Finite Group Databasebeta

← #10828569

Subgroups — #10828569

Order 768.

1002 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5583C211yesmaximal, characteristic
2C2×#4862C211yesmaximal, characteristic
3C2×#4811C211yesmaximal, characteristic
4#75968C211yesmaximal, characteristic
5C3×#1865C211yesmaximal, characteristic, complemented
6#71629C211yesmaximal, characteristic, complemented
7#70566C211yesmaximal, characteristic, complemented
8#70284C211yesmaximal, characteristic, complemented
9#70269C211yesmaximal, characteristic
10#70246C211yesmaximal, characteristic, complemented
11#69631C211yesmaximal, characteristic, complemented
12#68879C211yesmaximal, characteristic
13#68829C211yesmaximal, characteristic, complemented
14#68683C211yesmaximal, characteristic, complemented
15#68665C211yesmaximal, characteristic
16#24071—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×#946C2211yescharacteristic
19C2×#944C2211yescharacteristic
20#5583—24no—
21#5583—24no—
22C2×#898—21no—
23C2×#897C2211yescharacteristic
24C2×#892C2211yescharacteristic
25C2×D3×C4⋊C4C2211yescharacteristic
26C2×#890C2211yescharacteristic
27C2×#889—21no—
28C2×#876—21no—
29C2×#872C2211yescharacteristic
30C2×#870—21no—
31C2×C4×Dic6C2211yescharacteristic
32C2×C3×#106C2211yescharacteristic
33C2×C3×#103C2211yescharacteristic
34C2×C3×#103C2211yescharacteristic
35C3×#389C2211yescharacteristic
36C2×#835C2211yescharacteristic
37C2×#835C2211yescharacteristic
38C2×(C2×D3)⋊C8C2211yescharacteristic
39C2×(C2×D3)⋊C8C2211yescharacteristic
40C2×#818C2211yescharacteristic
41C2×#816C2211yescharacteristic
42C2×#816C2211yescharacteristic
43#4989—21no—
44#4984C2211yescharacteristic
45#4975—21no—
46#4972C2211yescharacteristic
47C2×#810C2211yescharacteristic
48C2×#810C2211yescharacteristic
49C2×#808C2211yescharacteristic
50C2×#807C2211yescharacteristic
51#4928C2211yescharacteristic
52#4862—24no—
53#4811—24no—
54#4668—21no—
55#4650—21no—
56C3×#340C2211yescharacteristic
57C3×#336C2211yescharacteristic
58#4551C2211yescharacteristic
59#4550C2211yescharacteristic
60#4550C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×#541—31no—
65C2×#484—31no—
66C2×#483—31no—
67#2317—31no—
68#2059—31no—
69#2059—31no—
70#2043—31no—
71#2043—31no—
72#1873—31no—
73#1873—31no—
74#1865—31no—
75#1865D311yescharacteristic, complemented
76#1863—31no—
77#1863—31no—
78#1818—31no—
79C22×C3×Q8C2311yescharacteristic
80C22×C4×D3—21no—
81C22×C4×D3C2311yescharacteristic
82C22×Dic6C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C3×C4×Q8—22no—
86C3×C4×Q8—22no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C42—21no—
93C2×C3×C42C2311yescharacteristic
94#946—22no—
95#946D414yes—
96#944—22no—
97#944—22no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4D411yescharacteristic
102C2×(C2×D3)⋊C4D411yescharacteristic
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4D411yescharacteristic
118C2×Dic3⋊C4D411yescharacteristic
119C2×C4×Dic3—21no—
120C2×C4×Dic3—21no—
121C2×C4×Dic3C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124#898—28no—
125#897—22no—
126#897D414yes—
127#892—22no—
128#892—22no—
129D3×C4⋊C4—22no—
130D3×C4⋊C4—22no—
131#890—22no—
132#890—22no—
133#889—28no—
134#876—28no—
135#872—22no—
136#872—22no—
137#870—28no—
138C4×Dic6—22no—
139C4×Dic6—22no—
140C3×#106—22no—
141C3×#103—22no—
142C3×#103—22no—
143C3×#95—21no—
144#835—22no—
145#835—22no—
146C3⋊#95—21no—
147C3⋊#95—21no—
148C3⋊#95—21no—
149(C2×D3)⋊C8—22no—
150(C2×D3)⋊C8—22no—
151#818—22no—
152#816—22no—
153#816—22no—
154#810—22no—
155#810—22no—
156#808—22no—
157#807—22no—
158#541—64no—
159#541—64no—
160C2×C4⋊Q8—31no—
161C2×#126—61no—
162C2×#126—61no—
163C2×#125—31no—
164C2×#125—61no—
165C2×#125—61no—
166C2×C22⋊Q8—31no—
167C2×C22⋊Q8—31no—
168C2×C4×Q8—31no—
169C2×C4×Q8C2×D311yescharacteristic
170C2×#117—31no—
171C2×#117—31no—
172C22×C4⋊C4—31no—
173#484—64no—
174#483—64no—
175C2×#107—31no—
176C2×#107—31no—
177C2×#106—31no—
178C2×#106C2×D311yescharacteristic
179C2×#103—31no—
180C2×#103—31no—
181C2×#103—31no—
182C2×#103—31no—
183C2×#103—31no—
184C2×#103—31no—
185C2×#103C2×D311yescharacteristic
186C2×#103C2×D311yescharacteristic
187C2×C22⋊C8—31no—
188C2×C22⋊C8—31no—
189#389—31no—
190#389—31no—
191#389—31no—
192#389C2×D311yescharacteristic
193#385—61no—
194#385—61no—
195#384—61no—
196#382—61no—
197#340—31no—
198#340—31no—
199#340—31no—
200#340C2×D311yescharacteristic
201#336—31no—
202#336—31no—
203#336—31no—
204#336C2×D311yescharacteristic
205C23×D3C2×D411yescharacteristic
206C2×C3×Q8—21no—
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8C2×D412yes—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2411yescharacteristic
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—21no—
223C22×Dic3C2×D411yescharacteristic
224C22×Dic3C2×D411yescharacteristic
225C22×Dic3C2×D411yescharacteristic
226C2×C4×D3—21no—
227C2×C4×D3—21no—
228C2×C4×D3—21no—
229C2×C4×D3—21no—
230C2×C4×D3—42no—
231C2×C4×D3—81no—
232C2×C4×D3—81no—
233C2×C4×D3C2×D412yes—
234C2×C4×D3C2×D412yes—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6—21no—
238C2×Dic6—21no—
239C2×Dic6—21no—
240C2×Dic6C2×D412yes—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C3×C4⋊C4—24no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—81no—
253C3×C4⋊C4C2×D412yes—
254C3×C4⋊C4#4112yes—
255C3×C42—24no—
256C3×C42—21no—
257C3×C42—21no—
258(C2×D3)⋊C4—24no—
259(C2×D3)⋊C4—22no—
260(C2×D3)⋊C4—22no—
261(C2×D3)⋊C4—24no—
262(C2×D3)⋊C4—24no—
263C4⋊Dic3—24no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—24no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—81no—
270C4⋊Dic3C2×D412yes—
271C4⋊Dic3#4112yes—
272Dic3⋊C4—24no—
273Dic3⋊C4—24no—
274Dic3⋊C4—24no—
275Dic3⋊C4—24no—
276Dic3⋊C4—22no—
277Dic3⋊C4—24no—
278Dic3⋊C4—22no—
279Dic3⋊C4—42no—
280Dic3⋊C4—81no—
281Dic3⋊C4—81no—
282C4×Dic3—24no—
283C4×Dic3—24no—
284C4×Dic3—21no—
285C4×Dic3—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—21no—
290C2×C3⋊C8—21no—
291C2×C3⋊C8—21no—
292C22×Q8—31no—
293C22×Q8C22×D311yescharacteristic
294C23×C4—31no—
295C23×C4—61no—
296C22×C8—31no—
297C22×C8—31no—
298C22×C8C22×D311yescharacteristic
299C22×C8C22×D311yescharacteristic
300C4⋊Q8—62no—
301C4⋊Q8—62no—
302#126—68no—
303#126—68no—
304#125—68no—
305#125—62no—
306#125—68no—
307#125—62no—
308C22⋊Q8—34no—
309C22⋊Q8—34no—
310C22⋊Q8—62no—
311C22⋊Q8—62no—
312C4×Q8—22no—
313C4×Q8—22no—
314C4×Q8—62no—
315C4×Q8—62no—
316#117—62no—
317#117—62no—
318#117—62no—
319#117—62no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—31no—
323C2×C4⋊C4—31no—
324C2×C4⋊C4—31no—
325C2×C4⋊C4—31no—
326C2×C4⋊C4—31no—
327C2×C4⋊C4—62no—
328C2×C4⋊C4—62no—
329C2×C4⋊C4—61no—
330C2×C4⋊C4—61no—
331C2×C4⋊C4—61no—
332C2×C4⋊C4—61no—
333C2×C4⋊C4—61no—
334C2×C4⋊C4—61no—
335C2×C4⋊C4—61no—
336C2×C4⋊C4—61no—
337C2×C4⋊C4—61no—
338C2×C4⋊C4—61no—
339C2×C4⋊C4—61no—
340C2×C4⋊C4C22×D311yescharacteristic
341C2×C4⋊C4C22×D311yescharacteristic
342C2×C4⋊C4C22×D311yescharacteristic
343C2×C22⋊C4—31no—
344C2×C22⋊C4—31no—
345C2×C22⋊C4—61no—
346C2×C22⋊C4—61no—
347C2×C22⋊C4—61no—
348C2×C42—21no—
349C2×C42—31no—
350C2×C42—61no—
351C2×C42—61no—
352C2×C42C22×D311yescharacteristic
353#107—62no—
354#107—62no—
355#106—22no—
356#106—62no—
357#103—22no—
358#103—22no—
359#103—62no—
360#103—62no—
361#103—62no—
362#103—62no—
363#103—62no—
364#103—62no—
365C22⋊C8—62no—
366C22⋊C8—62no—
367#95—21no—
368#95—61no—
369#95—61no—
370#95—61no—
371C23×C3C22×D411yescharacteristic
372C22×D3—21no—
373C22×D3—21no—
374C22×D3—21no—
375C22×D3—21no—
376C22×D3—21no—
377C22×D3—21no—
378C22×D3#12012yes—
379C3×Q8—22no—
380C3×Q8—22no—
381C3×Q8—41no—
382C3×Q8—41no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—22no—
386C2×C3×C4—22no—
387C2×C3×C4—22no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4C22×D411yescharacteristic
402C2×C3×C4C22×D411yescharacteristic
403C2×C3×C4C2×#4111yescharacteristic
404C2×C3×C4C2×#4111yescharacteristic
405C2×C3×C42−1+411yescharacteristic
406C2×C3×C42−1+411yescharacteristic
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—22no—
412C2×Dic3—22no—
413C2×Dic3—22no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3#12012yes—
428C4×D3—22no—
429C4×D3—22no—
430C4×D3—22no—
431C4×D3—22no—
432C4×D3—84no—
433Dic6—22no—
434Dic6—22no—
435Dic6—41no—
436Dic6—41no—
437C3×C8—41no—
438C3×C8—41no—
439C3⋊C8—41no—
440C3⋊C8—41no—
441C24—31no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—32no—
448C2×Q8—61no—
449C2×Q8—61no—
450C2×Q8—61no—
451C2×Q8—61no—
452C2×Q8—61no—
453C2×Q8D3×D412yes—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—31no—
459C22×C4—32no—
460C22×C4—32no—
461C22×C4—31no—
462C22×C4—31no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4—61no—
470C22×C4—61no—
471C22×C4—122no—
472C22×C4—241no—
473C22×C4—241no—
474C22×C4D3×D411yescharacteristic
475C22×C4D3×D411yescharacteristic
476C22×C4C23×D311yescharacteristic
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8—61no—
486C2×C8—61no—
487C2×C8—61no—
488C2×C8—61no—
489C4⋊C4—24no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—32no—
495C4⋊C4—32no—
496C4⋊C4—62no—
497C4⋊C4—62no—
498C4⋊C4—64no—
499C4⋊C4—64no—
500C4⋊C4—64no—
501C4⋊C4—64no—
502C4⋊C4—64no—
503C4⋊C4—64no—
504C4⋊C4—64no—
505C4⋊C4—61no—
506C4⋊C4—61no—
507C4⋊C4—61no—
508C4⋊C4—61no—
509C4⋊C4—81no—
510C4⋊C4—122no—
511C4⋊C4—241no—
512C4⋊C4—241no—
513C4⋊C4—241no—
514C4⋊C4#23512yes—
515C4⋊C4D3×D412yes—
516C22⋊C4—62no—
517C22⋊C4—62no—
518C22⋊C4—64no—
519C22⋊C4—64no—
520C22⋊C4—64no—
521C42—24no—
522C42—21no—
523C42—21no—
524C42—64no—
525C42—64no—
526C42—61no—
527C42—61no—
528C22×C3C2×#12011yescharacteristic
529C22×C3#52511yescharacteristic
530C22×C3#53611yescharacteristic
531C22×C3#57711yescharacteristic
532C22×C3#57711yescharacteristic
533C22×C3#57811yescharacteristic
534C22×C3#57811yescharacteristic
535C2×D3—22no—
536C2×D3—22no—
537C2×D3—22no—
538C2×D3—41no—
539C2×D3—41no—
540C2×D3—41no—
541C2×D3—41no—
542C3×C4—42no—
543C3×C4—42no—
544C3×C4—41no—
545C3×C4—41no—
546C3×C4—42no—
547C3×C4—81no—
548C3×C4#54711yescharacteristic
549C3×C4#54711yescharacteristic
550C3×C4#54711yescharacteristic
551C3×C4#54711yescharacteristic
552Dic3—22no—
553Dic3—42no—
554Dic3—42no—
555Dic3—41no—
556Dic3—42no—
557Dic3—41no—
558Dic3—81no—
559C23—32no—
560C23—61no—
561C23—61no—
562C23—61no—
563C23—61no—
564C23—61no—
565C23—61no—
566C23C2×D3×D411yescharacteristic
567Q8—22no—
568Q8—22no—
569Q8—41no—
570Q8—41no—
571Q8—62no—
572Q8—62no—
573Q8—121no—
574Q8—121no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—32no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—61no—
595C2×C4—61no—
596C2×C4—62no—
597C2×C4—62no—
598C2×C4—62no—
599C2×C4—62no—
600C2×C4—61no—
601C2×C4—61no—
602C2×C4—61no—
603C2×C4—61no—
604C2×C4—62no—
605C2×C4—62no—
606C2×C4—61no—
607C2×C4—61no—
608C2×C4—62no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—244no—
619C2×C4C2×#23511yescharacteristic
620C2×C4C2×D3×D411yescharacteristic
621C2×C4C2×D3×D411yescharacteristic
622C2×C4#100711yescharacteristic
623C2×C4D3×#4111yescharacteristic
624C2×C4#101011yescharacteristic
625C8—41no—
626C8—41no—
627C8—121no—
628C8—121no—
629C2×C3#240611yescharacteristic
630C2×C3#301011yescharacteristic
631C2×C3#301111yescharacteristic
632C2×C3#312011yescharacteristic
633C2×C3#312011yescharacteristic
634C2×C3#324211yescharacteristic
635C2×C3#324211yescharacteristic
636D3—42no—
637C22—62no—
638C22—62no—
639C22—62no—
640C22—121no—
641C22—121no—
642C22—121no—
643C22—121no—
644C22#554711yescharacteristic
645C22D3×#12011yescharacteristic
646C22#562811yescharacteristic
647C22#576211yescharacteristic
648C22#577111yescharacteristic
649C22#577811yescharacteristic
650C22#577911yescharacteristic
651C4—42no—
652C4—41no—
653C4—42no—
654C4—42no—
655C4—41no—
656C4—62no—
657C4—81no—
658C4—121no—
659C4—122no—
660C4—121no—
661C4—122no—
662C4—122no—
663C4—241no—
664C4#550711yescharacteristic
665C4#555111yescharacteristic
666C4#563111yescharacteristic
667C4#565511yescharacteristic
668C3#2407111yescharacteristic, complemented, Sylow
669C2—122no—
670C2#7460011yescharacteristic
671C2#7965711yescharacteristic
672C2#7984911yescharacteristic
673C2#8004111yescharacteristic
674C2#8023011yescharacteristic
675C2#8101911yescharacteristic
676C2#8124111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1865C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#1865D3
hypocenterC3#24071
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1865

Complements

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

NH ≅ G/Ntype
5. C3×#1865C2semidirect
6. #71629C2semidirect
7. #70566C2semidirect
8. #70284C2semidirect
10. #70246C2semidirect
11. #69631C2semidirect
13. #68829C2semidirect
14. #68683C2semidirect
75. #1865D3semidirect
668. C3#24071semidirect

Maximal subgroups

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

subgroupindex
C2×#55832
C2×#48622
C2×#48112
#759682
C3×#18652
#716292
#705662
#702842
#702692
#702462
#696312
#688792
#688292
#686832
#686652
#240713