Finite Group Databasebeta

← #14719650

Subgroups — #14719650

Order 2816.

1738 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622820C212yesmaximal, complemented
2#13066954C214yesmaximal, complemented
3#13066946C212yesmaximal, complemented
4#13066943C212yesmaximal, complemented
5#13066934C214yesmaximal, complemented
6C22×C4×D4×C11C211yesmaximal, characteristic
7C23×D4×C11C411yescharacteristic
8C24×C4×C11C411yescharacteristic
9C24×C4×C11C2211yescharacteristic
10C2×C4×D4×C11—22no—
11C2×C4×D4×C11—22no—
12C2×C4×D4×C11—22no—
13C2×C4×D4×C11—22no—
14C2×C4×D4×C11C412yes—
15C2×C4×D4×C11C412yes—
16C2×C4×D4×C11C2212yes—
17C2×C4×D4×C11C2212yes—
18C22×C11×C4⋊C4C2211yescharacteristic
19C22×C11×C22⋊C4C411yescharacteristic
20C22×C11×C22⋊C4C2211yescharacteristic
21C22×C42×C11C411yescharacteristic
22C22×#4318—22no—
23#10622820—22no—
24#10622820—22no—
25#10622820C2214yes—
26#10622820C2214yescomplemented
27C2×C22⋊(C11⋊C8)—22no—
28C2×C22⋊(C11⋊C8)C2214yescomplemented
29#10622166C2218yescomplemented
30#10622163C2218yescomplemented
31C25×C11C2×C411yescharacteristic
32C25×C11C2×C411yescharacteristic
33C22×D4×C11—22no—
34C22×D4×C11—22no—
35C22×D4×C11—28no—
36C22×D4×C11C2×C412yes—
37C22×D4×C11C2×C412yes—
38C23×C4×C11—21no—
39C23×C4×C11—21no—
40C23×C4×C11—21no—
41C23×C4×C11—21no—
42C23×C4×C11—44no—
43C23×C4×C11—44no—
44C23×C4×C11C2×C411yescharacteristic
45C23×C4×C11C2×C411yescharacteristic
46C23×C4×C11C2×C411yescharacteristic
47C23×C4×C11C2×C411yescharacteristic
48C23×C4×C11C2×C412yes—
49C23×C4×C11C2×C412yes—
50C23×C4×C11C2×C411yescharacteristic
51C23×C4×C11C2×C411yescharacteristic
52C23×C4×C11D412yes—
53C23×C4×C11D412yes—
54C23×C4×C11D412yes—
55C23×C4×C11D412yes—
56C23×C4×C11C2311yescharacteristic
57C23×C4×C11C2311yescharacteristic
58C23×C4×C11C2311yescharacteristic
59C4×D4×C11—24no—
60C4×D4×C11—24no—
61C4×D4×C11—24no—
62C4×D4×C11—24no—
63C4×D4×C11—24no—
64C4×D4×C11—24no—
65C4×D4×C11—24no—
66C4×D4×C11—24no—
67C2×C11×C4⋊C4—22no—
68C2×C11×C4⋊C4—22no—
69C2×C11×C4⋊C4C2×C412yes—
70C2×C11×C4⋊C4C2312yes—
71C2×C11×C22⋊C4—22no—
72C2×C11×C22⋊C4—22no—
73C2×C11×C22⋊C4—22no—
74C2×C11×C22⋊C4—22no—
75C2×C11×C22⋊C4C2×C412yes—
76C2×C11×C22⋊C4C2×C412yes—
77C2×C11×C22⋊C4C2×C412yes—
78C2×C11×C22⋊C4C2312yes—
79C2×C42×C11—21no—
80C2×C42×C11—21no—
81C2×C42×C11—21no—
82C2×C42×C11—21no—
83C2×C42×C11C2×C412yes—
84C2×C42×C11C2×C412yes—
85C2×#4318—24no—
86C2×#4318—24no—
87C2×#4318—48no—
88C22×C11⋊C8—24no—
89C22⋊(C11⋊C8)—24no—
90C22⋊(C11⋊C8)—24no—
91C22⋊(C11⋊C8)—24no—
92C22⋊(C11⋊C8)C2318yes—
93#17362—111nomaximal, Sylow
94C24×C11—21no—
95C24×C11—21no—
96C24×C11—24no—
97C24×C11—24no—
98C24×C11—42no—
99C24×C11—42no—
100C24×C11—42no—
101C24×C11—42no—
102C24×C11C22⋊C412yes—
103C24×C11C22⋊C412yes—
104C24×C11C22×C411yescharacteristic
105C24×C11C22×C411yescharacteristic
106C24×C11C22×C412yes—
107C24×C11C22×C411yescharacteristic
108C2×D4×C11—216no—
109C2×D4×C11—216no—
110C2×D4×C11—24no—
111C2×D4×C11—24no—
112C2×D4×C11—24no—
113C2×D4×C11—24no—
114C2×D4×C11—24no—
115C2×D4×C11M1618yescomplemented
116C22×C4×C11—21no—
117C22×C4×C11—21no—
118C22×C4×C11—28no—
119C22×C4×C11—24no—
120C22×C4×C11—24no—
121C22×C4×C11—21no—
122C22×C4×C11—21no—
123C22×C4×C11—21no—
124C22×C4×C11—21no—
125C22×C4×C11—22no—
126C22×C4×C11—22no—
127C22×C4×C11—22no—
128C22×C4×C11—22no—
129C22×C4×C11—42no—
130C22×C4×C11—42no—
131C22×C4×C11—44no—
132C22×C4×C11—44no—
133C22×C4×C11—42no—
134C22×C4×C11—42no—
135C22×C4×C11—44no—
136C22×C4×C11—42no—
137C22×C4×C11—44no—
138C22×C4×C11—44no—
139C22×C4×C11—44no—
140C22×C4×C11—44no—
141C22×C4×C11—44no—
142C22×C4×C11—44no—
143C22×C4×C11—42no—
144C22×C4×C11—42no—
145C22×C4×C11C22⋊C411yescharacteristic
146C22×C4×C11C22⋊C411yescharacteristic
147C22×C4×C11C22⋊C411yescharacteristic
148C22×C4×C11C22⋊C412yes—
149C22×C4×C11C22⋊C412yes—
150C22×C4×C11C22⋊C412yes—
151C22×C4×C11C22⋊C412yes—
152C22×C4×C11C22⋊C411yescharacteristic
153C22×C4×C11C22×C411yescharacteristic
154C22×C4×C11C22×C411yescharacteristic
155C22×C4×C11C22×C412yes—
156C22×C4×C11C22×C412yes—
157C22×C4×C11C22×C412yes—
158C22×C4×C11C22×C411yescharacteristic
159C22×C4×C11C2×D412yes—
160C22×C4×C11C2×D412yes—
161C22×C4×C11C2×D412yes—
162C22×C4×C11C2×D412yes—
163C22×C4×C11C2×D412yes—
164C22×C4×C11C2×D412yes—
165C22×C4×C11C2411yescharacteristic
166C11×C4⋊C4—24no—
167C11×C4⋊C4—24no—
168C11×C22⋊C4—24no—
169C11×C22⋊C4—24no—
170C11×C22⋊C4—24no—
171C11×C22⋊C4—24no—
172C42×C11—22no—
173C42×C11—22no—
174C42×C11—22no—
175C42×C11—22no—
176#4318—48no—
177#4318—48no—
178C2×C11⋊C8—28no—
179C2×C11⋊C8—48no—
180C22×C4×D4D1111yescharacteristic
181C2×#407—112no—
182#1465—114no—
183#1463—112no—
184#1462—112no—
185#1459—114no—
186C23×C11—22no—
187C23×C11—22no—
188C23×C11—22no—
189C23×C11—24no—
190C23×C11—24no—
191C23×C11—22no—
192C23×C11—22no—
193C23×C11—22no—
194C23×C11—24no—
195C23×C11—24no—
196C23×C11—24no—
197C23×C11—24no—
198C23×C11—42no—
199C23×C11—41no—
200C23×C11—41no—
201C23×C11—42no—
202C23×C11—41no—
203C23×C11—41no—
204C23×C11—42no—
205C23×C11—42no—
206C23×C11—42no—
207C23×C11—416no—
208C23×C11—416no—
209C23×C11—42no—
210C23×C11—42no—
211C23×C11—42no—
212C23×C11C2×C22⋊C412yes—
213C23×C11C2×C22⋊C412yes—
214C23×C11C2×C22⋊C411yescharacteristic
215C23×C11C2×C22⋊C411yescharacteristic
216C23×C11C2×M1614yes—
217C23×C11C2×M1614yescomplemented
218C23×C11C23×C411yescharacteristic
219D4×C11—216no—
220D4×C11—216no—
221C2×C4×C11—24no—
222C2×C4×C11—24no—
223C2×C4×C11—22no—
224C2×C4×C11—22no—
225C2×C4×C11—24no—
226C2×C4×C11—22no—
227C2×C4×C11—24no—
228C2×C4×C11—24no—
229C2×C4×C11—22no—
230C2×C4×C11—22no—
231C2×C4×C11—22no—
232C2×C4×C11—22no—
233C2×C4×C11—24no—
234C2×C4×C11—24no—
235C2×C4×C11—22no—
236C2×C4×C11—22no—
237C2×C4×C11—42no—
238C2×C4×C11—42no—
239C2×C4×C11—41no—
240C2×C4×C11—42no—
241C2×C4×C11—42no—
242C2×C4×C11—42no—
243C2×C4×C11—41no—
244C2×C4×C11—44no—
245C2×C4×C11—44no—
246C2×C4×C11—42no—
247C2×C4×C11—42no—
248C2×C4×C11—42no—
249C2×C4×C11—42no—
250C2×C4×C11—44no—
251C2×C4×C11—44no—
252C2×C4×C11—42no—
253C2×C4×C11—42no—
254C2×C4×C11#9914yes—
255C2×C4×C11C2×C22⋊C411yescharacteristic
256C2×C4×C11C2×C22⋊C411yescharacteristic
257C2×C4×C11C2×C22⋊C412yes—
258C2×C4×C11C2×C22⋊C411yescharacteristic
259C2×C4×C11C2×C22⋊C411yescharacteristic
260C2×C4×C11C2×M1614yescomplemented
261C2×C4×C11C22×D412yes—
262C11⋊C8—48no—
263C23×D4Dic1111yescharacteristic
264C24×C4Dic1111yescharacteristic
265C24×C4C2×D1111yescharacteristic
266C22×M16—222no—
267C2×C4×D4—22no—
268C2×C4×D4—22no—
269C2×C4×D4—22no—
270C2×C4×D4—22no—
271C2×C4×D4Dic1112yes—
272C2×C4×D4Dic1112yes—
273C2×C4×D4C2×D1112yes—
274C2×C4×D4C2×D1112yes—
275C22×C4⋊C4C2×D1111yescharacteristic
276C22×C22⋊C4Dic1111yescharacteristic
277C22×C22⋊C4C2×D1111yescharacteristic
278C22×C42Dic1111yescharacteristic
279#407—114no—
280#407—114no—
281#407—222no—
282#407—222no—
283C2×C22⋊C8—114no—
284C2×C22⋊C8—222no—
285#324—118no—
286#323—118no—
287C22×C11—28no—
288C22×C11—21no—
289C22×C11—21no—
290C22×C11—21no—
291C22×C11—21no—
292C22×C11—22no—
293C22×C11—21no—
294C22×C11—21no—
295C22×C11—22no—
296C22×C11—22no—
297C22×C11—24no—
298C22×C11—22no—
299C22×C11—22no—
300C22×C11—44no—
301C22×C11—48no—
302C22×C11—48no—
303C22×C11—44no—
304C22×C11—44no—
305C22×C11—44no—
306C22×C11—48no—
307C22×C11—48no—
308C22×C11—42no—
309C22×C11—42no—
310C22×C11#40714yes—
311C22×C11C2×#9912yes—
312C22×C11#41312yes—
313C22×C11C22×C22⋊C411yescharacteristic
314C22×C11C22×M1612yes—
315C4×C11—24no—
316C4×C11—21no—
317C4×C11—21no—
318C4×C11—22no—
319C4×C11—22no—
320C4×C11—42no—
321C4×C11—42no—
322C4×C11C2×#9912yes—
323C4×C11C2×#9912yes—
324C25C2×Dic1111yescharacteristic
325C25C2×Dic1111yescharacteristic
326C22×D4—22no—
327C22×D4—22no—
328C22×D4—28no—
329C22×D4C2×Dic1112yes—
330C22×D4C2×Dic1112yes—
331C23×C4—21no—
332C23×C4—21no—
333C23×C4—21no—
334C23×C4—21no—
335C23×C4—44no—
336C23×C4—44no—
337C23×C4C2×Dic1111yescharacteristic
338C23×C4C2×Dic1111yescharacteristic
339C23×C4C2×Dic1111yescharacteristic
340C23×C4C2×Dic1111yescharacteristic
341C23×C4C2×Dic1112yes—
342C23×C4C2×Dic1112yes—
343C23×C4C2×Dic1111yescharacteristic
344C23×C4C2×Dic1111yescharacteristic
345C23×C4C22⋊D1112yes—
346C23×C4C22⋊D1112yes—
347C23×C4C22⋊D1112yes—
348C23×C4C22⋊D1112yes—
349C23×C4C22×D1111yescharacteristic
350C23×C4C22×D1111yescharacteristic
351C23×C4C22×D1111yescharacteristic
352C2×M16—224no—
353C2×M16—224no—
354C2×M16—448no—
355C22×C8—224no—
356C4×D4—24no—
357C4×D4—24no—
358C4×D4—24no—
359C4×D4—24no—
360C4×D4—24no—
361C4×D4—24no—
362C4×D4—24no—
363C4×D4—24no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4C2×Dic1112yes—
367C2×C4⋊C4C22×D1112yes—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4C2×Dic1112yes—
373C2×C22⋊C4C2×Dic1112yes—
374C2×C22⋊C4C2×Dic1112yes—
375C2×C22⋊C4C22×D1112yes—
376C2×C42—21no—
377C2×C42—21no—
378C2×C42—21no—
379C2×C42—21no—
380C2×C42C2×Dic1112yes—
381C2×C42C2×Dic1112yes—
382C22⋊C8—118no—
383C22⋊C8—224no—
384C22⋊C8—224no—
385C22⋊C8—224no—
386C2×C11—22no—
387C2×C11—22no—
388C2×C11—22no—
389C2×C11—24no—
390C2×C11—42no—
391C2×C11—44no—
392C2×C11C2×#40711yescharacteristic
393C2×C11C22×#9911yescharacteristic
394C2×C11C2×#41311yescharacteristic
395C24—24no—
396C24—24no—
397C24—21no—
398C24—21no—
399C24—42no—
400C24—42no—
401C24—42no—
402C24—42no—
403C24C22⋊Dic1112yes—
404C24C22⋊Dic1112yes—
405C24C22×Dic1111yescharacteristic
406C24C22×Dic1112yes—
407C24C22×Dic1111yescharacteristic
408C24C22×Dic1111yescharacteristic
409C2×D4—216no—
410C2×D4—216no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4#431818yescomplemented
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—28no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—42no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4C22⋊Dic1111yescharacteristic
447C22×C4C22⋊Dic1111yescharacteristic
448C22×C4C22⋊Dic1112yes—
449C22×C4C22⋊Dic1111yescharacteristic
450C22×C4C22⋊Dic1112yes—
451C22×C4C22⋊Dic1112yes—
452C22×C4C22⋊Dic1112yes—
453C22×C4C22⋊Dic1111yescharacteristic
454C22×C4C22×Dic1112yes—
455C22×C4C22×Dic1111yescharacteristic
456C22×C4C22×Dic1111yescharacteristic
457C22×C4C22×Dic1112yes—
458C22×C4C22×Dic1112yes—
459C22×C4C22×Dic1111yescharacteristic
460C22×C4C2×C22⋊D1112yes—
461C22×C4C2×C22⋊D1112yes—
462C22×C4C2×C22⋊D1112yes—
463C22×C4C2×C22⋊D1112yes—
464C22×C4C2×C22⋊D1112yes—
465C22×C4C2×C22⋊D1112yes—
466C22×C4C23×D1111yescharacteristic
467M16—448no—
468M16—448no—
469C2×C8—228no—
470C2×C8—448no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C22⋊C4—24no—
474C22⋊C4—24no—
475C22⋊C4—24no—
476C22⋊C4—24no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—22no—
481C11#1736211yescharacteristic, complemented, Sylow
482C23—22no—
483C23—22no—
484C23—22no—
485C23—24no—
486C23—24no—
487C23—22no—
488C23—22no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—22no—
494C23—42no—
495C23—41no—
496C23—41no—
497C23—42no—
498C23—41no—
499C23—41no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—416no—
504C23—416no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23C2×#431814yes—
509C23C2×#431814yescomplemented
510C23C2×C22⋊Dic1112yes—
511C23C2×C22⋊Dic1112yes—
512C23C2×C22⋊Dic1111yescharacteristic
513C23C2×C22⋊Dic1111yescharacteristic
514C23C23×Dic1111yescharacteristic
515D4—216no—
516D4—216no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4#6689814yes—
551C2×C4C2×#431814yescomplemented
552C2×C4C2×C22⋊Dic1111yescharacteristic
553C2×C4C2×C22⋊Dic1111yescharacteristic
554C2×C4C2×C22⋊Dic1112yes—
555C2×C4C2×C22⋊Dic1111yescharacteristic
556C2×C4C2×C22⋊Dic1111yescharacteristic
557C2×C4C22×C22⋊D1112yes—
558C8—448no—
559C22—28no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—22no—
565C22—21no—
566C22—21no—
567C22—22no—
568C22—24no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—44no—
573C22—48no—
574C22—48no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—48no—
579C22—48no—
580C22—42no—
581C22—42no—
582C22#1062282014yes—
583C22C2×#6689812yes—
584C22#1062285012yes—
585C22C22×#431812yes—
586C22C22×C22⋊Dic1111yescharacteristic
587C4—24no—
588C4—22no—
589C4—21no—
590C4—21no—
591C4—22no—
592C4—42no—
593C4—42no—
594C4C2×#6689812yes—
595C4C2×#6689812yes—
596C2—22no—
597C2—22no—
598C2—22no—
599C2—24no—
600C2—42no—
601C2—44no—
602C2C2×#1062282011yescharacteristic
603C2C22×#6689811yescharacteristic
604C2C2×#1062285011yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊Dic11
commutatorC23×C11C23×C4
FrattiniC22×C4C23×D11
FittingC22×C4×D4×C11C2
socleC22×C11C22×C22⋊C4
prime radicalC23×D4×C11C4
cosocleC22×C4×C11C24
hypercenterC22×C4×D4D11
hypocenterC11#17362
LCSG ▷ C23×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#10622820C2semidirect
2. #13066954C2semidirect
3. #13066946C2semidirect
4. #13066943C2semidirect
5. #13066934C2semidirect
26. #10622820C22semidirect
28. C2×C22⋊(C11⋊C8)C22semidirect
29. #10622166C22semidirect
30. #10622163C22semidirect
115. C2×D4×C11M16semidirect
217. C23×C11C2×M16semidirect
260. C2×C4×C11C2×M16semidirect
416. C2×D4#4318semidirect
481. C11#17362semidirect
509. C23C2×#4318semidirect
551. C2×C4C2×#4318semidirect

Maximal subgroups

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

subgroupindex
C2×#106228202
#130669542
#130669462
#130669432
#130669342
C22×C4×D4×C112
#1736211