Finite Group Databasebeta

← #10858877

Subgroups — #10858877

Order 768.

738 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82748C211yesmaximal, characteristic
2#82249C211yesmaximal, characteristic
3C3×#1873C211yesmaximal, characteristic
4#71696C211yesmaximal, characteristic
5#71580C211yesmaximal, characteristic
6#70248C211yesmaximal, characteristic, complemented
7#70238C211yesmaximal, characteristic, complemented
8#70232C211yesmaximal, characteristic, complemented
9#70221C211yesmaximal, characteristic, complemented
10#69950C211yesmaximal, characteristic
11#69929C211yesmaximal, characteristic
12#68843C211yesmaximal, characteristic, complemented
13#68838C211yesmaximal, characteristic, complemented
14#68816C211yesmaximal, characteristic, complemented
15#68802C211yesmaximal, characteristic, complemented
16#26417—31nomaximal, Sylow
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C2×#926C2211yescharacteristic
21C23×C3⋊C8C2211yescharacteristic
22C2×#890C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×C22⋊C8C2211yescharacteristic
26#5229C2211yescharacteristic
27C2×#835C2211yescharacteristic
28C2×#835C2211yescharacteristic
29#5212C2211yescharacteristic
30C4×#217C2211yescharacteristic
31C2×#818C2211yescharacteristic
32C2×#817C2211yescharacteristic
33C2×Dic3⋊C8C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35#4984C2211yescharacteristic
36#4984C2211yescharacteristic
37#4980—21no—
38#4975C2211yescharacteristic
39Dic3×C4⋊C4—21no—
40Dic3×C4⋊C4C2211yescharacteristic
41#4972C2211yescharacteristic
42#4951—21no—
43#4951—21no—
44#4949—21no—
45#4948—21no—
46#4933—21no—
47#4928—21no—
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50#4551C2211yescharacteristic
51#4551C2211yescharacteristic
52#4550C2211yescharacteristic
53#4550C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2441—31no—
61#2362—31no—
62#2100—31no—
63#2086—31no—
64#2068—31no—
65#2026—31no—
66#1989—31no—
67#1919—31no—
68#1892—31no—
69#1873D311yescharacteristic
70#1865—31no—
71#1854—31no—
72#1850—31no—
73#1803—31no—
74#1735—31no—
75C22×C3×Q8C2311yescharacteristic
76C23×C3×C4C2311yescharacteristic
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C3×C22⋊Q8—22no—
80C3×C22⋊Q8Q814yes—
81C3×#117—22no—
82C3×#117—41no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C22⋊C4—21no—
88C2×C3×C22⋊C4—21no—
89C2×C3×C42—21no—
90C2×#217—21no—
91C2×#217—21no—
92C2×#217D411yescharacteristic
93C2×#217D411yescharacteristic
94Q8×Dic3—22no—
95Q8×Dic3—41no—
96#926—22no—
97#926—41no—
98C2×C4⋊Dic3—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×C4×Dic3—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3C2311yescharacteristic
114C2×C4×Dic3C2311yescharacteristic
115C22×C3⋊C8—22no—
116C22×C3⋊C8—41no—
117C22×C3⋊C8—41no—
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120#890—22no—
121#890—41no—
122C3×#103—22no—
123C3×#103—41no—
124C3×C22⋊C8—22no—
125#835—41no—
126#835—41no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95D411yescharacteristic
135C3⋊#95D411yescharacteristic
136#818—41no—
137#817—22no—
138Dic3⋊C8—41no—
139Dic3⋊C8—41no—
140C23×C8—31no—
141C2×#125—31no—
142C2×C22⋊Q8C2×D311yescharacteristic
143C2×C4×Q8—31no—
144C2×#117—31no—
145C2×#117C2×D311yescharacteristic
146C2×#107—31no—
147C2×#106—31no—
148C2×C4⋊C8—31no—
149C2×C4⋊C8—31no—
150C2×#103—31no—
151C2×#103—31no—
152C2×#103C2×D311yescharacteristic
153C2×#103C2×D311yescharacteristic
154C2×C22⋊C8C2×D311yescharacteristic
155#399—61no—
156#396—61no—
157#391—31no—
158#389—31no—
159#389—31no—
160#389—31no—
161#389—61no—
162#388—61no—
163#388—61no—
164#387—61no—
165#387—61no—
166#384—31no—
167#380—31no—
168C4×C4⋊C4—31no—
169C4×C4⋊C4—61no—
170C4×C22⋊C4—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340C2×D311yescharacteristic
180#340C2×D311yescharacteristic
181#336—31no—
182#336—31no—
183C24×C3C2×D411yescharacteristic
184C2×C3×Q8—21no—
185C2×C3×Q8—21no—
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8C2×Q812yes—
190C22×C3×C4—21no—
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×C4C2×D411yescharacteristic
199C22×C3×C4C2×Q812yes—
200C22×C3×C4#4112yes—
201C22×C3×C4C2411yescharacteristic
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3C2×D411yescharacteristic
207C22×Dic3C2×D411yescharacteristic
208C22×Dic3C2×D411yescharacteristic
209C22×Dic3C2×D411yescharacteristic
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C3×C4⋊C4—24no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4C2×Q812yes—
222C3×C4⋊C4#4112yes—
223C3×C22⋊C4—24no—
224C3×C22⋊C4—42no—
225C3×C42—42no—
226#217—22no—
227#217—41no—
228#217—42no—
229#217—81no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—42no—
238C4⋊Dic3—42no—
239C4⋊Dic3—42no—
240C4⋊Dic3—42no—
241C4⋊Dic3#4112yes—
242Dic3⋊C4—42no—
243Dic3⋊C4—42no—
244Dic3⋊C4—81no—
245Dic3⋊C4—81no—
246C4×Dic3—21no—
247C4×Dic3—21no—
248C4×Dic3—21no—
249C4×Dic3—21no—
250C4×Dic3—42no—
251C4×Dic3—42no—
252C4×Dic3—42no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—44no—
260C22×Q8C22×D311yescharacteristic
261C23×C4C22×D311yescharacteristic
262C22×C8—31no—
263C22×C8—31no—
264C22×C8—62no—
265C22×C8—121no—
266C22×C8—121no—
267C22×C8C22×D311yescharacteristic
268C22×C8C22×D311yescharacteristic
269#125—62no—
270#125—121no—
271C22⋊Q8—22no—
272C22⋊Q8Dic614yes—
273C4×Q8—62no—
274C4×Q8—121no—
275#117—22no—
276#117—41no—
277#117—62no—
278#117—121no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4C22×D311yescharacteristic
293C2×C4⋊C4C22×D311yescharacteristic
294C2×C4⋊C4C22×D311yescharacteristic
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—31no—
298C2×C22⋊C4—31no—
299C2×C22⋊C4—61no—
300C2×C22⋊C4—61no—
301C2×C42—21no—
302C2×C42—31no—
303C2×C42—31no—
304C2×C42—61no—
305C2×C42—61no—
306C2×C42—61no—
307#107—121no—
308#106—62no—
309C4⋊C8—121no—
310C4⋊C8—121no—
311#103—22no—
312#103—41no—
313#103—121no—
314#103—121no—
315C22⋊C8—22no—
316#95—31no—
317#95—31no—
318#95—61no—
319#95—61no—
320#95—61no—
321#95—61no—
322#95—61no—
323#95—61no—
324#95—61no—
325C23×C3—21no—
326C23×C3—21no—
327C23×C3—21no—
328C23×C3—21no—
329C23×C3—21no—
330C23×C3—21no—
331C23×C3C22⋊Q812yes—
332C23×C3C22×D411yescharacteristic
333C3×Q8—22no—
334C3×Q8—22no—
335C3×Q8—41no—
336C3×Q8—41no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
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×C4C22⋊Q812yes—
356C2×C3×C4C22×Q811yescharacteristic
357C2×C3×C4C2×#4111yescharacteristic
358C2×C3×C4C2×#4111yescharacteristic
359C2×C3×C4C2×#4111yescharacteristic
360C2×C3×C4H(5,𝔽2)11yescharacteristic
361C2×C3×C4H(5,𝔽2)11yescharacteristic
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—22no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
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#13512yes—
386C3×C8—41no—
387C3×C8—41no—
388C3⋊C8—41no—
389C3⋊C8—41no—
390C24D3×D411yescharacteristic
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8C2×Dic612yes—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—31no—
406C22×C4—31no—
407C22×C4—31no—
408C22×C4—31no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4—61no—
413C22×C4C2×Dic612yes—
414C22×C4D3×D411yescharacteristic
415C22×C4#23712yes—
416C22×C4C23×D311yescharacteristic
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—61no—
424C2×C8—61no—
425C2×C8—61no—
426C2×C8—61no—
427C2×C8—61no—
428C2×C8—61no—
429C2×C8—124no—
430C4⋊C4—24no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—32no—
436C4⋊C4—61no—
437C4⋊C4—61no—
438C4⋊C4—61no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—122no—
444C4⋊C4—122no—
445C4⋊C4—122no—
446C4⋊C4—122no—
447C4⋊C4—122no—
448C4⋊C4—122no—
449C4⋊C4—241no—
450C4⋊C4—241no—
451C4⋊C4C2×Dic612yes—
452C4⋊C4#23712yes—
453C22⋊C4—24no—
454C22⋊C4—42no—
455C22⋊C4—62no—
456C22⋊C4—121no—
457C22⋊C4—122no—
458C22⋊C4—241no—
459C42—42no—
460C42—61no—
461C42—61no—
462C42—61no—
463C42—61no—
464C42—122no—
465C42—122no—
466C42—122no—
467C22×C3—22no—
468C22×C3—22no—
469C22×C3—22no—
470C22×C3—41no—
471C22×C3—41no—
472C22×C3—41no—
473C22×C3—41no—
474C22×C3C2×C22⋊Q811yescharacteristic
475C22×C3#54611yescharacteristic
476C22×C3#54611yescharacteristic
477C22×C3C2×#13511yescharacteristic
478C22×C3#57511yescharacteristic
479C22×C3#57711yescharacteristic
480C22×C3#57811yescharacteristic
481C3×C4—22no—
482C3×C4—42no—
483C3×C4—42no—
484C3×C4—42no—
485C3×C4—81no—
486C3×C4#52911yescharacteristic
487C3×C4#53811yescharacteristic
488C3×C4#55411yescharacteristic
489C3×C4#55411yescharacteristic
490Dic3—22no—
491Dic3—41no—
492Dic3—41no—
493Dic3—42no—
494Dic3—41no—
495Dic3—81no—
496Dic3—81no—
497Dic3—81no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23#87812yes—
505C23C2×D3×D411yescharacteristic
506Q8—22no—
507Q8—22no—
508Q8—41no—
509Q8—41no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—32no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—61no—
530C2×C4—61no—
531C2×C4—61no—
532C2×C4—62no—
533C2×C4—61no—
534C2×C4—61no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—61no—
540C2×C4—61no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4#87812yes—
553C2×C4C22×Dic611yescharacteristic
554C2×C4C2×#23711yescharacteristic
555C2×C4C2×#23711yescharacteristic
556C2×C4#100411yescharacteristic
557C2×C4D3×#4111yescharacteristic
558C2×C4#100911yescharacteristic
559C8—41no—
560C8—41no—
561C8—121no—
562C8—121no—
563C2×C3—42no—
564C2×C3#245011yescharacteristic
565C2×C3#307611yescharacteristic
566C2×C3#308211yescharacteristic
567C2×C3#330111yescharacteristic
568C2×C3#330211yescharacteristic
569C2×C3#331511yescharacteristic
570C2×C3#331711yescharacteristic
571C22—22no—
572C22—22no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22C2×#87811yescharacteristic
579C22#559111yescharacteristic
580C22#565411yescharacteristic
581C22D3×#13511yescharacteristic
582C22#576811yescharacteristic
583C22#577811yescharacteristic
584C22#577911yescharacteristic
585C4—22no—
586C4—42no—
587C4—42no—
588C4—42no—
589C4—62no—
590C4—81no—
591C4—121no—
592C4—121no—
593C4—121no—
594C4—122no—
595C4—241no—
596C4—241no—
597C4—241no—
598C4#553911yescharacteristic
599C4#554311yescharacteristic
600C4#560011yescharacteristic
601C4#561711yescharacteristic
602C3#2641711yescharacteristic, complemented, Sylow
603C2—42no—
604C2#7471111yescharacteristic
605C2#7882211yescharacteristic
606C2#7895011yescharacteristic
607C2#8024811yescharacteristic
608C2#8025611yescharacteristic
609C2#8152511yescharacteristic
610C2#8154711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1873C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1873D3
hypocenterC3#26417
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1873

Complements

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

NH ≅ G/Ntype
6. #70248C2semidirect
7. #70238C2semidirect
8. #70232C2semidirect
9. #70221C2semidirect
12. #68843C2semidirect
13. #68838C2semidirect
14. #68816C2semidirect
15. #68802C2semidirect
602. C3#26417semidirect

Maximal subgroups

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

subgroupindex
#827482
#822492
C3×#18732
#716962
#715802
#702482
#702382
#702322
#702212
#699502
#699292
#688432
#688382
#688162
#688022
#264173