Finite Group Databasebeta

← #10797365

Subgroups — #10797365

Order 768.

974 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4881C211yesmaximal, characteristic
2D3×#389C211yesmaximal, characteristic
3C4×#892C211yesmaximal, characteristic
4C3×#1918C211yesmaximal, characteristic, complemented
5#70610C211yesmaximal, characteristic, complemented
6#69601C211yesmaximal, characteristic, complemented
7#69458C211yesmaximal, characteristic, complemented
8#68867C211yesmaximal, characteristic
9#68867C211yesmaximal, characteristic
10#68867C211yesmaximal, characteristic
11#68819C211yesmaximal, characteristic, complemented
12#68818C211yesmaximal, characteristic, complemented
13#68790C211yesmaximal, characteristic, complemented
14#68660C211yesmaximal, characteristic
15#68622C211yesmaximal, characteristic, complemented
16#21034—31nomaximal, Sylow
17C22×ΓL(1,𝔽25)C2211yescharacteristic
18C22×ΓL(1,𝔽25)C2211yescharacteristic
19C2×#892C2211yescharacteristic
20C2×#892C2211yescharacteristic
21C2×D3×C4⋊C4—21no—
22C2×D3×C4⋊C4C2211yescharacteristic
23C2×D3×C4⋊C4C2211yescharacteristic
24C2×C42×D3C2211yescharacteristic
25C2×C3×#107C2211yescharacteristic
26C3×#389C2211yescharacteristic
27C3×C4×C4⋊C4C2211yescharacteristic
28C2×#817C2211yescharacteristic
29#4984C2211yescharacteristic
30#4976—21no—
31Dic3×C4⋊C4C2211yescharacteristic
32Dic3×C4⋊C4C2211yescharacteristic
33#4972C2211yescharacteristic
34C2×#808C2211yescharacteristic
35C2×#807C2211yescharacteristic
36C4×(C2×D3)⋊C4—21no—
37C4×C4⋊Dic3C2211yescharacteristic
38C42×Dic3—21no—
39#4928C2211yescharacteristic
40#4881C418yescomplemented
41#4666—21no—
42D3×#95—21no—
43#4648—21no—
44#4647—21no—
45C3×#340C2211yescharacteristic
46C3×#340C2211yescharacteristic
47C3×#340C2211yescharacteristic
48C3×#336C2211yescharacteristic
49#4551C2211yescharacteristic
50#4551C2211yescharacteristic
51#4551C2211yescharacteristic
52#4550C2211yescharacteristic
53#4550C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61C2×#428—31no—
62C2×#389—31no—
63C4×#117—31no—
64#1921—31no—
65#1921—31no—
66#1920—31no—
67#1920—31no—
68#1919—31no—
69#1919—31no—
70#1918—31no—
71#1918D311yescharacteristic, complemented
72#1736—31no—
73#1736—31no—
74#1736—31no—
75#1728—31no—
76C22×C4×D3—21no—
77C22×C4×D3—21no—
78C22×C4×D3D411yescharacteristic
79C22×C4×D3D411yescharacteristic
80C22×C4×D3C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
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
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3C2311yescharacteristic
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×C4×Dic3—21no—
102C2×C4×Dic3—21no—
103C2×C4×Dic3—21no—
104C2×C4×Dic3—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3D411yescharacteristic
107C2×C4×Dic3D411yescharacteristic
108C2×C4×Dic3C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111C2×ΓL(1,𝔽25)—22no—
112C2×ΓL(1,𝔽25)—22no—
113C2×ΓL(1,𝔽25)—22no—
114C2×ΓL(1,𝔽25)—22no—
115C2×ΓL(1,𝔽25)—22no—
116C2×ΓL(1,𝔽25)C2×C414yes—
117#892—22no—
118#892—22no—
119#892C2×C414yes—
120#892Q814yes—
121D3×C4⋊C4—22no—
122D3×C4⋊C4—22no—
123D3×C4⋊C4—22no—
124D3×C4⋊C4—44no—
125D3×C4⋊C4C2×C414yes—
126C42×D3—22no—
127C42×D3—22no—
128C3×#107C2×C414yescomplemented
129C3×#95—21no—
130C3×#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137#817C2×C414yescomplemented
138#808C2×C414yescomplemented
139#807C2×C414yescomplemented
140C22×M16—31no—
141C22×M16—31no—
142C2×#117—31no—
143C2×#117—31no—
144C22×C4⋊C4—31no—
145C22×C4⋊C4—31no—
146C22×C4⋊C4—61no—
147C22×C42—31no—
148#428—38no—
149C2×#107—31no—
150C2×#107C2×D311yescharacteristic
151C2×#106—31no—
152C2×#106—31no—
153#389—31no—
154#389—31no—
155#389—31no—
156#389—61no—
157#389—61no—
158#389C2×D311yescharacteristic
159C4×C4⋊C4—31no—
160C4×C4⋊C4—31no—
161C4×C4⋊C4—31no—
162C4×C4⋊C4C2×D311yescharacteristic
163C4×C22⋊C4—61no—
164C4×C22⋊C4—61no—
165#376—61no—
166C2×#95—61no—
167C43—61no—
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340C2×D311yescharacteristic
178#340C2×D311yescharacteristic
179#340C2×D311yescharacteristic
180#336—31no—
181#336—31no—
182#336—31no—
183#336C2×D311yescharacteristic
184C23×D3C2×D411yescharacteristic
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2411yescharacteristic
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3C2×D411yescharacteristic
197C22×Dic3C2×D411yescharacteristic
198C22×Dic3C2×D411yescharacteristic
199C2×C4×D3—24no—
200C2×C4×D3—22no—
201C2×C4×D3—22no—
202C2×C4×D3—22no—
203C2×C4×D3—22no—
204C2×C4×D3—41no—
205C2×C4×D3—41no—
206C2×C4×D3—42no—
207C2×C4×D3—42no—
208C2×C4×D3—42no—
209C2×C4×D3—42no—
210C2×C4×D3—42no—
211C2×C4×D3C22×C412yes—
212C2×C4×D3C2×Q812yes—
213C2×C4×D3#4112yes—
214C2×C4×D3#4112yes—
215C2×C4×D3#4112yes—
216C2×C4×D3#4112yes—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8C22×C412yes—
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⋊C4C22×C412yes—
230C3×C4⋊C4C22×C412yes—
231C3×C4⋊C4C2×Q812yes—
232C3×C42—21no—
233C3×C42—21no—
234C3×C42—42no—
235(C2×D3)⋊C4—24no—
236(C2×D3)⋊C4—24no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—42no—
243C4⋊Dic3C22×C412yes—
244C4⋊Dic3C22×C412yes—
245C4⋊Dic3C2×Q812yes—
246Dic3⋊C4—24no—
247Dic3⋊C4—42no—
248Dic3⋊C4—42no—
249Dic3⋊C4—42no—
250C4×Dic3—21no—
251C4×Dic3—21no—
252C4×Dic3—24no—
253C4×Dic3—24no—
254C4×Dic3—22no—
255C4×Dic3—22no—
256C4×Dic3—42no—
257C4×Dic3—42no—
258C4×Dic3—42no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8C22×C412yes—
265ΓL(1,𝔽25)—28no—
266ΓL(1,𝔽25)—44no—
267C23×C4—31no—
268C23×C4—31no—
269C23×C4—31no—
270C23×C4—61no—
271C23×C4—61no—
272C2×M16—34no—
273C2×M16—62no—
274C2×M16—62no—
275C2×M16—62no—
276C2×M16—62no—
277C2×M16—62no—
278C22×C8—31no—
279C22×C8—31no—
280C22×C8C22×D311yescharacteristic
281C22×C8C22×D311yescharacteristic
282#117—34no—
283#117—34no—
284#117—62no—
285#117—62no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—34no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—62no—
293C2×C4⋊C4—62no—
294C2×C4⋊C4—62no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—124no—
301C2×C4⋊C4C22×D311yescharacteristic
302C2×C4⋊C4C22×D311yescharacteristic
303C2×C4⋊C4C22×D311yescharacteristic
304C2×C4⋊C4C22×D311yescharacteristic
305C2×C22⋊C4—61no—
306C2×C22⋊C4—61no—
307C2×C42—21no—
308C2×C42—31no—
309C2×C42—31no—
310C2×C42—31no—
311C2×C42—62no—
312C2×C42—62no—
313C2×C42—61no—
314C2×C42—61no—
315C2×C42—61no—
316C2×C42—61no—
317C2×C42—61no—
318C2×C42C22×D311yescharacteristic
319#107—34no—
320#107C4×D314yescomplemented
321#106—34no—
322#106—34no—
323#95—21no—
324#95—21no—
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×D3C4×D412yes—
337C22×D3C22⋊Q812yes—
338C22×D3#12312yes—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—41no—
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×C4AGL(1,ℤ/8)12yes—
357C2×C3×C4C23×C411yescharacteristic
358C2×C3×C4C22×Q811yescharacteristic
359C2×C3×C4C2×#4111yescharacteristic
360C2×C3×C4C2×#4111yescharacteristic
361C2×C3×C4C2×#4111yescharacteristic
362C2×C3×C4C2×#4111yescharacteristic
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—22no—
366C2×Dic3—22no—
367C2×Dic3—22no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—41no—
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×Dic3C4×D412yes—
381C2×Dic3C22⋊Q812yes—
382C2×Dic3#12312yes—
383C2×Dic3#13712yes—
384C4×D3—44no—
385C4×D3—44no—
386C4×D3—48no—
387C4×D3—84no—
388C4×D3#11714yes—
389C4×D3C4×Q814yes—
390C4×D3#12514yes—
391C4×D3#12614yes—
392C3×C8—22no—
393C3×C8—41no—
394C3⋊C8—22no—
395C3⋊C8—41no—
396C24—31no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—31no—
402C22×C4—32no—
403C22×C4—32no—
404C22×C4—32no—
405C22×C4—32no—
406C22×C4—32no—
407C22×C4—32no—
408C22×C4—31no—
409C22×C4—31no—
410C22×C4—62no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—64no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4—62no—
418C22×C4—61no—
419C22×C4—121no—
420C22×C4—121no—
421C22×C4—122no—
422C22×C4—122no—
423C22×C4—122no—
424C22×C4—122no—
425C22×C4—122no—
426C22×C4D3×D411yescharacteristic
427C22×C4D3×D411yescharacteristic
428C22×C4C23×D311yescharacteristic
429M16—68no—
430M16—124no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—32no—
437C2×C8—61no—
438C2×C8—61no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8—61no—
442C2×C8C2×C4×D312yes—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—32no—
449C4⋊C4—32no—
450C4⋊C4—32no—
451C4⋊C4—42no—
452C4⋊C4—64no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—122no—
459C4⋊C4—122no—
460C4⋊C4—122no—
461C4⋊C4—122no—
462C4⋊C4C2×C4×D312yes—
463C4⋊C4C2×C4×D312yes—
464C4⋊C4D3×Q812yes—
465C22⋊C4—64no—
466C22⋊C4—64no—
467C42—21no—
468C42—21no—
469C42—42no—
470C42—62no—
471C42—62no—
472C42—64no—
473C42—64no—
474C42—61no—
475C42—61no—
476C42—122no—
477C42—122no—
478C42—122no—
479C22×C3C2×C4×D411yescharacteristic
480C22×C3C2×C22⋊Q811yescharacteristic
481C22×C3C2×#12311yescharacteristic
482C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
483C22×C3C2×#13711yescharacteristic
484C22×C3#57511yescharacteristic
485C22×C3#57511yescharacteristic
486C2×D3—22no—
487C2×D3—22no—
488C2×D3—22no—
489C2×D3—22no—
490C2×D3—22no—
491C2×D3—22no—
492C2×D3#38714yes—
493C3×C4—22no—
494C3×C4—42no—
495C3×C4—41no—
496C3×C4—42no—
497C3×C4—42no—
498C3×C4—81no—
499C3×C4C2×#11711yescharacteristic
500C3×C4C2×C4×Q811yescharacteristic
501C3×C4C2×#12511yescharacteristic
502C3×C4C2×#12611yescharacteristic
503Dic3—22no—
504Dic3—42no—
505Dic3—41no—
506Dic3—42no—
507Dic3—42no—
508Dic3—81no—
509Dic3#38714yes—
510C23—32no—
511C23—32no—
512C23—32no—
513C23—61no—
514C23—61no—
515C23—61no—
516C23—61no—
517C23C2×D3×D411yescharacteristic
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—32no—
527C2×C4—32no—
528C2×C4—32no—
529C2×C4—34no—
530C2×C4—34no—
531C2×C4—34no—
532C2×C4—34no—
533C2×C4—32no—
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—41no—
543C2×C4—61no—
544C2×C4—61no—
545C2×C4—61no—
546C2×C4—61no—
547C2×C4—61no—
548C2×C4—62no—
549C2×C4—62no—
550C2×C4—62no—
551C2×C4—124no—
552C2×C4—124no—
553C2×C4—128no—
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—244no—
564C2×C4#91112yes—
565C2×C4C22×C4×D311yescharacteristic
566C2×C4C2×D3×Q811yescharacteristic
567C2×C4D3×#4111yescharacteristic
568C2×C4D3×#4111yescharacteristic
569C2×C4D3×#4111yescharacteristic
570C2×C4D3×#4111yescharacteristic
571C8—22no—
572C8—41no—
573C8—62no—
574C8—121no—
575C2×C3C2×#38711yescharacteristic
576C2×C3C4×AGL(1,ℤ/8)11yescharacteristic
577C2×C3C4×#13711yescharacteristic
578C2×C3#307911yescharacteristic
579C2×C3#308011yescharacteristic
580C2×C3#309411yescharacteristic
581C2×C3#309511yescharacteristic
582D3—24no—
583C22—34no—
584C22—62no—
585C22—62no—
586C22—62no—
587C22—62no—
588C22—62no—
589C22—62no—
590C22C4×D3×D411yescharacteristic
591C22D3×C22⋊Q811yescharacteristic
592C22D3×#12311yescharacteristic
593C22C2×#91111yescharacteristic
594C22#576811yescharacteristic
595C22D3×#13711yescharacteristic
596C22#577711yescharacteristic
597C4—22no—
598C4—34no—
599C4—42no—
600C4—41no—
601C4—42no—
602C4—42no—
603C4—62no—
604C4—81no—
605C4—122no—
606C4—121no—
607C4—122no—
608C4—122no—
609C4—241no—
610C4D3×#11711yescharacteristic
611C4C4×D3×Q811yescharacteristic
612C4D3×#12511yescharacteristic
613C4D3×#12611yescharacteristic
614C3#2103411yescharacteristic, complemented, Sylow
615C2—64no—
616C2D3×#38711yescharacteristic
617C2C4×#91111yescharacteristic
618C2#7982911yescharacteristic
619C2#8096611yescharacteristic
620C2#8123111yescharacteristic
621C2#8128311yescharacteristic
622C2#8151711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1918C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
cosocleC22×C3×C4C24
hypercenter#1918D3
hypocenterC3#21034
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1918

Complements

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

NH ≅ G/Ntype
4. C3×#1918C2semidirect
5. #70610C2semidirect
6. #69601C2semidirect
7. #69458C2semidirect
11. #68819C2semidirect
12. #68818C2semidirect
13. #68790C2semidirect
15. #68622C2semidirect
40. #4881C4semidirect
71. #1918D3semidirect
128. C3×#107C2×C4semidirect
137. #817C2×C4semidirect
138. #808C2×C4semidirect
139. #807C2×C4semidirect
320. #107C4×D3semidirect
614. C3#21034semidirect

Maximal subgroups

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

subgroupindex
C2×#48812
D3×#3892
C4×#8922
C3×#19182
#706102
#696012
#694582
#688672
#688192
#688182
#687902
#686602
#686222
#210343