Finite Group Databasebeta

← #10945116

Subgroups — #10945116

Order 768.

854 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5182C211yesmaximal, characteristic
2C4×Q8×Dic3C211yesmaximal, characteristic
3C2×#4793C211yesmaximal, characteristic
4#76607C211yesmaximal, characteristic
5C3×#1878C211yesmaximal, characteristic
6#72065C211yesmaximal, characteristic
7#71632C211yesmaximal, characteristic
8#70239C211yesmaximal, characteristic
9#70229C211yesmaximal, characteristic
10#70224C211yesmaximal, characteristic
11#69667C211yesmaximal, characteristic
12#68799C211yesmaximal, characteristic
13#68794C211yesmaximal, characteristic
14#68628C211yesmaximal, characteristic
15#68605C211yesmaximal, characteristic
16#33027—31nomaximal, Sylow
17C22×C3×Q16C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C2×Q8×Dic3C2211yescharacteristic
21C2×#944C2211yescharacteristic
22C22×#216C2211yescharacteristic
23C2×#888C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C3×#391C2211yescharacteristic
27#5229C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#835C2211yescharacteristic
31#5182—24no—
32C2×#818C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×#816C2211yescharacteristic
35C2×Dic3⋊C8C2211yescharacteristic
36C2×Dic3⋊C8C2211yescharacteristic
37Dic3×C4⋊C4—21no—
38Dic3×C4⋊C4—21no—
39Dic3×C4⋊C4C2211yescharacteristic
40Dic3×C4⋊C4C2211yescharacteristic
41#4973C2211yescharacteristic
42C2×#810C2211yescharacteristic
43C2×#807C2211yescharacteristic
44C4×C4⋊Dic3—21no—
45C4×C4⋊Dic3C2211yescharacteristic
46C42×Dic3—21no—
47C42×Dic3C2211yescharacteristic
48#4929C2211yescharacteristic
49#4793—24no—
50#4650—21no—
51#4650—21no—
52#4646—21no—
53#4646—21no—
54C3×#340C2211yescharacteristic
55C3×#336C2211yescharacteristic
56#4551C2211yescharacteristic
57#4550C2211yescharacteristic
58#4550C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#477—31no—
63C2×#462—31no—
64#2862—31no—
65C42×Q8—31no—
66#1987—31no—
67#1981—31no—
68#1928—31no—
69#1928—31no—
70#1928—31no—
71#1928—31no—
72#1878—31no—
73#1878—31no—
74#1878—31no—
75#1878D311yescharacteristic
76#1831—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×Q8C2311yescharacteristic
79C22×Dic6—21no—
80C22×Dic6C2311yescharacteristic
81C2×C3×Q16—22no—
82C2×C3×Q16—22no—
83C2×C3×Q16—22no—
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×C4×Q8—22no—
87C3×C4×Q8—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C42—21no—
92C2×C3×C42C2311yescharacteristic
93Q8×Dic3—22no—
94Q8×Dic3—22no—
95Q8×Dic3—22no—
96Q8×Dic3D414yes—
97#944—22no—
98#944—22no—
99C2×#216—22no—
100C2×#216—22no—
101C2×#216—22no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—21no—
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×Dic3—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3D411yescharacteristic
116C2×C4×Dic3D411yescharacteristic
117C2×C4×Dic3D411yescharacteristic
118C2×C4×Dic3D411yescharacteristic
119C2×C4×Dic3C2311yescharacteristic
120C2×C4×Dic3C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123#888—22no—
124#888—22no—
125C3×#103—22no—
126C3×#103—22no—
127C3×#95—21no—
128C3×#95—21no—
129#835—22no—
130#835—22no—
131#835—22no—
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—22no—
142#816—22no—
143#816—22no—
144Dic3⋊C8—22no—
145Dic3⋊C8—22no—
146#810—22no—
147#807—22no—
148C22×Q16—31no—
149C22×Q16C2×D311yescharacteristic
150C2×C4⋊Q8—31no—
151C2×C4⋊Q8—31no—
152C2×C4×Q8—31no—
153C2×C4×Q8—31no—
154C2×C4×Q8C2×D311yescharacteristic
155#477—64no—
156#462—64no—
157C2×#107—31no—
158C2×#107—31no—
159C2×C4⋊C8—31no—
160C2×C4⋊C8—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×#103—31no—
167C2×#103C2×D311yescharacteristic
168C2×#103C2×D311yescharacteristic
169#391—31no—
170#391—31no—
171#391—31no—
172#391—61no—
173#391—61no—
174#391C2×D311yescharacteristic
175#382—61no—
176#382—61no—
177C4×C4⋊C4—31no—
178C4×C4⋊C4—31no—
179C4×C4⋊C4—31no—
180C4×C4⋊C4—61no—
181C4×C4⋊C4—61no—
182C4×C4⋊C4—61no—
183C43—31no—
184C43—61no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340C2×D311yescharacteristic
189#336—31no—
190#336—31no—
191#336—31no—
192#336C2×D311yescharacteristic
193C2×C3×Q8—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×Q8C2×D412yes—
203C2×C3×Q8#4112yes—
204C2×C3×Q8#4112yes—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2411yescharacteristic
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3C2×D411yescharacteristic
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—42no—
227C2×Dic6—42no—
228C2×Dic6—42no—
229C3×Q16—44no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C3×C4⋊C4—24no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4#4112yes—
241C3×C42—24no—
242C3×C42—21no—
243C3×C42—21no—
244#216—44no—
245C4⋊Dic3—24no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—24no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3C2×D412yes—
252C4⋊Dic3#4112yes—
253Dic3⋊C4—42no—
254Dic3⋊C4—42no—
255Dic3⋊C4—42no—
256C4×Dic3—24no—
257C4×Dic3—21no—
258C4×Dic3—21no—
259C4×Dic3—24no—
260C4×Dic3—22no—
261C4×Dic3—41no—
262C4×Dic3—41no—
263C4×Dic3—42no—
264C4×Dic3—42no—
265C4×Dic3—42no—
266C4×Dic3Q1614yescomplemented
267C4×Dic3C2×D412yes—
268C4×Dic3#4112yes—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C22×Q8—31no—
276C22×Q8—61no—
277C22×Q8C22×D311yescharacteristic
278C22×Q8C22×D311yescharacteristic
279C2×Q16—22no—
280C2×Q16—22no—
281C2×Q16—22no—
282C2×Q16—62no—
283C2×Q16—62no—
284C2×Q16—62no—
285C22×C8—31no—
286C22×C8—31no—
287C22×C8C22×D311yescharacteristic
288C22×C8C22×D311yescharacteristic
289C4⋊Q8—62no—
290C4⋊Q8—62no—
291C4⋊Q8—62no—
292C4⋊Q8—62no—
293C4×Q8—22no—
294C4×Q8—22no—
295C4×Q8—34no—
296C4×Q8—62no—
297C4×Q8—62no—
298C4×Q8—62no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4C22×D311yescharacteristic
309C2×C4⋊C4C22×D311yescharacteristic
310C2×C42—21no—
311C2×C42—31no—
312C2×C42—31no—
313C2×C42—31no—
314C2×C42—31no—
315C2×C42—31no—
316C2×C42—31no—
317C2×C42—61no—
318C2×C42—61no—
319C2×C42—61no—
320C2×C42—61no—
321C2×C42—61no—
322C2×C42C22×D311yescharacteristic
323#107—62no—
324#107—62no—
325C4⋊C8—62no—
326C4⋊C8—62no—
327#103—22no—
328#103—22no—
329#103—62no—
330#103—62no—
331#103—62no—
332#103—62no—
333#103—62no—
334#103—62no—
335#95—21no—
336#95—21no—
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×C3C22×D411yescharacteristic
347C3×Q8—22no—
348C3×Q8—22no—
349C3×Q8—22no—
350C3×Q8—41no—
351C3×Q8—41no—
352C3×Q8—41no—
353C3×Q8—41no—
354C3×Q8#12414yes—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—21no—
358C2×C3×C4—22no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—22no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4C2×Q1612yes—
373C2×C3×C4C22×D411yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C2×C3×C4C2×#4111yescharacteristic
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—22no—
384C2×Dic3—21no—
385C2×Dic3—22no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3#12112yes—
399C2×Dic3#12112yes—
400C2×Dic3C2×Q1612yes—
401C2×Dic3C2×Q1612yes—
402C2×Dic3#13712yes—
403Dic6—41no—
404Dic6—41no—
405Dic6—41no—
406Dic6—41no—
407Dic6—84no—
408C3×C8—41no—
409C3×C8—41no—
410C3⋊C8—41no—
411C3⋊C8—41no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—61no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×Q8—61no—
426C2×Q8—61no—
427C2×Q8—122no—
428C2×Q8—122no—
429C2×Q8—122no—
430C2×Q8#23512yes—
431C2×Q8D3×D412yes—
432C2×Q8#23712yes—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—31no—
438C22×C4—31no—
439C22×C4—31no—
440C22×C4—31no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4D3×D411yescharacteristic
446C22×C4D3×D411yescharacteristic
447C22×C4C23×D311yescharacteristic
448Q16—44no—
449Q16—124no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—61no—
457C2×C8—61no—
458C2×C8—61no—
459C2×C8—61no—
460C2×C8—61no—
461C2×C8—61no—
462C4⋊C4—24no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—32no—
467C4⋊C4—32no—
468C4⋊C4—64no—
469C4⋊C4—64no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—122no—
475C4⋊C4—122no—
476C4⋊C4—122no—
477C4⋊C4#23712yes—
478C42—24no—
479C42—21no—
480C42—21no—
481C42—34no—
482C42—32no—
483C42—32no—
484C42—62no—
485C42—64no—
486C42—64no—
487C42—61no—
488C42—61no—
489C42—121no—
490C42—121no—
491C42—122no—
492C42—122no—
493C42—122no—
494C22×C3C2×#12111yescharacteristic
495C22×C3#52511yescharacteristic
496C22×C3#52511yescharacteristic
497C22×C3C22×Q1611yescharacteristic
498C22×C3C2×#13711yescharacteristic
499C22×C3#57511yescharacteristic
500C22×C3#57511yescharacteristic
501C3×C4—22no—
502C3×C4—24no—
503C3×C4—41no—
504C3×C4—42no—
505C3×C4—81no—
506C3×C4—81no—
507C3×C4C2×#12411yescharacteristic
508C3×C4#52911yescharacteristic
509C3×C4#52911yescharacteristic
510C3×C4#53211yescharacteristic
511Dic3—22no—
512Dic3—22no—
513Dic3—41no—
514Dic3—42no—
515Dic3—42no—
516Dic3—81no—
517Dic3—81no—
518Dic3#46214yes—
519C23C2×D3×D411yescharacteristic
520Q8—22no—
521Q8—22no—
522Q8—22no—
523Q8—41no—
524Q8—41no—
525Q8—41no—
526Q8—41no—
527Q8—121no—
528Q8—121no—
529Q8—121no—
530Q8—121no—
531Q8—244no—
532Q8#88514yes—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—32no—
543C2×C4—32no—
544C2×C4—32no—
545C2×C4—32no—
546C2×C4—32no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—61no—
556C2×C4—61no—
557C2×C4—61no—
558C2×C4—61no—
559C2×C4—61no—
560C2×C4—61no—
561C2×C4—62no—
562C2×C4—61no—
563C2×C4—62no—
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×C4D3×Q1612yes—
575C2×C4C2×#23511yescharacteristic
576C2×C4C2×D3×D411yescharacteristic
577C2×C4C2×#23711yescharacteristic
578C2×C4C2×#23711yescharacteristic
579C2×C4D3×#4111yescharacteristic
580C2×C4D3×#4111yescharacteristic
581C8—41no—
582C8—41no—
583C8—121no—
584C8—121no—
585C2×C3#285011yescharacteristic
586C2×C3C2×#46211yescharacteristic
587C2×C3#304011yescharacteristic
588C2×C3#310411yescharacteristic
589C2×C3#310411yescharacteristic
590C2×C3#311611yescharacteristic
591C2×C3#311611yescharacteristic
592C22#554711yescharacteristic
593C22#558611yescharacteristic
594C22D3×#12111yescharacteristic
595C22C2×D3×Q1611yescharacteristic
596C22#576811yescharacteristic
597C22D3×#13711yescharacteristic
598C22#577711yescharacteristic
599C4—22no—
600C4—24no—
601C4—34no—
602C4—41no—
603C4—42no—
604C4—62no—
605C4—62no—
606C4—81no—
607C4—81no—
608C4—122no—
609C4—121no—
610C4—122no—
611C4—241no—
612C4—241no—
613C4C2×#88511yescharacteristic
614C4#553811yescharacteristic
615C4#559611yescharacteristic
616C4#564911yescharacteristic
617C3#3302711yescharacteristic, complemented, Sylow
618C2#7486911yescharacteristic
619C2#7972211yescharacteristic
620C2#7984811yescharacteristic
621C2#8017211yescharacteristic
622C2#8021511yescharacteristic
623C2D3×#46211yescharacteristic
624C2#8059911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1878C2
socle, prime radicalC23×C3C22×D4
hypercenter#1878D3
hypocenterC3#33027
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1878

Complements

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

NH ≅ G/Ntype
266. C4×Dic3Q16semidirect
617. C3#33027semidirect

Maximal subgroups

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

subgroupindex
C2×#51822
C4×Q8×Dic32
C2×#47932
#766072
C3×#18782
#720652
#716322
#702392
#702292
#702242
#696672
#687992
#687942
#686282
#686052
#330273