Finite Group Databasebeta

← #10814351

Subgroups — #10814351

Order 768.

874 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Dic3×SD16C211yesmaximal, characteristic
2Q8×Dic3⋊C4C211yesmaximal, characteristic, complemented
3#76302C211yesmaximal, characteristic
4C3×#1880C211yesmaximal, characteristic
5#72048C211yesmaximal, characteristic
6#71631C211yesmaximal, characteristic, complemented
7#70221C211yesmaximal, characteristic, complemented
8#70220C211yesmaximal, characteristic, complemented
9#70188C211yesmaximal, characteristic
10#70153C211yesmaximal, characteristic
11#69645C211yesmaximal, characteristic
12#68847C211yesmaximal, characteristic, complemented
13#68816C211yesmaximal, characteristic, complemented
14#68652C211yesmaximal, characteristic, complemented
15#68622C211yesmaximal, characteristic, complemented
16#22759—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C2×Q8×Dic3C2211yescharacteristic
21C2×D4×Dic3C2211yescharacteristic
22C2×#877—21no—
23C2×C3×#103C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C3×#390C2211yescharacteristic
26C2×#835C2211yescharacteristic
27C2×#835C2211yescharacteristic
28#5221C2211yescharacteristic
29C2×#832C2211yescharacteristic
30C2×#832C2211yescharacteristic
31Dic3×SD16C418yescomplemented
32C2×#817C2211yescharacteristic
33C2×#815C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35C2×C8×Dic3C2211yescharacteristic
36#4975—21no—
37#4975C2211yescharacteristic
38#4975C2211yescharacteristic
39Dic3×C4⋊C4C2211yescharacteristic
40#4972—21no—
41#4972C2211yescharacteristic
42#4945—21no—
43#4944—21no—
44#4942—21no—
45C4×C4⋊Dic3C2211yescharacteristic
46C4×Dic3⋊C4—21no—
47C4×Dic3⋊C4C2211yescharacteristic
48#4928—21no—
49#4928C2211yescharacteristic
50C2×#803C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#336C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4550C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61C2×C4×SD16—31no—
62#2354—31no—
63Q8×C4⋊C4—31no—
64#1936—31no—
65#1936—31no—
66#1919—31no—
67#1919—31no—
68#1880—31no—
69#1880D311yescharacteristic
70#1834—31no—
71#1803—31no—
72#1802—31no—
73#1770—31no—
74#1763—31no—
75#1763—31no—
76C22×C3×Q8C2311yescharacteristic
77C22×C3×D4C2311yescharacteristic
78C23×Dic3—21no—
79C2×C3×SD16—22no—
80C2×C3×SD16—22no—
81C2×C3×SD16C2×C414yes—
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C3×C4×Q8—22no—
85C3×C4×Q8—41no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C22⋊C4—21no—
90C2×C3×C22⋊C4—21no—
91C2×C3×C42—21no—
92C2×C3×C42C2311yescharacteristic
93C2×#217—21no—
94C2×#217—21no—
95C2×#217—21no—
96Q8×Dic3—22no—
97Q8×Dic3—22no—
98Q8×Dic3—41no—
99Q8×Dic3C2×C414yes—
100D4×Dic3—22no—
101D4×Dic3C2×C414yes—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3C2311yescharacteristic
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⋊C4D411yescharacteristic
115C2×Dic3⋊C4D411yescharacteristic
116C2×C4×Dic3—21no—
117C2×C4×Dic3—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3—21no—
120C2×C4×Dic3C2311yescharacteristic
121C2×C4×Dic3C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124#877—44no—
125C3×#103—22no—
126C3×#102—41no—
127#835—22no—
128#835C2×C414yes—
129#832—41no—
130#832C2×C414yes—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95D411yescharacteristic
136C3⋊#95D411yescharacteristic
137#817C2×C414yes—
138#815—41no—
139Dic3⋊C8—22no—
140C8×Dic3C2×C414yes—
141#803—41no—
142C22×SD16C2×D311yescharacteristic
143C2×C4×Q8—31no—
144C2×C4×Q8—31no—
145C2×C4×Q8C2×D311yescharacteristic
146C2×C4×D4—31no—
147C2×#117—61no—
148C4×SD16—38no—
149C2×#106—31no—
150C2×C4⋊C8—31no—
151C2×#103—31no—
152C2×#103—31no—
153C2×#103C2×D311yescharacteristic
154C2×#102—31no—
155C2×#102—31no—
156C2×#102C2×D311yescharacteristic
157C2×#97—31no—
158C2×#97—31no—
159C2×C4×C8—31no—
160#390—31no—
161#390C2×D311yescharacteristic
162#389—31no—
163#389—31no—
164#389—61no—
165#389—61no—
166#388—61no—
167#388—61no—
168#384—31no—
169#384—31no—
170#384—61no—
171#380—61no—
172C4×C4⋊C4—31no—
173C4×C4⋊C4—31no—
174C4×C4⋊C4—31no—
175C4×C4⋊C4—61no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#340—31no—
182#340—31no—
183#340C2×D311yescharacteristic
184#336—31no—
185#336C2×D311yescharacteristic
186C24×C3—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8C22×C412yes—
191C2×C3×Q8#4112yes—
192C2×C3×Q8#4112yes—
193C2×C3×D4—21no—
194C2×C3×D4—21no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4C22×C412yes—
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—24no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3—42no—
211C22×Dic3—42no—
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C22×Dic3C2×D411yescharacteristic
215C22×Dic3C2×D411yescharacteristic
216C3×SD16—28no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8C22×C412yes—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—42no—
228C3×C22⋊C4—42no—
229C3×C22⋊C4—81no—
230C3×C42—21no—
231C3×C42—21no—
232C3×C42—42no—
233#217—24no—
234#217—42no—
235#217—81no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—24no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—42no—
244C4⋊Dic3C22×C412yes—
245C4⋊Dic3C22×C412yes—
246Dic3⋊C4—22no—
247Dic3⋊C4—22no—
248Dic3⋊C4—42no—
249Dic3⋊C4—42no—
250Dic3⋊C4—81no—
251Dic3⋊C4—81no—
252Dic3⋊C4—81no—
253Dic3⋊C4—81no—
254C4×Dic3—21no—
255C4×Dic3—21no—
256C4×Dic3—24no—
257C4×Dic3—42no—
258C4×Dic3—42no—
259C4×Dic3—42no—
260C4×Dic3C22×C412yes—
261C4×Dic3#4112yes—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8C22×C412yes—
268C22×Q8C22×D311yescharacteristic
269C22×D4C22×D311yescharacteristic
270C23×C4—61no—
271C2×SD16—22no—
272C2×SD16—22no—
273C2×SD16C4×D314yes—
274C22×C8—31no—
275C22×C8—31no—
276C22×C8C22×D311yescharacteristic
277C22×C8C22×D311yescharacteristic
278C4×Q8—22no—
279C4×Q8—34no—
280C4×Q8—41no—
281C4×Q8—62no—
282C4×Q8—62no—
283C4×Q8—121no—
284C4×D4—34no—
285C4×D4—62no—
286#117—124no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—31no—
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⋊C4C22×D311yescharacteristic
303C2×C4⋊C4C22×D311yescharacteristic
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—61no—
307C2×C22⋊C4—61no—
308C2×C22⋊C4—61no—
309C2×C42—21no—
310C2×C42—31no—
311C2×C42—31no—
312C2×C42—61no—
313C2×C42—61no—
314C2×C42—61no—
315C2×C42—61no—
316C2×C42C22×D311yescharacteristic
317#106—34no—
318C4⋊C8—62no—
319#103—22no—
320#103—34no—
321#103—62no—
322#102—34no—
323#102—41no—
324#102—121no—
325#97—121no—
326#97—121no—
327C4×C8—34no—
328#95—31no—
329#95—31no—
330#95—61no—
331#95—61no—
332#95—61no—
333#95—61no—
334C23×C3—22no—
335C23×C3—41no—
336C23×C3—41no—
337C23×C3—41no—
338C23×C3—41no—
339C23×C3—41no—
340C23×C3—41no—
341C23×C3C22×D411yescharacteristic
342C3×Q8—22no—
343C3×Q8—22no—
344C3×Q8—22no—
345C3×Q8#11714yescomplemented
346C3×D4—22no—
347C3×D4—22no—
348C3×D4—41no—
349C3×D4—41no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
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×C4C23×C411yescharacteristic
366C2×C3×C4C2×#4111yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368C2×C3×C4C2×#4111yescharacteristic
369C2×C3×C4H(5,𝔽2)11yescharacteristic
370C2×C3×C42−1+411yescharacteristic
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—22no—
374C2×Dic3—22no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—48no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3C4×D412yes—
393C2×Dic3C4×D412yes—
394C2×Dic3AGL(1,ℤ/8)12yes—
395C2×Dic3#13712yes—
396C3×C8—22no—
397C3×C8—41no—
398C3⋊C8—22no—
399C3⋊C8—41no—
400C24—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8C2×C4×D312yes—
405C2×Q8#23712yes—
406C2×Q8#23712yes—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4C2×C4×D312yes—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—31no—
417C22×C4—31no—
418C22×C4—31no—
419C22×C4—31no—
420C22×C4—64no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—122no—
426C22×C4—122no—
427C22×C4D3×D411yescharacteristic
428C22×C4D3×D411yescharacteristic
429C22×C4C23×D311yescharacteristic
430SD16—28no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—32no—
437C2×C8—61no—
438C2×C8—61no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8—61no—
442C2×C8C2×C4×D312yes—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—32no—
448C4⋊C4—32no—
449C4⋊C4—42no—
450C4⋊C4—62no—
451C4⋊C4—62no—
452C4⋊C4—64no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—61no—
459C4⋊C4—122no—
460C4⋊C4—122no—
461C4⋊C4—122no—
462C4⋊C4—241no—
463C4⋊C4—241no—
464C4⋊C4—241no—
465C4⋊C4—241no—
466C22⋊C4—42no—
467C22⋊C4—64no—
468C22⋊C4—81no—
469C22⋊C4—122no—
470C22⋊C4—241no—
471C42—21no—
472C42—21no—
473C42—32no—
474C42—32no—
475C42—42no—
476C42—64no—
477C42—61no—
478C42—61no—
479C42—122no—
480C42—122no—
481C42—122no—
482C22×C3—42no—
483C22×C3—42no—
484C22×C3—42no—
485C22×C3—81no—
486C22×C3—81no—
487C22×C3—81no—
488C22×C3—81no—
489C22×C3C2×C4×D411yescharacteristic
490C22×C3#54611yescharacteristic
491C22×C3#54811yescharacteristic
492C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
493C22×C3C2×#13711yescharacteristic
494C22×C3#57711yescharacteristic
495C22×C3#57711yescharacteristic
496C3×C4—24no—
497C3×C4—41no—
498C3×C4—41no—
499C3×C4—81no—
500C3×C4—81no—
501C3×C4C2×#11711yescharacteristic
502C3×C4#52011yescharacteristic
503C3×C4#55211yescharacteristic
504C3×C4#55311yescharacteristic
505Dic3—22no—
506Dic3—41no—
507Dic3—41no—
508Dic3—42no—
509Dic3—42no—
510Dic3—81no—
511Dic3—81no—
512Dic3#44014yes—
513C23—22no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23C2×D3×D411yescharacteristic
521Q8—22no—
522Q8—22no—
523Q8—22no—
524Q8#87714yescomplemented
525D4—22no—
526D4—22no—
527D4—41no—
528D4—41no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—32no—
539C2×C4—32no—
540C2×C4—32no—
541C2×C4—32no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
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—61no—
557C2×C4—61no—
558C2×C4—128no—
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×C4C22×C4×D311yescharacteristic
570C2×C4C2×#23711yescharacteristic
571C2×C4C2×#23711yescharacteristic
572C2×C4#100411yescharacteristic
573C2×C4D3×#4111yescharacteristic
574C2×C4#101011yescharacteristic
575C8—22no—
576C8—41no—
577C8—62no—
578C8—121no—
579C2×C3—82no—
580C2×C3#235211yescharacteristic
581C2×C3C2×#44011yescharacteristic
582C2×C3#295211yescharacteristic
583C2×C3#332911yescharacteristic
584C2×C3#333211yescharacteristic
585C2×C3#336811yescharacteristic
586C2×C3#336911yescharacteristic
587C22—42no—
588C22—42no—
589C22—42no—
590C22—81no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22C4×D3×D411yescharacteristic
595C22#559111yescharacteristic
596C22#559811yescharacteristic
597C22#576211yescharacteristic
598C22#577011yescharacteristic
599C22D3×AGL(1,ℤ/8)11yescharacteristic
600C22D3×#13711yescharacteristic
601C4—24no—
602C4—34no—
603C4—41no—
604C4—41no—
605C4—62no—
606C4—81no—
607C4—81no—
608C4—122no—
609C4—122no—
610C4—121no—
611C4—121no—
612C4—241no—
613C4—241no—
614C4C2×#87711yescharacteristic
615C4#554611yescharacteristic
616C4#560111yescharacteristic
617C4#565011yescharacteristic
618C3#2275911yescharacteristic, complemented, Sylow
619C2—82no—
620C2#7467311yescharacteristic
621C2#7965011yescharacteristic
622C2D3×#44011yescharacteristic
623C2#8005611yescharacteristic
624C2#8011711yescharacteristic
625C2#8037211yescharacteristic
626C2#8058211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1880C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
cosocleC22×C3×C4C24
hypercenter#1880D3
hypocenterC3#22759
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1880

Complements

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

NH ≅ G/Ntype
2. Q8×Dic3⋊C4C2semidirect
6. #71631C2semidirect
7. #70221C2semidirect
8. #70220C2semidirect
12. #68847C2semidirect
13. #68816C2semidirect
14. #68652C2semidirect
15. #68622C2semidirect
31. Dic3×SD16C4semidirect
345. C3×Q8#117semidirect
524. Q8#877semidirect
618. C3#22759semidirect

Maximal subgroups

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

subgroupindex
C2×Dic3×SD162
Q8×Dic3⋊C42
#763022
C3×#18802
#720482
#716312
#702212
#702202
#701882
#701532
#696452
#688472
#688162
#686522
#686222
#227593