Finite Group Databasebeta

← #10858374

Subgroups — #10858374

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
1C2×C3×#474C211yesmaximal, characteristic
2#83235C211yesmaximal, characteristic, complemented
3D4×Dic3⋊C4C211yesmaximal, characteristic
4C2×#4757C211yesmaximal, characteristic
5#71782C211yesmaximal, characteristic, complemented
6#71612C211yesmaximal, characteristic
7#70644C211yesmaximal, characteristic, complemented
8#70642C211yesmaximal, characteristic, complemented
9#70619C211yesmaximal, characteristic, complemented
10#70614C211yesmaximal, characteristic, complemented
11#70488C211yesmaximal, characteristic, complemented
12#70448C211yesmaximal, characteristic, complemented
13#70171C211yesmaximal, characteristic
14#70170C211yesmaximal, characteristic
15#70167C211yesmaximal, characteristic
16#26384—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×D4×Dic3C2211yescharacteristic
20C2×D4×Dic3C2211yescharacteristic
21C22×#214C2211yescharacteristic
22C22×Dic3⋊C4—21no—
23C2×#887C2211yescharacteristic
24C3×#474—24no—
25C2×C3×#107C2211yescharacteristic
26C2×C3×#107C2211yescharacteristic
27C2×C3×C4⋊C8C2211yescharacteristic
28C2×C3×#102C2211yescharacteristic
29C2×C3×#102C2211yescharacteristic
30#5230—21no—
31#5229—21no—
32C2×#832C2211yescharacteristic
33#5210—21no—
34C2×#817C2211yescharacteristic
35C2×#816C2211yescharacteristic
36C2×#816C2211yescharacteristic
37C2×#815C2211yescharacteristic
38#4984C2211yescharacteristic
39#4982—21no—
40#4978—21no—
41#4975C2211yescharacteristic
42#4973C2211yescharacteristic
43#4972C2211yescharacteristic
44#4972C2211yescharacteristic
45C2×#810C2211yescharacteristic
46C2×#807C2211yescharacteristic
47C2×#807C2211yescharacteristic
48#4946—21no—
49#4942—21no—
50C4×Dic3⋊C4C2211yescharacteristic
51#4929C2211yescharacteristic
52#4928C2211yescharacteristic
53#4757—24no—
54#4551C2211yescharacteristic
55#4550C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#474D311yescharacteristic
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#1863—31no—
76#1862—31no—
77C22×C3×Q8—21no—
78C22×C3×D4C2311yescharacteristic
79C23×C3×C4—21no—
80C23×Dic3—21no—
81C23×Dic3—21no—
82C22×Dic6C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C3×C4⋊Q8—22no—
86C3×C4⋊Q8—41no—
87C3×C4×D4—22no—
88C3×C4×D4—41no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C22⋊C4—21no—
94C2×C3×C42C2311yescharacteristic
95C2×#217—21no—
96C2×#217—21no—
97D4×Dic3—22no—
98D4×Dic3—22no—
99D4×Dic3—22no—
100D4×Dic3—41no—
101C2×#214—22no—
102C2×#214—22no—
103C2×#214Q814yescomplemented
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—44no—
115C2×Dic3⋊C4D411yescharacteristic
116C2×Dic3⋊C4D411yescharacteristic
117C2×C4×Dic3—21no—
118C2×C4×Dic3C2311yescharacteristic
119C2×C4×Dic3C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122#887—22no—
123#887—41no—
124C3×#107—22no—
125C3×#107—22no—
126C3×C4⋊C8—22no—
127C3×#102—22no—
128C3×#102D414yes—
129#832—22no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95D411yescharacteristic
139C3⋊#95D411yescharacteristic
140#817—22no—
141#816—22no—
142#816—22no—
143#815—22no—
144#810—41no—
145#807—22no—
146#807—41no—
147C22×SD16—31no—
148C2×C4⋊Q8C2×D311yescharacteristic
149C2×C4×Q8—31no—
150C2×C4×D4—31no—
151C2×C4×D4—31no—
152C2×C4×D4C2×D311yescharacteristic
153C22×C4⋊C4—61no—
154#474—24no—
155#440—64no—
156C2×#107—31no—
157C2×#107—31no—
158C2×#107C2×D311yescharacteristic
159C2×#107C2×D311yescharacteristic
160C2×#106—31no—
161C2×C4⋊C8C2×D311yescharacteristic
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#102—31no—
166C2×#102C2×D311yescharacteristic
167C2×#102C2×D311yescharacteristic
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#389—31no—
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#340—31no—
188#336—31no—
189#336—31no—
190#336—31no—
191#336—31no—
192C24×C3—21no—
193C2×C3×Q8—42no—
194C2×C3×Q8—42no—
195C2×C3×Q8—81no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4C2×D412yes—
200C2×C3×D4C2×Q812yes—
201C2×C3×D4#4112yes—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—42no—
206C22×C3×C4—42no—
207C22×C3×C4—81no—
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2411yescharacteristic
211C22×Dic3—24no—
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3—42no—
217C22×Dic3—42no—
218C22×Dic3—42no—
219C22×Dic3—42no—
220C22×Dic3—81no—
221C22×Dic3C2×D411yescharacteristic
222C22×Dic3C2×D411yescharacteristic
223C22×Dic3C2×D411yescharacteristic
224C22×Dic3C2×D411yescharacteristic
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6C2×Q812yes—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8C2×D412yes—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—42no—
243C3×C4⋊C4C2×D412yes—
244C3×C22⋊C4—42no—
245C3×C42—21no—
246C3×C42—21no—
247#217—24no—
248#217—42no—
249#214—28no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—81no—
256C4⋊Dic3#4112yes—
257Dic3⋊C4—22no—
258Dic3⋊C4—22no—
259Dic3⋊C4—42no—
260Dic3⋊C4—42no—
261Dic3⋊C4—42no—
262Dic3⋊C4—42no—
263Dic3⋊C4—81no—
264Dic3⋊C4—81no—
265C4×Dic3—21no—
266C4×Dic3—21no—
267C4×Dic3—21no—
268C4×Dic3—21no—
269C4×Dic3—42no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8C2×Q812yes—
276C22×Q8—21no—
277C22×Q8—31no—
278C22×D4C22×D311yescharacteristic
279C23×C4—21no—
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—22no—
290C4⋊Q8—41no—
291C4×Q8—62no—
292C4×Q8—121no—
293C4×D4—22no—
294C4×D4—41no—
295C4×D4—62no—
296C4×D4—62no—
297C4×D4—62no—
298C4×D4—121no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—61no—
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—21no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C42—31no—
320C2×C42—31no—
321C2×C42—61no—
322C2×C42C22×D311yescharacteristic
323#107—22no—
324#107—22no—
325#107—62no—
326#107—121no—
327#106—62no—
328C4⋊C8—22no—
329#103—62no—
330#103—62no—
331#103—121no—
332#102—22no—
333#102—62no—
334#102#6714yes—
335#97—62no—
336#95—31no—
337#95—31no—
338#95—61no—
339#95—61no—
340#95—61no—
341#95—61no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346C23×C3—22no—
347C23×C3—22no—
348C23×C3—22no—
349C23×C3—41no—
350C23×C3—41no—
351C23×C3—41no—
352C23×C3—41no—
353C23×C3C22×D411yescharacteristic
354C3×Q8—84no—
355C3×D4—22no—
356C3×D4—22no—
357C3×D4—22no—
358C3×D4C22⋊Q814yes—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—22no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
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—84no—
374C2×C3×C4AGL(1,ℤ/8)12yes—
375C2×C3×C4C22×D411yescharacteristic
376C2×C3×C4C22×Q811yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C2×C3×C4C2×#4111yescharacteristic
379C2×C3×C4H(5,𝔽2)11yescharacteristic
380C2×C3×C42−1+411yescharacteristic
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—22no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—22no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—48no—
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×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—84no—
406C2×Dic3#13712yes—
407Dic6—22no—
408Dic6—22no—
409Dic6—41no—
410Dic6—41no—
411C3×C8—22no—
412C3×C8—41no—
413C3⋊C8—22no—
414C3⋊C8—41no—
415C24—21no—
416C2×Q8—32no—
417C2×Q8—42no—
418C2×Q8—42no—
419C2×Q8—61no—
420C2×Q8—61no—
421C2×Q8—61no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8—81no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4D3×Q812yes—
429C2×D4#23912yes—
430C2×D4C2×#6712yes—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—31no—
435C22×C4—31no—
436C22×C4—31no—
437C22×C4—31no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—64no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—81no—
446C22×C4—122no—
447C22×C4—122no—
448C22×C4—122no—
449C22×C4—122no—
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—32no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8—61no—
464C2×C8—61no—
465C2×C8—61no—
466C2×C8C2×#6712yes—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—32no—
473C4⋊C4—42no—
474C4⋊C4—62no—
475C4⋊C4—62no—
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—122no—
485C4⋊C4—241no—
486C4⋊C4—241no—
487C4⋊C4—241no—
488C4⋊C4C2×#6712yes—
489C22⋊C4—42no—
490C22⋊C4—64no—
491C22⋊C4—122no—
492C42—21no—
493C42—21no—
494C42—61no—
495C42—61no—
496C42—61no—
497C42—61no—
498C42—122no—
499C22×C3—24no—
500C22×C3—42no—
501C22×C3—42no—
502C22×C3—42no—
503C22×C3—42no—
504C22×C3—42no—
505C22×C3—42no—
506C22×C3#52511yescharacteristic
507C22×C3D4211yescharacteristic
508C22×C3D4×Q811yescharacteristic
509C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
510C22×C3C2×#13711yescharacteristic
511C22×C3#57611yescharacteristic
512C22×C3#57811yescharacteristic
513C3×C4—22no—
514C3×C4—42no—
515C3×C4—41no—
516C3×C4—81no—
517C3×C4—81no—
518C3×C4C2×C22⋊Q811yescharacteristic
519C3×C4#53911yescharacteristic
520C3×C4#54711yescharacteristic
521C3×C4#55411yescharacteristic
522Dic3—22no—
523Dic3—42no—
524Dic3—41no—
525Dic3—42no—
526Dic3—41no—
527Dic3—41no—
528Dic3—81no—
529Dic3—81no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23C2×D3×D411yescharacteristic
538Q8—62no—
539Q8—62no—
540Q8—84no—
541Q8—121no—
542Q8—121no—
543D4—22no—
544D4—22no—
545D4—22no—
546D4#94614yes—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—32no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—62no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—62no—
575C2×C4—84no—
576C2×C4—128no—
577C2×C4—121no—
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#91112yes—
589C2×C4#100411yescharacteristic
590C2×C4C2×D3×Q811yescharacteristic
591C2×C4C2×#23911yescharacteristic
592C2×C4D3×#4111yescharacteristic
593C2×C4#101011yescharacteristic
594C2×C4C22×#6711yescharacteristic
595C8—22no—
596C8—41no—
597C8—62no—
598C8—121no—
599C2×C3—44no—
600C2×C3#244911yescharacteristic
601C2×C3#311011yescharacteristic
602C2×C3#312411yescharacteristic
603C2×C3#326411yescharacteristic
604C2×C3#327611yescharacteristic
605C2×C3#338511yescharacteristic
606C2×C3#338611yescharacteristic
607C22—24no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22#562711yescharacteristic
615C22#563211yescharacteristic
616C22C2×#91111yescharacteristic
617C22#575711yescharacteristic
618C22D3×#13711yescharacteristic
619C22#577911yescharacteristic
620C22D4×#6711yescharacteristic
621C4—22no—
622C4—41no—
623C4—42no—
624C4—62no—
625C4—81no—
626C4—81no—
627C4—121no—
628C4—122no—
629C4—121no—
630C4—122no—
631C4—121no—
632C4—241no—
633C4—241no—
634C4#563611yescharacteristic
635C4#564511yescharacteristic
636C4C2×#94611yescharacteristic
637C4#582411yescharacteristic
638C3#2638411yescharacteristic, complemented, Sylow
639C2—44no—
640C2#8095011yescharacteristic
641C2#8095311yescharacteristic
642C2#8096011yescharacteristic
643C2#8096111yescharacteristic
644C2#8276411yescharacteristic
645C2#8383711yescharacteristic
646C2#8399611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#474C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenterC2×#474D3
hypocenterC3#26384
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
2. #83235C2semidirect
5. #71782C2semidirect
7. #70644C2semidirect
8. #70642C2semidirect
9. #70619C2semidirect
10. #70614C2semidirect
11. #70488C2semidirect
12. #70448C2semidirect
103. C2×#214Q8semidirect
638. C3#26384semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4742
#832352
D4×Dic3⋊C42
C2×#47572
#717822
#716122
#706442
#706422
#706192
#706142
#704882
#704482
#701712
#701702
#701672
#263843