharacteristic
23C2×#894—21no—
24C2×#893—21no—
25C2×#893C411yescharacteristic
26C2×#892—21no—
27C2×#887C411yescharacteristic
28C2×#875C2211yescharacteristic
29C2×C42×D3C2211yescharacteristic
30C3×#432C2211yescharacteristic
31C3×C4×M16C2211yescharacteristic
32#5126—21no—
33#5118—21no—
34#4926C2211yescharacteristic
35C4×#208C2211yescharacteristic
36#4835—21no—
37D3×C4⋊C8—21no—
38D3×C22⋊C8—21no—
39#4719—21no—
40#4705C2212yescomplemented
41#4705C2212yescomplemented
42#4705C2212yescomplemented
43D3×#97C2212yescomplemented
44#4685—21no—
45#4685—21no—
46C3×#330C2214yescomplemented
47#4543C2214yescomplemented
48#4464C2214yescomplemented
49#4464C2214yescomplemented
50C2×#529—31no—
51#2920—31no—
52C2×#432—31no—
53#1535—34no—
54#1532—34no—
55#1528—31no—
56#1528D311yescharacteristic, complemented
57#1527—31no—
58#1527—31no—
59C22×D12—21no—
60C22×D12C2×C411yescharacteristic
61C22×C4×D3—21no—
62C22×C4×D3C2×C411yescharacteristic
63C22×C4×D3D411yescharacteristic
64C22×C4×D3D411yescharacteristic
65C22×Dic6C2×C411yescharacteristic
66C2×C3×M16—21no—
67C3×#125—21no—
68C3×#125—21no—
69C3×#125C2×C412yes—
70C3×#125C2312yes—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4C2×C411yescharacteristic
74C2×C3×C4⋊C4C2×C411yescharacteristic
75C2×C3×C42C2311yescharacteristic
76C2×(C2×D3)⋊C4—21no—
77C2×(C2×D3)⋊C4—21no—
78C2×(C2×D3)⋊C4—21no—
79C2×(C2×D3)⋊C4—21no—
80C2×(C2×D3)⋊C4—21no—
81C2×C4⋊Dic3—21no—
82C2×Dic3⋊C4—21no—
83C2×C4×Dic3—21no—
84C2×C4×Dic3C2×C411yescharacteristic
85C2×C4×Dic3D411yescharacteristic
86C2×C4×Dic3D411yescharacteristic
87C2×#208—21no—
88C2×D3×C8—22no—
89#898—22no—
90#898—22no—
91#898—22no—
92#898—22no—
93#897—21no—
94#897—21no—
95#897C2×C412yes—
96#897C2×C412yes—
97#895—21no—
98#895—21no—
99#895C2×C412yes—
100#895C2×C412yes—
101#894—22no—
102#894—22no—
103#894—22no—
104#894—22no—
105#893—22no—
106#893—22no—
107#893—22no—
108#893—22no—
109#893—21no—
110#893—21no—
111#893C2×C412yes—
112#893C2×C412yes—
113#892—22no—
114#892—22no—
115#892—22no—
116#892—22no—
117#887—21no—
118#887—21no—
119#887C2×C412yes—
120#887C2×C412yes—
121#875—21no—
122#875—21no—
123#875C2×C412yes—
124#875C2312yes—
125C42×D3C2×C411yescharacteristic
126C42×D3C2×C411yescharacteristic
127C42×D3D411yescharacteristic
128C42×D3D411yescharacteristic
129C42×D3D411yescharacteristic
130C42×D3D411yescharacteristic
131C42×D3C2311yescharacteristic
132C42×D3C2311yescharacteristic
133C3×C4⋊C8—21no—
134C3×C22⋊C8—21no—
135C3×#97C2312yes—
136C3×#97C2312yes—
137C3×C4×C8—21no—
138#830—21no—
139(C2×D3)⋊C8—22no—
140(C2×D3)⋊C8—22no—
141(C2×D3)⋊C8—22no—
142#815—22no—
143Dic3⋊C8—22no—
144Dic3⋊C8—22no—
145Dic3⋊C8—22no—
146C4⋊(C3⋊C8)—21no—
147#803C2312yes—
148#803C2312yes—
149C4×C3⋊C8—21no—
150#529—32no—
151#529—32no—
152#529—32no—
153#529—32no—
154#529—61no—
155#529—61no—
156#529—61no—
157#529—61no—
158C2×#126—61no—
159C2×#125C2×D311yescharacteristic
160C2×#124—31no—
161C2×#123—61no—
162C2×C22⋊Q8—31no—
163C2×#121—31no—
164C2×C4×Q8—31no—
165C2×C4×D4—31no—
166C2×C4×D4—61no—
167C2×#117—61no—
168C22×C42—31no—
169#432—31no—
170#432—32no—
171#432—32no—
172#432—32no—
173#432—61no—
174#432—61no—
175#432C2×D311yescharacteristic
176#431—61no—
177#431—61no—
178#423—61no—
179C2×C4⋊C8—61no—
180#407—61no—
181C2×C22⋊C8—61no—
182C4×M16—31no—
183C4×M16C2×D311yescharacteristic
184C2×#97—32no—
185#330—34no—
186#330C2×D314yescomplemented
187#329—34no—
188#329—34no—
189C23×D3—21no—
190C23×D3C22⋊C411yescharacteristic
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4C22×C411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×Dic3—21no—
197C22×Dic3C22⋊C411yescharacteristic
198C22×Dic3C22⋊C411yescharacteristic
199C22×Dic3C22⋊C411yescharacteristic
200C2×D12—22no—
201C2×D12—22no—
202C2×D12—21no—
203C2×D12—44no—
204C2×D12—42no—
205C2×D12C22×C412yes—
206C2×C4×D3—22no—
207C2×C4×D3—22no—
208C2×C4×D3—42no—
209C2×C4×D3—42no—
210C2×C4×D3—42no—
211C2×C4×D3—44no—
212C2×C4×D3C22⋊C411yescharacteristic
213C2×C4×D3C22⋊C411yescharacteristic
214C2×C4×D3C22⋊C411yescharacteristic
215C2×C4×D3C22⋊C411yescharacteristic
216C2×C4×D3C22⋊C411yescharacteristic
217C2×C4×D3C22⋊C411yescharacteristic
218C2×C4×D3C22×C411yescharacteristic
219C2×C4×D3C22×C411yescharacteristic
220C2×C4×D3C2×D411yescharacteristic
221C2×C4×D3C2×D411yescharacteristic
222C2×C4×D3C2×D411yescharacteristic
223C2×C4×D3C2×D411yescharacteristic
224C2×Dic6—21no—
225C2×Dic6—42no—
226C2×Dic6C22×C412yes—
227C3×M16—42no—
228C2×C3×C8—22no—
229C2×C3×C8—22no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—22no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—22no—
235C3×C4⋊C4—22no—
236C3×C4⋊C4C22×C412yes—
237C3×C4⋊C4C22×C412yes—
238C3×C42C22×C411yescharacteristic
239C3×C42C2×D411yescharacteristic
240C3×C42C2×D411yescharacteristic
241C3×C42C2411yescharacteristic
242(C2×D3)⋊C4—22no—
243(C2×D3)⋊C4—22no—
244(C2×D3)⋊C4—22no—
245(C2×D3)⋊C4—22no—
246(C2×D3)⋊C4—22no—
247(C2×D3)⋊C4—22no—
248(C2×D3)⋊C4—22no—
249(C2×D3)⋊C4—22no—
250(C2×D3)⋊C4—22no—
251(C2×D3)⋊C4—22no—
252C4⋊Dic3—22no—
253C4⋊Dic3—22no—
254Dic3⋊C4—22no—
255Dic3⋊C4—22no—
256C4×Dic3—22no—
257C4×Dic3—22no—
258C4×Dic3C22⋊C411yescharacteristic
259C4×Dic3C22⋊C411yescharacteristic
260C4×Dic3C22⋊C411yescharacteristic
261C4×Dic3C22⋊C411yescharacteristic
262C4×Dic3C22⋊C411yescharacteristic
263C4×Dic3C22⋊C411yescharacteristic
264C4×Dic3C22×C411yescharacteristic
265C4×Dic3C22×C411yescharacteristic
266C4×Dic3C2×D411yescharacteristic
267C4×Dic3C2×D411yescharacteristic
268C4×Dic3C2×D411yescharacteristic
269C4×Dic3C2×D411yescharacteristic
270#208—42no—
271C2×C3⋊C8—22no—
272C2×C3⋊C8—22no—
273D3×C8—44no—
274C22×Q8—31no—
275C22×D4—31no—
276C22×D4—61no—
277C23×C4—31no—
278C23×C4—31no—
279C23×C4—31no—
280C23×C4—61no—
281C2×M16—21no—
282C2×M16—61no—
283C22×C8—62no—
284#126—62no—
285#126—62no—
286#126—62no—
287#126—62no—
288#125—21no—
289#125—21no—
290#125C4×D312yes—
291#125C22×D312yes—
292#124—32no—
293#124—32no—
294#124—61no—
295#124—61no—
296#123—62no—
297#123—62no—
298#123—62no—
299#123—62no—
300C22⋊Q8—32no—
301C22⋊Q8—32no—
302C22⋊Q8—61no—
303C22⋊Q8—61no—
304#121—32no—
305#121—32no—
306#121—61no—
307#121—61no—
308C4×Q8—32no—
309C4×Q8—32no—
310C4×Q8—61no—
311C4×Q8—61no—
312C4×D4—32no—
313C4×D4—32no—
314C4×D4—61no—
315C4×D4—61no—
316C4×D4—62no—
317C4×D4—62no—
318C4×D4—62no—
319C4×D4—62no—
320#117—62no—
321#117—62no—
322#117—62no—
323#117—62no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—61no—
327C2×C4⋊C4—61no—
328C2×C4⋊C4C4×D311yescharacteristic
329C2×C4⋊C4C4×D311yescharacteristic
330C2×C22⋊C4—61no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C42—31no—
336C2×C42—31no—
337C2×C42—31no—
338C2×C42—31no—
339C2×C42—31no—
340C2×C42—31no—
341C2×C42—31no—
342C2×C42—31no—
343C2×C42—31no—
344C2×C42—31no—
345C2×C42—31no—
346C2×C42—61no—
347C2×C42C22×D311yescharacteristic
348C4⋊C8—21no—
349C4⋊C8—62no—
350C4⋊C8—62no—
351C4⋊C8—62no—
352C4⋊C8—61no—
353C22⋊C8—21no—
354C22⋊C8—62no—
355C22⋊C8—62no—
356C22⋊C8—62no—
357C22⋊C8—61no—
358#97—32no—
359#97—32no—
360#97—62no—
361#97C22×D312yes—
362#97C22×D312yes—
363C4×C8—21no—
364C4×C8—61no—
365C23×C3C2×C22⋊C411yescharacteristic
366C22×D3—22no—
367C22×D3—22no—
368C22×D3—22no—
369C22×D3—41no—
370C22×D3—42no—
371C22×D3—42no—
372C22×D3—42no—
373C22×D3C2×C22⋊C411yescharacteristic
374C22×D3C2×C22⋊C411yescharacteristic
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—41no—
380C2×C3×C4—42no—
381C2×C3×C4—42no—
382C2×C3×C4C2×C22⋊C411yescharacteristic
383C2×C3×C4C2×C22⋊C411yescharacteristic
384C2×C3×C4C2×C22⋊C411yescharacteristic
385C2×C3×C4C23×C411yescharacteristic
386C2×C3×C4C22×D411yescharacteristic
387C2×C3×C4C22×D411yescharacteristic
388C2×Dic3—22no—
389C2×Dic3—22no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—42no—
393C2×Dic3C2×C22⋊C411yescharacteristic
394C2×Dic3C2×C22⋊C411yescharacteristic
395C2×Dic3C2×C22⋊C411yescharacteristic
396C2×Dic3C2×C22⋊C411yescharacteristic
397C2×Dic3C2×C22⋊C411yescharacteristic
398C2×Dic3C2×C22⋊C411yescharacteristic
399D12—42no—
400D12—44no—
401D12—44no—
402D12—42no—
403C4×D3—22no—
404C4×D3—22no—
405C4×D3—22no—
406C4×D3—22no—
407C4×D3—41no—
408C4×D3—41no—
409C4×D3—41no—
410C4×D3—41no—
411C4×D3—41no—
412C4×D3—41no—
413C4×D3—41no—
414C4×D3—41no—
415C4×D3—44no—
416C4×D3—44no—
417Dic6—42no—
418Dic6—42no—
419C3×C8—42no—
420C3×C8—42no—
421C3⋊C8—42no—
422C3⋊C8—42no—
423C24—31no—
424C24—61no—
425C2×Q8—32no—
426C2×Q8—61no—
427C2×Q8—122no—
428C2×D4—32no—
429C2×D4—62no—
430C2×D4—62no—
431C2×D4—61no—
432C2×D4—122no—
433C2×D4—124no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—31no—
437C22×C4—31no—
438C22×C4—31no—
439C22×C4—31no—
440C22×C4—31no—
441C22×C4—31no—
442C22×C4—31no—
443C22×C4—31no—
444C22×C4—31no—
445C22×C4—31no—
446C22×C4—31no—
447C22×C4—31no—
448C22×C4—31no—
449C22×C4—31no—
450C22×C4—31no—
451C22×C4—62no—
452C22×C4—62no—
453C22×C4—61no—
454C22×C4—122no—
455C22×C4—122no—
456C22×C4—122no—
457C22×C4—124no—
458C22×C4C2×C4×D311yescharacteristic
459C22×C4D3×D411yescharacteristic
460C22×C4D3×D411yescharacteristic
461M16—42no—
462M16—122no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—62no—
466C2×C8—62no—
467C2×C8—124no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—62no—
475C4⋊C4—62no—
476C4⋊C4—62no—
477C4⋊C4—62no—
478C4⋊C4C2×C4×D312yes—
479C4⋊C4C2×C4×D312yes—
480C22⋊C4—62no—
481C22⋊C4—62no—
482C22⋊C4—62no—
483C22⋊C4—62no—
484C22⋊C4—62no—
485C22⋊C4—62no—
486C22⋊C4—62no—
487C22⋊C4—62no—
488C22⋊C4—62no—
489C22⋊C4—62no—
490C42—31no—
491C42—31no—
492C42—31no—
493C42—31no—
494C42—31no—
495C42—31no—
496C42—31no—
497C42—31no—
498C42—31no—
499C42—31no—
500C42—31no—
501C42—31no—
502C42—62no—
503C42—62no—
504C42C2×C4×D311yescharacteristic
505C42D3×D411yescharacteristic
506C42D3×D411yescharacteristic
507C42C23×D311yescharacteristic
508C22×C3—21no—
509C22×C3—21no—
510C22×C3#42112yes—
511C22×C3C22×C22⋊C411yescharacteristic
512C2×D3—21no—
513C2×D3—21no—
514C2×D3—21no—
515C2×D3—21no—
516C2×D3—21no—
517C2×D3—44no—
518C2×D3—42no—
519C2×D3—42no—
520C2×D3—42no—
521C2×D3—84no—
522C2×D3#42112yes—
523C3×C4—21no—
524C3×C4—21no—
525C3×C4—21no—
526C3×C4—21no—
527C3×C4—21no—
528C3×C4—42no—
529C3×C4—42no—
530C3×C4#41312yes—
531Dic3—21no—
532Dic3—21no—
533Dic3—21no—
534Dic3—21no—
535Dic3—21no—
536Dic3—21no—
537Dic3—42no—
538C23—31no—
539C23—31no—
540C23—62no—
541C23—62no—
542C23—62no—
543C23—121no—
544C23—122no—
545C23—122no—
546C23—122no—
547C23D3×C22⋊C411yescharacteristic
548Q8—122no—
549Q8—122no—
550D4—122no—
551D4—124no—
552D4—124no—
553D4—122no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—31no—
559C2×C4—31no—
560C2×C4—31no—
561C2×C4—31no—
562C2×C4—31no—
563C2×C4—31no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—62no—
568C2×C4—62no—
569C2×C4—62no—
570C2×C4—62no—
571C2×C4—62no—
572C2×C4—62no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—124no—
582C2×C4—124no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—122no—
586C2×C4D3×C22⋊C411yescharacteristic
587C2×C4D3×C22⋊C411yescharacteristic
588C2×C4D3×C22⋊C411yescharacteristic
589C2×C4C22×C4×D311yescharacteristic
590C2×C4C2×D3×D411yescharacteristic
591C2×C4C2×D3×D411yescharacteristic
592C8—42no—
593C8—42no—
594C8—122no—
595C8—122no—
596C2×C3—22no—
597C2×C3C2×#41311yescharacteristic
598C2×C3C2×#42111yescharacteristic
599C2×C3C2×#42111yescharacteristic
600D3—22no—
601D3—41no—
602D3—82no—
603C22—21no—
604C22—21no—
605C22—32no—
606C22—61no—
607C22—61no—
608C22—61no—
609C22—61no—
610C22—61no—
611C22—124no—
612C22—122no—
613C22—122no—
614C22—122no—
615C22—244no—
616C22#482112yes—
617C22C2×D3×C22⋊C411yescharacteristic
618C4—21no—
619C4—21no—
620C4—21no—
621C4—21no—
622C4—21no—
623C4—42no—
624C4—42no—
625C4—61no—
626C4—61no—
627C4—61no—
628C4—61no—
629C4—61no—
630C4—61no—
631C4—122no—
632C4#475112yes—
633C3#1766811yescharacteristic, complemented, Sylow
634C2—22no—
635C2—62no—
636C2—121no—
637C2—242no—
638C2C2×#475111yescharacteristic
639C2C2×#482111yescharacteristic
640C2D3×#42111yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1528C2
socleC22×C3C22×C22⋊C4
prime radicalC2×#895C4
cosocleC3×C42C24
hypercenter#1528D3
hypocenterC3#17668
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1528

Complements

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

NH ≅ G/Ntype
2. D3×#432C2semidirect
3. #78517C2semidirect
4. C3×#1528C2semidirect
5. #71735C2semidirect
6. #70804C2semidirect
7. #69860C2semidirect
8. #68368C2semidirect
9. #68366C2semidirect
40. #4705C22semidirect
41. #4705C22semidirect
42. #4705C22semidirect
43. D3×#97C22semidirect
46. C3×#330C22semidirect
47. #4543C22semidirect
48. #4464C22semidirect
49. #4464C22semidirect
56. #1528D3semidirect
186. #330C2×D3semidirect
633. C3#17668semidirect

Maximal subgroups

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

subgroupindex
C2×#56912
D3×#4322
#785172
C3×#15282
#717352
#708042
#698602
#683682
#683662
#176683