Finite Group Databasebeta

← #10824634

Subgroups — #10824634

Order 768.

978 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Dic3×SD16C211yesmaximal, characteristic, complemented
2Dic3×C22⋊Q8C211yesmaximal, characteristic, complemented
3Dic3×#121C211yesmaximal, characteristic
4C2×#4718C211yesmaximal, characteristic, complemented
5C3×#2000C211yesmaximal, characteristic
6#72051C211yesmaximal, characteristic, complemented
7#71694C211yesmaximal, characteristic, complemented
8#71680C211yesmaximal, characteristic
9#70261C211yesmaximal, characteristic, complemented
10Dic3×#103C211yesmaximal, characteristic, complemented
11#70193C211yesmaximal, characteristic, complemented
12Dic3×#102C211yesmaximal, characteristic, complemented
13#69940C211yesmaximal, characteristic, complemented
14#68840C211yesmaximal, characteristic, complemented
15#68784C211yesmaximal, characteristic, complemented
16#23700—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×C3×C22⋊Q8C2211yescharacteristic
19C2×C3×#121C2211yescharacteristic
20C2×Q8×Dic3C2211yescharacteristic
21C2×D4×Dic3—21no—
22C2×D4×Dic3C2211yescharacteristic
23C22×C4×Dic3C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×C22⋊C8C2211yescharacteristic
27#5229C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×#835C2211yescharacteristic
30#5221C2211yescharacteristic
31#5220—21no—
32C2×#832C2211yescharacteristic
33C2×#832C2211yescharacteristic
34#5212C2211yescharacteristic
35#5212C2211yescharacteristic
36C2×#830C2211yescharacteristic
37Dic3×SD16—24no—
38C2×#817C2211yescharacteristic
39C2×Dic3⋊C8C2211yescharacteristic
40C2×Dic3⋊C8C2211yescharacteristic
41C2×C8×Dic3C2211yescharacteristic
42C2×C8×Dic3C2211yescharacteristic
43#4984C2211yescharacteristic
44#4984C2211yescharacteristic
45#4980—21no—
46Dic3×C4⋊C4—21no—
47Dic3×C4⋊C4C2211yescharacteristic
48Dic3×C4⋊C4C2211yescharacteristic
49C2×#808C2211yescharacteristic
50C2×#808C2211yescharacteristic
51#4951—21no—
52#4950—21no—
53Dic3×C22⋊C4—21no—
54Dic3×C22⋊C4—21no—
55#4718—24no—
56C3×#340C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×C4×SD16—31no—
63C2×#431—31no—
64C4×C22⋊Q8—31no—
65C4×#121—31no—
66#2088—31no—
67#2077—31no—
68#2066—31no—
69#2065—31no—
70#2055—31no—
71#2035—31no—
72#2034—31no—
73#2000D311yescharacteristic
74C4×#106—31no—
75C4×#103—31no—
76C4×#102—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×D4—21no—
79C22×C3×D4C2311yescharacteristic
80C23×C3×C4C2311yescharacteristic
81C23×Dic3—21no—
82C23×Dic3D411yescharacteristic
83C23×Dic3D411yescharacteristic
84C2×C3×SD16—22no—
85C2×C3×SD16—22no—
86C2×C3×SD16—22no—
87C22×C3×C8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C3×C22⋊Q8—22no—
90C3×C22⋊Q8—22no—
91C3×#121—22no—
92C3×#121—22no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×#217—21no—
99C2×#217—21no—
100C2×#217—21no—
101C2×#217—21no—
102C2×#217—21no—
103Q8×Dic3—22no—
104Q8×Dic3—22no—
105D4×Dic3—22no—
106D4×Dic3—22no—
107D4×Dic3—44no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3—21no—
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4×Dic3—22no—
116C2×C4×Dic3—22no—
117C2×C4×Dic3—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3—21no—
120C2×C4×Dic3D411yescharacteristic
121C2×C4×Dic3D411yescharacteristic
122C2×C4×Dic3C2311yescharacteristic
123C2×C4×Dic3C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125C22×C3⋊C8C2311yescharacteristic
126C3×#103—22no—
127C3×#102—22no—
128C3×C22⋊C8—22no—
129#835—22no—
130#835—22no—
131#832—22no—
132#832—22no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141#830—22no—
142#817—22no—
143Dic3⋊C8—22no—
144Dic3⋊C8—22no—
145C8×Dic3—22no—
146C8×Dic3D414yescomplemented
147#808—22no—
148#808—22no—
149C22×SD16C2×D311yescharacteristic
150C2×C22⋊Q8C2×D311yescharacteristic
151C2×#121C2×D311yescharacteristic
152C2×C4×Q8—31no—
153C2×C4×D4—31no—
154C2×C4×D4—61no—
155C22×C42—31no—
156C4×SD16—64no—
157#431—64no—
158C2×#106—31no—
159C2×#106—31no—
160C2×#106—31no—
161C2×C4⋊C8—31no—
162C2×C4⋊C8—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103C2×D311yescharacteristic
166C2×#102—31no—
167C2×#102—31no—
168C2×#102C2×D311yescharacteristic
169C2×C22⋊C8—31no—
170C2×C22⋊C8C2×D311yescharacteristic
171C2×C4×C8—31no—
172C2×C4×C8—31no—
173#391—31no—
174#390—31no—
175#389—31no—
176#389—31no—
177#388—61no—
178#387—61no—
179#386—61no—
180#385—61no—
181#380—31no—
182#380—31no—
183C4×C4⋊C4—31no—
184C4×C4⋊C4—31no—
185C4×C4⋊C4—61no—
186C4×C22⋊C4—61no—
187C4×C22⋊C4—61no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193#340C2×D311yescharacteristic
194C24×C3—21no—
195C24×C3C2×D411yescharacteristic
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8#4112yes—
202C2×C3×D4—24no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—42no—
209C2×C3×D4—42no—
210C2×C3×D4#4112yes—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
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×C4C2×D411yescharacteristic
221C22×C3×C4C2411yescharacteristic
222C22×Dic3—22no—
223C22×Dic3—22no—
224C22×Dic3—22no—
225C22×Dic3—21no—
226C22×Dic3—21no—
227C22×Dic3—21no—
228C22×Dic3—21no—
229C22×Dic3—41no—
230C22×Dic3—41no—
231C22×Dic3—42no—
232C22×Dic3—42no—
233C22×Dic3—42no—
234C22×Dic3SD1614yescomplemented
235C22×Dic3C2×D411yescharacteristic
236C22×Dic3C2×D411yescharacteristic
237C22×Dic3C2×D411yescharacteristic
238C22×Dic3C2×D411yescharacteristic
239C3×SD16—28no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8C2×D412yes—
246C3×C4⋊C4—24no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4#4112yes—
250C3×C4⋊C4#4112yes—
251C3×C22⋊C4—24no—
252C3×C22⋊C4—24no—
253#217—42no—
254#217—42no—
255#217—42no—
256#217—42no—
257#217—42no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—42no—
267C4⋊Dic3—42no—
268C4⋊Dic3—42no—
269C4×Dic3—21no—
270C4×Dic3—21no—
271C4×Dic3—22no—
272C4×Dic3—22no—
273C4×Dic3—42no—
274C4×Dic3—42no—
275C4×Dic3—42no—
276C4×Dic3C2×D412yes—
277C4×Dic3#4112yes—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8C2×D412yes—
284C22×Q8C22×D311yescharacteristic
285C22×D4—21no—
286C22×D4C22×D311yescharacteristic
287C23×C4—31no—
288C23×C4—31no—
289C23×C4—61no—
290C23×C4C22×D311yescharacteristic
291C2×SD16—22no—
292C2×SD16—22no—
293C2×SD16—22no—
294C22×C8—31no—
295C22×C8—31no—
296C22×C8C22×D311yescharacteristic
297C22×C8C22×D311yescharacteristic
298C22⋊Q8—22no—
299C22⋊Q8—22no—
300#121—22no—
301#121—22no—
302C4×Q8—62no—
303C4×Q8—62no—
304C4×D4—62no—
305C4×D4—62no—
306C4×D4—124no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C4⋊C4C22×D311yescharacteristic
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C42—31no—
325C2×C42—31no—
326C2×C42—31no—
327C2×C42—31no—
328C2×C42—62no—
329C2×C42—62no—
330C2×C42—61no—
331C2×C42—61no—
332C2×C42—61no—
333#106—62no—
334#106—62no—
335#106—62no—
336C4⋊C8—62no—
337C4⋊C8—62no—
338#103—22no—
339#103—62no—
340#103—62no—
341#102—22no—
342#102—62no—
343#102—62no—
344C22⋊C8—22no—
345C22⋊C8—62no—
346C4×C8—34no—
347C4×C8—62no—
348#95—61no—
349#95—61no—
350#95—61no—
351#95—61no—
352#95—61no—
353#95—61no—
354#95—61no—
355#95—61no—
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—21no—
359C23×C3—21no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—22no—
363C23×C3—41no—
364C23×C3—41no—
365C23×C3—41no—
366C23×C3—41no—
367C23×C3—41no—
368C23×C3—41no—
369C23×C3C2×SD1612yes—
370C23×C3C22×D411yescharacteristic
371C3×Q8—22no—
372C3×Q8—22no—
373C3×Q8—41no—
374C3×Q8—41no—
375C3×D4—22no—
376C3×D4—22no—
377C3×D4—48no—
378C3×D4—41no—
379C3×D4—41no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—22no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—21no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4C22×D411yescharacteristic
397C2×C3×C4C2×#4111yescharacteristic
398C2×C3×C4C2×#4111yescharacteristic
399C2×C3×C4C2×#4111yescharacteristic
400C2×C3×C4C2×#4111yescharacteristic
401C2×C3×C4C2×#4111yescharacteristic
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—28no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—44no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—84no—
424C2×Dic3#12112yes—
425C2×Dic3#12112yes—
426C2×Dic3C2×SD1612yes—
427C2×Dic3C2×SD1612yes—
428C2×Dic3#13512yes—
429C3×C8—22no—
430C3×C8—41no—
431C3⋊C8—22no—
432C3⋊C8—41no—
433C24—21no—
434C24D3×D411yescharacteristic
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8#23712yes—
441C2×D4—24no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4#23712yes—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—31no—
460C22×C4—34no—
461C22×C4—31no—
462C22×C4—31no—
463C22×C4—31no—
464C22×C4—62no—
465C22×C4—62no—
466C22×C4—62no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4—61no—
470C22×C4—61no—
471C22×C4—122no—
472C22×C4—122no—
473C22×C4—122no—
474C22×C4—121no—
475C22×C4—121no—
476C22×C4D3×D411yescharacteristic
477C22×C4C23×D311yescharacteristic
478SD16—28no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—32no—
485C2×C8—61no—
486C2×C8—61no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8—61no—
490C2×C8D3×D412yes—
491C4⋊C4—24no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—61no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—61no—
498C4⋊C4—61no—
499C4⋊C4—61no—
500C4⋊C4—61no—
501C4⋊C4—61no—
502C4⋊C4—122no—
503C4⋊C4—122no—
504C4⋊C4—122no—
505C4⋊C4#23712yes—
506C4⋊C4#23712yes—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—122no—
510C22⋊C4—122no—
511C22⋊C4—122no—
512C22⋊C4—122no—
513C22⋊C4—122no—
514C42—32no—
515C42—32no—
516C42—62no—
517C42—62no—
518C42—61no—
519C42—61no—
520C42—122no—
521C42—122no—
522C42—122no—
523C22×C3—22no—
524C22×C3—22no—
525C22×C3—22no—
526C22×C3—41no—
527C22×C3—41no—
528C22×C3—41no—
529C22×C3—41no—
530C22×C3—42no—
531C22×C3—42no—
532C22×C3—42no—
533C22×C3—81no—
534C22×C3—81no—
535C22×C3—81no—
536C22×C3—81no—
537C22×C3C2×#12111yescharacteristic
538C22×C3#52511yescharacteristic
539C22×C3#52511yescharacteristic
540C22×C3C22×SD1611yescharacteristic
541C22×C3C2×#13511yescharacteristic
542C22×C3#57511yescharacteristic
543C22×C3#57511yescharacteristic
544C3×C4—42no—
545C3×C4—42no—
546C3×C4—41no—
547C3×C4—42no—
548C3×C4#52511yescharacteristic
549C3×C4#52911yescharacteristic
550C3×C4#52911yescharacteristic
551C3×C4#53211yescharacteristic
552Dic3—22no—
553Dic3—22no—
554Dic3—41no—
555Dic3—81no—
556Dic3—81no—
557Dic3—81no—
558Dic3—81no—
559Dic3#46514yes—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—22no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23D3×SD1612yes—
574C23C2×D3×D411yescharacteristic
575Q8—22no—
576Q8—22no—
577Q8—41no—
578Q8—41no—
579D4—22no—
580D4—22no—
581D4—48no—
582D4—41no—
583D4—41no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—32no—
591C2×C4—32no—
592C2×C4—32no—
593C2×C4—32no—
594C2×C4—32no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—68no—
606C2×C4—61no—
607C2×C4—61no—
608C2×C4—61no—
609C2×C4—61no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—124no—
614C2×C4—121no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4—121no—
623C2×C4—121no—
624C2×C4—121no—
625C2×C4—121no—
626C2×C4—244no—
627C2×C4C2×D3×D411yescharacteristic
628C2×C4C2×#23711yescharacteristic
629C2×C4C2×#23711yescharacteristic
630C2×C4C2×#23711yescharacteristic
631C2×C4C2×#23711yescharacteristic
632C2×C4D3×#4111yescharacteristic
633C8—22no—
634C8—41no—
635C8—62no—
636C8—121no—
637C2×C3—42no—
638C2×C3—82no—
639C2×C3#239211yescharacteristic
640C2×C3C2×#46511yescharacteristic
641C2×C3#305411yescharacteristic
642C2×C3#310111yescharacteristic
643C2×C3#310211yescharacteristic
644C2×C3#310211yescharacteristic
645C2×C3#310311yescharacteristic
646C22—22no—
647C22—22no—
648C22—22no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22#558611yescharacteristic
661C22#558611yescharacteristic
662C22D3×#12111yescharacteristic
663C22C2×D3×SD1611yescharacteristic
664C22D3×#13511yescharacteristic
665C22#577311yescharacteristic
666C22#577711yescharacteristic
667C4—34no—
668C4—42no—
669C4—41no—
670C4—42no—
671C4—42no—
672C4—62no—
673C4—62no—
674C4—121no—
675C4—241no—
676C4—241no—
677C4—241no—
678C4—241no—
679C4#558611yescharacteristic
680C4#560011yescharacteristic
681C4#560011yescharacteristic
682C4#571611yescharacteristic
683C3#2370011yescharacteristic, complemented, Sylow
684C2—42no—
685C2—82no—
686C2#7666611yescharacteristic
687C2#8002811yescharacteristic
688C2#8006811yescharacteristic
689C2#8013211yescharacteristic
690C2#8021211yescharacteristic
691C2D3×#46511yescharacteristic
692C2#8080411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2000C2
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
hypercenter#2000D3
hypocenterC3#23700
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2000

Complements

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

NH ≅ G/Ntype
1. C2×Dic3×SD16C2semidirect
2. Dic3×C22⋊Q8C2semidirect
4. C2×#4718C2semidirect
6. #72051C2semidirect
7. #71694C2semidirect
9. #70261C2semidirect
10. Dic3×#103C2semidirect
11. #70193C2semidirect
12. Dic3×#102C2semidirect
13. #69940C2semidirect
14. #68840C2semidirect
15. #68784C2semidirect
146. C8×Dic3D4semidirect
234. C22×Dic3SD16semidirect
683. C3#23700semidirect

Maximal subgroups

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

subgroupindex
C2×Dic3×SD162
Dic3×C22⋊Q82
Dic3×#1212
C2×#47182
C3×#20002
#720512
#716942
#716802
#702612
Dic3×#1032
#701932
Dic3×#1022
#699402
#688402
#687842
#237003