Finite Group Databasebeta

← #10897633

Subgroups — #10897633

Order 768.

748 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76484C211yesmaximal, characteristic
2#73625C211yesmaximal, characteristic, complemented
3C3×#1927C211yesmaximal, characteristic, complemented
4#70590C211yesmaximal, characteristic, complemented
5#70584C211yesmaximal, characteristic
6#70279C211yesmaximal, characteristic
7#70278C211yesmaximal, characteristic
8#70239C211yesmaximal, characteristic, complemented
9#70237C211yesmaximal, characteristic, complemented
10#69690C211yesmaximal, characteristic, complemented
11#69463C211yesmaximal, characteristic
12#68896C211yesmaximal, characteristic
13#68811C211yesmaximal, characteristic, complemented
14#68686C211yesmaximal, characteristic
15#68608C211yesmaximal, characteristic, complemented
16#29383—31nomaximal, Sylow
17C2×#947C2211yescharacteristic
18C2×Q8×Dic3C2211yescharacteristic
19C22×#216C2211yescharacteristic
20C22×#204C2211yescharacteristic
21C2×#893C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C2×C3×#106C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C3×#391C2211yescharacteristic
27C3×C4×C4⋊C4C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#816C2211yescharacteristic
31#4975C2211yescharacteristic
32#4973C2211yescharacteristic
33#4973C2211yescharacteristic
34#4972C2211yescharacteristic
35#4972C2211yescharacteristic
36C2×#810C2211yescharacteristic
37C2×#809C2211yescharacteristic
38C2×#809C2211yescharacteristic
39C2×#808C2211yescharacteristic
40#4939C2211yescharacteristic
41#4933—21no—
42#4933—21no—
43#4932C2211yescharacteristic
44#4669—21no—
45#4669—21no—
46#4668—21no—
47#4668—21no—
48#4651—21no—
49#4650—21no—
50C3×#340C2211yescharacteristic
51C3×#336C2211yescharacteristic
52#4551C2211yescharacteristic
53#4550C2211yescharacteristic
54#4550C2211yescharacteristic
55#4550C2211yescharacteristic
56#4550C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2326—31no—
61#2320—31no—
62#2100—31no—
63#2096—31no—
64#2042—31no—
65#2015—31no—
66#1927D311yescharacteristic, complemented
67#1926—31no—
68#1880—31no—
69#1878—31no—
70#1878—31no—
71#1877—31no—
72#1865—31no—
73#1864—31no—
74#1840—31no—
75C22×C3×Q8C2311yescharacteristic
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×Dic6C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C42—21no—
85C2×C3×C42C2311yescharacteristic
86#947—22no—
87#947—22no—
88Q8×Dic3—41no—
89Q8×Dic3—41no—
90C2×#216—22no—
91C2×#216—22no—
92C2×#216—41no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×C4×Dic3—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3C2311yescharacteristic
109C2×C4×Dic3C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112C2×#204—22no—
113C2×#204—22no—
114C2×#204—41no—
115#893—41no—
116#893—41no—
117C4×Dic6—22no—
118C4×Dic6—22no—
119C3×#106—41no—
120C3×#103—22no—
121C3×#103—41no—
122C3×#95—21no—
123C3×#95—21no—
124C3×#95—21no—
125#835—22no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95D411yescharacteristic
133C3⋊#95D411yescharacteristic
134C3⋊#95D411yescharacteristic
135C3⋊#95D411yescharacteristic
136#821—22no—
137#816—41no—
138#810—41no—
139#809—22no—
140#809—41no—
141#808—41no—
142C22×Q16—31no—
143C22×SD16—31no—
144C2×#124—31no—
145C2×C4×Q8—31no—
146C2×C4×Q8—31no—
147C2×C4×D4—31no—
148C2×#106—31no—
149C2×#106C2×D311yescharacteristic
150C2×#103—31no—
151C2×#103—31no—
152C2×#103—31no—
153C2×#103C2×D311yescharacteristic
154C2×#103C2×D311yescharacteristic
155C2×#102—31no—
156C2×#102—31no—
157C2×#102—31no—
158#391—31no—
159#391—31no—
160#391C2×D311yescharacteristic
161#390—31no—
162#389—31no—
163#389—31no—
164#388—61no—
165#388—61no—
166#387—61no—
167#387—61no—
168#385—61no—
169#385—61no—
170#384—31no—
171#383—61no—
172#382—61no—
173C4×C4⋊C4C2×D311yescharacteristic
174#376—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340C2×D311yescharacteristic
179#336—31no—
180#336—31no—
181#336—31no—
182#336—31no—
183#336—31no—
184#336C2×D311yescharacteristic
185C23×D3—21no—
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8—21no—
191C2×C3×Q8#4112yes—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2411yescharacteristic
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3C2×D411yescharacteristic
203C22×Dic3C2×D411yescharacteristic
204C22×Dic3C2×D411yescharacteristic
205C22×Dic3C2×D411yescharacteristic
206C2×D12—21no—
207C2×D12—21no—
208C2×D12—21no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12#4112yes—
212C2×C4×D3—42no—
213C2×C4×D3—81no—
214C2×C4×D3—81no—
215C2×Dic6—21no—
216C2×Dic6—21no—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6#4112yes—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C42—21no—
234C3×C42—21no—
235C3×C42—81no—
236#216—44no—
237(C2×D3)⋊C4—24no—
238(C2×D3)⋊C4—24no—
239(C2×D3)⋊C4—42no—
240(C2×D3)⋊C4—81no—
241(C2×D3)⋊C4—81no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—42no—
246C4⋊Dic3#4112yes—
247Dic3⋊C4—24no—
248Dic3⋊C4—81no—
249Dic3⋊C4—81no—
250Dic3⋊C4—81no—
251Dic3⋊C4—81no—
252C4×Dic3—24no—
253C4×Dic3—21no—
254C4×Dic3—21no—
255C4×Dic3—21no—
256C4×Dic3—21no—
257C4×Dic3—42no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264#204—44no—
265C22×Q8—31no—
266C22×Q8C22×D311yescharacteristic
267C22×D4—31no—
268C23×C4—61no—
269C2×Q16—62no—
270C2×Q16—62no—
271C2×Q16—121no—
272C2×SD16—62no—
273C2×SD16—62no—
274C2×SD16—121no—
275C22×C8—31no—
276C22×C8—31no—
277C22×C8C22×D311yescharacteristic
278C22×C8C22×D311yescharacteristic
279#124—62no—
280#124—62no—
281C4×Q8—62no—
282C4×Q8—62no—
283C4×Q8—121no—
284C4×Q8—121no—
285C4×D4—121no—
286C4×D4—121no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4C22×D311yescharacteristic
296C2×C4⋊C4C22×D311yescharacteristic
297C2×C4⋊C4C22×D311yescharacteristic
298C2×C22⋊C4—61no—
299C2×C22⋊C4—61no—
300C2×C22⋊C4—61no—
301C2×C22⋊C4—61no—
302C2×C22⋊C4—61no—
303C2×C42—21no—
304C2×C42—31no—
305C2×C42—31no—
306C2×C42—61no—
307C2×C42—61no—
308C2×C42C22×D311yescharacteristic
309#106—41no—
310#106—121no—
311#103—22no—
312#103—41no—
313#103—62no—
314#103—121no—
315#103—121no—
316#102—62no—
317#102—62no—
318#102—121no—
319#95—21no—
320#95—21no—
321#95—21no—
322#95—31no—
323#95—31no—
324#95—31no—
325#95—31no—
326#95—61no—
327#95—61no—
328#95—61no—
329#95—61no—
330#95—61no—
331#95—61no—
332C23×C3C22×D411yescharacteristic
333C22×D3—22no—
334C22×D3—41no—
335C22×D3—41no—
336C22×D3—41no—
337C22×D3—41no—
338C22×D3—41no—
339C22×D3—41no—
340C3×Q8—22no—
341C3×Q8—22no—
342C3×Q8—41no—
343C3×Q8—41no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—22no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4C2×#4111yescharacteristic
363C2×C3×C4C2×#4111yescharacteristic
364C2×C3×C4C2×#4111yescharacteristic
365C2×C3×C4C2×#4111yescharacteristic
366C2×C3×C4H(5,𝔽2)11yescharacteristic
367C2×C3×C4H(5,𝔽2)11yescharacteristic
368C2×Dic3—22no—
369C2×Dic3—22no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389D12—22no—
390D12—22no—
391D12—41no—
392D12—41no—
393C4×D3—84no—
394Dic6—22no—
395Dic6—22no—
396Dic6—41no—
397Dic6—41no—
398C3×C8—41no—
399C3×C8—41no—
400C3⋊C8—41no—
401C3⋊C8—41no—
402C24—61no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—32no—
409C2×Q8—61no—
410C2×Q8—61no—
411C2×Q8—61no—
412C2×Q8—61no—
413C2×Q8—61no—
414C2×Q8#23512yes—
415C2×D4—32no—
416C2×D4—61no—
417C2×D4—61no—
418C2×D4—61no—
419C2×D4—61no—
420C2×D4—61no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—31no—
426C22×C4—31no—
427C22×C4—31no—
428C22×C4—31no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4—61no—
432C22×C4—122no—
433C22×C4—241no—
434C22×C4—241no—
435C22×C4D3×D411yescharacteristic
436C22×C4D3×D411yescharacteristic
437C22×C4C23×D311yescharacteristic
438Q16—124no—
439SD16—124no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—61no—
447C2×C8—61no—
448C2×C8—61no—
449C2×C8—61no—
450C2×C8—61no—
451C2×C8—61no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—32no—
459C4⋊C4—64no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4—122no—
464C4⋊C4—241no—
465C4⋊C4—241no—
466C4⋊C4—241no—
467C4⋊C4—241no—
468C22⋊C4—64no—
469C22⋊C4—64no—
470C22⋊C4—122no—
471C22⋊C4—241no—
472C22⋊C4—241no—
473C42—21no—
474C42—21no—
475C42—64no—
476C42—61no—
477C42—61no—
478C42—61no—
479C42—61no—
480C42—81no—
481C42—122no—
482C22×C3#52511yescharacteristic
483C22×C3#52511yescharacteristic
484C22×C3#53511yescharacteristic
485C22×C3#57711yescharacteristic
486C22×C3#57711yescharacteristic
487C22×C3#57711yescharacteristic
488C22×C3#57811yescharacteristic
489C2×D3—42no—
490C2×D3—42no—
491C2×D3—42no—
492C2×D3—81no—
493C2×D3—81no—
494C2×D3—81no—
495C2×D3—81no—
496C3×C4—42no—
497C3×C4—41no—
498C3×C4—41no—
499C3×C4—81no—
500C3×C4—81no—
501C3×C4—81no—
502C3×C4#55111yescharacteristic
503C3×C4#55111yescharacteristic
504C3×C4#55311yescharacteristic
505C3×C4#55311yescharacteristic
506Dic3—42no—
507Dic3—41no—
508Dic3—41no—
509Dic3—42no—
510Dic3—41no—
511Dic3—41no—
512Dic3—81no—
513C23—62no—
514C23—121no—
515C23—121no—
516C23—121no—
517C23—121no—
518C23—121no—
519C23—121no—
520C23C2×D3×D411yescharacteristic
521Q8—22no—
522Q8—22no—
523Q8—41no—
524Q8—41no—
525Q8—62no—
526Q8—62no—
527Q8—121no—
528Q8—121no—
529D4—62no—
530D4—62no—
531D4—121no—
532D4—121no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
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—41no—
551C2×C4—62no—
552C2×C4—62no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—61no—
557C2×C4—61no—
558C2×C4—61no—
559C2×C4—61no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—244no—
573C2×C4C2×#23511yescharacteristic
574C2×C4#100411yescharacteristic
575C2×C4D3×#4111yescharacteristic
576C2×C4D3×#4111yescharacteristic
577C2×C4D3×#4111yescharacteristic
578C2×C4#100911yescharacteristic
579C8—41no—
580C8—41no—
581C8—121no—
582C8—121no—
583C2×C3#259811yescharacteristic
584C2×C3#312011yescharacteristic
585C2×C3#312111yescharacteristic
586C2×C3#312511yescharacteristic
587C2×C3#312511yescharacteristic
588C2×C3#322211yescharacteristic
589C2×C3#322511yescharacteristic
590D3—82no—
591C22—122no—
592C22—122no—
593C22—122no—
594C22—241no—
595C22—241no—
596C22—241no—
597C22—241no—
598C22#554711yescharacteristic
599C22#560711yescharacteristic
600C22#565611yescharacteristic
601C22#576211yescharacteristic
602C22#577011yescharacteristic
603C22#577811yescharacteristic
604C22#577911yescharacteristic
605C4—41no—
606C4—41no—
607C4—42no—
608C4—81no—
609C4—81no—
610C4—81no—
611C4—122no—
612C4—121no—
613C4—121no—
614C4—122no—
615C4—121no—
616C4—121no—
617C4—241no—
618C4#552711yescharacteristic
619C4#554011yescharacteristic
620C4#567911yescharacteristic
621C4#570911yescharacteristic
622C3#2938311yescharacteristic, complemented, Sylow
623C2—242no—
624C2#7589511yescharacteristic
625C2#7969811yescharacteristic
626C2#7984911yescharacteristic
627C2#8043711yescharacteristic
628C2#8060811yescharacteristic
629C2#8141711yescharacteristic
630C2#8149711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1927C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1927D3
hypocenterC3#29383
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1927

Complements

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

NH ≅ G/Ntype
2. #73625C2semidirect
3. C3×#1927C2semidirect
4. #70590C2semidirect
8. #70239C2semidirect
9. #70237C2semidirect
10. #69690C2semidirect
13. #68811C2semidirect
15. #68608C2semidirect
66. #1927D3semidirect
622. C3#29383semidirect

Maximal subgroups

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

subgroupindex
#764842
#736252
C3×#19272
#705902
#705842
#702792
#702782
#702392
#702372
#696902
#694632
#688962
#688112
#686862
#686082
#293833