Finite Group Databasebeta

← #14769489

Subgroups — #14769489

Order 2816.

1602 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#4314C211yesmaximal, characteristic
2C2×#10622502C212yesmaximal, complemented
3#13071448C211yesmaximal, characteristic, complemented
4#13071436C211yesmaximal, characteristic
5#13069495C212yesmaximal, complemented
6#13068972C212yesmaximal, complemented
7#13068951C212yesmaximal, complemented
8#13068550C211yesmaximal, characteristic, complemented
9#13068546C211yesmaximal, characteristic
10#13068392C211yesmaximal, characteristic, complemented
11C11×#1808C211yesmaximal, characteristic, complemented
12C23×D44C2211yescharacteristic
13C23×Dic22C2211yescharacteristic
14C23×C8×C11C2211yescharacteristic
15C22×#4314—28no—
16C22×#4314C418yescomplemented
17C22×#4314C2211yescharacteristic
18C22×#4314C2211yescharacteristic
19C22×#4314C2211yescharacteristic
20C22×#4314C2211yescharacteristic
21C2×#66946C2212yes—
22C2×#66940C2212yescomplemented
23C2×C22⋊C4×D11—21no—
24C2×C11×#107C2212yescomplemented
25C11×#380C2211yescharacteristic
26C11×#380C2211yescharacteristic
27C4×C22⋊Dic11C2211yescharacteristic
28C2×C22⋊(C11⋊C8)C2211yescharacteristic
29C2×#66879C2212yescomplemented
30#10622601C2212yes—
31#10622587C2212yescomplemented
32C2×#66874C2212yescomplemented
33C2×#66874C2212yescomplemented
34C2×#66873C2212yes—
35C2×#66873C2212yes—
36#10622568—21no—
37#10622555—21no—
38C22⋊C4×Dic11—21no—
39#10622502—28no—
40#10622278—22no—
41#10622259—22no—
42C11×#340C2212yescomplemented
43#10622188C2212yescomplemented
44C24×D11—21no—
45C23×C4×C11C2311yescharacteristic
46C23×Dic11—21no—
47C22×D44—24no—
48C22×D44—21no—
49C22×D44—22no—
50C22×D44—42no—
51C22×D44C2×C414yes—
52C22×D44C2311yescharacteristic
53C22×D44C2311yescharacteristic
54C22×C4×D11—22no—
55C22×Dic22—24no—
56C22×Dic22—21no—
57C22×Dic22—22no—
58C22×Dic22—42no—
59C22×Dic22C2×C414yescomplemented
60C22×Dic22C2311yescharacteristic
61C22×Dic22C2311yescharacteristic
62C22×C8×C11—24no—
63C22×C8×C11C2×C414yescomplemented
64C22×C8×C11C2311yescharacteristic
65C22×C8×C11C2311yescharacteristic
66C2×C11×C4⋊C4C2312yes—
67C2×C11×C4⋊C4C2312yes—
68C2×C11×C22⋊C4—21no—
69C2×C11×C22⋊C4—21no—
70C2×C22⋊Dic11D411yescharacteristic
71C2×C22⋊Dic11D411yescharacteristic
72C2×(C2×D11)⋊C4—22no—
73C2×(C2×D11)⋊C4—22no—
74C2×(C2×D11)⋊C4—22no—
75C2×Dic11⋊C4—22no—
76C2×C4×Dic11—22no—
77C2×C4×Dic11C2312yes—
78C22×C11⋊C8C2312yes—
79C2×#4314—24no—
80C2×#4314—24no—
81C2×#4314—24no—
82C2×#4314—232no—
83C2×#4314C2×C414yes—
84C2×#4314C2×C414yes—
85C2×#4314C2×C414yescomplemented
86C2×#4314C2×C414yescomplemented
87C2×#4314D418yes—
88#66946—24no—
89#66946—42no—
90#66940—24no—
91#66940—42no—
92C22⋊C4×D11—44no—
93C11×#107—24no—
94C11×#95—21no—
95C11×#95—21no—
96C11⋊#95—22no—
97C11⋊#95—22no—
98C11⋊#95D412yes—
99C22⋊(C11⋊C8)—22no—
100#66879—24no—
101#66874—24no—
102#66874—42no—
103#66873—24no—
104#66873—42no—
105#22437—111nomaximal, Sylow
106C24×C11C2×D411yescharacteristic
107C23×D11—21no—
108C23×D11—21no—
109C23×D11—24no—
110C23×D11—44no—
111C23×D11—42no—
112C23×D11—42no—
113C23×D11—82no—
114C22×C4×C11—22no—
115C22×C4×C11—22no—
116C22×C4×C11—22no—
117C22×C4×C11—22no—
118C22×C4×C11C22×C412yes—
119C22×C4×C11C2×D411yescharacteristic
120C22×C4×C11#4112yes—
121C22×C4×C11C2411yescharacteristic
122C22×Dic11—24no—
123C22×Dic11—21no—
124C22×Dic11—21no—
125C22×Dic11—44no—
126C22×Dic11C2×D412yes—
127C22×Dic11C2×D412yes—
128C2×D44—28no—
129C2×D44—24no—
130C2×D44—22no—
131C2×D44—21no—
132C2×D44—21no—
133C2×D44—22no—
134C2×D44—22no—
135C2×D44—44no—
136C2×D44—44no—
137C2×D44—48no—
138C2×D44C22×C412yes—
139C2×D44C22×C412yes—
140C2×D44C2×D414yes—
141C2×C4×D11—44no—
142C2×C4×D11—44no—
143C2×C4×D11—82no—
144C2×Dic22—28no—
145C2×Dic22—24no—
146C2×Dic22—22no—
147C2×Dic22—21no—
148C2×Dic22—21no—
149C2×Dic22—22no—
150C2×Dic22—22no—
151C2×Dic22—44no—
152C2×Dic22—48no—
153C2×Dic22—44no—
154C2×Dic22C22×C412yes—
155C2×Dic22C22×C412yes—
156C2×Dic22C2×D414yes—
157C2×C8×C11—28no—
158C2×C8×C11—22no—
159C2×C8×C11C22×C412yes—
160C2×C8×C11C22×C412yes—
161C2×C8×C11C2×D414yes—
162C11×C4⋊C4—22no—
163C11×C4⋊C4—22no—
164C11×C4⋊C4—22no—
165C11×C4⋊C4—22no—
166C11×C22⋊C4—42no—
167C11×C22⋊C4—42no—
168C22⋊Dic11—22no—
169C22⋊Dic11—22no—
170(C2×D11)⋊C4—44no—
171(C2×D11)⋊C4—44no—
172(C2×D11)⋊C4—82no—
173Dic11⋊C4—44no—
174C4×Dic11—22no—
175C4×Dic11—22no—
176C4×Dic11—44no—
177C2×C11⋊C8—22no—
178C2×C11⋊C8—22no—
179C2×C11⋊C8—22no—
180#4314—28no—
181#4314—28no—
182#4314—28no—
183#4314C22⋊C4116yescomplemented
184C23×SD16—111no—
185C2×#440—112no—
186Q8×C22⋊C4—111no—
187D4×C22⋊C4—111no—
188#1973—112no—
189#1880—112no—
190#1877—112no—
191#1808D1111yescharacteristic, complemented
192#1787—111no—
193#1786—111no—
194#1752—111no—
195C23×C11—22no—
196C23×C11—22no—
197C23×C11C4×D412yes—
198C23×C11AGL(1,ℤ/8)12yes—
199C23×C11#13712yes—
200C23×C11C22×D411yescharacteristic
201C22×D11—22no—
202C22×D11—24no—
203C22×D11—22no—
204C22×D11—48no—
205C22×D11—41no—
206C22×D11—41no—
207C22×D11—42no—
208C22×D11—42no—
209C22×D11—41no—
210C22×D11—41no—
211C22×D11—42no—
212C22×D11—48no—
213C22×D11—84no—
214C22×D11—88no—
215C22×D11—88no—
216C2×C4×C11—22no—
217C2×C4×C11—22no—
218C2×C4×C11—22no—
219C2×C4×C11—24no—
220C2×C4×C11—42no—
221C2×C4×C11—42no—
222C2×C4×C11—42no—
223C2×C4×C11—42no—
224C2×C4×C11—42no—
225C2×C4×C11—42no—
226C2×C4×C11C4×D412yes—
227C2×C4×C11C23×C411yescharacteristic
228C2×C4×C11C22×D412yes—
229C2×C4×C11C2×#4111yescharacteristic
230C2×C4×C11H(5,𝔽2)12yes—
231C2×Dic11—22no—
232C2×Dic11—24no—
233C2×Dic11—22no—
234C2×Dic11—22no—
235C2×Dic11—22no—
236C2×Dic11—22no—
237C2×Dic11—22no—
238C2×Dic11—22no—
239C2×Dic11—22no—
240C2×Dic11—48no—
241C2×Dic11—42no—
242D44—22no—
243D44—22no—
244D44—22no—
245D44—24no—
246D44—48no—
247D44—42no—
248D44C2×C22⋊C414yes—
249C4×D11—88no—
250Dic22—22no—
251Dic22—24no—
252Dic22—22no—
253Dic22—22no—
254Dic22—48no—
255Dic22—42no—
256Dic22C2×C22⋊C414yescomplemented
257C8×C11—22no—
258C8×C11C2×C22⋊C414yescomplemented
259C11⋊C8—42no—
260C23×Q8—111no—
261C23×D4—111no—
262C22×SD16—111no—
263C22×SD16—111no—
264C22×SD16—111no—
265C22×SD16—111no—
266C22×SD16—118no—
267C22×SD16—228no—
268C23×C8C2×D1111yescharacteristic
269C2×C4×Q8—112no—
270C2×C4×D4—112no—
271C22×C22⋊C4—221no—
272#440—228no—
273C2×#107C2×D1112yescomplemented
274C2×#103—112no—
275C2×#103—112no—
276C2×#102—112no—
277C2×#102—112no—
278C2×C22⋊C8—111no—
279C2×#97—112no—
280#391—112no—
281#391—222no—
282#390—112no—
283#386—222no—
284#380—221no—
285#380C2×D1111yescharacteristic
286#380C2×D1111yescharacteristic
287#379—221no—
288C4×C22⋊C4—111no—
289C4×C22⋊C4—221no—
290#340C2×D1112yescomplemented
291#336—112no—
292C22×C11—22no—
293C22×C11—22no—
294C22×C11—24no—
295C22×C11—24no—
296C22×C11#44014yescomplemented
297C22×C11C2×C4×D411yescharacteristic
298C22×C11D4212yes—
299C22×C11C2×AGL(1,ℤ/8)11yescharacteristic
300C22×C11C2×#13711yescharacteristic
301C22×C11#57712yes—
302C2×D11—24no—
303C2×D11—42no—
304C2×D11—42no—
305C2×D11—44no—
306C2×D11—44no—
307C2×D11—42no—
308C2×D11—42no—
309C2×D11—42no—
310C2×D11—816no—
311C2×D11—82no—
312C2×D11—82no—
313C4×C11—22no—
314C4×C11—82no—
315C4×C11—82no—
316C4×C11C22×C22⋊C411yescharacteristic
317C4×C11#51811yescharacteristic
318C4×C11#54612yes—
319Dic11—24no—
320Dic11—42no—
321Dic11—42no—
322Dic11—42no—
323C25—221no—
324C22×Q8—114no—
325C22×Q8—111no—
326C22×Q8—111no—
327C22×Q8—224no—
328C22×Q8—221no—
329C22×Q8—222no—
330C22×Q8—442no—
331C22×D4—114no—
332C22×D4—111no—
333C22×D4—111no—
334C22×D4—224no—
335C22×D4—221no—
336C22×D4—222no—
337C22×D4—442no—
338C23×C4—222no—
339C23×C4—221no—
340C23×C4C22×D1111yescharacteristic
341C2×SD16—114no—
342C2×SD16—114no—
343C2×SD16—114no—
344C2×SD16—114no—
345C2×SD16—118no—
346C2×SD16—2232no—
347C2×SD16—224no—
348C2×SD16—224no—
349C2×SD16—224no—
350C22×C8—24no—
351C22×C8—112no—
352C22×C8C4×D1114yescomplemented
353C22×C8C22×D1111yescharacteristic
354C22×C8C22×D1111yescharacteristic
355C4×Q8—224no—
356C4×Q8—442no—
357C4×D4—224no—
358C4×D4—442no—
359C2×C4⋊C4—222no—
360C2×C4⋊C4C22×D1112yes—
361C2×C4⋊C4C22×D1112yes—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—111no—
365C2×C22⋊C4—111no—
366C2×C22⋊C4—222no—
367C2×C22⋊C4—222no—
368C2×C22⋊C4—222no—
369C2×C22⋊C4—444no—
370C2×C42—112no—
371C2×C42—222no—
372#107—24no—
373#103—224no—
374#103—442no—
375#102—224no—
376#102—442no—
377C22⋊C8—222no—
378#97—224no—
379#95—21no—
380#95—21no—
381#95—112no—
382#95—222no—
383#95—222no—
384C2×C11—24no—
385C2×C11D4×C22⋊C411yescharacteristic
386C2×C11C2×#44011yescharacteristic
387C2×C11#294911yescharacteristic
388C2×C11#326512yes—
389C2×C11#326612yes—
390D11—44no—
391D11—82no—
392C24—221no—
393C24—221no—
394C24—224no—
395C24—444no—
396C24—442no—
397C24—442no—
398C24—882no—
399C24D4×D1111yescharacteristic
400C2×Q8—112no—
401C2×Q8—112no—
402C2×Q8—114no—
403C2×Q8—228no—
404C2×Q8—224no—
405C2×Q8—221no—
406C2×Q8—221no—
407C2×Q8—222no—
408C2×Q8—222no—
409C2×Q8—222no—
410C2×Q8—444no—
411C2×Q8—444no—
412C2×Q8—448no—
413C2×D4—112no—
414C2×D4—112no—
415C2×D4—114no—
416C2×D4—228no—
417C2×D4—224no—
418C2×D4—222no—
419C2×D4—221no—
420C2×D4—221no—
421C2×D4—222no—
422C2×D4—222no—
423C2×D4—444no—
424C2×D4—448no—
425C2×D4—444no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—112no—
431C22×C4—112no—
432C22×C4—221no—
433C22×C4—224no—
434C22×C4—221no—
435C22×C4—444no—
436C22×C4—444no—
437C22×C4—444no—
438C22×C4—882no—
439C22×C4C2×C4×D1112yes—
440C22×C4D4×D1111yescharacteristic
441C22×C4#434312yes—
442C22×C4C23×D1111yescharacteristic
443SD16—1116no—
444SD16—228no—
445SD16—228no—
446SD16—228no—
447C2×C8—22no—
448C2×C8—28no—
449C2×C8—222no—
450C2×C8—222no—
451C2×C8—222no—
452C2×C8C2×C4×D1112yes—
453C2×C8C2×C4×D1112yes—
454C2×C8D4×D1114yes—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—444no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C22⋊C4—222no—
463C22⋊C4—222no—
464C22⋊C4—444no—
465C22⋊C4—444no—
466C22⋊C4—882no—
467C42—222no—
468C42—222no—
469C42—444no—
470C11#2243711yescharacteristic, complemented, Sylow
471C23—22no—
472C23—22no—
473C23—222no—
474C23—224no—
475C23—222no—
476C23—448no—
477C23—441no—
478C23—441no—
479C23—442no—
480C23—448no—
481C23—441no—
482C23—441no—
483C23—442no—
484C23—442no—
485C23—884no—
486C23—888no—
487C23—888no—
488C23#6694612yes—
489C23#6696412yes—
490C23#6697112yes—
491C23C2×D4×D1111yescharacteristic
492Q8—114no—
493Q8—222no—
494Q8—224no—
495Q8—222no—
496Q8—222no—
497Q8—448no—
498Q8—442no—
499D4—114no—
500D4—222no—
501D4—222no—
502D4—222no—
503D4—224no—
504D4—448no—
505D4—442no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—222no—
517C2×C4—224no—
518C2×C4—222no—
519C2×C4—222no—
520C2×C4—222no—
521C2×C4—222no—
522C2×C4—222no—
523C2×C4—222no—
524C2×C4—222no—
525C2×C4—448no—
526C2×C4—442no—
527C2×C4—888no—
528C2×C4#6694612yes—
529C2×C4C22×C4×D1111yescharacteristic
530C2×C4C2×D4×D1112yes—
531C2×C4C2×#434311yescharacteristic
532C2×C4#6704212yes—
533C8—22no—
534C8—442no—
535C8C22⋊C4×D1114yescomplemented
536C22—24no—
537C22—24no—
538C22—22no—
539C22—22no—
540C22—224no—
541C22—444no—
542C22—444no—
543C22—442no—
544C22—442no—
545C22—442no—
546C22—442no—
547C22—442no—
548C22—8816no—
549C22—882no—
550C22—882no—
551C22#1062250214yescomplemented
552C22C2×#6694611yescharacteristic
553C22#1062314012yes—
554C22C2×#6696411yescharacteristic
555C22C2×#6697111yescharacteristic
556C22#1062330112yes—
557C4—22no—
558C4—82no—
559C4—82no—
560C4—224no—
561C4—442no—
562C4—442no—
563C4—442no—
564C4C2×C22⋊C4×D1111yescharacteristic
565C4#1062305211yescharacteristic
566C4#1062316112yes—
567C2—24no—
568C2—444no—
569C2—882no—
570C2#1307143611yescharacteristic
571C2C2×#1062250211yescharacteristic
572C2#1307747411yescharacteristic
573C2#1308010812yes—
574C2#1308012312yes—
subgroupG/H
centerC23C2×D4×D11
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingC11×#1808C2
socleC23×C11C22×D4
prime radicalC23×D44C22
cosocleC22×C4×C11C24
hypercenter#1808D11
hypocenterC11#22437
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1808

Complements

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

NH ≅ G/Ntype
2. C2×#10622502C2semidirect
3. #13071448C2semidirect
5. #13069495C2semidirect
6. #13068972C2semidirect
7. #13068951C2semidirect
8. #13068550C2semidirect
10. #13068392C2semidirect
11. C11×#1808C2semidirect
16. C22×#4314C4semidirect
22. C2×#66940C22semidirect
24. C2×C11×#107C22semidirect
29. C2×#66879C22semidirect
31. #10622587C22semidirect
32. C2×#66874C22semidirect
33. C2×#66874C22semidirect
42. C11×#340C22semidirect
43. #10622188C22semidirect
59. C22×Dic22C2×C4semidirect
63. C22×C8×C11C2×C4semidirect
85. C2×#4314C2×C4semidirect
86. C2×#4314C2×C4semidirect
183. #4314C22⋊C4semidirect
191. #1808D11semidirect
256. Dic22C2×C22⋊C4semidirect
258. C8×C11C2×C22⋊C4semidirect
273. C2×#107C2×D11semidirect
290. #340C2×D11semidirect
296. C22×C11#440semidirect
352. C22×C8C4×D11semidirect
470. C11#22437semidirect
535. C8C22⋊C4×D11semidirect
551. C22#10622502semidirect

Maximal subgroups

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

subgroupindex
C23×#43142
C2×#106225022
#130714482
#130714362
#130694952
#130689722
#130689512
#130685502
#130685462
#130683922
C11×#18082
#2243711