Finite Group Databasebeta

← #10849264

Subgroups — #10849264

Order 768.

1018 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#556C211yesmaximal, characteristic
2C2×#5056C211yesmaximal, characteristic
3#82483C211yesmaximal, characteristic, complemented
4C2×#5022C211yesmaximal, characteristic, complemented
5#71781C211yesmaximal, characteristic, complemented
6#71781C211yesmaximal, characteristic, complemented
7#71722C211yesmaximal, characteristic
8#71722C211yesmaximal, characteristic
9#71721C211yesmaximal, characteristic
10#71634C211yesmaximal, characteristic, complemented
11C4×#835C211yesmaximal, characteristic, complemented
12#71629C211yesmaximal, characteristic, complemented
13#71628C211yesmaximal, characteristic, complemented
14#71607C211yesmaximal, characteristic
15#71606C211yesmaximal, characteristic
16#25654—31nomaximal, Sylow
17C3×#556—24no—
18C3×#556—24no—
19C2×C3×C4⋊Q8C2211yescharacteristic
20C2×C3×#126—21no—
21C2×C3×#126—21no—
22C2×C3×#124C2211yescharacteristic
23C2×C3×#124C2211yescharacteristic
24C2×C3×C22⋊Q8—21no—
25C2×C3×C4×Q8C2211yescharacteristic
26C2×C3×C4×Q8C2211yescharacteristic
27C2×C3×C4×Q8C2211yescharacteristic
28C2×C3×C4×D4C2211yescharacteristic
29C2×C3×#117—21no—
30C2×#870C2211yescharacteristic
31#5230—21no—
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#835C2211yescharacteristic
35C2×#835C2211yescharacteristic
36C2×#835C2211yescharacteristic
37C2×#835C2211yescharacteristic
38C2×#835C2211yescharacteristic
39C2×#835C2211yescharacteristic
40C2×#832C2211yescharacteristic
41C2×#832C2211yescharacteristic
42C2×#832C2211yescharacteristic
43C2×#832C2211yescharacteristic
44#5056—24no—
45#5022—24no—
46#4983—21no—
47#4976—21no—
48#4976—21no—
49C2×#808C2211yescharacteristic
50C2×#807C2211yescharacteristic
51C4×C4⋊Dic3C2211yescharacteristic
52#4928C2211yescharacteristic
53#4928C2211yescharacteristic
54#4928C2211yescharacteristic
55#4928C2211yescharacteristic
56C2×C4⋊(C3⋊C8)C2211yescharacteristic
57C2×C4×C3⋊C8C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61#4524C2211yescharacteristic
62#4524C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×#556D311yescharacteristic
65C2×#488—31no—
66C2×#479—31no—
67#2584—31no—
68#2043—31no—
69#2043—31no—
70#2042—31no—
71#2042—31no—
72#1984—31no—
73#1983—31no—
74#1865—31no—
75#1864—31no—
76#1863—31no—
77#1862—31no—
78C4×#103—31no—
79C22×C3×Q8D411yescharacteristic
80C22×C3×Q8C2311yescharacteristic
81C22×C3×Q8C2311yescharacteristic
82C22×C3×D4C2311yescharacteristic
83C23×C3×C4—21no—
84C3×C4⋊Q8—22no—
85C3×C4⋊Q8—22no—
86C3×#126—28no—
87C3×#126—28no—
88C3×#124—22no—
89C3×#124—22no—
90C3×#124—22no—
91C3×#124D414yes—
92C3×C22⋊Q8—28no—
93C3×C4×Q8—22no—
94C3×C4×Q8—22no—
95C3×C4×Q8—22no—
96C3×C4×Q8—22no—
97C3×C4×Q8—22no—
98C3×C4×Q8D414yes—
99C3×C4×D4—22no—
100C3×C4×D4—22no—
101C3×#117—28no—
102C2×C3×C4⋊C4—21no—
103C2×C3×C4⋊C4—21no—
104C2×C3×C4⋊C4—21no—
105C2×C3×C4⋊C4—21no—
106C2×C3×C4⋊C4D411yescharacteristic
107C2×C3×C4⋊C4D411yescharacteristic
108C2×C3×C4⋊C4D411yescharacteristic
109C2×C3×C4⋊C4C2311yescharacteristic
110C2×C3×C22⋊C4—21no—
111C2×C3×C22⋊C4—21no—
112C2×C3×C22⋊C4—21no—
113C2×C3×C22⋊C4—21no—
114C2×C3×C22⋊C4—21no—
115C2×C3×C42—21no—
116C2×C3×C42—21no—
117C2×C3×C42C2311yescharacteristic
118C2×C3×C42C2311yescharacteristic
119C2×C3×C42C2311yescharacteristic
120C2×C4⋊Dic3C2311yescharacteristic
121C2×C4⋊Dic3C2311yescharacteristic
122C2×C4⋊Dic3C2311yescharacteristic
123C2×C4⋊Dic3C2311yescharacteristic
124C2×Dic3⋊C4—21no—
125C2×Dic3⋊C4—21no—
126C2×Dic3⋊C4—21no—
127C2×Dic3⋊C4—21no—
128C2×Dic3⋊C4—21no—
129C2×Dic3⋊C4—21no—
130C2×C4×Dic3—21no—
131C22×C3⋊C8C2311yescharacteristic
132C22×C3⋊C8C2311yescharacteristic
133C22×C3⋊C8C2311yescharacteristic
134C22×C3⋊C8C2311yescharacteristic
135#870—22no—
136#870—22no—
137#835—22no—
138#835—22no—
139#835—22no—
140#835—22no—
141#835—22no—
142#835—22no—
143#835—22no—
144#835—22no—
145#832—22no—
146#832—22no—
147#832—22no—
148#832—22no—
149C3⋊#95—21no—
150C3⋊#95—21no—
151C3⋊#95—21no—
152C3⋊#95—21no—
153C3⋊#95—21no—
154#808—22no—
155#807—22no—
156C4⋊(C3⋊C8)—22no—
157C4×C3⋊C8—22no—
158#556—24no—
159#556—24no—
160C2×C4⋊Q8C2×D311yescharacteristic
161C2×#126—21no—
162C2×#126—21no—
163C2×#125—31no—
164C2×#124C2×D311yescharacteristic
165C2×#124C2×D311yescharacteristic
166C2×C22⋊Q8—21no—
167C2×C4×Q8C2×D311yescharacteristic
168C2×C4×Q8C2×D311yescharacteristic
169C2×C4×Q8C2×D311yescharacteristic
170C2×C4×D4C2×D311yescharacteristic
171C2×#117—21no—
172#488—64no—
173#479—64no—
174C2×#107—31no—
175C2×#106—31no—
176C2×C4⋊C8—31no—
177C2×#103—31no—
178C2×#103—31no—
179C2×#103—31no—
180C2×#103—31no—
181C2×#103—31no—
182C2×#103—31no—
183C2×#103—31no—
184C2×#103—31no—
185C2×#102—31no—
186C2×#102—31no—
187C2×#102—31no—
188C2×#102—31no—
189C2×C4×C8—31no—
190#398—61no—
191#395—61no—
192#389—31no—
193#389—31no—
194#389—31no—
195#389—31no—
196#389—61no—
197#389—61no—
198C4×C4⋊C4—31no—
199#340—31no—
200#340—31no—
201#336—31no—
202#336—31no—
203#336—31no—
204#336—31no—
205C24×C3—21no—
206C2×C3×Q8—22no—
207C2×C3×Q8—22no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C2×C3×Q8—21no—
214C2×C3×Q8—21no—
215C2×C3×Q8—21no—
216C2×C3×Q8—21no—
217C2×C3×Q8—21no—
218C2×C3×Q8—41no—
219C2×C3×Q8C2×D412yes—
220C2×C3×Q8C2×D412yes—
221C2×C3×D4—21no—
222C2×C3×D4—21no—
223C2×C3×D4—21no—
224C2×C3×D4—21no—
225C2×C3×D4—21no—
226C2×C3×D4C2×D412yes—
227C22×C3×C4—24no—
228C22×C3×C4—21no—
229C22×C3×C4—21no—
230C22×C3×C4—21no—
231C22×C3×C4—42no—
232C22×C3×C4—42no—
233C22×C3×C4C2×D411yescharacteristic
234C22×C3×C4C2×D411yescharacteristic
235C22×C3×C4C2×D411yescharacteristic
236C22×C3×C4C2×D411yescharacteristic
237C22×C3×C4C2×D411yescharacteristic
238C22×C3×C4C2×D411yescharacteristic
239C22×C3×C4C2411yescharacteristic
240C22×Dic3—21no—
241C22×Dic3—21no—
242C22×Dic3—21no—
243C22×Dic3—21no—
244C3×C4⋊C4—24no—
245C3×C4⋊C4—24no—
246C3×C4⋊C4—24no—
247C3×C4⋊C4—24no—
248C3×C4⋊C4—22no—
249C3×C4⋊C4—22no—
250C3×C4⋊C4—22no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4C2×D412yes—
253C3×C22⋊C4—24no—
254C3×C22⋊C4—24no—
255C3×C22⋊C4—24no—
256C3×C22⋊C4—24no—
257C3×C22⋊C4—24no—
258C3×C42—24no—
259C3×C42—24no—
260C3×C42—21no—
261C3×C42—21no—
262C3×C42—21no—
263C3×C42—21no—
264C3×C42C2×D412yes—
265C3×C42C2×D412yes—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3#4112yes—
273C4⋊Dic3#4112yes—
274Dic3⋊C4—42no—
275Dic3⋊C4—42no—
276Dic3⋊C4—42no—
277Dic3⋊C4—42no—
278Dic3⋊C4—81no—
279Dic3⋊C4—81no—
280C4×Dic3—81no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—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—
292C2×C3⋊C8—21no—
293C22×Q8#6711yescharacteristic
294C22×Q8C22×D311yescharacteristic
295C22×Q8C22×D311yescharacteristic
296C22×D4C22×D311yescharacteristic
297C23×C4—21no—
298C22×C8—31no—
299C22×C8—31no—
300C22×C8—31no—
301C22×C8—31no—
302C4⋊Q8—22no—
303C4⋊Q8—22no—
304#126—28no—
305#126—28no—
306#125—62no—
307#125—62no—
308#124—22no—
309#124—22no—
310#124—22no—
311#124#6714yes—
312C22⋊Q8—28no—
313C4×Q8—22no—
314C4×Q8—22no—
315C4×Q8—22no—
316C4×Q8—22no—
317C4×Q8—22no—
318C4×Q8#6714yes—
319C4×D4—22no—
320C4×D4—22no—
321#117—28no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—31no—
327C2×C4⋊C4—31no—
328C2×C4⋊C4—31no—
329C2×C4⋊C4—31no—
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#6711yescharacteristic
337C2×C4⋊C4#6711yescharacteristic
338C2×C4⋊C4#6711yescharacteristic
339C2×C4⋊C4C22×D311yescharacteristic
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—61no—
348C2×C42C22×D311yescharacteristic
349C2×C42C22×D311yescharacteristic
350C2×C42C22×D311yescharacteristic
351#107—62no—
352#106—62no—
353C4⋊C8—62no—
354#103—62no—
355#103—62no—
356#103—62no—
357#103—62no—
358#103—62no—
359#103—62no—
360#103—62no—
361#103—62no—
362#102—62no—
363#102—62no—
364#102—62no—
365#102—62no—
366C4×C8—62no—
367#95—61no—
368#95—61no—
369#95—61no—
370#95—61no—
371#95—61no—
372C23×C3—22no—
373C23×C3—41no—
374C23×C3—41no—
375C23×C3—41no—
376C23×C3—41no—
377C23×C3—41no—
378C23×C3—41no—
379C23×C3C22×D411yescharacteristic
380C3×Q8—22no—
381C3×Q8—22no—
382C3×Q8—22no—
383C3×Q8—22no—
384C3×Q8—44no—
385C3×Q8—41no—
386C3×Q8—41no—
387C3×Q8—41no—
388C3×Q8—41no—
389C3×D4—22no—
390C3×D4—22no—
391C3×D4—41no—
392C3×D4—41no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
400C2×C3×C4—22no—
401C2×C3×C4—21no—
402C2×C3×C4—22no—
403C2×C3×C4—22no—
404C2×C3×C4—21no—
405C2×C3×C4—21no—
406C2×C3×C4—21no—
407C2×C3×C4—21no—
408C2×C3×C4—21no—
409C2×C3×C4—21no—
410C2×C3×C4—21no—
411C2×C3×C4—48no—
412C2×C3×C4—41no—
413C2×C3×C4—41no—
414C2×C3×C4—41no—
415C2×C3×C4—41no—
416C2×C3×C4—41no—
417C2×C3×C4—41no—
418C2×C3×C4#12012yes—
419C2×C3×C4#12012yes—
420C2×C3×C4#13512yes—
421C2×C3×C4C22×D411yescharacteristic
422C2×C3×C4C22×D411yescharacteristic
423C2×C3×C4C2×#4111yescharacteristic
424C2×C3×C4C2×#4111yescharacteristic
425C2×C3×C42−1+411yescharacteristic
426C2×C3×C42−1+411yescharacteristic
427C2×Dic3—22no—
428C2×Dic3—22no—
429C2×Dic3—41no—
430C2×Dic3—41no—
431C2×Dic3—41no—
432C2×Dic3—41no—
433C2×Dic3—41no—
434C2×Dic3—41no—
435C2×Dic3—41no—
436C2×Dic3—41no—
437C2×Dic3—41no—
438C2×Dic3—41no—
439C3⋊C8—41no—
440C3⋊C8—41no—
441C3⋊C8—41no—
442C3⋊C8—41no—
443C24—21no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—41no—
457C2×Q8C2×#6712yes—
458C2×Q8C2×#6712yes—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4C2×#6712yes—
465C22×C4—24no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—61no—
472C22×C4—61no—
473C22×C4—61no—
474C22×C4—61no—
475C22×C4C2×#6711yescharacteristic
476C22×C4C2×#6711yescharacteristic
477C22×C4C2×#6711yescharacteristic
478C22×C4C2×#6711yescharacteristic
479C22×C4C2×#6711yescharacteristic
480C22×C4C2×#6711yescharacteristic
481C22×C4C23×D311yescharacteristic
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8—61no—
486C2×C8—61no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8—61no—
490C2×C8—61no—
491C2×C8—61no—
492C2×C8—61no—
493C2×C8—61no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—32no—
503C4⋊C4—32no—
504C4⋊C4—61no—
505C4⋊C4—61no—
506C4⋊C4—61no—
507C4⋊C4—61no—
508C4⋊C4—61no—
509C4⋊C4—61no—
510C4⋊C4—122no—
511C4⋊C4—122no—
512C4⋊C4—122no—
513C4⋊C4—122no—
514C4⋊C4—241no—
515C4⋊C4—241no—
516C4⋊C4C2×#6712yes—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—24no—
521C22⋊C4—24no—
522C42—24no—
523C42—24no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—241no—
529C42C2×#6712yes—
530C42C2×#6712yes—
531C22×C3—42no—
532C22×C3—42no—
533C22×C3—42no—
534C22×C3—81no—
535C22×C3—81no—
536C22×C3—81no—
537C22×C3—81no—
538C22×C3C2×#12011yescharacteristic
539C22×C3#54811yescharacteristic
540C22×C3#54811yescharacteristic
541C22×C3C2×#13511yescharacteristic
542C22×C3#57511yescharacteristic
543C22×C3#57711yescharacteristic
544C22×C3#57811yescharacteristic
545C3×C4—22no—
546C3×C4—22no—
547C3×C4—22no—
548C3×C4—42no—
549C3×C4—42no—
550C3×C4—41no—
551C3×C4—42no—
552C3×C4—41no—
553C3×C4—41no—
554C3×C4#53211yescharacteristic
555C3×C4#53611yescharacteristic
556C3×C4#54711yescharacteristic
557C3×C4#54711yescharacteristic
558Dic3—42no—
559Dic3—42no—
560Dic3—81no—
561Dic3—81no—
562C23—22no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23C22×#6711yescharacteristic
570Q8—22no—
571Q8—22no—
572Q8—22no—
573Q8—22no—
574Q8—44no—
575Q8—41no—
576Q8—41no—
577Q8—41no—
578Q8—41no—
579D4—22no—
580D4—22no—
581D4—41no—
582D4—41no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—48no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—62no—
609C2×C4—62no—
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—121no—
619C2×C4—121no—
620C2×C4#95012yes—
621C2×C4#95312yes—
622C2×C4#95312yes—
623C2×C4D3×#4111yescharacteristic
624C2×C4D3×#4111yescharacteristic
625C2×C4#101011yescharacteristic
626C2×C4#101011yescharacteristic
627C2×C4C22×#6711yescharacteristic
628C2×C4C22×#6711yescharacteristic
629C8—121no—
630C8—121no—
631C8—121no—
632C8—121no—
633C2×C3—82no—
634C2×C3#243211yescharacteristic
635C2×C3#300211yescharacteristic
636C2×C3#301311yescharacteristic
637C2×C3#336011yescharacteristic
638C2×C3#336411yescharacteristic
639C2×C3#337611yescharacteristic
640C2×C3#337711yescharacteristic
641C22—42no—
642C22—42no—
643C22—42no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22C2×#95011yescharacteristic
649C22#582211yescharacteristic
650C22#583111yescharacteristic
651C22#583111yescharacteristic
652C22#583711yescharacteristic
653C22#583811yescharacteristic
654C22C2×#95311yescharacteristic
655C4—22no—
656C4—22no—
657C4—22no—
658C4—41no—
659C4—42no—
660C4—42no—
661C4—42no—
662C4—41no—
663C4—41no—
664C4—122no—
665C4—122no—
666C4—241no—
667C4—241no—
668C4#582411yescharacteristic
669C4#582411yescharacteristic
670C4#582511yescharacteristic
671C4#583211yescharacteristic
672C3#2565411yescharacteristic, complemented, Sylow
673C2—82no—
674C2#8366711yescharacteristic
675C2#8366911yescharacteristic
676C2#8367611yescharacteristic
677C2#8367811yescharacteristic
678C2#8408011yescharacteristic
679C2#8409311yescharacteristic
680C2#8411411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#556C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenterC2×#556D3
hypocenterC3#25654
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
3. #82483C2semidirect
4. C2×#5022C2semidirect
5. #71781C2semidirect
6. #71781C2semidirect
10. #71634C2semidirect
11. C4×#835C2semidirect
12. #71629C2semidirect
13. #71628C2semidirect
672. C3#25654semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5562
C2×#50562
#824832
C2×#50222
#717812
#717222
#717212
#716342
C4×#8352
#716292
#716282
#716072
#716062
#256543