Finite Group Databasebeta

← #10868045

Subgroups — #10868045

Order 768.

706 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76237C211yesmaximal, characteristic
2#75984C211yesmaximal, characteristic
3C3×#1942C211yesmaximal, characteristic, complemented
4#70643C211yesmaximal, characteristic, complemented
5#70635C211yesmaximal, characteristic
6#70633C211yesmaximal, characteristic
7#70615C211yesmaximal, characteristic, complemented
8#69785C211yesmaximal, characteristic
9C4×#816C211yesmaximal, characteristic, complemented
10#69632C211yesmaximal, characteristic, complemented
11#69600C211yesmaximal, characteristic, complemented
12#68914C211yesmaximal, characteristic, complemented
13#68825C211yesmaximal, characteristic
14#68819C211yesmaximal, characteristic, complemented
15#68813C211yesmaximal, characteristic
16#27043—31nomaximal, Sylow
17C2×#898—21no—
18C2×#897C2211yescharacteristic
19C2×#892C2211yescharacteristic
20C2×C4×Dic6C2211yescharacteristic
21C2×C3×#107C2211yescharacteristic
22C2×C3×#107C2211yescharacteristic
23C2×C3×C4×C8C2211yescharacteristic
24C3×#389C2211yescharacteristic
25C3×#389C2211yescharacteristic
26C2×(C2×D3)⋊C8C2211yescharacteristic
27C2×(C2×D3)⋊C8C2211yescharacteristic
28C2×#816C2211yescharacteristic
29C2×#816C2211yescharacteristic
30C2×Dic3⋊C8C2211yescharacteristic
31#4990—21no—
32#4987C2211yescharacteristic
33#4987C2211yescharacteristic
34#4984C2211yescharacteristic
35#4978—21no—
36#4975C2211yescharacteristic
37Dic3×C4⋊C4—21no—
38Dic3×C4⋊C4C2211yescharacteristic
39#4973C2211yescharacteristic
40C2×#810C2211yescharacteristic
41C2×#810C2211yescharacteristic
42C2×#808C2211yescharacteristic
43C2×#807C2211yescharacteristic
44C4×(C2×D3)⋊C4C2211yescharacteristic
45C4×C4⋊Dic3C2211yescharacteristic
46#4675—21no—
47#4669—21no—
48#4660—21no—
49#4654—21no—
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4550C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2496—31no—
61#2362—31no—
62#2072—31no—
63#2068—31no—
64#2036—31no—
65#2036—31no—
66#1947—31no—
67#1942D311yescharacteristic, complemented
68#1923—31no—
69#1920—31no—
70#1918—31no—
71#1865—31no—
72#1854—31no—
73#1825—31no—
74C4×#103—31no—
75C22×C4×D3C2311yescharacteristic
76C22×Dic6C2311yescharacteristic
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C42C2311yescharacteristic
86C2×(C2×D3)⋊C4—21no—
87C2×(C2×D3)⋊C4—21no—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4D411yescharacteristic
91C2×(C2×D3)⋊C4D411yescharacteristic
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×C4×Dic3—21no—
101C2×C4×Dic3—21no—
102C2×C4×Dic3—21no—
103C2×C4×Dic3—21no—
104C2×C4×Dic3C2311yescharacteristic
105C22×C3⋊C8C2311yescharacteristic
106C22×C3⋊C8C2311yescharacteristic
107#898—44no—
108#897—22no—
109#897—22no—
110#892—22no—
111#892—41no—
112C4×Dic6—22no—
113C4×Dic6—41no—
114C3×#107—22no—
115C3×#107—41no—
116C3×C4×C8—41no—
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(C2×D3)⋊C8—22no—
129(C2×D3)⋊C8—41no—
130#816—22no—
131#816—22no—
132Dic3⋊C8—22no—
133#810—22no—
134#810—41no—
135#808—22no—
136#807—22no—
137C2×#126—61no—
138C2×C22⋊Q8—31no—
139C2×C4×Q8—31no—
140C2×#117—31no—
141C2×#107—31no—
142C2×#107C2×D311yescharacteristic
143C2×#107C2×D311yescharacteristic
144C2×#106—31no—
145C2×C4⋊C8—31no—
146C2×#103—31no—
147C2×#103—31no—
148C2×#103—31no—
149C2×#103—31no—
150C2×C22⋊C8—31no—
151C2×C22⋊C8—31no—
152C2×C4×C8C2×D311yescharacteristic
153#400—61no—
154#397—61no—
155#391—31no—
156#389—31no—
157#389C2×D311yescharacteristic
158#389C2×D311yescharacteristic
159#388—61no—
160#388—61no—
161#387—61no—
162#387—61no—
163#384—31no—
164#380—31no—
165#380—31no—
166C4×C4⋊C4—31no—
167C4×C4⋊C4—31no—
168C4×C4⋊C4—61no—
169C4×C22⋊C4—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340C2×D311yescharacteristic
177#340C2×D311yescharacteristic
178#336—31no—
179#336—31no—
180C23×D3C2×D411yescharacteristic
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2411yescharacteristic
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3—21no—
191C22×Dic3—21no—
192C22×Dic3C2×D411yescharacteristic
193C22×Dic3C2×D411yescharacteristic
194C22×Dic3C2×D411yescharacteristic
195C2×C4×D3—21no—
196C2×C4×D3—21no—
197C2×C4×D3—21no—
198C2×C4×D3—21no—
199C2×C4×D3—21no—
200C2×C4×D3—21no—
201C2×Dic6—21no—
202C2×Dic6—21no—
203C2×Dic6—21no—
204C2×Dic6—21no—
205C2×Dic6—21no—
206C2×Dic6#4112yes—
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8#4112yes—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—42no—
221C3×C4⋊C4—42no—
222C3×C4⋊C4#4112yes—
223C3×C42—21no—
224C3×C42—21no—
225(C2×D3)⋊C4—24no—
226(C2×D3)⋊C4—22no—
227(C2×D3)⋊C4—22no—
228(C2×D3)⋊C4—42no—
229(C2×D3)⋊C4—42no—
230(C2×D3)⋊C4—42no—
231C4⋊Dic3—24no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—42no—
238C4⋊Dic3#4112yes—
239Dic3⋊C4—42no—
240Dic3⋊C4—42no—
241Dic3⋊C4—42no—
242C4×Dic3—21no—
243C4×Dic3—21no—
244C4×Dic3—42no—
245C4×Dic3—42no—
246C4×Dic3—42no—
247C4×Dic3—42no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
254C22×Q8—31no—
255C23×C4—31no—
256C22×C8—31no—
257C22×C8—31no—
258C22×C8C22×D311yescharacteristic
259C22×C8C22×D311yescharacteristic
260#126—124no—
261C22⋊Q8—62no—
262C22⋊Q8—62no—
263C4×Q8—62no—
264C4×Q8—121no—
265#117—62no—
266#117—121no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—31no—
271C2×C4⋊C4—31no—
272C2×C4⋊C4—61no—
273C2×C4⋊C4—61no—
274C2×C4⋊C4—61no—
275C2×C4⋊C4—61no—
276C2×C4⋊C4—61no—
277C2×C4⋊C4C22×D311yescharacteristic
278C2×C4⋊C4C22×D311yescharacteristic
279C2×C4⋊C4C22×D311yescharacteristic
280C2×C4⋊C4C22×D311yescharacteristic
281C2×C22⋊C4—31no—
282C2×C22⋊C4—31no—
283C2×C22⋊C4—61no—
284C2×C22⋊C4—61no—
285C2×C22⋊C4—61no—
286C2×C22⋊C4—61no—
287C2×C42—31no—
288C2×C42—61no—
289C2×C42—61no—
290C2×C42—61no—
291C2×C42—61no—
292C2×C42C22×D311yescharacteristic
293#107—22no—
294#107—41no—
295#107—62no—
296#106—62no—
297C4⋊C8—62no—
298#103—62no—
299#103—62no—
300#103—62no—
301#103—121no—
302C22⋊C8—62no—
303C22⋊C8—121no—
304C4×C8—41no—
305#95—21no—
306#95—21no—
307#95—31no—
308#95—31no—
309#95—61no—
310#95—61no—
311#95—61no—
312#95—61no—
313#95—61no—
314#95—61no—
315#95—61no—
316C23×C3C22×D411yescharacteristic
317C22×D3—21no—
318C22×D3—21no—
319C22×D3—21no—
320C22×D3—21no—
321C22×D3—21no—
322C22×D3—21no—
323C22×D3—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—22no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—41no—
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×C4AGL(1,ℤ/8)12yes—
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—21no—
350C2×Dic3—21no—
351C2×Dic3—22no—
352C2×Dic3—22no—
353C2×Dic3—21no—
354C2×Dic3—21no—
355C2×Dic3—21no—
356C2×Dic3—21no—
357C2×Dic3—21no—
358C2×Dic3—41no—
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#13512yes—
369C4×D3—41no—
370C4×D3—41no—
371C4×D3—41no—
372C4×D3—41no—
373Dic6—22no—
374Dic6—22no—
375Dic6—41no—
376Dic6—41no—
377C3×C8—22no—
378C3×C8—41no—
379C3⋊C8—41no—
380C3⋊C8—41no—
381C24—31no—
382C2×Q8—32no—
383C2×Q8—61no—
384C2×Q8—61no—
385C2×Q8—61no—
386C2×Q8—61no—
387C2×Q8—61no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—31no—
393C22×C4—31no—
394C22×C4—31no—
395C22×C4—61no—
396C22×C4—61no—
397C22×C4—61no—
398C22×C4—61no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4D3×D411yescharacteristic
406C22×C4D3×D411yescharacteristic
407C22×C4C23×D311yescharacteristic
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—61no—
414C2×C8—61no—
415C2×C8—61no—
416C2×C8—61no—
417C2×C8—61no—
418C2×C8—61no—
419C2×C8#23912yes—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—32no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—64no—
431C4⋊C4—61no—
432C4⋊C4—61no—
433C4⋊C4—61no—
434C4⋊C4—61no—
435C4⋊C4—61no—
436C4⋊C4—122no—
437C4⋊C4—122no—
438C4⋊C4—122no—
439C4⋊C4—122no—
440C4⋊C4#23512yes—
441C22⋊C4—62no—
442C22⋊C4—64no—
443C22⋊C4—62no—
444C22⋊C4—122no—
445C22⋊C4—122no—
446C22⋊C4—122no—
447C42—21no—
448C42—21no—
449C42—61no—
450C42—61no—
451C42—122no—
452C42—122no—
453C42—122no—
454C42—122no—
455C22×C3#52511yescharacteristic
456C22×C3#54611yescharacteristic
457C22×C3#54711yescharacteristic
458C22×C3C2×#13511yescharacteristic
459C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
460C22×C3#57811yescharacteristic
461C22×C3#57811yescharacteristic
462C2×D3—41no—
463C2×D3—41no—
464C2×D3—41no—
465C2×D3—41no—
466C2×D3—41no—
467C2×D3—41no—
468C2×D3—41no—
469C3×C4—22no—
470C3×C4—42no—
471C3×C4—41no—
472C3×C4—81no—
473C3×C4—81no—
474C3×C4—81no—
475C3×C4#52911yescharacteristic
476C3×C4#53911yescharacteristic
477C3×C4#55311yescharacteristic
478C3×C4#55611yescharacteristic
479Dic3—22no—
480Dic3—41no—
481Dic3—42no—
482Dic3—41no—
483Dic3—42no—
484Dic3—81no—
485Dic3—81no—
486C23—61no—
487C23—61no—
488C23—61no—
489C23—61no—
490C23—61no—
491C23—61no—
492C23—61no—
493C23C2×D3×D411yescharacteristic
494Q8—62no—
495Q8—62no—
496Q8—121no—
497Q8—121no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—32no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—61no—
517C2×C4—61no—
518C2×C4—61no—
519C2×C4—61no—
520C2×C4—61no—
521C2×C4—61no—
522C2×C4—62no—
523C2×C4—61no—
524C2×C4—62no—
525C2×C4—61no—
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×C4—121no—
536C2×C4—121no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4#91112yes—
541C2×C4C2×#23511yescharacteristic
542C2×C4#100411yescharacteristic
543C2×C4C2×#23911yescharacteristic
544C2×C4#100711yescharacteristic
545C2×C4D3×#4111yescharacteristic
546C2×C4D3×#4111yescharacteristic
547C8—22no—
548C8—41no—
549C8—121no—
550C8—121no—
551C2×C3#246611yescharacteristic
552C2×C3#310611yescharacteristic
553C2×C3#312711yescharacteristic
554C2×C3#330211yescharacteristic
555C2×C3#330811yescharacteristic
556C2×C3#333611yescharacteristic
557C2×C3#335311yescharacteristic
558D3—81no—
559C22—121no—
560C22—121no—
561C22—121no—
562C22—121no—
563C22—121no—
564C22—121no—
565C22—121no—
566C22#554711yescharacteristic
567C22#561011yescharacteristic
568C22#572611yescharacteristic
569C22C2×#91111yescharacteristic
570C22#576611yescharacteristic
571C22D3×#13511yescharacteristic
572C22#577111yescharacteristic
573C4—22no—
574C4—41no—
575C4—42no—
576C4—62no—
577C4—81no—
578C4—81no—
579C4—81no—
580C4—121no—
581C4—122no—
582C4—121no—
583C4—122no—
584C4—241no—
585C4—241no—
586C4#551411yescharacteristic
587C4#556811yescharacteristic
588C4#563811yescharacteristic
589C4#569111yescharacteristic
590C3#2704311yescharacteristic, complemented, Sylow
591C2—241no—
592C2#7601911yescharacteristic
593C2#7959311yescharacteristic
594C2#7972411yescharacteristic
595C2#8073811yescharacteristic
596C2#8077611yescharacteristic
597C2#8192411yescharacteristic
598C2#8195011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1942C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#1942D3
hypocenterC3#27043
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1942

Complements

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

NH ≅ G/Ntype
3. C3×#1942C2semidirect
4. #70643C2semidirect
7. #70615C2semidirect
9. C4×#816C2semidirect
10. #69632C2semidirect
11. #69600C2semidirect
12. #68914C2semidirect
14. #68819C2semidirect
67. #1942D3semidirect
590. C3#27043semidirect

Maximal subgroups

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

subgroupindex
#762372
#759842
C3×#19422
#706432
#706352
#706332
#706152
#697852
C4×#8162
#696322
#696002
#689142
#688252
#688192
#688132
#270433