Finite Group Databasebeta

← #10812464

Subgroups — #10812464

Order 768.

934 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4802C211yesmaximal, characteristic
2#76614C211yesmaximal, characteristic
3#73634C211yesmaximal, characteristic
4C3×#1927C211yesmaximal, characteristic, complemented
5#70588C211yesmaximal, characteristic, complemented
6#70578C211yesmaximal, characteristic
7#70266C211yesmaximal, characteristic
8#70254C211yesmaximal, characteristic, complemented
9#70224C211yesmaximal, characteristic, complemented
10#69675C211yesmaximal, characteristic, complemented
11#69475C211yesmaximal, characteristic, complemented
12#68873C211yesmaximal, characteristic
13#68781C211yesmaximal, characteristic, complemented
14#68683C211yesmaximal, characteristic, complemented
15#68650C211yesmaximal, characteristic
16#22581—31nomaximal, Sylow
17C22×D3×Q8C2211yescharacteristic
18C22×ΓL(1,𝔽25)C2211yescharacteristic
19C2×#892C2211yescharacteristic
20C2×#887C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C2×C3×#106C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C3×#391C2211yescharacteristic
26C3×C4×C4⋊C4C2211yescharacteristic
27#5229C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×(C2×D3)⋊C8C2211yescharacteristic
31C2×#818C2211yescharacteristic
32C2×#816C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×C8×Dic3C2211yescharacteristic
35#4987C2211yescharacteristic
36#4987C2211yescharacteristic
37Dic3×C4⋊C4C2211yescharacteristic
38#4972C2211yescharacteristic
39C2×#810C2211yescharacteristic
40C2×#810C2211yescharacteristic
41#4940—21no—
42C4×(C2×D3)⋊C4C2211yescharacteristic
43C4×Dic3⋊C4—21no—
44#4928C2211yescharacteristic
45C2×#803C2211yescharacteristic
46#4802C418yescomplemented
47#4669—21no—
48#4667—21no—
49#4666—21no—
50#4651—21no—
51#4646—21no—
52#4646—21no—
53C3×#340C2211yescharacteristic
54C3×#336C2211yescharacteristic
55#4551C2211yescharacteristic
56#4550C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61C2×#419—31no—
62#2347—31no—
63Q8×C22⋊C4—31no—
64#1935—31no—
65#1933—31no—
66#1928—31no—
67#1927D311yescharacteristic, complemented
68#1865—31no—
69#1863—31no—
70#1825—31no—
71#1809—31no—
72#1787—31no—
73#1763—31no—
74C4×#103—31no—
75#1753—31no—
76C22×C3×Q8C2311yescharacteristic
77C2×D3×Q8—22no—
78C2×D3×Q8—22no—
79C2×D3×Q8—22no—
80C2×D3×Q8—41no—
81C2×D3×Q8—41no—
82C2×D3×Q8C2×C414yes—
83C22×C4×D3—21no—
84C22×C4×D3C2311yescharacteristic
85C22×Dic6—21no—
86C22×Dic6C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C42—21no—
93C2×C3×C42C2311yescharacteristic
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4D411yescharacteristic
98C2×(C2×D3)⋊C4D411yescharacteristic
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×C4×Dic3—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C2×ΓL(1,𝔽25)—41no—
114C2×ΓL(1,𝔽25)—41no—
115C2×ΓL(1,𝔽25)C2×C414yes—
116#892—22no—
117#892C2×C414yes—
118#887—22no—
119#887—41no—
120C4×Dic6—22no—
121C4×Dic6—41no—
122C3×#106—41no—
123C3×#103—41no—
124C3×#103C2×C414yes—
125C3×#95—21no—
126C3×#95—21no—
127C3×#95—21no—
128#835—41no—
129#835C2×C414yes—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95D411yescharacteristic
138C3⋊#95D411yescharacteristic
139(C2×D3)⋊C8—22no—
140#818—41no—
141#816—22no—
142#816C2×C414yes—
143C8×Dic3—41no—
144#810—22no—
145#810C2×C414yes—
146#803—41no—
147C23×Q8—31no—
148C22×M16—31no—
149C2×C4×Q8—31no—
150C2×C4×Q8—31no—
151C2×#117—31no—
152C2×#107—31no—
153C2×#106C2×D311yescharacteristic
154#419—38no—
155C2×#103—31no—
156C2×#103—31no—
157C2×#103—31no—
158C2×#103—31no—
159C2×#103—31no—
160C2×#103—31no—
161C2×#103C2×D311yescharacteristic
162C2×#103C2×D311yescharacteristic
163C2×C22⋊C8—31no—
164C2×#97—31no—
165C2×C4×C8—31no—
166#391—31no—
167#391—61no—
168#391—61no—
169#391C2×D311yescharacteristic
170#389—31no—
171#389—31no—
172#388—61no—
173#388—61no—
174#383—61no—
175#380—31no—
176#380—31no—
177#380—61no—
178C4×C4⋊C4—31no—
179C4×C4⋊C4—61no—
180C4×C4⋊C4C2×D311yescharacteristic
181C4×C22⋊C4—31no—
182C4×C22⋊C4—61no—
183#340—31no—
184#340—31no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340C2×D311yescharacteristic
189#336—31no—
190#336C2×D311yescharacteristic
191C23×D3C2×D411yescharacteristic
192C2×C3×Q8—21no—
193C2×C3×Q8—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8C22×C412yes—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2411yescharacteristic
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3C2×D411yescharacteristic
210C22×Dic3C2×D411yescharacteristic
211C22×Dic3C2×D411yescharacteristic
212D3×Q8—28no—
213D3×Q8—44no—
214D3×Q8—44no—
215D3×Q8—44no—
216C2×C4×D3—21no—
217C2×C4×D3—21no—
218C2×C4×D3—21no—
219C2×C4×D3—21no—
220C2×C4×D3—24no—
221C2×C4×D3—42no—
222C2×C4×D3—42no—
223C2×C4×D3C22×C412yes—
224C2×C4×D3#4112yes—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—24no—
229C2×Dic6—42no—
230C2×Dic6—42no—
231C2×Dic6C22×C412yes—
232C2×Dic6#4112yes—
233C2×Dic6#4112yes—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8C22×C412yes—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4C22×C412yes—
246C3×C42—21no—
247C3×C42—21no—
248C3×C42—81no—
249(C2×D3)⋊C4—24no—
250(C2×D3)⋊C4—22no—
251(C2×D3)⋊C4—41no—
252(C2×D3)⋊C4—42no—
253(C2×D3)⋊C4—81no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3C22×C412yes—
260Dic3⋊C4—42no—
261Dic3⋊C4—42no—
262Dic3⋊C4—42no—
263Dic3⋊C4—42no—
264C4×Dic3—21no—
265C4×Dic3—21no—
266C4×Dic3—24no—
267C4×Dic3—42no—
268C4×Dic3—42no—
269C4×Dic3—81no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8C22×C412yes—
276ΓL(1,𝔽25)—44no—
277C22×Q8—34no—
278C22×Q8—31no—
279C22×Q8—61no—
280C22×Q8—62no—
281C22×Q8—62no—
282C22×Q8—62no—
283C22×Q8—121no—
284C22×Q8—121no—
285C22×Q8C22×D311yescharacteristic
286C23×C4—31no—
287C23×C4—61no—
288C2×M16—34no—
289C2×M16—121no—
290C2×M16—121no—
291C22×C8—31no—
292C22×C8—31no—
293C22×C8C22×D311yescharacteristic
294C22×C8C22×D311yescharacteristic
295C4×Q8—62no—
296C4×Q8—62no—
297C4×Q8—121no—
298C4×Q8—121no—
299#117—34no—
300#117—62no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4C22×D311yescharacteristic
309C2×C4⋊C4C22×D311yescharacteristic
310C2×C4⋊C4C22×D311yescharacteristic
311C2×C22⋊C4—31no—
312C2×C22⋊C4—31no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
316C2×C42—21no—
317C2×C42—31no—
318C2×C42—61no—
319C2×C42—61no—
320C2×C42—61no—
321C2×C42—61no—
322C2×C42C22×D311yescharacteristic
323#107—121no—
324#106—41no—
325#103—34no—
326#103—34no—
327#103—34no—
328#103—41no—
329#103—62no—
330#103—62no—
331#103—121no—
332#103C4×D314yes—
333C22⋊C8—62no—
334#97—121no—
335C4×C8—121no—
336#95—21no—
337#95—21no—
338#95—21no—
339#95—31no—
340#95—31no—
341#95—61no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346#95—61no—
347#95—61no—
348C23×C3C22×D411yescharacteristic
349C22×D3—21no—
350C22×D3—21no—
351C22×D3—21no—
352C22×D3—21no—
353C22×D3—21no—
354C22×D3—21no—
355C22×D3C4×D412yes—
356C3×Q8—22no—
357C3×Q8—22no—
358C3×Q8—41no—
359C3×Q8—41no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4C23×C411yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380C2×C3×C4C2×#4111yescharacteristic
381C2×C3×C4C2×#4111yescharacteristic
382C2×C3×C4H(5,𝔽2)11yescharacteristic
383C2×C3×C4H(5,𝔽2)11yescharacteristic
384C2×Dic3—22no—
385C2×Dic3—21no—
386C2×Dic3—22no—
387C2×Dic3—22no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—22no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3C4×D412yes—
404C2×Dic3#13712yes—
405C4×D3—22no—
406C4×D3—22no—
407C4×D3—22no—
408C4×D3—22no—
409C4×D3—48no—
410Dic6—22no—
411Dic6—22no—
412Dic6—22no—
413Dic6—48no—
414Dic6#11714yes—
415C3×C8—22no—
416C3×C8—41no—
417C3⋊C8—22no—
418C3⋊C8—41no—
419C24—31no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—32no—
426C2×Q8—32no—
427C2×Q8—32no—
428C2×Q8—64no—
429C2×Q8—61no—
430C2×Q8—61no—
431C2×Q8—61no—
432C2×Q8—68no—
433C2×Q8—122no—
434C2×Q8—122no—
435C2×Q8—124no—
436C2×Q8—124no—
437C2×Q8—124no—
438C2×Q8C2×C4×D312yes—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—31no—
444C22×C4—31no—
445C22×C4—32no—
446C22×C4—32no—
447C22×C4—31no—
448C22×C4—64no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—61no—
457C22×C4—122no—
458C22×C4—122no—
459C22×C4D3×D411yescharacteristic
460C22×C4D3×D411yescharacteristic
461C22×C4C23×D311yescharacteristic
462M16—124no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—32no—
469C2×C8—61no—
470C2×C8—61no—
471C2×C8—61no—
472C2×C8—61no—
473C2×C8—61no—
474C2×C8C2×C4×D312yes—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—32no—
481C4⋊C4—61no—
482C4⋊C4—61no—
483C4⋊C4—61no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C4⋊C4—122no—
487C4⋊C4—122no—
488C4⋊C4—122no—
489C4⋊C4—122no—
490C4⋊C4C2×C4×D312yes—
491C22⋊C4—62no—
492C22⋊C4—64no—
493C22⋊C4—121no—
494C22⋊C4—122no—
495C22⋊C4—241no—
496C42—21no—
497C42—21no—
498C42—64no—
499C42—61no—
500C42—61no—
501C42—81no—
502C42—122no—
503C42—122no—
504C42—241no—
505C22×C3C2×C4×D411yescharacteristic
506C22×C3#54611yescharacteristic
507C22×C3#54611yescharacteristic
508C22×C3C2×#13711yescharacteristic
509C22×C3#57511yescharacteristic
510C22×C3#57711yescharacteristic
511C22×C3#57811yescharacteristic
512C2×D3—22no—
513C2×D3—22no—
514C2×D3—22no—
515C2×D3—41no—
516C2×D3—41no—
517C2×D3—41no—
518C2×D3—41no—
519C3×C4—41no—
520C3×C4—42no—
521C3×C4—42no—
522C3×C4—41no—
523C3×C4—81no—
524C3×C4—81no—
525C3×C4C2×#11711yescharacteristic
526C3×C4#51811yescharacteristic
527C3×C4#55111yescharacteristic
528C3×C4#55111yescharacteristic
529Dic3—24no—
530Dic3—22no—
531Dic3—22no—
532Dic3—42no—
533Dic3—41no—
534Dic3—81no—
535Dic3—81no—
536C23—32no—
537C23—61no—
538C23—61no—
539C23—61no—
540C23—61no—
541C23—61no—
542C23—61no—
543C23C2×D3×D411yescharacteristic
544Q8—22no—
545Q8—22no—
546Q8—34no—
547Q8—41no—
548Q8—41no—
549Q8—62no—
550Q8—62no—
551Q8—62no—
552Q8—128no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—32no—
562C2×C4—32no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—62no—
574C2×C4—61no—
575C2×C4—61no—
576C2×C4—62no—
577C2×C4—61no—
578C2×C4—62no—
579C2×C4—61no—
580C2×C4—62no—
581C2×C4—62no—
582C2×C4—62no—
583C2×C4—62no—
584C2×C4—61no—
585C2×C4—61no—
586C2×C4—62no—
587C2×C4—61no—
588C2×C4—128no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—121no—
597C2×C4C22×C4×D311yescharacteristic
598C2×C4#100411yescharacteristic
599C2×C4D3×#4111yescharacteristic
600C2×C4D3×#4111yescharacteristic
601C2×C4D3×#4111yescharacteristic
602C2×C4#100911yescharacteristic
603C8—22no—
604C8—41no—
605C8—62no—
606C8—121no—
607C2×C3#234211yescharacteristic
608C2×C3C4×#13711yescharacteristic
609C2×C3#294711yescharacteristic
610C2×C3#331311yescharacteristic
611C2×C3#332011yescharacteristic
612C2×C3#332611yescharacteristic
613C2×C3#332811yescharacteristic
614D3—42no—
615C22—62no—
616C22—62no—
617C22—62no—
618C22—121no—
619C22—121no—
620C22—121no—
621C22—121no—
622C22C4×D3×D411yescharacteristic
623C22#561211yescharacteristic
624C22#565411yescharacteristic
625C22#576211yescharacteristic
626C22#577111yescharacteristic
627C22D3×#13711yescharacteristic
628C22#577711yescharacteristic
629C4—41no—
630C4—42no—
631C4—42no—
632C4—41no—
633C4—62no—
634C4—64no—
635C4—62no—
636C4—81no—
637C4—81no—
638C4—122no—
639C4—121no—
640C4—241no—
641C4—241no—
642C4D3×#11711yescharacteristic
643C4#554511yescharacteristic
644C4#567911yescharacteristic
645C4#570511yescharacteristic
646C3#2258111yescharacteristic, complemented, Sylow
647C2—122no—
648C2#7583211yescharacteristic
649C2#7964811yescharacteristic
650C2#7982911yescharacteristic
651C2#8045311yescharacteristic
652C2#8061811yescharacteristic
653C2#8143411yescharacteristic
654C2#8155411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1927C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
cosocleC22×C3×C4C24
hypercenter#1927D3
hypocenterC3#22581
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1927

Complements

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

NH ≅ G/Ntype
4. C3×#1927C2semidirect
5. #70588C2semidirect
8. #70254C2semidirect
9. #70224C2semidirect
10. #69675C2semidirect
11. #69475C2semidirect
13. #68781C2semidirect
14. #68683C2semidirect
46. #4802C4semidirect
67. #1927D3semidirect
646. C3#22581semidirect

Maximal subgroups

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

subgroupindex
C2×#48022
#766142
#736342
C3×#19272
#705882
#705782
#702662
#702542
#702242
#696752
#694752
#688732
#687812
#686832
#686502
#225813