Finite Group Databasebeta

← #14914101

Subgroups — #14914101

Order 2816.

3038 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10623343C211yesmaximal, characteristic, complemented
2C22×#66979C211yesmaximal, characteristic
3#13071647C214yesmaximal, complemented
4#13071625C214yesmaximal, complemented
5#13071485C214yesmaximal, complemented
6#13071465C214yesmaximal
7#13071300C214yesmaximal, complemented
8#13071234C211yesmaximal, characteristic, complemented
9#13071202C211yesmaximal, characteristic, complemented
10C22⋊Q8×Dic11C214yesmaximal, complemented
11#13070999C211yesmaximal, characteristic
12#13070973C211yesmaximal, characteristic, complemented
13C22×C11×C22⋊Q8C211yesmaximal, characteristic
14C23×Q8×C11C2211yescharacteristic
15C24×C4×C11C2211yescharacteristic
16C2×C11×C22⋊Q8C4116yescomplemented
17C2×C11×C22⋊Q8C2212yes—
18C2×C11×C22⋊Q8C2212yes—
19C2×C11×C22⋊Q8C2212yes—
20C2×C11×C22⋊Q8C2212yes—
21C22×C11×C4⋊C4C2211yescharacteristic
22C22×C11×C4⋊C4C2212yes—
23C22×C11×C22⋊C4C2212yes—
24C22×C22⋊Dic11C2212yes—
25#10623343—28no—
26C2×Q8×Dic11C2214yescomplemented
27C2×Q8×Dic11C2214yescomplemented
28C2×#66979—28no—
29C2×#66979C2212yes—
30C2×#66979C2211yescharacteristic
31C2×#66979C2211yescharacteristic
32C2×#66979C2214yes—
33C2×#66979C2212yes—
34C2×#66979C2211yescharacteristic
35C2×#66979C2212yes—
36C2×#66979C2211yescharacteristic
37C22×C4⋊Dic11C2212yes—
38C22×C4×Dic11C2212yes—
39#10622860C2212yes—
40#10622843C2214yescomplemented
41#10622843C2214yescomplemented
42#10622825C2212yes—
43#10622825C2214yes—
44#10622825C2212yes—
45#10622824C2214yes—
46#10622824C2212yes—
47C4×C22⋊Dic11C2212yes—
48C4×C22⋊Dic11C2212yes—
49C4×C22⋊Dic11C2214yes—
50#10622598C2214yescomplemented
51#10622598C2218yescomplemented
52#10622598C2214yescomplemented
53#10622594C2218yescomplemented
54#10622594C2218yescomplemented
55C4⋊C4×Dic11C2214yescomplemented
56C4⋊C4×Dic11C2218yescomplemented
57C4⋊C4×Dic11C2214yescomplemented
58#10622564C2218yes—
59#10622564C2218yes—
60#10622563C2218yes—
61C22⋊C4×Dic11C2218yes—
62C25×C11C2311yescharacteristic
63C22×Q8×C11—24no—
64C22×Q8×C11—24no—
65C22×Q8×C11C2×C418yescomplemented
66C22×Q8×C11C2312yes—
67C22×Q8×C11C2312yes—
68C23×C4×C11—24no—
69C23×C4×C11—24no—
70C23×C4×C11C2×C418yes—
71C23×C4×C11C2311yescharacteristic
72C23×C4×C11C2311yescharacteristic
73C23×C4×C11C2311yescharacteristic
74C23×C4×C11C2312yes—
75C23×C4×C11C2311yescharacteristic
76C23×C4×C11C2311yescharacteristic
77C23×C4×C11C2312yes—
78C23×C4×C11C2311yescharacteristic
79C23×Dic11C2314yes—
80C11×C22⋊Q8C2×C4116yes—
81C11×C22⋊Q8C2×C4116yes—
82C11×C22⋊Q8C2×C4116yescomplemented
83C11×C22⋊Q8C2×C4116yescomplemented
84C2×C11×C4⋊C4C2×C4116yescomplemented
85C2×C11×C4⋊C4C2×C418yescomplemented
86C2×C11×C4⋊C4C2314yes—
87C2×C11×C4⋊C4C2312yes—
88C2×C11×C4⋊C4C2312yes—
89C2×C11×C4⋊C4C2314yes—
90C2×C11×C22⋊C4C2×C4116yes—
91C2×C11×C22⋊C4C2314yes—
92C2×C11×C22⋊C4C2314yes—
93C2×C22⋊Dic11—28no—
94C2×C22⋊Dic11C2312yes—
95C2×C22⋊Dic11C2314yes—
96C2×C22⋊Dic11C2312yes—
97C2×C22⋊Dic11C2312yes—
98C2×C22⋊Dic11C2314yes—
99C2×C22⋊Dic11C2314yes—
100C2×C22⋊Dic11C2312yes—
101Q8×Dic11—216no—
102Q8×Dic11—216no—
103#66979—24no—
104#66979—24no—
105#66979—24no—
106#66979—28no—
107#66979—216no—
108#66979—28no—
109#66979—28no—
110#66979D418yes—
111C2×C4⋊Dic11—28no—
112C2×C4⋊Dic11C2312yes—
113C2×C4⋊Dic11C2314yes—
114C2×C4⋊Dic11C2318yes—
115C2×C4⋊Dic11C2312yes—
116C2×C4⋊Dic11C2312yes—
117C2×C4⋊Dic11C2318yes—
118C2×C4⋊Dic11C2312yes—
119C2×C4×Dic11—28no—
120C2×C4×Dic11C2312yes—
121C2×C4×Dic11C2314yes—
122C2×C4×Dic11C2318yes—
123C2×C4×Dic11C2312yes—
124C2×C4×Dic11C2312yes—
125C2×C4×Dic11C2318yes—
126C2×C4×Dic11C2312yes—
127C11⋊#95C2318yes—
128C11⋊#95C2318yes—
129C11⋊#95C2318yes—
130C11⋊#95C2318yes—
131C11⋊#95C2314yes—
132C11⋊#95C2314yes—
133#34203—111nomaximal, Sylow
134C24×C11—22no—
135C24×C11—22no—
136C24×C11—24no—
137C24×C11—42no—
138C24×C11C22×C414yes—
139C24×C11C2411yescharacteristic
140C24×C11C2411yescharacteristic
141C24×C11C2411yescharacteristic
142C2×Q8×C11—232no—
143C2×Q8×C11—24no—
144C2×Q8×C11—24no—
145C2×Q8×C11—24no—
146C2×Q8×C11—24no—
147C2×Q8×C11C22×C418yes—
148C2×Q8×C11C22×C418yes—
149C22×C4×C11—22no—
150C22×C4×C11—24no—
151C22×C4×C11—22no—
152C22×C4×C11—24no—
153C22×C4×C11—232no—
154C22×C4×C11—22no—
155C22×C4×C11—22no—
156C22×C4×C11—22no—
157C22×C4×C11—22no—
158C22×C4×C11—22no—
159C22×C4×C11—22no—
160C22×C4×C11—22no—
161C22×C4×C11—22no—
162C22×C4×C11—44no—
163C22×C4×C11C22×C418yes—
164C22×C4×C11C22×C414yes—
165C22×C4×C11C22×C414yes—
166C22×C4×C11C22×C414yes—
167C22×C4×C11C22×C414yes—
168C22×C4×C11C22×C414yes—
169C22×C4×C11C22×C418yes—
170C22×C4×C11C22×C414yes—
171C22×C4×C11C2×D414yes—
172C22×C4×C11#4114yes—
173C22×C4×C11C2411yescharacteristic
174C22×C4×C11C2414yes—
175C22×C4×C11C2414yes—
176C22×C4×C11C2411yescharacteristic
177C22×C4×C11C2411yescharacteristic
178C22×C4×C11C2411yescharacteristic
179C22×Dic11—28no—
180C22×Dic11—28no—
181C22×Dic11—44no—
182C22×Dic11C2414yes—
183C22×Dic11C2414yes—
184C22×Dic11C2414yes—
185C22×Dic11C2414yes—
186C11×C4⋊C4C22×C418yes—
187C11×C4⋊C4C22×C418yes—
188C11×C4⋊C4C22×C4116yes—
189C11×C4⋊C4C22×C4116yes—
190C11×C22⋊C4C22×C4116yes—
191C11×C22⋊C4C22×C4116yes—
192C22⋊Dic11—28no—
193C22⋊Dic11—24no—
194C22⋊Dic11—24no—
195C22⋊Dic11—24no—
196C22⋊Dic11—28no—
197C22⋊Dic11—44no—
198C22⋊Dic11C2×D418yes—
199C4⋊Dic11—28no—
200C4⋊Dic11—216no—
201C4⋊Dic11—24no—
202C4⋊Dic11—24no—
203C4⋊Dic11—24no—
204C4⋊Dic11—48no—
205C4⋊Dic11C2×D418yes—
206C4×Dic11—28no—
207C4×Dic11—216no—
208C4×Dic11—24no—
209C4×Dic11—24no—
210C4×Dic11—24no—
211C4×Dic11—48no—
212C4×Dic11C2×D418yes—
213C22×C22⋊Q8D1111yescharacteristic
214C2×#519—111no—
215C22×#117—111no—
216#2362—114no—
217#2360—114no—
218#2344—114no—
219#2342—114no—
220#2323—114no—
221#2316—111no—
222#2310—111no—
223C4×C22⋊Q8—114no—
224#2280—111no—
225#2274—111no—
226C23×C11—21no—
227C23×C11—21no—
228C23×C11—22no—
229C23×C11—22no—
230C23×C11—22no—
231C23×C11—28no—
232C23×C11—28no—
233C23×C11—21no—
234C23×C11—21no—
235C23×C11—22no—
236C23×C11—22no—
237C23×C11—44no—
238C23×C11—48no—
239C23×C11—48no—
240C23×C11C23×C412yes—
241C23×C11C23×C412yes—
242C23×C11C23×C412yes—
243C23×C11C22×D412yes—
244C23×C11C2×#4112yes—
245C23×C112−1+414yes—
246C23×C11C2511yescharacteristic
247Q8×C11—216no—
248Q8×C11—216no—
249C2×C4×C11—28no—
250C2×C4×C11—216no—
251C2×C4×C11—21no—
252C2×C4×C11—24no—
253C2×C4×C11—21no—
254C2×C4×C11—21no—
255C2×C4×C11—21no—
256C2×C4×C11—24no—
257C2×C4×C11—28no—
258C2×C4×C11—28no—
259C2×C4×C11—28no—
260C2×C4×C11—28no—
261C2×C4×C11—28no—
262C2×C4×C11—48no—
263C2×C4×C11C4×D4116yes—
264C2×C4×C11C23×C418yes—
265C2×C4×C11C23×C418yes—
266C2×C4×C11C23×C412yes—
267C2×C4×C11C23×C412yes—
268C2×C4×C11C23×C412yes—
269C2×C4×C11C23×C412yes—
270C2×C4×C11C22×D412yes—
271C2×C4×C11C22×D412yes—
272C2×C4×C11C2×#4112yes—
273C2×C4×C11C2×#4112yes—
274C2×Dic11—24no—
275C2×Dic11—24no—
276C2×Dic11—24no—
277C2×Dic11—24no—
278C2×Dic11—24no—
279C2×Dic11—24no—
280C2×Dic11—24no—
281C2×Dic11—24no—
282C2×Dic11—24no—
283C2×Dic11—24no—
284C2×Dic11—24no—
285C2×Dic11—416no—
286C2×Dic11C22×D418yes—
287C23×Q8C2×D1111yescharacteristic
288C24×C4C2×D1111yescharacteristic
289C2×C22⋊Q8Dic11116yescomplemented
290C2×C22⋊Q8C2×D1112yes—
291C2×C22⋊Q8C2×D1112yes—
292C2×C22⋊Q8C2×D1112yes—
293C2×C22⋊Q8C2×D1112yes—
294#519—228no—
295C2×C4×Q8—114no—
296C2×C4×Q8—114no—
297C2×#117—111no—
298C2×#117—111no—
299C2×#117—111no—
300C2×#117—111no—
301C2×#117—112no—
302C2×#117—114no—
303C2×#117—112no—
304C2×#117—112no—
305C2×#117—228no—
306C22×C4⋊C4—112no—
307C22×C4⋊C4C2×D1111yescharacteristic
308C22×C4⋊C4C2×D1112yes—
309C22×C22⋊C4—112no—
310C22×C22⋊C4C2×D1112yes—
311C22×C42—112no—
312#391—114no—
313#391—114no—
314#389—114no—
315#389—114no—
316#389—118no—
317#388—118no—
318#388—118no—
319#387—118no—
320#387—118no—
321#385—118no—
322#381—114no—
323#381—112no—
324#380—114no—
325#380—112no—
326#380—112no—
327#379—112no—
328C4×C4⋊C4—114no—
329C4×C4⋊C4—118no—
330C4×C4⋊C4—114no—
331C4×C22⋊C4—114no—
332C4×C22⋊C4—112no—
333C4×C22⋊C4—112no—
334C4×C22⋊C4—118no—
335C22×C11—22no—
336C22×C11—22no—
337C22×C11—22no—
338C22×C11—22no—
339C22×C11—24no—
340C22×C11—24no—
341C22×C11—24no—
342C22×C11—24no—
343C22×C11—22no—
344C22×C11—22no—
345C22×C11—22no—
346C22×C11—416no—
347C22×C11—42no—
348C22×C11—42no—
349C22×C11C2×C4×D414yes—
350C22×C11#51914yescomplemented
351C22×C11C24×C411yescharacteristic
352C22×C11C23×D411yescharacteristic
353C22×C11C22×#4111yescharacteristic
354C22×C11C2×2−1+412yes—
355C22×C11#58512yes—
356C4×C11—22no—
357C4×C11—22no—
358C4×C11—28no—
359C4×C11—28no—
360C4×C11C2×C4×D414yes—
361C4×C11C2×C4×D414yes—
362Dic11—28no—
363Dic11—44no—
364Dic11—44no—
365Dic11—44no—
366C25C22×D1111yescharacteristic
367C22×Q8—24no—
368C22×Q8—24no—
369C22×Q8C2×Dic1118yescomplemented
370C22×Q8C22×D1112yes—
371C22×Q8C22×D1112yes—
372C23×C4—24no—
373C23×C4—24no—
374C23×C4—114no—
375C23×C4C2×Dic1118yes—
376C23×C4C22×D1112yes—
377C23×C4C22×D1111yescharacteristic
378C23×C4C22×D1111yescharacteristic
379C23×C4C22×D1111yescharacteristic
380C23×C4C22×D1111yescharacteristic
381C23×C4C22×D1111yescharacteristic
382C23×C4C22×D1112yes—
383C23×C4C22×D1111yescharacteristic
384C22⋊Q8C2×Dic11116yes—
385C22⋊Q8C2×Dic11116yes—
386C22⋊Q8C2×Dic11116yescomplemented
387C22⋊Q8C2×Dic11116yescomplemented
388C4×Q8—2216no—
389C4×Q8—2216no—
390#117—118no—
391#117—228no—
392#117—228no—
393#117—2216no—
394#117—228no—
395#117—224no—
396#117—224no—
397#117—224no—
398C2×C4⋊C4—112no—
399C2×C4⋊C4—114no—
400C2×C4⋊C4—112no—
401C2×C4⋊C4—112no—
402C2×C4⋊C4—118no—
403C2×C4⋊C4—112no—
404C2×C4⋊C4—118no—
405C2×C4⋊C4—228no—
406C2×C4⋊C4C2×Dic11116yescomplemented
407C2×C4⋊C4C2×Dic1118yescomplemented
408C2×C4⋊C4C22×D1114yes—
409C2×C4⋊C4C22×D1112yes—
410C2×C4⋊C4C22×D1112yes—
411C2×C4⋊C4C22×D1114yes—
412C2×C22⋊C4—112no—
413C2×C22⋊C4—112no—
414C2×C22⋊C4—114no—
415C2×C22⋊C4—112no—
416C2×C22⋊C4—112no—
417C2×C22⋊C4—114no—
418C2×C22⋊C4—114no—
419C2×C22⋊C4—228no—
420C2×C22⋊C4C2×Dic11116yes—
421C2×C22⋊C4C22×D1114yes—
422C2×C22⋊C4C22×D1114yes—
423C2×C42—112no—
424C2×C42—114no—
425C2×C42—112no—
426C2×C42—112no—
427C2×C42—118no—
428C2×C42—112no—
429C2×C42—118no—
430C2×C42—228no—
431#95—114no—
432#95—118no—
433#95—114no—
434#95—118no—
435#95—118no—
436#95—118no—
437C2×C11—24no—
438C2×C11—24no—
439C2×C11—42no—
440C2×C11C22×C4×D411yescharacteristic
441C2×C11C2×#51911yescharacteristic
442C2×C11#342811yescharacteristic
443C2×C11#348912yes—
444C2×C11#349512yes—
445C24—22no—
446C24—22no—
447C24—24no—
448C24—42no—
449C24C22×Dic1114yes—
450C24C23×D1111yescharacteristic
451C24C23×D1111yescharacteristic
452C24C23×D1111yescharacteristic
453C2×Q8—232no—
454C2×Q8—24no—
455C2×Q8—24no—
456C2×Q8—24no—
457C2×Q8—24no—
458C2×Q8C22×Dic1118yes—
459C2×Q8C22×Dic1118yes—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—232no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—44no—
474C22×C4—114no—
475C22×C4—114no—
476C22×C4—114no—
477C22×C4—114no—
478C22×C4—228no—
479C22×C4—228no—
480C22×C4—444no—
481C22×C4D4×D1114yes—
482C22×C4#434114yes—
483C22×C4C22×Dic1118yes—
484C22×C4C22×Dic1114yes—
485C22×C4C22×Dic1114yes—
486C22×C4C22×Dic1114yes—
487C22×C4C22×Dic1114yes—
488C22×C4C22×Dic1114yes—
489C22×C4C22×Dic1118yes—
490C22×C4C22×Dic1114yes—
491C22×C4C23×D1111yescharacteristic
492C22×C4C23×D1114yes—
493C22×C4C23×D1114yes—
494C22×C4C23×D1111yescharacteristic
495C22×C4C23×D1111yescharacteristic
496C22×C4C23×D1111yescharacteristic
497C4⋊C4—118no—
498C4⋊C4—228no—
499C4⋊C4—2216no—
500C4⋊C4—224no—
501C4⋊C4—224no—
502C4⋊C4—224no—
503C4⋊C4—448no—
504C4⋊C4C22×Dic1118yes—
505C4⋊C4C22×Dic1118yes—
506C4⋊C4C22×Dic11116yes—
507C4⋊C4C22×Dic11116yes—
508C22⋊C4—118no—
509C22⋊C4—228no—
510C22⋊C4—224no—
511C22⋊C4—224no—
512C22⋊C4—224no—
513C22⋊C4—228no—
514C22⋊C4—444no—
515C22⋊C4C22×Dic11116yes—
516C22⋊C4C22×Dic11116yes—
517C42—118no—
518C42—228no—
519C42—2216no—
520C42—224no—
521C42—224no—
522C42—224no—
523C42—448no—
524C11#3420311yescharacteristic, complemented, Sylow
525C23—21no—
526C23—21no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—28no—
531C23—28no—
532C23—21no—
533C23—21no—
534C23—22no—
535C23—22no—
536C23—44no—
537C23—48no—
538C23—48no—
539C23C2×D4×D1112yes—
540C23C2×#434112yes—
541C23#6704014yes—
542C23C23×Dic1112yes—
543C23C23×Dic1112yes—
544C23C23×Dic1112yes—
545C23C24×D1111yescharacteristic
546Q8—216no—
547Q8—216no—
548C2×C4—28no—
549C2×C4—216no—
550C2×C4—21no—
551C2×C4—24no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—48no—
562C2×C4—118no—
563C2×C4—224no—
564C2×C4—224no—
565C2×C4—224no—
566C2×C4—224no—
567C2×C4—224no—
568C2×C4—224no—
569C2×C4—224no—
570C2×C4—224no—
571C2×C4—224no—
572C2×C4—224no—
573C2×C4—224no—
574C2×C4—4416no—
575C2×C4D4×Dic11116yes—
576C2×C4C2×D4×D1112yes—
577C2×C4C2×D4×D1112yes—
578C2×C4C2×#434112yes—
579C2×C4C2×#434112yes—
580C2×C4C23×Dic1118yes—
581C2×C4C23×Dic1118yes—
582C2×C4C23×Dic1112yes—
583C2×C4C23×Dic1112yes—
584C2×C4C23×Dic1112yes—
585C2×C4C23×Dic1112yes—
586C22—24no—
587C22—24no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—24no—
593C22—24no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—416no—
598C22—42no—
599C22—42no—
600C22C2×D4×Dic1114yes—
601C22#1062334314yescomplemented
602C22C22×D4×D1111yescharacteristic
603C22C22×#434111yescharacteristic
604C22C2×#6704012yes—
605C22#1062345312yes—
606C22C24×Dic1111yescharacteristic
607C4—22no—
608C4—22no—
609C4—28no—
610C4—28no—
611C4—228no—
612C4—444no—
613C4—444no—
614C4—444no—
615C4C2×D4×Dic1114yes—
616C4C2×D4×Dic1114yes—
617C2—24no—
618C2—24no—
619C2—42no—
620C2C22×D4×Dic1111yescharacteristic
621C2C2×#1062334311yescharacteristic
622C2#1308181911yescharacteristic
623C2#1308229212yes—
624C2#1308236212yes—
subgroupG/H
center, FrattiniC23C24×D11
commutatorC22×C11C24×C4
FittingC22×C11×C22⋊Q8C2
socle, cosocleC23×C11C25
prime radicalC25×C11C23
hypercenterC22×C22⋊Q8D11
hypocenterC11#34203
LCSG ▷ C22×C11 ▷ C11
UCSC1 ◁ C23 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#10623343C2semidirect
3. #13071647C2semidirect
4. #13071625C2semidirect
5. #13071485C2semidirect
7. #13071300C2semidirect
8. #13071234C2semidirect
9. #13071202C2semidirect
10. C22⋊Q8×Dic11C2semidirect
12. #13070973C2semidirect
16. C2×C11×C22⋊Q8C4semidirect
26. C2×Q8×Dic11C22semidirect
27. C2×Q8×Dic11C22semidirect
40. #10622843C22semidirect
41. #10622843C22semidirect
50. #10622598C22semidirect
51. #10622598C22semidirect
52. #10622598C22semidirect
53. #10622594C22semidirect
54. #10622594C22semidirect
55. C4⋊C4×Dic11C22semidirect
56. C4⋊C4×Dic11C22semidirect
57. C4⋊C4×Dic11C22semidirect
65. C22×Q8×C11C2×C4semidirect
82. C11×C22⋊Q8C2×C4semidirect
83. C11×C22⋊Q8C2×C4semidirect
84. C2×C11×C4⋊C4C2×C4semidirect
85. C2×C11×C4⋊C4C2×C4semidirect
289. C2×C22⋊Q8Dic11semidirect
350. C22×C11#519semidirect
369. C22×Q8C2×Dic11semidirect
386. C22⋊Q8C2×Dic11semidirect
387. C22⋊Q8C2×Dic11semidirect
406. C2×C4⋊C4C2×Dic11semidirect
407. C2×C4⋊C4C2×Dic11semidirect
524. C11#34203semidirect
601. C22#10623343semidirect

Maximal subgroups

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

subgroupindex
C2×#106233432
C22×#669792
#130716472
#130716252
#130714852
#130714652
#130713002
#130712342
#130712022
C22⋊Q8×Dic112
#130709992
#130709732
C22×C11×C22⋊Q82
#3420311