Finite Group Databasebeta

← #10824545

Subgroups — #10824545

Order 768.

892 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Dic3×SD16C211yesmaximal, characteristic, complemented
2C2×#5152C211yesmaximal, characteristic
3#82953C211yesmaximal, characteristic, complemented
4#76295C211yesmaximal, characteristic, complemented
5C3×#2012C211yesmaximal, characteristic
6#72051C211yesmaximal, characteristic, complemented
7#72040C211yesmaximal, characteristic
8#71722C211yesmaximal, characteristic, complemented
9#70247C211yesmaximal, characteristic, complemented
10#70220C211yesmaximal, characteristic, complemented
11#70179C211yesmaximal, characteristic, complemented
12Dic3×#102C211yesmaximal, characteristic, complemented
13#69640C211yesmaximal, characteristic, complemented
14#68640C211yesmaximal, characteristic, complemented
15#68604C211yesmaximal, characteristic, complemented
16#23693—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C22×C3×D8C2211yescharacteristic
19C2×C3×#124C2211yescharacteristic
20C2×Q8×Dic3C2211yescharacteristic
21C2×D4×Dic3C2211yescharacteristic
22C2×D4×Dic3C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C3×#390C2211yescharacteristic
26#5229C2211yescharacteristic
27C2×#835C2211yescharacteristic
28C2×#835C2211yescharacteristic
29#5221C2211yescharacteristic
30#5221C2211yescharacteristic
31C2×#832C2211yescharacteristic
32C2×#832C2211yescharacteristic
33C2×#832C2211yescharacteristic
34C2×#832C2211yescharacteristic
35Dic3×SD16—24no—
36#5152—24no—
37C2×#817C2211yescharacteristic
38C2×#817C2211yescharacteristic
39C2×#815C2211yescharacteristic
40C2×Dic3⋊C8C2211yescharacteristic
41C2×Dic3⋊C8C2211yescharacteristic
42C2×C8×Dic3C2211yescharacteristic
43Dic3×C4⋊C4C2211yescharacteristic
44#4972C2211yescharacteristic
45#4951—21no—
46#4951—21no—
47#4950—21no—
48#4950—21no—
49#4945—21no—
50#4944—21no—
51#4943—21no—
52#4942—21no—
53#4935C2211yescharacteristic
54#4932C2211yescharacteristic
55#4928C2211yescharacteristic
56C3×#336C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#442—31no—
63C2×C4×SD16—31no—
64#2326—31no—
65#2319—31no—
66#2084—31no—
67#2071—31no—
68#2066—31no—
69#2065—31no—
70#2055—31no—
71#2054—31no—
72#2042—31no—
73#2012D311yescharacteristic
74#1768—31no—
75#1763—31no—
76C4×#102—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C22×C3×D4C2311yescharacteristic
80C23×Dic3—21no—
81C23×Dic3—21no—
82C2×C3×SD16—22no—
83C2×C3×SD16—22no—
84C2×C3×SD16—22no—
85C2×C3×D8—22no—
86C2×C3×D8—22no—
87C2×C3×D8—22no—
88C22×C3×C8C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C3×#124—22no—
91C3×#124—22no—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C42C2311yescharacteristic
98C2×#217—21no—
99C2×#217—21no—
100C2×#217—21no—
101C2×#217—21no—
102C2×#217—21no—
103C2×#217—21no—
104Q8×Dic3—22no—
105Q8×Dic3—22no—
106D4×Dic3—22no—
107D4×Dic3—22no—
108D4×Dic3—22no—
109D4×Dic3D414yes—
110C2×C4⋊Dic3—21no—
111C2×C4⋊Dic3—21no—
112C2×C4⋊Dic3—21no—
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4—21no—
119C2×C4×Dic3—21no—
120C2×C4×Dic3—21no—
121C2×C4×Dic3C2311yescharacteristic
122C2×C4×Dic3C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125C3×#103—22no—
126C3×#102—22no—
127#835—22no—
128#835—22no—
129#832—22no—
130#832—22no—
131#832—22no—
132#832—22no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95D411yescharacteristic
139C3⋊#95D411yescharacteristic
140C3⋊#95D411yescharacteristic
141C3⋊#95D411yescharacteristic
142#817—22no—
143#817—22no—
144#815—22no—
145Dic3⋊C8—22no—
146Dic3⋊C8—22no—
147C8×Dic3—22no—
148C22×SD16C2×D311yescharacteristic
149C22×D8C2×D311yescharacteristic
150C2×#124C2×D311yescharacteristic
151C2×C4×Q8—31no—
152C2×C4×D4—31no—
153C2×C4×D4—31no—
154#442—64no—
155C4×SD16—64no—
156C2×#106—31no—
157C2×#106—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8—31no—
160C2×#103—31no—
161C2×#103—31no—
162C2×#103C2×D311yescharacteristic
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102C2×D311yescharacteristic
168C2×#97—31no—
169C2×C4×C8—31no—
170#391—31no—
171#390—31no—
172#390—31no—
173#390C2×D311yescharacteristic
174#389—31no—
175#389—31no—
176#388—61no—
177#388—61no—
178#388—61no—
179#388—61no—
180#385—61no—
181#385—61no—
182#384—31no—
183#381—61no—
184#380—61no—
185C4×C4⋊C4—31no—
186#376—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#336—31no—
192#336C2×D311yescharacteristic
193C24×C3—21no—
194C24×C3—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8#4112yes—
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—21no—
209C2×C3×D4—21no—
210C2×C3×D4—21no—
211C2×C3×D4C2×D412yes—
212C2×C3×D4#4112yes—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2411yescharacteristic
218C22×Dic3—21no—
219C22×Dic3—24no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—21no—
223C22×Dic3—42no—
224C22×Dic3—81no—
225C22×Dic3—81no—
226C22×Dic3—81no—
227C22×Dic3—81no—
228C22×Dic3C2×D411yescharacteristic
229C22×Dic3C2×D411yescharacteristic
230C22×Dic3C2×D411yescharacteristic
231C22×Dic3C2×D411yescharacteristic
232C3×SD16—28no—
233C3×D8—28no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8#4112yes—
239C2×C3×C8#4112yes—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C22⋊C4—24no—
243C3×C22⋊C4—24no—
244C3×C22⋊C4—81no—
245C3×C22⋊C4—81no—
246C3×C42—21no—
247C3×C42—21no—
248#217—24no—
249#217—42no—
250#217—81no—
251#217—81no—
252#217—81no—
253#217—81no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—42no—
262C4⋊Dic3—81no—
263C4⋊Dic3—81no—
264C4⋊Dic3C2×D412yes—
265Dic3⋊C4—81no—
266Dic3⋊C4—81no—
267C4×Dic3—21no—
268C4×Dic3—21no—
269C4×Dic3—42no—
270C4×Dic3—81no—
271C4×Dic3C2×D412yes—
272C4×Dic3#4112yes—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C22×Q8C22×D311yescharacteristic
280C22×D4C22×D311yescharacteristic
281C22×D4C22×D311yescharacteristic
282C23×C4—61no—
283C23×C4—61no—
284C2×SD16—22no—
285C2×SD16—22no—
286C2×SD16—22no—
287C2×D8—22no—
288C2×D8—22no—
289C2×D8—22no—
290C22×C8—31no—
291C22×C8—31no—
292C22×C8C22×D311yescharacteristic
293C22×C8C22×D311yescharacteristic
294#124—22no—
295#124—22no—
296C4×Q8—62no—
297C4×Q8—62no—
298C4×D4—34no—
299C4×D4—62no—
300C4×D4—62no—
301C4×D4—62no—
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⋊C4C22×D311yescharacteristic
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C42—31no—
323C2×C42—31no—
324C2×C42—61no—
325C2×C42—61no—
326C2×C42C22×D311yescharacteristic
327#106—62no—
328#106—62no—
329C4⋊C8—62no—
330C4⋊C8—62no—
331#103—22no—
332#103—62no—
333#103—62no—
334#102—22no—
335#102—62no—
336#102—62no—
337#102—62no—
338#102—62no—
339#97—62no—
340C4×C8—62no—
341#95—31no—
342#95—31no—
343#95—31no—
344#95—31no—
345#95—61no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350C23×C3—22no—
351C23×C3—22no—
352C23×C3—41no—
353C23×C3—41no—
354C23×C3—41no—
355C23×C3—41no—
356C23×C3—41no—
357C23×C3—41no—
358C23×C3—41no—
359C23×C3—41no—
360C23×C3—41no—
361C23×C3—41no—
362C23×C3—41no—
363C23×C3—41no—
364C23×C3C22×D411yescharacteristic
365C3×Q8—22no—
366C3×Q8—22no—
367C3×Q8—41no—
368C3×Q8—41no—
369C3×D4—22no—
370C3×D4—22no—
371C3×D4—22no—
372C3×D4—22no—
373C3×D4—41no—
374C3×D4—41no—
375C3×D4—41no—
376C3×D4—41no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—22no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4C22×D411yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392C2×C3×C4C2×#4111yescharacteristic
393C2×C3×C4C2×#4111yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—22no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—84no—
418C2×Dic3—84no—
419C2×Dic3#12112yes—
420C2×Dic3#12112yes—
421C3×C8—22no—
422C3×C8—22no—
423C3⋊C8—41no—
424C3⋊C8—41no—
425C24—21no—
426C24—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8#23712yes—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4D3×D412yes—
444C2×D4#23712yes—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—31no—
448C22×C4—31no—
449C22×C4—31no—
450C22×C4—31no—
451C22×C4—64no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—122no—
457C22×C4—241no—
458C22×C4—241no—
459C22×C4—241no—
460C22×C4—241no—
461C22×C4D3×D411yescharacteristic
462C22×C4D3×D411yescharacteristic
463C22×C4C23×D311yescharacteristic
464SD16—28no—
465D8—28no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—61no—
471C2×C8—61no—
472C2×C8—61no—
473C2×C8—61no—
474C2×C8—61no—
475C2×C8—61no—
476C2×C8#23712yes—
477C2×C8#23712yes—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—32no—
481C4⋊C4—61no—
482C4⋊C4—61no—
483C4⋊C4—61no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—122no—
489C4⋊C4—241no—
490C4⋊C4—241no—
491C4⋊C4—241no—
492C4⋊C4—241no—
493C22⋊C4—24no—
494C22⋊C4—24no—
495C22⋊C4—64no—
496C22⋊C4—81no—
497C22⋊C4—81no—
498C22⋊C4—122no—
499C22⋊C4—241no—
500C22⋊C4—241no—
501C22⋊C4—241no—
502C22⋊C4—241no—
503C42—21no—
504C42—21no—
505C42—32no—
506C42—32no—
507C42—61no—
508C42—61no—
509C42—122no—
510C42—241no—
511C22×C3—42no—
512C22×C3—42no—
513C22×C3—42no—
514C22×C3—42no—
515C22×C3—42no—
516C22×C3—42no—
517C22×C3—81no—
518C22×C3—81no—
519C22×C3—81no—
520C22×C3—81no—
521C22×C3—81no—
522C22×C3—81no—
523C22×C3—81no—
524C22×C3—81no—
525C22×C3C2×#12111yescharacteristic
526C22×C3#52511yescharacteristic
527C22×C3#53211yescharacteristic
528C22×C3#57611yescharacteristic
529C22×C3#57711yescharacteristic
530C22×C3#57711yescharacteristic
531C22×C3#57711yescharacteristic
532C3×C4—41no—
533C3×C4—41no—
534C3×C4—42no—
535C3×C4—81no—
536C3×C4#52511yescharacteristic
537C3×C4#52511yescharacteristic
538C3×C4#52911yescharacteristic
539C3×C4#52911yescharacteristic
540Dic3—22no—
541Dic3—22no—
542Dic3—41no—
543Dic3—41no—
544Dic3—42no—
545Dic3—81no—
546Dic3—81no—
547Dic3—81no—
548C23—22no—
549C23—22no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23C2×D3×D411yescharacteristic
563Q8—22no—
564Q8—22no—
565Q8—41no—
566Q8—41no—
567D4—22no—
568D4—22no—
569D4—22no—
570D4—22no—
571D4—41no—
572D4—41no—
573D4—41no—
574D4—41no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—32no—
583C2×C4—32no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—61no—
590C2×C4—61no—
591C2×C4—61no—
592C2×C4—61no—
593C2×C4—61no—
594C2×C4—61no—
595C2×C4—61no—
596C2×C4—61no—
597C2×C4—61no—
598C2×C4—61no—
599C2×C4—62no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4—244no—
612C2×C4—244no—
613C2×C4C2×D3×D411yescharacteristic
614C2×C4C2×#23711yescharacteristic
615C2×C4C2×#23711yescharacteristic
616C2×C4C2×#23711yescharacteristic
617C2×C4C2×#23711yescharacteristic
618C2×C4D3×#4111yescharacteristic
619C8—22no—
620C8—22no—
621C8—121no—
622C8—121no—
623C2×C3—82no—
624C2×C3—82no—
625C2×C3#239211yescharacteristic
626C2×C3#304211yescharacteristic
627C2×C3#304511yescharacteristic
628C2×C3#311911yescharacteristic
629C2×C3#312511yescharacteristic
630C2×C3#317311yescharacteristic
631C2×C3#317911yescharacteristic
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22#558611yescharacteristic
647C22D3×#12111yescharacteristic
648C22#571611yescharacteristic
649C22#577011yescharacteristic
650C22#577011yescharacteristic
651C22#577411yescharacteristic
652C22#577511yescharacteristic
653C4—42no—
654C4—41no—
655C4—41no—
656C4—62no—
657C4—62no—
658C4—81no—
659C4—121no—
660C4—122no—
661C4—121no—
662C4—241no—
663C4—241no—
664C4—241no—
665C4#558611yescharacteristic
666C4#558611yescharacteristic
667C4#560011yescharacteristic
668C4#560011yescharacteristic
669C3#2369311yescharacteristic, complemented, Sylow
670C2—82no—
671C2—82no—
672C2#7666611yescharacteristic
673C2#8007311yescharacteristic
674C2#8007511yescharacteristic
675C2#8053711yescharacteristic
676C2#8053811yescharacteristic
677C2#8181011yescharacteristic
678C2#8183711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2012C2
socleC23×C3C22×D4
prime radicalC22×C3×D8C22
hypercenter#2012D3
hypocenterC3#23693
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2012

Complements

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

NH ≅ G/Ntype
1. C2×Dic3×SD16C2semidirect
3. #82953C2semidirect
4. #76295C2semidirect
6. #72051C2semidirect
8. #71722C2semidirect
9. #70247C2semidirect
10. #70220C2semidirect
11. #70179C2semidirect
12. Dic3×#102C2semidirect
13. #69640C2semidirect
14. #68640C2semidirect
15. #68604C2semidirect
669. C3#23693semidirect

Maximal subgroups

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

subgroupindex
C2×Dic3×SD162
C2×#51522
#829532
#762952
C3×#20122
#720512
#720402
#717222
#702472
#702202
#701792
Dic3×#1022
#696402
#686402
#686042
#236933