Finite Group Databasebeta

← #10858377

Subgroups — #10858377

Order 768.

894 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82470C211yesmaximal, characteristic, complemented
2C2×#5026C211yesmaximal, characteristic
3C2×#4890C211yesmaximal, characteristic
4#76202C211yesmaximal, characteristic
5C3×#1863C211yesmaximal, characteristic, complemented
6#70619C211yesmaximal, characteristic, complemented
7#70276C211yesmaximal, characteristic
8#70276C211yesmaximal, characteristic
9#70261C211yesmaximal, characteristic, complemented
10#70249C211yesmaximal, characteristic, complemented
11#69621C211yesmaximal, characteristic, complemented
12#68895C211yesmaximal, characteristic
13#68826C211yesmaximal, characteristic, complemented
14#68670C211yesmaximal, characteristic
15#68656C211yesmaximal, characteristic, complemented
16#26384—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C22×#215C2211yescharacteristic
19C2×#893C2211yescharacteristic
20C2×#893C2211yescharacteristic
21C2×D3×C4⋊C4—21no—
22C2×#888C2211yescharacteristic
23C2×C4×D12C2211yescharacteristic
24C2×C3×#107C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C3×#389C2211yescharacteristic
28#5229C2211yescharacteristic
29#5229C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×#821C2211yescharacteristic
33C2×#818C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35#5026—24no—
36#4989—21no—
37#4984C2211yescharacteristic
38#4984C2211yescharacteristic
39#4982—21no—
40#4979—21no—
41#4975C2211yescharacteristic
42C2×#809C2211yescharacteristic
43C2×#808C2211yescharacteristic
44C2×#807C2211yescharacteristic
45C2×#807C2211yescharacteristic
46#4933—21no—
47C4×Dic3⋊C4C2211yescharacteristic
48#4929—21no—
49#4928C2211yescharacteristic
50C2×#803C2211yescharacteristic
51#4890—24no—
52#4668—21no—
53#4667—21no—
54C3×#340C2211yescharacteristic
55C3×#336C2211yescharacteristic
56#4551C2211yescharacteristic
57#4550C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#474—31no—
63C2×#440—31no—
64#2451—31no—
65D4×C4⋊C4—31no—
66#2095—31no—
67#2083—31no—
68#2066—31no—
69#2057—31no—
70#1944—31no—
71#1917—31no—
72#1877—31no—
73#1877—31no—
74#1864—31no—
75#1863D311yescharacteristic, complemented
76#1862—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×D12C2311yescharacteristic
79C22×C4×D3—21no—
80C22×C4×D3—21no—
81C22×C4×D3—21no—
82C22×Dic6—21no—
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C3×C4×Q8—22no—
86C3×C4×Q8—41no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C42—21no—
93C2×C3×C42C2311yescharacteristic
94C2×#215—22no—
95C2×#215—22no—
96C2×#215Q814yescomplemented
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3—21no—
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×Dic3⋊C4D411yescharacteristic
110C2×Dic3⋊C4D411yescharacteristic
111C2×C4×Dic3C2311yescharacteristic
112C2×C4×Dic3C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115#893—22no—
116#893—22no—
117#893—22no—
118#893—41no—
119D3×C4⋊C4—44no—
120#888—22no—
121#888—41no—
122C4×D12—22no—
123C4×D12—41no—
124C3×#107—22no—
125C3×#103—22no—
126C3×#103—41no—
127C3×#95—21no—
128#835—22no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95D411yescharacteristic
137C3⋊#95D411yescharacteristic
138#821—22no—
139#821—22no—
140#818—41no—
141Dic3⋊C8—22no—
142#809D414yes—
143#808—22no—
144#807—22no—
145#807—22no—
146#803—22no—
147C22×SD16—31no—
148C2×C4⋊Q8—31no—
149C2×C4×Q8C2×D311yescharacteristic
150C2×C4×D4—31no—
151C2×C4×D4—31no—
152C2×C4×D4—31no—
153C22×C4⋊C4—61no—
154#474—64no—
155#440—64no—
156C2×#107—31no—
157C2×#107—31no—
158C2×#107—31no—
159C2×#107C2×D311yescharacteristic
160C2×#106—31no—
161C2×C4⋊C8—31no—
162C2×#103—31no—
163C2×#103C2×D311yescharacteristic
164C2×#103C2×D311yescharacteristic
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#97—31no—
169#398—61no—
170#395—61no—
171#391—31no—
172#391—31no—
173#391—61no—
174#389—31no—
175#389—31no—
176#389—31no—
177#389C2×D311yescharacteristic
178#387—61no—
179#385—61no—
180#385—61no—
181#384—31no—
182#380—61no—
183C4×C4⋊C4—31no—
184#340—31no—
185#340—31no—
186#340—31no—
187#340C2×D311yescharacteristic
188#336—31no—
189#336—31no—
190#336—31no—
191#336C2×D311yescharacteristic
192C23×D3—21no—
193C2×C3×Q8—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8C2×Q812yes—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2411yescharacteristic
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3C2×D411yescharacteristic
210C22×Dic3C2×D411yescharacteristic
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C2×D12—21no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12C2×D412yes—
217C2×D12C2×Q812yes—
218C2×D12#4112yes—
219C2×C4×D3—24no—
220C2×C4×D3—42no—
221C2×C4×D3—42no—
222C2×C4×D3—42no—
223C2×C4×D3—42no—
224C2×C4×D3—42no—
225C2×C4×D3—42no—
226C2×C4×D3—81no—
227C2×C4×D3—81no—
228C2×Dic6—42no—
229C2×Dic6—42no—
230C2×Dic6—81no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—42no—
242C3×C4⋊C4—42no—
243C3×C4⋊C4C2×D412yes—
244C3×C4⋊C4#4112yes—
245C3×C42—21no—
246C3×C42—21no—
247C3×C42—42no—
248#215—28no—
249(C2×D3)⋊C4—24no—
250(C2×D3)⋊C4—42no—
251(C2×D3)⋊C4—42no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—42no—
259C4⋊Dic3—81no—
260Dic3⋊C4—22no—
261Dic3⋊C4—22no—
262Dic3⋊C4—42no—
263Dic3⋊C4—42no—
264Dic3⋊C4—81no—
265Dic3⋊C4—81no—
266C4×Dic3—21no—
267C4×Dic3—21no—
268C4×Dic3—21no—
269C4×Dic3—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8C2×D412yes—
275C2×C3⋊C8C2×Q812yes—
276C22×Q8—61no—
277C22×Q8C22×D311yescharacteristic
278C22×D4—31no—
279C23×C4—61no—
280C23×C4—61no—
281C23×C4—61no—
282C2×SD16—34no—
283C2×SD16—62no—
284C2×SD16—62no—
285C22×C8—31no—
286C22×C8—31no—
287C22×C8C22×D311yescharacteristic
288C22×C8C22×D311yescharacteristic
289C4⋊Q8—62no—
290C4⋊Q8—121no—
291C4×Q8—22no—
292C4×Q8—41no—
293C4×D4—62no—
294C4×D4—62no—
295C4×D4—62no—
296C4×D4—62no—
297C4×D4—121no—
298C4×D4—121no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—124no—
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C42—21no—
320C2×C42—31no—
321C2×C42—31no—
322C2×C42C22×D311yescharacteristic
323#107—22no—
324#107—62no—
325#107—62no—
326#107—121no—
327#106—62no—
328C4⋊C8—62no—
329#103—22no—
330#103—41no—
331#103—62no—
332#102—34no—
333#102—62no—
334#102—62no—
335#97—62no—
336#95—21no—
337#95—31no—
338#95—31no—
339#95—61no—
340#95—61no—
341#95—61no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346C23×C3C22×D411yescharacteristic
347C22×D3—22no—
348C22×D3—22no—
349C22×D3—22no—
350C22×D3—41no—
351C22×D3—41no—
352C22×D3—41no—
353C22×D3—41no—
354C3×Q8—22no—
355C3×Q8—22no—
356C3×Q8—41no—
357C3×Q8—41no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—22no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—41no—
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×C4C22×D411yescharacteristic
377C2×C3×C4C22×Q811yescharacteristic
378C2×C3×C4C2×#4111yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380C2×C3×C4H(5,𝔽2)11yescharacteristic
381C2×C3×C42−1+411yescharacteristic
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3AGL(1,ℤ/8)12yes—
402C2×Dic3#13712yes—
403D12—22no—
404D12—22no—
405D12—22no—
406D12C22⋊Q814yes—
407C4×D3—48no—
408C4×D3—84no—
409C4×D3—84no—
410Dic6—84no—
411C3×C8—41no—
412C3×C8—41no—
413C3⋊C8—22no—
414C3⋊C8—22no—
415C24—61no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—122no—
422C2×Q8—122no—
423C2×Q8—241no—
424C2×Q8D3×Q812yes—
425C2×D4—32no—
426C2×D4—32no—
427C2×D4—32no—
428C2×D4—61no—
429C2×D4—61no—
430C2×D4—61no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—31no—
436C22×C4—31no—
437C22×C4—31no—
438C22×C4—31no—
439C22×C4—64no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—122no—
444C22×C4—122no—
445C22×C4—122no—
446C22×C4—122no—
447C22×C4—122no—
448C22×C4—122no—
449C22×C4—241no—
450C22×C4—241no—
451C22×C4D3×D411yescharacteristic
452C22×C4D3×D411yescharacteristic
453C22×C4C23×D311yescharacteristic
454SD16—68no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—32no—
462C2×C8—32no—
463C2×C8—61no—
464C2×C8—61no—
465C2×C8—61no—
466C2×C8—61no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—62no—
474C4⋊C4—62no—
475C4⋊C4—61no—
476C4⋊C4—61no—
477C4⋊C4—61no—
478C4⋊C4—61no—
479C4⋊C4—61no—
480C4⋊C4—61no—
481C4⋊C4—122no—
482C4⋊C4—122no—
483C4⋊C4—122no—
484C4⋊C4—241no—
485C4⋊C4—241no—
486C4⋊C4—241no—
487C4⋊C4#23512yes—
488C4⋊C4D3×D412yes—
489C22⋊C4—64no—
490C22⋊C4—122no—
491C22⋊C4—122no—
492C42—21no—
493C42—21no—
494C42—42no—
495C42—61no—
496C42—61no—
497C42—61no—
498C42—61no—
499C22×C3#52511yescharacteristic
500C22×C3D4211yescharacteristic
501C22×C3D4×Q811yescharacteristic
502C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
503C22×C3C2×#13711yescharacteristic
504C22×C3#57611yescharacteristic
505C22×C3#57811yescharacteristic
506C2×D3—24no—
507C2×D3—42no—
508C2×D3—42no—
509C2×D3—42no—
510C2×D3—42no—
511C2×D3—42no—
512C2×D3—42no—
513C3×C4—42no—
514C3×C4—42no—
515C3×C4—42no—
516C3×C4—41no—
517C3×C4—41no—
518C3×C4—81no—
519C3×C4C2×C22⋊Q811yescharacteristic
520C3×C4#53911yescharacteristic
521C3×C4#54711yescharacteristic
522C3×C4#55411yescharacteristic
523Dic3—22no—
524Dic3—22no—
525Dic3—41no—
526Dic3—41no—
527Dic3—81no—
528Dic3—81no—
529Dic3—81no—
530C23—62no—
531C23—62no—
532C23—62no—
533C23—121no—
534C23—121no—
535C23—121no—
536C23—121no—
537C23C2×D3×D411yescharacteristic
538Q8—22no—
539Q8—22no—
540Q8—41no—
541Q8—41no—
542Q8—244no—
543D4—34no—
544D4—62no—
545D4—62no—
546D4—62no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—32no—
557C2×C4—32no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—61no—
575C2×C4—61no—
576C2×C4—61no—
577C2×C4—128no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—244no—
588C2×C4—244no—
589C2×C4C2×#23511yescharacteristic
590C2×C4C2×D3×D411yescharacteristic
591C2×C4#100411yescharacteristic
592C2×C4C2×D3×Q811yescharacteristic
593C2×C4D3×#4111yescharacteristic
594C2×C4#101011yescharacteristic
595C8—41no—
596C8—41no—
597C8—62no—
598C8—62no—
599C2×C3#244911yescharacteristic
600C2×C3#311011yescharacteristic
601C2×C3#312411yescharacteristic
602C2×C3#326411yescharacteristic
603C2×C3#327611yescharacteristic
604C2×C3#338511yescharacteristic
605C2×C3#338611yescharacteristic
606D3—44no—
607C22—64no—
608C22—122no—
609C22—122no—
610C22—122no—
611C22—122no—
612C22—122no—
613C22—122no—
614C22#554711yescharacteristic
615C22#559011yescharacteristic
616C22#563211yescharacteristic
617C22#576611yescharacteristic
618C22#576911yescharacteristic
619C22D3×AGL(1,ℤ/8)11yescharacteristic
620C22D3×#13711yescharacteristic
621C4—41no—
622C4—42no—
623C4—42no—
624C4—41no—
625C4—42no—
626C4—62no—
627C4—62no—
628C4—81no—
629C4—121no—
630C4—121no—
631C4—241no—
632C4—241no—
633C4—241no—
634C4#551111yescharacteristic
635C4#555111yescharacteristic
636C4D3×C22⋊Q811yescharacteristic
637C4#566011yescharacteristic
638C3#2638411yescharacteristic, complemented, Sylow
639C2—124no—
640C2#7460911yescharacteristic
641C2#7974911yescharacteristic
642C2#7977911yescharacteristic
643C2#8012011yescharacteristic
644C2#8020611yescharacteristic
645C2#8110311yescharacteristic
646C2#8122311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1863C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1863D3
hypocenterC3#26384
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1863

Complements

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

NH ≅ G/Ntype
1. #82470C2semidirect
5. C3×#1863C2semidirect
6. #70619C2semidirect
9. #70261C2semidirect
10. #70249C2semidirect
11. #69621C2semidirect
13. #68826C2semidirect
15. #68656C2semidirect
75. #1863D3semidirect
96. C2×#215Q8semidirect
638. C3#26384semidirect

Maximal subgroups

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

subgroupindex
#824702
C2×#50262
C2×#48902
#762022
C3×#18632
#706192
#702762
#702612
#702492
#696212
#688952
#688262
#686702
#686562
#263843