Finite Group Databasebeta

← #10844892

Subgroups — #10844892

Order 768.

802 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#461C211yesmaximal, characteristic
2#83244C211yesmaximal, characteristic, complemented
3#82388C211yesmaximal, characteristic
4C2×#4828C211yesmaximal, characteristic, complemented
5#72051C211yesmaximal, characteristic
6#72051C211yesmaximal, characteristic
7#72050C211yesmaximal, characteristic
8#72049C211yesmaximal, characteristic
9#71786C211yesmaximal, characteristic, complemented
10#71609C211yesmaximal, characteristic
11#70455C211yesmaximal, characteristic, complemented
12#70256C211yesmaximal, characteristic, complemented
13#70254C211yesmaximal, characteristic, complemented
14#70248C211yesmaximal, characteristic, complemented
15#70247C211yesmaximal, characteristic, complemented
16#25308—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C22×C3×SD16C2211yescharacteristic
19C2×C3×C4⋊Q8C2211yescharacteristic
20C2×C3×C4×D4C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C3×#461—24no—
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26#5229C2211yescharacteristic
27#5229C2211yescharacteristic
28#5229C2211yescharacteristic
29#5229C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34#5221C2211yescharacteristic
35#5221C2211yescharacteristic
36C2×#832C2211yescharacteristic
37C2×#832C2211yescharacteristic
38#5210—21no—
39#5210—21no—
40C2×#818C2211yescharacteristic
41C2×#817C2211yescharacteristic
42C2×#815C2211yescharacteristic
43C2×Dic3⋊C8C2211yescharacteristic
44C2×Dic3⋊C8C2211yescharacteristic
45C2×C8×Dic3C2211yescharacteristic
46#4985—21no—
47#4985—21no—
48#4977—21no—
49#4973—21no—
50#4972C2211yescharacteristic
51#4972C2211yescharacteristic
52#4949—21no—
53#4947—21no—
54#4936C2211yescharacteristic
55C4×C4⋊Dic3C2211yescharacteristic
56C2×#803C2211yescharacteristic
57#4828—24no—
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#461D311yescharacteristic
63C2×#446—31no—
64#2580—31no—
65#2568—31no—
66#2068—31no—
67#2066—31no—
68#2055—31no—
69#2055—31no—
70#1973—31no—
71#1968—31no—
72#1960—31no—
73#1949—31no—
74#1935—31no—
75#1933—31no—
76#1762—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×Q8C2311yescharacteristic
79C22×C3×D4C2311yescharacteristic
80C23×C3×C4—21no—
81C22×Dic6D411yescharacteristic
82C2×C3×SD16—22no—
83C2×C3×SD16—22no—
84C2×C3×SD16—22no—
85C2×C3×SD16—22no—
86C2×C3×SD16—41no—
87C2×C3×SD16—41no—
88C22×C3×C8C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C3×C4⋊Q8—22no—
91C3×C4⋊Q8—22no—
92C3×C4×D4—22no—
93C3×C4×D4—22no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C22⋊C4—21no—
98C2×C3×C42C2311yescharacteristic
99C2×#217—21no—
100C2×#217—21no—
101C2×#217—21no—
102C2×#217—21no—
103C2×C4⋊Dic3D411yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4D411yescharacteristic
111C2×Dic3⋊C4D411yescharacteristic
112C2×C4×Dic3—21no—
113C2×C4×Dic3C2311yescharacteristic
114C2×C4×Dic3C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C4×Dic6—22no—
118C4×Dic6—22no—
119C3×C4⋊C8D414yes—
120C3×#103—22no—
121C3×#103—22no—
122#835—22no—
123#835—22no—
124#835—41no—
125#835—41no—
126#832—22no—
127#832—22no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141#818—41no—
142#817—41no—
143#815—22no—
144Dic3⋊C8—22no—
145Dic3⋊C8—22no—
146C8×Dic3—22no—
147#803—22no—
148C22×SD16C2×D311yescharacteristic
149C22×SD16C2×D311yescharacteristic
150C2×C4⋊Q8C2×D311yescharacteristic
151C2×C4×Q8—31no—
152C2×C4×D4C2×D311yescharacteristic
153#461—24no—
154#446—64no—
155C2×#107—31no—
156C2×#106—31no—
157C2×C4⋊C8—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8C2×D311yescharacteristic
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103C2×D311yescharacteristic
165C2×#103C2×D311yescharacteristic
166C2×#102—31no—
167C2×#102—31no—
168C2×#97—31no—
169C2×#97—31no—
170C2×C4×C8—31no—
171#400—61no—
172#400—61no—
173#399—61no—
174#393—61no—
175#391—31no—
176#391—31no—
177#391—31no—
178#391—31no—
179#391—61no—
180#390—31no—
181#390—31no—
182#389—31no—
183#389—31no—
184#385—61no—
185#385—61no—
186#382—31no—
187#382—61no—
188C4×C4⋊C4—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193C24×C3—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C2×C3×D4—21no—
211C2×C3×D4#4112yes—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—42no—
216C22×C3×C4—81no—
217C22×C3×C4—81no—
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2411yescharacteristic
221C22×Dic3—21no—
222C22×Dic3—21no—
223C22×Dic3—21no—
224C22×Dic3—21no—
225C22×Dic3C2×D411yescharacteristic
226C22×Dic3C2×D411yescharacteristic
227C22×Dic3C2×D411yescharacteristic
228C22×Dic3C2×D411yescharacteristic
229C2×Dic6—22no—
230C2×Dic6—22no—
231C2×Dic6—41no—
232C3×SD16—44no—
233C3×SD16—44no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8C2×D412yes—
239C2×C3×C8C2×D412yes—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—42no—
243C3×C4⋊C4—42no—
244C3×C22⋊C4—42no—
245C3×C42C2×D412yes—
246C3×C42#4112yes—
247#217—42no—
248#217—42no—
249#217—81no—
250#217—81no—
251C4⋊Dic3—22no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260Dic3⋊C4—22no—
261Dic3⋊C4—22no—
262Dic3⋊C4—81no—
263Dic3⋊C4—81no—
264C4×Dic3—21no—
265C4×Dic3—21no—
266C4×Dic3—21no—
267C4×Dic3—21no—
268C4×Dic3—42no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8#4112yes—
275C22×Q8—31no—
276C22×Q8C22×D311yescharacteristic
277C22×Q8C22×D311yescharacteristic
278C22×D4C22×D311yescharacteristic
279C23×C4—21no—
280C2×SD16—22no—
281C2×SD16—22no—
282C2×SD16—22no—
283C2×SD16—22no—
284C2×SD16—41no—
285C2×SD16—41no—
286C22×C8—31no—
287C22×C8—31no—
288C22×C8C22×D311yescharacteristic
289C22×C8C22×D311yescharacteristic
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4×Q8—62no—
293C4×Q8—62no—
294C4×D4—22no—
295C4×D4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4C22×D311yescharacteristic
308C2×C22⋊C4—21no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C42—31no—
314C2×C42—31no—
315C2×C42—61no—
316C2×C42C22×D311yescharacteristic
317#107—121no—
318#106—121no—
319C4⋊C8—62no—
320C4⋊C8—62no—
321C4⋊C8#6714yes—
322#103—22no—
323#103—22no—
324#103—62no—
325#103—62no—
326#103—121no—
327#103—121no—
328#102—62no—
329#102—62no—
330#97—62no—
331#97—62no—
332C4×C8—62no—
333#95—61no—
334#95—61no—
335#95—61no—
336#95—61no—
337#95—61no—
338#95—61no—
339#95—61no—
340#95—61no—
341#95—61no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346C23×C3—22no—
347C23×C3—41no—
348C23×C3—41no—
349C23×C3—41no—
350C23×C3—41no—
351C23×C3—41no—
352C23×C3—41no—
353C23×C3C22×D411yescharacteristic
354C3×Q8—41no—
355C3×Q8—41no—
356C3×Q8—41no—
357C3×Q8—41no—
358C3×Q8—41no—
359C3×Q8—41no—
360C3×Q8—41no—
361C3×Q8—41no—
362C3×D4—22no—
363C3×D4—22no—
364C3×D4—41no—
365C3×D4—41no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—84no—
380C2×C3×C4#12112yes—
381C2×C3×C4#12112yes—
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C2×C3×C4H(5,𝔽2)11yescharacteristic
387C2×C3×C42−1+411yescharacteristic
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
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—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3#13712yes—
412Dic6—44no—
413C3×C8—22no—
414C3×C8—22no—
415C3⋊C8—22no—
416C3⋊C8—41no—
417C24—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—62no—
431C2×Q8—62no—
432C2×Q8—121no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4#23712yes—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—31no—
443C22×C4—31no—
444C22×C4—31no—
445C22×C4—31no—
446C22×C4—42no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—81no—
452C22×C4—81no—
453C22×C4D3×D411yescharacteristic
454C22×C4D3×D411yescharacteristic
455C22×C4C23×D311yescharacteristic
456SD16—44no—
457SD16—44no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—32no—
463C2×C8—61no—
464C2×C8—61no—
465C2×C8—61no—
466C2×C8—61no—
467C2×C8—61no—
468C2×C8C2×#6712yes—
469C2×C8C2×#6712yes—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—62no—
475C4⋊C4—62no—
476C4⋊C4—62no—
477C4⋊C4—61no—
478C4⋊C4—61no—
479C4⋊C4—61no—
480C4⋊C4—61no—
481C4⋊C4—61no—
482C4⋊C4—61no—
483C4⋊C4—61no—
484C4⋊C4—61no—
485C4⋊C4—241no—
486C4⋊C4—241no—
487C22⋊C4—42no—
488C22⋊C4—122no—
489C22⋊C4—122no—
490C22⋊C4—241no—
491C22⋊C4—241no—
492C42—61no—
493C42—61no—
494C42—61no—
495C42—61no—
496C42—122no—
497C42#23712yes—
498C42C2×#6712yes—
499C22×C3—42no—
500C22×C3—42no—
501C22×C3—42no—
502C22×C3—81no—
503C22×C3—81no—
504C22×C3—81no—
505C22×C3—81no—
506C22×C3C2×#12111yescharacteristic
507C22×C3#54811yescharacteristic
508C22×C3#55011yescharacteristic
509C22×C3C2×#13711yescharacteristic
510C22×C3#57511yescharacteristic
511C22×C3#57711yescharacteristic
512C22×C3#57711yescharacteristic
513C3×C4—22no—
514C3×C4—22no—
515C3×C4—81no—
516C3×C4—81no—
517C3×C4—81no—
518C3×C4#52911yescharacteristic
519C3×C4#54211yescharacteristic
520C3×C4#54611yescharacteristic
521C3×C4#54811yescharacteristic
522Dic3—22no—
523Dic3—41no—
524Dic3—41no—
525Dic3—41no—
526Dic3—81no—
527Dic3—81no—
528Dic3—81no—
529Dic3—81no—
530C23—22no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23C2×D3×D411yescharacteristic
538Q8—41no—
539Q8—41no—
540Q8—41no—
541Q8—41no—
542Q8—41no—
543Q8—41no—
544Q8—41no—
545Q8—41no—
546Q8—124no—
547D4—22no—
548D4—22no—
549D4—41no—
550D4—41no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—32no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—61no—
575C2×C4—61no—
576C2×C4—84no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4#93812yes—
590C2×C4#93812yes—
591C2×C4C2×#23711yescharacteristic
592C2×C4C2×#23711yescharacteristic
593C2×C4#100411yescharacteristic
594C2×C4D3×#4111yescharacteristic
595C2×C4#101011yescharacteristic
596C2×C4C22×#6711yescharacteristic
597C8—22no—
598C8—22no—
599C8—62no—
600C8—121no—
601C2×C3—82no—
602C2×C3#242611yescharacteristic
603C2×C3#305611yescharacteristic
604C2×C3#305711yescharacteristic
605C2×C3#337211yescharacteristic
606C2×C3#337611yescharacteristic
607C2×C3#340011yescharacteristic
608C2×C3#340111yescharacteristic
609C22—42no—
610C22—42no—
611C22—42no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22#559811yescharacteristic
617C22#559911yescharacteristic
618C22#576211yescharacteristic
619C22#576211yescharacteristic
620C22D3×#13711yescharacteristic
621C22#577711yescharacteristic
622C22C2×#93811yescharacteristic
623C4—22no—
624C4—22no—
625C4—62no—
626C4—81no—
627C4—81no—
628C4—81no—
629C4—121no—
630C4—121no—
631C4—121no—
632C4—241no—
633C4—241no—
634C4—241no—
635C4—241no—
636C4#560011yescharacteristic
637C4#560311yescharacteristic
638C4#580611yescharacteristic
639C4#583111yescharacteristic
640C3#2530811yescharacteristic, complemented, Sylow
641C2—82no—
642C2#8041811yescharacteristic
643C2#8041911yescharacteristic
644C2#8042311yescharacteristic
645C2#8042411yescharacteristic
646C2#8264611yescharacteristic
647C2#8386311yescharacteristic
648C2#8398511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#461C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenterC2×#461D3
hypocenterC3#25308
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
2. #83244C2semidirect
4. C2×#4828C2semidirect
9. #71786C2semidirect
11. #70455C2semidirect
12. #70256C2semidirect
13. #70254C2semidirect
14. #70248C2semidirect
15. #70247C2semidirect
640. C3#25308semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4612
#832442
#823882
C2×#48282
#720512
#720502
#720492
#717862
#716092
#704552
#702562
#702542
#702482
#702472
#253083