Finite Group Databasebeta

← #10805000

Subgroups — #10805000

Order 768.

1158 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#529C211yesmaximal, characteristic
2#82062C211yesmaximal, characteristic, complemented
3#82060C211yesmaximal, characteristic, complemented
4#71705C214yesmaximal, complemented
5#71701C212yesmaximal, complemented
6#71686C212yesmaximal
7#71619C214yesmaximal, complemented
8#21875—31nomaximal, Sylow
9C3×#529—21no—
10C3×#529—21no—
11C3×#529—21no—
12C3×#529—21no—
13C3×#529—21no—
14C3×#529—21no—
15C3×#529—21no—
16C3×#529—21no—
17C2×C3×#126C412yescomplemented
18C2×C3×#125C411yescharacteristic, complemented
19C2×C3×#124C411yescharacteristic, complemented
20C2×C3×#123C2212yes—
21C2×C3×C22⋊Q8C2211yescharacteristic
22C2×C3×#121C2211yescharacteristic
23C2×C3×C4×Q8C411yescharacteristic, complemented
24C2×C3×C4×D4C411yescharacteristic, complemented
25C2×C3×C4×D4C2212yes—
26C2×C3×#117C412yescomplemented
27C22×C3×C42C2211yescharacteristic
28C2×#926C2212yes—
29C2×#834C2218yes—
30#5212C2212yes—
31#5212C2212yes—
32#5212C2212yes—
33#5211C2212yes—
34C4×#217C2212yes—
35#4937—21no—
36#4936—21no—
37#4936—21no—
38#4935—21no—
39C4×C4⋊Dic3—21no—
40C4×C4⋊Dic3—21no—
41C4×C4⋊Dic3—21no—
42#4932—21no—
43#4526C2218yes—
44C2×#529D311yescharacteristic
45#2285—31no—
46#2283—31no—
47#1996—32no—
48#1964—32no—
49#1899—34no—
50#1799—34no—
51C22×C3×Q8C2×C411yescharacteristic
52C22×C3×D4C2×C411yescharacteristic
53C22×C3×D4C2312yes—
54C23×C3×C4C2×C412yes—
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C3×#126—22no—
59C3×#126—22no—
60C3×#126—22no—
61C3×#126—22no—
62C3×#125—21no—
63C3×#125—21no—
64C3×#125—21no—
65C3×#125—21no—
66C3×#124—21no—
67C3×#124—21no—
68C3×#124—21no—
69C3×#124—21no—
70C3×#123—22no—
71C3×#123—22no—
72C3×#123—22no—
73C3×#123—22no—
74C3×C22⋊Q8—21no—
75C3×C22⋊Q8—21no—
76C3×C22⋊Q8—21no—
77C3×C22⋊Q8—21no—
78C3×#121—21no—
79C3×#121—21no—
80C3×#121—21no—
81C3×#121—21no—
82C3×C4×Q8—21no—
83C3×C4×Q8—21no—
84C3×C4×Q8—21no—
85C3×C4×Q8—21no—
86C3×C4×D4—22no—
87C3×C4×D4—22no—
88C3×C4×D4—22no—
89C3×C4×D4—22no—
90C3×C4×D4—21no—
91C3×C4×D4—21no—
92C3×C4×D4—21no—
93C3×C4×D4—21no—
94C3×#117—22no—
95C3×#117—22no—
96C3×#117—22no—
97C3×#117—22no—
98C2×C3×C4⋊C4C2×C412yes—
99C2×C3×C4⋊C4C2×C412yes—
100C2×C3×C4⋊C4C2×C411yescharacteristic
101C2×C3×C4⋊C4C2×C412yes—
102C2×C3×C4⋊C4C2×C411yescharacteristic
103C2×C3×C4⋊C4C2×C412yes—
104C2×C3×C22⋊C4C2×C412yes—
105C2×C3×C22⋊C4C2×C412yes—
106C2×C3×C22⋊C4C2×C412yes—
107C2×C3×C22⋊C4C2×C412yes—
108C2×C3×C22⋊C4C2312yes—
109C2×C3×C42C2×C412yes—
110C2×C3×C42C2×C411yescharacteristic
111C2×C3×C42C2×C411yescharacteristic
112C2×C3×C42C2×C411yescharacteristic
113C2×C3×C42C2×C411yescharacteristic
114C2×C3×C42D411yescharacteristic
115C2×C3×C42D411yescharacteristic
116C2×C3×C42D411yescharacteristic
117C2×C3×C42D411yescharacteristic
118C2×C3×C42D411yescharacteristic
119C2×C3×C42D411yescharacteristic
120C2×C3×C42D411yescharacteristic
121C2×C3×C42D411yescharacteristic
122C2×#217C2314yes—
123C2×#217C2314yes—
124#926—44no—
125C2×C4⋊Dic3—22no—
126C2×C4⋊Dic3—22no—
127C2×C4⋊Dic3—22no—
128C2×C4⋊Dic3—22no—
129C2×C4×Dic3—22no—
130C2×C4×Dic3—22no—
131#834—48no—
132C3⋊#95—22no—
133C3⋊#95—22no—
134C3⋊#95—22no—
135C3⋊#95—22no—
136C3⋊#95—22no—
137C3⋊#95—22no—
138#529—21no—
139#529—21no—
140#529—21no—
141#529—21no—
142#529—21no—
143#529—21no—
144#529—21no—
145#529—21no—
146C2×#126Dic312yescomplemented
147C2×#125Dic311yescharacteristic, complemented
148C2×#124Dic311yescharacteristic, complemented
149C2×#123C2×D312yes—
150C2×C22⋊Q8C2×D311yescharacteristic
151C2×#121C2×D311yescharacteristic
152C2×C4×Q8Dic311yescharacteristic, complemented
153C2×C4×D4Dic311yescharacteristic, complemented
154C2×C4×D4C2×D312yes—
155C2×#117—32no—
156C2×#117Dic312yescomplemented
157C22×C42C2×D311yescharacteristic
158C2×#99—38no—
159#384—61no—
160#383—61no—
161#382—61no—
162#382—61no—
163#381—32no—
164#380—32no—
165#380—32no—
166#380—32no—
167C4×C4⋊C4—61no—
168C4×C4⋊C4—61no—
169C4×C4⋊C4—61no—
170C4×C22⋊C4—32no—
171#376—61no—
172#342—38no—
173C24×C3C22×C412yes—
174C24×C3C2411yescharacteristic
175C2×C3×Q8—21no—
176C2×C3×Q8—21no—
177C2×C3×Q8—42no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—24no—
181C2×C3×D4—24no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4—42no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—22no—
189C22×C3×C4—42no—
190C22×C3×C4—42no—
191C22×C3×C4—44no—
192C22×C3×C4C22×C412yes—
193C22×C3×C4C22×C411yescharacteristic
194C22×C3×C4C22×C411yescharacteristic
195C22×C3×C4C22×C411yescharacteristic
196C22×C3×C4C22×C411yescharacteristic
197C22×C3×C4C22×C412yes—
198C22×C3×C4C22×C412yes—
199C22×C3×C4C22×C411yescharacteristic
200C22×C3×C4C22×C411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×Dic3—24no—
214C22×Dic3—24no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—22no—
218C3×C4⋊C4—22no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—22no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4—22no—
227C3×C22⋊C4—22no—
228C3×C22⋊C4—22no—
229C3×C22⋊C4—22no—
230C3×C22⋊C4—22no—
231C3×C22⋊C4—22no—
232C3×C22⋊C4—22no—
233C3×C22⋊C4—22no—
234C3×C22⋊C4—22no—
235C3×C22⋊C4—22no—
236C3×C22⋊C4—22no—
237C3×C42—22no—
238C3×C42—22no—
239C3×C42C22⋊C411yescharacteristic
240C3×C42C22⋊C411yescharacteristic
241C3×C42C22⋊C411yescharacteristic
242C3×C42C22⋊C411yescharacteristic
243C3×C42C22⋊C411yescharacteristic
244C3×C42C22⋊C411yescharacteristic
245C3×C42C22⋊C411yescharacteristic
246C3×C42C22⋊C411yescharacteristic
247C3×C42C22⋊C411yescharacteristic
248C3×C42C22⋊C411yescharacteristic
249C3×C42C22⋊C411yescharacteristic
250C3×C42C22⋊C411yescharacteristic
251C3×C42C22⋊C411yescharacteristic
252C3×C42C22⋊C411yescharacteristic
253C3×C42C22⋊C411yescharacteristic
254C3×C42C22⋊C411yescharacteristic
255#217—24no—
256#217—28no—
257#217—24no—
258C4⋊Dic3—44no—
259C4⋊Dic3—44no—
260C4⋊Dic3—44no—
261C4⋊Dic3—44no—
262C4×Dic3—44no—
263C4×Dic3—44no—
264C22×Q8C2×Dic311yescharacteristic
265C22×D4C2×Dic311yescharacteristic
266C22×D4C22×D312yes—
267C23×C4C2×Dic312yes—
268C23×C4C22×D311yescharacteristic
269C23×C4C22×D311yescharacteristic
270C23×C4C22×D311yescharacteristic
271#126—22no—
272#126—22no—
273#126—22no—
274#126—22no—
275#125—21no—
276#125—21no—
277#125—21no—
278#125—21no—
279#124—21no—
280#124—21no—
281#124—21no—
282#124—21no—
283#123—22no—
284#123—22no—
285#123—22no—
286#123—22no—
287C22⋊Q8—21no—
288C22⋊Q8—21no—
289C22⋊Q8—21no—
290C22⋊Q8—21no—
291#121—21no—
292#121—21no—
293#121—21no—
294#121—21no—
295C4×Q8—21no—
296C4×Q8—21no—
297C4×Q8—21no—
298C4×Q8—21no—
299C4×D4—22no—
300C4×D4—22no—
301C4×D4—22no—
302C4×D4—22no—
303C4×D4—21no—
304C4×D4—21no—
305C4×D4—21no—
306C4×D4—21no—
307#117—22no—
308#117—22no—
309#117—22no—
310#117—22no—
311#117—124no—
312C2×C4⋊C4—62no—
313C2×C4⋊C4—62no—
314C2×C4⋊C4—62no—
315C2×C4⋊C4—62no—
316C2×C4⋊C4C2×Dic312yes—
317C2×C4⋊C4C2×Dic312yes—
318C2×C4⋊C4C2×Dic311yescharacteristic
319C2×C4⋊C4C2×Dic312yes—
320C2×C4⋊C4C2×Dic311yescharacteristic
321C2×C4⋊C4C2×Dic312yes—
322C2×C22⋊C4—34no—
323C2×C22⋊C4—34no—
324C2×C22⋊C4C2×Dic312yes—
325C2×C22⋊C4C2×Dic312yes—
326C2×C22⋊C4C2×Dic312yes—
327C2×C22⋊C4C2×Dic312yes—
328C2×C22⋊C4C22×D312yes—
329C2×C42—62no—
330C2×C42—62no—
331C2×C42C2×Dic312yes—
332C2×C42C2×Dic311yescharacteristic
333C2×C42C2×Dic311yescharacteristic
334C2×C42C2×Dic311yescharacteristic
335C2×C42C2×Dic311yescharacteristic
336C2×C42#6711yescharacteristic
337C2×C42#6711yescharacteristic
338C2×C42#6711yescharacteristic
339C2×C42#6711yescharacteristic
340C2×C42#6711yescharacteristic
341C2×C42#6711yescharacteristic
342C2×C42#6711yescharacteristic
343C2×C42#6711yescharacteristic
344#99—128no—
345#95—62no—
346#95—62no—
347#95—62no—
348#95—62no—
349#95—62no—
350#95—62no—
351C23×C3—22no—
352C23×C3—24no—
353C23×C3—41no—
354C23×C3—41no—
355C23×C3—42no—
356C23×C3—42no—
357C23×C3C23×C411yescharacteristic
358C23×C3C22×D411yescharacteristic
359C23×C3C22×D411yescharacteristic
360C23×C3H(5,𝔽2)12yes—
361C23×C32−1+412yes—
362C3×Q8—42no—
363C3×Q8—42no—
364C3×D4—42no—
365C3×D4—42no—
366C3×D4—42no—
367C3×D4—44no—
368C3×D4—42no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—22no—
380C2×C3×C4—22no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—24no—
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—44no—
394C2×C3×C4—44no—
395C2×C3×C4—42no—
396C2×C3×C4—42no—
397C2×C3×C4C2×C22⋊C411yescharacteristic
398C2×C3×C4C2×C22⋊C411yescharacteristic
399C2×C3×C4C2×C22⋊C411yescharacteristic
400C2×C3×C4C2×C22⋊C411yescharacteristic
401C2×C3×C4C2×C22⋊C411yescharacteristic
402C2×C3×C4C2×C22⋊C411yescharacteristic
403C2×C3×C4C2×C22⋊C411yescharacteristic
404C2×C3×C4C2×C22⋊C411yescharacteristic
405C2×C3×C4C2×C22⋊C411yescharacteristic
406C2×C3×C4C2×C22⋊C411yescharacteristic
407C2×C3×C4C2×C22⋊C411yescharacteristic
408C2×C3×C4C2×C22⋊C411yescharacteristic
409C2×Dic3—44no—
410C2×Dic3—44no—
411C2×Dic3—44no—
412C2×Dic3—44no—
413C2×Dic3—44no—
414C2×Dic3—44no—
415C24C22×Dic312yes—
416C24C23×D311yescharacteristic
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—42no—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—42no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—44no—
434C22×C4—64no—
435C22×C4—64no—
436C22×C4C22×Dic312yes—
437C22×C4C22×Dic311yescharacteristic
438C22×C4C22×Dic311yescharacteristic
439C22×C4C22×Dic311yescharacteristic
440C22×C4C22×Dic311yescharacteristic
441C22×C4C22×Dic312yes—
442C22×C4C22×Dic312yes—
443C22×C4C22×Dic311yescharacteristic
444C22×C4C22×Dic311yescharacteristic
445C22×C4C2×#6711yescharacteristic
446C22×C4C2×#6711yescharacteristic
447C22×C4C2×#6711yescharacteristic
448C22×C4C2×#6711yescharacteristic
449C22×C4C2×#6711yescharacteristic
450C22×C4C2×#6711yescharacteristic
451C22×C4C2×#6711yescharacteristic
452C22×C4C2×#6711yescharacteristic
453C22×C4C2×#6711yescharacteristic
454C22×C4C2×#6711yescharacteristic
455C22×C4C2×#6711yescharacteristic
456C22×C4C2×#6711yescharacteristic
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—124no—
470C4⋊C4—124no—
471C4⋊C4—124no—
472C4⋊C4—124no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—64no—
484C22⋊C4—68no—
485C22⋊C4—64no—
486C42—22no—
487C42—22no—
488C42—124no—
489C42—124no—
490C42#21711yescharacteristic
491C42#21711yescharacteristic
492C42#21711yescharacteristic
493C42#21711yescharacteristic
494C42#21711yescharacteristic
495C42#21711yescharacteristic
496C42#21711yescharacteristic
497C42#21711yescharacteristic
498C42#21711yescharacteristic
499C42#21711yescharacteristic
500C42#21711yescharacteristic
501C42#21711yescharacteristic
502C42#21711yescharacteristic
503C42#21711yescharacteristic
504C42#21711yescharacteristic
505C42#21711yescharacteristic
506C22×C3—21no—
507C22×C3—21no—
508C22×C3—21no—
509C22×C3—21no—
510C22×C3—21no—
511C22×C3—21no—
512C22×C3—42no—
513C22×C3—42no—
514C22×C3—42no—
515C22×C3—42no—
516C22×C3—42no—
517C22×C3—84no—
518C22×C3C22×C22⋊C411yescharacteristic
519C22×C3#52012yes—
520C22×C3#53511yescharacteristic
521C22×C3#53611yescharacteristic
522C22×C3#53712yes—
523C3×C4—21no—
524C3×C4—21no—
525C3×C4—21no—
526C3×C4—21no—
527C3×C4—21no—
528C3×C4—21no—
529C3×C4—21no—
530C3×C4—21no—
531C3×C4—21no—
532C3×C4—21no—
533C3×C4—42no—
534C3×C4—42no—
535C3×C4—42no—
536C3×C4#41012yes—
537C3×C4#41012yes—
538Dic3—84no—
539Dic3—84no—
540C23—22no—
541C23—24no—
542C23—41no—
543C23—41no—
544C23—42no—
545C23—42no—
546C23#100912yes—
547C23#101012yes—
548C23C23×Dic311yescharacteristic
549C23C22×#6711yescharacteristic
550C23C22×#6711yescharacteristic
551Q8—42no—
552Q8—42no—
553D4—42no—
554D4—42no—
555D4—42no—
556D4—44no—
557D4—42no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—124no—
587C2×C4—124no—
588C2×C4—124no—
589C2×C4—124no—
590C2×C4—124no—
591C2×C4—124no—
592C2×C4C2×#21711yescharacteristic
593C2×C4C2×#21711yescharacteristic
594C2×C4C2×#21711yescharacteristic
595C2×C4C2×#21711yescharacteristic
596C2×C4C2×#21711yescharacteristic
597C2×C4C2×#21711yescharacteristic
598C2×C4C2×#21711yescharacteristic
599C2×C4C2×#21711yescharacteristic
600C2×C4C2×#21711yescharacteristic
601C2×C4C2×#21711yescharacteristic
602C2×C4C2×#21711yescharacteristic
603C2×C4C2×#21711yescharacteristic
604C2×C3—21no—
605C2×C3—22no—
606C2×C3—22no—
607C2×C3—21no—
608C2×C3—82no—
609C2×C3#231111yescharacteristic
610C2×C3C2×#41011yescharacteristic
611C2×C3C2×#41011yescharacteristic
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—84no—
624C22#582511yescharacteristic
625C22#582712yes—
626C22#583011yescharacteristic
627C22#583312yes—
628C22C22×#21711yescharacteristic
629C4—21no—
630C4—21no—
631C4—21no—
632C4—21no—
633C4—21no—
634C4—21no—
635C4—21no—
636C4—21no—
637C4—21no—
638C4—21no—
639C4—42no—
640C4—42no—
641C4—42no—
642C4—244no—
643C4—244no—
644C4#524012yes—
645C4#524012yes—
646C3#2187511yescharacteristic, complemented, Sylow
647C2—21no—
648C2—22no—
649C2—22no—
650C2—21no—
651C2—82no—
652C2C2×#524011yescharacteristic
653C2C2×#524011yescharacteristic
654C2#8408711yescharacteristic
subgroupG/H
centerC22C22×#217
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC2×C3×#529C2
socleC22×C3C22×C22⋊C4
prime radicalC2×C3×#121C22
cosocleC24×C3C24
hypercenterC2×#529D3
hypocenterC3#21875
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C42 ◁ C2×#529

Complements

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

NH ≅ G/Ntype
2. #82062C2semidirect
3. #82060C2semidirect
4. #71705C2semidirect
5. #71701C2semidirect
7. #71619C2semidirect
17. C2×C3×#126C4semidirect
18. C2×C3×#125C4semidirect
19. C2×C3×#124C4semidirect
23. C2×C3×C4×Q8C4semidirect
24. C2×C3×C4×D4C4semidirect
26. C2×C3×#117C4semidirect
146. C2×#126Dic3semidirect
147. C2×#125Dic3semidirect
148. C2×#124Dic3semidirect
152. C2×C4×Q8Dic3semidirect
153. C2×C4×D4Dic3semidirect
156. C2×#117Dic3semidirect
646. C3#21875semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5292
#820622
#820602
#717052
#717012
#716862
#716192
#218753