Finite Group Databasebeta

← #10795121

Subgroups — #10795121

Order 768.

1866 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×D4C211yesmaximal, characteristic
2#84027C212yesmaximal, complemented
3#72130C212yesmaximal
4#71619C214yesmaximal, complemented
5C4×#834C214yesmaximal, complemented
6#71542C212yesmaximal, complemented
7#20722—31nomaximal, Sylow
8C23×C3×D4C2211yescharacteristic
9C24×C3×C4C411yescharacteristic, complemented
10C24×C3×C4C2211yescharacteristic
11C2×C3×C4×D4—22no—
12C2×C3×C4×D4—22no—
13C2×C3×C4×D4—22no—
14C2×C3×C4×D4—22no—
15C2×C3×C4×D4C412yescomplemented
16C2×C3×C4×D4C412yescomplemented
17C2×C3×C4×D4C2212yes—
18C2×C3×C4×D4C2212yes—
19C22×C3×C4⋊C4C411yescharacteristic, complemented
20C22×C3×C22⋊C4C411yescharacteristic, complemented
21C22×C3×C22⋊C4C2211yescharacteristic
22C22×C3×C42C411yescharacteristic, complemented
23C2×#926—22no—
24#5247C2214yes—
25C2×#834C2218yes—
26#5212—22no—
27#5212C2214yescomplemented
28#5211—22no—
29C4×#217—22no—
30C4×#217C2214yescomplemented
31#4526C2218yescomplemented
32C22×C4×D4D311yescharacteristic
33#2280—32no—
34#1819—32no—
35#1799—34no—
36#1781—32no—
37C4×#99—34no—
38C25×C3C2×C411yescharacteristic
39C25×C3C2311yescharacteristic
40C22×C3×D4—22no—
41C22×C3×D4—22no—
42C22×C3×D4—28no—
43C22×C3×D4C2×C412yes—
44C22×C3×D4C2312yes—
45C23×C3×C4—21no—
46C23×C3×C4—21no—
47C23×C3×C4—21no—
48C23×C3×C4—21no—
49C23×C3×C4—44no—
50C23×C3×C4—44no—
51C23×C3×C4C2×C411yescharacteristic
52C23×C3×C4C2×C411yescharacteristic
53C23×C3×C4C2×C411yescharacteristic
54C23×C3×C4C2×C412yes—
55C23×C3×C4C2×C412yes—
56C23×C3×C4C2×C411yescharacteristic
57C23×C3×C4C2×C411yescharacteristic
58C23×C3×C4C2×C411yescharacteristic
59C23×C3×C4C2×C411yescharacteristic
60C23×C3×C4D412yes—
61C23×C3×C4D412yes—
62C23×C3×C4D412yes—
63C23×C3×C4D412yes—
64C23×C3×C4C2311yescharacteristic
65C23×C3×C4C2311yescharacteristic
66C3×C4×D4—24no—
67C3×C4×D4—24no—
68C3×C4×D4—24no—
69C3×C4×D4—24no—
70C3×C4×D4—24no—
71C3×C4×D4—24no—
72C3×C4×D4—24no—
73C3×C4×D4—24no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4C2×C412yes—
77C2×C3×C4⋊C4C2×C412yes—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4C2×C412yes—
83C2×C3×C22⋊C4C2×C412yes—
84C2×C3×C22⋊C4C2×C412yes—
85C2×C3×C22⋊C4C2312yes—
86C2×C3×C42—21no—
87C2×C3×C42—21no—
88C2×C3×C42—21no—
89C2×C3×C42—21no—
90C2×C3×C42C2×C412yes—
91C2×C3×C42C2×C412yes—
92C2×#217—24no—
93C2×#217—24no—
94C2×#217C2318yes—
95#926—48no—
96C2×C4⋊Dic3—24no—
97C2×C4×Dic3—24no—
98#834—216no—
99C3⋊#95—24no—
100C3⋊#95—24no—
101C23×D4C2×D311yescharacteristic
102C24×C4Dic311yescharacteristic, complemented
103C24×C4C2×D311yescharacteristic
104C2×C4×D4—22no—
105C2×C4×D4—22no—
106C2×C4×D4—22no—
107C2×C4×D4—22no—
108C2×C4×D4Dic312yescomplemented
109C2×C4×D4Dic312yescomplemented
110C2×C4×D4C2×D312yes—
111C2×C4×D4C2×D312yes—
112C2×#117—62no—
113C22×C4⋊C4Dic311yescharacteristic, complemented
114C22×C22⋊C4Dic311yescharacteristic, complemented
115C22×C22⋊C4C2×D311yescharacteristic
116C22×C42Dic311yescharacteristic, complemented
117C2×#99—38no—
118#381—62no—
119#380—34no—
120#380—62no—
121#379—34no—
122C4×C22⋊C4—34no—
123C4×C22⋊C4—62no—
124#342—38no—
125C24×C3—21no—
126C24×C3—21no—
127C24×C3—24no—
128C24×C3—24no—
129C24×C3—42no—
130C24×C3—42no—
131C24×C3—42no—
132C24×C3—42no—
133C24×C3C22×C411yescharacteristic
134C24×C3C22×C411yescharacteristic
135C24×C3C22×C412yes—
136C24×C3C2×D412yes—
137C24×C3C2×D412yes—
138C24×C3C2411yescharacteristic
139C2×C3×D4—216no—
140C2×C3×D4—216no—
141C2×C3×D4—24no—
142C2×C3×D4—24no—
143C2×C3×D4—24no—
144C2×C3×D4—24no—
145C2×C3×D4—24no—
146C2×C3×D4#4118yes—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—28no—
150C22×C3×C4—24no—
151C22×C3×C4—24no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—42no—
161C22×C3×C4—42no—
162C22×C3×C4—44no—
163C22×C3×C4—44no—
164C22×C3×C4—42no—
165C22×C3×C4—42no—
166C22×C3×C4—42no—
167C22×C3×C4—44no—
168C22×C3×C4—44no—
169C22×C3×C4—44no—
170C22×C3×C4—44no—
171C22×C3×C4—44no—
172C22×C3×C4—44no—
173C22×C3×C4—44no—
174C22×C3×C4—42no—
175C22×C3×C4—42no—
176C22×C3×C4C22⋊C411yescharacteristic
177C22×C3×C4C22⋊C411yescharacteristic
178C22×C3×C4C22⋊C411yescharacteristic
179C22×C3×C4C22⋊C412yes—
180C22×C3×C4C22⋊C412yes—
181C22×C3×C4C22⋊C412yes—
182C22×C3×C4C22⋊C412yes—
183C22×C3×C4C22⋊C412yes—
184C22×C3×C4C22⋊C412yes—
185C22×C3×C4C22⋊C411yescharacteristic
186C22×C3×C4C22×C411yescharacteristic
187C22×C3×C4C22×C411yescharacteristic
188C22×C3×C4C22×C412yes—
189C22×C3×C4C22×C412yes—
190C22×C3×C4C22×C412yes—
191C22×C3×C4C22×C411yescharacteristic
192C22×C3×C4C22×C411yescharacteristic
193C22×C3×C4C2×D412yes—
194C22×C3×C4C2×D412yes—
195C22×C3×C4C2×D412yes—
196C22×C3×C4C2×D412yes—
197C22×Dic3—28no—
198C3×C4⋊C4—24no—
199C3×C4⋊C4—24no—
200C3×C22⋊C4—24no—
201C3×C22⋊C4—24no—
202C3×C22⋊C4—24no—
203C3×C22⋊C4—24no—
204C3×C42—22no—
205C3×C42—22no—
206C3×C42—22no—
207C3×C42—22no—
208#217—216no—
209#217—48no—
210#217—48no—
211C4⋊Dic3—48no—
212C4×Dic3—48no—
213C25C2×Dic311yescharacteristic
214C25C22×D311yescharacteristic
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—28no—
218C22×D4C2×Dic312yes—
219C22×D4C22×D312yes—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4C2×Dic311yescharacteristic
227C23×C4C2×Dic311yescharacteristic
228C23×C4C2×Dic311yescharacteristic
229C23×C4C2×Dic311yescharacteristic
230C23×C4C2×Dic312yes—
231C23×C4C2×Dic312yes—
232C23×C4C2×Dic311yescharacteristic
233C23×C4C2×Dic311yescharacteristic
234C23×C4C2×Dic311yescharacteristic
235C23×C4#6712yes—
236C23×C4#6712yes—
237C23×C4#6712yes—
238C23×C4#6712yes—
239C23×C4C22×D311yescharacteristic
240C23×C4C22×D311yescharacteristic
241C4×D4—24no—
242C4×D4—24no—
243C4×D4—24no—
244C4×D4—24no—
245C4×D4—24no—
246C4×D4—24no—
247C4×D4—24no—
248C4×D4—24no—
249#117—128no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—64no—
253C2×C4⋊C4C2×Dic312yes—
254C2×C4⋊C4C2×Dic312yes—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—38no—
260C2×C22⋊C4—64no—
261C2×C22⋊C4—64no—
262C2×C22⋊C4C2×Dic312yes—
263C2×C22⋊C4C2×Dic312yes—
264C2×C22⋊C4C2×Dic312yes—
265C2×C22⋊C4C22×D312yes—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—64no—
271C2×C42C2×Dic312yes—
272C2×C42C2×Dic312yes—
273#99—616no—
274#95—64no—
275#95—64no—
276C23×C3—22no—
277C23×C3—22no—
278C23×C3—22no—
279C23×C3—24no—
280C23×C3—24no—
281C23×C3—22no—
282C23×C3—22no—
283C23×C3—22no—
284C23×C3—24no—
285C23×C3—24no—
286C23×C3—24no—
287C23×C3—24no—
288C23×C3—41no—
289C23×C3—42no—
290C23×C3—41no—
291C23×C3—41no—
292C23×C3—42no—
293C23×C3—41no—
294C23×C3—42no—
295C23×C3—42no—
296C23×C3—42no—
297C23×C3—416no—
298C23×C3—416no—
299C23×C3—42no—
300C23×C3—42no—
301C23×C3—42no—
302C23×C3C2×C22⋊C412yes—
303C23×C3C2×C22⋊C411yescharacteristic
304C23×C3C2×C22⋊C411yescharacteristic
305C23×C3#11714yescomplemented
306C23×C3C23×C411yescharacteristic
307C23×C3C22×D412yes—
308C23×C3C2×#4114yes—
309C3×D4—216no—
310C3×D4—216no—
311C2×C3×C4—24no—
312C2×C3×C4—24no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—24no—
317C2×C3×C4—24no—
318C2×C3×C4—24no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—44no—
335C2×C3×C4—44no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—44no—
341C2×C3×C4—44no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4#9914yescomplemented
345C2×C3×C4C2×C22⋊C411yescharacteristic
346C2×C3×C4C2×C22⋊C411yescharacteristic
347C2×C3×C4C2×C22⋊C412yes—
348C2×C3×C4C2×C22⋊C412yes—
349C2×C3×C4C2×C22⋊C411yescharacteristic
350C2×C3×C4C2×C22⋊C411yescharacteristic
351C2×C3×C4#11714yescomplemented
352C2×Dic3—48no—
353C2×Dic3—48no—
354C2×Dic3—48no—
355C24—24no—
356C24—21no—
357C24—21no—
358C24—24no—
359C24—42no—
360C24—42no—
361C24—42no—
362C24—42no—
363C24C22×Dic311yescharacteristic
364C24C22×Dic312yes—
365C24C22×Dic311yescharacteristic
366C24C2×#6712yes—
367C24C2×#6712yes—
368C24C23×D311yescharacteristic
369C2×D4—216no—
370C2×D4—216no—
371C2×D4—24no—
372C2×D4—24no—
373C2×D4—24no—
374C2×D4—24no—
375C2×D4—24no—
376C2×D4#23518yes—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—28no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—68no—
407C22×C4#21711yescharacteristic
408C22×C4#21711yescharacteristic
409C22×C4#21712yes—
410C22×C4#21712yes—
411C22×C4#21711yescharacteristic
412C22×C4#21712yes—
413C22×C4#21712yes—
414C22×C4#21712yes—
415C22×C4#21712yes—
416C22×C4#21711yescharacteristic
417C22×C4C22×Dic312yes—
418C22×C4C22×Dic311yescharacteristic
419C22×C4C22×Dic311yescharacteristic
420C22×C4C22×Dic312yes—
421C22×C4C22×Dic312yes—
422C22×C4C22×Dic311yescharacteristic
423C22×C4C22×Dic311yescharacteristic
424C22×C4C2×#6712yes—
425C22×C4C2×#6712yes—
426C22×C4C2×#6712yes—
427C22×C4C2×#6712yes—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—128no—
431C22⋊C4—24no—
432C22⋊C4—24no—
433C22⋊C4—24no—
434C22⋊C4—24no—
435C22⋊C4—616no—
436C22⋊C4—128no—
437C22⋊C4—128no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—128no—
443C22×C3—28no—
444C22×C3—22no—
445C22×C3—21no—
446C22×C3—21no—
447C22×C3—22no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—21no—
451C22×C3—21no—
452C22×C3—22no—
453C22×C3—22no—
454C22×C3—22no—
455C22×C3—24no—
456C22×C3—44no—
457C22×C3—48no—
458C22×C3—48no—
459C22×C3—44no—
460C22×C3—44no—
461C22×C3—44no—
462C22×C3—48no—
463C22×C3—48no—
464C22×C3—42no—
465C22×C3—42no—
466C22×C3C2×#9912yes—
467C22×C3#41012yes—
468C22×C3C22×C22⋊C411yescharacteristic
469C22×C3C2×#11712yes—
470C22×C3#52514yes—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—22no—
474C3×C4—22no—
475C3×C4—24no—
476C3×C4—42no—
477C3×C4—42no—
478C3×C4C2×#9912yes—
479C3×C4C2×#9912yes—
480Dic3—88no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—22no—
489C23—22no—
490C23—24no—
491C23—24no—
492C23—22no—
493C23—41no—
494C23—42no—
495C23—41no—
496C23—41no—
497C23—42no—
498C23—41no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—416no—
503C23—416no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23#92614yescomplemented
508C23C2×#21712yes—
509C23C2×#21711yescharacteristic
510C23C2×#21711yescharacteristic
511C23C2×#23514yes—
512C23C23×Dic311yescharacteristic
513C23C22×#6712yes—
514D4—216no—
515D4—216no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—128no—
550C2×C4—128no—
551C2×C4—128no—
552C2×C4#83414yescomplemented
553C2×C4#92614yescomplemented
554C2×C4C2×#21711yescharacteristic
555C2×C4C2×#21711yescharacteristic
556C2×C4C2×#21712yes—
557C2×C4C2×#21712yes—
558C2×C4C2×#21711yescharacteristic
559C2×C4C2×#21711yescharacteristic
560C2×C3—24no—
561C2×C3—22no—
562C2×C3—22no—
563C2×C3—22no—
564C2×C3—42no—
565C2×C3—44no—
566C2×C3#228011yescharacteristic
567C2×C3C22×#9911yescharacteristic
568C2×C3C2×#41011yescharacteristic
569C22—28no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—22no—
575C22—21no—
576C22—21no—
577C22—22no—
578C22—24no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—44no—
583C22—48no—
584C22—48no—
585C22—44no—
586C22—48no—
587C22—48no—
588C22—44no—
589C22—44no—
590C22—42no—
591C22—42no—
592C22C2×#83412yes—
593C22#524012yes—
594C22C2×#92612yes—
595C22#579114yes—
596C22C22×#21711yescharacteristic
597C4—24no—
598C4—21no—
599C4—21no—
600C4—22no—
601C4—22no—
602C4—42no—
603C4—42no—
604C4—248no—
605C4C2×#83412yes—
606C4C2×#83412yes—
607C3#2072211yescharacteristic, complemented, Sylow
608C2—24no—
609C2—22no—
610C2—22no—
611C2—22no—
612C2—42no—
613C2—44no—
614C2#8402711yescharacteristic
615C2C22×#83411yescharacteristic
616C2C2×#524011yescharacteristic
subgroupG/H
centerC2×C4C2×#217
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC22×C3×C4×D4C2
socleC22×C3C22×C22⋊C4
prime radicalC23×C3×D4C22
cosocleC24×C3C24
hypercenterC22×C4×D4D3
hypocenterC3#20722
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
2. #84027C2semidirect
4. #71619C2semidirect
5. C4×#834C2semidirect
6. #71542C2semidirect
9. C24×C3×C4C4semidirect
15. C2×C3×C4×D4C4semidirect
16. C2×C3×C4×D4C4semidirect
19. C22×C3×C4⋊C4C4semidirect
20. C22×C3×C22⋊C4C4semidirect
22. C22×C3×C42C4semidirect
27. #5212C22semidirect
30. C4×#217C22semidirect
31. #4526C22semidirect
102. C24×C4Dic3semidirect
108. C2×C4×D4Dic3semidirect
109. C2×C4×D4Dic3semidirect
113. C22×C4⋊C4Dic3semidirect
114. C22×C22⋊C4Dic3semidirect
116. C22×C42Dic3semidirect
305. C23×C3#117semidirect
344. C2×C3×C4#99semidirect
351. C2×C3×C4#117semidirect
507. C23#926semidirect
552. C2×C4#834semidirect
553. C2×C4#926semidirect
607. C3#20722semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×D42
#840272
#721302
#716192
C4×#8342
#715422
#207223