Finite Group Databasebeta

← #10824470

Subgroups — #10824470

Order 768.

956 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5152C211yesmaximal, characteristic, complemented
2C2×Dic3×D8C211yesmaximal, characteristic, complemented
3#83172C211yesmaximal, characteristic, complemented
4#76295C211yesmaximal, characteristic, complemented
5C3×#2011C211yesmaximal, characteristic
6#72040C211yesmaximal, characteristic, complemented
7#72040C211yesmaximal, characteristic, complemented
8#71762C211yesmaximal, characteristic, complemented
9#70180C211yesmaximal, characteristic, complemented
10#70179C211yesmaximal, characteristic, complemented
11#70152C211yesmaximal, characteristic, complemented
12Dic3×#102C211yesmaximal, characteristic, complemented
13#69641C211yesmaximal, characteristic, complemented
14#68641C211yesmaximal, characteristic, complemented
15#68604C211yesmaximal, characteristic, complemented
16#23688—31nomaximal, Sylow
17C22×C3×D8C2211yescharacteristic
18C22×C3×D8C2211yescharacteristic
19C2×C3×#127C2211yescharacteristic
20C2×D4×Dic3C2211yescharacteristic
21C2×D4×Dic3C2211yescharacteristic
22C2×D4×Dic3C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C3×#390C2211yescharacteristic
26#5221C2211yescharacteristic
27#5221C2211yescharacteristic
28#5221C2211yescharacteristic
29#5220—21no—
30#5220—21no—
31#5220—21no—
32#5220—21no—
33C2×#832C2211yescharacteristic
34C2×#832C2211yescharacteristic
35C2×#832C2211yescharacteristic
36C2×#832C2211yescharacteristic
37C2×#832C2211yescharacteristic
38C2×#832C2211yescharacteristic
39#5152—24no—
40Dic3×D8—24no—
41C2×#818C2211yescharacteristic
42C2×#817C2211yescharacteristic
43C2×#815C2211yescharacteristic
44C2×Dic3⋊C8C2211yescharacteristic
45C2×Dic3⋊C8C2211yescharacteristic
46C2×C8×Dic3C2211yescharacteristic
47Dic3×C4⋊C4C2211yescharacteristic
48#4972C2211yescharacteristic
49#4945—21no—
50#4944—21no—
51#4943—21no—
52#4942—21no—
53#4936C2211yescharacteristic
54#4932C2211yescharacteristic
55#4928C2211yescharacteristic
56C3×#336C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#442—31no—
63C2×C4×D8—31no—
64#2328—31no—
65#2319—31no—
66#2085—31no—
67#2073—31no—
68#2067—31no—
69#2065—31no—
70#2054—31no—
71#2054—31no—
72#2041—31no—
73#2011D311yescharacteristic
74#1768—31no—
75#1762—31no—
76C4×#102—31no—
77C22×C3×D4—21no—
78C22×C3×D4—21no—
79C22×C3×D4C2311yescharacteristic
80C22×C3×D4C2311yescharacteristic
81C22×C3×D4C2311yescharacteristic
82C23×Dic3—21no—
83C23×Dic3—21no—
84C23×Dic3—21no—
85C2×C3×D8—22no—
86C2×C3×D8—22no—
87C2×C3×D8—22no—
88C2×C3×D8—22no—
89C2×C3×D8—22no—
90C2×C3×D8—22no—
91C22×C3×C8C2311yescharacteristic
92C22×C3×C8C2311yescharacteristic
93C3×#127—22no—
94C3×#127—22no—
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C42C2311yescharacteristic
99C2×#217—21no—
100C2×#217—21no—
101C2×#217—21no—
102C2×#217—21no—
103C2×#217—21no—
104C2×#217—21no—
105C2×#217—21no—
106C2×#217—21no—
107C2×#217—21no—
108D4×Dic3—22no—
109D4×Dic3—22no—
110D4×Dic3—22no—
111D4×Dic3—22no—
112D4×Dic3—22no—
113D4×Dic3D414yescomplemented
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×C4⋊Dic3C2311yescharacteristic
118C2×Dic3⋊C4—21no—
119C2×Dic3⋊C4—21no—
120C2×C4×Dic3—21no—
121C2×C4×Dic3C2311yescharacteristic
122C2×C4×Dic3C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125C3×#102—22no—
126C3×#102—22no—
127#832—22no—
128#832—22no—
129#832—22no—
130#832—22no—
131#832—22no—
132#832—22no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95D411yescharacteristic
139C3⋊#95D411yescharacteristic
140C3⋊#95D411yescharacteristic
141C3⋊#95D411yescharacteristic
142#818—22no—
143#817—22no—
144#815—22no—
145Dic3⋊C8—22no—
146Dic3⋊C8—22no—
147C8×Dic3—22no—
148C22×D8C2×D311yescharacteristic
149C22×D8C2×D311yescharacteristic
150C2×#127C2×D311yescharacteristic
151C2×C4×D4—31no—
152C2×C4×D4—31no—
153C2×C4×D4—31no—
154#442—64no—
155C4×D8—64no—
156C2×#107—31no—
157C2×#106—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8—31no—
160C2×#102—31no—
161C2×#102—31no—
162C2×#102—31no—
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102C2×D311yescharacteristic
167C2×#102C2×D311yescharacteristic
168C2×#97—31no—
169C2×C4×C8—31no—
170#390—31no—
171#390—31no—
172#390—31no—
173#390C2×D311yescharacteristic
174#389—31no—
175#389—31no—
176#388—61no—
177#388—61no—
178#386—61no—
179#386—61no—
180#386—61no—
181#386—61no—
182#382—31no—
183#381—61no—
184#380—61no—
185C4×C4⋊C4—31no—
186#376—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#336—31no—
192#336C2×D311yescharacteristic
193C24×C3—21no—
194C24×C3—21no—
195C24×C3—21no—
196C2×C3×D4—24no—
197C2×C3×D4—24no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C2×C3×D4—21no—
211C2×C3×D4—21no—
212C2×C3×D4—21no—
213C2×C3×D4—81no—
214C2×C3×D4—81no—
215C2×C3×D4—81no—
216C2×C3×D4—81no—
217C2×C3×D4C2×D412yes—
218C2×C3×D4#4112yes—
219C2×C3×D4#4112yes—
220C22×C3×C4—21no—
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2411yescharacteristic
224C22×Dic3—21no—
225C22×Dic3—24no—
226C22×Dic3—21no—
227C22×Dic3—21no—
228C22×Dic3—21no—
229C22×Dic3—42no—
230C22×Dic3—42no—
231C22×Dic3—81no—
232C22×Dic3—81no—
233C22×Dic3—81no—
234C22×Dic3—81no—
235C22×Dic3—81no—
236C22×Dic3—81no—
237C22×Dic3C2×D411yescharacteristic
238C22×Dic3C2×D411yescharacteristic
239C22×Dic3C2×D411yescharacteristic
240C22×Dic3C2×D411yescharacteristic
241C3×D8—28no—
242C3×D8—28no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8#4112yes—
248C2×C3×C8#4112yes—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C22⋊C4—81no—
252C3×C22⋊C4—81no—
253C3×C42—21no—
254C3×C42—21no—
255#217—24no—
256#217—42no—
257#217—42no—
258#217—81no—
259#217—81no—
260#217—81no—
261#217—81no—
262#217—81no—
263#217—81no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3C2×D412yes—
272Dic3⋊C4—81no—
273Dic3⋊C4—81no—
274C4×Dic3—21no—
275C4×Dic3—21no—
276C4×Dic3—81no—
277C4×Dic3C2×D412yes—
278C4×Dic3#4112yes—
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×D4—21no—
286C22×D4—21no—
287C22×D4C22×D311yescharacteristic
288C22×D4C22×D311yescharacteristic
289C22×D4C22×D311yescharacteristic
290C23×C4—61no—
291C23×C4—61no—
292C23×C4—61no—
293C2×D8—22no—
294C2×D8—22no—
295C2×D8—22no—
296C2×D8—22no—
297C2×D8—22no—
298C2×D8—22no—
299C22×C8—31no—
300C22×C8—31no—
301C22×C8C22×D311yescharacteristic
302C22×C8C22×D311yescharacteristic
303#127—22no—
304#127—22no—
305C4×D4—34no—
306C4×D4—62no—
307C4×D4—62no—
308C4×D4—62no—
309C4×D4—62no—
310C4×D4—62no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4C22×D311yescharacteristic
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C22⋊C4—61no—
328C2×C22⋊C4—61no—
329C2×C42—31no—
330C2×C42—31no—
331C2×C42—61no—
332C2×C42C22×D311yescharacteristic
333#107—62no—
334#106—62no—
335C4⋊C8—62no—
336C4⋊C8—62no—
337#102—22no—
338#102—22no—
339#102—62no—
340#102—62no—
341#102—62no—
342#102—62no—
343#102—62no—
344#102—62no—
345#97—62no—
346C4×C8—62no—
347#95—31no—
348#95—31no—
349#95—31no—
350#95—31no—
351#95—61no—
352#95—61no—
353#95—61no—
354#95—61no—
355#95—61no—
356C23×C3—22no—
357C23×C3—22no—
358C23×C3—22no—
359C23×C3—41no—
360C23×C3—41no—
361C23×C3—41no—
362C23×C3—41no—
363C23×C3—41no—
364C23×C3—41no—
365C23×C3—41no—
366C23×C3—41no—
367C23×C3—41no—
368C23×C3—41no—
369C23×C3—41no—
370C23×C3—41no—
371C23×C3—41no—
372C23×C3—41no—
373C23×C3—41no—
374C23×C3—41no—
375C23×C3—41no—
376C23×C3—41no—
377C23×C3C22×D411yescharacteristic
378C3×D4—22no—
379C3×D4—22no—
380C3×D4—22no—
381C3×D4—22no—
382C3×D4—22no—
383C3×D4—22no—
384C3×D4—41no—
385C3×D4—41no—
386C3×D4—41no—
387C3×D4—41no—
388C3×D4—41no—
389C3×D4—41no—
390C3×D4—84no—
391C3×D4—84no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4C22×D411yescharacteristic
402C2×C3×C4C2×#4111yescharacteristic
403C2×C3×C4C2×#4111yescharacteristic
404C2×C3×C4C2×#4111yescharacteristic
405C2×C3×C4C2×#4111yescharacteristic
406C2×C3×C4C2×#4111yescharacteristic
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—22no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3—41no—
429C2×Dic3—84no—
430C2×Dic3—84no—
431C2×Dic3—84no—
432C2×Dic3#12112yes—
433C2×Dic3#12112yes—
434C3×C8—22no—
435C3×C8—22no—
436C3⋊C8—41no—
437C3⋊C8—41no—
438C24—21no—
439C24—21no—
440C24—21no—
441C2×D4—24no—
442C2×D4—24no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—81no—
459C2×D4—81no—
460C2×D4—81no—
461C2×D4—81no—
462C2×D4D3×D412yes—
463C2×D4#23712yes—
464C2×D4#23712yes—
465C22×C4—21no—
466C22×C4—31no—
467C22×C4—31no—
468C22×C4—31no—
469C22×C4—31no—
470C22×C4—64no—
471C22×C4—61no—
472C22×C4—61no—
473C22×C4—61no—
474C22×C4—61no—
475C22×C4—122no—
476C22×C4—122no—
477C22×C4—241no—
478C22×C4—241no—
479C22×C4—241no—
480C22×C4—241no—
481C22×C4—241no—
482C22×C4—241no—
483C22×C4D3×D411yescharacteristic
484C22×C4D3×D411yescharacteristic
485C22×C4C23×D311yescharacteristic
486D8—28no—
487D8—28no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—61no—
493C2×C8—61no—
494C2×C8—61no—
495C2×C8—61no—
496C2×C8—61no—
497C2×C8—61no—
498C2×C8#23712yes—
499C2×C8#23712yes—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—32no—
503C4⋊C4—61no—
504C4⋊C4—61no—
505C4⋊C4—61no—
506C4⋊C4—61no—
507C4⋊C4—61no—
508C4⋊C4—61no—
509C4⋊C4—61no—
510C4⋊C4—241no—
511C4⋊C4—241no—
512C22⋊C4—64no—
513C22⋊C4—81no—
514C22⋊C4—81no—
515C22⋊C4—122no—
516C22⋊C4—122no—
517C22⋊C4—241no—
518C22⋊C4—241no—
519C22⋊C4—241no—
520C22⋊C4—241no—
521C22⋊C4—241no—
522C22⋊C4—241no—
523C42—21no—
524C42—21no—
525C42—32no—
526C42—32no—
527C42—61no—
528C42—61no—
529C42—241no—
530C22×C3—42no—
531C22×C3—42no—
532C22×C3—42no—
533C22×C3—42no—
534C22×C3—42no—
535C22×C3—42no—
536C22×C3—42no—
537C22×C3—42no—
538C22×C3—42no—
539C22×C3—81no—
540C22×C3—81no—
541C22×C3—81no—
542C22×C3—81no—
543C22×C3—81no—
544C22×C3—81no—
545C22×C3—81no—
546C22×C3—81no—
547C22×C3—81no—
548C22×C3—81no—
549C22×C3—81no—
550C22×C3—81no—
551C22×C3C2×#12111yescharacteristic
552C22×C3#52511yescharacteristic
553C22×C3#53211yescharacteristic
554C22×C3#57611yescharacteristic
555C22×C3#57611yescharacteristic
556C22×C3#57611yescharacteristic
557C22×C3#57711yescharacteristic
558C3×C4—41no—
559C3×C4—41no—
560C3×C4—81no—
561C3×C4#52511yescharacteristic
562C3×C4#52511yescharacteristic
563C3×C4#52911yescharacteristic
564C3×C4#52911yescharacteristic
565Dic3—22no—
566Dic3—22no—
567Dic3—41no—
568Dic3—41no—
569Dic3—42no—
570Dic3—81no—
571Dic3—81no—
572Dic3—81no—
573C23—22no—
574C23—22no—
575C23—22no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23—41no—
591C23—41no—
592C23—41no—
593C23—41no—
594C23C2×D3×D411yescharacteristic
595D4—22no—
596D4—22no—
597D4—22no—
598D4—22no—
599D4—22no—
600D4—22no—
601D4—41no—
602D4—41no—
603D4—41no—
604D4—41no—
605D4—41no—
606D4—41no—
607D4—84no—
608D4—84no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—32no—
616C2×C4—32no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—61no—
621C2×C4—61no—
622C2×C4—61no—
623C2×C4—61no—
624C2×C4—61no—
625C2×C4—61no—
626C2×C4—61no—
627C2×C4—61no—
628C2×C4—61no—
629C2×C4—61no—
630C2×C4—62no—
631C2×C4—121no—
632C2×C4—121no—
633C2×C4—121no—
634C2×C4—121no—
635C2×C4—121no—
636C2×C4—121no—
637C2×C4—121no—
638C2×C4—121no—
639C2×C4—121no—
640C2×C4—121no—
641C2×C4—121no—
642C2×C4—244no—
643C2×C4—244no—
644C2×C4—244no—
645C2×C4C2×D3×D411yescharacteristic
646C2×C4C2×#23711yescharacteristic
647C2×C4C2×#23711yescharacteristic
648C2×C4C2×#23711yescharacteristic
649C2×C4C2×#23711yescharacteristic
650C2×C4D3×#4111yescharacteristic
651C8—22no—
652C8—22no—
653C8—121no—
654C8—121no—
655C2×C3—82no—
656C2×C3—82no—
657C2×C3—82no—
658C2×C3#239211yescharacteristic
659C2×C3#304111yescharacteristic
660C2×C3#304511yescharacteristic
661C2×C3#311811yescharacteristic
662C2×C3#312211yescharacteristic
663C2×C3#317611yescharacteristic
664C2×C3#317911yescharacteristic
665C22—42no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—81no—
675C22—81no—
676C22—81no—
677C22—81no—
678C22—81no—
679C22—81no—
680C22—81no—
681C22—81no—
682C22—81no—
683C22—81no—
684C22—81no—
685C22—81no—
686C22#558611yescharacteristic
687C22D3×#12111yescharacteristic
688C22#571611yescharacteristic
689C22#575711yescharacteristic
690C22#575711yescharacteristic
691C22#577411yescharacteristic
692C22#577511yescharacteristic
693C4—41no—
694C4—41no—
695C4—62no—
696C4—62no—
697C4—81no—
698C4—121no—
699C4—121no—
700C4—122no—
701C4—241no—
702C4—241no—
703C4—241no—
704C4#558611yescharacteristic
705C4#558611yescharacteristic
706C4#560011yescharacteristic
707C4#560011yescharacteristic
708C3#2368811yescharacteristic, complemented, Sylow
709C2—82no—
710C2—82no—
711C2—82no—
712C2#7666611yescharacteristic
713C2#7999211yescharacteristic
714C2#7999311yescharacteristic
715C2#8033411yescharacteristic
716C2#8033511yescharacteristic
717C2#8177611yescharacteristic
718C2#8183711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#2011C2
socleC23×C3C22×D4
hypercenter#2011D3
hypocenterC3#23688
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2011

Complements

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

NH ≅ G/Ntype
1. C2×#5152C2semidirect
2. C2×Dic3×D8C2semidirect
3. #83172C2semidirect
4. #76295C2semidirect
6. #72040C2semidirect
7. #72040C2semidirect
8. #71762C2semidirect
9. #70180C2semidirect
10. #70179C2semidirect
11. #70152C2semidirect
12. Dic3×#102C2semidirect
13. #69641C2semidirect
14. #68641C2semidirect
15. #68604C2semidirect
113. D4×Dic3D4semidirect
708. C3#23688semidirect

Maximal subgroups

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

subgroupindex
C2×#51522
C2×Dic3×D82
#831722
#762952
C3×#20112
#720402
#717622
#701802
#701792
#701522
Dic3×#1022
#696412
#686412
#686042
#236883