Finite Group Databasebeta

← #10868310

Subgroups — #10868310

Order 768.

672 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76491C211yesmaximal, characteristic
2#74234C211yesmaximal, characteristic
3C3×#2074C211yesmaximal, characteristic, complemented
4#70275C211yesmaximal, characteristic
5#70274C211yesmaximal, characteristic
6#70236C211yesmaximal, characteristic, complemented
7#70220C211yesmaximal, characteristic, complemented
8#69682C211yesmaximal, characteristic, complemented
9#69540C211yesmaximal, characteristic, complemented
10#68877C211yesmaximal, characteristic
11#68860C211yesmaximal, characteristic, complemented
12#68813C211yesmaximal, characteristic
13#68789C211yesmaximal, characteristic, complemented
14#68624C211yesmaximal, characteristic, complemented
15#68616C211yesmaximal, characteristic
16#27061—31nomaximal, Sylow
17C2×#946C2211yescharacteristic
18C2×Q8×Dic3C2211yescharacteristic
19C2×#892C2211yescharacteristic
20C2×C3×#103C2211yescharacteristic
21C2×C3×#103C2211yescharacteristic
22C3×#391C2211yescharacteristic
23C3×#382C2211yescharacteristic
24C2×#835C2211yescharacteristic
25C2×#835C2211yescharacteristic
26C2×(C2×D3)⋊C8C2211yescharacteristic
27C2×(C2×D3)⋊C8C2211yescharacteristic
28C2×#818C2211yescharacteristic
29C2×#815C2211yescharacteristic
30#4987C2211yescharacteristic
31#4987C2211yescharacteristic
32#4984C2211yescharacteristic
33Dic3×C4⋊C4C2211yescharacteristic
34#4972C2211yescharacteristic
35C2×#808C2211yescharacteristic
36C2×#808C2211yescharacteristic
37C2×#807C2211yescharacteristic
38C4×(C2×D3)⋊C4C2211yescharacteristic
39C4×C4⋊Dic3C2211yescharacteristic
40#4928C2211yescharacteristic
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42#4675—21no—
43#4670—21no—
44#4670—21no—
45#4669—21no—
46C3⋊#401—21no—
47#4655—21no—
48#4654—21no—
49#4649—21no—
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#336C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4550C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2494—31no—
61#2362—31no—
62#2074D311yescharacteristic, complemented
63#2066—31no—
64#2036—31no—
65#2035—31no—
66#1949—31no—
67#1943—31no—
68#1924—31no—
69#1921—31no—
70#1920—31no—
71#1863—31no—
72#1854—31no—
73#1825—31no—
74#1763—31no—
75C22×C3×Q8C2311yescharacteristic
76C22×C4×D3C2311yescharacteristic
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C42C2311yescharacteristic
83#946—22no—
84#946—22no—
85Q8×Dic3—22no—
86Q8×Dic3—41no—
87C2×(C2×D3)⋊C4—21no—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4D411yescharacteristic
92C2×(C2×D3)⋊C4D411yescharacteristic
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×C4×Dic3—21no—
103C2×C4×Dic3—21no—
104C2×C4×Dic3—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109#892—22no—
110#892—41no—
111C3×#103—22no—
112C3×#103—22no—
113C3×#95—21no—
114C3×#95—21no—
115C3×#95—21no—
116C3×#95—21no—
117#835—22no—
118#835—41no—
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(C2×D3)⋊C8—22no—
129(C2×D3)⋊C8—41no—
130#818—22no—
131#815—41no—
132#808—22no—
133#808—41no—
134#807—22no—
135C4⋊(C3⋊C8)—22no—
136C2×C22⋊Q8—31no—
137C2×C4×Q8—31no—
138C2×#117—31no—
139C2×#107—31no—
140C2×#107—31no—
141C2×#106—31no—
142C2×#106—31no—
143C2×C4⋊C8—31no—
144C2×#103—31no—
145C2×#103—31no—
146C2×#103C2×D311yescharacteristic
147C2×#103C2×D311yescharacteristic
148C2×C22⋊C8—31no—
149C2×C22⋊C8—31no—
150C2×#97—31no—
151#401—61no—
152#397—61no—
153#391C2×D311yescharacteristic
154#389—31no—
155#389—31no—
156#389—31no—
157#388—61no—
158#388—61no—
159#388—61no—
160#387—61no—
161#387—61no—
162#382C2×D311yescharacteristic
163#380—31no—
164#380—31no—
165C4×C4⋊C4—31no—
166C4×C4⋊C4—31no—
167C4×C4⋊C4—61no—
168C4×C22⋊C4—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340C2×D311yescharacteristic
176#340C2×D311yescharacteristic
177#336—31no—
178#336C2×D311yescharacteristic
179C23×D3C2×D411yescharacteristic
180C2×C3×Q8—21no—
181C2×C3×Q8—21no—
182C2×C3×Q8—21no—
183C2×C3×Q8—21no—
184C2×C3×Q8—21no—
185C2×C3×Q8#4112yes—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2411yescharacteristic
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3C2×D411yescharacteristic
198C22×Dic3C2×D411yescharacteristic
199C22×Dic3C2×D411yescharacteristic
200C2×C4×D3—21no—
201C2×C4×D3—21no—
202C2×C4×D3—21no—
203C2×C4×D3—21no—
204C2×C4×D3—21no—
205C2×C4×D3—21no—
206C2×C3×C8—21no—
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C42—21no—
219C3×C42—21no—
220(C2×D3)⋊C4—24no—
221(C2×D3)⋊C4—41no—
222(C2×D3)⋊C4—41no—
223(C2×D3)⋊C4—42no—
224(C2×D3)⋊C4—42no—
225(C2×D3)⋊C4—81no—
226C4⋊Dic3—21no—
227C4⋊Dic3—21no—
228C4⋊Dic3—21no—
229C4⋊Dic3—21no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—42no—
233C4⋊Dic3—42no—
234C4⋊Dic3#4112yes—
235C4⋊Dic3#4112yes—
236Dic3⋊C4—24no—
237Dic3⋊C4—42no—
238Dic3⋊C4—81no—
239C4×Dic3—21no—
240C4×Dic3—21no—
241C4×Dic3—42no—
242C4×Dic3—42no—
243C4×Dic3—42no—
244C4×Dic3—81no—
245C2×C3⋊C8—21no—
246C2×C3⋊C8—21no—
247C2×C3⋊C8—21no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8#4112yes—
251C22×Q8C22×D311yescharacteristic
252C23×C4—31no—
253C22×C8—31no—
254C22×C8—31no—
255C22×C8C22×D311yescharacteristic
256C22×C8C22×D311yescharacteristic
257C22⋊Q8—62no—
258C22⋊Q8—62no—
259C4×Q8—62no—
260C4×Q8—121no—
261#117—62no—
262#117—121no—
263C2×C4⋊C4—31no—
264C2×C4⋊C4—31no—
265C2×C4⋊C4—31no—
266C2×C4⋊C4—31no—
267C2×C4⋊C4—61no—
268C2×C4⋊C4—61no—
269C2×C4⋊C4—61no—
270C2×C4⋊C4—61no—
271C2×C4⋊C4—61no—
272C2×C4⋊C4C22×D311yescharacteristic
273C2×C4⋊C4C22×D311yescharacteristic
274C2×C4⋊C4C22×D311yescharacteristic
275C2×C22⋊C4—31no—
276C2×C22⋊C4—31no—
277C2×C22⋊C4—61no—
278C2×C22⋊C4—61no—
279C2×C22⋊C4—61no—
280C2×C22⋊C4—61no—
281C2×C42—31no—
282C2×C42—61no—
283C2×C42—61no—
284C2×C42—61no—
285C2×C42—61no—
286C2×C42C22×D311yescharacteristic
287#107—62no—
288#107—62no—
289#106—62no—
290#106—121no—
291C4⋊C8—62no—
292#103—22no—
293#103—22no—
294#103—62no—
295#103—121no—
296C22⋊C8—62no—
297C22⋊C8—121no—
298#97—121no—
299#95—21no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—31no—
304#95—31no—
305#95—61no—
306#95—61no—
307#95—61no—
308#95—61no—
309#95—61no—
310#95—61no—
311#95—61no—
312C23×C3C22×D411yescharacteristic
313C22×D3—21no—
314C22×D3—21no—
315C22×D3—21no—
316C22×D3—21no—
317C22×D3—21no—
318C22×D3—21no—
319C22×D3—21no—
320C3×Q8—22no—
321C3×Q8—22no—
322C3×Q8—41no—
323C3×Q8—41no—
324C2×C3×C4—22no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4C2×#4111yescharacteristic
343C2×C3×C4C2×#4111yescharacteristic
344C2×C3×C4C2×#4111yescharacteristic
345C2×C3×C4C2×#4111yescharacteristic
346C2×C3×C4H(5,𝔽2)11yescharacteristic
347C2×C3×C42−1+411yescharacteristic
348C2×Dic3—21no—
349C2×Dic3—22no—
350C2×Dic3—21no—
351C2×Dic3—21no—
352C2×Dic3—22no—
353C2×Dic3—21no—
354C2×Dic3—21no—
355C2×Dic3—21no—
356C2×Dic3—21no—
357C2×Dic3—21no—
358C2×Dic3—21no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C4×D3—41no—
370C4×D3—41no—
371C4×D3—41no—
372C4×D3—41no—
373C3×C8—41no—
374C3×C8—41no—
375C3⋊C8—22no—
376C3⋊C8—41no—
377C24—31no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8#23712yes—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—31no—
389C22×C4—31no—
390C22×C4—31no—
391C22×C4—61no—
392C22×C4—61no—
393C22×C4—61no—
394C22×C4—61no—
395C22×C4—61no—
396C22×C4—61no—
397C22×C4—61no—
398C22×C4—61no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4D3×D411yescharacteristic
402C22×C4D3×D411yescharacteristic
403C22×C4C23×D311yescharacteristic
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—32no—
411C2×C8—61no—
412C2×C8—61no—
413C2×C8—61no—
414C2×C8—61no—
415C2×C8—61no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—32no—
423C4⋊C4—32no—
424C4⋊C4—64no—
425C4⋊C4—61no—
426C4⋊C4—61no—
427C4⋊C4—61no—
428C4⋊C4—61no—
429C4⋊C4—61no—
430C4⋊C4—61no—
431C4⋊C4—122no—
432C4⋊C4—122no—
433C4⋊C4—122no—
434C4⋊C4—241no—
435C22⋊C4—64no—
436C22⋊C4—121no—
437C22⋊C4—121no—
438C22⋊C4—122no—
439C22⋊C4—122no—
440C22⋊C4—241no—
441C42—21no—
442C42—21no—
443C42—61no—
444C42—61no—
445C42—122no—
446C42—122no—
447C42—122no—
448C42—241no—
449C22×C3#52511yescharacteristic
450C22×C3#54611yescharacteristic
451C22×C3#54711yescharacteristic
452C22×C3#57511yescharacteristic
453C22×C3#57511yescharacteristic
454C22×C3#57711yescharacteristic
455C22×C3#57811yescharacteristic
456C2×D3—41no—
457C2×D3—41no—
458C2×D3—41no—
459C2×D3—41no—
460C2×D3—41no—
461C2×D3—41no—
462C2×D3—41no—
463C3×C4—41no—
464C3×C4—41no—
465C3×C4—42no—
466C3×C4—81no—
467C3×C4—81no—
468C3×C4—81no—
469C3×C4#52911yescharacteristic
470C3×C4#53911yescharacteristic
471C3×C4#55311yescharacteristic
472C3×C4#55611yescharacteristic
473Dic3—42no—
474Dic3—41no—
475Dic3—41no—
476Dic3—42no—
477Dic3—41no—
478Dic3—81no—
479Dic3—81no—
480C23—61no—
481C23—61no—
482C23—61no—
483C23—61no—
484C23—61no—
485C23—61no—
486C23—61no—
487C23C2×D3×D411yescharacteristic
488Q8—22no—
489Q8—22no—
490Q8—41no—
491Q8—41no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—61no—
511C2×C4—61no—
512C2×C4—61no—
513C2×C4—62no—
514C2×C4—61no—
515C2×C4—61no—
516C2×C4—61no—
517C2×C4—61no—
518C2×C4—61no—
519C2×C4—62no—
520C2×C4—61no—
521C2×C4—121no—
522C2×C4—121no—
523C2×C4—121no—
524C2×C4—121no—
525C2×C4—121no—
526C2×C4—121no—
527C2×C4—121no—
528C2×C4—121no—
529C2×C4—121no—
530C2×C4—121no—
531C2×C4—121no—
532C2×C4—121no—
533C2×C4—121no—
534C2×C4—121no—
535C2×C4C2×#23711yescharacteristic
536C2×C4D3×#4111yescharacteristic
537C2×C4D3×#4111yescharacteristic
538C2×C4D3×#4111yescharacteristic
539C2×C4#100911yescharacteristic
540C2×C4#101011yescharacteristic
541C8—41no—
542C8—41no—
543C8—62no—
544C8—121no—
545C2×C3#246611yescharacteristic
546C2×C3#311111yescharacteristic
547C2×C3#312111yescharacteristic
548C2×C3#330711yescharacteristic
549C2×C3#332711yescharacteristic
550C2×C3#334911yescharacteristic
551C2×C3#335011yescharacteristic
552D3—81no—
553C22—121no—
554C22—121no—
555C22—121no—
556C22—121no—
557C22—121no—
558C22—121no—
559C22—121no—
560C22#560611yescharacteristic
561C22#565511yescharacteristic
562C22#567611yescharacteristic
563C22#576811yescharacteristic
564C22#577111yescharacteristic
565C22#577711yescharacteristic
566C22#577811yescharacteristic
567C4—41no—
568C4—41no—
569C4—42no—
570C4—81no—
571C4—81no—
572C4—81no—
573C4—121no—
574C4—122no—
575C4—121no—
576C4—122no—
577C4—121no—
578C4—241no—
579C4—241no—
580C4#564911yescharacteristic
581C4#566511yescharacteristic
582C4#570211yescharacteristic
583C4#570911yescharacteristic
584C3#2706111yescharacteristic, complemented, Sylow
585C2—241no—
586C2#7747111yescharacteristic
587C2#8057011yescharacteristic
588C2#8057311yescharacteristic
589C2#8150911yescharacteristic
590C2#8153911yescharacteristic
591C2#8165911yescharacteristic
592C2#8174811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2074C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#2074D3
hypocenterC3#27061
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2074

Complements

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

NH ≅ G/Ntype
3. C3×#2074C2semidirect
6. #70236C2semidirect
7. #70220C2semidirect
8. #69682C2semidirect
9. #69540C2semidirect
11. #68860C2semidirect
13. #68789C2semidirect
14. #68624C2semidirect
62. #2074D3semidirect
584. C3#27061semidirect

Maximal subgroups

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

subgroupindex
#764912
#742342
C3×#20742
#702752
#702742
#702362
#702202
#696822
#695402
#688772
#688602
#688132
#687892
#686242
#686162
#270613