Finite Group Databasebeta

← #10853227

Subgroups — #10853227

Order 768.

696 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82239C211yesmaximal, characteristic
2#73901C211yesmaximal, characteristic
3C3×#1825C211yesmaximal, characteristic
4#71581C211yesmaximal, characteristic
5#69951C211yesmaximal, characteristic
6#69927C211yesmaximal, characteristic
7#69513C211yesmaximal, characteristic
8#68845C211yesmaximal, characteristic, complemented
9#68838C211yesmaximal, characteristic, complemented
10#68829C211yesmaximal, characteristic, complemented
11#68821C211yesmaximal, characteristic, complemented
12#68820C211yesmaximal, characteristic, complemented
13#68807C211yesmaximal, characteristic, complemented
14#68801C211yesmaximal, characteristic, complemented
15#68798C211yesmaximal, characteristic, complemented
16#26017—31nomaximal, Sylow
17C2×C3×#117C2211yescharacteristic
18C22×C4⋊Dic3C2211yescharacteristic
19C22×#208C2211yescharacteristic
20C2×#890C2211yescharacteristic
21C2×C3×C22⋊C8C2211yescharacteristic
22C3×#380C2211yescharacteristic
23#5212C2211yescharacteristic
24C4×#217C2211yescharacteristic
25C2×#818C2211yescharacteristic
26C2×#818C2211yescharacteristic
27C2×Dic3⋊C8C2211yescharacteristic
28C2×Dic3⋊C8C2211yescharacteristic
29#4984C2211yescharacteristic
30#4984C2211yescharacteristic
31#4984C2211yescharacteristic
32#4977C2211yescharacteristic
33Dic3×C4⋊C4C2211yescharacteristic
34Dic3×C4⋊C4C2211yescharacteristic
35#4972C2211yescharacteristic
36C2×#808C2211yescharacteristic
37C2×#807C2211yescharacteristic
38#4951—21no—
39#4951—21no—
40#4950—21no—
41#4942—21no—
42#4936—21no—
43#4932—21no—
44#4661—21no—
45#4660—21no—
46C3×#340C2211yescharacteristic
47C3×#340C2211yescharacteristic
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#2442—31no—
61#2346—31no—
62#2101—31no—
63#2086—31no—
64#2086—31no—
65#2059—31no—
66#1989—31no—
67#1949—31no—
68#1921—31no—
69#1921—31no—
70#1852—31no—
71#1851—31no—
72#1825D311yescharacteristic
73#1809—31no—
74#1736—31no—
75C23×C3×C4C2311yescharacteristic
76C23×Dic3—21no—
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C3×#117—22no—
80C3×#117Q814yes—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4—21no—
87C2×C3×C42—21no—
88C2×#217—21no—
89C2×#217D411yescharacteristic
90C2×#217D411yescharacteristic
91C2×C4⋊Dic3—22no—
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3—41no—
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×C4×Dic3—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3C2311yescharacteristic
107C2×C4×Dic3C2311yescharacteristic
108C2×#208—22no—
109C2×#208—22no—
110C2×#208—41no—
111C22×C3⋊C8C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113#890—22no—
114#890—41no—
115C3×C22⋊C8—22no—
116C3×#95—21no—
117C3⋊#95—21no—
118C3⋊#95—21no—
119C3⋊#95—21no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95D411yescharacteristic
127C3⋊#95D411yescharacteristic
128#818—22no—
129#818—41no—
130Dic3⋊C8—22no—
131Dic3⋊C8—41no—
132#808—41no—
133#807—22no—
134C22×M16—31no—
135C2×#125—31no—
136C2×#117C2×D311yescharacteristic
137C22×C4⋊C4—31no—
138C2×#107—31no—
139C2×#107—31no—
140C2×#107—31no—
141C2×#106—31no—
142C2×C4⋊C8—31no—
143C2×C4⋊C8—31no—
144C2×C22⋊C8C2×D311yescharacteristic
145#400—61no—
146#398—61no—
147#389—31no—
148#389—31no—
149#389—31no—
150#389—31no—
151#388—61no—
152#388—61no—
153#385—61no—
154#382—31no—
155#382—61no—
156#380—31no—
157#380—61no—
158#380C2×D311yescharacteristic
159C4×C4⋊C4—31no—
160C4×C4⋊C4—31no—
161C4×C22⋊C4—31no—
162#376—61no—
163#340—31no—
164#340—31no—
165#340—31no—
166#340—31no—
167#340—31no—
168#340—31no—
169#340—31no—
170#340—31no—
171#340C2×D311yescharacteristic
172#340C2×D311yescharacteristic
173#340C2×D311yescharacteristic
174#340C2×D311yescharacteristic
175#336—31no—
176#336—31no—
177C24×C3C2×D411yescharacteristic
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×Q812yes—
188C22×C3×C4#4112yes—
189C22×C3×C4C2411yescharacteristic
190C22×Dic3—21no—
191C22×Dic3—21no—
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3—42no—
195C22×Dic3—42no—
196C22×Dic3—81no—
197C22×Dic3C2×D411yescharacteristic
198C22×Dic3C2×D411yescharacteristic
199C22×Dic3C2×D411yescharacteristic
200C22×Dic3C2×D411yescharacteristic
201C2×C3×C8—21no—
202C2×C3×C8—21no—
203C2×C3×C8—21no—
204C2×C3×C8—21no—
205C2×C3×C8—21no—
206C2×C3×C8—21no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4C2×Q812yes—
214C3×C4⋊C4C2×Q812yes—
215C3×C22⋊C4—24no—
216C3×C22⋊C4—42no—
217C3×C42—24no—
218#217—41no—
219#217—41no—
220#217—81no—
221C4⋊Dic3—21no—
222C4⋊Dic3—21no—
223C4⋊Dic3—21no—
224C4⋊Dic3—21no—
225C4⋊Dic3—21no—
226C4⋊Dic3—21no—
227C4⋊Dic3—21no—
228C4⋊Dic3—42no—
229C4⋊Dic3—42no—
230C4⋊Dic3—81no—
231C4⋊Dic3—81no—
232C4⋊Dic3—81no—
233C4⋊Dic3#4112yes—
234Dic3⋊C4—42no—
235Dic3⋊C4—81no—
236C4×Dic3—21no—
237C4×Dic3—21no—
238C4×Dic3—21no—
239C4×Dic3—21no—
240C4×Dic3—42no—
241C4×Dic3—42no—
242#208—44no—
243C2×C3⋊C8—21no—
244C2×C3⋊C8—21no—
245C2×C3⋊C8—21no—
246C2×C3⋊C8—21no—
247C2×C3⋊C8—21no—
248C2×C3⋊C8#4112yes—
249C23×C4—61no—
250C23×C4C22×D311yescharacteristic
251C2×M16—62no—
252C2×M16—62no—
253C2×M16—121no—
254C22×C8—31no—
255C22×C8—31no—
256C22×C8C22×D311yescharacteristic
257C22×C8C22×D311yescharacteristic
258#125—62no—
259#125—121no—
260#117—22no—
261#117Dic614yes—
262C2×C4⋊C4—31no—
263C2×C4⋊C4—31no—
264C2×C4⋊C4—31no—
265C2×C4⋊C4—31no—
266C2×C4⋊C4—62no—
267C2×C4⋊C4—61no—
268C2×C4⋊C4—61no—
269C2×C4⋊C4—61no—
270C2×C4⋊C4—61no—
271C2×C4⋊C4—61no—
272C2×C4⋊C4—61no—
273C2×C4⋊C4—61no—
274C2×C4⋊C4—121no—
275C2×C4⋊C4C22×D311yescharacteristic
276C2×C4⋊C4C22×D311yescharacteristic
277C2×C4⋊C4C22×D311yescharacteristic
278C2×C4⋊C4C22×D311yescharacteristic
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—31no—
282C2×C22⋊C4—31no—
283C2×C22⋊C4—61no—
284C2×C42—21no—
285C2×C42—31no—
286C2×C42—31no—
287C2×C42—61no—
288C2×C42—61no—
289#107—62no—
290#107—62no—
291#107—121no—
292#106—121no—
293C4⋊C8—62no—
294C4⋊C8—121no—
295C22⋊C8—22no—
296#95—21no—
297#95—31no—
298#95—31no—
299#95—61no—
300#95—61no—
301#95—61no—
302#95—61no—
303#95—61no—
304#95—61no—
305#95—61no—
306#95—61no—
307#95—61no—
308C23×C3—21no—
309C23×C3—21no—
310C23×C3—21no—
311C23×C3—21no—
312C23×C3—21no—
313C23×C3—21no—
314C23×C3C22⋊Q812yes—
315C23×C3C22×D411yescharacteristic
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4C22⋊Q812yes—
335C2×C3×C4C22×Q811yescharacteristic
336C2×C3×C4C2×#4111yescharacteristic
337C2×C3×C4C2×#4111yescharacteristic
338C2×C3×C4C2×#4111yescharacteristic
339C2×C3×C4H(5,𝔽2)11yescharacteristic
340C2×C3×C42−1+411yescharacteristic
341C2×Dic3—21no—
342C2×Dic3—21no—
343C2×Dic3—22no—
344C2×Dic3—21no—
345C2×Dic3—21no—
346C2×Dic3—21no—
347C2×Dic3—21no—
348C2×Dic3—21no—
349C2×Dic3—21no—
350C2×Dic3—21no—
351C2×Dic3—21no—
352C2×Dic3—21no—
353C2×Dic3—21no—
354C2×Dic3—41no—
355C2×Dic3—41no—
356C2×Dic3—41no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3—84no—
366C3×C8—41no—
367C3×C8—41no—
368C3⋊C8—22no—
369C3⋊C8—41no—
370C24D3×D411yescharacteristic
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—31no—
380C22×C4—31no—
381C22×C4—31no—
382C22×C4—31no—
383C22×C4—61no—
384C22×C4—61no—
385C22×C4—61no—
386C22×C4—61no—
387C22×C4—122no—
388C22×C4—122no—
389C22×C4—241no—
390C22×C4C2×Dic612yes—
391C22×C4D3×D411yescharacteristic
392C22×C4#23712yes—
393C22×C4C23×D311yescharacteristic
394M16—124no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—32no—
402C2×C8—61no—
403C2×C8—61no—
404C2×C8—61no—
405C2×C8—61no—
406C2×C8—61no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—32no—
414C4⋊C4—61no—
415C4⋊C4—61no—
416C4⋊C4—61no—
417C4⋊C4—61no—
418C4⋊C4—61no—
419C4⋊C4—61no—
420C4⋊C4—61no—
421C4⋊C4—122no—
422C4⋊C4—122no—
423C4⋊C4—122no—
424C4⋊C4—241no—
425C4⋊C4—241no—
426C4⋊C4—241no—
427C4⋊C4—241no—
428C4⋊C4C2×Dic612yes—
429C4⋊C4C2×Dic612yes—
430C22⋊C4—24no—
431C22⋊C4—42no—
432C22⋊C4—121no—
433C22⋊C4—121no—
434C22⋊C4—241no—
435C42—24no—
436C42—61no—
437C42—61no—
438C42—61no—
439C42—61no—
440C42—122no—
441C42—122no—
442C22×C3—22no—
443C22×C3—22no—
444C22×C3—22no—
445C22×C3—41no—
446C22×C3—41no—
447C22×C3—41no—
448C22×C3—41no—
449C22×C3C2×C22⋊Q811yescharacteristic
450C22×C3#54611yescharacteristic
451C22×C3#54711yescharacteristic
452C22×C3#57511yescharacteristic
453C22×C3#57511yescharacteristic
454C22×C3#57611yescharacteristic
455C22×C3#57811yescharacteristic
456C3×C4—22no—
457C3×C4—42no—
458C3×C4—42no—
459C3×C4—81no—
460C3×C4—81no—
461C3×C4#53311yescharacteristic
462C3×C4#54411yescharacteristic
463C3×C4#55111yescharacteristic
464C3×C4#55211yescharacteristic
465Dic3—41no—
466Dic3—41no—
467Dic3—41no—
468Dic3—42no—
469Dic3—41no—
470Dic3—81no—
471Dic3—81no—
472Dic3—81no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23#87812yes—
480C23C2×D3×D411yescharacteristic
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—61no—
500C2×C4—61no—
501C2×C4—62no—
502C2×C4—61no—
503C2×C4—61no—
504C2×C4—61no—
505C2×C4—61no—
506C2×C4—61no—
507C2×C4—61no—
508C2×C4—61no—
509C2×C4—61no—
510C2×C4—61no—
511C2×C4—61no—
512C2×C4—121no—
513C2×C4—121no—
514C2×C4—121no—
515C2×C4—121no—
516C2×C4—121no—
517C2×C4—121no—
518C2×C4—121no—
519C2×C4—121no—
520C2×C4—121no—
521C2×C4—121no—
522C2×C4—121no—
523C2×C4—244no—
524C2×C4#87812yes—
525C2×C4C22×Dic611yescharacteristic
526C2×C4C2×#23711yescharacteristic
527C2×C4D3×#4111yescharacteristic
528C2×C4D3×#4111yescharacteristic
529C2×C4#100911yescharacteristic
530C2×C4#101011yescharacteristic
531C8—41no—
532C8—41no—
533C8—62no—
534C8—121no—
535C2×C3—42no—
536C2×C3#243811yescharacteristic
537C2×C3#308011yescharacteristic
538C2×C3#308311yescharacteristic
539C2×C3#331911yescharacteristic
540C2×C3#332711yescharacteristic
541C2×C3#334711yescharacteristic
542C2×C3#335211yescharacteristic
543C22—22no—
544C22—22no—
545C22—22no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22C2×#87811yescharacteristic
551C22#565411yescharacteristic
552C22#565511yescharacteristic
553C22#576911yescharacteristic
554C22#577111yescharacteristic
555C22#577311yescharacteristic
556C22#577711yescharacteristic
557C4—22no—
558C4—42no—
559C4—42no—
560C4—81no—
561C4—81no—
562C4—121no—
563C4—121no—
564C4—121no—
565C4—122no—
566C4—121no—
567C4—241no—
568C4—241no—
569C4—241no—
570C4#551711yescharacteristic
571C4#551911yescharacteristic
572C4#565011yescharacteristic
573C4#565811yescharacteristic
574C3#2601711yescharacteristic, complemented, Sylow
575C2—42no—
576C2#7406711yescharacteristic
577C2#7882311yescharacteristic
578C2#7887111yescharacteristic
579C2#8139311yescharacteristic
580C2#8141111yescharacteristic
581C2#8151111yescharacteristic
582C2#8152811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1825C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1825D3
hypocenterC3#26017
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1825

Complements

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

NH ≅ G/Ntype
8. #68845C2semidirect
9. #68838C2semidirect
10. #68829C2semidirect
11. #68821C2semidirect
12. #68820C2semidirect
13. #68807C2semidirect
14. #68801C2semidirect
15. #68798C2semidirect
574. C3#26017semidirect

Maximal subgroups

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

subgroupindex
#822392
#739012
C3×#18252
#715812
#699512
#699272
#695132
#688452
#688382
#688292
#688212
#688202
#688072
#688012
#687982
#260173