Finite Group Databasebeta

← #10867869

Subgroups — #10867869

Order 768.

792 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82636C211yesmaximal, characteristic
2#73924C211yesmaximal, characteristic, complemented
3C3×#2034C211yesmaximal, characteristic
4#71680C211yesmaximal, characteristic
5#70563C211yesmaximal, characteristic, complemented
6#70560C211yesmaximal, characteristic, complemented
7#70179C211yesmaximal, characteristic, complemented
8#70170C211yesmaximal, characteristic, complemented
9#70165C211yesmaximal, characteristic, complemented
10Dic3×#102C211yesmaximal, characteristic, complemented
11#69928C211yesmaximal, characteristic, complemented
12#69913C211yesmaximal, characteristic, complemented
13#69501C211yesmaximal, characteristic, complemented
14#68818C211yesmaximal, characteristic, complemented
15#68817C211yesmaximal, characteristic, complemented
16#27032—31nomaximal, Sylow
17C2×C3×#121C2211yescharacteristic
18C2×D4×Dic3—21no—
19C2×D4×Dic3C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C2×C3×#106C2211yescharacteristic
22C2×C3×#102C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×C22⋊C8C2211yescharacteristic
25C3×#380C2211yescharacteristic
26#5221C2211yescharacteristic
27#5220—21no—
28C2×#832C2211yescharacteristic
29C2×#832C2211yescharacteristic
30#5212C2211yescharacteristic
31C4×#217C2211yescharacteristic
32C2×#830C2211yescharacteristic
33C2×#817C2211yescharacteristic
34C2×#817C2211yescharacteristic
35C2×Dic3⋊C8C2211yescharacteristic
36C2×C8×Dic3C2211yescharacteristic
37#4984C2211yescharacteristic
38#4984C2211yescharacteristic
39#4975C2211yescharacteristic
40Dic3×C4⋊C4C2211yescharacteristic
41Dic3×C4⋊C4C2211yescharacteristic
42#4972C2211yescharacteristic
43C2×#808C2211yescharacteristic
44#4951—21no—
45#4950—21no—
46#4948—21no—
47#4944—21no—
48#4659—21no—
49#4652—21no—
50C3×#340C2211yescharacteristic
51#4551C2211yescharacteristic
52#4551C2211yescharacteristic
53#4550C2211yescharacteristic
54#4550C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2497—31no—
61#2361—31no—
62#2071—31no—
63#2065—31no—
64#2034—31no—
65#2034D311yescharacteristic
66#1948—31no—
67#1941—31no—
68#1923—31no—
69#1920—31no—
70#1919—31no—
71#1864—31no—
72#1853—31no—
73#1823—31no—
74C4×#102—31no—
75C22×C3×D4—21no—
76C22×C3×D4C2311yescharacteristic
77C23×C3×C4C2311yescharacteristic
78C23×Dic3—21no—
79C23×Dic3—21no—
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×#121—22no—
83C3×#121—22no—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C22⋊C4—21no—
88C2×C3×C22⋊C4—21no—
89C2×#217—21no—
90C2×#217—21no—
91C2×#217—21no—
92C2×#217—21no—
93C2×#217D411yescharacteristic
94C2×#217D411yescharacteristic
95D4×Dic3—22no—
96D4×Dic3—41no—
97D4×Dic3—44no—
98C2×C4⋊Dic3—22no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3—41no—
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3C2311yescharacteristic
113C2×C4×Dic3C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C3×#106—22no—
117C3×#102—22no—
118C3×#102—22no—
119C3×C22⋊C8—22no—
120C3×#95—21no—
121#832—22no—
122#832—41no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95D411yescharacteristic
130C3⋊#95D411yescharacteristic
131#830—41no—
132#817—22no—
133#817—41no—
134Dic3⋊C8—22no—
135C8×Dic3—41no—
136#808—22no—
137C2×#121C2×D311yescharacteristic
138C2×C4×D4—31no—
139C2×C4×D4—61no—
140C22×C4⋊C4—31no—
141C2×#106—31no—
142C2×#106—31no—
143C2×#106—31no—
144C2×#106C2×D311yescharacteristic
145C2×C4⋊C8—31no—
146C2×#102—31no—
147C2×#102—31no—
148C2×#102C2×D311yescharacteristic
149C2×#102C2×D311yescharacteristic
150C2×C22⋊C8—31no—
151C2×C22⋊C8C2×D311yescharacteristic
152C2×C4×C8—31no—
153#400—61no—
154#396—61no—
155#390—31no—
156#389—31no—
157#389—31no—
158#389—31no—
159#388—61no—
160#388—61no—
161#387—61no—
162#386—61no—
163#385—61no—
164#384—31no—
165#380—31no—
166#380C2×D311yescharacteristic
167C4×C4⋊C4—31no—
168C4×C4⋊C4—31no—
169C4×C22⋊C4—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340C2×D311yescharacteristic
178#336—31no—
179#336—31no—
180C24×C3—21no—
181C24×C3C2×D411yescharacteristic
182C2×C3×D4—24no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—42no—
189C2×C3×D4—42no—
190C2×C3×D4#4112yes—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2411yescharacteristic
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—42no—
207C22×Dic3—42no—
208C22×Dic3—42no—
209C22×Dic3—42no—
210C22×Dic3—81no—
211C22×Dic3—81no—
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C22×Dic3C2×D411yescharacteristic
215C22×Dic3C2×D411yescharacteristic
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8#4112yes—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4#4112yes—
228C3×C22⋊C4—24no—
229C3×C22⋊C4—81no—
230#217—41no—
231#217—42no—
232#217—42no—
233#217—81no—
234#217—81no—
235#217SD1614yes—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—42no—
244C4⋊Dic3—42no—
245C4⋊Dic3—81no—
246C4⋊Dic3#4112yes—
247Dic3⋊C4—42no—
248Dic3⋊C4—81no—
249Dic3⋊C4—81no—
250C4×Dic3—21no—
251C4×Dic3—21no—
252C4×Dic3—21no—
253C4×Dic3—21no—
254C4×Dic3—42no—
255C4×Dic3—81no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C22×D4—21no—
263C22×D4C22×D311yescharacteristic
264C23×C4—61no—
265C23×C4—61no—
266C23×C4C22×D311yescharacteristic
267C22×C8—31no—
268C22×C8—31no—
269C22×C8C22×D311yescharacteristic
270C22×C8C22×D311yescharacteristic
271#121—22no—
272#121—22no—
273C4×D4—62no—
274C4×D4—121no—
275C4×D4—124no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—62no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—121no—
288C2×C4⋊C4C22×D311yescharacteristic
289C2×C4⋊C4C22×D311yescharacteristic
290C2×C4⋊C4C22×D311yescharacteristic
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—31no—
294C2×C22⋊C4—31no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4—61no—
299C2×C42—31no—
300C2×C42—31no—
301C2×C42—61no—
302C2×C42—61no—
303#106—22no—
304#106—62no—
305#106—62no—
306#106—121no—
307C4⋊C8—62no—
308#102—22no—
309#102—22no—
310#102—62no—
311#102—121no—
312C22⋊C8—22no—
313C22⋊C8—121no—
314C4×C8—121no—
315#95—21no—
316#95—31no—
317#95—31no—
318#95—61no—
319#95—61no—
320#95—61no—
321#95—61no—
322#95—61no—
323#95—61no—
324C23×C3—21no—
325C23×C3—21no—
326C23×C3—21no—
327C23×C3—21no—
328C23×C3—21no—
329C23×C3—21no—
330C23×C3—22no—
331C23×C3—41no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3—41no—
335C23×C3—41no—
336C23×C3—41no—
337C23×C3C2×SD1612yes—
338C23×C3C22×D411yescharacteristic
339C3×D4—22no—
340C3×D4—22no—
341C3×D4—48no—
342C3×D4—41no—
343C3×D4—41no—
344C2×C3×C4—21no—
345C2×C3×C4—22no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4C2×#4111yescharacteristic
361C2×C3×C4C2×#4111yescharacteristic
362C2×C3×C4C2×#4111yescharacteristic
363C2×C3×C4C2×#4111yescharacteristic
364C2×C3×C4H(5,𝔽2)11yescharacteristic
365C2×C3×C42−1+411yescharacteristic
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—22no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—84no—
389C2×Dic3—84no—
390C2×Dic3C2×SD1612yes—
391C2×Dic3C2×SD1612yes—
392C3×C8—22no—
393C3×C8—41no—
394C3⋊C8—41no—
395C3⋊C8—41no—
396C24—21no—
397C24D3×D411yescharacteristic
398C2×D4—24no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4#23712yes—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—31no—
417C22×C4—31no—
418C22×C4—31no—
419C22×C4—31no—
420C22×C4—61no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—122no—
425C22×C4—122no—
426C22×C4—122no—
427C22×C4—122no—
428C22×C4—241no—
429C22×C4—241no—
430C22×C4D3×D411yescharacteristic
431C22×C4C23×D311yescharacteristic
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—61no—
438C2×C8—61no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8—61no—
442C2×C8—61no—
443C2×C8#23712yes—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—32no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C4⋊C4—122no—
458C4⋊C4—122no—
459C4⋊C4—122no—
460C4⋊C4—241no—
461C4⋊C4—241no—
462C4⋊C4—241no—
463C4⋊C4#23712yes—
464C22⋊C4—24no—
465C22⋊C4—34no—
466C22⋊C4—81no—
467C22⋊C4—122no—
468C22⋊C4—121no—
469C22⋊C4—122no—
470C22⋊C4—241no—
471C22⋊C4—241no—
472C42—61no—
473C42—61no—
474C42—61no—
475C42—61no—
476C42—122no—
477C42—241no—
478C22×C3—22no—
479C22×C3—22no—
480C22×C3—22no—
481C22×C3—41no—
482C22×C3—41no—
483C22×C3—41no—
484C22×C3—41no—
485C22×C3—42no—
486C22×C3—42no—
487C22×C3—42no—
488C22×C3—81no—
489C22×C3—81no—
490C22×C3—81no—
491C22×C3—81no—
492C22×C3#52511yescharacteristic
493C22×C3#54611yescharacteristic
494C22×C3#54711yescharacteristic
495C22×C3C22×SD1611yescharacteristic
496C22×C3#57511yescharacteristic
497C22×C3#57611yescharacteristic
498C22×C3#57711yescharacteristic
499C3×C4—42no—
500C3×C4—41no—
501C3×C4—81no—
502C3×C4—81no—
503C3×C4#52911yescharacteristic
504C3×C4#53911yescharacteristic
505C3×C4#55311yescharacteristic
506C3×C4#55611yescharacteristic
507Dic3—22no—
508Dic3—22no—
509Dic3—41no—
510Dic3—42no—
511Dic3—41no—
512Dic3—81no—
513Dic3—81no—
514Dic3—81no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—22no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23D3×SD1612yes—
529C23C2×D3×D411yescharacteristic
530D4—22no—
531D4—22no—
532D4—48no—
533D4—41no—
534D4—41no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—32no—
540C2×C4—32no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—61no—
557C2×C4—61no—
558C2×C4—62no—
559C2×C4—61no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—244no—
576C2×C4—244no—
577C2×C4C2×#23711yescharacteristic
578C2×C4C2×#23711yescharacteristic
579C2×C4C2×#23711yescharacteristic
580C2×C4D3×#4111yescharacteristic
581C2×C4#100911yescharacteristic
582C2×C4#101011yescharacteristic
583C8—22no—
584C8—41no—
585C8—121no—
586C8—121no—
587C2×C3—42no—
588C2×C3—82no—
589C2×C3#246611yescharacteristic
590C2×C3#310111yescharacteristic
591C2×C3#311911yescharacteristic
592C2×C3#329511yescharacteristic
593C2×C3#332111yescharacteristic
594C2×C3#333511yescharacteristic
595C2×C3#335511yescharacteristic
596C22—22no—
597C22—22no—
598C22—22no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22#558611yescharacteristic
611C22#561111yescharacteristic
612C22#565411yescharacteristic
613C22C2×D3×SD1611yescharacteristic
614C22#577011yescharacteristic
615C22#577311yescharacteristic
616C22#577411yescharacteristic
617C4—42no—
618C4—41no—
619C4—62no—
620C4—62no—
621C4—81no—
622C4—81no—
623C4—122no—
624C4—121no—
625C4—121no—
626C4—241no—
627C4—241no—
628C4—241no—
629C4#560011yescharacteristic
630C4#561811yescharacteristic
631C4#566511yescharacteristic
632C4#567111yescharacteristic
633C3#2703211yescharacteristic, complemented, Sylow
634C2—42no—
635C2—82no—
636C2#7691911yescharacteristic
637C2#8007311yescharacteristic
638C2#8013211yescharacteristic
639C2#8064811yescharacteristic
640C2#8086111yescharacteristic
641C2#8134211yescharacteristic
642C2#8138611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2034C2
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
hypercenter#2034D3
hypocenterC3#27032
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2034

Complements

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

NH ≅ G/Ntype
2. #73924C2semidirect
5. #70563C2semidirect
6. #70560C2semidirect
7. #70179C2semidirect
8. #70170C2semidirect
9. #70165C2semidirect
10. Dic3×#102C2semidirect
11. #69928C2semidirect
12. #69913C2semidirect
13. #69501C2semidirect
14. #68818C2semidirect
15. #68817C2semidirect
633. C3#27032semidirect

Maximal subgroups

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

subgroupindex
#826362
#739242
C3×#20342
#716802
#705632
#705602
#701792
#701702
#701652
Dic3×#1022
#699282
#699132
#695012
#688182
#688172
#270323