Finite Group Databasebeta

← #10824380

Subgroups — #10824380

Order 768.

956 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4872C211yesmaximal, characteristic
2C2×#4788C211yesmaximal, characteristic, complemented
3#76480C211yesmaximal, characteristic, complemented
4#75961C211yesmaximal, characteristic, complemented
5C3×#2068C211yesmaximal, characteristic, complemented
6#70620C211yesmaximal, characteristic, complemented
7#70481C211yesmaximal, characteristic
8Dic3×C4⋊C8C211yesmaximal, characteristic, complemented
9#70277C211yesmaximal, characteristic
10#70245C211yesmaximal, characteristic, complemented
11Dic3×#103C211yesmaximal, characteristic, complemented
12#69680C211yesmaximal, characteristic, complemented
13#69629C211yesmaximal, characteristic, complemented
14#68903C211yesmaximal, characteristic, complemented
15#68780C211yesmaximal, characteristic, complemented
16#23680—31nomaximal, Sylow
17C2×#947C2211yescharacteristic
18C2×Q8×Dic3C2211yescharacteristic
19C22×#215C2211yescharacteristic
20C22×D24C2211yescharacteristic
21C2×#893C2211yescharacteristic
22C2×#893C2211yescharacteristic
23C2×#892—21no—
24C2×C3×#107C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×#103C2211yescharacteristic
28C3×#391C2211yescharacteristic
29C3×#389C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×#821C2211yescharacteristic
33C2×C8×Dic3C2211yescharacteristic
34C2×C8×Dic3C2211yescharacteristic
35#4990—21no—
36#4988C2211yescharacteristic
37#4984C2211yescharacteristic
38#4977C2211yescharacteristic
39Dic3×C4⋊C4C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#809C2211yescharacteristic
43C2×#808C2211yescharacteristic
44#4939C2211yescharacteristic
45#4939C2211yescharacteristic
46C42×Dic3C2211yescharacteristic
47C2×C4⋊(C3⋊C8)C2211yescharacteristic
48#4872—24no—
49#4788—24no—
50#4670—21no—
51#4670—21no—
52#4670—21no—
53#4666—21no—
54#4666—21no—
55#4666—21no—
56C3×#340C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×C4×SD16—31no—
63C2×C4×D8—31no—
64C4×#124—31no—
65#2293—31no—
66#2085—31no—
67#2073—31no—
68#2068D311yescharacteristic, complemented
69#2065—31no—
70#2055—31no—
71#2054—31no—
72#2042—31no—
73#2012—31no—
74C4×C4⋊C8—31no—
75C4×#103—31no—
76C4×#102—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×D12C2311yescharacteristic
79C22×D12C2311yescharacteristic
80C22×C4×D3—21no—
81C22×C4×D3—21no—
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C42C2311yescharacteristic
89#947—22no—
90#947—22no—
91Q8×Dic3—22no—
92Q8×Dic3—22no—
93C2×#215—22no—
94C2×#215—22no—
95C2×#215—22no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4×Dic3—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3D411yescharacteristic
113C2×C4×Dic3D411yescharacteristic
114C2×C4×Dic3D411yescharacteristic
115C2×C4×Dic3D411yescharacteristic
116C2×C4×Dic3C2311yescharacteristic
117C2×C4×Dic3C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C2×D24—22no—
121C2×D24—22no—
122C2×D24—22no—
123#893—22no—
124#893—22no—
125#893—22no—
126#893D414yes—
127#892—44no—
128C3×#107—22no—
129C3×C4⋊C8—22no—
130C3×#103—22no—
131C3×#103—22no—
132C3×#95—21no—
133C3×#95—21no—
134C3×#95—21no—
135#835—22no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140#821—22no—
141#821—22no—
142C8×Dic3—22no—
143C8×Dic3—22no—
144#809—22no—
145#809—22no—
146#809—22no—
147#808—22no—
148C4⋊(C3⋊C8)—22no—
149C22×SD16—31no—
150C22×D8—31no—
151C2×#124—31no—
152C2×C4×Q8—31no—
153C2×C4×D4—31no—
154C2×C4×D4—31no—
155C2×#117—61no—
156C4×SD16—64no—
157C4×D8—64no—
158C2×#107C2×D311yescharacteristic
159C2×#106—31no—
160C2×C4⋊C8—31no—
161C2×C4⋊C8C2×D311yescharacteristic
162C2×#103—31no—
163C2×#103C2×D311yescharacteristic
164C2×#103C2×D311yescharacteristic
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×#102—31no—
170C2×C4×C8—31no—
171C2×C4×C8—31no—
172#391C2×D311yescharacteristic
173#390—31no—
174#390—31no—
175#390—31no—
176#389—31no—
177#389C2×D311yescharacteristic
178#388—61no—
179#388—61no—
180#388—61no—
181#388—61no—
182#382—31no—
183C4×C4⋊C4—31no—
184C4×C22⋊C4—61no—
185C4×C22⋊C4—61no—
186C4×C22⋊C4—61no—
187C43—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340C2×D311yescharacteristic
192#336—31no—
193#336—31no—
194C23×D3—21no—
195C23×D3—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8#4112yes—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2411yescharacteristic
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C22×Dic3C2×D411yescharacteristic
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12C2×D412yes—
226C2×D12#4112yes—
227C2×C4×D3—24no—
228C2×C4×D3—42no—
229C2×C4×D3—42no—
230C2×C4×D3—42no—
231C2×C4×D3—42no—
232C2×C4×D3—42no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8#4112yes—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—42no—
245C3×C4⋊C4C2×D412yes—
246C3×C42—21no—
247C3×C42—21no—
248#215—28no—
249(C2×D3)⋊C4—24no—
250(C2×D3)⋊C4—24no—
251(C2×D3)⋊C4—24no—
252(C2×D3)⋊C4—42no—
253(C2×D3)⋊C4—42no—
254(C2×D3)⋊C4—42no—
255(C2×D3)⋊C4—42no—
256(C2×D3)⋊C4—42no—
257(C2×D3)⋊C4—42no—
258(C2×D3)⋊C4—42no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—42no—
264C4⋊Dic3—42no—
265C4×Dic3—22no—
266C4×Dic3—21no—
267C4×Dic3—21no—
268C4×Dic3—22no—
269C4×Dic3—22no—
270C4×Dic3—41no—
271C4×Dic3—42no—
272C4×Dic3—42no—
273C4×Dic3C2×D412yes—
274C4×Dic3#4112yes—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8#4112yes—
281D24—28no—
282C22×Q8C22×D311yescharacteristic
283C22×D4—31no—
284C22×D4—31no—
285C23×C4—61no—
286C23×C4—61no—
287C2×SD16—62no—
288C2×SD16—62no—
289C2×SD16—62no—
290C2×D8—62no—
291C2×D8—62no—
292C2×D8—62no—
293C22×C8—31no—
294C22×C8—31no—
295C22×C8C22×D311yescharacteristic
296C22×C8C22×D311yescharacteristic
297#124—62no—
298#124—62no—
299C4×Q8—62no—
300C4×Q8—62no—
301C4×D4—34no—
302C4×D4—62no—
303C4×D4—62no—
304C4×D4—62no—
305#117—124no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4C22×D311yescharacteristic
312C2×C4⋊C4C22×D311yescharacteristic
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
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×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C42—31no—
325C2×C42—31no—
326C2×C42—31no—
327C2×C42—31no—
328C2×C42—31no—
329C2×C42—31no—
330C2×C42—61no—
331C2×C42—61no—
332C2×C42C22×D311yescharacteristic
333#107—22no—
334#106—62no—
335C4⋊C8—22no—
336C4⋊C8—62no—
337#103—22no—
338#103—22no—
339#103—62no—
340#102—62no—
341#102—62no—
342#102—62no—
343#102—62no—
344#102—62no—
345C4×C8—62no—
346C4×C8—62no—
347#95—21no—
348#95—21no—
349#95—21no—
350#95—61no—
351#95—61no—
352#95—61no—
353#95—61no—
354C23×C3C22×D411yescharacteristic
355C22×D3—22no—
356C22×D3—22no—
357C22×D3—41no—
358C22×D3—41no—
359C22×D3—41no—
360C22×D3—41no—
361C22×D3—41no—
362C22×D3—41no—
363C22×D3—41no—
364C22×D3—41no—
365C22×D3—41no—
366C22×D3—41no—
367C22×D3—41no—
368C22×D3—41no—
369C3×Q8—22no—
370C3×Q8—22no—
371C3×Q8—41no—
372C3×Q8—41no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4#13512yes—
391C2×C3×C4C22×D411yescharacteristic
392C2×C3×C4C2×#4111yescharacteristic
393C2×C3×C4C2×#4111yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396C2×C3×C4C2×#4111yescharacteristic
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3#12112yes—
412C2×Dic3#12112yes—
413C2×Dic3#13512yes—
414C2×Dic3AGL(1,ℤ/8)12yes—
415D12—22no—
416D12—22no—
417D12—22no—
418D12—22no—
419D12—41no—
420D12—41no—
421D12—41no—
422D12—41no—
423C4×D3—48no—
424C4×D3—84no—
425C3×C8—22no—
426C3×C8—41no—
427C3⋊C8—22no—
428C3⋊C8—41no—
429C24—61no—
430C24—61no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8#23912yes—
437C2×D4—32no—
438C2×D4—32no—
439C2×D4—61no—
440C2×D4—61no—
441C2×D4—61no—
442C2×D4—61no—
443C2×D4—61no—
444C2×D4—61no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4—61no—
448C2×D4—61no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—31no—
454C22×C4—31no—
455C22×C4—31no—
456C22×C4—31no—
457C22×C4—64no—
458C22×C4—61no—
459C22×C4—61no—
460C22×C4—122no—
461C22×C4—122no—
462C22×C4—122no—
463C22×C4—122no—
464C22×C4—122no—
465C22×C4D3×D411yescharacteristic
466C22×C4D3×D411yescharacteristic
467C22×C4C23×D311yescharacteristic
468SD16—68no—
469D8—68no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—32no—
476C2×C8—61no—
477C2×C8—61no—
478C2×C8—61no—
479C2×C8—61no—
480C2×C8—61no—
481C2×C8#23712yes—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—42no—
488C4⋊C4—61no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C4⋊C4—122no—
493C4⋊C4—122no—
494C4⋊C4D3×D412yes—
495C22⋊C4—64no—
496C22⋊C4—64no—
497C22⋊C4—64no—
498C22⋊C4—122no—
499C22⋊C4—122no—
500C22⋊C4—122no—
501C22⋊C4—122no—
502C22⋊C4—122no—
503C22⋊C4—122no—
504C22⋊C4—122no—
505C42—21no—
506C42—21no—
507C42—32no—
508C42—32no—
509C42—62no—
510C42—62no—
511C42—62no—
512C42—61no—
513C42—61no—
514C42—121no—
515C42—122no—
516C42—122no—
517C22×C3C2×#12111yescharacteristic
518C22×C3#52511yescharacteristic
519C22×C3#53211yescharacteristic
520C22×C3C2×#13511yescharacteristic
521C22×C3C2×#13511yescharacteristic
522C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
523C22×C3#57511yescharacteristic
524C2×D3—42no—
525C2×D3—42no—
526C2×D3—42no—
527C2×D3—42no—
528C2×D3—42no—
529C2×D3—42no—
530C2×D3—81no—
531C2×D3—81no—
532C2×D3—81no—
533C2×D3—81no—
534C2×D3—81no—
535C2×D3—81no—
536C2×D3—81no—
537C2×D3—81no—
538C3×C4—22no—
539C3×C4—42no—
540C3×C4—41no—
541C3×C4—42no—
542C3×C4—81no—
543C3×C4—81no—
544C3×C4#52511yescharacteristic
545C3×C4#52511yescharacteristic
546C3×C4#52911yescharacteristic
547C3×C4#52911yescharacteristic
548Dic3—22no—
549Dic3—41no—
550Dic3—41no—
551Dic3—81no—
552Dic3—81no—
553Dic3#46314yes—
554C23—62no—
555C23—62no—
556C23—121no—
557C23—121no—
558C23—121no—
559C23—121no—
560C23—121no—
561C23—121no—
562C23—121no—
563C23—121no—
564C23—121no—
565C23—121no—
566C23—121no—
567C23—121no—
568C23C2×D3×D411yescharacteristic
569Q8—22no—
570Q8—22no—
571Q8—41no—
572Q8—41no—
573D4—62no—
574D4—62no—
575D4—62no—
576D4—62no—
577D4—121no—
578D4—121no—
579D4—121no—
580D4—121no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—32no—
589C2×C4—32no—
590C2×C4—32no—
591C2×C4—32no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—61no—
603C2×C4—61no—
604C2×C4—61no—
605C2×C4—61no—
606C2×C4—61no—
607C2×C4—61no—
608C2×C4—61no—
609C2×C4—61no—
610C2×C4—128no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—244no—
618C2×C4#91912yes—
619C2×C4C2×D3×D411yescharacteristic
620C2×C4C2×#23711yescharacteristic
621C2×C4C2×#23911yescharacteristic
622C2×C4D3×#4111yescharacteristic
623C2×C4D3×#4111yescharacteristic
624C2×C4D3×#4111yescharacteristic
625C8—22no—
626C8—41no—
627C8—62no—
628C8—121no—
629C2×C3#239211yescharacteristic
630C2×C3C2×#46311yescharacteristic
631C2×C3#303511yescharacteristic
632C2×C3#310211yescharacteristic
633C2×C3#310611yescharacteristic
634C2×C3#316811yescharacteristic
635C2×C3#317111yescharacteristic
636D3—82no—
637D3—82no—
638C22—122no—
639C22—122no—
640C22—122no—
641C22—122no—
642C22—122no—
643C22—122no—
644C22—241no—
645C22—241no—
646C22—241no—
647C22—241no—
648C22—241no—
649C22—241no—
650C22—241no—
651C22—241no—
652C22#559611yescharacteristic
653C22D3×#12111yescharacteristic
654C22#562711yescharacteristic
655C22C2×#91911yescharacteristic
656C22D3×#13511yescharacteristic
657C22D3×AGL(1,ℤ/8)11yescharacteristic
658C22#577711yescharacteristic
659C4—22no—
660C4—34no—
661C4—41no—
662C4—42no—
663C4—42no—
664C4—62no—
665C4—81no—
666C4—81no—
667C4—121no—
668C4—121no—
669C4—241no—
670C4—241no—
671C4#560611yescharacteristic
672C4#562411yescharacteristic
673C4#562711yescharacteristic
674C4#570611yescharacteristic
675C3#2368011yescharacteristic, complemented, Sylow
676C2—242no—
677C2—242no—
678C2#7728111yescharacteristic
679C2#8047511yescharacteristic
680C2D3×#46311yescharacteristic
681C2#8074011yescharacteristic
682C2#8078411yescharacteristic
683C2#8109311yescharacteristic
684C2#8120411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2068C2
socleC23×C3C22×D4
prime radicalC22×D24C22
hypercenter#2068D3
hypocenterC3#23680
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2068

Complements

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

NH ≅ G/Ntype
2. C2×#4788C2semidirect
3. #76480C2semidirect
4. #75961C2semidirect
5. C3×#2068C2semidirect
6. #70620C2semidirect
8. Dic3×C4⋊C8C2semidirect
10. #70245C2semidirect
11. Dic3×#103C2semidirect
12. #69680C2semidirect
13. #69629C2semidirect
14. #68903C2semidirect
15. #68780C2semidirect
68. #2068D3semidirect
675. C3#23680semidirect

Maximal subgroups

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

subgroupindex
C2×#48722
C2×#47882
#764802
#759612
C3×#20682
#706202
#704812
Dic3×C4⋊C82
#702772
#702452
Dic3×#1032
#696802
#696292
#689032
#687802
#236803