Finite Group Databasebeta

← #10900954

Subgroups — #10900954

Order 768.

1014 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×D3×#125C211yesmaximal, characteristic
2C2×#4811C211yesmaximal, characteristic
3C2×#4805C211yesmaximal, characteristic
4#76663C211yesmaximal, characteristic
5C3×#2026C211yesmaximal, characteristic, complemented
6#71739C211yesmaximal, characteristic, complemented
7#70232C211yesmaximal, characteristic, complemented
8#70232C211yesmaximal, characteristic, complemented
9#69669C211yesmaximal, characteristic, complemented
10#68872C211yesmaximal, characteristic
11#68871C211yesmaximal, characteristic
12#68829C211yesmaximal, characteristic, complemented
13#68828C211yesmaximal, characteristic, complemented
14#68664C211yesmaximal, characteristic
15#68634C211yesmaximal, characteristic, complemented
16#29643—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C2×#946C2211yescharacteristic
19D3×#125—24no—
20D3×#125—24no—
21C2×D3×C4⋊C4—21no—
22C2×D3×C4⋊C4—21no—
23C2×D3×C4⋊C4C2211yescharacteristic
24C2×D3×C4⋊C4C2211yescharacteristic
25C2×#890C2211yescharacteristic
26C2×#890C2211yescharacteristic
27C2×#889—21no—
28C2×#889—21no—
29C2×C42×D3C2211yescharacteristic
30C2×#870C2211yescharacteristic
31C2×C3×#103C2211yescharacteristic
32C2×C3×#103C2211yescharacteristic
33C3×#391C2211yescharacteristic
34#5229C2211yescharacteristic
35C2×#835C2211yescharacteristic
36C2×#835C2211yescharacteristic
37C2×(C2×D3)⋊C8C2211yescharacteristic
38C2×(C2×D3)⋊C8C2211yescharacteristic
39C2×#818C2211yescharacteristic
40C2×#817C2211yescharacteristic
41#4987C2211yescharacteristic
42#4984C2211yescharacteristic
43C2×#808C2211yescharacteristic
44C2×#807C2211yescharacteristic
45#4928C2211yescharacteristic
46#4811—24no—
47#4805—24no—
48#4668—21no—
49#4664—21no—
50#4654—21no—
51#4652—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
64C22×#125—31no—
65C2×#484—31no—
66C2×#483—31no—
67#2855—31no—
68#2059—31no—
69#2059—31no—
70#2059—31no—
71#2059—31no—
72#2026—31no—
73#2026—31no—
74#2026—31no—
75#2026D311yescharacteristic, complemented
76#1824—31no—
77#1823—31no—
78#1817—31no—
79C22×C3×Q8C2311yescharacteristic
80C22×C4×D3—21no—
81C22×C4×D3—21no—
82C22×C4×D3D411yescharacteristic
83C22×C4×D3D411yescharacteristic
84C22×C4×D3C2311yescharacteristic
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⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C42C2311yescharacteristic
95#946—22no—
96#946—22no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×C4×Dic3D411yescharacteristic
115C2×C4×Dic3D411yescharacteristic
116C2×C4×Dic3C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119D3×C4⋊C4—28no—
120D3×C4⋊C4—28no—
121D3×C4⋊C4—22no—
122D3×C4⋊C4—22no—
123D3×C4⋊C4D414yes—
124D3×C4⋊C4D414yes—
125#890—22no—
126#890—22no—
127#890—22no—
128#890—22no—
129#889—28no—
130#889—28no—
131C42×D3—22no—
132C42×D3—22no—
133#870—22no—
134#870—22no—
135C3×#103—22no—
136C3×#103—22no—
137C3×#95—21no—
138C3×#95—21no—
139#835—22no—
140#835—22no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144C3⋊#95—21no—
145C3⋊#95—21no—
146(C2×D3)⋊C8—22no—
147(C2×D3)⋊C8—22no—
148#818—22no—
149#817—22no—
150#808—22no—
151#807—22no—
152C2×#125—31no—
153C2×#125—31no—
154C2×#125—31no—
155C2×#125—61no—
156C2×#125—61no—
157C2×#125—64no—
158C2×#125—64no—
159C2×#125C2×D311yescharacteristic
160C2×C22⋊Q8—31no—
161C22×C4⋊C4—31no—
162C22×C4⋊C4—31no—
163C22×C4⋊C4—61no—
164C22×C4⋊C4—61no—
165C22×C42—31no—
166#484—64no—
167#483—64no—
168C2×#107—31no—
169C2×#107—31no—
170C2×#106—31no—
171C2×#106—31no—
172C2×#103—31no—
173C2×#103—31no—
174C2×#103C2×D311yescharacteristic
175C2×#103C2×D311yescharacteristic
176C2×C22⋊C8—31no—
177C2×C22⋊C8—31no—
178#391—31no—
179#391C2×D311yescharacteristic
180#389—31no—
181#389—31no—
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
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2411yescharacteristic
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220C2×C4×D3—21no—
221C2×C4×D3—21no—
222C2×C4×D3—21no—
223C2×C4×D3—21no—
224C2×C4×D3—22no—
225C2×C4×D3—22no—
226C2×C4×D3—24no—
227C2×C4×D3—24no—
228C2×C4×D3—22no—
229C2×C4×D3—22no—
230C2×C4×D3—41no—
231C2×C4×D3—41no—
232C2×C4×D3—42no—
233C2×C4×D3—42no—
234C2×C4×D3—42no—
235C2×C4×D3—42no—
236C2×C4×D3C2×D412yes—
237C2×C4×D3C2×D412yes—
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⋊C4C2×D412yes—
251C3×C4⋊C4C2×D412yes—
252C3×C42—21no—
253C3×C42—21no—
254(C2×D3)⋊C4—42no—
255(C2×D3)⋊C4—42no—
256C4⋊Dic3—24no—
257C4⋊Dic3—24no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—42no—
265C4⋊Dic3C2×D412yes—
266C4⋊Dic3C2×D412yes—
267Dic3⋊C4—24no—
268Dic3⋊C4—24no—
269Dic3⋊C4—24no—
270Dic3⋊C4—24no—
271Dic3⋊C4—24no—
272Dic3⋊C4—24no—
273Dic3⋊C4—42no—
274Dic3⋊C4—42no—
275C4×Dic3—21no—
276C4×Dic3—21no—
277C4×Dic3—22no—
278C4×Dic3—22no—
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×Q8C22×D311yescharacteristic
286C23×C4—31no—
287C23×C4—31no—
288C23×C4—31no—
289C23×C4—61no—
290C23×C4—61no—
291C22×C8—31no—
292C22×C8—31no—
293C22×C8C22×D311yescharacteristic
294C22×C8C22×D311yescharacteristic
295#125—22no—
296#125—22no—
297#125—62no—
298#125—62no—
299#125—62no—
300#125—62no—
301#125—68no—
302#125—68no—
303#125—62no—
304#125—62no—
305C22⋊Q8—62no—
306C22⋊Q8—62no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—34no—
312C2×C4⋊C4—34no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—62no—
316C2×C4⋊C4—62no—
317C2×C4⋊C4—68no—
318C2×C4⋊C4—68no—
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⋊C4—61no—
326C2×C4⋊C4—61no—
327C2×C4⋊C4—61no—
328C2×C4⋊C4—61no—
329C2×C4⋊C4—61no—
330C2×C4⋊C4C22×D311yescharacteristic
331C2×C4⋊C4C22×D311yescharacteristic
332C2×C4⋊C4C22×D311yescharacteristic
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C42—31no—
336C2×C42—31no—
337C2×C42—31no—
338C2×C42—62no—
339C2×C42—62no—
340C2×C42C22×D311yescharacteristic
341#107—62no—
342#107—62no—
343#106—62no—
344#106—62no—
345#103—22no—
346#103—22no—
347#103—62no—
348#103—62no—
349C22⋊C8—62no—
350C22⋊C8—62no—
351#95—21no—
352#95—21no—
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#12012yes—
364C22×D3#13512yes—
365C22×D3#13712yes—
366C3×Q8—41no—
367C3×Q8—41no—
368C3×Q8—41no—
369C3×Q8—41no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
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×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4C22×D411yescharacteristic
389C2×C3×C4C22×D411yescharacteristic
390C2×C3×C42−1+411yescharacteristic
391C2×C3×C42−1+411yescharacteristic
392C2×C3×C42−1+411yescharacteristic
393C2×C3×C42−1+411yescharacteristic
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—22no—
399C2×Dic3—22no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3#12012yes—
415C4×D3—22no—
416C4×D3—22no—
417C4×D3—22no—
418C4×D3—22no—
419C4×D3—44no—
420C4×D3—48no—
421C4×D3—44no—
422C4×D3—48no—
423C3×C8—41no—
424C3×C8—41no—
425C3⋊C8—41no—
426C3⋊C8—41no—
427C24—31no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—31no—
439C22×C4—32no—
440C22×C4—32no—
441C22×C4—31no—
442C22×C4—31no—
443C22×C4—62no—
444C22×C4—62no—
445C22×C4—64no—
446C22×C4—64no—
447C22×C4—62no—
448C22×C4—62no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—61no—
457C22×C4—121no—
458C22×C4—121no—
459C22×C4—122no—
460C22×C4—122no—
461C22×C4—122no—
462C22×C4—122no—
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—32no—
485C4⋊C4—32no—
486C4⋊C4—64no—
487C4⋊C4—64no—
488C4⋊C4—64no—
489C4⋊C4—64no—
490C4⋊C4—64no—
491C4⋊C4—64no—
492C4⋊C4—64no—
493C4⋊C4—64no—
494C4⋊C4—61no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—61no—
498C4⋊C4—61no—
499C4⋊C4—61no—
500C4⋊C4—122no—
501C4⋊C4—122no—
502C4⋊C4—122no—
503C4⋊C4D3×D412yes—
504C4⋊C4D3×D412yes—
505C22⋊C4—122no—
506C22⋊C4—122no—
507C42—21no—
508C42—21no—
509C42—62no—
510C42—62no—
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—22no—
524C2×D3—22no—
525C2×D3—22no—
526C2×D3#45214yes—
527C3×C4—41no—
528C3×C4—42no—
529C3×C4—41no—
530C3×C4—42no—
531C3×C4—81no—
532C3×C4—81no—
533C3×C4#53611yescharacteristic
534C3×C4#53611yescharacteristic
535C3×C4#53611yescharacteristic
536C3×C4#53611yescharacteristic
537Dic3—22no—
538Dic3—41no—
539Dic3—42no—
540Dic3—41no—
541Dic3—42no—
542Dic3—81no—
543Dic3—81no—
544C23—32no—
545C23—32no—
546C23—32no—
547C23—61no—
548C23—61no—
549C23—61no—
550C23—61no—
551C23C2×D3×D411yescharacteristic
552Q8—41no—
553Q8—41no—
554Q8—41no—
555Q8—41no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—32no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—61no—
576C2×C4—61no—
577C2×C4—62no—
578C2×C4—62no—
579C2×C4—62no—
580C2×C4—62no—
581C2×C4—61no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—61no—
585C2×C4—61no—
586C2×C4—61no—
587C2×C4—62no—
588C2×C4—62no—
589C2×C4—124no—
590C2×C4—128no—
591C2×C4—124no—
592C2×C4—128no—
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#101011yescharacteristic
607C2×C4#101011yescharacteristic
608C2×C4#101011yescharacteristic
609C8—41no—
610C8—41no—
611C8—121no—
612C8—121no—
613C2×C3#261411yescharacteristic
614C2×C3C2×#45211yescharacteristic
615C2×C3#300511yescharacteristic
616C2×C3#323211yescharacteristic
617C2×C3#323211yescharacteristic
618C2×C3#323611yescharacteristic
619C2×C3#323611yescharacteristic
620D3—24no—
621C22—34no—
622C22—62no—
623C22—62no—
624C22—62no—
625C22—62no—
626C22—62no—
627C22—62no—
628C22D3×#12011yescharacteristic
629C22#562811yescharacteristic
630C22#567511yescharacteristic
631C22D3×#13511yescharacteristic
632C22#576811yescharacteristic
633C22D3×#13711yescharacteristic
634C22#577711yescharacteristic
635C4—41no—
636C4—42no—
637C4—41no—
638C4—42no—
639C4—62no—
640C4—81no—
641C4—81no—
642C4—121no—
643C4—122no—
644C4—121no—
645C4—122no—
646C4—241no—
647C4—241no—
648C4#562811yescharacteristic
649C4#562811yescharacteristic
650C4#567511yescharacteristic
651C4#567511yescharacteristic
652C3#2964311yescharacteristic, complemented, Sylow
653C2—64no—
654C2#7687011yescharacteristic
655C2D3×#45211yescharacteristic
656C2#8022711yescharacteristic
657C2#8123711yescharacteristic
658C2#8123811yescharacteristic
659C2#8164811yescharacteristic
660C2#8174511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2026C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#2026D3
hypocenterC3#29643
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2026

Complements

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

NH ≅ G/Ntype
5. C3×#2026C2semidirect
6. #71739C2semidirect
7. #70232C2semidirect
8. #70232C2semidirect
9. #69669C2semidirect
12. #68829C2semidirect
13. #68828C2semidirect
15. #68634C2semidirect
75. #2026D3semidirect
652. C3#29643semidirect

Maximal subgroups

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

subgroupindex
C2×D3×#1252
C2×#48112
C2×#48052
#766632
C3×#20262
#717392
#702322
#696692
#688722
#688712
#688292
#688282
#686642
#686342
#296433