Finite Group Databasebeta

← #10794593

Subgroups — #10794593

Order 768.

1462 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5546C211yesmaximal, characteristic
2C2×#4740C212yesmaximal
3#73869C212yesmaximal, complemented
4C3×#1799C211yesmaximal, characteristic, complemented
5#71619C211yesmaximal, characteristic, complemented
6#70028C212yesmaximal, complemented
7#69530C212yesmaximal, complemented
8#68959C212yesmaximal, complemented
9#68951C212yesmaximal, complemented
10#20634—31nomaximal, Sylow
11C22×#237C2211yescharacteristic
12C2×C3×C4×D4C2211yescharacteristic
13C2×D4×Dic3C2211yescharacteristic
14C2×C4×#67C411yescharacteristic, complemented
15C2×C4×#67C2211yescharacteristic
16C22×Dic3⋊C4C411yescharacteristic, complemented
17C22×Dic3⋊C4C2211yescharacteristic
18#5546—24no—
19#5546—24no—
20C2×D3×C4⋊C4C411yescharacteristic, complemented
21C2×#887C411yescharacteristic, complemented
22C2×#67⋊C4C411yescharacteristic, complemented
23C2×#67⋊C4C2211yescharacteristic
24C2×#877C411yescharacteristic, complemented
25C2×#877C2214yes—
26C2×#877C2211yescharacteristic
27C2×#872C411yescharacteristic, complemented
28C2×C4×Dic6C411yescharacteristic, complemented
29C2×C3×#99C2212yes—
30C3×#380C2212yes—
31C2×#834C2212yes—
32#5212C2212yes—
33#4976—24no—
34C2×#806C2214yes—
35#4942C2214yes—
36#4740—28no—
37#4647—24no—
38C3×#342C2212yes—
39#4527C2214yes—
40#4526C2212yes—
41C2×#520—31no—
42C2×#410—32no—
43#2316—32no—
44#1799—32no—
45#1799—32no—
46#1799—32no—
47#1799—31no—
48#1799D311yescharacteristic, complemented
49#1740—32no—
50C22×C3×D4C2311yescharacteristic
51C23×C3×C4C2×C411yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C22×#67C2×C411yescharacteristic
54C22×#67C2311yescharacteristic
55C23×Dic3C2×C411yescharacteristic
56C23×Dic3C2×C411yescharacteristic
57C23×Dic3C2311yescharacteristic
58C23×Dic3C2311yescharacteristic
59C2×#237—24no—
60C2×#237—24no—
61C2×#237D414yes—
62C2×#237Q814yes—
63C22×C4×D3C2×C411yescharacteristic
64C22×C4×D3C2×C411yescharacteristic
65C22×C4×D3C2×C411yescharacteristic
66C22×Dic6C2×C411yescharacteristic
67C3×C4×D4—22no—
68C3×C4×D4—22no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4C2×C411yescharacteristic
71C2×C3×C22⋊C4C2×C411yescharacteristic
72C2×C3×C22⋊C4C2314yes—
73C2×C3×C22⋊C4C2311yescharacteristic
74C2×C3×C42C2×C411yescharacteristic
75C2×#217C2×C411yescharacteristic
76C2×#217C2314yes—
77C2×#217C2311yescharacteristic
78D4×Dic3—24no—
79C4×#67—22no—
80C4×#67—22no—
81C4×#67—22no—
82C4×#67—22no—
83C2×(C2×D3)⋊C4C2×C411yescharacteristic
84C2×(C2×D3)⋊C4C2×C411yescharacteristic
85C2×C4⋊Dic3—22no—
86C2×C4⋊Dic3C2×C411yescharacteristic
87C2×Dic3⋊C4—22no—
88C2×Dic3⋊C4—22no—
89C2×Dic3⋊C4—24no—
90C2×Dic3⋊C4—24no—
91C2×Dic3⋊C4C2×C411yescharacteristic
92C2×Dic3⋊C4C2×C411yescharacteristic
93C2×Dic3⋊C4C2×C411yescharacteristic
94C2×Dic3⋊C4C2×C411yescharacteristic
95C2×Dic3⋊C4C2×C411yescharacteristic
96C2×Dic3⋊C4C2×C411yescharacteristic
97C2×Dic3⋊C4C2×C411yescharacteristic
98C2×Dic3⋊C4C2×C411yescharacteristic
99C2×Dic3⋊C4D414yes—
100C2×Dic3⋊C4D414yes—
101C2×C4×Dic3—24no—
102C2×C4×Dic3C2×C411yescharacteristic
103C2×C4×Dic3C2×C411yescharacteristic
104C2×C4×Dic3C2×C411yescharacteristic
105C2×C4×Dic3C2×C411yescharacteristic
106C2×C4×Dic3C2×C411yescharacteristic
107D3×C4⋊C4—24no—
108#887—24no—
109#67⋊C4—24no—
110#67⋊C4—24no—
111#877—24no—
112#877—24no—
113#877—28no—
114#877—28no—
115#872—22no—
116#872—22no—
117C4×Dic6—22no—
118C4×Dic6—22no—
119C3×#99—24no—
120C3×#95—22no—
121#834—24no—
122C3⋊#95—22no—
123C3⋊#95—24no—
124#806—28no—
125C22×#41—31no—
126#520—64no—
127#520—64no—
128C2×C4×Q8—31no—
129C2×C4×Q8—31no—
130C2×C4×D4—31no—
131C2×C4×D4—31no—
132C2×C4×D4—31no—
133C2×C4×D4—31no—
134C2×C4×D4—31no—
135C2×C4×D4C2×D311yescharacteristic
136C2×#117—31no—
137C2×#117—31no—
138C2×#117—31no—
139C2×#117—34no—
140C22×C4⋊C4—31no—
141C22×C4⋊C4—31no—
142C22×C4⋊C4—31no—
143#410—68no—
144C2×#99—32no—
145C2×#99—34no—
146C2×#99C2×D312yes—
147#389—64no—
148#389—64no—
149#380—32no—
150#380—34no—
151#380C2×D312yes—
152#342—32no—
153#342—34no—
154#342C2×D312yes—
155C24×C3C22×C411yescharacteristic
156C24×C3C2411yescharacteristic
157C23×D3C22×C411yescharacteristic
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4C2×D412yes—
161C2×C3×D4C2×Q812yes—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—24no—
165C22×C3×C4—81no—
166C22×C3×C4—81no—
167C22×C3×C4C22×C411yescharacteristic
168C22×C3×C4C22×C411yescharacteristic
169C22×C3×C4C22×C411yescharacteristic
170C22×C3×C4C22×C411yescharacteristic
171C22×C3×C4C22×C411yescharacteristic
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×D412yes—
174C2×#67—22no—
175C2×#67—22no—
176C2×#67—22no—
177C2×#67—22no—
178C2×#67C4⋊C414yescomplemented
179C2×#67C2×D412yes—
180C2×#67C2×Q812yes—
181C22×Dic3—22no—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3—24no—
185C22×Dic3—81no—
186C22×Dic3—81no—
187C22×Dic3C4⋊C414yescomplemented
188C22×Dic3C22×C411yescharacteristic
189C22×Dic3C22×C411yescharacteristic
190C22×Dic3C22×C411yescharacteristic
191C22×Dic3C22×C411yescharacteristic
192C22×Dic3C22×C411yescharacteristic
193C22×Dic3C22×C411yescharacteristic
194C22×Dic3C22×C411yescharacteristic
195C22×Dic3C2×D414yes—
196C22×Dic3C2×D414yes—
197C22×Dic3C2×D412yes—
198C22×Dic3C2×Q812yes—
199C22×Dic3#4114yes—
200#237—216no—
201#237—216no—
202C2×C4×D3—22no—
203C2×C4×D3—22no—
204C2×C4×D3—22no—
205C2×C4×D3—22no—
206C2×C4×D3—81no—
207C2×C4×D3—81no—
208C2×C4×D3C4⋊C414yescomplemented
209C2×Dic6—22no—
210C2×Dic6—22no—
211C2×Dic6C4⋊C414yescomplemented
212C3×C4⋊C4—22no—
213C3×C4⋊C4—82no—
214C3×C22⋊C4—22no—
215C3×C22⋊C4—22no—
216C3×C22⋊C4—24no—
217C3×C22⋊C4—24no—
218C3×C42—21no—
219C3×C42—21no—
220#217—22no—
221#217—22no—
222#217—24no—
223#217—24no—
224(C2×D3)⋊C4—21no—
225(C2×D3)⋊C4—21no—
226(C2×D3)⋊C4—22no—
227C4⋊Dic3—22no—
228C4⋊Dic3—82no—
229Dic3⋊C4—21no—
230Dic3⋊C4—21no—
231Dic3⋊C4—22no—
232Dic3⋊C4—22no—
233Dic3⋊C4—22no—
234Dic3⋊C4—48no—
235Dic3⋊C4—84no—
236Dic3⋊C4C22⋊C412yes—
237Dic3⋊C4C22⋊C412yes—
238Dic3⋊C4C22⋊C412yes—
239Dic3⋊C4C22⋊C412yes—
240Dic3⋊C4C22⋊C414yes—
241Dic3⋊C4C22⋊C414yes—
242C4×Dic3—22no—
243C4×Dic3—22no—
244C4×Dic3—22no—
245C4×Dic3—22no—
246C4×Dic3—22no—
247C4×Dic3—48no—
248C2×#41—34no—
249C2×#41—34no—
250C2×#41—64no—
251C2×#41—64no—
252C22×Q8—31no—
253C22×D4—31no—
254C22×D4—31no—
255C22×D4C22×D311yescharacteristic
256C23×C4—31no—
257C23×C4—31no—
258C23×C4—31no—
259C23×C4—31no—
260C23×C4—31no—
261C23×C4—31no—
262C23×C4—31no—
263C23×C4C4×D311yescharacteristic
264C23×C4C22×D311yescharacteristic
265C4×Q8—62no—
266C4×Q8—62no—
267C4×Q8—64no—
268C4×D4—22no—
269C4×D4—22no—
270C4×D4—64no—
271C4×D4—64no—
272C4×D4—62no—
273C4×D4—62no—
274C4×D4—64no—
275C4×D4—62no—
276C4×D4—62no—
277#117—62no—
278#117—62no—
279#117—64no—
280#117—64no—
281#117—68no—
282#117—68no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—34no—
289C2×C4⋊C4—34no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—64no—
296C2×C4⋊C4—62no—
297C2×C4⋊C4—62no—
298C2×C4⋊C4—64no—
299C2×C4⋊C4—64no—
300C2×C4⋊C4—62no—
301C2×C4⋊C4C4×D311yescharacteristic
302C2×C22⋊C4—31no—
303C2×C22⋊C4—31no—
304C2×C22⋊C4—31no—
305C2×C22⋊C4—31no—
306C2×C22⋊C4—34no—
307C2×C22⋊C4C4×D311yescharacteristic
308C2×C22⋊C4C22×D314yes—
309C2×C22⋊C4C22×D311yescharacteristic
310C2×C42—31no—
311C2×C42—31no—
312C2×C42—31no—
313C2×C42—31no—
314C2×C42—31no—
315C2×C42—64no—
316C2×C42C4×D311yescharacteristic
317#99—24no—
318#99—68no—
319#99—64no—
320#95—22no—
321#95—64no—
322#95—62no—
323C23×C3—21no—
324C23×C3—21no—
325C23×C3—41no—
326C23×C3—41no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3C2×C4⋊C412yes—
330C23×C3C23×C411yescharacteristic
331C23×C3C22×D412yes—
332C23×C3C22×D411yescharacteristic
333C23×C3C22×Q811yescharacteristic
334C23×C3C2×#4112yes—
335C22×D3—21no—
336C22×D3—21no—
337C22×D3—41no—
338C22×D3—41no—
339C22×D3C2×C4⋊C412yes—
340C3×D4—24no—
341C3×D4—24no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—44no—
353C2×C3×C4—44no—
354C2×C3×C4—44no—
355C2×C3×C4—81no—
356C2×C3×C4—81no—
357C2×C3×C4C2×C22⋊C412yes—
358C2×C3×C4C2×C22⋊C411yescharacteristic
359C2×C3×C4C2×C22⋊C411yescharacteristic
360C2×C3×C4C2×C4⋊C412yes—
361#67—24no—
362#67—24no—
363#67—28no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—28no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—44no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—44no—
375C2×Dic3—44no—
376C2×Dic3—44no—
377C2×Dic3—82no—
378C2×Dic3C2×C22⋊C412yes—
379C2×Dic3C2×C22⋊C412yes—
380C2×Dic3C2×C22⋊C412yes—
381C2×Dic3C2×C22⋊C412yes—
382C2×Dic3C2×C4⋊C412yes—
383C2×Dic3C2×C4⋊C412yes—
384C2×Dic3C2×C4⋊C412yes—
385C2×Dic3#11712yes—
386C2×Dic3#11712yes—
387C2×Dic3#12118yes—
388C2×Dic3C22⋊Q818yes—
389C4×D3—28no—
390C4×D3—82no—
391C4×D3—82no—
392Dic6—28no—
393C24—31no—
394C24C2×C4×D311yescharacteristic
395C24C23×D311yescharacteristic
396#41—616no—
397#41—616no—
398C2×Q8—34no—
399C2×Q8—62no—
400C2×Q8—62no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—34no—
404C2×D4—32no—
405C2×D4—32no—
406C2×D4—62no—
407C2×D4—62no—
408C2×D4—62no—
409C2×D4—62no—
410C2×D4D3×D412yes—
411C2×D4D3×Q812yes—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—24no—
415C22×C4—31no—
416C22×C4—31no—
417C22×C4—34no—
418C22×C4—32no—
419C22×C4—32no—
420C22×C4—34no—
421C22×C4—34no—
422C22×C4—31no—
423C22×C4—31no—
424C22×C4—31no—
425C22×C4—34no—
426C22×C4—34no—
427C22×C4—31no—
428C22×C4—31no—
429C22×C4—62no—
430C22×C4—62no—
431C22×C4—62no—
432C22×C4—62no—
433C22×C4—62no—
434C22×C4—62no—
435C22×C4—62no—
436C22×C4—64no—
437C22×C4—81no—
438C22×C4—81no—
439C22×C4—241no—
440C22×C4—241no—
441C22×C4—241no—
442C22×C4—241no—
443C22×C4C2×C4×D311yescharacteristic
444C22×C4C2×C4×D311yescharacteristic
445C22×C4C2×C4×D311yescharacteristic
446C22×C4C2×C4×D311yescharacteristic
447C22×C4C2×C4×D311yescharacteristic
448C22×C4D3×D412yes—
449C22×C4D3×D412yes—
450C4⋊C4—22no—
451C4⋊C4—32no—
452C4⋊C4—32no—
453C4⋊C4—34no—
454C4⋊C4—32no—
455C4⋊C4—32no—
456C4⋊C4—34no—
457C4⋊C4—62no—
458C4⋊C4—62no—
459C4⋊C4—62no—
460C4⋊C4—62no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4—82no—
464C4⋊C4—128no—
465C4⋊C4—244no—
466C4⋊C4—242no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—62no—
472C22⋊C4—62no—
473C22⋊C4—62no—
474C22⋊C4—61no—
475C22⋊C4—61no—
476C22⋊C4—64no—
477C22⋊C4—64no—
478C42—21no—
479C42—21no—
480C42—62no—
481C42—62no—
482C42—62no—
483C42—62no—
484C42—62no—
485C42—128no—
486C22×C3—22no—
487C22×C3—21no—
488C22×C3—22no—
489C22×C3—21no—
490C22×C3—22no—
491C22×C3—41no—
492C22×C3—41no—
493C22×C3—42no—
494C22×C3—42no—
495C22×C3—42no—
496C22×C3—41no—
497C22×C3—41no—
498C22×C3C22×C22⋊C411yescharacteristic
499C22×C3C22×C4⋊C411yescharacteristic
500C22×C3C2×#11711yescharacteristic
501C22×C3C2×#12112yes—
502C22×C3C2×C22⋊Q812yes—
503C2×D3—22no—
504C2×D3—42no—
505C2×D3—41no—
506C2×D3—41no—
507C2×D3—42no—
508C3×C4—22no—
509C3×C4—21no—
510C3×C4—21no—
511C3×C4—22no—
512C3×C4—41no—
513C3×C4—41no—
514C3×C4—84no—
515Dic3—22no—
516Dic3—22no—
517Dic3—22no—
518Dic3—41no—
519Dic3—41no—
520Dic3—84no—
521Dic3#38014yes—
522Dic3#38014yes—
523C23—21no—
524C23—21no—
525C23—32no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—61no—
531C23—61no—
532C23—121no—
533C23—121no—
534C23D3×C4⋊C412yes—
535C23C22×C4×D311yescharacteristic
536C23C2×D3×D412yes—
537C23C2×D3×D411yescharacteristic
538C23C2×D3×Q811yescharacteristic
539C23D3×#4112yes—
540Q8—68no—
541D4—24no—
542D4—24no—
543D4—68no—
544D4—64no—
545D4—64no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—32no—
551C2×C4—32no—
552C2×C4—32no—
553C2×C4—32no—
554C2×C4—32no—
555C2×C4—32no—
556C2×C4—38no—
557C2×C4—38no—
558C2×C4—32no—
559C2×C4—32no—
560C2×C4—32no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—68no—
571C2×C4—68no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—61no—
575C2×C4—61no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—124no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—124no—
584C2×C4—124no—
585C2×C4—124no—
586C2×C4—242no—
587C2×C4—242no—
588C2×C4—242no—
589C2×C4D3×C22⋊C411yescharacteristic
590C2×C4D3×C22⋊C411yescharacteristic
591C2×C4D3×C22⋊C412yes—
592C2×C4D3×C4⋊C412yes—
593C2×C3—22no—
594C2×C3—21no—
595C2×C3—21no—
596C2×C3—22no—
597C2×C3—42no—
598C2×C3C2×#38011yescharacteristic
599C2×C3#288311yescharacteristic
600C2×C3#288411yescharacteristic
601D3—42no—
602C22—22no—
603C22—21no—
604C22—22no—
605C22—21no—
606C22—22no—
607C22—41no—
608C22—41no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—41no—
613C22—41no—
614C22—62no—
615C22—122no—
616C22—122no—
617C22—121no—
618C22—121no—
619C22C2×D3×C22⋊C411yescharacteristic
620C22C2×D3×C4⋊C411yescharacteristic
621C22D3×#11711yescharacteristic
622C22D3×#12112yes—
623C22D3×C22⋊Q812yes—
624C4—22no—
625C4—21no—
626C4—21no—
627C4—22no—
628C4—34no—
629C4—34no—
630C4—41no—
631C4—41no—
632C4—62no—
633C4—62no—
634C4—62no—
635C4—84no—
636C4—121no—
637C4—121no—
638C4—244no—
639C3#2063411yescharacteristic, complemented, Sylow
640C2—22no—
641C2—21no—
642C2—21no—
643C2—22no—
644C2—42no—
645C2—122no—
646C2D3×#38011yescharacteristic
647C2#7868811yescharacteristic
648C2#7868911yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC3×#1799C2
socleC22×C3C22×C22⋊C4
prime radicalC22×#237C22
cosocleC24×C3C24
hypercenter#1799D3
hypocenterC3#20634
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
3. #73869C2semidirect
4. C3×#1799C2semidirect
5. #71619C2semidirect
6. #70028C2semidirect
7. #69530C2semidirect
8. #68959C2semidirect
9. #68951C2semidirect
14. C2×C4×#67C4semidirect
16. C22×Dic3⋊C4C4semidirect
20. C2×D3×C4⋊C4C4semidirect
21. C2×#887C4semidirect
22. C2×#67⋊C4C4semidirect
24. C2×#877C4semidirect
27. C2×#872C4semidirect
28. C2×C4×Dic6C4semidirect
48. #1799D3semidirect
178. C2×#67C4⋊C4semidirect
187. C22×Dic3C4⋊C4semidirect
208. C2×C4×D3C4⋊C4semidirect
211. C2×Dic6C4⋊C4semidirect
639. C3#20634semidirect

Maximal subgroups

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

subgroupindex
C2×#55462
C2×#47402
#738692
C3×#17992
#716192
#700282
#695302
#689592
#689512
#206343