Finite Group Databasebeta

← #10801693

Subgroups — #10801693

Order 768.

1054 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4813C211yesmaximal, characteristic
2C2×#4790C211yesmaximal, characteristic, complemented
3C2×C4×D3×C8C211yesmaximal, characteristic
4#76479C211yesmaximal, characteristic
5#75965C211yesmaximal, characteristic
6C3×#1933C211yesmaximal, characteristic, complemented
7#70270C211yesmaximal, characteristic
8Dic3×#103C211yesmaximal, characteristic, complemented
9#69767C211yesmaximal, characteristic, complemented
10#69677C211yesmaximal, characteristic, complemented
11#69619C211yesmaximal, characteristic, complemented
12#68882C211yesmaximal, characteristic
13#68869C211yesmaximal, characteristic
14#68785C211yesmaximal, characteristic, complemented
15#68781C211yesmaximal, characteristic, complemented
16C4×#467—31nomaximal, Sylow
17C2×#946C2211yescharacteristic
18C2×Q8×Dic3C2211yescharacteristic
19C22×#216C2211yescharacteristic
20C22×D3×C8C2211yescharacteristic
21C22×D3×C8C2211yescharacteristic
22C2×#897C2211yescharacteristic
23C2×#887C2211yescharacteristic
24C2×C42×D3C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×C4×C8C2211yescharacteristic
28C3×#391C2211yescharacteristic
29C3×#389C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×#816C2211yescharacteristic
33C2×C8×Dic3C2211yescharacteristic
34C2×C8×Dic3C2211yescharacteristic
35#4989—21no—
36#4987C2211yescharacteristic
37#4987C2211yescharacteristic
38Dic3×C4⋊C4—21no—
39Dic3×C4⋊C4C2211yescharacteristic
40Dic3×C4⋊C4C2211yescharacteristic
41C2×#810C2211yescharacteristic
42C2×#807C2211yescharacteristic
43C2×#807C2211yescharacteristic
44C2×#807C2211yescharacteristic
45#4929C2211yescharacteristic
46#4928C2211yescharacteristic
47C2×C4×C3⋊C8C2211yescharacteristic
48#4813C418yescomplemented
49#4790—24no—
50C4×D3×C8—24no—
51#4668—21no—
52#4666—21no—
53#4666—21no—
54#4655—21no—
55#4654—21no—
56#4649—21no—
57C3×#340C2211yescharacteristic
58C3×#340C2211yescharacteristic
59#4551C2211yescharacteristic
60#4551C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63C2×#467—31no—
64C2×C4×Q16—31no—
65C22×C4×C8—31no—
66C4×C22⋊Q8—31no—
67C4×C22⋊Q8—31no—
68#1969—31no—
69#1942—31no—
70#1933—31no—
71#1933D311yescharacteristic, complemented
72#1892—31no—
73#1892—31no—
74#1808—31no—
75C4×#107—31no—
76C4×#103—31no—
77C4×#103—31no—
78C22×C3×Q8C2311yescharacteristic
79C22×C4×D3D411yescharacteristic
80C22×C4×D3D411yescharacteristic
81C22×C4×D3C2311yescharacteristic
82C22×Dic6C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C42C2311yescharacteristic
90#946—22no—
91#946C2×C414yes—
92Q8×Dic3—22no—
93Q8×Dic3—22no—
94C2×#216—22no—
95C2×#216—22no—
96C2×#216C2×C414yescomplemented
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⋊Dic3—21no—
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3D411yescharacteristic
114C2×C4×Dic3D411yescharacteristic
115C2×C4×Dic3C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C2×D3×C8—22no—
119C2×D3×C8—22no—
120C2×D3×C8—22no—
121C2×D3×C8—22no—
122C2×D3×C8—22no—
123C2×D3×C8C2×C414yes—
124#897—22no—
125#897C2×C414yes—
126#887—22no—
127#887—22no—
128C42×D3—22no—
129C42×D3—22no—
130C3×#103—22no—
131C3×#103C2×C414yescomplemented
132C3×C4×C8—22no—
133C3×#95—21no—
134C3×#95—21no—
135C3×#95—21no—
136#835—22no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144#816—22no—
145#816C2×C414yescomplemented
146C8×Dic3—22no—
147C8×Dic3D414yes—
148#810—22no—
149#807—22no—
150#807—22no—
151#807C2×C414yescomplemented
152C4×C3⋊C8—22no—
153C22×Q16—31no—
154C23×C8—31no—
155C23×C8—31no—
156C2×C22⋊Q8—31no—
157C2×C22⋊Q8—31no—
158C2×C4×Q8—31no—
159C2×C4×Q8—31no—
160C22×C42—31no—
161#467—38no—
162C4×Q16—64no—
163C2×#107—31no—
164C2×#107—31no—
165C2×#107—31no—
166C2×#103—31no—
167C2×#103—31no—
168C2×#103—31no—
169C2×#103—31no—
170C2×#103C2×D311yescharacteristic
171C2×#103C2×D311yescharacteristic
172C2×C4×C8—31no—
173C2×C4×C8—31no—
174C2×C4×C8—31no—
175C2×C4×C8—64no—
176C2×C4×C8C2×D311yescharacteristic
177#391—31no—
178#391C2×D311yescharacteristic
179#389—31no—
180#389C2×D311yescharacteristic
181#387—61no—
182#387—61no—
183#385—61no—
184#385—61no—
185#380—31no—
186#380—31no—
187C4×C4⋊C4—31no—
188C4×C4⋊C4—31no—
189C4×C4⋊C4—61no—
190C4×C4⋊C4—61no—
191C4×C22⋊C4—61no—
192C4×C22⋊C4—61no—
193#340—31no—
194#340—31no—
195#340—31no—
196#340—31no—
197#340C2×D311yescharacteristic
198#340C2×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×Q8C22×C412yes—
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—22no—
221C2×C4×D3—22no—
222C2×C4×D3—21no—
223C2×C4×D3—21no—
224C2×C4×D3—21no—
225C2×C4×D3—21no—
226C2×C4×D3—22no—
227C2×C4×D3—22no—
228C2×C4×D3—22no—
229C2×C4×D3Q1614yescomplemented
230C2×C4×D3C22×C412yes—
231C2×C4×D3#4112yes—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6C22×C412yes—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8C22×C412yes—
243C2×C3×C8C2×D412yes—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—42no—
250C3×C4⋊C4C22×C412yes—
251C3×C42—21no—
252C3×C42—21no—
253#216—28no—
254(C2×D3)⋊C4—24no—
255(C2×D3)⋊C4—24no—
256(C2×D3)⋊C4—42no—
257(C2×D3)⋊C4—42no—
258C4⋊Dic3—21no—
259C4⋊Dic3—24no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—42no—
265C4⋊Dic3C22×C412yes—
266Dic3⋊C4—24no—
267Dic3⋊C4—42no—
268Dic3⋊C4—42no—
269C4×Dic3—22no—
270C4×Dic3—22no—
271C4×Dic3—42no—
272C4×Dic3—42no—
273C4×Dic3—42no—
274C4×Dic3—42no—
275C4×Dic3C2×D412yes—
276C4×Dic3#4112yes—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8C22×C412yes—
280C2×C3⋊C8C2×D412yes—
281C2×C3⋊C8#4112yes—
282C2×C3⋊C8#4112yes—
283D3×C8—28no—
284D3×C8—28no—
285C22×Q8—31no—
286C22×Q8C22×D311yescharacteristic
287C23×C4—31no—
288C23×C4—31no—
289C23×C4—31no—
290C2×Q16—34no—
291C2×Q16—62no—
292C2×Q16—62no—
293C22×C8—31no—
294C22×C8—31no—
295C22×C8—34no—
296C22×C8—62no—
297C22×C8—62no—
298C22×C8—62no—
299C22×C8—62no—
300C22×C8—62no—
301C22×C8C22×D311yescharacteristic
302C22×C8C22×D311yescharacteristic
303C22⋊Q8—34no—
304C22⋊Q8—34no—
305C22⋊Q8—62no—
306C22⋊Q8—62no—
307C4×Q8—62no—
308C4×Q8—62no—
309C4×Q8—62no—
310C4×Q8—62no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4—61no—
318C2×C4⋊C4—61no—
319C2×C4⋊C4—61no—
320C2×C4⋊C4C22×D311yescharacteristic
321C2×C4⋊C4C22×D311yescharacteristic
322C2×C4⋊C4C22×D311yescharacteristic
323C2×C22⋊C4—61no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C42—31no—
328C2×C42—31no—
329C2×C42—31no—
330C2×C42—62no—
331C2×C42—62no—
332C2×C42—61no—
333C2×C42—61no—
334C2×C42—61no—
335C2×C42—61no—
336C2×C42C22×D311yescharacteristic
337#107—34no—
338#107—62no—
339#107—62no—
340#103—22no—
341#103—34no—
342#103—62no—
343#103—62no—
344#103—62no—
345#103C4×D314yescomplemented
346C4×C8—22no—
347C4×C8—34no—
348C4×C8—62no—
349C4×C8—62no—
350#95—21no—
351#95—21no—
352#95—21no—
353#95—61no—
354#95—61no—
355#95—61no—
356#95—61no—
357#95—61no—
358#95—61no—
359#95—61no—
360C23×C3C22×D411yescharacteristic
361C22×D3—21no—
362C22×D3—21no—
363C22×D3—21no—
364C22×D3—21no—
365C22×D3C4×D412yes—
366C22×D3C2×Q1612yes—
367C22×D3#13512yes—
368C3×Q8—22no—
369C3×Q8—22no—
370C3×Q8—41no—
371C3×Q8—41no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
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×C4C2×Q1612yes—
389C2×C3×C4#13512yes—
390C2×C3×C4C23×C411yescharacteristic
391C2×C3×C4C22×D411yescharacteristic
392C2×C3×C4C2×#4111yescharacteristic
393C2×C3×C4C2×#4111yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—22no—
401C2×Dic3—22no—
402C2×Dic3—41no—
403C2×Dic3—41no—
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×Dic3C4×D412yes—
413C2×Dic3#12112yes—
414C2×Dic3#12112yes—
415C2×Dic3C2×Q1612yes—
416C2×Dic3#13512yes—
417C4×D3—28no—
418C4×D3—22no—
419C4×D3—22no—
420C4×D3—22no—
421C4×D3—22no—
422C4×D3—28no—
423Dic6—22no—
424Dic6—22no—
425Dic6—41no—
426Dic6—41no—
427C3×C8—22no—
428C3×C8—22no—
429C3⋊C8—22no—
430C3⋊C8C4×D414yes—
431C24—31no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—32no—
438C2×Q8—61no—
439C2×Q8—61no—
440C2×Q8—61no—
441C2×Q8—61no—
442C2×Q8—61no—
443C2×Q8C2×C4×D312yes—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—31no—
449C22×C4—31no—
450C22×C4—31no—
451C22×C4—34no—
452C22×C4—32no—
453C22×C4—32no—
454C22×C4—62no—
455C22×C4—62no—
456C22×C4—62no—
457C22×C4—62no—
458C22×C4—62no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4—61no—
462C22×C4—61no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4D3×D411yescharacteristic
468C22×C4D3×D411yescharacteristic
469C22×C4C23×D311yescharacteristic
470Q16—68no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—32no—
476C2×C8—32no—
477C2×C8—32no—
478C2×C8—32no—
479C2×C8—68no—
480C2×C8—68no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8C2×C4×D312yes—
484C2×C8D3×D412yes—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—32no—
491C4⋊C4—42no—
492C4⋊C4—64no—
493C4⋊C4—64no—
494C4⋊C4—61no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—61no—
498C4⋊C4—61no—
499C4⋊C4—122no—
500C4⋊C4—122no—
501C4⋊C4—122no—
502C4⋊C4C2×C4×D312yes—
503C22⋊C4—64no—
504C22⋊C4—64no—
505C22⋊C4—122no—
506C22⋊C4—122no—
507C42—21no—
508C42—21no—
509C42—32no—
510C42—32no—
511C42—62no—
512C42—62no—
513C42—122no—
514C42—122no—
515C42—122no—
516C42—122no—
517C22×C3C2×C4×D411yescharacteristic
518C22×C3C2×#12111yescharacteristic
519C22×C3#53211yescharacteristic
520C22×C3C22×Q1611yescharacteristic
521C22×C3C2×#13511yescharacteristic
522C22×C3C2×#13511yescharacteristic
523C22×C3C2×#13511yescharacteristic
524C2×D3—22no—
525C2×D3—22no—
526C2×D3—22no—
527C2×D3—22no—
528C2×D3—22no—
529C2×D3—22no—
530C2×D3C4×Q1614yescomplemented
531C3×C4—22no—
532C3×C4—22no—
533C3×C4—42no—
534C3×C4—42no—
535C3×C4—81no—
536C3×C4—81no—
537C3×C4C2×C4×D411yescharacteristic
538C3×C4C4×#4111yescharacteristic
539C3×C4#52511yescharacteristic
540C3×C4#52911yescharacteristic
541Dic3—22no—
542Dic3—22no—
543Dic3—42no—
544Dic3—42no—
545Dic3—81no—
546Dic3—81no—
547Dic3#46714yescomplemented
548C23—32no—
549C23—32no—
550C23—32no—
551C23—61no—
552C23—61no—
553C23—61no—
554C23—61no—
555C23C2×D3×D411yescharacteristic
556Q8—22no—
557Q8—22no—
558Q8—41no—
559Q8—41no—
560Q8—62no—
561Q8—62no—
562Q8—121no—
563Q8—121no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—32no—
571C2×C4—32no—
572C2×C4—32no—
573C2×C4—32no—
574C2×C4—32no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—68no—
586C2×C4—68no—
587C2×C4—61no—
588C2×C4—61no—
589C2×C4—61no—
590C2×C4—61no—
591C2×C4—62no—
592C2×C4—62no—
593C2×C4—62no—
594C2×C4—62no—
595C2×C4—62no—
596C2×C4—62no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4#91612yes—
608C2×C4D3×Q1612yes—
609C2×C4C22×C4×D311yescharacteristic
610C2×C4C2×D3×D411yescharacteristic
611C2×C4D3×#4111yescharacteristic
612C2×C4D3×#4111yescharacteristic
613C2×C4D3×#4111yescharacteristic
614C2×C4D3×#4111yescharacteristic
615C8—22no—
616C8—22no—
617C8—34no—
618C8—62no—
619C2×C3C4×#12111yescharacteristic
620C2×C3C2×C4×Q1611yescharacteristic
621C2×C3C4×#13511yescharacteristic
622C2×C3C2×#46711yescharacteristic
623C2×C3#305011yescharacteristic
624C2×C3#316811yescharacteristic
625C2×C3#317011yescharacteristic
626D3—24no—
627C22—34no—
628C22—62no—
629C22—62no—
630C22—62no—
631C22—62no—
632C22—62no—
633C22—62no—
634C22C4×D3×D411yescharacteristic
635C22D3×#12111yescharacteristic
636C22#566811yescharacteristic
637C22C2×#91611yescharacteristic
638C22C2×D3×Q1611yescharacteristic
639C22D3×#13511yescharacteristic
640C22D3×#13511yescharacteristic
641C4—22no—
642C4—22no—
643C4—34no—
644C4—42no—
645C4—42no—
646C4—62no—
647C4—62no—
648C4—81no—
649C4—81no—
650C4—122no—
651C4—122no—
652C4—241no—
653C4—241no—
654C4#552211yescharacteristic
655C4C4×D3×D411yescharacteristic
656C4#565611yescharacteristic
657C4#570611yescharacteristic
658C3C4×#46711yescharacteristic, complemented, Sylow
659C2—64no—
660C2#7545211yescharacteristic
661C2C4×#91611yescharacteristic
662C2C4×D3×Q1611yescharacteristic
663C2#8065611yescharacteristic
664C2D3×#46711yescharacteristic
665C2#8159311yescharacteristic
666C2#8162811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1933C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
cosocleC22×C3×C4C24
hypercenter#1933D3
hypocenterC3C4×#467
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1933

Complements

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

NH ≅ G/Ntype
2. C2×#4790C2semidirect
6. C3×#1933C2semidirect
8. Dic3×#103C2semidirect
9. #69767C2semidirect
10. #69677C2semidirect
11. #69619C2semidirect
14. #68785C2semidirect
15. #68781C2semidirect
48. #4813C4semidirect
71. #1933D3semidirect
96. C2×#216C2×C4semidirect
131. C3×#103C2×C4semidirect
145. #816C2×C4semidirect
151. #807C2×C4semidirect
229. C2×C4×D3Q16semidirect
345. #103C4×D3semidirect
530. C2×D3C4×Q16semidirect
547. Dic3#467semidirect
658. C3C4×#467semidirect

Maximal subgroups

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

subgroupindex
C2×#48132
C2×#47902
C2×C4×D3×C82
#764792
#759652
C3×#19332
#702702
Dic3×#1032
#697672
#696772
#696192
#688822
#688692
#687852
#687812
C4×#4673