Finite Group Databasebeta

← #14781582

Subgroups — #14781582

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
1#13077179C211yesmaximal, characteristic, complemented
2#13077135C211yesmaximal, characteristic, complemented
3C2×#10622828C211yesmaximal, characteristic, complemented
4C2×#10622850C211yesmaximal, characteristic, complemented
5C22×#66897C211yesmaximal, characteristic, complemented
6#13069874C214yesmaximal, complemented
7#13069869C214yesmaximal, complemented
8#13068343C211yesmaximal, characteristic, complemented
9C22×C11×AGL(1,ℤ/8)C211yesmaximal, characteristic
10C22×C11×#41C2211yescharacteristic
11C23×D4×C11C2211yescharacteristic
12C2×C11×AGL(1,ℤ/8)—24no—
13C2×C11×AGL(1,ℤ/8)—24no—
14C2×C11×AGL(1,ℤ/8)—22no—
15C2×C11×AGL(1,ℤ/8)—22no—
16C22×C11×SD16C2212yes—
17C22×C11×D8C2212yes—
18C22×C11×M16C2211yescharacteristic
19C2×#66979C2211yescharacteristic
20C22×#4318C2211yescharacteristic
21C22×#4318C2211yescharacteristic
22C22×#4318C2211yescharacteristic
23#10622850—24no—
24#10622846—22no—
25#10622845—22no—
26C2×#66900C2212yescomplemented
27#10622838C2212yescomplemented
28C2×#66897—24no—
29C2×#66897C2214yes—
30C2×#66897C2212yescomplemented
31#10622828—22no—
32#10622828C2214yes—
33#10622828C2212yescomplemented
34C2×#66896—22no—
35#10622732—22no—
36#10622731—22no—
37#10622715C2214yescomplemented
38#10622190C2214yescomplemented
39C25×C11—21no—
40C2×C11×#41—22no—
41C2×C11×#41—21no—
42C2×C11×#41—21no—
43C2×C11×#41—21no—
44C2×C11×#41—21no—
45C2×C11×#41—22no—
46C2×C11×#41—22no—
47C2×C11×#41D414yes—
48C22×Q8×C11C2311yescharacteristic
49C22×D4×C11—21no—
50C22×D4×C11—21no—
51C22×D4×C11—21no—
52C22×D4×C11—22no—
53C22×D4×C11—21no—
54C22×D4×C11—21no—
55C22×D4×C11—22no—
56C22×D4×C11—21no—
57C22×D4×C11—22no—
58C22×D4×C11D414yes—
59C22×D4×C11C2312yes—
60C22×D4×C11C2311yescharacteristic
61C23×C4×C11—21no—
62C23×C4×C11C2311yescharacteristic
63C11×AGL(1,ℤ/8)—216no—
64C11×AGL(1,ℤ/8)—216no—
65C2×C11×SD16—24no—
66C2×C11×SD16—24no—
67C2×C11×SD16—24no—
68C2×C11×D8—24no—
69C2×C11×D8—24no—
70C2×C11×D8—24no—
71C2×C11×M16—22no—
72C2×C11×M16—22no—
73C2×C11×M16D412yes—
74C2×C11×M16D412yes—
75C22×C8×C11C2312yes—
76C2×C22⋊Dic11—21no—
77#66979—21no—
78#66979—21no—
79#66979D412yes—
80#66979C2312yes—
81C2×C4⋊Dic11D412yes—
82C2×C4×Dic11—21no—
83C2×#4318—21no—
84C2×#4318—21no—
85C2×#4318—21no—
86C2×#4318—22no—
87C2×#4318—22no—
88C2×#4318—22no—
89C2×#4318—22no—
90C2×#4318—22no—
91C2×#4318—22no—
92C2×#4318C2312yes—
93C2×#4318C2312yes—
94C2×#4318C2312yes—
95C22×C11⋊C8—21no—
96C22×C11⋊C8—21no—
97C22×C11⋊C8—21no—
98#66900—24no—
99#66899—24no—
100#66897—28no—
101#66897—24no—
102#66896—28no—
103#66896—24no—
104Dic11⋊C8—28no—
105#23521—111nomaximal, Sylow
106C24×C11—24no—
107C24×C11—21no—
108C24×C11—22no—
109C24×C11—41no—
110C24×C11—41no—
111C24×C11—42no—
112C24×C11—84no—
113C24×C11#4111yescharacteristic
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—24no—
130C2×D4×C11—24no—
131C2×D4×C11—24no—
132C2×D4×C11—24no—
133C2×D4×C11—24no—
134C2×D4×C11—22no—
135C2×D4×C11—21no—
136C2×D4×C11—21no—
137C2×D4×C11—21no—
138C2×D4×C11—22no—
139C2×D4×C11—22no—
140C2×D4×C11—22no—
141C2×D4×C11—21no—
142C2×D4×C11—22no—
143C2×D4×C11—21no—
144C2×D4×C11—41no—
145C2×D4×C11—41no—
146C2×D4×C11—42no—
147C2×D4×C11—41no—
148C2×D4×C11—41no—
149C2×D4×C11—42no—
150C2×D4×C11—42no—
151C2×D4×C11—42no—
152C2×D4×C11—42no—
153C2×D4×C11—42no—
154C2×D4×C11C2×D412yes—
155C2×D4×C11C2×D414yes—
156C22×C4×C11—21no—
157C22×C4×C11—24no—
158C22×C4×C11—41no—
159C22×C4×C11—41no—
160C22×C4×C11—42no—
161C22×C4×C11C2×D412yes—
162C22×C4×C11C2×D412yes—
163C22×C4×C11C2×D411yescharacteristic
164C22×C4×C11C2×D411yescharacteristic
165C22×C4×C11C2×D411yescharacteristic
166C22×C4×C11C2×D411yescharacteristic
167C22×C4×C11#4111yescharacteristic
168C22×C4×C11#4112yes—
169C22×C4×C11#4112yes—
170C22×C4×C11C2411yescharacteristic
171C22×Dic11—22no—
172C11×SD16—216no—
173C11×D8—216no—
174C11×M16—24no—
175C11×M16—24no—
176C2×C8×C11—22no—
177C2×C8×C11—22no—
178C2×C8×C11C2×D414yes—
179C22⋊Dic11—22no—
180C22⋊Dic11—22no—
181C4⋊Dic11—22no—
182C4⋊Dic11C2×D414yes—
183C4×Dic11—22no—
184C4×Dic11—22no—
185#4318—22no—
186#4318—22no—
187#4318—22no—
188#4318—22no—
189#4318—22no—
190#4318—22no—
191#4318—42no—
192#4318—42no—
193#4318—42no—
194C2×C11⋊C8—22no—
195C2×C11⋊C8—22no—
196C2×C11⋊C8—22no—
197C2×C11⋊C8—42no—
198C2×C11⋊C8—42no—
199C2×C11⋊C8—42no—
200C22×AGL(1,ℤ/8)D1111yescharacteristic
201#2905—111no—
202#2896—111no—
203C2×#418—111no—
204C2×#413—111no—
205C22×#100—111no—
206#2046—114no—
207#2045—114no—
208#1738—111no—
209C23×C11—22no—
210C23×C11—24no—
211C23×C11—42no—
212C23×C11—41no—
213C23×C11—42no—
214C23×C11—41no—
215C23×C11—42no—
216C23×C11—44no—
217C23×C11—44no—
218C23×C11—82no—
219C23×C11—84no—
220C23×C11—84no—
221C23×C11—84no—
222C23×C11—88no—
223C23×C11—88no—
224C23×C11#12111yescharacteristic
225C23×C11#12111yescharacteristic
226C23×C11#12312yes—
227C23×C11#12312yes—
228C23×C11C2×#4111yescharacteristic
229Q8×C11—22no—
230Q8×C11—22no—
231Q8×C11—41no—
232Q8×C11—41no—
233D4×C11—22no—
234D4×C11—24no—
235D4×C11—24no—
236D4×C11—22no—
237D4×C11—44no—
238D4×C11—44no—
239D4×C11—42no—
240D4×C11—41no—
241D4×C11—42no—
242D4×C11—41no—
243D4×C11—44no—
244D4×C11—44no—
245C2×C4×C11—22no—
246C2×C4×C11—41no—
247C2×C4×C11—41no—
248C2×C4×C11—44no—
249C2×C4×C11—44no—
250C2×C4×C11#12012yes—
251C2×C4×C11#12012yes—
252C2×C4×C11#12111yescharacteristic
253C2×C4×C11#12111yescharacteristic
254C2×C4×C11#12112yes—
255C2×C4×C11#12112yes—
256C2×C4×C11#12112yes—
257C2×C4×C11#12112yes—
258C2×C4×C11#12312yes—
259C2×C4×C11#12312yes—
260C2×C4×C11#12412yes—
261C2×C4×C11#12412yes—
262C2×C4×C11C22×D411yescharacteristic
263C2×C4×C11C22×D411yescharacteristic
264C2×C4×C11C22×D411yescharacteristic
265C2×C4×C11C22×D411yescharacteristic
266C2×C4×C11C2×#4111yescharacteristic
267C2×C4×C11C2×#4111yescharacteristic
268C2×Dic11—24no—
269C2×Dic11—82no—
270C8×C11—24no—
271C11⋊C8—42no—
272C11⋊C8—42no—
273C11⋊C8—42no—
274C22×#41C2×D1111yescharacteristic
275C23×D4C2×D1111yescharacteristic
276C2×AGL(1,ℤ/8)—24no—
277C2×AGL(1,ℤ/8)—24no—
278C2×AGL(1,ℤ/8)—22no—
279C2×AGL(1,ℤ/8)—22no—
280C22×SD16C2×D1112yes—
281C22×D8C2×D1112yes—
282C22×M16—111no—
283C22×M16—111no—
284C22×M16—111no—
285C22×M16C2×D1111yescharacteristic
286C2×#117—111no—
287#424—114no—
288#424—222no—
289#423—222no—
290#419—112no—
291#418—114no—
292#418—112no—
293#418—222no—
294#417—222no—
295#416—222no—
296C2×#102—222no—
297#413—224no—
298C2×#101—112no—
299C2×#100—114no—
300C2×#100—112no—
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—44no—
310C22×C11—82no—
311C22×C11—84no—
312C22×C11—82no—
313C22×C11—88no—
314C22×C11—84no—
315C22×C11—88no—
316C22×C11—82no—
317C22×C11—84no—
318C22×C11#39412yes—
319C22×C11#39412yes—
320C22×C11C2×#12111yescharacteristic
321C22×C11C2×#12311yescharacteristic
322C22×C11C2×#12311yescharacteristic
323C4×C11—42no—
324C4×C11#39412yes—
325C4×C11#39412yes—
326C4×C11C2×#12011yescharacteristic
327C4×C11C2×#12111yescharacteristic
328C4×C11C2×#12111yescharacteristic
329C4×C11C2×#12411yescharacteristic
330Dic11—44no—
331C25—21no—
332C2×#41—22no—
333C2×#41—21no—
334C2×#41—21no—
335C2×#41—22no—
336C2×#41—22no—
337C2×#41—21no—
338C2×#41—21no—
339C2×#41C22⋊D1114yes—
340C22×Q8C22×D1111yescharacteristic
341C22×D4—21no—
342C22×D4—21no—
343C22×D4—21no—
344C22×D4—22no—
345C22×D4—21no—
346C22×D4—22no—
347C22×D4—22no—
348C22×D4—21no—
349C22×D4—21no—
350C22×D4C22⋊D1114yes—
351C22×D4C22×D1111yescharacteristic
352C22×D4C22×D1112yes—
353C23×C4—21no—
354C23×C4C22×D1111yescharacteristic
355AGL(1,ℤ/8)—216no—
356AGL(1,ℤ/8)—216no—
357C2×SD16—24no—
358C2×SD16—24no—
359C2×SD16—24no—
360C2×D8—24no—
361C2×D8—24no—
362C2×D8—24no—
363C2×M16—22no—
364C2×M16—22no—
365C2×M16—112no—
366C2×M16—112no—
367C2×M16—112no—
368C2×M16—221no—
369C2×M16—221no—
370C2×M16—221no—
371C2×M16—222no—
372C2×M16—222no—
373C2×M16—222no—
374C2×M16—222no—
375C2×M16—222no—
376C2×M16—222no—
377C2×M16C22⋊D1112yes—
378C2×M16C22⋊D1112yes—
379C22×C8—221no—
380C22×C8—221no—
381C22×C8—221no—
382C22×C8C22×D1112yes—
383#117—112no—
384#117—112no—
385#117—221no—
386#117—221no—
387C2×C4⋊C4—112no—
388C2×C22⋊C4—221no—
389C2×C42—221no—
390C4⋊C8—228no—
391#103—224no—
392#102—228no—
393#102—224no—
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×C11—84no—
403C2×C11C2×#39411yescharacteristic
404C2×C11#306812yes—
405C24—24no—
406C24—21no—
407C24—22no—
408C24—42no—
409C24—41no—
410C24—41no—
411C24—84no—
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×C22⋊D1112yes—
427C2×D4—24no—
428C2×D4—24no—
429C2×D4—24no—
430C2×D4—24no—
431C2×D4—24no—
432C2×D4—24no—
433C2×D4—22no—
434C2×D4—21no—
435C2×D4—22no—
436C2×D4—21no—
437C2×D4—22no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—42no—
444C2×D4—41no—
445C2×D4—41no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—41no—
450C2×D4—41no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4C2×C22⋊D1112yes—
454C2×D4C2×C22⋊D1114yes—
455C22×C4—21no—
456C22×C4—24no—
457C22×C4—42no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—222no—
461C22×C4D4×D1111yescharacteristic
462C22×C4D4×D1111yescharacteristic
463C22×C4#434111yescharacteristic
464C22×C4#434112yes—
465C22×C4#434112yes—
466C22×C4C2×C22⋊D1112yes—
467C22×C4C2×C22⋊D1112yes—
468C22×C4C2×C22⋊D1111yescharacteristic
469C22×C4C2×C22⋊D1111yescharacteristic
470C22×C4C23×D1111yescharacteristic
471SD16—216no—
472D8—216no—
473M16—24no—
474M16—24no—
475M16—222no—
476M16—222no—
477M16—222no—
478M16—222no—
479M16—222no—
480M16—222no—
481M16—442no—
482M16—442no—
483M16—442no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—222no—
487C2×C8—222no—
488C2×C8—222no—
489C2×C8—442no—
490C2×C8—442no—
491C2×C8—442no—
492C2×C8C2×C22⋊D1114yes—
493C4⋊C4—114no—
494C4⋊C4—222no—
495C22⋊C4—222no—
496C22⋊C4—222no—
497C42—222no—
498C42—222no—
499C11#2352111yescharacteristic, complemented, Sylow
500C23—22no—
501C23—24no—
502C23—42no—
503C23—41no—
504C23—42no—
505C23—41no—
506C23—42no—
507C23—44no—
508C23—44no—
509C23—82no—
510C23—84no—
511C23—84no—
512C23—84no—
513C23—88no—
514C23—88no—
515C23#6698812yes—
516C23#6698812yes—
517C23#6699111yescharacteristic
518C23#6699111yescharacteristic
519C23C2×#434111yescharacteristic
520Q8—22no—
521Q8—22no—
522Q8—41no—
523Q8—41no—
524D4—24no—
525D4—22no—
526D4—22no—
527D4—24no—
528D4—44no—
529D4—44no—
530D4—42no—
531D4—41no—
532D4—42no—
533D4—41no—
534D4—44no—
535D4—44no—
536C2×C4—22no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—224no—
542C2×C4—882no—
543C2×C4#6698812yes—
544C2×C4#6698812yes—
545C2×C4#6698912yes—
546C2×C4#6698912yes—
547C2×C4#6699111yescharacteristic
548C2×C4#6699111yescharacteristic
549C2×C4#6699212yes—
550C2×C4#6699212yes—
551C2×C4#6699212yes—
552C2×C4#6699212yes—
553C2×C4#6700612yes—
554C2×C4#6700612yes—
555C2×C4C2×D4×D1111yescharacteristic
556C2×C4C2×#434111yescharacteristic
557C2×C4C2×#434111yescharacteristic
558C2×C4C22×C22⋊D1111yescharacteristic
559C2×C4C22×C22⋊D1111yescharacteristic
560C2×C4C22×C22⋊D1111yescharacteristic
561C8—24no—
562C8—442no—
563C8—442no—
564C8—442no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—44no—
572C22—84no—
573C22—82no—
574C22—84no—
575C22—88no—
576C22—88no—
577C22—82no—
578C22—84no—
579C22—82no—
580C22#1062283512yes—
581C22#1062283512yes—
582C22C2×#6698811yescharacteristic
583C22C2×#6698811yescharacteristic
584C22C2×#6699111yescharacteristic
585C4—42no—
586C4—444no—
587C4#1062283512yes—
588C4#1062283512yes—
589C4C2×#6698911yescharacteristic
590C4C2×#6699211yescharacteristic
591C4C2×#6699211yescharacteristic
592C4C2×#6700611yescharacteristic
593C2—21no—
594C2—21no—
595C2—22no—
596C2—22no—
597C2—84no—
598C2—82no—
599C2C2×#1062283511yescharacteristic
600C2#1307834012yes—
subgroupG/H
centerC22C2×#66991
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
Fitting, prime radicalC22×C11×AGL(1,ℤ/8)C2
socleC22×C11C2×#121
hypercenterC22×AGL(1,ℤ/8)D11
hypocenterC11#23521
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #13077179C2semidirect
2. #13077135C2semidirect
3. C2×#10622828C2semidirect
4. C2×#10622850C2semidirect
5. C22×#66897C2semidirect
6. #13069874C2semidirect
7. #13069869C2semidirect
8. #13068343C2semidirect
26. C2×#66900C22semidirect
27. #10622838C22semidirect
30. C2×#66897C22semidirect
33. #10622828C22semidirect
37. #10622715C22semidirect
38. #10622190C22semidirect
499. C11#23521semidirect

Maximal subgroups

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

subgroupindex
#130771792
#130771352
C2×#106228282
C2×#106228502
C22×#668972
#130698742
#130698692
#130683432
C22×C11×AGL(1,ℤ/8)2
#2352111