Finite Group Databasebeta

← #14781584

Subgroups — #14781584

Order 2816.

1454 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#67002C211yesmaximal, characteristic
2#13077180C211yesmaximal, characteristic, complemented
3C2×#10622614C211yesmaximal, characteristic, complemented
4C2×#10622850C211yesmaximal, characteristic, complemented
5C22×#66887C211yesmaximal, characteristic, complemented
6#13069876C214yesmaximal, complemented
7#13069871C214yesmaximal, complemented
8#13068342C211yesmaximal, characteristic, complemented
9C11×#2896C211yesmaximal, characteristic, complemented
10C22×C11×#41C2211yescharacteristic
11C23×D44C2211yescharacteristic
12C22×C11×M16C2211yescharacteristic
13C2×C11×#117C2211yescharacteristic
14C2×#67002—24no—
15C2×#67002—24no—
16C2×#67002—22no—
17C2×#67002—22no—
18C22×#4325C2212yes—
19C22×#4323C2212yes—
20C22×#4318C2211yescharacteristic
21C22×#4318C2211yescharacteristic
22C22×#4318C2211yescharacteristic
23C11×#419C2212yescomplemented
24C11×#418C2212yescomplemented
25C11×#417—22no—
26C11×#416—22no—
27#10622850—24no—
28C2×#66900C2212yescomplemented
29C2×#66897C2212yescomplemented
30C2×#66887—24no—
31C2×#66887C2214yes—
32#10622614—22no—
33#10622614C2214yes—
34#10622612—22no—
35#10622612C2214yescomplemented
36C2×#66873—22no—
37#10622537—22no—
38#10622196C2214yescomplemented
39C24×D11—21no—
40C2×C11×#41—22no—
41C2×C11×#41—21no—
42C2×C11×#41—21no—
43C2×C11×#41—22no—
44C2×C11×#41—21no—
45C2×C11×#41—21no—
46C2×C11×#41—22no—
47C2×C11×#41D414yes—
48C22×Q8×C11C2311yescharacteristic
49C22×D4×C11—21no—
50C22×D4×C11C2311yescharacteristic
51C23×C4×C11—21no—
52C23×C4×C11C2311yescharacteristic
53C22×D44—21no—
54C22×D44—22no—
55C22×D44—21no—
56C22×D44—21no—
57C22×D44—21no—
58C22×D44—22no—
59C22×D44—22no—
60C22×D44—21no—
61C22×D44D414yes—
62C22×D44C2312yes—
63C2×C11×M16—21no—
64C2×C11×M16—22no—
65C2×C11×M16—22no—
66C2×C11×M16C2312yes—
67C22×C8×C11—21no—
68C11×#117—21no—
69C11×#117—21no—
70C11×#117D412yes—
71C11×#117C2312yes—
72C2×C11×C4⋊C4D412yes—
73C2×C11×C22⋊C4—21no—
74C2×C42×C11—21no—
75#67002—216no—
76#67002—216no—
77C2×#4325—24no—
78C2×#4325—24no—
79C2×#4325—24no—
80C2×#4323—24no—
81C2×#4323—24no—
82C2×#4323—24no—
83C2×#4318—22no—
84C2×#4318—22no—
85C2×#4318—21no—
86C2×#4318—21no—
87C2×#4318—22no—
88C2×#4318—22no—
89C2×#4318—22no—
90C2×#4318—22no—
91C2×#4318D412yes—
92C2×#4318D412yes—
93C2×#4318C2312yes—
94C2×#4318C2312yes—
95C22×C11⋊C8—21no—
96C22×C11⋊C8—21no—
97C22×C11⋊C8C2312yes—
98C11×#103—24no—
99C11×#102—24no—
100#66900—24no—
101#66897—24no—
102#66887—28no—
103#66873—28no—
104C4⋊(C11⋊C8)—28no—
105#23521—111nomaximal, Sylow
106C24×C11—21no—
107C24×C11#4111yescharacteristic
108C23×D11—24no—
109C23×D11—22no—
110C23×D11—41no—
111C23×D11—41no—
112C23×D11—42no—
113C23×D11—84no—
114C11×#41—24no—
115C11×#41—24no—
116C11×#41—24no—
117C11×#41—24no—
118C11×#41—42no—
119C11×#41—42no—
120C11×#41—42no—
121C11×#41—42no—
122C2×Q8×C11—21no—
123C2×Q8×C11—21no—
124C2×Q8×C11—21no—
125C2×Q8×C11—21no—
126C2×Q8×C11—21no—
127C2×Q8×C11C2×D412yes—
128C2×D4×C11—24no—
129C2×D4×C11—21no—
130C2×D4×C11—21no—
131C2×D4×C11—21no—
132C2×D4×C11—21no—
133C2×D4×C11—21no—
134C2×D4×C11—41no—
135C2×D4×C11—41no—
136C2×D4×C11—42no—
137C2×D4×C11C2×D412yes—
138C22×C4×C11—21no—
139C22×C4×C11—22no—
140C22×C4×C11—24no—
141C22×C4×C11—41no—
142C22×C4×C11—41no—
143C22×C4×C11—42no—
144C22×C4×C11C2×D412yes—
145C22×C4×C11C2×D412yes—
146C22×C4×C11C2×D411yescharacteristic
147C22×C4×C11C2×D411yescharacteristic
148C22×C4×C11C2×D411yescharacteristic
149C22×C4×C11C2×D411yescharacteristic
150C22×C4×C11#4111yescharacteristic
151C22×C4×C11#4112yes—
152C22×C4×C11#4112yes—
153C22×C4×C11C2411yescharacteristic
154C2×D44—24no—
155C2×D44—24no—
156C2×D44—24no—
157C2×D44—24no—
158C2×D44—24no—
159C2×D44—22no—
160C2×D44—22no—
161C2×D44—22no—
162C2×D44—22no—
163C2×D44—22no—
164C2×D44—42no—
165C2×D44—42no—
166C2×D44—41no—
167C2×D44—41no—
168C2×D44—42no—
169C2×D44—42no—
170C2×D44—42no—
171C2×D44C2×D414yes—
172C11×M16—22no—
173C11×M16—22no—
174C11×M16—42no—
175C2×C8×C11—22no—
176C2×C8×C11—42no—
177C11×C4⋊C4—22no—
178C11×C4⋊C4C2×D414yes—
179C11×C22⋊C4—22no—
180C11×C22⋊C4—22no—
181C42×C11—22no—
182C42×C11—22no—
183#4325—216no—
184#4323—216no—
185#4318—24no—
186#4318—24no—
187#4318—22no—
188#4318—22no—
189#4318—22no—
190#4318—22no—
191#4318—42no—
192#4318—42no—
193C2×C11⋊C8—22no—
194C2×C11⋊C8—22no—
195C2×C11⋊C8—22no—
196C2×C11⋊C8—22no—
197C2×C11⋊C8—42no—
198C2×C11⋊C8—42no—
199C2×C11⋊C8C2×D414yes—
200C22×AGL(1,ℤ/8)—111no—
201#2905—111no—
202#2896D1111yescharacteristic, complemented
203C2×#418—111no—
204C2×#413—111no—
205C22×#100—111no—
206#2046—114no—
207#2045—114no—
208#1738—111no—
209C23×C11—22no—
210C23×C11—42no—
211C23×C11—41no—
212C23×C11—41no—
213C23×C11—82no—
214C23×C11#12111yescharacteristic
215C23×C11#12111yescharacteristic
216C23×C11#12312yes—
217C23×C11#12312yes—
218C23×C11C2×#4111yescharacteristic
219C22×D11—24no—
220C22×D11—42no—
221C22×D11—42no—
222C22×D11—44no—
223C22×D11—44no—
224C22×D11—84no—
225C22×D11—84no—
226C22×D11—88no—
227C22×D11—84no—
228C22×D11—88no—
229Q8×C11—22no—
230Q8×C11—22no—
231Q8×C11—41no—
232Q8×C11—41no—
233D4×C11—22no—
234D4×C11—22no—
235D4×C11—44no—
236D4×C11—44no—
237D4×C11—41no—
238D4×C11—41no—
239C2×C4×C11—24no—
240C2×C4×C11—22no—
241C2×C4×C11—41no—
242C2×C4×C11—41no—
243C2×C4×C11—44no—
244C2×C4×C11—44no—
245C2×C4×C11—82no—
246C2×C4×C11#12012yes—
247C2×C4×C11#12012yes—
248C2×C4×C11#12112yes—
249C2×C4×C11#12112yes—
250C2×C4×C11#12112yes—
251C2×C4×C11#12112yes—
252C2×C4×C11#12111yescharacteristic
253C2×C4×C11#12111yescharacteristic
254C2×C4×C11#12312yes—
255C2×C4×C11#12312yes—
256C2×C4×C11#12412yes—
257C2×C4×C11#12412yes—
258C2×C4×C11C22×D411yescharacteristic
259C2×C4×C11C22×D411yescharacteristic
260C2×C4×C11C22×D411yescharacteristic
261C2×C4×C11C22×D411yescharacteristic
262C2×C4×C11C2×#4111yescharacteristic
263C2×C4×C11C2×#4111yescharacteristic
264D44—24no—
265D44—24no—
266D44—42no—
267D44—42no—
268D44—44no—
269D44—44no—
270C8×C11—42no—
271C11⋊C8—24no—
272C11⋊C8—42no—
273C11⋊C8—42no—
274C22×#41C2×D1111yescharacteristic
275C23×D4—111no—
276C2×AGL(1,ℤ/8)—222no—
277C2×AGL(1,ℤ/8)—222no—
278C2×AGL(1,ℤ/8)—224no—
279C2×AGL(1,ℤ/8)—224no—
280C22×SD16—112no—
281C22×D8—112no—
282C22×M16—111no—
283C22×M16—111no—
284C22×M16—111no—
285C22×M16C2×D1111yescharacteristic
286C2×#117C2×D1111yescharacteristic
287#424—114no—
288#424—222no—
289#423—222no—
290#419C2×D1112yescomplemented
291#418—114no—
292#418—222no—
293#418C2×D1112yescomplemented
294#417—22no—
295#416—22no—
296C2×#102—222no—
297#413—224no—
298C2×#101—112no—
299C2×#100—112no—
300C2×#100—114no—
301C2×#100—224no—
302#341—114no—
303C22×C11—22no—
304C22×C11—22no—
305C22×C11—22no—
306C22×C11—42no—
307C22×C11—42no—
308C22×C11—42no—
309C22×C11—82no—
310C22×C11—82no—
311C22×C11—82no—
312C22×C11#39412yes—
313C22×C11#39412yes—
314C22×C11C2×#12111yescharacteristic
315C22×C11C2×#12311yescharacteristic
316C22×C11C2×#12311yescharacteristic
317C2×D11—44no—
318C2×D11—88no—
319C2×D11—84no—
320C2×D11—88no—
321C2×D11—84no—
322C2×D11—84no—
323C4×C11—42no—
324C4×C11—44no—
325C4×C11#39412yes—
326C4×C11#39412yes—
327C4×C11C2×#12011yescharacteristic
328C4×C11C2×#12111yescharacteristic
329C4×C11C2×#12111yescharacteristic
330C4×C11C2×#12411yescharacteristic
331C25—221no—
332C2×#41—22no—
333C2×#41—22no—
334C2×#41—22no—
335C2×#41—21no—
336C2×#41—21no—
337C2×#41—21no—
338C2×#41—21no—
339C2×#41D4414yes—
340C22×Q8C22×D1111yescharacteristic
341C22×D4—21no—
342C22×D4—114no—
343C22×D4—112no—
344C22×D4—222no—
345C22×D4—222no—
346C22×D4—221no—
347C22×D4—221no—
348C22×D4—222no—
349C22×D4—221no—
350C22×D4—221no—
351C22×D4—221no—
352C22×D4C22×D1111yescharacteristic
353C23×C4—21no—
354C23×C4C22×D1111yescharacteristic
355AGL(1,ℤ/8)—2216no—
356AGL(1,ℤ/8)—2216no—
357C2×SD16—224no—
358C2×SD16—224no—
359C2×SD16—224no—
360C2×D8—224no—
361C2×D8—224no—
362C2×D8—224no—
363C2×M16—21no—
364C2×M16—22no—
365C2×M16—22no—
366C2×M16—112no—
367C2×M16—112no—
368C2×M16—112no—
369C2×M16—112no—
370C2×M16—222no—
371C2×M16—222no—
372C2×M16—221no—
373C2×M16—221no—
374C2×M16—222no—
375C2×M16—222no—
376C2×M16—222no—
377C2×M16—222no—
378C2×M16C22×D1112yes—
379C22×C8—21no—
380C22×C8—112no—
381C22×C8—221no—
382C22×C8—221no—
383#117—21no—
384#117—21no—
385#117C22⋊D1112yes—
386#117C22×D1112yes—
387C2×C4⋊C4C22⋊D1112yes—
388C2×C22⋊C4—21no—
389C2×C42—21no—
390C4⋊C8—228no—
391#103—24no—
392#102—24no—
393#102—228no—
394#101—224no—
395#100—228no—
396#100—224no—
397C2×C11—21no—
398C2×C11—21no—
399C2×C11—22no—
400C2×C11—22no—
401C2×C11—82no—
402C2×C11C2×#39411yescharacteristic
403C2×C11#306812yes—
404D11—84no—
405C24—21no—
406C24—224no—
407C24—222no—
408C24—442no—
409C24—441no—
410C24—441no—
411C24—884no—
412C24#434111yescharacteristic
413#41—24no—
414#41—24no—
415#41—24no—
416#41—24no—
417#41—42no—
418#41—42no—
419#41—42no—
420#41—42no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8C2×D4412yes—
427C2×D4—24no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—42no—
434C2×D4—41no—
435C2×D4—41no—
436C2×D4—114no—
437C2×D4—224no—
438C2×D4—224no—
439C2×D4—224no—
440C2×D4—224no—
441C2×D4—224no—
442C2×D4—222no—
443C2×D4—222no—
444C2×D4—222no—
445C2×D4—222no—
446C2×D4—222no—
447C2×D4—442no—
448C2×D4—442no—
449C2×D4—442no—
450C2×D4—442no—
451C2×D4—442no—
452C2×D4—441no—
453C2×D4—441no—
454C2×D4C2×D4412yes—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—42no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4C2×D4412yes—
462C22×C4#433912yes—
463C22×C4D4×D1112yes—
464C22×C4D4×D1111yescharacteristic
465C22×C4D4×D1111yescharacteristic
466C22×C4#434111yescharacteristic
467C22×C4#434112yes—
468C22×C4C2×C22⋊D1111yescharacteristic
469C22×C4C2×C22⋊D1111yescharacteristic
470C22×C4C23×D1111yescharacteristic
471SD16—2216no—
472D8—2216no—
473M16—22no—
474M16—22no—
475M16—42no—
476M16—224no—
477M16—224no—
478M16—222no—
479M16—222no—
480M16—222no—
481M16—222no—
482M16—442no—
483M16—442no—
484C2×C8—22no—
485C2×C8—42no—
486C2×C8—114no—
487C2×C8—222no—
488C2×C8—222no—
489C2×C8—222no—
490C2×C8—222no—
491C2×C8—442no—
492C2×C8—442no—
493C4⋊C4—22no—
494C4⋊C4C2×C22⋊D1114yes—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C42—22no—
498C42—22no—
499C11#2352111yescharacteristic, complemented, Sylow
500C23—22no—
501C23—42no—
502C23—41no—
503C23—41no—
504C23—82no—
505C23—224no—
506C23—442no—
507C23—442no—
508C23—444no—
509C23—444no—
510C23—884no—
511C23—884no—
512C23—884no—
513C23—888no—
514C23—888no—
515C23#6693612yes—
516C23#6693912yes—
517C23#6699211yescharacteristic
518C23#6699211yescharacteristic
519C23C2×#434111yescharacteristic
520Q8—22no—
521Q8—22no—
522Q8—41no—
523Q8—41no—
524D4—22no—
525D4—22no—
526D4—44no—
527D4—44no—
528D4—41no—
529D4—41no—
530D4—224no—
531D4—224no—
532D4—442no—
533D4—442no—
534D4—444no—
535D4—444no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—82no—
543C2×C4#6693512yes—
544C2×C4#6693512yes—
545C2×C4#6693612yes—
546C2×C4#6693812yes—
547C2×C4#6693812yes—
548C2×C4#6693912yes—
549C2×C4#6698312yes—
550C2×C4#6698312yes—
551C2×C4#6699112yes—
552C2×C4#6699112yes—
553C2×C4#6699211yescharacteristic
554C2×C4#6699211yescharacteristic
555C2×C4C22×D4411yescharacteristic
556C2×C4C2×#433911yescharacteristic
557C2×C4C2×D4×D1111yescharacteristic
558C2×C4C2×D4×D1111yescharacteristic
559C2×C4C2×#434111yescharacteristic
560C2×C4C22×C22⋊D1111yescharacteristic
561C8—42no—
562C8—224no—
563C8—442no—
564C8—442no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—82no—
572C22—82no—
573C22—82no—
574C22—444no—
575C22—884no—
576C22—884no—
577C22—888no—
578C22—888no—
579C22—884no—
580C22#1062257412yes—
581C22#1062257412yes—
582C22C2×#6693611yescharacteristic
583C22C2×#6693911yescharacteristic
584C22C2×#6699211yescharacteristic
585C4—44no—
586C4—42no—
587C4#1062257412yes—
588C4#1062257412yes—
589C4C2×#6693511yescharacteristic
590C4C2×#6693811yescharacteristic
591C4C2×#6698311yescharacteristic
592C4C2×#6699111yescharacteristic
593C2—21no—
594C2—21no—
595C2—22no—
596C2—22no—
597C2—82no—
598C2—884no—
599C2C2×#1062257411yescharacteristic
600C2#1307834612yes—
subgroupG/H
centerC22C2×#66992
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
FittingC11×#2896C2
socleC22×C11C2×#121
prime radicalC22×#67002C2
hypercenter#2896D11
hypocenterC11#23521
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2896

Complements

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

NH ≅ G/Ntype
2. #13077180C2semidirect
3. C2×#10622614C2semidirect
4. C2×#10622850C2semidirect
5. C22×#66887C2semidirect
6. #13069876C2semidirect
7. #13069871C2semidirect
8. #13068342C2semidirect
9. C11×#2896C2semidirect
23. C11×#419C22semidirect
24. C11×#418C22semidirect
28. C2×#66900C22semidirect
29. C2×#66897C22semidirect
35. #10622612C22semidirect
38. #10622196C22semidirect
202. #2896D11semidirect
290. #419C2×D11semidirect
293. #418C2×D11semidirect
499. C11#23521semidirect

Maximal subgroups

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

subgroupindex
C22×#670022
#130771802
C2×#106226142
C2×#106228502
C22×#668872
#130698762
#130698712
#130683422
C11×#28962
#2352111