Finite Group Databasebeta

← #10849249

Subgroups — #10849249

Order 768.

1082 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#559C211yesmaximal, characteristic
2C2×#5056C211yesmaximal, characteristic, complemented
3#82483C211yesmaximal, characteristic, complemented
4C2×#5009C211yesmaximal, characteristic, complemented
5#71762C211yesmaximal, characteristic
6#71762C211yesmaximal, characteristic
7#71722C211yesmaximal, characteristic, complemented
8#71722C211yesmaximal, characteristic, complemented
9#71721C211yesmaximal, characteristic, complemented
10#71634C211yesmaximal, characteristic, complemented
11#71629C211yesmaximal, characteristic, complemented
12#71628C211yesmaximal, characteristic, complemented
13C4×#832C211yesmaximal, characteristic, complemented
14#71607C211yesmaximal, characteristic, complemented
15#71606C211yesmaximal, characteristic, complemented
16#25653—31nomaximal, Sylow
17C3×#559—24no—
18C3×#559—24no—
19C2×C3×#127C2211yescharacteristic
20C2×C3×#124—21no—
21C2×C3×#124C2211yescharacteristic
22C2×C3×#124C2211yescharacteristic
23C2×C3×#123—21no—
24C2×C3×#123—21no—
25C2×C3×C4×Q8C2211yescharacteristic
26C2×C3×C4×Q8C2211yescharacteristic
27C2×C3×C4×D4—21no—
28C2×C3×C4×D4C2211yescharacteristic
29C2×C3×C4×D4C2211yescharacteristic
30C2×#870C2211yescharacteristic
31#5230—21no—
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#835C2211yescharacteristic
35C2×#835C2211yescharacteristic
36C2×#832C2211yescharacteristic
37C2×#832C2211yescharacteristic
38C2×#832C2211yescharacteristic
39C2×#832C2211yescharacteristic
40C2×#832C2211yescharacteristic
41C2×#832C2211yescharacteristic
42C2×#832C2211yescharacteristic
43C2×#832C2211yescharacteristic
44#5056—24no—
45#5009—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×#559D311yescharacteristic
65C2×#488—31no—
66C2×#478—31no—
67#2584—31no—
68#2042—31no—
69#2042—31no—
70#2041—31no—
71#2041—31no—
72#1984—31no—
73#1983—31no—
74#1865—31no—
75#1864—31no—
76#1863—31no—
77#1862—31no—
78C4×#102—31no—
79C22×C3×Q8D411yescharacteristic
80C22×C3×Q8C2311yescharacteristic
81C22×C3×D4—21no—
82C22×C3×D4—21no—
83C22×C3×D4C2311yescharacteristic
84C22×C3×D4C2311yescharacteristic
85C23×C3×C4—21no—
86C23×C3×C4—21no—
87C3×#127—22no—
88C3×#127—22no—
89C3×#124—28no—
90C3×#124—22no—
91C3×#124—22no—
92C3×#124—22no—
93C3×#124D414yes—
94C3×#123—28no—
95C3×#123—28no—
96C3×C4×Q8—22no—
97C3×C4×Q8—22no—
98C3×C4×Q8—22no—
99C3×C4×Q8—22no—
100C3×C4×D4—28no—
101C3×C4×D4—22no—
102C3×C4×D4—22no—
103C3×C4×D4—22no—
104C3×C4×D4D414yes—
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×C22⋊C4—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#832—22no—
142#832—22no—
143#832—22no—
144#832—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#559—24no—
159#559—24no—
160C2×#127C2×D311yescharacteristic
161C2×#125—31no—
162C2×#124—21no—
163C2×#124C2×D311yescharacteristic
164C2×#124C2×D311yescharacteristic
165C2×#123—21no—
166C2×#123—21no—
167C2×C4×Q8C2×D311yescharacteristic
168C2×C4×Q8C2×D311yescharacteristic
169C2×C4×D4—21no—
170C2×C4×D4C2×D311yescharacteristic
171C2×C4×D4C2×D311yescharacteristic
172#488—64no—
173#478—64no—
174C2×#107—31no—
175C2×#106—31no—
176C2×C4⋊C8—31no—
177C2×#103—31no—
178C2×#103—31no—
179C2×#103—31no—
180C2×#103—31no—
181C2×#102—31no—
182C2×#102—31no—
183C2×#102—31no—
184C2×#102—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—
206C24×C3—21no—
207C2×C3×Q8—22no—
208C2×C3×Q8—22no—
209C2×C3×Q8—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C2×C3×Q8—21no—
214C2×C3×Q8—41no—
215C2×C3×Q8C2×D412yes—
216C2×C3×D4—24no—
217C2×C3×D4—24no—
218C2×C3×D4—21no—
219C2×C3×D4—21no—
220C2×C3×D4—21no—
221C2×C3×D4—21no—
222C2×C3×D4—21no—
223C2×C3×D4—21no—
224C2×C3×D4—21no—
225C2×C3×D4—21no—
226C2×C3×D4—21no—
227C2×C3×D4—21no—
228C2×C3×D4—42no—
229C2×C3×D4—42no—
230C2×C3×D4—81no—
231C2×C3×D4—81no—
232C2×C3×D4C2×D412yes—
233C2×C3×D4C2×D412yes—
234C22×C3×C4—21no—
235C22×C3×C4—21no—
236C22×C3×C4—24no—
237C22×C3×C4—24no—
238C22×C3×C4—81no—
239C22×C3×C4—81no—
240C22×C3×C4—81no—
241C22×C3×C4—81no—
242C22×C3×C4C2×D411yescharacteristic
243C22×C3×C4C2×D411yescharacteristic
244C22×C3×C4C2×D411yescharacteristic
245C22×C3×C4C2×D411yescharacteristic
246C22×C3×C4C2×D411yescharacteristic
247C22×C3×C4C2×D411yescharacteristic
248C22×C3×C4C2411yescharacteristic
249C22×Dic3—21no—
250C22×Dic3—21no—
251C22×Dic3—21no—
252C22×Dic3—21no—
253C3×C4⋊C4—24no—
254C3×C4⋊C4—22no—
255C3×C4⋊C4—22no—
256C3×C4⋊C4—22no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4C2×D412yes—
259C3×C22⋊C4—24no—
260C3×C22⋊C4—24no—
261C3×C22⋊C4—24no—
262C3×C22⋊C4—24no—
263C3×C22⋊C4—24no—
264C3×C22⋊C4—24no—
265C3×C42—24no—
266C3×C42—21no—
267C3×C42—21no—
268C3×C42—21no—
269C3×C42—21no—
270C3×C42C2×D412yes—
271C3×C42C2×D412yes—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274C4⋊Dic3—21no—
275C4⋊Dic3—21no—
276C4⋊Dic3—21no—
277C4⋊Dic3—21no—
278C4⋊Dic3#4112yes—
279C4⋊Dic3#4112yes—
280Dic3⋊C4—42no—
281Dic3⋊C4—42no—
282Dic3⋊C4—42no—
283Dic3⋊C4—42no—
284Dic3⋊C4—81no—
285Dic3⋊C4—81no—
286C4×Dic3—81no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—21no—
290C2×C3⋊C8—21no—
291C2×C3⋊C8—21no—
292C2×C3⋊C8—21no—
293C2×C3⋊C8—21no—
294C2×C3⋊C8—21no—
295C2×C3⋊C8—21no—
296C2×C3⋊C8—21no—
297C2×C3⋊C8—21no—
298C2×C3⋊C8—21no—
299C22×Q8#6711yescharacteristic
300C22×Q8C22×D311yescharacteristic
301C22×D4—21no—
302C22×D4—21no—
303C22×D4C22×D311yescharacteristic
304C22×D4C22×D311yescharacteristic
305C23×C4—21no—
306C23×C4—21no—
307C22×C8—31no—
308C22×C8—31no—
309C22×C8—31no—
310C22×C8—31no—
311#127—22no—
312#127—22no—
313#125—62no—
314#125—62no—
315#124—28no—
316#124—22no—
317#124—22no—
318#124—22no—
319#124#6714yes—
320#123—28no—
321#123—28no—
322C4×Q8—22no—
323C4×Q8—22no—
324C4×Q8—22no—
325C4×Q8—22no—
326C4×D4—28no—
327C4×D4—22no—
328C4×D4—22no—
329C4×D4—22no—
330C4×D4#6714yes—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—31no—
333C2×C4⋊C4—31no—
334C2×C4⋊C4—31no—
335C2×C4⋊C4—31no—
336C2×C4⋊C4—61no—
337C2×C4⋊C4—61no—
338C2×C4⋊C4—61no—
339C2×C4⋊C4—61no—
340C2×C4⋊C4—61no—
341C2×C4⋊C4—61no—
342C2×C4⋊C4#6711yescharacteristic
343C2×C4⋊C4#6711yescharacteristic
344C2×C4⋊C4#6711yescharacteristic
345C2×C4⋊C4C22×D311yescharacteristic
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C42—21no—
353C2×C42—61no—
354C2×C42C22×D311yescharacteristic
355C2×C42C22×D311yescharacteristic
356C2×C42C22×D311yescharacteristic
357#107—62no—
358#106—62no—
359C4⋊C8—62no—
360#103—62no—
361#103—62no—
362#103—62no—
363#103—62no—
364#102—62no—
365#102—62no—
366#102—62no—
367#102—62no—
368#102—62no—
369#102—62no—
370#102—62no—
371#102—62no—
372C4×C8—62no—
373#95—61no—
374#95—61no—
375#95—61no—
376#95—61no—
377#95—61no—
378C23×C3—22no—
379C23×C3—22no—
380C23×C3—41no—
381C23×C3—41no—
382C23×C3—41no—
383C23×C3—41no—
384C23×C3—41no—
385C23×C3—41no—
386C23×C3—41no—
387C23×C3—41no—
388C23×C3—41no—
389C23×C3—41no—
390C23×C3—41no—
391C23×C3—41no—
392C23×C3C22×D411yescharacteristic
393C3×Q8—22no—
394C3×Q8—22no—
395C3×Q8—44no—
396C3×Q8—41no—
397C3×Q8—41no—
398C3×D4—22no—
399C3×D4—22no—
400C3×D4—22no—
401C3×D4—22no—
402C3×D4—48no—
403C3×D4—41no—
404C3×D4—41no—
405C3×D4—41no—
406C3×D4—41no—
407C3×D4—84no—
408C2×C3×C4—21no—
409C2×C3×C4—21no—
410C2×C3×C4—21no—
411C2×C3×C4—21no—
412C2×C3×C4—21no—
413C2×C3×C4—21no—
414C2×C3×C4—22no—
415C2×C3×C4—21no—
416C2×C3×C4—21no—
417C2×C3×C4—22no—
418C2×C3×C4—21no—
419C2×C3×C4—21no—
420C2×C3×C4—21no—
421C2×C3×C4—21no—
422C2×C3×C4—21no—
423C2×C3×C4—21no—
424C2×C3×C4—21no—
425C2×C3×C4—41no—
426C2×C3×C4—41no—
427C2×C3×C4—41no—
428C2×C3×C4—41no—
429C2×C3×C4—84no—
430C2×C3×C4—84no—
431C2×C3×C4#12012yes—
432C2×C3×C4#12012yes—
433C2×C3×C4#13512yes—
434C2×C3×C4C22×D411yescharacteristic
435C2×C3×C4C22×D411yescharacteristic
436C2×C3×C4C2×#4111yescharacteristic
437C2×C3×C4C2×#4111yescharacteristic
438C2×C3×C42−1+411yescharacteristic
439C2×C3×C42−1+411yescharacteristic
440C2×Dic3—22no—
441C2×Dic3—22no—
442C2×Dic3—41no—
443C2×Dic3—41no—
444C2×Dic3—41no—
445C2×Dic3—41no—
446C2×Dic3—41no—
447C2×Dic3—41no—
448C2×Dic3—41no—
449C2×Dic3—41no—
450C2×Dic3—41no—
451C2×Dic3—41no—
452C3⋊C8—41no—
453C3⋊C8—41no—
454C3⋊C8—41no—
455C3⋊C8—41no—
456C24—21no—
457C24—21no—
458C2×Q8—22no—
459C2×Q8—22no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—41no—
466C2×Q8C2×#6712yes—
467C2×D4—24no—
468C2×D4—24no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—42no—
480C2×D4—42no—
481C2×D4—81no—
482C2×D4—81no—
483C2×D4C2×#6712yes—
484C2×D4C2×#6712yes—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—24no—
488C22×C4—24no—
489C22×C4—61no—
490C22×C4—61no—
491C22×C4—61no—
492C22×C4—61no—
493C22×C4—81no—
494C22×C4—81no—
495C22×C4—81no—
496C22×C4—81no—
497C22×C4C2×#6711yescharacteristic
498C22×C4C2×#6711yescharacteristic
499C22×C4C2×#6711yescharacteristic
500C22×C4C2×#6711yescharacteristic
501C22×C4C2×#6711yescharacteristic
502C22×C4C2×#6711yescharacteristic
503C22×C4C23×D311yescharacteristic
504C2×C8—61no—
505C2×C8—61no—
506C2×C8—61no—
507C2×C8—61no—
508C2×C8—61no—
509C2×C8—61no—
510C2×C8—61no—
511C2×C8—61no—
512C2×C8—61no—
513C2×C8—61no—
514C2×C8—61no—
515C2×C8—61no—
516C4⋊C4—24no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—21no—
521C4⋊C4—32no—
522C4⋊C4—32no—
523C4⋊C4—61no—
524C4⋊C4—61no—
525C4⋊C4—61no—
526C4⋊C4—61no—
527C4⋊C4—61no—
528C4⋊C4—61no—
529C4⋊C4—122no—
530C4⋊C4—122no—
531C4⋊C4—122no—
532C4⋊C4—122no—
533C4⋊C4—241no—
534C4⋊C4—241no—
535C4⋊C4C2×#6712yes—
536C22⋊C4—24no—
537C22⋊C4—24no—
538C22⋊C4—24no—
539C22⋊C4—24no—
540C22⋊C4—24no—
541C22⋊C4—24no—
542C42—24no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42—241no—
548C42C2×#6712yes—
549C42C2×#6712yes—
550C22×C3—42no—
551C22×C3—42no—
552C22×C3—42no—
553C22×C3—42no—
554C22×C3—42no—
555C22×C3—42no—
556C22×C3—81no—
557C22×C3—81no—
558C22×C3—81no—
559C22×C3—81no—
560C22×C3—81no—
561C22×C3—81no—
562C22×C3—81no—
563C22×C3—81no—
564C22×C3C2×#12011yescharacteristic
565C22×C3#54811yescharacteristic
566C22×C3#54811yescharacteristic
567C22×C3C2×#13511yescharacteristic
568C22×C3#57511yescharacteristic
569C22×C3#57611yescharacteristic
570C22×C3#57711yescharacteristic
571C3×C4—22no—
572C3×C4—22no—
573C3×C4—22no—
574C3×C4—42no—
575C3×C4—41no—
576C3×C4—41no—
577C3×C4—41no—
578C3×C4—42no—
579C3×C4#53211yescharacteristic
580C3×C4#53611yescharacteristic
581C3×C4#54711yescharacteristic
582C3×C4#54711yescharacteristic
583Dic3—42no—
584Dic3—42no—
585Dic3—81no—
586Dic3—81no—
587C23—22no—
588C23—22no—
589C23—41no—
590C23—41no—
591C23—41no—
592C23—41no—
593C23—41no—
594C23—41no—
595C23—41no—
596C23—41no—
597C23—41no—
598C23—41no—
599C23—41no—
600C23—41no—
601C23C22×#6711yescharacteristic
602Q8—22no—
603Q8—22no—
604Q8—44no—
605Q8—41no—
606Q8—41no—
607D4—22no—
608D4—22no—
609D4—22no—
610D4—22no—
611D4—48no—
612D4—41no—
613D4—41no—
614D4—41no—
615D4—41no—
616D4—84no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—22no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—22no—
633C2×C4—21no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—62no—
639C2×C4—62no—
640C2×C4—84no—
641C2×C4—84no—
642C2×C4—121no—
643C2×C4—121no—
644C2×C4—121no—
645C2×C4—121no—
646C2×C4—121no—
647C2×C4—121no—
648C2×C4—121no—
649C2×C4—121no—
650C2×C4—121no—
651C2×C4—121no—
652C2×C4#95012yes—
653C2×C4#95312yes—
654C2×C4#95312yes—
655C2×C4D3×#4111yescharacteristic
656C2×C4D3×#4111yescharacteristic
657C2×C4#101011yescharacteristic
658C2×C4#101011yescharacteristic
659C2×C4C22×#6711yescharacteristic
660C2×C4C22×#6711yescharacteristic
661C8—121no—
662C8—121no—
663C8—121no—
664C8—121no—
665C2×C3—82no—
666C2×C3—82no—
667C2×C3#243211yescharacteristic
668C2×C3#300211yescharacteristic
669C2×C3#301211yescharacteristic
670C2×C3#336111yescharacteristic
671C2×C3#336511yescharacteristic
672C2×C3#337011yescharacteristic
673C2×C3#337111yescharacteristic
674C22—42no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—81no—
681C22—81no—
682C22—81no—
683C22—81no—
684C22—81no—
685C22—81no—
686C22—81no—
687C22—81no—
688C22C2×#95011yescharacteristic
689C22#582211yescharacteristic
690C22#583111yescharacteristic
691C22#583111yescharacteristic
692C22#583511yescharacteristic
693C22#583611yescharacteristic
694C22C2×#95311yescharacteristic
695C4—22no—
696C4—22no—
697C4—22no—
698C4—41no—
699C4—42no—
700C4—41no—
701C4—41no—
702C4—42no—
703C4—122no—
704C4—122no—
705C4—241no—
706C4—241no—
707C4#582411yescharacteristic
708C4#582411yescharacteristic
709C4#582511yescharacteristic
710C4#583211yescharacteristic
711C3#2565311yescharacteristic, complemented, Sylow
712C2—82no—
713C2—82no—
714C2#8371311yescharacteristic
715C2#8371411yescharacteristic
716C2#8371711yescharacteristic
717C2#8371811yescharacteristic
718C2#8408011yescharacteristic
719C2#8409311yescharacteristic
720C2#8410211yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#559C2
socleC23×C3C22×D4
prime radicalC2×C3×#127C22
hypercenterC2×#559D3
hypocenterC3#25653
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#559

Complements

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

NH ≅ G/Ntype
2. C2×#5056C2semidirect
3. #82483C2semidirect
4. C2×#5009C2semidirect
7. #71722C2semidirect
8. #71722C2semidirect
9. #71721C2semidirect
10. #71634C2semidirect
11. #71629C2semidirect
12. #71628C2semidirect
13. C4×#832C2semidirect
14. #71607C2semidirect
15. #71606C2semidirect
711. C3#25653semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5592
C2×#50562
#824832
C2×#50092
#717622
#717222
#717212
#716342
#716292
#716282
C4×#8322
#716072
#716062
#256533