Finite Group Databasebeta

← #10900980

Subgroups — #10900980

Order 768.

1002 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5688C211yesmaximal, characteristic
2C2×#4888C211yesmaximal, characteristic
3C2×#4868C211yesmaximal, characteristic
4#76287C211yesmaximal, characteristic
5C3×#2059C211yesmaximal, characteristic, complemented
6#71744C211yesmaximal, characteristic, complemented
7#70617C211yesmaximal, characteristic, complemented
8#70565C211yesmaximal, characteristic, complemented
9#69606C211yesmaximal, characteristic, complemented
10#68910C211yesmaximal, characteristic, complemented
11#68910C211yesmaximal, characteristic, complemented
12#68873C211yesmaximal, characteristic
13#68813C211yesmaximal, characteristic
14#68690C211yesmaximal, characteristic, complemented
15#68664C211yesmaximal, characteristic
16#29645—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18#5688—24no—
19#5688—24no—
20C2×#897C2211yescharacteristic
21C2×#896—21no—
22C2×#896—21no—
23C2×#892C2211yescharacteristic
24C2×#892C2211yescharacteristic
25C2×#888C2211yescharacteristic
26C2×#888C2211yescharacteristic
27C2×#887—21no—
28C2×#887—21no—
29C2×C42×D3C2211yescharacteristic
30C2×#870C2211yescharacteristic
31C2×C3×#107C2211yescharacteristic
32C2×C3×#106C2211yescharacteristic
33C3×#389C2211yescharacteristic
34C2×(C2×D3)⋊C8C2211yescharacteristic
35C2×(C2×D3)⋊C8C2211yescharacteristic
36C2×#816C2211yescharacteristic
37C2×#816C2211yescharacteristic
38#4989—21no—
39#4987C2211yescharacteristic
40#4980—21no—
41#4973C2211yescharacteristic
42#4972C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#810C2211yescharacteristic
45C2×#808C2211yescharacteristic
46C2×#807C2211yescharacteristic
47#4929C2211yescharacteristic
48#4888—24no—
49#4868—24no—
50#4664—21no—
51#4655—21no—
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#336C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4550C2211yescharacteristic
58#4550C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×#533—31no—
65C2×#485—31no—
66C2×#485—31no—
67#2855—31no—
68#2059—31no—
69#2059D311yescharacteristic, complemented
70#2058—31no—
71#2057—31no—
72#2026—31no—
73#2026—31no—
74#2024—31no—
75#2024—31no—
76#1825—31no—
77#1825—31no—
78#1817—31no—
79C22×C4×D3D411yescharacteristic
80C22×C4×D3D411yescharacteristic
81C22×C4×D3C2311yescharacteristic
82C22×Dic6—21no—
83C22×Dic6—21no—
84C22×Dic6C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C3×#125—22no—
88C3×#125—22no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C42C2311yescharacteristic
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3D411yescharacteristic
113C2×C4×Dic3D411yescharacteristic
114C2×C4×Dic3C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117#897—22no—
118#897—22no—
119#896—28no—
120#896—28no—
121#892—22no—
122#892—22no—
123#892D414yes—
124#892D414yes—
125#888—22no—
126#888—22no—
127#888—22no—
128#888—22no—
129#887—28no—
130#887—28no—
131C42×D3—22no—
132C42×D3—22no—
133#870—22no—
134#870—22no—
135C3×#107—22no—
136C3×#106—22no—
137C3×#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144(C2×D3)⋊C8—22no—
145(C2×D3)⋊C8—22no—
146#816—22no—
147#816—22no—
148#810—22no—
149#810—22no—
150#808—22no—
151#807—22no—
152#533—64no—
153#533—64no—
154C2×C4⋊Q8—31no—
155C2×C4⋊Q8—31no—
156C2×#125—31no—
157C2×#125C2×D311yescharacteristic
158C2×C22⋊Q8—31no—
159C2×C22⋊Q8—61no—
160C2×C22⋊Q8—61no—
161C2×C4×Q8—61no—
162C2×C4×Q8—61no—
163C2×#117—31no—
164C2×#117—31no—
165C22×C42—31no—
166#485—64no—
167#485—64no—
168C2×#107—31no—
169C2×#107C2×D311yescharacteristic
170C2×#106—31no—
171C2×#106C2×D311yescharacteristic
172C2×#103—31no—
173C2×#103—31no—
174C2×#103—31no—
175C2×#103—31no—
176C2×C22⋊C8—31no—
177C2×C22⋊C8—31no—
178#391—31no—
179#391—31no—
180#389—31no—
181#389C2×D311yescharacteristic
182#387—61no—
183#387—61no—
184#385—61no—
185#381—61no—
186#380—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193#340C2×D311yescharacteristic
194#340C2×D311yescharacteristic
195#336—31no—
196#336—31no—
197#336—31no—
198#336C2×D311yescharacteristic
199C23×D3C2×D411yescharacteristic
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2411yescharacteristic
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C2×C4×D3—21no—
215C2×C4×D3—21no—
216C2×C4×D3—21no—
217C2×C4×D3—21no—
218C2×C4×D3—22no—
219C2×C4×D3—22no—
220C2×C4×D3—22no—
221C2×C4×D3—22no—
222C2×C4×D3—41no—
223C2×C4×D3—41no—
224C2×C4×D3C2×D412yes—
225C2×C4×D3C2×D412yes—
226C2×Dic6—24no—
227C2×Dic6—24no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—42no—
235C2×Dic6—42no—
236C2×Dic6—42no—
237C2×Dic6—42no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C3×C4⋊C4—24no—
245C3×C4⋊C4—24no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—42no—
253C3×C4⋊C4C2×D412yes—
254C3×C4⋊C4C2×D412yes—
255C3×C42—21no—
256C3×C42—21no—
257(C2×D3)⋊C4—24no—
258(C2×D3)⋊C4—24no—
259(C2×D3)⋊C4—42no—
260(C2×D3)⋊C4—42no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—42no—
266C4⋊Dic3C2×D412yes—
267C4⋊Dic3C2×D412yes—
268Dic3⋊C4—24no—
269Dic3⋊C4—24no—
270Dic3⋊C4—24no—
271Dic3⋊C4—24no—
272Dic3⋊C4—42no—
273C4×Dic3—21no—
274C4×Dic3—21no—
275C4×Dic3—22no—
276C4×Dic3—22no—
277C4×Dic3—24no—
278C4×Dic3—24no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C22×Q8—31no—
286C22×Q8—61no—
287C22×Q8—61no—
288C23×C4—31no—
289C23×C4—31no—
290C23×C4—31no—
291C22×C8—31no—
292C22×C8—31no—
293C22×C8C22×D311yescharacteristic
294C22×C8C22×D311yescharacteristic
295C4⋊Q8—62no—
296C4⋊Q8—62no—
297C4⋊Q8—62no—
298C4⋊Q8—62no—
299#125—22no—
300#125—22no—
301#125—62no—
302#125—62no—
303C22⋊Q8—68no—
304C22⋊Q8—68no—
305C22⋊Q8—62no—
306C22⋊Q8—62no—
307C4×Q8—68no—
308C4×Q8—68no—
309#117—34no—
310#117—34no—
311#117—62no—
312#117—62no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—61no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4—61no—
324C2×C4⋊C4—61no—
325C2×C4⋊C4C22×D311yescharacteristic
326C2×C4⋊C4C22×D311yescharacteristic
327C2×C4⋊C4C22×D311yescharacteristic
328C2×C4⋊C4C22×D311yescharacteristic
329C2×C22⋊C4—61no—
330C2×C22⋊C4—61no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C42—31no—
334C2×C42—31no—
335C2×C42—31no—
336C2×C42—62no—
337C2×C42—62no—
338C2×C42—61no—
339C2×C42—61no—
340C2×C42C22×D311yescharacteristic
341#107—22no—
342#107—62no—
343#106—22no—
344#106—62no—
345#103—62no—
346#103—62no—
347#103—62no—
348#103—62no—
349C22⋊C8—62no—
350C22⋊C8—62no—
351#95—21no—
352#95—61no—
353#95—61no—
354#95—61no—
355#95—61no—
356#95—61no—
357#95—61no—
358C23×C3C22×D411yescharacteristic
359C22×D3—21no—
360C22×D3—21no—
361C22×D3—21no—
362C22×D3—21no—
363C22×D3—21no—
364C22×D3—21no—
365C22×D3#12012yes—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4C22×D411yescharacteristic
385C2×C3×C4C22×D411yescharacteristic
386C2×C3×C42−1+411yescharacteristic
387C2×C3×C42−1+411yescharacteristic
388C2×C3×C42−1+411yescharacteristic
389C2×C3×C42−1+411yescharacteristic
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—22no—
397C2×Dic3—22no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3#12012yes—
409C2×Dic3#13512yes—
410C2×Dic3#13712yes—
411C4×D3—22no—
412C4×D3—22no—
413C4×D3—22no—
414C4×D3—22no—
415C4×D3—44no—
416C4×D3—44no—
417Dic6—48no—
418Dic6—48no—
419Dic6—41no—
420Dic6—41no—
421Dic6—41no—
422Dic6—41no—
423C3×C8—41no—
424C3×C8—41no—
425C3⋊C8—41no—
426C3⋊C8—41no—
427C24—31no—
428C2×Q8—64no—
429C2×Q8—64no—
430C2×Q8—61no—
431C2×Q8—61no—
432C2×Q8—61no—
433C2×Q8—61no—
434C2×Q8—61no—
435C2×Q8—61no—
436C2×Q8—122no—
437C2×Q8—122no—
438C2×Q8—122no—
439C2×Q8—122no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—31no—
445C22×C4—31no—
446C22×C4—31no—
447C22×C4—32no—
448C22×C4—32no—
449C22×C4—62no—
450C22×C4—62no—
451C22×C4—62no—
452C22×C4—62no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—61no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4—121no—
462C22×C4—121no—
463C22×C4D3×D411yescharacteristic
464C22×C4D3×D411yescharacteristic
465C22×C4C23×D311yescharacteristic
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—61no—
473C2×C8—61no—
474C2×C8—61no—
475C2×C8—61no—
476C2×C8—61no—
477C2×C8—61no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—32no—
487C4⋊C4—32no—
488C4⋊C4—42no—
489C4⋊C4—64no—
490C4⋊C4—64no—
491C4⋊C4—64no—
492C4⋊C4—64no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—122no—
498C4⋊C4—122no—
499C4⋊C4D3×D412yes—
500C4⋊C4D3×D412yes—
501C22⋊C4—64no—
502C22⋊C4—64no—
503C22⋊C4—122no—
504C22⋊C4—122no—
505C42—21no—
506C42—21no—
507C42—62no—
508C42—62no—
509C42—64no—
510C42—64no—
511C42—61no—
512C42—61no—
513C22×C3C2×#12011yescharacteristic
514C22×C3#53611yescharacteristic
515C22×C3#53611yescharacteristic
516C22×C3C2×#13511yescharacteristic
517C22×C3C2×#13711yescharacteristic
518C22×C3#57511yescharacteristic
519C22×C3#57511yescharacteristic
520C2×D3—22no—
521C2×D3—22no—
522C2×D3—22no—
523C2×D3—41no—
524C2×D3—41no—
525C2×D3—41no—
526C2×D3—41no—
527C3×C4—42no—
528C3×C4—41no—
529C3×C4—42no—
530C3×C4—41no—
531C3×C4—81no—
532C3×C4—81no—
533C3×C4#53611yescharacteristic
534C3×C4#53611yescharacteristic
535C3×C4#53611yescharacteristic
536C3×C4#53611yescharacteristic
537Dic3—22no—
538Dic3—22no—
539Dic3—22no—
540Dic3—42no—
541Dic3—42no—
542Dic3—81no—
543Dic3—81no—
544C23—32no—
545C23—61no—
546C23—61no—
547C23—61no—
548C23—61no—
549C23—61no—
550C23—61no—
551C23C2×D3×D411yescharacteristic
552Q8—128no—
553Q8—128no—
554Q8—121no—
555Q8—121no—
556Q8—121no—
557Q8—121no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—32no—
567C2×C4—32no—
568C2×C4—32no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—61no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—61no—
585C2×C4—62no—
586C2×C4—62no—
587C2×C4—62no—
588C2×C4—62no—
589C2×C4—62no—
590C2×C4—62no—
591C2×C4—124no—
592C2×C4—124no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4C2×D3×D411yescharacteristic
604C2×C4C2×D3×D411yescharacteristic
605C2×C4#100711yescharacteristic
606C2×C4#100711yescharacteristic
607C2×C4#100711yescharacteristic
608C2×C4#101011yescharacteristic
609C8—41no—
610C8—41no—
611C8—121no—
612C8—121no—
613C2×C3#261411yescharacteristic
614C2×C3#300211yescharacteristic
615C2×C3#300711yescharacteristic
616C2×C3#323111yescharacteristic
617C2×C3#323311yescharacteristic
618C2×C3#323711yescharacteristic
619C2×C3#324011yescharacteristic
620D3—42no—
621C22—62no—
622C22—62no—
623C22—62no—
624C22—121no—
625C22—121no—
626C22—121no—
627C22—121no—
628C22D3×#12011yescharacteristic
629C22#562811yescharacteristic
630C22#572411yescharacteristic
631C22D3×#13511yescharacteristic
632C22#576811yescharacteristic
633C22#577311yescharacteristic
634C22D3×#13711yescharacteristic
635C4—41no—
636C4—42no—
637C4—41no—
638C4—42no—
639C4—62no—
640C4—62no—
641C4—62no—
642C4—81no—
643C4—81no—
644C4—122no—
645C4—122no—
646C4—241no—
647C4—241no—
648C4#562811yescharacteristic
649C4#562811yescharacteristic
650C4#572411yescharacteristic
651C4#572411yescharacteristic
652C3#2964511yescharacteristic, complemented, Sylow
653C2—122no—
654C2#7725611yescharacteristic
655C2#8008211yescharacteristic
656C2#8022811yescharacteristic
657C2#8117511yescharacteristic
658C2#8123611yescharacteristic
659C2#8192111yescharacteristic
660C2#8198111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2059C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#2059D3
hypocenterC3#29645
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2059

Complements

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

NH ≅ G/Ntype
5. C3×#2059C2semidirect
6. #71744C2semidirect
7. #70617C2semidirect
8. #70565C2semidirect
9. #69606C2semidirect
10. #68910C2semidirect
11. #68910C2semidirect
14. #68690C2semidirect
69. #2059D3semidirect
652. C3#29645semidirect

Maximal subgroups

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

subgroupindex
C2×#56882
C2×#48882
C2×#48682
#762872
C3×#20592
#717442
#706172
#705652
#696062
#689102
#688732
#688132
#686902
#686642
#296453