Finite Group Databasebeta

← #10824798

Subgroups — #10824798

Order 768.

932 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5152C211yesmaximal, characteristic, complemented
2#82633C211yesmaximal, characteristic, complemented
3#82633C211yesmaximal, characteristic, complemented
4C2×#4720C211yesmaximal, characteristic, complemented
5C3×#1999C211yesmaximal, characteristic
6#72040C211yesmaximal, characteristic, complemented
7#71680C211yesmaximal, characteristic, complemented
8#71679C211yesmaximal, characteristic, complemented
9#70194C211yesmaximal, characteristic, complemented
10#70193C211yesmaximal, characteristic, complemented
11Dic3×#102C211yesmaximal, characteristic, complemented
12Dic3×#102C211yesmaximal, characteristic, complemented
13#69940C211yesmaximal, characteristic, complemented
14#68840C211yesmaximal, characteristic, complemented
15#68786C211yesmaximal, characteristic, complemented
16#23712—31nomaximal, Sylow
17C22×C3×D8C2211yescharacteristic
18C2×C3×#121C2211yescharacteristic
19C2×C3×#121C2211yescharacteristic
20C2×D4×Dic3C2211yescharacteristic
21C2×D4×Dic3C2211yescharacteristic
22C2×#926C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×C22⋊C8C2211yescharacteristic
26#5221—21no—
27#5221—21no—
28#5221C2211yescharacteristic
29#5221C2211yescharacteristic
30#5220—21no—
31#5220—21no—
32C2×#832C2211yescharacteristic
33C2×#832C2211yescharacteristic
34C2×#832C2211yescharacteristic
35C2×#832C2211yescharacteristic
36#5212C2211yescharacteristic
37#5212C2211yescharacteristic
38C2×#830C2211yescharacteristic
39#5152—24no—
40C2×#817C2211yescharacteristic
41C2×#815C2211yescharacteristic
42C2×Dic3⋊C8C2211yescharacteristic
43C2×Dic3⋊C8C2211yescharacteristic
44C2×C8×Dic3C2211yescharacteristic
45#4984C2211yescharacteristic
46#4984C2211yescharacteristic
47Dic3×C4⋊C4C2211yescharacteristic
48Dic3×C4⋊C4C2211yescharacteristic
49C2×#808C2211yescharacteristic
50C2×#807C2211yescharacteristic
51#4951—21no—
52#4951—21no—
53#4950—21no—
54#4950—21no—
55#4720—24no—
56C3×#340C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#442—31no—
63C2×#433—31no—
64#2324—31no—
65#2324—31no—
66#2088—31no—
67#2077—31no—
68#2067—31no—
69#2065—31no—
70#2054—31no—
71#2034—31no—
72#2033—31no—
73#1999D311yescharacteristic
74#1776—31no—
75C4×#102—31no—
76C4×#102—31no—
77C22×C3×D4—21no—
78C22×C3×D4—21no—
79C22×C3×D4C2311yescharacteristic
80C22×C3×D4C2311yescharacteristic
81C23×C3×C4C2311yescharacteristic
82C23×Dic3—21no—
83C23×Dic3—21no—
84C2×C3×D8—22no—
85C2×C3×D8—22no—
86C2×C3×D8—22no—
87C22×C3×C8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C3×#121—22no—
90C3×#121—22no—
91C3×#121—22no—
92C3×#121—22no—
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×#217—21no—
98C2×#217—21no—
99C2×#217—21no—
100C2×#217—21no—
101C2×#217—21no—
102C2×#217—21no—
103C2×#217—21no—
104C2×#217—21no—
105C2×#217D411yescharacteristic
106C2×#217D411yescharacteristic
107D4×Dic3—22no—
108D4×Dic3—22no—
109D4×Dic3—22no—
110D4×Dic3—22no—
111#926—22no—
112#926—22no—
113C2×C4⋊Dic3—21no—
114C2×C4⋊Dic3—21no—
115C2×C4⋊Dic3D411yescharacteristic
116C2×C4⋊Dic3D411yescharacteristic
117C2×C4⋊Dic3C2311yescharacteristic
118C2×C4⋊Dic3C2311yescharacteristic
119C2×C4⋊Dic3C2311yescharacteristic
120C2×C4⋊Dic3C2311yescharacteristic
121C2×C4×Dic3—21no—
122C2×C4×Dic3—21no—
123C2×C4×Dic3C2311yescharacteristic
124C2×C4×Dic3C2311yescharacteristic
125C22×C3⋊C8C2311yescharacteristic
126C22×C3⋊C8C2311yescharacteristic
127C3×#102—22no—
128C3×#102—22no—
129C3×C22⋊C8—22no—
130#832—22no—
131#832—22no—
132#832—22no—
133#832—22no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140#830—22no—
141#817—22no—
142#815D414yescomplemented
143Dic3⋊C8—22no—
144Dic3⋊C8—22no—
145C8×Dic3—22no—
146#808—22no—
147#807—22no—
148C22×D8C2×D311yescharacteristic
149C2×#121C2×D311yescharacteristic
150C2×#121C2×D311yescharacteristic
151C2×C4×D4—31no—
152C2×C4×D4—31no—
153C2×#117—31no—
154#442—64no—
155#433—64no—
156C2×#107—31no—
157C2×#106—31no—
158C2×#106—31no—
159C2×C4⋊C8—31no—
160C2×C4⋊C8—31no—
161C2×#102—31no—
162C2×#102—31no—
163C2×#102—31no—
164C2×#102—31no—
165C2×#102C2×D311yescharacteristic
166C2×#102C2×D311yescharacteristic
167C2×C22⋊C8—31no—
168C2×C22⋊C8C2×D311yescharacteristic
169C2×#97—31no—
170C2×C4×C8—31no—
171#390—31no—
172#390—31no—
173#390—61no—
174#390—61no—
175#389—31no—
176#389—31no—
177#388—61no—
178#388—61no—
179#386—61no—
180#386—61no—
181#385—61no—
182#385—61no—
183#380—31no—
184#380—31no—
185C4×C4⋊C4—31no—
186C4×C4⋊C4—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340C2×D311yescharacteristic
193C24×C3—21no—
194C24×C3—21no—
195C24×C3C2×D411yescharacteristic
196C2×C3×D4—24no—
197C2×C3×D4—24no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—81no—
209C2×C3×D4—81no—
210C2×C3×D4—81no—
211C2×C3×D4—81no—
212C2×C3×D4#4112yes—
213C2×C3×D4#4112yes—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2411yescharacteristic
224C22×Dic3—21no—
225C22×Dic3—21no—
226C22×Dic3—21no—
227C22×Dic3—21no—
228C22×Dic3—42no—
229C22×Dic3—42no—
230C22×Dic3—81no—
231C22×Dic3—81no—
232C22×Dic3—81no—
233C22×Dic3—81no—
234C22×Dic3C2×D411yescharacteristic
235C22×Dic3C2×D411yescharacteristic
236C22×Dic3C2×D411yescharacteristic
237C22×Dic3C2×D411yescharacteristic
238C3×D8—28no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8C2×D412yes—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4#4112yes—
248C3×C4⋊C4#4112yes—
249C3×C22⋊C4—24no—
250C3×C22⋊C4—24no—
251#217—22no—
252#217—22no—
253#217—42no—
254#217—42no—
255#217—81no—
256#217—81no—
257#217—81no—
258#217—81no—
259#217—81no—
260#217—81no—
261C4⋊Dic3—22no—
262C4⋊Dic3—22no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—81no—
272C4⋊Dic3—81no—
273C4×Dic3—21no—
274C4×Dic3—21no—
275C4×Dic3—81no—
276C4×Dic3—81no—
277C4×Dic3C2×D412yes—
278C4×Dic3#4112yes—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8C2×D412yes—
285C22×D4—21no—
286C22×D4—21no—
287C22×D4C22×D311yescharacteristic
288C22×D4C22×D311yescharacteristic
289C23×C4—61no—
290C23×C4—61no—
291C23×C4C22×D311yescharacteristic
292C2×D8—22no—
293C2×D8—22no—
294C2×D8—22no—
295C22×C8—31no—
296C22×C8—31no—
297C22×C8C22×D311yescharacteristic
298C22×C8C22×D311yescharacteristic
299#121—22no—
300#121—22no—
301#121—22no—
302#121—22no—
303C4×D4—62no—
304C4×D4—62no—
305C4×D4—62no—
306C4×D4—62no—
307#117—62no—
308#117—62no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4C22×D311yescharacteristic
318C2×C4⋊C4C22×D311yescharacteristic
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—31no—
322C2×C22⋊C4—31no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C22⋊C4—61no—
328C2×C22⋊C4—61no—
329C2×C22⋊C4—61no—
330C2×C22⋊C4—61no—
331C2×C42—31no—
332C2×C42—31no—
333C2×C42—61no—
334C2×C42—61no—
335#107—62no—
336#106—62no—
337#106—62no—
338C4⋊C8—62no—
339C4⋊C8—62no—
340#102—22no—
341#102—22no—
342#102—62no—
343#102—62no—
344#102—62no—
345#102—62no—
346C22⋊C8—22no—
347C22⋊C8—62no—
348#97—34no—
349C4×C8—62no—
350#95—61no—
351#95—61no—
352#95—61no—
353#95—61no—
354#95—61no—
355#95—61no—
356C23×C3—21no—
357C23×C3—22no—
358C23×C3—22no—
359C23×C3—21no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—21no—
363C23×C3—21no—
364C23×C3—21no—
365C23×C3—41no—
366C23×C3—41no—
367C23×C3—41no—
368C23×C3—41no—
369C23×C3—41no—
370C23×C3—41no—
371C23×C3—41no—
372C23×C3—41no—
373C23×C3—41no—
374C23×C3—41no—
375C23×C3—41no—
376C23×C3—41no—
377C23×C3C22×D411yescharacteristic
378C3×D4—22no—
379C3×D4—22no—
380C3×D4—22no—
381C3×D4—22no—
382C3×D4—41no—
383C3×D4—41no—
384C3×D4—41no—
385C3×D4—41no—
386C3×D4—84no—
387C3×D4—84no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—22no—
392C2×C3×C4—22no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4C22×D411yescharacteristic
402C2×C3×C4C2×#4111yescharacteristic
403C2×C3×C4C2×#4111yescharacteristic
404C2×C3×C4C2×#4111yescharacteristic
405C2×C3×C4C2×#4111yescharacteristic
406C2×C3×C4C2×#4111yescharacteristic
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3—41no—
429C2×Dic3—84no—
430C2×Dic3—84no—
431C2×Dic3#12112yes—
432C2×Dic3#12112yes—
433C3×C8—22no—
434C3×C8—41no—
435C3⋊C8—22no—
436C3⋊C8—41no—
437C24—21no—
438C24—21no—
439C24D3×D411yescharacteristic
440C2×D4—24no—
441C2×D4—24no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—81no—
453C2×D4—81no—
454C2×D4—81no—
455C2×D4—81no—
456C2×D4#23712yes—
457C2×D4#23712yes—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—31no—
467C22×C4—31no—
468C22×C4—31no—
469C22×C4—31no—
470C22×C4—61no—
471C22×C4—61no—
472C22×C4—61no—
473C22×C4—61no—
474C22×C4—122no—
475C22×C4—122no—
476C22×C4—241no—
477C22×C4—241no—
478C22×C4—241no—
479C22×C4—241no—
480C22×C4D3×D411yescharacteristic
481C22×C4C23×D311yescharacteristic
482D8—28no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—32no—
489C2×C8—61no—
490C2×C8—61no—
491C2×C8—61no—
492C2×C8—61no—
493C2×C8—61no—
494C2×C8D3×D412yes—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—62no—
498C4⋊C4—62no—
499C4⋊C4—61no—
500C4⋊C4—61no—
501C4⋊C4—61no—
502C4⋊C4—61no—
503C4⋊C4—61no—
504C4⋊C4—61no—
505C4⋊C4—61no—
506C4⋊C4—61no—
507C4⋊C4—241no—
508C4⋊C4—241no—
509C4⋊C4#23712yes—
510C4⋊C4#23712yes—
511C22⋊C4—24no—
512C22⋊C4—24no—
513C22⋊C4—62no—
514C22⋊C4—62no—
515C22⋊C4—122no—
516C22⋊C4—122no—
517C22⋊C4—241no—
518C22⋊C4—241no—
519C22⋊C4—241no—
520C22⋊C4—241no—
521C22⋊C4—241no—
522C22⋊C4—241no—
523C42—32no—
524C42—32no—
525C42—61no—
526C42—61no—
527C42—241no—
528C42—241no—
529C22×C3—41no—
530C22×C3—42no—
531C22×C3—41no—
532C22×C3—42no—
533C22×C3—41no—
534C22×C3—42no—
535C22×C3—41no—
536C22×C3—42no—
537C22×C3—41no—
538C22×C3—42no—
539C22×C3—42no—
540C22×C3—41no—
541C22×C3—41no—
542C22×C3—81no—
543C22×C3—81no—
544C22×C3—81no—
545C22×C3—81no—
546C22×C3—81no—
547C22×C3—81no—
548C22×C3—81no—
549C22×C3—81no—
550C22×C3C2×#12111yescharacteristic
551C22×C3#52511yescharacteristic
552C22×C3#52511yescharacteristic
553C22×C3#57611yescharacteristic
554C22×C3#57611yescharacteristic
555C22×C3#57711yescharacteristic
556C22×C3#57711yescharacteristic
557C3×C4—42no—
558C3×C4—42no—
559C3×C4—41no—
560C3×C4#52511yescharacteristic
561C3×C4#52911yescharacteristic
562C3×C4#52911yescharacteristic
563C3×C4#53211yescharacteristic
564Dic3—22no—
565Dic3—22no—
566Dic3—41no—
567Dic3—41no—
568Dic3—81no—
569Dic3—81no—
570Dic3—81no—
571Dic3—81no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—22no—
577C23—22no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23—41no—
591C23—41no—
592C23—41no—
593C23C2×D3×D411yescharacteristic
594D4—22no—
595D4—22no—
596D4—22no—
597D4—22no—
598D4—41no—
599D4—41no—
600D4—41no—
601D4—41no—
602D4—84no—
603D4—84no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—21no—
609C2×C4—32no—
610C2×C4—32no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—61no—
620C2×C4—61no—
621C2×C4—61no—
622C2×C4—61no—
623C2×C4—61no—
624C2×C4—61no—
625C2×C4—61no—
626C2×C4—61no—
627C2×C4—61no—
628C2×C4—61no—
629C2×C4—121no—
630C2×C4—121no—
631C2×C4—121no—
632C2×C4—121no—
633C2×C4—121no—
634C2×C4—121no—
635C2×C4—121no—
636C2×C4—121no—
637C2×C4—121no—
638C2×C4—121no—
639C2×C4—121no—
640C2×C4—121no—
641C2×C4—244no—
642C2×C4—244no—
643C2×C4C2×D3×D411yescharacteristic
644C2×C4C2×#23711yescharacteristic
645C2×C4C2×#23711yescharacteristic
646C2×C4C2×#23711yescharacteristic
647C2×C4C2×#23711yescharacteristic
648C2×C4D3×#4111yescharacteristic
649C8—22no—
650C8—41no—
651C8—62no—
652C8—121no—
653C2×C3—81no—
654C2×C3—82no—
655C2×C3—82no—
656C2×C3#239211yescharacteristic
657C2×C3#306211yescharacteristic
658C2×C3#306211yescharacteristic
659C2×C3#311811yescharacteristic
660C2×C3#311911yescharacteristic
661C2×C3#312211yescharacteristic
662C2×C3#312511yescharacteristic
663C22—41no—
664C22—41no—
665C22—42no—
666C22—41no—
667C22—42no—
668C22—41no—
669C22—42no—
670C22—42no—
671C22—41no—
672C22—42no—
673C22—41no—
674C22—42no—
675C22—41no—
676C22—81no—
677C22—81no—
678C22—81no—
679C22—81no—
680C22—81no—
681C22—81no—
682C22—81no—
683C22—81no—
684C22#558611yescharacteristic
685C22#558611yescharacteristic
686C22D3×#12111yescharacteristic
687C22#575711yescharacteristic
688C22#577011yescharacteristic
689C22#577411yescharacteristic
690C22#577511yescharacteristic
691C4—42no—
692C4—42no—
693C4—41no—
694C4—62no—
695C4—62no—
696C4—121no—
697C4—121no—
698C4—241no—
699C4—241no—
700C4—241no—
701C4—241no—
702C4#558611yescharacteristic
703C4#560011yescharacteristic
704C4#560011yescharacteristic
705C4#571611yescharacteristic
706C3#2371211yescharacteristic, complemented, Sylow
707C2—81no—
708C2—82no—
709C2—82no—
710C2#7666611yescharacteristic
711C2#7999211yescharacteristic
712C2#7999311yescharacteristic
713C2#8007311yescharacteristic
714C2#8007511yescharacteristic
715C2#8071811yescharacteristic
716C2#8086711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#1999C2
socleC23×C3C22×D4
hypercenter#1999D3
hypocenterC3#23712
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1999

Complements

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

NH ≅ G/Ntype
1. C2×#5152C2semidirect
2. #82633C2semidirect
3. #82633C2semidirect
4. C2×#4720C2semidirect
6. #72040C2semidirect
7. #71680C2semidirect
8. #71679C2semidirect
9. #70194C2semidirect
10. #70193C2semidirect
11. Dic3×#102C2semidirect
12. Dic3×#102C2semidirect
13. #69940C2semidirect
14. #68840C2semidirect
15. #68786C2semidirect
142. #815D4semidirect
706. C3#23712semidirect

Maximal subgroups

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

subgroupindex
C2×#51522
#826332
C2×#47202
C3×#19992
#720402
#716802
#716792
#701942
#701932
Dic3×#1022
#699402
#688402
#687862
#237123