Finite Group Databasebeta

← #10858124

Subgroups — #10858124

Order 768.

804 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
1C2×#4881C211yesmaximal, characteristic
2C2×#4796C211yesmaximal, characteristic, complemented
3#76491C211yesmaximal, characteristic
4#76003C211yesmaximal, characteristic
5C3×#2068C211yesmaximal, characteristic, complemented
6#70618C211yesmaximal, characteristic, complemented
7#70478C211yesmaximal, characteristic
8#70455C211yesmaximal, characteristic, complemented
9#70271C211yesmaximal, characteristic
10#70268C211yesmaximal, characteristic
11#70238C211yesmaximal, characteristic, complemented
12#69686C211yesmaximal, characteristic, complemented
13#69601C211yesmaximal, characteristic, complemented
14#68877C211yesmaximal, characteristic
15#68808C211yesmaximal, characteristic, complemented
16#26367—31nomaximal, Sylow
17C2×#946C2211yescharacteristic
18C2×Q8×Dic3C2211yescharacteristic
19C22×ΓL(1,𝔽25)C2211yescharacteristic
20C2×#892C2211yescharacteristic
21C2×D3×C4⋊C4C2211yescharacteristic
22C2×#890C2211yescharacteristic
23C2×C3×#107C2211yescharacteristic
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×#817C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35#4987—21no—
36#4987C2211yescharacteristic
37#4984C2211yescharacteristic
38#4981—21no—
39#4972C2211yescharacteristic
40C2×#808C2211yescharacteristic
41C2×#808C2211yescharacteristic
42C2×#807C2211yescharacteristic
43C4×(C2×D3)⋊C4C2211yescharacteristic
44#4936C2211yescharacteristic
45C4×C4⋊Dic3C2211yescharacteristic
46#4881—24no—
47#4796—24no—
48#4670—21no—
49#4669—21no—
50#4668—21no—
51#4659—21no—
52#4655—21no—
53#4649—21no—
54C3×#340C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#479—31no—
63C2×#428—31no—
64#2442—31no—
65#2362—31no—
66#2097—31no—
67#2087—31no—
68#2068D311yescharacteristic, complemented
69#2059—31no—
70#1949—31no—
71#1919—31no—
72#1894—31no—
73#1872—31no—
74#1865—31no—
75#1854—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⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
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×Dic3—21no—
110C2×C4×Dic3C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C2×ΓL(1,𝔽25)—22no—
114C2×ΓL(1,𝔽25)—22no—
115C2×ΓL(1,𝔽25)—22no—
116#892—22no—
117#892—41no—
118D3×C4⋊C4—22no—
119D3×C4⋊C4—41no—
120#890—22no—
121#890—41no—
122C3×#107—22no—
123C3×C4⋊C8—22no—
124C3×#103—22no—
125C3×#103—22no—
126C3×#95—21no—
127C3×#95—21no—
128C3×#95—21no—
129#835—22no—
130#835—41no—
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—22no—
141#817—22no—
142Dic3⋊C8—22no—
143#808—41no—
144#808D414yes—
145#807—22no—
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×#107C2×D311yescharacteristic
157C2×#106—31no—
158C2×#106—31no—
159C2×#106—31no—
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#400—61no—
168#398—61no—
169#391C2×D311yescharacteristic
170#389—31no—
171#389—31no—
172#389C2×D311yescharacteristic
173#388—61no—
174#388—61no—
175#387—61no—
176#385—61no—
177#382—31no—
178#380—31no—
179#380—61no—
180C4×C4⋊C4—31no—
181C4×C4⋊C4—61no—
182C4×C22⋊C4—31no—
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—81no—
233C3×C4⋊C4C2×D412yes—
234C3×C42—21no—
235C3×C42—21no—
236(C2×D3)⋊C4—24no—
237(C2×D3)⋊C4—41no—
238(C2×D3)⋊C4—41no—
239(C2×D3)⋊C4—42no—
240(C2×D3)⋊C4—42no—
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—81no—
250C4⋊Dic3C2×D412yes—
251C4⋊Dic3C2×Q812yes—
252Dic3⋊C4—24no—
253Dic3⋊C4—42no—
254Dic3⋊C4—42no—
255Dic3⋊C4—42no—
256C4×Dic3—21no—
257C4×Dic3—21no—
258C4×Dic3—42no—
259C4×Dic3—42no—
260C4×Dic3—42no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8C2×D412yes—
267ΓL(1,𝔽25)—28no—
268C22×Q8C22×D311yescharacteristic
269C23×C4—31no—
270C23×C4—61no—
271C2×M16—62no—
272C2×M16—62no—
273C2×M16—62no—
274C22×C8—31no—
275C22×C8—31no—
276C22×C8C22×D311yescharacteristic
277C22×C8C22×D311yescharacteristic
278#125—62no—
279#125—121no—
280C22⋊Q8—34no—
281C22⋊Q8—62no—
282C4×Q8—62no—
283C4×Q8—121no—
284#117—62no—
285#117—121no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4—62no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—121no—
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—31no—
309C2×C42—61no—
310C2×C42—61no—
311C2×C42—61no—
312C2×C42C22×D311yescharacteristic
313#107—22no—
314#107—62no—
315#107—62no—
316#106—34no—
317#106—62no—
318#106—121no—
319C4⋊C8—22no—
320C4⋊C8—62no—
321#103—22no—
322#103—22no—
323#103—62no—
324#103—121no—
325C22⋊C8—62no—
326#95—21no—
327#95—21no—
328#95—21no—
329#95—31no—
330#95—31no—
331#95—61no—
332#95—61no—
333#95—61no—
334#95—61no—
335#95—61no—
336#95—61no—
337C23×C3C22×D411yescharacteristic
338C22×D3—21no—
339C22×D3—21no—
340C22×D3—21no—
341C22×D3—21no—
342C22×D3—21no—
343C22×D3—21no—
344C22×D3C22⋊Q812yes—
345C3×Q8—22no—
346C3×Q8—22no—
347C3×Q8—41no—
348C3×Q8—41no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—41no—
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×C4C22×D411yescharacteristic
368C2×C3×C4C22×Q811yescharacteristic
369C2×C3×C4C2×#4111yescharacteristic
370C2×C3×C4C2×#4111yescharacteristic
371C2×C3×C4H(5,𝔽2)11yescharacteristic
372C2×C3×C42−1+411yescharacteristic
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—22no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—22no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—41no—
384C2×Dic3—41no—
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×Dic3C22⋊Q812yes—
394C4×D3—22no—
395C4×D3—22no—
396C4×D3—22no—
397C4×D3—22no—
398C4×D3—84no—
399C3×C8—22no—
400C3×C8—41no—
401C3⋊C8—22no—
402C3⋊C8—41no—
403C24—31no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8D3×Q812yes—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—31no—
415C22×C4—32no—
416C22×C4—32no—
417C22×C4—31no—
418C22×C4—31no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—122no—
428C22×C4—122no—
429C22×C4—241no—
430C22×C4D3×D411yescharacteristic
431C22×C4D3×D411yescharacteristic
432C22×C4C23×D311yescharacteristic
433M16—68no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—32no—
440C2×C8—61no—
441C2×C8—61no—
442C2×C8—61no—
443C2×C8—61no—
444C2×C8—61no—
445C2×C8#23712yes—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—32no—
452C4⋊C4—32no—
453C4⋊C4—64no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4—81no—
461C4⋊C4—122no—
462C4⋊C4—122no—
463C4⋊C4—122no—
464C4⋊C4—122no—
465C4⋊C4—122no—
466C4⋊C4—241no—
467C4⋊C4D3×D412yes—
468C22⋊C4—64no—
469C22⋊C4—121no—
470C22⋊C4—121no—
471C22⋊C4—122no—
472C22⋊C4—122no—
473C42—21no—
474C42—21no—
475C42—61no—
476C42—61no—
477C42—122no—
478C42—122no—
479C42—122no—
480C22×C3C2×C22⋊Q811yescharacteristic
481C22×C3D4211yescharacteristic
482C22×C3#54711yescharacteristic
483C22×C3#57511yescharacteristic
484C22×C3#57511yescharacteristic
485C22×C3#57711yescharacteristic
486C22×C3#57811yescharacteristic
487C2×D3—22no—
488C2×D3—22no—
489C2×D3—22no—
490C2×D3—41no—
491C2×D3—41no—
492C2×D3—41no—
493C2×D3—41no—
494C3×C4—41no—
495C3×C4—41no—
496C3×C4—42no—
497C3×C4—42no—
498C3×C4—81no—
499C3×C4—81no—
500C3×C4#52511yescharacteristic
501C3×C4#53911yescharacteristic
502C3×C4D4×Q811yescharacteristic
503C3×C4#55411yescharacteristic
504Dic3—22no—
505Dic3—42no—
506Dic3—41no—
507Dic3—42no—
508Dic3—41no—
509Dic3—81no—
510Dic3—81no—
511C23—32no—
512C23—61no—
513C23—61no—
514C23—61no—
515C23—61no—
516C23—61no—
517C23—61no—
518C23C2×D3×D411yescharacteristic
519Q8—22no—
520Q8—22no—
521Q8—41no—
522Q8—41no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—32no—
532C2×C4—41no—
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—62no—
545C2×C4—62no—
546C2×C4—62no—
547C2×C4—62no—
548C2×C4—61no—
549C2×C4—61no—
550C2×C4—61no—
551C2×C4—61no—
552C2×C4—61no—
553C2×C4—61no—
554C2×C4—62no—
555C2×C4—62no—
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—121no—
566C2×C4—244no—
567C2×C4C2×D3×D411yescharacteristic
568C2×C4C2×#23711yescharacteristic
569C2×C4C2×D3×Q811yescharacteristic
570C2×C4D3×#4111yescharacteristic
571C2×C4#100911yescharacteristic
572C2×C4#101011yescharacteristic
573C8—22no—
574C8—41no—
575C8—62no—
576C8—121no—
577C2×C3#244911yescharacteristic
578C2×C3#308011yescharacteristic
579C2×C3#308211yescharacteristic
580C2×C3#326911yescharacteristic
581C2×C3#327511yescharacteristic
582C2×C3#334911yescharacteristic
583C2×C3#335111yescharacteristic
584D3—42no—
585C22—62no—
586C22—62no—
587C22—62no—
588C22—121no—
589C22—121no—
590C22—121no—
591C22—121no—
592C22#560911yescharacteristic
593C22#561111yescharacteristic
594C22D3×C22⋊Q811yescharacteristic
595C22#576611yescharacteristic
596C22#576811yescharacteristic
597C22#577511yescharacteristic
598C22#577711yescharacteristic
599C4—42no—
600C4—41no—
601C4—41no—
602C4—42no—
603C4—62no—
604C4—81no—
605C4—81no—
606C4—122no—
607C4—121no—
608C4—122no—
609C4—121no—
610C4—241no—
611C4—241no—
612C4#560611yescharacteristic
613C4#563211yescharacteristic
614C4#563511yescharacteristic
615C4#570711yescharacteristic
616C3#2636711yescharacteristic, complemented, Sylow
617C2—122no—
618C2#7741911yescharacteristic
619C2#8050011yescharacteristic
620C2#8055611yescharacteristic
621C2#8075611yescharacteristic
622C2#8080011yescharacteristic
623C2#8111011yescharacteristic
624C2#8123111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2068C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#2068D3
hypocenterC3#26367
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2068

Complements

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

NH ≅ G/Ntype
2. C2×#4796C2semidirect
5. C3×#2068C2semidirect
6. #70618C2semidirect
8. #70455C2semidirect
11. #70238C2semidirect
12. #69686C2semidirect
13. #69601C2semidirect
15. #68808C2semidirect
68. #2068D3semidirect
616. C3#26367semidirect

Maximal subgroups

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

subgroupindex
C2×#48812
C2×#47962
#764912
#760032
C3×#20682
#706182
#704782
#704552
#702712
#702682
#702382
#696862
#696012
#688772
#688082
#263673