Finite Group Databasebeta

← #10798375

Subgroups — #10798375

Order 768.

938 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Dic3×D8C211yesmaximal, characteristic
2#82317C211yesmaximal, characteristic, complemented
3#76295C211yesmaximal, characteristic, complemented
4C3×#1879C211yesmaximal, characteristic
5#72037C211yesmaximal, characteristic
6#71607C211yesmaximal, characteristic, complemented
7#70174C211yesmaximal, characteristic, complemented
8#70169C211yesmaximal, characteristic, complemented
9#70152C211yesmaximal, characteristic, complemented
10#70152C211yesmaximal, characteristic, complemented
11#69637C211yesmaximal, characteristic, complemented
12#68814C211yesmaximal, characteristic, complemented
13#68784C211yesmaximal, characteristic, complemented
14#68621C211yesmaximal, characteristic, complemented
15#68598C211yesmaximal, characteristic, complemented
16#21134—31nomaximal, Sylow
17C22×C3×D8C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×D4×Dic3C2211yescharacteristic
20C2×D4×Dic3C2211yescharacteristic
21C2×D4×Dic3C2211yescharacteristic
22C2×C3×#102C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C3×#390C2211yescharacteristic
25#5221C2211yescharacteristic
26C2×#832C2211yescharacteristic
27C2×#832C2211yescharacteristic
28C2×#832C2211yescharacteristic
29C2×#832C2211yescharacteristic
30C4×#217—21no—
31C2×C3⋊#95—21no—
32Dic3×D8C418yescomplemented
33C2×#818C2211yescharacteristic
34C2×#815C2211yescharacteristic
35C2×C8×Dic3C2211yescharacteristic
36Dic3×C4⋊C4C2211yescharacteristic
37Dic3×C4⋊C4C2211yescharacteristic
38Dic3×C4⋊C4C2211yescharacteristic
39#4972C2211yescharacteristic
40C2×#808C2211yescharacteristic
41C2×#808C2211yescharacteristic
42#4945—21no—
43#4944—21no—
44#4943—21no—
45#4942—21no—
46Dic3×C22⋊C4—21no—
47Dic3×C22⋊C4—21no—
48C4×C4⋊Dic3C2211yescharacteristic
49#4932C2211yescharacteristic
50#4928C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#336C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61C2×C4×D8—31no—
62#2319—31no—
63#2275—31no—
64#1926—31no—
65#1926—31no—
66#1918—31no—
67#1918—31no—
68#1879—31no—
69#1879D311yescharacteristic
70#1864—31no—
71#1864—31no—
72C4×#106—31no—
73#1762—31no—
74#1762—31no—
75#1746—31no—
76C22×C3×D4C2311yescharacteristic
77C22×C3×D4C2311yescharacteristic
78C23×C3×C4—21no—
79C23×Dic3—21no—
80C23×Dic3—21no—
81C23×Dic3—21no—
82C2×C3×D8—22no—
83C2×C3×D8—22no—
84C2×C3×D8C2×C414yes—
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C3×C4×D4—22no—
88C3×C4×D4—22no—
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C42C2311yescharacteristic
95C2×#217—21no—
96C2×#217—21no—
97C2×#217—21no—
98C2×#217—21no—
99C2×#217—21no—
100D4×Dic3—22no—
101D4×Dic3—22no—
102D4×Dic3—22no—
103D4×Dic3—22no—
104D4×Dic3C2×C414yes—
105D4×Dic3C2×C414yescomplemented
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3C2311yescharacteristic
117C2×C4×Dic3C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C3×#102—22no—
121C3×#102—22no—
122#832—22no—
123#832—22no—
124#832C2×C414yes—
125#832C2×C414yescomplemented
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95D411yescharacteristic
133C3⋊#95D411yescharacteristic
134C3⋊#95D411yescharacteristic
135C3⋊#95D411yescharacteristic
136#818C2×C414yescomplemented
137#815—22no—
138C8×Dic3C2×C414yescomplemented
139#808—22no—
140#808—22no—
141C22×D8C2×D311yescharacteristic
142C2×C4×D4—31no—
143C2×C4×D4—31no—
144C2×C4×D4—31no—
145C2×C4×D4C2×D311yescharacteristic
146C4×D8—38no—
147C2×#107—31no—
148C2×#106—31no—
149C2×#106—31no—
150C2×#102—31no—
151C2×#102—31no—
152C2×#102—31no—
153C2×#102—31no—
154C2×#102C2×D311yescharacteristic
155C2×#102C2×D311yescharacteristic
156C2×#97—31no—
157C2×C4×C8—31no—
158#390—31no—
159#390C2×D311yescharacteristic
160#389—31no—
161#389—31no—
162#388—61no—
163#388—61no—
164#381—61no—
165#380—61no—
166C4×C4⋊C4—31no—
167C4×C4⋊C4—31no—
168C4×C4⋊C4—31no—
169C4×C4⋊C4—31no—
170C4×C22⋊C4—61no—
171C4×C22⋊C4—61no—
172C4×C22⋊C4—61no—
173#376—31no—
174C2×#95—61no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340C2×D311yescharacteristic
181#336—31no—
182#336—31no—
183#336—31no—
184#336C2×D311yescharacteristic
185C24×C3—21no—
186C24×C3—21no—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4—21no—
190C2×C3×D4—21no—
191C2×C3×D4—21no—
192C2×C3×D4—21no—
193C2×C3×D4—21no—
194C2×C3×D4—21no—
195C2×C3×D4C22×C412yes—
196C2×C3×D4C22×C412yes—
197C2×C3×D4#4112yes—
198C2×C3×D4#4112yes—
199C22×C3×C4—21no—
200C22×C3×C4—24no—
201C22×C3×C4—21no—
202C22×C3×C4—81no—
203C22×C3×C4—81no—
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2411yescharacteristic
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3—24no—
211C22×Dic3—24no—
212C22×Dic3—24no—
213C22×Dic3—21no—
214C22×Dic3—81no—
215C22×Dic3—81no—
216C22×Dic3—81no—
217C22×Dic3—81no—
218C22×Dic3—81no—
219C22×Dic3—81no—
220C22×Dic3C2×D411yescharacteristic
221C22×Dic3C2×D411yescharacteristic
222C22×Dic3C2×D411yescharacteristic
223C22×Dic3C2×D411yescharacteristic
224C3×D8—28no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8C22×C412yes—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4#4112yes—
235C3×C22⋊C4—24no—
236C3×C22⋊C4—81no—
237C3×C22⋊C4—81no—
238C3×C42—21no—
239C3×C42—21no—
240#217—24no—
241#217—24no—
242#217—24no—
243#217—81no—
244#217—81no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3C22×C412yes—
251C4⋊Dic3C22×C412yes—
252C4⋊Dic3#4112yes—
253Dic3⋊C4—81no—
254Dic3⋊C4—81no—
255C4×Dic3—21no—
256C4×Dic3—21no—
257C4×Dic3—81no—
258C4×Dic3—81no—
259C4×Dic3—81no—
260C4×Dic3—81no—
261C4×Dic3C22×C412yes—
262C4×Dic3#4112yes—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8C22×C412yes—
269C22×D4C22×D311yescharacteristic
270C22×D4C22×D311yescharacteristic
271C23×C4—21no—
272C23×C4—61no—
273C23×C4—61no—
274C23×C4—61no—
275C2×D8—22no—
276C2×D8—22no—
277C2×D8C4×D314yes—
278C22×C8—31no—
279C22×C8—31no—
280C22×C8C22×D311yescharacteristic
281C22×C8C22×D311yescharacteristic
282C4×D4—22no—
283C4×D4—22no—
284C4×D4—34no—
285C4×D4—34no—
286C4×D4—62no—
287C4×D4—62no—
288C4×D4—62no—
289C4×D4—62no—
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⋊C4C22×D311yescharacteristic
297C2×C4⋊C4C22×D311yescharacteristic
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—61no—
302C2×C22⋊C4—61no—
303C2×C22⋊C4—61no—
304C2×C22⋊C4—61no—
305C2×C22⋊C4—61no—
306C2×C42—31no—
307C2×C42—31no—
308C2×C42—61no—
309C2×C42—61no—
310C2×C42—61no—
311C2×C42—61no—
312C2×C42C22×D311yescharacteristic
313#107—34no—
314#106—62no—
315#106—62no—
316#102—22no—
317#102—22no—
318#102—34no—
319#102—34no—
320#102—62no—
321#102—62no—
322#97—62no—
323C4×C8—34no—
324#95—31no—
325#95—31no—
326#95—31no—
327#95—31no—
328#95—61no—
329#95—61no—
330#95—61no—
331#95—61no—
332#95—61no—
333#95—61no—
334C23×C3—22no—
335C23×C3—22no—
336C23×C3—22no—
337C23×C3—22no—
338C23×C3—41no—
339C23×C3—41no—
340C23×C3—41no—
341C23×C3—41no—
342C23×C3—41no—
343C23×C3—41no—
344C23×C3—41no—
345C23×C3—41no—
346C23×C3—41no—
347C23×C3—41no—
348C23×C3C22×D411yescharacteristic
349C3×D4—22no—
350C3×D4—22no—
351C3×D4—22no—
352C3×D4—22no—
353C3×D4—22no—
354C3×D4—41no—
355C3×D4—41no—
356C3×D4#11714yes—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—22no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—84no—
370C2×C3×C4C23×C411yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—22no—
381C2×Dic3—22no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—22no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—84no—
399C2×Dic3—84no—
400C2×Dic3—84no—
401C2×Dic3C4×D412yes—
402C2×Dic3C4×D412yes—
403C3×C8—22no—
404C3×C8—41no—
405C3⋊C8—22no—
406C3⋊C8—41no—
407C24—21no—
408C24—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4C2×C4×D312yes—
418C2×D4C2×C4×D312yes—
419C2×D4#23712yes—
420C2×D4#23712yes—
421C22×C4—21no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—31no—
425C22×C4—31no—
426C22×C4—31no—
427C22×C4—31no—
428C22×C4—64no—
429C22×C4—64no—
430C22×C4—61no—
431C22×C4—61no—
432C22×C4—61no—
433C22×C4—64no—
434C22×C4—61no—
435C22×C4—81no—
436C22×C4—81no—
437C22×C4—241no—
438C22×C4—241no—
439C22×C4—241no—
440C22×C4—241no—
441C22×C4—241no—
442C22×C4—241no—
443C22×C4D3×D411yescharacteristic
444C22×C4D3×D411yescharacteristic
445C22×C4C23×D311yescharacteristic
446D8—28no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—32no—
453C2×C8—61no—
454C2×C8—61no—
455C2×C8—61no—
456C2×C8—61no—
457C2×C8—61no—
458C2×C8C2×C4×D312yes—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—32no—
463C4⋊C4—32no—
464C4⋊C4—32no—
465C4⋊C4—61no—
466C4⋊C4—61no—
467C4⋊C4—61no—
468C4⋊C4—61no—
469C4⋊C4—61no—
470C4⋊C4—241no—
471C4⋊C4—241no—
472C4⋊C4#23712yes—
473C22⋊C4—24no—
474C22⋊C4—64no—
475C22⋊C4—64no—
476C22⋊C4—64no—
477C22⋊C4—81no—
478C22⋊C4—81no—
479C22⋊C4—241no—
480C22⋊C4—241no—
481C42—21no—
482C42—21no—
483C42—32no—
484C42—32no—
485C42—61no—
486C42—61no—
487C42—241no—
488C42—241no—
489C42—241no—
490C42—241no—
491C22×C3—24no—
492C22×C3—42no—
493C22×C3—42no—
494C22×C3—42no—
495C22×C3—42no—
496C22×C3—42no—
497C22×C3—42no—
498C22×C3—42no—
499C22×C3—42no—
500C22×C3—42no—
501C22×C3—81no—
502C22×C3—81no—
503C22×C3—81no—
504C22×C3—81no—
505C22×C3C2×C4×D411yescharacteristic
506C22×C3#52511yescharacteristic
507C22×C3#52511yescharacteristic
508C22×C3#57611yescharacteristic
509C22×C3#57611yescharacteristic
510C22×C3#57711yescharacteristic
511C22×C3#57711yescharacteristic
512C3×C4—41no—
513C3×C4—42no—
514C3×C4—41no—
515C3×C4—81no—
516C3×C4C2×#11711yescharacteristic
517C3×C4C4×#4111yescharacteristic
518C3×C4#52911yescharacteristic
519C3×C4#52911yescharacteristic
520Dic3—22no—
521Dic3—22no—
522Dic3—41no—
523Dic3—42no—
524Dic3—41no—
525Dic3—42no—
526Dic3—42no—
527Dic3—81no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23C2×D3×D411yescharacteristic
543D4—22no—
544D4—22no—
545D4—22no—
546D4—22no—
547D4—22no—
548D4—41no—
549D4—41no—
550D4#87714yes—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—32no—
559C2×C4—32no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
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—62no—
574C2×C4—62no—
575C2×C4—61no—
576C2×C4—61no—
577C2×C4—62no—
578C2×C4—84no—
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—121no—
588C2×C4—244no—
589C2×C4—244no—
590C2×C4—244no—
591C2×C4C22×C4×D311yescharacteristic
592C2×C4C2×#23711yescharacteristic
593C2×C4C2×#23711yescharacteristic
594C2×C4C2×#23711yescharacteristic
595C2×C4D3×#4111yescharacteristic
596C2×C4D3×#4111yescharacteristic
597C8—22no—
598C8—41no—
599C8—62no—
600C8—121no—
601C2×C3—44no—
602C2×C3—82no—
603C2×C3#229011yescharacteristic
604C2×C3#294811yescharacteristic
605C2×C3#295211yescharacteristic
606C2×C3#311811yescharacteristic
607C2×C3#311811yescharacteristic
608C2×C3#311911yescharacteristic
609C2×C3#311911yescharacteristic
610C22—24no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22C4×D3×D411yescharacteristic
625C22#558611yescharacteristic
626C22#560611yescharacteristic
627C22#575711yescharacteristic
628C22#576911yescharacteristic
629C22#577511yescharacteristic
630C22#577511yescharacteristic
631C4—41no—
632C4—42no—
633C4—41no—
634C4—62no—
635C4—62no—
636C4—81no—
637C4—121no—
638C4—122no—
639C4—121no—
640C4—122no—
641C4—122no—
642C4—241no—
643C4C2×#87711yescharacteristic
644C4C4×#23711yescharacteristic
645C4#560011yescharacteristic
646C4#564911yescharacteristic
647C3#2113411yescharacteristic, complemented, Sylow
648C2—44no—
649C2—82no—
650C2#7465711yescharacteristic
651C2#7958811yescharacteristic
652C2#7986811yescharacteristic
653C2#7999211yescharacteristic
654C2#8007211yescharacteristic
655C2#8031611yescharacteristic
656C2#8050811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1879C2
socleC23×C3C22×D4
prime radicalC22×C3×D8C22
cosocleC22×C3×C4C24
hypercenter#1879D3
hypocenterC3#21134
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1879

Complements

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

NH ≅ G/Ntype
2. #82317C2semidirect
3. #76295C2semidirect
6. #71607C2semidirect
7. #70174C2semidirect
8. #70169C2semidirect
9. #70152C2semidirect
10. #70152C2semidirect
11. #69637C2semidirect
12. #68814C2semidirect
13. #68784C2semidirect
14. #68621C2semidirect
15. #68598C2semidirect
32. Dic3×D8C4semidirect
105. D4×Dic3C2×C4semidirect
125. #832C2×C4semidirect
136. #818C2×C4semidirect
138. C8×Dic3C2×C4semidirect
647. C3#21134semidirect

Maximal subgroups

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

subgroupindex
C2×Dic3×D82
#823172
#762952
C3×#18792
#720372
#716072
#701742
#701692
#701522
#696372
#688142
#687842
#686212
#685982
#211343