Finite Group Databasebeta

← #10824363

Subgroups — #10824363

Order 768.

848 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Dic3×Q16C211yesmaximal, characteristic
2C2×#4790C211yesmaximal, characteristic
3#76481C211yesmaximal, characteristic
4#76477C211yesmaximal, characteristic
5C3×#2056C211yesmaximal, characteristic
6#72073C211yesmaximal, characteristic
7#70489C211yesmaximal, characteristic
8Dic3×C4⋊C8C211yesmaximal, characteristic
9#70284C211yesmaximal, characteristic
10#70233C211yesmaximal, characteristic
11Dic3×#103C211yesmaximal, characteristic
12#69683C211yesmaximal, characteristic
13#69681C211yesmaximal, characteristic
14#68834C211yesmaximal, characteristic
15#68781C211yesmaximal, characteristic
16#23679—31nomaximal, Sylow
17C22×C3×Q16C2211yescharacteristic
18C2×Q8×Dic3C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C2×#944C2211yescharacteristic
21C22×#216C2211yescharacteristic
22C2×#887C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C3×#391C2211yescharacteristic
27C3×#391C2211yescharacteristic
28#5229C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#835C2211yescharacteristic
32Dic3×Q16—24no—
33C2×#818C2211yescharacteristic
34C2×#816C2211yescharacteristic
35C2×#816C2211yescharacteristic
36C2×C8×Dic3C2211yescharacteristic
37C2×C8×Dic3C2211yescharacteristic
38#4975C2211yescharacteristic
39Dic3×C4⋊C4C2211yescharacteristic
40#4972C2211yescharacteristic
41C2×#810C2211yescharacteristic
42C2×#807C2211yescharacteristic
43C42×Dic3C2211yescharacteristic
44#4929C2211yescharacteristic
45#4928C2211yescharacteristic
46C2×C4⋊(C3⋊C8)C2211yescharacteristic
47#4790—24no—
48#4654—21no—
49#4653—21no—
50#4649—21no—
51#4649—21no—
52#4649—21no—
53#4648—21no—
54#4647—21no—
55#4647—21no—
56C3×#340C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×C4×Q16—31no—
63C2×C4×Q16—31no—
64C4×C4⋊Q8—31no—
65#2295—31no—
66#2083—31no—
67#2072—31no—
68#2068—31no—
69#2068—31no—
70#2056—31no—
71#2056D311yescharacteristic
72#2043—31no—
73#2016—31no—
74C4×C4⋊C8—31no—
75C4×#103—31no—
76C4×#103—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×Q8C2311yescharacteristic
79C22×Dic6C2311yescharacteristic
80C2×C3×Q16—22no—
81C2×C3×Q16—22no—
82C2×C3×Q16—22no—
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C42C2311yescharacteristic
88Q8×Dic3—22no—
89Q8×Dic3—22no—
90Q8×Dic3—22no—
91Q8×Dic3D414yes—
92#944—22no—
93#944—22no—
94C2×#216—22no—
95C2×#216—22no—
96C2×#216—22no—
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
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×Dic3⋊C4—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3D411yescharacteristic
114C2×C4×Dic3D411yescharacteristic
115C2×C4×Dic3D411yescharacteristic
116C2×C4×Dic3D411yescharacteristic
117C2×C4×Dic3C2311yescharacteristic
118C2×C4×Dic3C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121#887—22no—
122#887—22no—
123C3×C4⋊C8—22no—
124C3×#103—22no—
125C3×#103—22no—
126C3×#95—21no—
127C3×#95—21no—
128C3×#95—21no—
129C3×#95—21no—
130#835—22no—
131#835—22no—
132#835—22no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140#818—22no—
141#816—22no—
142#816—22no—
143C8×Dic3—22no—
144C8×Dic3—22no—
145#810—22no—
146#807—22no—
147C4⋊(C3⋊C8)—22no—
148C22×Q16—31no—
149C22×Q16C2×D311yescharacteristic
150C2×C4⋊Q8—31no—
151C2×C4×Q8—31no—
152C2×C4×Q8—31no—
153C2×C4×Q8—31no—
154C4×Q16—64no—
155C4×Q16—64no—
156C2×#107—31no—
157C2×#107—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8C2×D311yescharacteristic
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×#103C2×D311yescharacteristic
167C2×#103C2×D311yescharacteristic
168C2×C4×C8—31no—
169C2×C4×C8—31no—
170#391—31no—
171#391—31no—
172#391C2×D311yescharacteristic
173#391C2×D311yescharacteristic
174#389—31no—
175#389—31no—
176#389—61no—
177#389—61no—
178#387—61no—
179#387—61no—
180#384—31no—
181C4×C4⋊C4—31no—
182C4×C4⋊C4—61no—
183C4×C4⋊C4—61no—
184C4×C4⋊C4—61no—
185#376—61no—
186C43—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340C2×D311yescharacteristic
191#336—31no—
192#336—31no—
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×Q8—21no—
203C2×C3×Q8C2×D412yes—
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#4112yes—
226C3×Q16—28no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8#4112yes—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C42—21no—
238C3×C42—21no—
239#216—28no—
240C4⋊Dic3—21no—
241C4⋊Dic3—24no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—42no—
247C4⋊Dic3C2×D412yes—
248Dic3⋊C4—24no—
249Dic3⋊C4—24no—
250Dic3⋊C4—42no—
251Dic3⋊C4—42no—
252Dic3⋊C4—42no—
253Dic3⋊C4—42no—
254Dic3⋊C4—42no—
255C4×Dic3—21no—
256C4×Dic3—21no—
257C4×Dic3—24no—
258C4×Dic3—22no—
259C4×Dic3—22no—
260C4×Dic3—41no—
261C4×Dic3—42no—
262C4×Dic3—42no—
263C4×Dic3—42no—
264C4×Dic3Q1614yescomplemented
265C4×Dic3C2×D412yes—
266C4×Dic3#4112yes—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8#4112yes—
273C22×Q8—31no—
274C22×Q8C22×D311yescharacteristic
275C22×Q8C22×D311yescharacteristic
276C2×Q16—22no—
277C2×Q16—22no—
278C2×Q16—22no—
279C2×Q16—62no—
280C2×Q16—62no—
281C2×Q16—62no—
282C22×C8—31no—
283C22×C8—31no—
284C22×C8C22×D311yescharacteristic
285C22×C8C22×D311yescharacteristic
286C4⋊Q8—62no—
287C4⋊Q8—62no—
288C4×Q8—34no—
289C4×Q8—62no—
290C4×Q8—62no—
291C4×Q8—62no—
292C4×Q8—62no—
293C4×Q8—62no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C4⋊C4C22×D311yescharacteristic
308C2×C42—31no—
309C2×C42—31no—
310C2×C42—31no—
311C2×C42—31no—
312C2×C42—31no—
313C2×C42—31no—
314C2×C42—61no—
315C2×C42—61no—
316C2×C42—61no—
317C2×C42—61no—
318C2×C42C22×D311yescharacteristic
319#107—62no—
320#107—62no—
321C4⋊C8—22no—
322C4⋊C8—62no—
323#103—22no—
324#103—22no—
325#103—62no—
326#103—62no—
327#103—62no—
328#103—62no—
329#103—62no—
330#103—62no—
331C4×C8—62no—
332C4×C8—62no—
333#95—21no—
334#95—21no—
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—
344C23×C3C22×D411yescharacteristic
345C3×Q8—22no—
346C3×Q8—22no—
347C3×Q8—22no—
348C3×Q8—22no—
349C3×Q8—41no—
350C3×Q8—41no—
351C3×Q8—41no—
352C3×Q8—41no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
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×C4C2×Q1612yes—
371C2×C3×C4C22×D411yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—22no—
384C2×Dic3—22no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3#12112yes—
396C2×Dic3#12112yes—
397C2×Dic3C2×Q1612yes—
398C2×Dic3C2×Q1612yes—
399C2×Dic3#13512yes—
400C2×Dic3#13712yes—
401Dic6—22no—
402Dic6—22no—
403Dic6—41no—
404Dic6—41no—
405C3×C8—22no—
406C3×C8—41no—
407C3⋊C8—22no—
408C3⋊C8—41no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—32no—
420C2×Q8—61no—
421C2×Q8—61no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×Q8D3×D412yes—
426C2×Q8#23712yes—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—31no—
432C22×C4—31no—
433C22×C4—31no—
434C22×C4—31no—
435C22×C4—61no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4D3×D411yescharacteristic
440C22×C4D3×D411yescharacteristic
441C22×C4C23×D311yescharacteristic
442Q16—28no—
443Q16—68no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—32no—
450C2×C8—61no—
451C2×C8—61no—
452C2×C8—61no—
453C2×C8—61no—
454C2×C8—61no—
455C2×C8#23712yes—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—32no—
461C4⋊C4—64no—
462C4⋊C4—64no—
463C4⋊C4—64no—
464C4⋊C4—61no—
465C4⋊C4—61no—
466C4⋊C4—61no—
467C4⋊C4—61no—
468C4⋊C4—61no—
469C4⋊C4—122no—
470C4⋊C4—122no—
471C4⋊C4—122no—
472C4⋊C4—122no—
473C4⋊C4—122no—
474C4⋊C4—122no—
475C42—21no—
476C42—21no—
477C42—34no—
478C42—32no—
479C42—32no—
480C42—62no—
481C42—62no—
482C42—64no—
483C42—61no—
484C42—61no—
485C42—121no—
486C42—122no—
487C42—122no—
488C42—122no—
489C22×C3C2×#12111yescharacteristic
490C22×C3#52511yescharacteristic
491C22×C3#53211yescharacteristic
492C22×C3C22×Q1611yescharacteristic
493C22×C3C2×#13511yescharacteristic
494C22×C3C2×#13711yescharacteristic
495C22×C3#57511yescharacteristic
496C3×C4—22no—
497C3×C4—42no—
498C3×C4—42no—
499C3×C4—41no—
500C3×C4—81no—
501C3×C4—81no—
502C3×C4#52511yescharacteristic
503C3×C4#52511yescharacteristic
504C3×C4#52911yescharacteristic
505C3×C4#52911yescharacteristic
506Dic3—22no—
507Dic3—22no—
508Dic3—22no—
509Dic3—42no—
510Dic3—42no—
511Dic3—81no—
512Dic3—81no—
513Dic3#46214yes—
514C23C2×D3×D411yescharacteristic
515Q8—22no—
516Q8—22no—
517Q8—22no—
518Q8—22no—
519Q8—41no—
520Q8—41no—
521Q8—41no—
522Q8—41no—
523Q8—62no—
524Q8—62no—
525Q8—121no—
526Q8—121no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—32no—
535C2×C4—32no—
536C2×C4—32no—
537C2×C4—32no—
538C2×C4—32no—
539C2×C4—32no—
540C2×C4—41no—
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—61no—
551C2×C4—61no—
552C2×C4—61no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—62no—
557C2×C4—62no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4D3×Q1612yes—
569C2×C4C2×D3×D411yescharacteristic
570C2×C4C2×#23711yescharacteristic
571C2×C4C2×#23711yescharacteristic
572C2×C4D3×#4111yescharacteristic
573C2×C4D3×#4111yescharacteristic
574C2×C4D3×#4111yescharacteristic
575C8—22no—
576C8—41no—
577C8—62no—
578C8—121no—
579C2×C3#239211yescharacteristic
580C2×C3C2×#46211yescharacteristic
581C2×C3#303511yescharacteristic
582C2×C3#310411yescharacteristic
583C2×C3#310811yescharacteristic
584C2×C3#317011yescharacteristic
585C2×C3#317211yescharacteristic
586C22#559611yescharacteristic
587C22D3×#12111yescharacteristic
588C22#560611yescharacteristic
589C22C2×D3×Q1611yescharacteristic
590C22D3×#13511yescharacteristic
591C22D3×#13711yescharacteristic
592C22#577711yescharacteristic
593C4—22no—
594C4—34no—
595C4—42no—
596C4—42no—
597C4—41no—
598C4—62no—
599C4—62no—
600C4—62no—
601C4—81no—
602C4—81no—
603C4—122no—
604C4—122no—
605C4—241no—
606C4—241no—
607C4#558611yescharacteristic
608C4#564911yescharacteristic
609C4#565611yescharacteristic
610C4#568011yescharacteristic
611C3#2367911yescharacteristic, complemented, Sylow
612C2#7689011yescharacteristic
613C2#8045611yescharacteristic
614C2D3×#46211yescharacteristic
615C2#8056911yescharacteristic
616C2#8059711yescharacteristic
617C2#8074111yescharacteristic
618C2#8088811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2056C2
socle, prime radicalC23×C3C22×D4
hypercenter#2056D3
hypocenterC3#23679
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2056

Complements

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

NH ≅ G/Ntype
264. C4×Dic3Q16semidirect
611. C3#23679semidirect

Maximal subgroups

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

subgroupindex
C2×Dic3×Q162
C2×#47902
#764812
#764772
C3×#20562
#720732
#704892
Dic3×C4⋊C82
#702842
#702332
Dic3×#1032
#696832
#696812
#688342
#687812
#236793