Finite Group Databasebeta

← #10763267

Subgroups — #10763267

Order 768.

984 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5680C211yesmaximal, characteristic
2#79331C211yesmaximal, characteristic, complemented
3D3×#432C211yesmaximal, characteristic, complemented
4C3×#1533C211yesmaximal, characteristic, complemented
5#71719C211yesmaximal, characteristic, complemented
6#70814C211yesmaximal, characteristic, complemented
7#70814C211yesmaximal, characteristic, complemented
8#68346C214yesmaximal, complemented
9#68344C214yesmaximal, complemented
10#17669—31nomaximal, Sylow
11C2×C3×#124C2211yescharacteristic
12C2×#946C411yescharacteristic
13C2×Q8×Dic3C411yescharacteristic
14C2×#938C411yescharacteristic
15C2×D4×Dic3C411yescharacteristic
16#5680—21no—
17#5680—21no—
18#5680—21no—
19#5680—21no—
20#5680C412yes—
21#5680C412yes—
22#5680C2212yes—
23#5680C2212yes—
24C2×#883—21no—
25C2×#67⋊C4—21no—
26C2×#879—21no—
27C2×#877—21no—
28C2×C42×D3C2211yescharacteristic
29C2×#870C2211yescharacteristic
30C3×#432C2211yescharacteristic
31C3×#432C2211yescharacteristic
32#4926C2211yescharacteristic
33#4926C2211yescharacteristic
34#4837—21no—
35#4835—21no—
36#4833—21no—
37D3×C4⋊C8—21no—
38#4726—21no—
39D3×C22⋊C8—21no—
40#4719—21no—
41#4718—21no—
42#4707C2212yescomplemented
43#4705C2212yescomplemented
44#4705C2212yescomplemented
45D3×#97C2212yescomplemented
46C3×#329C2214yescomplemented
47#4540C2214yescomplemented
48#4465C2214yescomplemented
49#4465C2214yescomplemented
50C2×#529—31no—
51#2920—31no—
52C2×#432—31no—
53#1534—31no—
54#1534—31no—
55#1533—31no—
56#1533D311yescharacteristic, complemented
57#1532—34no—
58#1529—34no—
59C22×C3×Q8C2×C411yescharacteristic
60C22×C3×D4C2×C411yescharacteristic
61C22×#67—21no—
62C23×Dic3—21no—
63C22×C4×D3C2×C411yescharacteristic
64C22×C4×D3D411yescharacteristic
65C22×C4×D3D411yescharacteristic
66C3×#124—21no—
67C3×#124—21no—
68C3×#124C2×C412yes—
69C3×#124C2312yes—
70C2×C3×C22⋊C4—21no—
71C2×C3×C22⋊C4—21no—
72C2×C3×C42C2311yescharacteristic
73C2×#217—21no—
74C2×#217—21no—
75#946—21no—
76#946—21no—
77#946C2×C412yes—
78#946C2×C412yes—
79Q8×Dic3—21no—
80Q8×Dic3—21no—
81Q8×Dic3C2×C412yes—
82Q8×Dic3C2×C412yes—
83#938—21no—
84#938—21no—
85#938C2×C412yes—
86#938C2×C412yes—
87D4×Dic3—21no—
88D4×Dic3—21no—
89D4×Dic3C2×C412yes—
90D4×Dic3C2×C412yes—
91C2×(C2×D3)⋊C4—21no—
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3C2×C411yescharacteristic
94C2×C4⋊Dic3C2×C411yescharacteristic
95C2×Dic3⋊C4—21no—
96C2×Dic3⋊C4—21no—
97C2×Dic3⋊C4—21no—
98C2×C4×Dic3—21no—
99C2×C4×Dic3C2×C411yescharacteristic
100C2×C4×Dic3D411yescharacteristic
101C2×C4×Dic3D411yescharacteristic
102C2×ΓL(1,𝔽25)—22no—
103C2×D3×C8—22no—
104#883—22no—
105#883—22no—
106#883—22no—
107#883—22no—
108#67⋊C4—22no—
109#67⋊C4—22no—
110#67⋊C4—22no—
111#67⋊C4—22no—
112#879—22no—
113#879—22no—
114#879—22no—
115#879—22no—
116#877—22no—
117#877—22no—
118#877—22no—
119#877—22no—
120C42×D3C2×C411yescharacteristic
121C42×D3C2×C411yescharacteristic
122C42×D3D411yescharacteristic
123C42×D3D411yescharacteristic
124C42×D3D411yescharacteristic
125C42×D3D411yescharacteristic
126C42×D3C2311yescharacteristic
127C42×D3C2311yescharacteristic
128#870—21no—
129#870—21no—
130#870C2×C412yes—
131#870C2312yes—
132C3×C4⋊C8—21no—
133C3×C4⋊C8—21no—
134C3×C22⋊C8—21no—
135C3×C22⋊C8—21no—
136C3×#97C2312yes—
137C3×#97C2312yes—
138#830—21no—
139#830—21no—
140(C2×D3)⋊C8—22no—
141(C2×D3)⋊C8—22no—
142#815—22no—
143Dic3⋊C8—22no—
144Dic3⋊C8—22no—
145C8×Dic3—22no—
146C4⋊(C3⋊C8)—21no—
147C4⋊(C3⋊C8)—21no—
148#803C2312yes—
149#803C2312yes—
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×#125—31no—
160C2×#124C2×D311yescharacteristic
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—31no—
171#432—32no—
172#432—32no—
173#432—61no—
174#432—61no—
175#432C2×D311yescharacteristic
176#432C2×D311yescharacteristic
177#431—61no—
178#431—61no—
179#423—61no—
180C2×C4⋊C8—61no—
181#407—61no—
182C2×C22⋊C8—61no—
183C4×M16—32no—
184C2×#97—32no—
185#330—34no—
186#330—34no—
187#329—34no—
188#329C2×D314yescomplemented
189C24×C3—21no—
190C23×D3C22⋊C411yescharacteristic
191C2×C3×Q8—21no—
192C2×C3×Q8—42no—
193C2×C3×Q8C22×C412yes—
194C2×C3×D4—21no—
195C2×C3×D4—42no—
196C2×C3×D4C22×C412yes—
197C22×C3×C4—21no—
198C22×C3×C4C22×C411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C2×#67—22no—
202C2×#67—22no—
203C2×#67—44no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—22no—
207C22×Dic3—22no—
208C22×Dic3—44no—
209C22×Dic3C22⋊C411yescharacteristic
210C22×Dic3C22⋊C411yescharacteristic
211C22×Dic3C22⋊C411yescharacteristic
212C2×C4×D3—42no—
213C2×C4×D3—42no—
214C2×C4×D3—42no—
215C2×C4×D3C22⋊C411yescharacteristic
216C2×C4×D3C22⋊C411yescharacteristic
217C2×C4×D3C22⋊C411yescharacteristic
218C2×C4×D3C22⋊C411yescharacteristic
219C2×C4×D3C22⋊C411yescharacteristic
220C2×C4×D3C22⋊C411yescharacteristic
221C2×C4×D3C22×C411yescharacteristic
222C2×C4×D3C22×C411yescharacteristic
223C2×C4×D3C2×D411yescharacteristic
224C2×C4×D3C2×D411yescharacteristic
225C2×C4×D3C2×D411yescharacteristic
226C2×C4×D3C2×D411yescharacteristic
227C2×C3×C8—22no—
228C2×C3×C8—22no—
229C3×C22⋊C4—22no—
230C3×C22⋊C4—22no—
231C3×C22⋊C4—22no—
232C3×C22⋊C4—22no—
233C3×C42C22×C411yescharacteristic
234C3×C42C2×D411yescharacteristic
235C3×C42C2×D411yescharacteristic
236C3×C42C2411yescharacteristic
237#217—22no—
238#217—22no—
239#217—22no—
240#217—22no—
241(C2×D3)⋊C4—22no—
242(C2×D3)⋊C4—22no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—22no—
246C4⋊Dic3—22no—
247C4⋊Dic3C22×C412yes—
248C4⋊Dic3C22×C412yes—
249Dic3⋊C4—22no—
250Dic3⋊C4—22no—
251Dic3⋊C4—22no—
252Dic3⋊C4—22no—
253Dic3⋊C4—22no—
254Dic3⋊C4—22no—
255C4×Dic3—22no—
256C4×Dic3—22no—
257C4×Dic3C22⋊C411yescharacteristic
258C4×Dic3C22⋊C411yescharacteristic
259C4×Dic3C22⋊C411yescharacteristic
260C4×Dic3C22⋊C411yescharacteristic
261C4×Dic3C22⋊C411yescharacteristic
262C4×Dic3C22⋊C411yescharacteristic
263C4×Dic3C22×C411yescharacteristic
264C4×Dic3C22×C411yescharacteristic
265C4×Dic3C2×D411yescharacteristic
266C4×Dic3C2×D411yescharacteristic
267C4×Dic3C2×D411yescharacteristic
268C4×Dic3C2×D411yescharacteristic
269C2×C3⋊C8—22no—
270C2×C3⋊C8—22no—
271ΓL(1,𝔽25)—44no—
272D3×C8—44no—
273C22×Q8C4×D311yescharacteristic
274C22×D4—61no—
275C22×D4C4×D311yescharacteristic
276C23×C4—31no—
277C23×C4—31no—
278C23×C4—31no—
279C23×C4—61no—
280C2×M16—62no—
281C22×C8—62no—
282#126—62no—
283#126—62no—
284#126—62no—
285#126—62no—
286#125—32no—
287#125—32no—
288#125—61no—
289#125—61no—
290#124—21no—
291#124—21no—
292#124C4×D312yes—
293#124C22×D312yes—
294#123—62no—
295#123—62no—
296#123—62no—
297#123—62no—
298C22⋊Q8—32no—
299C22⋊Q8—32no—
300C22⋊Q8—61no—
301C22⋊Q8—61no—
302#121—32no—
303#121—32no—
304#121—61no—
305#121—61no—
306C4×Q8—32no—
307C4×Q8—32no—
308C4×Q8—61no—
309C4×Q8—61no—
310C4×D4—32no—
311C4×D4—32no—
312C4×D4—61no—
313C4×D4—61no—
314C4×D4—62no—
315C4×D4—62no—
316C4×D4—62no—
317C4×D4—62no—
318#117—62no—
319#117—62no—
320#117—62no—
321#117—62no—
322C2×C4⋊C4—31no—
323C2×C4⋊C4—31no—
324C2×C4⋊C4—61no—
325C2×C4⋊C4—61no—
326C2×C4⋊C4—61no—
327C2×C4⋊C4—61no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—61no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C42—31no—
334C2×C42—31no—
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—61no—
345C2×C42C22×D311yescharacteristic
346C4⋊C8—21no—
347C4⋊C8—21no—
348C4⋊C8—62no—
349C4⋊C8—62no—
350C4⋊C8—61no—
351C4⋊C8—61no—
352C22⋊C8—21no—
353C22⋊C8—21no—
354C22⋊C8—62no—
355C22⋊C8—62no—
356C22⋊C8—61no—
357C22⋊C8—61no—
358#97—32no—
359#97—32no—
360#97—62no—
361#97C22×D312yes—
362#97C22×D312yes—
363C4×C8—62no—
364C23×C3—22no—
365C23×C3—42no—
366C23×C3—42no—
367C23×C3—42no—
368C23×C3C2×C22⋊C411yescharacteristic
369C22×D3—22no—
370C22×D3—22no—
371C22×D3—41no—
372C22×D3C2×C22⋊C411yescharacteristic
373C22×D3C2×C22⋊C411yescharacteristic
374C3×Q8—42no—
375C3×Q8—42no—
376C3×D4—42no—
377C3×D4—42no—
378C2×C3×C4—22no—
379C2×C3×C4—22no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—42no—
383C2×C3×C4C2×C22⋊C411yescharacteristic
384C2×C3×C4C2×C22⋊C411yescharacteristic
385C2×C3×C4C2×C22⋊C411yescharacteristic
386C2×C3×C4C23×C411yescharacteristic
387C2×C3×C4C22×D411yescharacteristic
388C2×C3×C4C22×D411yescharacteristic
389#67—44no—
390#67—44no—
391C2×Dic3—22no—
392C2×Dic3—22no—
393C2×Dic3—22no—
394C2×Dic3—22no—
395C2×Dic3—41no—
396C2×Dic3—44no—
397C2×Dic3—42no—
398C2×Dic3—42no—
399C2×Dic3—44no—
400C2×Dic3C2×C22⋊C411yescharacteristic
401C2×Dic3C2×C22⋊C411yescharacteristic
402C2×Dic3C2×C22⋊C411yescharacteristic
403C2×Dic3C2×C22⋊C411yescharacteristic
404C2×Dic3C2×C22⋊C411yescharacteristic
405C2×Dic3C2×C22⋊C411yescharacteristic
406C4×D3—22no—
407C4×D3—22no—
408C4×D3—22no—
409C4×D3—22no—
410C4×D3—41no—
411C4×D3—41no—
412C4×D3—41no—
413C4×D3—41no—
414C4×D3—41no—
415C4×D3—41no—
416C4×D3—41no—
417C4×D3—41no—
418C3×C8—42no—
419C3×C8—42no—
420C3⋊C8—42no—
421C3⋊C8—42no—
422C24—21no—
423C24—31no—
424C2×Q8—21no—
425C2×Q8—42no—
426C2×Q8C2×C4×D312yes—
427C2×D4—21no—
428C2×D4—42no—
429C2×D4—62no—
430C2×D4—62no—
431C2×D4—124no—
432C2×D4C2×C4×D312yes—
433C22×C4—21no—
434C22×C4—31no—
435C22×C4—31no—
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—62no—
450C22×C4—62no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—122no—
454C22×C4—122no—
455C22×C4—122no—
456C22×C4—124no—
457C22×C4C2×C4×D311yescharacteristic
458C22×C4D3×D411yescharacteristic
459C22×C4D3×D411yescharacteristic
460M16—124no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—62no—
464C2×C8—62no—
465C2×C8—124no—
466C4⋊C4—32no—
467C4⋊C4—32no—
468C4⋊C4—62no—
469C4⋊C4—62no—
470C4⋊C4—62no—
471C4⋊C4—62no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—62no—
475C4⋊C4—62no—
476C4⋊C4—62no—
477C4⋊C4—62no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—62no—
483C22⋊C4—62no—
484C22⋊C4—62no—
485C22⋊C4—62no—
486C22⋊C4—62no—
487C22⋊C4—62no—
488C42—31no—
489C42—31no—
490C42—31no—
491C42—31no—
492C42—31no—
493C42—31no—
494C42—31no—
495C42—31no—
496C42—31no—
497C42—31no—
498C42—31no—
499C42—31no—
500C42—62no—
501C42—62no—
502C42C2×C4×D311yescharacteristic
503C42D3×D411yescharacteristic
504C42D3×D411yescharacteristic
505C42C23×D311yescharacteristic
506C22×C3—21no—
507C22×C3—21no—
508C22×C3—42no—
509C22×C3—42no—
510C22×C3—42no—
511C22×C3—84no—
512C22×C3#41312yes—
513C22×C3C22×C22⋊C411yescharacteristic
514C2×D3—21no—
515C2×D3—21no—
516C2×D3—21no—
517C2×D3—21no—
518C2×D3—21no—
519C2×D3—44no—
520C2×D3#42112yes—
521C3×C4—21no—
522C3×C4—21no—
523C3×C4—21no—
524C3×C4—21no—
525C3×C4—21no—
526C3×C4—21no—
527C3×C4—42no—
528Dic3—21no—
529Dic3—21no—
530Dic3—21no—
531Dic3—21no—
532Dic3—21no—
533Dic3—42no—
534Dic3—42no—
535Dic3#42112yes—
536C23—22no—
537C23—31no—
538C23—31no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—62no—
543C23—62no—
544C23—121no—
545C23D3×C22⋊C411yescharacteristic
546Q8—42no—
547Q8—42no—
548D4—42no—
549D4—42no—
550D4—124no—
551D4—124no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—31no—
555C2×C4—31no—
556C2×C4—31no—
557C2×C4—31no—
558C2×C4—31no—
559C2×C4—31no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—62no—
564C2×C4—62no—
565C2×C4—62no—
566C2×C4—62no—
567C2×C4—62no—
568C2×C4—62no—
569C2×C4—62no—
570C2×C4—62no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—124no—
580C2×C4—121no—
581C2×C4—124no—
582C2×C4—122no—
583C2×C4—122no—
584C2×C4D3×C22⋊C411yescharacteristic
585C2×C4D3×C22⋊C411yescharacteristic
586C2×C4D3×C22⋊C411yescharacteristic
587C2×C4C22×C4×D311yescharacteristic
588C2×C4C2×D3×D411yescharacteristic
589C2×C4C2×D3×D411yescharacteristic
590C8—42no—
591C8—42no—
592C8—122no—
593C8—122no—
594C2×C3—22no—
595C2×C3—82no—
596C2×C3C2×#41311yescharacteristic
597C2×C3C2×#42111yescharacteristic
598C2×C3C2×#42111yescharacteristic
599D3—22no—
600D3—41no—
601C22—21no—
602C22—21no—
603C22—32no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—61no—
608C22—61no—
609C22—61no—
610C22—61no—
611C22—61no—
612C22—84no—
613C22—124no—
614C22#474512yes—
615C22C2×D3×C22⋊C411yescharacteristic
616C4—21no—
617C4—21no—
618C4—21no—
619C4—21no—
620C4—21no—
621C4—21no—
622C4—32no—
623C4—42no—
624C4—61no—
625C4—61no—
626C4—61no—
627C4—61no—
628C4—61no—
629C4—122no—
630C4—122no—
631C3#1766911yescharacteristic, complemented, Sylow
632C2—22no—
633C2—62no—
634C2—82no—
635C2—121no—
636C2C2×#474511yescharacteristic
637C2D3×#42111yescharacteristic
638C2D3×#42111yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1533C2
socleC22×C3C22×C22⋊C4
prime radicalC2×#938C4
cosocleC3×C42C24
hypercenter#1533D3
hypocenterC3#17669
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1533

Complements

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

NH ≅ G/Ntype
2. #79331C2semidirect
3. D3×#432C2semidirect
4. C3×#1533C2semidirect
5. #71719C2semidirect
6. #70814C2semidirect
7. #70814C2semidirect
8. #68346C2semidirect
9. #68344C2semidirect
42. #4707C22semidirect
43. #4705C22semidirect
44. #4705C22semidirect
45. D3×#97C22semidirect
46. C3×#329C22semidirect
47. #4540C22semidirect
48. #4465C22semidirect
49. #4465C22semidirect
56. #1533D3semidirect
188. #329C2×D3semidirect
631. C3#17669semidirect

Maximal subgroups

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

subgroupindex
C2×#56802
#793312
D3×#4322
C3×#15332
#717192
#708142
#683462
#683442
#176693