Finite Group Databasebeta

← #10859050

Subgroups — #10859050

Order 768.

784 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
1#83233C211yesmaximal, characteristic
2#73647C211yesmaximal, characteristic
3C3×#1988C211yesmaximal, characteristic, complemented
4#71794C211yesmaximal, characteristic, complemented
5#70488C211yesmaximal, characteristic, complemented
6#70478C211yesmaximal, characteristic
7#69458C211yesmaximal, characteristic, complemented
8#68906C211yesmaximal, characteristic, complemented
9#68904C211yesmaximal, characteristic, complemented
10#68884C211yesmaximal, characteristic
11#68878C211yesmaximal, characteristic
12#68876C211yesmaximal, characteristic
13#68867C211yesmaximal, characteristic
14#68843C211yesmaximal, characteristic, complemented
15#68840C211yesmaximal, characteristic, complemented
16#26429—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×#946—21no—
19C22×ΓL(1,𝔽25)C2211yescharacteristic
20C2×#897C2211yescharacteristic
21C2×#892C2211yescharacteristic
22C2×D3×C4⋊C4C2211yescharacteristic
23C2×#887C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C3×C4×C4⋊C4C2211yescharacteristic
26#5230—21no—
27C2×(C2×D3)⋊C8C2211yescharacteristic
28C2×#817C2211yescharacteristic
29C2×#816C2211yescharacteristic
30C2×#816C2211yescharacteristic
31C2×Dic3⋊C8C2211yescharacteristic
32#4989—21no—
33#4987C2211yescharacteristic
34#4984C2211yescharacteristic
35#4984C2211yescharacteristic
36#4982—21no—
37#4975C2211yescharacteristic
38C2×#810C2211yescharacteristic
39C2×#810C2211yescharacteristic
40C2×#807C2211yescharacteristic
41#4940—21no—
42C4×(C2×D3)⋊C4C2211yescharacteristic
43C4×C4⋊Dic3—21no—
44#4929C2211yescharacteristic
45#4928C2211yescharacteristic
46#4670—21no—
47#4653—21no—
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52#4551C2211yescharacteristic
53#4550C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2440—31no—
61#2362—31no—
62#2100—31no—
63#2088—31no—
64#2066—31no—
65#2024—31no—
66#1988D311yescharacteristic, complemented
67#1918—31no—
68#1894—31no—
69#1872—31no—
70#1863—31no—
71#1854—31no—
72#1851—31no—
73#1803—31no—
74#1736—31no—
75C22×C3×Q8—21no—
76C22×C4×D3—21no—
77C22×C4×D3C2311yescharacteristic
78C22×Dic6C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C3×C4⋊Q8—22no—
82C3×C4⋊Q8—41no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C42—21no—
89C2×C3×C42C2311yescharacteristic
90#946—44no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4D411yescharacteristic
95C2×(C2×D3)⋊C4D411yescharacteristic
96C2×C4⋊Dic3—21no—
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×C4×Dic3—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112C2×ΓL(1,𝔽25)—22no—
113C2×ΓL(1,𝔽25)—41no—
114C2×ΓL(1,𝔽25)—41no—
115#897—22no—
116#897Q814yescomplemented
117#892—22no—
118#892—41no—
119D3×C4⋊C4—22no—
120D3×C4⋊C4—41no—
121#887—22no—
122#887—41no—
123C3×C4⋊C8—41no—
124C3×#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95D411yescharacteristic
132C3⋊#95D411yescharacteristic
133(C2×D3)⋊C8—22no—
134#817—41no—
135#816—41no—
136#816—41no—
137Dic3⋊C8—41no—
138#810—22no—
139#810—41no—
140#807—22no—
141C22×M16—31no—
142C2×C4⋊Q8C2×D311yescharacteristic
143C2×C22⋊Q8—31no—
144C2×C22⋊Q8—61no—
145C2×C4×Q8—31no—
146C2×#117—31no—
147C22×C4⋊C4—31no—
148C2×#107—31no—
149C2×#106—31no—
150C2×C4⋊C8—31no—
151C2×C4⋊C8C2×D311yescharacteristic
152C2×#103—31no—
153C2×#103—31no—
154C2×#103—31no—
155C2×#103—31no—
156C2×C22⋊C8—31no—
157#398—61no—
158#395—61no—
159#391—31no—
160#389—31no—
161#389—31no—
162#389—31no—
163#388—61no—
164#388—61no—
165#387—61no—
166#385—61no—
167#384—31no—
168#380—31no—
169C4×C4⋊C4—61no—
170C4×C4⋊C4C2×D311yescharacteristic
171C4×C22⋊C4—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340C2×D311yescharacteristic
179#340C2×D311yescharacteristic
180#340C2×D311yescharacteristic
181#340C2×D311yescharacteristic
182#336—31no—
183#336—31no—
184C23×D3C2×D411yescharacteristic
185C2×C3×Q8—42no—
186C2×C3×Q8—42no—
187C2×C3×Q8—81no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2411yescharacteristic
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3C2×D411yescharacteristic
200C22×Dic3C2×D411yescharacteristic
201C22×Dic3C2×D411yescharacteristic
202C2×C4×D3—21no—
203C2×C4×D3—21no—
204C2×C4×D3—21no—
205C2×C4×D3—21no—
206C2×C4×D3—42no—
207C2×C4×D3—42no—
208C2×C4×D3—81no—
209C2×C4×D3C2×Q812yes—
210C2×C4×D3#4112yes—
211C2×Dic6—21no—
212C2×Dic6—21no—
213C2×Dic6—21no—
214C2×Dic6—21no—
215C2×Dic6—21no—
216C2×Dic6C2×Q812yes—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—42no—
229C3×C4⋊C4C2×Q812yes—
230C3×C4⋊C4#4112yes—
231C3×C4⋊C4#4112yes—
232C3×C42—21no—
233C3×C42—21no—
234C3×C42—42no—
235(C2×D3)⋊C4—24no—
236(C2×D3)⋊C4—22no—
237(C2×D3)⋊C4—22no—
238(C2×D3)⋊C4—42no—
239(C2×D3)⋊C4—42no—
240C4⋊Dic3—24no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—42no—
248C4⋊Dic3—42no—
249Dic3⋊C4—42no—
250Dic3⋊C4—42no—
251Dic3⋊C4—42no—
252Dic3⋊C4—42no—
253Dic3⋊C4—81no—
254C4×Dic3—21no—
255C4×Dic3—21no—
256C4×Dic3—42no—
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—
264ΓL(1,𝔽25)—44no—
265C22×Q8—21no—
266C22×Q8—31no—
267C23×C4—31no—
268C23×C4—61no—
269C2×M16—62no—
270C2×M16—121no—
271C2×M16—121no—
272C22×C8—31no—
273C22×C8—31no—
274C22×C8C22×D311yescharacteristic
275C22×C8C22×D311yescharacteristic
276C4⋊Q8—22no—
277C4⋊Q8—41no—
278C22⋊Q8—34no—
279C22⋊Q8—62no—
280C22⋊Q8—124no—
281C4×Q8—62no—
282C4×Q8—121no—
283#117—62no—
284#117—121no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—62no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—121no—
299C2×C4⋊C4C22×D311yescharacteristic
300C2×C4⋊C4C22×D311yescharacteristic
301C2×C4⋊C4C22×D311yescharacteristic
302C2×C4⋊C4C22×D311yescharacteristic
303C2×C22⋊C4—31no—
304C2×C22⋊C4—31no—
305C2×C22⋊C4—61no—
306C2×C22⋊C4—61no—
307C2×C22⋊C4—61no—
308C2×C42—21no—
309C2×C42—31no—
310C2×C42—61no—
311C2×C42—61no—
312C2×C42C22×D311yescharacteristic
313#107—62no—
314#106—121no—
315C4⋊C8—41no—
316C4⋊C8—121no—
317#103—62no—
318#103—121no—
319#103—121no—
320#103—121no—
321C22⋊C8—62no—
322#95—21no—
323#95—31no—
324#95—31no—
325#95—61no—
326#95—61no—
327#95—61no—
328#95—61no—
329#95—61no—
330#95—61no—
331C23×C3C22×D411yescharacteristic
332C22×D3—21no—
333C22×D3—21no—
334C22×D3—21no—
335C22×D3—21no—
336C22×D3—21no—
337C22×D3—21no—
338C22×D3C22⋊Q812yes—
339C3×Q8—84no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
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×C4AGL(1,ℤ/8)12yes—
358C2×C3×C4C22×Q811yescharacteristic
359C2×C3×C4C2×#4111yescharacteristic
360C2×C3×C4C2×#4111yescharacteristic
361C2×C3×C4C2×#4111yescharacteristic
362C2×C3×C4H(5,𝔽2)11yescharacteristic
363C2×C3×C4H(5,𝔽2)11yescharacteristic
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—22no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3C22⋊Q812yes—
384C2×Dic3#13712yes—
385C4×D3—22no—
386C4×D3—22no—
387C4×D3—22no—
388C4×D3—22no—
389C4×D3—84no—
390Dic6—22no—
391Dic6—22no—
392Dic6—41no—
393Dic6—41no—
394C3×C8—41no—
395C3×C8—41no—
396C3⋊C8—41no—
397C3⋊C8—41no—
398C24—31no—
399C2×Q8—32no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8—61no—
403C2×Q8—61no—
404C2×Q8—61no—
405C2×Q8—61no—
406C2×Q8—61no—
407C2×Q8—81no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—31no—
413C22×C4—31no—
414C22×C4—32no—
415C22×C4—32no—
416C22×C4—31no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—122no—
426C22×C4—122no—
427C22×C4—241no—
428C22×C4D3×D411yescharacteristic
429C22×C4D3×D411yescharacteristic
430C22×C4C23×D311yescharacteristic
431M16—124no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—61no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8—61no—
442C2×C8—61no—
443C2×C8—61no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—42no—
450C4⋊C4—64no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C4⋊C4—122no—
458C4⋊C4—122no—
459C4⋊C4—122no—
460C4⋊C4—122no—
461C4⋊C4—122no—
462C4⋊C4—122no—
463C4⋊C4—241no—
464C4⋊C4#23512yes—
465C4⋊C4D3×Q812yes—
466C4⋊C4#23912yes—
467C22⋊C4—62no—
468C22⋊C4—64no—
469C22⋊C4—62no—
470C22⋊C4—122no—
471C22⋊C4—122no—
472C42—21no—
473C42—21no—
474C42—42no—
475C42—61no—
476C42—61no—
477C42—122no—
478C42—122no—
479C22×C3C2×C22⋊Q811yescharacteristic
480C22×C3#54611yescharacteristic
481C22×C3#54611yescharacteristic
482C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
483C22×C3C2×#13711yescharacteristic
484C22×C3#57711yescharacteristic
485C22×C3#57811yescharacteristic
486C2×D3—22no—
487C2×D3—22no—
488C2×D3—22no—
489C2×D3—41no—
490C2×D3—41no—
491C2×D3—41no—
492C2×D3—41no—
493C3×C4—22no—
494C3×C4—41no—
495C3×C4—42no—
496C3×C4—42no—
497C3×C4—42no—
498C3×C4—81no—
499C3×C4#52911yescharacteristic
500C3×C4#53811yescharacteristic
501C3×C4#55411yescharacteristic
502C3×C4#55411yescharacteristic
503Dic3—22no—
504Dic3—22no—
505Dic3—41no—
506Dic3—42no—
507Dic3—81no—
508Dic3—81no—
509Dic3—81no—
510C23—32no—
511C23—61no—
512C23—61no—
513C23—61no—
514C23—61no—
515C23—61no—
516C23—61no—
517C23C2×D3×D411yescharacteristic
518Q8—62no—
519Q8—62no—
520Q8—84no—
521Q8—121no—
522Q8—121no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—32no—
532C2×C4—32no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—61no—
543C2×C4—61no—
544C2×C4—61no—
545C2×C4—61no—
546C2×C4—62no—
547C2×C4—62no—
548C2×C4—62no—
549C2×C4—62no—
550C2×C4—61no—
551C2×C4—61no—
552C2×C4—62no—
553C2×C4—61no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—244no—
566C2×C4#91112yes—
567C2×C4C2×#23511yescharacteristic
568C2×C4#100411yescharacteristic
569C2×C4C2×D3×Q811yescharacteristic
570C2×C4C2×#23911yescharacteristic
571C2×C4D3×#4111yescharacteristic
572C2×C4#100911yescharacteristic
573C8—41no—
574C8—41no—
575C8—121no—
576C8—121no—
577C2×C3#245011yescharacteristic
578C2×C3#307911yescharacteristic
579C2×C3#308211yescharacteristic
580C2×C3#331011yescharacteristic
581C2×C3#331611yescharacteristic
582C2×C3#332811yescharacteristic
583C2×C3#333011yescharacteristic
584D3—42no—
585C22—62no—
586C22—62no—
587C22—62no—
588C22—121no—
589C22—121no—
590C22—121no—
591C22—121no—
592C22#549011yescharacteristic
593C22D3×C22⋊Q811yescharacteristic
594C22#563011yescharacteristic
595C22C2×#91111yescharacteristic
596C22#576211yescharacteristic
597C22D3×#13711yescharacteristic
598C22#577911yescharacteristic
599C4—22no—
600C4—42no—
601C4—42no—
602C4—41no—
603C4—42no—
604C4—62no—
605C4—62no—
606C4—81no—
607C4—122no—
608C4—121no—
609C4—241no—
610C4—241no—
611C4—241no—
612C4#557911yescharacteristic
613C4#558011yescharacteristic
614C4#567811yescharacteristic
615C4#573011yescharacteristic
616C3#2642911yescharacteristic, complemented, Sylow
617C2—122no—
618C2#7662811yescharacteristic
619C2#7876211yescharacteristic
620C2#7897211yescharacteristic
621C2#8096611yescharacteristic
622C2#8098511yescharacteristic
623C2#8101311yescharacteristic
624C2#8102411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1988C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#1988D3
hypocenterC3#26429
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1988

Complements

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

NH ≅ G/Ntype
3. C3×#1988C2semidirect
4. #71794C2semidirect
5. #70488C2semidirect
7. #69458C2semidirect
8. #68906C2semidirect
9. #68904C2semidirect
14. #68843C2semidirect
15. #68840C2semidirect
66. #1988D3semidirect
116. #897Q8semidirect
616. C3#26429semidirect

Maximal subgroups

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

subgroupindex
#832332
#736472
C3×#19882
#717942
#704882
#704782
#694582
#689062
#689042
#688842
#688782
#688762
#688672
#688432
#688402
#264293