Finite Group Databasebeta

← #10858049

Subgroups — #10858049

Order 768.

812 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4861C211yesmaximal, characteristic
2C2×#4796C211yesmaximal, characteristic, complemented
3#76615C211yesmaximal, characteristic
4#76000C211yesmaximal, characteristic
5C3×#2066C211yesmaximal, characteristic, complemented
6#70566C211yesmaximal, characteristic, complemented
7#70478C211yesmaximal, characteristic
8#70453C211yesmaximal, characteristic, complemented
9#70271C211yesmaximal, characteristic
10#70269C211yesmaximal, characteristic
11#70236C211yesmaximal, characteristic, complemented
12#69684C211yesmaximal, characteristic, complemented
13#69601C211yesmaximal, characteristic, complemented
14#68874C211yesmaximal, characteristic
15#68807C211yesmaximal, characteristic, complemented
16#26362—31nomaximal, Sylow
17C2×#946C2211yescharacteristic
18C2×Q8×Dic3C2211yescharacteristic
19C22×ΓL(1,𝔽25)C2211yescharacteristic
20C2×#892C2211yescharacteristic
21C2×D3×C4⋊C4C2211yescharacteristic
22C2×#890C2211yescharacteristic
23C2×C3×#106C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C3×#391C2211yescharacteristic
28C3×#389C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×(C2×D3)⋊C8C2211yescharacteristic
32C2×#818C2211yescharacteristic
33C2×#818C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35#4994—21no—
36#4987C2211yescharacteristic
37#4984—21no—
38#4984C2211yescharacteristic
39#4972C2211yescharacteristic
40C2×#808C2211yescharacteristic
41C2×#807C2211yescharacteristic
42C2×#807C2211yescharacteristic
43C4×(C2×D3)⋊C4C2211yescharacteristic
44#4935C2211yescharacteristic
45C4×C4⋊Dic3C2211yescharacteristic
46#4861—24no—
47#4796—24no—
48#4677—21no—
49#4668—21no—
50#4666—21no—
51#4653—21no—
52#4652—21no—
53#4647—21no—
54C3×#340C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#479—31no—
63C2×#428—31no—
64#2441—31no—
65#2360—31no—
66#2095—31no—
67#2086—31no—
68#2066D311yescharacteristic, complemented
69#2059—31no—
70#1947—31no—
71#1919—31no—
72#1894—31no—
73#1873—31no—
74#1863—31no—
75#1852—31no—
76#1851—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C4×D3—21no—
79C22×C4×D3C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C42C2311yescharacteristic
87#946—22no—
88#946Q814yes—
89Q8×Dic3—22no—
90Q8×Dic3—41no—
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⋊Dic3—21no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C2×ΓL(1,𝔽25)—22no—
116C2×ΓL(1,𝔽25)—22no—
117C2×ΓL(1,𝔽25)—22no—
118#892—22no—
119#892—41no—
120D3×C4⋊C4—22no—
121D3×C4⋊C4—41no—
122#890—22no—
123#890—41no—
124C3×#106—22no—
125C3×C4⋊C8—22no—
126C3×#103—22no—
127C3×#103—22no—
128C3×#95—21no—
129C3×#95—21no—
130C3×#95—21no—
131#835—22no—
132#835—41no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95D411yescharacteristic
138C3⋊#95D411yescharacteristic
139(C2×D3)⋊C8—22no—
140#818—22no—
141#818—22no—
142Dic3⋊C8—22no—
143#808D414yes—
144#807—22no—
145#807—41no—
146C22×M16—31no—
147C2×#125—31no—
148C2×C22⋊Q8—31no—
149C2×C4×Q8—31no—
150C2×#117—31no—
151C22×C4⋊C4—31no—
152#479—64no—
153#428—64no—
154C2×#107—31no—
155C2×#107—31no—
156C2×#107—31no—
157C2×#107—31no—
158C2×#106—31no—
159C2×#106C2×D311yescharacteristic
160C2×C4⋊C8—31no—
161C2×C4⋊C8C2×D311yescharacteristic
162C2×#103—31no—
163C2×#103—31no—
164C2×#103C2×D311yescharacteristic
165C2×#103C2×D311yescharacteristic
166C2×C22⋊C8—31no—
167#399—61no—
168#396—61no—
169#391C2×D311yescharacteristic
170#389—31no—
171#389—31no—
172#389—61no—
173#389—61no—
174#389C2×D311yescharacteristic
175#387—61no—
176#387—61no—
177#385—61no—
178#384—31no—
179#380—31no—
180C4×C4⋊C4—31no—
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#336—31no—
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×Q8C2×Q812yes—
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
212C2×C4×D3—21no—
213C2×C4×D3—21no—
214C2×C4×D3—21no—
215C2×C4×D3—21no—
216C2×C4×D3—42no—
217C2×C4×D3—42no—
218C2×C4×D3—81no—
219C2×C4×D3C2×Q812yes—
220C2×C4×D3#4112yes—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8#4112yes—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—42no—
233C3×C4⋊C4C2×D412yes—
234C3×C42—21no—
235C3×C42—21no—
236(C2×D3)⋊C4—24no—
237(C2×D3)⋊C4—22no—
238(C2×D3)⋊C4—41no—
239(C2×D3)⋊C4—42no—
240(C2×D3)⋊C4—81no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—42no—
248C4⋊Dic3—42no—
249C4⋊Dic3—42no—
250C4⋊Dic3—81no—
251C4⋊Dic3—81no—
252C4⋊Dic3C2×D412yes—
253C4⋊Dic3C2×Q812yes—
254Dic3⋊C4—24no—
255Dic3⋊C4—42no—
256Dic3⋊C4—42no—
257Dic3⋊C4—81no—
258C4×Dic3—21no—
259C4×Dic3—21no—
260C4×Dic3—42no—
261C4×Dic3—42no—
262C4×Dic3—81no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8C2×D412yes—
269ΓL(1,𝔽25)—28no—
270C22×Q8C22×D311yescharacteristic
271C23×C4—31no—
272C23×C4—61no—
273C2×M16—62no—
274C2×M16—62no—
275C2×M16—62no—
276C22×C8—31no—
277C22×C8—31no—
278C22×C8C22×D311yescharacteristic
279C22×C8C22×D311yescharacteristic
280#125—62no—
281#125—121no—
282C22⋊Q8—34no—
283C22⋊Q8—62no—
284C4×Q8—62no—
285C4×Q8—121no—
286#117—62no—
287#117—121no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—62no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—121no—
304C2×C4⋊C4C22×D311yescharacteristic
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C22⋊C4—31no—
308C2×C22⋊C4—31no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C42—31no—
313C2×C42—61no—
314C2×C42—61no—
315C2×C42—61no—
316C2×C42C22×D311yescharacteristic
317#107—62no—
318#107—62no—
319#107—62no—
320#107—121no—
321#106—22no—
322#106—34no—
323C4⋊C8—22no—
324C4⋊C8—62no—
325#103—22no—
326#103—22no—
327#103—62no—
328#103—121no—
329C22⋊C8—62no—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—31no—
334#95—31no—
335#95—61no—
336#95—61no—
337#95—61no—
338#95—61no—
339C23×C3C22×D411yescharacteristic
340C22×D3—21no—
341C22×D3—21no—
342C22×D3—21no—
343C22×D3—21no—
344C22×D3—21no—
345C22×D3—21no—
346C22×D3C22⋊Q812yes—
347C3×Q8—22no—
348C3×Q8—22no—
349C3×Q8—41no—
350C3×Q8—41no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4#13712yes—
369C2×C3×C4C22×D411yescharacteristic
370C2×C3×C4C22×Q811yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4H(5,𝔽2)11yescharacteristic
374C2×C3×C42−1+411yescharacteristic
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—22no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—22no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3C22⋊Q812yes—
396C4×D3—22no—
397C4×D3—22no—
398C4×D3—22no—
399C4×D3—22no—
400C4×D3—84no—
401C3×C8—22no—
402C3×C8—41no—
403C3⋊C8—22no—
404C3⋊C8—41no—
405C24—31no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8D3×Q812yes—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—31no—
417C22×C4—32no—
418C22×C4—32no—
419C22×C4—31no—
420C22×C4—31no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4—61no—
429C22×C4—122no—
430C22×C4—122no—
431C22×C4—241no—
432C22×C4D3×D411yescharacteristic
433C22×C4D3×D411yescharacteristic
434C22×C4C23×D311yescharacteristic
435M16—68no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—32no—
442C2×C8—61no—
443C2×C8—61no—
444C2×C8—61no—
445C2×C8—61no—
446C2×C8—61no—
447C2×C8#23712yes—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—32no—
454C4⋊C4—32no—
455C4⋊C4—42no—
456C4⋊C4—64no—
457C4⋊C4—61no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4—122no—
464C4⋊C4—122no—
465C4⋊C4—122no—
466C4⋊C4—122no—
467C4⋊C4—122no—
468C4⋊C4—241no—
469C4⋊C4—241no—
470C4⋊C4—241no—
471C4⋊C4D3×D412yes—
472C22⋊C4—64no—
473C22⋊C4—62no—
474C22⋊C4—121no—
475C22⋊C4—122no—
476C22⋊C4—241no—
477C42—21no—
478C42—21no—
479C42—61no—
480C42—61no—
481C42—122no—
482C42—122no—
483C42—241no—
484C22×C3C2×C22⋊Q811yescharacteristic
485C22×C3D4211yescharacteristic
486C22×C3#54711yescharacteristic
487C22×C3C2×#13711yescharacteristic
488C22×C3#57511yescharacteristic
489C22×C3#57611yescharacteristic
490C22×C3#57811yescharacteristic
491C2×D3—22no—
492C2×D3—22no—
493C2×D3—22no—
494C2×D3—41no—
495C2×D3—41no—
496C2×D3—41no—
497C2×D3—41no—
498C3×C4—22no—
499C3×C4—41no—
500C3×C4—42no—
501C3×C4—42no—
502C3×C4—81no—
503C3×C4—81no—
504C3×C4#52511yescharacteristic
505C3×C4#53911yescharacteristic
506C3×C4D4×Q811yescharacteristic
507C3×C4#55411yescharacteristic
508Dic3—22no—
509Dic3—42no—
510Dic3—42no—
511Dic3—41no—
512Dic3—41no—
513Dic3—81no—
514Dic3—81no—
515C23—32no—
516C23—61no—
517C23—61no—
518C23—61no—
519C23—61no—
520C23—61no—
521C23—61no—
522C23C2×D3×D411yescharacteristic
523Q8—22no—
524Q8—22no—
525Q8—41no—
526Q8—41no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—32no—
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—41no—
544C2×C4—41no—
545C2×C4—61no—
546C2×C4—61no—
547C2×C4—62no—
548C2×C4—62no—
549C2×C4—62no—
550C2×C4—62no—
551C2×C4—61no—
552C2×C4—61no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—61no—
557C2×C4—62no—
558C2×C4—62no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—244no—
570C2×C4#91512yes—
571C2×C4C2×D3×D411yescharacteristic
572C2×C4C2×#23711yescharacteristic
573C2×C4C2×D3×Q811yescharacteristic
574C2×C4D3×#4111yescharacteristic
575C2×C4#100911yescharacteristic
576C2×C4#101011yescharacteristic
577C8—22no—
578C8—41no—
579C8—62no—
580C8—121no—
581C2×C3#244911yescharacteristic
582C2×C3#307811yescharacteristic
583C2×C3#308311yescharacteristic
584C2×C3#326811yescharacteristic
585C2×C3#327611yescharacteristic
586C2×C3#334611yescharacteristic
587C2×C3#335211yescharacteristic
588D3—42no—
589C22—62no—
590C22—62no—
591C22—62no—
592C22—121no—
593C22—121no—
594C22—121no—
595C22—121no—
596C22#560911yescharacteristic
597C22#561111yescharacteristic
598C22D3×C22⋊Q811yescharacteristic
599C22C2×#91511yescharacteristic
600C22#576911yescharacteristic
601C22#577711yescharacteristic
602C22#577911yescharacteristic
603C4—22no—
604C4—41no—
605C4—42no—
606C4—42no—
607C4—62no—
608C4—81no—
609C4—81no—
610C4—121no—
611C4—122no—
612C4—121no—
613C4—122no—
614C4—241no—
615C4—241no—
616C4#560611yescharacteristic
617C4#563211yescharacteristic
618C4#563511yescharacteristic
619C4#570711yescharacteristic
620C3#2636211yescharacteristic, complemented, Sylow
621C2—122no—
622C2#7741911yescharacteristic
623C2#8044711yescharacteristic
624C2#8062811yescharacteristic
625C2#8065211yescharacteristic
626C2#8090011yescharacteristic
627C2#8101111yescharacteristic
628C2#8123411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2066C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#2066D3
hypocenterC3#26362
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2066

Complements

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

NH ≅ G/Ntype
2. C2×#4796C2semidirect
5. C3×#2066C2semidirect
6. #70566C2semidirect
8. #70453C2semidirect
11. #70236C2semidirect
12. #69684C2semidirect
13. #69601C2semidirect
15. #68807C2semidirect
68. #2066D3semidirect
620. C3#26362semidirect

Maximal subgroups

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

subgroupindex
C2×#48612
C2×#47962
#766152
#760002
C3×#20662
#705662
#704782
#704532
#702712
#702692
#702362
#696842
#696012
#688742
#688072
#263623