Finite Group Databasebeta

← #10858875

Subgroups — #10858875

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#82243C211yesmaximal, characteristic
2#73900C211yesmaximal, characteristic
3C3×#1854C211yesmaximal, characteristic
4#71580C211yesmaximal, characteristic
5#70643C211yesmaximal, characteristic, complemented
6#70621C211yesmaximal, characteristic, complemented
7#70588C211yesmaximal, characteristic, complemented
8#70557C211yesmaximal, characteristic, complemented
9#69950C211yesmaximal, characteristic
10#69931C211yesmaximal, characteristic
11#69514C211yesmaximal, characteristic
12#68911C211yesmaximal, characteristic, complemented
13#68904C211yesmaximal, characteristic, complemented
14#68811C211yesmaximal, characteristic, complemented
15#68801C211yesmaximal, characteristic, complemented
16#26417—31nomaximal, Sylow
17C2×C3×#117C2211yescharacteristic
18C2×#926C2211yescharacteristic
19C2×#924C2211yescharacteristic
20C23×C3⋊C8C2211yescharacteristic
21C2×#890C2211yescharacteristic
22C2×#887C2211yescharacteristic
23C2×C3×#107C2211yescharacteristic
24C2×C3×#106C2211yescharacteristic
25C2×C3×C22⋊C8C2211yescharacteristic
26C3×#380C2211yescharacteristic
27C4×#217C2211yescharacteristic
28C2×#816C2211yescharacteristic
29C2×#816C2211yescharacteristic
30C2×Dic3⋊C8C2211yescharacteristic
31C2×Dic3⋊C8C2211yescharacteristic
32#4984C2211yescharacteristic
33#4975C2211yescharacteristic
34Dic3×C4⋊C4C2211yescharacteristic
35#4973C2211yescharacteristic
36#4972C2211yescharacteristic
37#4972C2211yescharacteristic
38C2×#810C2211yescharacteristic
39C2×#810C2211yescharacteristic
40#4949—21no—
41#4948—21no—
42#4945—21no—
43#4944—21no—
44#4933—21no—
45C4×Dic3⋊C4—21no—
46#4655—21no—
47#4647—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#1873—31no—
70#1865—31no—
71#1854D311yescharacteristic
72#1850—31no—
73#1803—31no—
74#1735—31no—
75C23×C3×C4C2311yescharacteristic
76C22×Dic6C2311yescharacteristic
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C3×#117—22no—
80C3×#117—41no—
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×#217—21no—
90C2×#217D411yescharacteristic
91C2×#217D411yescharacteristic
92#926—22no—
93#926—41no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99#924—22no—
100#924Q814yes—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3C2311yescharacteristic
112C2×C4×Dic3C2311yescharacteristic
113C22×C3⋊C8—22no—
114C22×C3⋊C8—41no—
115C22×C3⋊C8—41no—
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118#890—22no—
119#890—41no—
120#887—22no—
121#887—41no—
122C3×#107—41no—
123C3×#106—22no—
124C3×C22⋊C8—22no—
125C3×#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95D411yescharacteristic
133C3⋊#95D411yescharacteristic
134#816—22no—
135#816—41no—
136Dic3⋊C8—41no—
137Dic3⋊C8—41no—
138#810—41no—
139#810—41no—
140C23×C8—31no—
141C2×#125—31no—
142C2×C22⋊Q8—31no—
143C2×C4×Q8—31no—
144C2×#117—31no—
145C2×#117C2×D311yescharacteristic
146C2×#107C2×D311yescharacteristic
147C2×#106C2×D311yescharacteristic
148C2×C4⋊C8—31no—
149C2×C4⋊C8—31no—
150C2×#103—31no—
151C2×#103—31no—
152C2×#103—31no—
153C2×#103—31no—
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#380C2×D311yescharacteristic
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
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×Q812yes—
194C22×C3×C4#4112yes—
195C22×C3×C4C2411yescharacteristic
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3C2×D411yescharacteristic
201C22×Dic3C2×D411yescharacteristic
202C22×Dic3C2×D411yescharacteristic
203C22×Dic3C2×D411yescharacteristic
204C2×Dic6—21no—
205C2×Dic6—21no—
206C2×Dic6—21no—
207C2×Dic6—21no—
208C2×Dic6—21no—
209C2×Dic6C2×Q812yes—
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—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4#4112yes—
224C3×C22⋊C4—42no—
225C3×C22⋊C4—81no—
226C3×C42—42no—
227#217—24no—
228#217—22no—
229#217—41no—
230#217—42no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—42no—
236C4⋊Dic3—81no—
237C4⋊Dic3C2×Q812yes—
238C4⋊Dic3#4112yes—
239Dic3⋊C4—24no—
240Dic3⋊C4—42no—
241Dic3⋊C4—42no—
242Dic3⋊C4—42no—
243Dic3⋊C4—42no—
244Dic3⋊C4—42no—
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×Q8—31no—
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—34no—
272C22⋊Q8—62no—
273C4×Q8—62no—
274C4×Q8—121no—
275#117—22no—
276#117—41no—
277#117—62no—
278#117—121no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4—61no—
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⋊C4C22×D311yescharacteristic
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—41no—
308#106—22no—
309C4⋊C8—121no—
310C4⋊C8—121no—
311#103—62no—
312#103—121no—
313#103—121no—
314#103—121no—
315C22⋊C8—22no—
316#95—21no—
317#95—31no—
318#95—31no—
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
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4C22⋊Q812yes—
352C2×C3×C4C22×Q811yescharacteristic
353C2×C3×C4C2×#4111yescharacteristic
354C2×C3×C4C2×#4111yescharacteristic
355C2×C3×C4C2×#4111yescharacteristic
356C2×C3×C4H(5,𝔽2)11yescharacteristic
357C2×C3×C4H(5,𝔽2)11yescharacteristic
358C2×Dic3—21no—
359C2×Dic3—21no—
360C2×Dic3—22no—
361C2×Dic3—22no—
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—22no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3#13512yes—
382Dic6—22no—
383Dic6—22no—
384Dic6—41no—
385Dic6—41no—
386C3×C8—41no—
387C3×C8—41no—
388C3⋊C8—41no—
389C3⋊C8—41no—
390C24D3×D411yescharacteristic
391C2×Q8—32no—
392C2×Q8—61no—
393C2×Q8—61no—
394C2×Q8—61no—
395C2×Q8—61no—
396C2×Q8—61no—
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×C4#23512yes—
414C22×C4D3×D411yescharacteristic
415C22×C4D3×Q812yes—
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—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—32no—
438C4⋊C4—32no—
439C4⋊C4—64no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—122no—
445C4⋊C4—122no—
446C4⋊C4—122no—
447C4⋊C4—122no—
448C4⋊C4—122no—
449C4⋊C4—122no—
450C4⋊C4—241no—
451C4⋊C4—241no—
452C4⋊C4#23712yes—
453C22⋊C4—42no—
454C22⋊C4—62no—
455C22⋊C4—64no—
456C22⋊C4—81no—
457C22⋊C4—121no—
458C22⋊C4—122no—
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—81no—
484C3×C4—81no—
485C3×C4—81no—
486C3×C4#52911yescharacteristic
487C3×C4#53811yescharacteristic
488C3×C4#55411yescharacteristic
489C3×C4#55411yescharacteristic
490Dic3—22no—
491Dic3—42no—
492Dic3—42no—
493Dic3—41no—
494Dic3—42no—
495Dic3—41no—
496Dic3—41no—
497Dic3—81no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23#89612yes—
505C23C2×D3×D411yescharacteristic
506Q8—62no—
507Q8—62no—
508Q8—121no—
509Q8—121no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—32no—
518C2×C4—41no—
519C2×C4—41no—
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—61no—
533C2×C4—62no—
534C2×C4—62no—
535C2×C4—62no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—61no—
540C2×C4—61no—
541C2×C4—61no—
542C2×C4—61no—
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#89612yes—
553C2×C4C2×#23511yescharacteristic
554C2×C4C2×#23711yescharacteristic
555C2×C4#100411yescharacteristic
556C2×C4C2×D3×Q811yescharacteristic
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×#89611yescharacteristic
579C22#559111yescharacteristic
580C22#565411yescharacteristic
581C22D3×#13511yescharacteristic
582C22#576811yescharacteristic
583C22#577511yescharacteristic
584C22#577911yescharacteristic
585C4—22no—
586C4—42no—
587C4—62no—
588C4—81no—
589C4—81no—
590C4—81no—
591C4—121no—
592C4—122no—
593C4—121no—
594C4—121no—
595C4—122no—
596C4—122no—
597C4—241no—
598C4#553811yescharacteristic
599C4#556011yescharacteristic
600C4#563511yescharacteristic
601C4#563611yescharacteristic
602C3#2641711yescharacteristic, complemented, Sylow
603C2—42no—
604C2#7442311yescharacteristic
605C2#7919611yescharacteristic
606C2#7924611yescharacteristic
607C2#8010311yescharacteristic
608C2#8025311yescharacteristic
609C2#8140911yescharacteristic
610C2#8154411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1854C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1854D3
hypocenterC3#26417
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
5. #70643C2semidirect
6. #70621C2semidirect
7. #70588C2semidirect
8. #70557C2semidirect
12. #68911C2semidirect
13. #68904C2semidirect
14. #68811C2semidirect
15. #68801C2semidirect
602. C3#26417semidirect

Maximal subgroups

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

subgroupindex
#822432
#739002
C3×#18542
#715802
#706432
#706212
#705882
#705572
#699502
#699312
#695142
#689112
#689042
#688112
#688012
#264173