Finite Group Databasebeta

← #14759278

Subgroups — #14759278

Order 2816.

1598 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#4318C211yesmaximal, characteristic, complemented
2C2×#10622624C212yesmaximal, complemented
3C2×D4×C11⋊C8C211yesmaximal, characteristic, complemented
4#13068914C212yesmaximal, complemented
5#13068909C211yesmaximal, characteristic, complemented
6#13068801C212yesmaximal, complemented
7#13068796C211yesmaximal, characteristic, complemented
8#13068526C211yesmaximal, characteristic, complemented
9#13068400C212yesmaximal, complemented
10#13068394C211yesmaximal, characteristic, complemented
11C4×C11×#120C211yesmaximal, characteristic
12C24×C4×C11C2211yescharacteristic
13C2×C11×#120C411yescharacteristic
14C2×C4×D4×C11C2211yescharacteristic
15C2×C4×D4×C11C2212yes—
16C22×#4318—28no—
17C22×#4318—24no—
18C22×#4318C2212yes—
19C22×#4318C2212yes—
20C23×C11⋊C8C2211yescharacteristic
21C23×C11⋊C8C2211yescharacteristic
22C23×C11⋊C8C2211yescharacteristic, complemented
23C11×#386C411yescharacteristic
24C11×#386C412yes—
25C11×#385C411yescharacteristic
26C11×#385C412yes—
27C11×#379C411yescharacteristic
28C4×C11×C22⋊C4C2212yes—
29C4×C11×C22⋊C4C2211yescharacteristic
30C2×C22⋊(C11⋊C8)C2212yes—
31C2×C22⋊(C11⋊C8)C2211yescharacteristic
32C2×C22⋊(C11⋊C8)C2211yescharacteristic
33C2×C22⋊(C11⋊C8)C2212yes—
34C2×C22⋊(C11⋊C8)C2211yescharacteristic, complemented
35C2×C22⋊(C11⋊C8)C2212yescomplemented
36#10622624—28no—
37D4×C11⋊C8—24no—
38C2×C4⋊(C11⋊C8)C2211yescharacteristic, complemented
39C2×C4⋊(C11⋊C8)C2212yescomplemented
40C2×#66867C2212yescomplemented
41C2×C4×C11⋊C8C2211yescharacteristic, complemented
42#10622162C2212yescomplemented
43#10622162C2212yescomplemented
44#10622162C2212yescomplemented
45C25×C11C2×C411yescharacteristic
46C22×D4×C11C2×C412yes—
47C22×D4×C11C2×C411yescharacteristic
48C23×C4×C11—24no—
49C23×C4×C11—22no—
50C23×C4×C11—24no—
51C23×C4×C11—22no—
52C23×C4×C11C2×C412yes—
53C23×C4×C11C2×C411yescharacteristic
54C23×C4×C11C2311yescharacteristic
55C23×C4×C11C2311yescharacteristic
56C23×C4×C11C2312yes—
57C11×#120—24no—
58C4×D4×C11—22no—
59C4×D4×C11—24no—
60C4×D4×C11—22no—
61C4×D4×C11—24no—
62C2×C11×C4⋊C4C2×C412yes—
63C2×C11×C4⋊C4C2×C411yescharacteristic
64C2×C11×C22⋊C4C2×C411yescharacteristic
65C2×C11×C22⋊C4C2×C411yescharacteristic
66C2×C11×C22⋊C4C2×C412yes—
67C2×C11×C22⋊C4C2×C412yes—
68C2×C11×C22⋊C4C2×C411yescharacteristic
69C2×C11×C22⋊C4C2×C411yescharacteristic
70C2×C11×C22⋊C4C2×C412yes—
71C2×C11×C22⋊C4C2×C412yes—
72C2×C42×C11C2311yescharacteristic
73C2×C42×C11C2312yes—
74C2×#4318—232no—
75C2×#4318—24no—
76C2×#4318—24no—
77C2×#4318—24no—
78C2×#4318—24no—
79C2×#4318—24no—
80C2×#4318D418yescomplemented
81C22×C11⋊C8—22no—
82C22×C11⋊C8—24no—
83C22×C11⋊C8—24no—
84C22×C11⋊C8—22no—
85C22×C11⋊C8—24no—
86C22×C11⋊C8D414yescomplemented
87C22×C11⋊C8C2311yescharacteristic
88C22×C11⋊C8C2312yes—
89C22×C11⋊C8C2311yescharacteristic
90C22×C11⋊C8C2312yes—
91C22×C11⋊C8C2311yescharacteristic
92C22×C11⋊C8C2311yescharacteristic
93C11×#95C2×C412yes—
94C11×#95C2×C412yes—
95C11×#95C2×C412yes—
96C22⋊(C11⋊C8)—22no—
97C22⋊(C11⋊C8)—24no—
98C22⋊(C11⋊C8)—22no—
99C22⋊(C11⋊C8)—24no—
100C22⋊(C11⋊C8)—22no—
101C22⋊(C11⋊C8)—24no—
102C4⋊(C11⋊C8)—22no—
103C4⋊(C11⋊C8)—24no—
104#66867—24no—
105C4×C11⋊C8—22no—
106#21439—111nomaximal, Sylow
107C24×C11—24no—
108C24×C11—22no—
109C24×C11—22no—
110C24×C11—22no—
111C24×C11—42no—
112C24×C11C22×C412yes—
113C24×C11C22×C411yescharacteristic
114C24×C11C22×C411yescharacteristic
115C2×D4×C11—22no—
116C2×D4×C11—22no—
117C2×D4×C11—24no—
118C2×D4×C11—24no—
119C2×D4×C11—42no—
120C2×D4×C11M1614yescomplemented
121C22×C4×C11—28no—
122C22×C4×C11—28no—
123C22×C4×C11—216no—
124C22×C4×C11—24no—
125C22×C4×C11—22no—
126C22×C4×C11—24no—
127C22×C4×C11—24no—
128C22×C4×C11—24no—
129C22×C4×C11—22no—
130C22×C4×C11—22no—
131C22×C4×C11—22no—
132C22×C4×C11—22no—
133C22×C4×C11—22no—
134C22×C4×C11—22no—
135C22×C4×C11—21no—
136C22×C4×C11—22no—
137C22×C4×C11—21no—
138C22×C4×C11—22no—
139C22×C4×C11C22×C412yes—
140C22×C4×C11C22×C411yescharacteristic
141C22×C4×C11C22×C412yes—
142C22×C4×C11C22×C411yescharacteristic
143C22×C4×C11C22×C411yescharacteristic
144C22×C4×C11C22×C412yes—
145C22×C4×C11C22×C411yescharacteristic
146C22×C4×C11C2×D412yes—
147C22×C4×C11C2×D414yes—
148C22×C4×C11#4114yes—
149C22×C4×C11#4112yes—
150C22×C4×C11C2411yescharacteristic
151C11×C4⋊C4—22no—
152C11×C4⋊C4—24no—
153C11×C22⋊C4—24no—
154C11×C22⋊C4—22no—
155C11×C22⋊C4—22no—
156C11×C22⋊C4—22no—
157C11×C22⋊C4—22no—
158C11×C22⋊C4—24no—
159C11×C22⋊C4—24no—
160C11×C22⋊C4—42no—
161C42×C11—21no—
162C42×C11—22no—
163C42×C11—21no—
164C42×C11—22no—
165#4318—216no—
166#4318—216no—
167C2×C11⋊C8—28no—
168C2×C11⋊C8—28no—
169C2×C11⋊C8—22no—
170C2×C11⋊C8—22no—
171C2×C11⋊C8—22no—
172C2×C11⋊C8—21no—
173C2×C11⋊C8—22no—
174C2×C11⋊C8—21no—
175C2×C11⋊C8—22no—
176C2×C11⋊C8—22no—
177C2×C11⋊C8—22no—
178C2×C11⋊C8—22no—
179C2×C11⋊C8—44no—
180C2×C11⋊C8C2×D414yes—
181C2×C11⋊C8C2×D412yes—
182C2×C11⋊C8C2×D412yes—
183C2×C11⋊C8C2×D414yes—
184C23×M16—111no—
185C2×#435—112no—
186C2×C8×D4—111no—
187C4×#120D1111yescharacteristic
188#1870—112no—
189#1869—111no—
190#1851—112no—
191#1850—111no—
192#1779—111no—
193#1753—112no—
194#1752—111no—
195C23×C11—22no—
196C23×C11—22no—
197C23×C11—28no—
198C23×C11—21no—
199C23×C11—22no—
200C23×C11—22no—
201C23×C11—22no—
202C23×C11—22no—
203C23×C11—21no—
204C23×C11—22no—
205C23×C11—22no—
206C23×C11—22no—
207C23×C11—22no—
208C23×C11—48no—
209C23×C11—48no—
210C23×C11—44no—
211C23×C11—44no—
212C23×C11C4×D414yes—
213C23×C11C4×D412yes—
214C23×C11C2×M1612yes—
215C23×C11C2×M1612yescomplemented
216C23×C11#13112yes—
217C23×C11C23×C411yescharacteristic
218D4×C11—28no—
219D4×C11—48no—
220C2×C4×C11—28no—
221C2×C4×C11—216no—
222C2×C4×C11—21no—
223C2×C4×C11—22no—
224C2×C4×C11—22no—
225C2×C4×C11—22no—
226C2×C4×C11—21no—
227C2×C4×C11—22no—
228C2×C4×C11—24no—
229C2×C4×C11—22no—
230C2×C4×C11—24no—
231C2×C4×C11—24no—
232C2×C4×C11—24no—
233C2×C4×C11—24no—
234C2×C4×C11—21no—
235C2×C4×C11—22no—
236C2×C4×C11—22no—
237C2×C4×C11—21no—
238C2×C4×C11—22no—
239C2×C4×C11—22no—
240C2×C4×C11—22no—
241C2×C4×C11—22no—
242C2×C4×C11—22no—
243C2×C4×C11—22no—
244C2×C4×C11—41no—
245C2×C4×C11—41no—
246C2×C4×C11—42no—
247C2×C4×C11C4×D412yes—
248C2×C4×C11C4×D414yes—
249C2×C4×C11C2×M1612yescomplemented
250C2×C4×C11C22×D411yescharacteristic
251C2×C4×C11C22×D412yes—
252C2×C4×C11C2×#4111yescharacteristic
253C2×C4×C11C2×#4112yes—
254C11⋊C8—24no—
255C11⋊C8—22no—
256C11⋊C8—41no—
257C11⋊C8—41no—
258C11⋊C8—42no—
259C11⋊C8#12014yescomplemented
260C24×C4C2×D1111yescharacteristic
261C22×M16—112no—
262C22×M16—112no—
263C22×M16—224no—
264C22×M16—228no—
265C23×C8—111no—
266C23×C8—111no—
267C23×C8—111no—
268C2×#120Dic1111yescharacteristic
269C2×C4×D4C2×D1111yescharacteristic
270C2×C4×D4C2×D1112yes—
271#435—228no—
272C8×D4—224no—
273C2×C4⋊C8—112no—
274C2×C4⋊C8—111no—
275C2×C22⋊C8—111no—
276C2×C22⋊C8—112no—
277C2×C22⋊C8—112no—
278C2×C22⋊C8—111no—
279C2×C22⋊C8—111no—
280C2×C22⋊C8—112no—
281C2×#97—112no—
282C2×C4×C8—111no—
283#386Dic1111yescharacteristic
284#386Dic1112yes—
285#385Dic1112yes—
286#385Dic1111yescharacteristic
287#379Dic1111yescharacteristic
288C4×C22⋊C4C2×D1111yescharacteristic
289C4×C22⋊C4C2×D1112yes—
290#336—112no—
291#336—112no—
292#336—112no—
293C22×C11—22no—
294C22×C11—24no—
295C22×C11—24no—
296C22×C11—22no—
297C22×C11—22no—
298C22×C11—24no—
299C22×C11—24no—
300C22×C11—24no—
301C22×C11—24no—
302C22×C11—416no—
303C22×C11—42no—
304C22×C11—41no—
305C22×C11—42no—
306C22×C11—42no—
307C22×C11—42no—
308C22×C11—42no—
309C22×C11—41no—
310C22×C11#43514yescomplemented
311C22×C11C2×C4×D411yescharacteristic
312C22×C11C2×C4×D412yes—
313C22×C11C22×M1611yescharacteristic
314C22×C11C2×#13111yescharacteristic
315C22×C11#56812yes—
316C4×C11—22no—
317C4×C11—22no—
318C4×C11—24no—
319C4×C11—41no—
320C4×C11—42no—
321C4×C11—42no—
322C4×C11—41no—
323C4×C11C2×#12011yescharacteristic
324C4×C11#52511yescharacteristic
325C4×C11#52512yes—
326C25C2×Dic1111yescharacteristic
327C22×D4C2×Dic1112yes—
328C22×D4C2×Dic1111yescharacteristic
329C23×C4—24no—
330C23×C4—22no—
331C23×C4—24no—
332C23×C4—22no—
333C23×C4C2×Dic1112yes—
334C23×C4C2×Dic1111yescharacteristic
335C23×C4C22×D1111yescharacteristic
336C23×C4C22×D1111yescharacteristic
337C23×C4C22×D1112yes—
338C2×M16—118no—
339C2×M16—224no—
340C2×M16—224no—
341C2×M16—224no—
342C2×M16—224no—
343C2×M16—2232no—
344C2×M16—224no—
345C22×C8—111no—
346C22×C8—114no—
347C22×C8—111no—
348C22×C8—112no—
349C22×C8—111no—
350C22×C8—111no—
351C22×C8—112no—
352C22×C8—224no—
353C22×C8—224no—
354C22×C8—222no—
355C22×C8—224no—
356C22×C8—222no—
357#120—24no—
358C4×D4—22no—
359C4×D4—24no—
360C4×D4—22no—
361C4×D4—24no—
362C2×C4⋊C4C2×Dic1112yes—
363C2×C4⋊C4C2×Dic1111yescharacteristic
364C2×C22⋊C4C2×Dic1112yes—
365C2×C22⋊C4C2×Dic1111yescharacteristic
366C2×C22⋊C4C2×Dic1111yescharacteristic
367C2×C22⋊C4C2×Dic1112yes—
368C2×C22⋊C4C2×Dic1111yescharacteristic
369C2×C22⋊C4C2×Dic1112yes—
370C2×C22⋊C4C2×Dic1111yescharacteristic
371C2×C22⋊C4C2×Dic1112yes—
372C2×C42C22×D1111yescharacteristic
373C2×C42C22×D1112yes—
374C4⋊C8—222no—
375C4⋊C8—224no—
376C22⋊C8—222no—
377C22⋊C8—224no—
378C22⋊C8—222no—
379C22⋊C8—224no—
380C22⋊C8—222no—
381C22⋊C8—224no—
382#97—224no—
383C4×C8—222no—
384#95C2×Dic1112yes—
385#95C2×Dic1112yes—
386#95C2×Dic1112yes—
387C2×C11—24no—
388C2×C11—42no—
389C2×C11—44no—
390C2×C11C4×#12011yescharacteristic
391C2×C11C2×#43511yescharacteristic
392C2×C11#293011yescharacteristic
393C2×C11#293312yes—
394C2×C11D4×M1612yes—
395C24—24no—
396C24—22no—
397C24—22no—
398C24—22no—
399C24—42no—
400C24C22×Dic1111yescharacteristic
401C24C22×Dic1112yes—
402C24C22×Dic1111yescharacteristic
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—42no—
408C2×D4#431814yescomplemented
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—216no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4D4×D1112yes—
428C22×C4D4×D1114yes—
429C22×C4#434114yes—
430C22×C4#434112yes—
431C22×C4C22×Dic1112yes—
432C22×C4C22×Dic1111yescharacteristic
433C22×C4C22×Dic1112yes—
434C22×C4C22×Dic1111yescharacteristic
435C22×C4C22×Dic1111yescharacteristic
436C22×C4C22×Dic1111yescharacteristic
437C22×C4C22×Dic1112yes—
438C22×C4C23×D1111yescharacteristic
439M16—2216no—
440M16—2216no—
441C2×C8—112no—
442C2×C8—114no—
443C2×C8—112no—
444C2×C8—114no—
445C2×C8—228no—
446C2×C8—228no—
447C2×C8—222no—
448C2×C8—222no—
449C2×C8—222no—
450C2×C8—221no—
451C2×C8—222no—
452C2×C8—221no—
453C2×C8—222no—
454C2×C8—222no—
455C2×C8—222no—
456C2×C8—222no—
457C2×C8—444no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C22⋊C4—22no—
461C22⋊C4—24no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—24no—
465C22⋊C4—22no—
466C22⋊C4—24no—
467C22⋊C4—42no—
468C42—21no—
469C42—22no—
470C42—21no—
471C42—22no—
472C11#2143911yescharacteristic, complemented, Sylow
473C23—22no—
474C23—22no—
475C23—28no—
476C23—22no—
477C23—21no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—21no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—48no—
487C23—48no—
488C23—44no—
489C23—44no—
490C23C2×#431812yes—
491C23C2×#431812yescomplemented
492C23D4×Dic1114yes—
493C23D4×Dic1112yes—
494C23#6700112yes—
495C23C23×Dic1111yescharacteristic
496D4—28no—
497D4—48no—
498C2×C4—28no—
499C2×C4—216no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4C2×#431812yescomplemented
526C2×C4D4×Dic1112yes—
527C2×C4D4×Dic1114yes—
528C2×C4C2×D4×D1111yescharacteristic
529C2×C4C2×D4×D1112yes—
530C2×C4C2×#434111yescharacteristic
531C2×C4C2×#434112yes—
532C8—114no—
533C8—222no—
534C8—224no—
535C8—441no—
536C8—442no—
537C8—441no—
538C22—22no—
539C22—22no—
540C22—24no—
541C22—24no—
542C22—24no—
543C22—22no—
544C22—24no—
545C22—24no—
546C22—24no—
547C22—416no—
548C22—41no—
549C22—42no—
550C22—42no—
551C22—41no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22#1062262414yescomplemented
556C22C22×#431811yescharacteristic
557C22C2×D4×Dic1111yescharacteristic
558C22C2×D4×Dic1112yes—
559C22C2×#6700111yescharacteristic
560C22#1062335012yes—
561C4—22no—
562C4—24no—
563C4—22no—
564C4—41no—
565C4—41no—
566C4—42no—
567C4—42no—
568C4#1062311711yescharacteristic
569C4#1062311712yes—
570C4D11×#12011yescharacteristic
571C2—24no—
572C2—44no—
573C2—42no—
574C2#120×Dic1111yescharacteristic
575C2C2×#1062262411yescharacteristic
576C2#1307732011yescharacteristic
577C2#1307734312yes—
578C2D4×#431812yes—
subgroupG/H
center, FrattiniC22×C4C23×D11
commutator, socleC23×C11C23×C4
FittingC4×C11×#120C2
prime radicalC2×C11×#120C4
cosocleC22×C4×C11C24
hypercenterC4×#120D11
hypocenterC11#21439
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C22×C4 ◁ C4×#120

Complements

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

NH ≅ G/Ntype
1. C23×#4318C2semidirect
2. C2×#10622624C2semidirect
3. C2×D4×C11⋊C8C2semidirect
4. #13068914C2semidirect
5. #13068909C2semidirect
6. #13068801C2semidirect
7. #13068796C2semidirect
8. #13068526C2semidirect
9. #13068400C2semidirect
10. #13068394C2semidirect
22. C23×C11⋊C8C22semidirect
34. C2×C22⋊(C11⋊C8)C22semidirect
35. C2×C22⋊(C11⋊C8)C22semidirect
38. C2×C4⋊(C11⋊C8)C22semidirect
39. C2×C4⋊(C11⋊C8)C22semidirect
40. C2×#66867C22semidirect
41. C2×C4×C11⋊C8C22semidirect
42. #10622162C22semidirect
43. #10622162C22semidirect
44. #10622162C22semidirect
80. C2×#4318D4semidirect
86. C22×C11⋊C8D4semidirect
120. C2×D4×C11M16semidirect
215. C23×C11C2×M16semidirect
249. C2×C4×C11C2×M16semidirect
259. C11⋊C8#120semidirect
310. C22×C11#435semidirect
408. C2×D4#4318semidirect
472. C11#21439semidirect
491. C23C2×#4318semidirect
525. C2×C4C2×#4318semidirect
555. C22#10622624semidirect

Maximal subgroups

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

subgroupindex
C23×#43182
C2×#106226242
C2×D4×C11⋊C82
#130689142
#130689092
#130688012
#130687962
#130685262
#130684002
#130683942
C4×C11×#1202
#2143911