Finite Group Databasebeta

← #10824622

Subgroups — #10824622

Order 768.

894 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4833C211yesmaximal, characteristic
2C2×#4790C211yesmaximal, characteristic, complemented
3#76479C211yesmaximal, characteristic
4#75965C211yesmaximal, characteristic
5C3×#2068C211yesmaximal, characteristic, complemented
6#70635C211yesmaximal, characteristic
7Dic3×#107C211yesmaximal, characteristic, complemented
8#70489C211yesmaximal, characteristic, complemented
9#70273C211yesmaximal, characteristic
10#70244C211yesmaximal, characteristic
11Dic3×#103C211yesmaximal, characteristic, complemented
12#69681C211yesmaximal, characteristic, complemented
13#69623C211yesmaximal, characteristic, complemented
14#68890C211yesmaximal, characteristic
15#68781C211yesmaximal, characteristic, complemented
16#23698—31nomaximal, Sylow
17C2×#946C2211yescharacteristic
18C2×Q8×Dic3C2211yescharacteristic
19C22×#216C2211yescharacteristic
20C2×#897C2211yescharacteristic
21C2×#887C2211yescharacteristic
22C2×C42×D3C2211yescharacteristic
23C2×C3×#107C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C3×#391C2211yescharacteristic
28C3×#389C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×(C2×D3)⋊C8C2211yescharacteristic
31C2×(C2×D3)⋊C8C2211yescharacteristic
32C2×#818C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×#816C2211yescharacteristic
35C2×C8×Dic3C2211yescharacteristic
36C2×C8×Dic3C2211yescharacteristic
37#4989—21no—
38#4987C2211yescharacteristic
39#4987C2211yescharacteristic
40Dic3×C4⋊C4—21no—
41Dic3×C4⋊C4C2211yescharacteristic
42Dic3×C4⋊C4C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#807C2211yescharacteristic
45#4929C2211yescharacteristic
46#4928C2211yescharacteristic
47C2×C4⋊(C3⋊C8)C2211yescharacteristic
48#4833—24no—
49#4790—24no—
50#4668—21no—
51#4666—21no—
52#4666—21no—
53#4655—21no—
54#4654—21no—
55#4649—21no—
56C3×#340C2211yescharacteristic
57#4551C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×C4×Q16—31no—
63C2×#431—31no—
64C4×C22⋊Q8—31no—
65C4×C22⋊Q8—31no—
66#2087—31no—
67#2076—31no—
68#2068—31no—
69#2068D311yescharacteristic, complemented
70#2056—31no—
71#2036—31no—
72#2036—31no—
73#2001—31no—
74C4×#107—31no—
75C4×#103—31no—
76C4×#103—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C4×D3D411yescharacteristic
79C22×C4×D3D411yescharacteristic
80C22×C4×D3C2311yescharacteristic
81C22×Dic6C2311yescharacteristic
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#946—22no—
90#946—22no—
91Q8×Dic3—22no—
92Q8×Dic3—22no—
93C2×#216—22no—
94C2×#216—22no—
95C2×#216—22no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3D411yescharacteristic
113C2×C4×Dic3D411yescharacteristic
114C2×C4×Dic3C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117#897—22no—
118#897—22no—
119#887—22no—
120#887—22no—
121C42×D3—22no—
122C42×D3—22no—
123C3×#107—22no—
124C3×C4⋊C8—22no—
125C3×#103—22no—
126C3×#103—22no—
127C3×#95—21no—
128C3×#95—21no—
129C3×#95—21no—
130#835—22no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138(C2×D3)⋊C8—22no—
139(C2×D3)⋊C8—22no—
140#818—22no—
141#816—22no—
142#816—22no—
143C8×Dic3—22no—
144C8×Dic3D414yes—
145#810—22no—
146#807—22no—
147C4⋊(C3⋊C8)—22no—
148C22×Q16—31no—
149C2×C22⋊Q8—31no—
150C2×C22⋊Q8—31no—
151C2×C4×Q8—31no—
152C2×C4×Q8—31no—
153C22×C42—31no—
154C4×Q16—64no—
155#431—64no—
156C2×#107—31no—
157C2×#107—31no—
158C2×#107C2×D311yescharacteristic
159C2×C4⋊C8—31no—
160C2×C4⋊C8C2×D311yescharacteristic
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103C2×D311yescharacteristic
166C2×#103C2×D311yescharacteristic
167C2×C22⋊C8—31no—
168C2×C22⋊C8—31no—
169C2×C4×C8—31no—
170C2×C4×C8—31no—
171#391—31no—
172#391C2×D311yescharacteristic
173#389—31no—
174#389C2×D311yescharacteristic
175#387—61no—
176#387—61no—
177#385—61no—
178#385—61no—
179#380—31no—
180#380—31no—
181C4×C4⋊C4—31no—
182C4×C4⋊C4—31no—
183C4×C4⋊C4—61no—
184C4×C4⋊C4—61no—
185C4×C22⋊C4—61no—
186C4×C22⋊C4—61no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340C2×D311yescharacteristic
193C23×D3C2×D411yescharacteristic
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8#4112yes—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2411yescharacteristic
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C2×C4×D3—22no—
215C2×C4×D3—22no—
216C2×C4×D3—21no—
217C2×C4×D3—21no—
218C2×C4×D3—21no—
219C2×C4×D3—21no—
220C2×C4×D3—21no—
221C2×C4×D3—21no—
222C2×C4×D3—22no—
223C2×C4×D3—41no—
224C2×C4×D3—41no—
225C2×C4×D3Q1614yescomplemented
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6#4112yes—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8C2×D412yes—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—42no—
244C3×C4⋊C4#4112yes—
245C3×C42—21no—
246C3×C42—21no—
247#216—28no—
248(C2×D3)⋊C4—24no—
249(C2×D3)⋊C4—24no—
250(C2×D3)⋊C4—42no—
251(C2×D3)⋊C4—42no—
252C4⋊Dic3—24no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—42no—
259C4⋊Dic3#4112yes—
260Dic3⋊C4—24no—
261Dic3⋊C4—42no—
262Dic3⋊C4—42no—
263C4×Dic3—22no—
264C4×Dic3—22no—
265C4×Dic3—42no—
266C4×Dic3—42no—
267C4×Dic3—42no—
268C4×Dic3—42no—
269C4×Dic3C2×D412yes—
270C4×Dic3#4112yes—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8C2×D412yes—
277C22×Q8—31no—
278C22×Q8C22×D311yescharacteristic
279C23×C4—31no—
280C23×C4—31no—
281C23×C4—31no—
282C2×Q16—62no—
283C2×Q16—62no—
284C2×Q16—62no—
285C22×C8—31no—
286C22×C8—31no—
287C22×C8C22×D311yescharacteristic
288C22×C8C22×D311yescharacteristic
289C22⋊Q8—62no—
290C22⋊Q8—62no—
291C22⋊Q8—62no—
292C22⋊Q8—62no—
293C4×Q8—62no—
294C4×Q8—62no—
295C4×Q8—62no—
296C4×Q8—62no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C4⋊C4C22×D311yescharacteristic
308C2×C4⋊C4C22×D311yescharacteristic
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C42—31no—
314C2×C42—31no—
315C2×C42—31no—
316C2×C42—62no—
317C2×C42—62no—
318C2×C42—61no—
319C2×C42—61no—
320C2×C42—61no—
321C2×C42—61no—
322C2×C42C22×D311yescharacteristic
323#107—22no—
324#107—62no—
325#107—62no—
326C4⋊C8—22no—
327C4⋊C8—62no—
328#103—22no—
329#103—22no—
330#103—62no—
331#103—62no—
332#103—62no—
333#103—62no—
334C22⋊C8—62no—
335C22⋊C8—62no—
336C4×C8—34no—
337C4×C8—62no—
338#95—21no—
339#95—21no—
340#95—21no—
341#95—61no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346#95—61no—
347#95—61no—
348C23×C3C22×D411yescharacteristic
349C22×D3—21no—
350C22×D3—21no—
351C22×D3—21no—
352C22×D3—21no—
353C22×D3—21no—
354C22×D3—21no—
355C22×D3C2×Q1612yes—
356C3×Q8—22no—
357C3×Q8—22no—
358C3×Q8—41no—
359C3×Q8—41no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4C2×Q1612yes—
378C2×C3×C4C22×D411yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380C2×C3×C4C2×#4111yescharacteristic
381C2×C3×C4C2×#4111yescharacteristic
382C2×C3×C4C2×#4111yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—22no—
389C2×Dic3—21no—
390C2×Dic3—22no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3#12112yes—
402C2×Dic3#12112yes—
403C2×Dic3C2×Q1612yes—
404C2×Dic3#13512yes—
405C4×D3—28no—
406C4×D3—44no—
407C4×D3—41no—
408C4×D3—41no—
409C4×D3—41no—
410C4×D3—41no—
411Dic6—22no—
412Dic6—22no—
413Dic6—41no—
414Dic6—41no—
415C3×C8—22no—
416C3×C8—41no—
417C3⋊C8—22no—
418C3⋊C8—41no—
419C24—31no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—32no—
426C2×Q8—61no—
427C2×Q8—61no—
428C2×Q8—61no—
429C2×Q8—61no—
430C2×Q8—61no—
431C2×Q8#23912yes—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—31no—
437C22×C4—31no—
438C22×C4—34no—
439C22×C4—31no—
440C22×C4—62no—
441C22×C4—62no—
442C22×C4—62no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—121no—
454C22×C4—121no—
455C22×C4D3×D411yescharacteristic
456C22×C4D3×D411yescharacteristic
457C22×C4C23×D311yescharacteristic
458Q16—68no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—32no—
465C2×C8—61no—
466C2×C8—61no—
467C2×C8—61no—
468C2×C8—61no—
469C2×C8—61no—
470C2×C8D3×D412yes—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—32no—
477C4⋊C4—42no—
478C4⋊C4—64no—
479C4⋊C4—64no—
480C4⋊C4—61no—
481C4⋊C4—61no—
482C4⋊C4—61no—
483C4⋊C4—61no—
484C4⋊C4—61no—
485C4⋊C4—122no—
486C4⋊C4—122no—
487C4⋊C4—122no—
488C4⋊C4#23712yes—
489C22⋊C4—64no—
490C22⋊C4—64no—
491C22⋊C4—122no—
492C22⋊C4—122no—
493C42—21no—
494C42—21no—
495C42—32no—
496C42—32no—
497C42—62no—
498C42—62no—
499C42—122no—
500C42—122no—
501C42—122no—
502C42—122no—
503C22×C3C2×#12111yescharacteristic
504C22×C3#52511yescharacteristic
505C22×C3#52511yescharacteristic
506C22×C3C22×Q1611yescharacteristic
507C22×C3C2×#13511yescharacteristic
508C22×C3#57511yescharacteristic
509C22×C3#57511yescharacteristic
510C2×D3—22no—
511C2×D3—22no—
512C2×D3—22no—
513C2×D3—41no—
514C2×D3—41no—
515C2×D3—41no—
516C2×D3—41no—
517C3×C4—22no—
518C3×C4—42no—
519C3×C4—42no—
520C3×C4—41no—
521C3×C4—81no—
522C3×C4—81no—
523C3×C4#52511yescharacteristic
524C3×C4#52911yescharacteristic
525C3×C4#52911yescharacteristic
526C3×C4#53211yescharacteristic
527Dic3—22no—
528Dic3—42no—
529Dic3—42no—
530Dic3—41no—
531Dic3—81no—
532Dic3—81no—
533Dic3#46714yes—
534C23—32no—
535C23—61no—
536C23—61no—
537C23—61no—
538C23—61no—
539C23—61no—
540C23—61no—
541C23C2×D3×D411yescharacteristic
542Q8—22no—
543Q8—22no—
544Q8—41no—
545Q8—41no—
546Q8—62no—
547Q8—62no—
548Q8—121no—
549Q8—121no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—32no—
558C2×C4—32no—
559C2×C4—32no—
560C2×C4—32no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—68no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—61no—
575C2×C4—62no—
576C2×C4—61no—
577C2×C4—62no—
578C2×C4—61no—
579C2×C4—124no—
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—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4D3×Q1612yes—
595C2×C4C2×D3×D411yescharacteristic
596C2×C4C2×#23711yescharacteristic
597C2×C4C2×#23911yescharacteristic
598C2×C4D3×#4111yescharacteristic
599C2×C4D3×#4111yescharacteristic
600C2×C4D3×#4111yescharacteristic
601C8—22no—
602C8—41no—
603C8—62no—
604C8—121no—
605C2×C3#239211yescharacteristic
606C2×C3C2×#46711yescharacteristic
607C2×C3#305411yescharacteristic
608C2×C3#310211yescharacteristic
609C2×C3#310211yescharacteristic
610C2×C3#310411yescharacteristic
611C2×C3#310411yescharacteristic
612D3—42no—
613C22—62no—
614C22—62no—
615C22—62no—
616C22—121no—
617C22—121no—
618C22—121no—
619C22—121no—
620C22D3×#12111yescharacteristic
621C22#560611yescharacteristic
622C22#562711yescharacteristic
623C22C2×D3×Q1611yescharacteristic
624C22D3×#13511yescharacteristic
625C22#577311yescharacteristic
626C22#577711yescharacteristic
627C4—22no—
628C4—34no—
629C4—42no—
630C4—41no—
631C4—42no—
632C4—62no—
633C4—81no—
634C4—81no—
635C4—122no—
636C4—121no—
637C4—122no—
638C4—241no—
639C4—241no—
640C4#559611yescharacteristic
641C4#562411yescharacteristic
642C4#562711yescharacteristic
643C4#570611yescharacteristic
644C3#2369811yescharacteristic, complemented, Sylow
645C2—122no—
646C2#7728111yescharacteristic
647C2#8045611yescharacteristic
648C2#8050411yescharacteristic
649C2D3×#46711yescharacteristic
650C2#8080411yescharacteristic
651C2#8109211yescharacteristic
652C2#8120411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2068C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#2068D3
hypocenterC3#23698
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×#4790C2semidirect
5. C3×#2068C2semidirect
7. Dic3×#107C2semidirect
8. #70489C2semidirect
11. Dic3×#103C2semidirect
12. #69681C2semidirect
13. #69623C2semidirect
15. #68781C2semidirect
69. #2068D3semidirect
225. C2×C4×D3Q16semidirect
644. C3#23698semidirect

Maximal subgroups

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

subgroupindex
C2×#48332
C2×#47902
#764792
#759652
C3×#20682
#706352
Dic3×#1072
#704892
#702732
#702442
Dic3×#1032
#696812
#696232
#688902
#687812
#236983