Finite Group Databasebeta

← #14781122

Subgroups — #14781122

Order 2816.

1250 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10623050C211yesmaximal, characteristic, complemented
2C2×#10622746C211yesmaximal, characteristic
3C2×#10622615C211yesmaximal, characteristic
4#13071209C211yesmaximal, characteristic
5#13069766C212yesmaximal, complemented
6#13069761C212yesmaximal, complemented
7#13069756C212yesmaximal, complemented
8#13069751C212yesmaximal, complemented
9C2×#10622167C211yesmaximal, characteristic, complemented
10#13068314C211yesmaximal, characteristic, complemented
11C11×#1736C211yesmaximal, characteristic, complemented
12C22×#4339C2211yescharacteristic
13C22×C11×M16C2211yescharacteristic
14C2×C11×#117C2211yescharacteristic
15C22×C11×C4⋊C4C2211yescharacteristic
16C2×#66979C2211yescharacteristic
17C22×C4⋊Dic11C2211yescharacteristic
18C23×C11⋊C8C2211yescharacteristic
19#10623050—24no—
20#10623050—24no—
21C2×#66943C2212yes—
22C2×#66941C2212yes—
23C2×#66931—22no—
24C2×#66923—21no—
25C2×#66922—21no—
26#10622826—22no—
27#10622746—24no—
28C2×#66885C2212yes—
29C2×#66880C2212yes—
30#10622615—24no—
31#10622601C2212yes—
32#10622600—21no—
33#10622600—21no—
34#10622587C2212yes—
35C2×#66874C2211yescharacteristic
36C2×#66874C2211yescharacteristic
37C2×#66873C2211yescharacteristic
38C2×#66873C2211yescharacteristic
39C11×#340C2212yes—
40C11×#340C2212yes—
41#10622188C2212yes—
42#10622188C2212yes—
43#10622167C2212yes—
44#10622167C2212yes—
45C23×C4×C11—21no—
46C23×C4×C11C2311yescharacteristic
47C22×C22⋊D11—21no—
48C23×Dic11—21no—
49C2×#4339—22no—
50C2×#4339—22no—
51C2×#4339—22no—
52C2×#4339—22no—
53C2×#4339—22no—
54C2×#4339Q814yescomplemented
55C22×D44C2311yescharacteristic
56C22×C4×D11—21no—
57C22×Dic22—22no—
58C22×Dic22C2311yescharacteristic
59C2×C11×M16—22no—
60C2×C11×M16—22no—
61C2×C11×M16—22no—
62C22×C8×C11C2312yes—
63C11×#117—22no—
64C11×#117D414yes—
65C2×C11×C4⋊C4—22no—
66C2×C11×C4⋊C4—22no—
67C2×C11×C4⋊C4—21no—
68C2×C11×C4⋊C4C2312yes—
69C2×C11×C4⋊C4C2311yescharacteristic
70C2×C11×C4⋊C4C2311yescharacteristic
71C2×C11×C22⋊C4—21no—
72C2×C42×C11—21no—
73C2×C22⋊Dic11D412yes—
74C2×(C2×D11)⋊C4—22no—
75C2×(C2×D11)⋊C4—22no—
76#66979—22no—
77#66979—22no—
78C2×C4⋊Dic11—21no—
79C2×C4⋊Dic11—22no—
80C2×C4⋊Dic11—22no—
81C2×C4⋊Dic11D411yescharacteristic
82C2×C4⋊Dic11D411yescharacteristic
83C2×C4⋊Dic11D414yes—
84C2×C4⋊Dic11C2312yes—
85C2×Dic11⋊C4—22no—
86C2×Dic11⋊C4—22no—
87C2×C4×Dic11C2312yes—
88C22×C11⋊C8—22no—
89C22×C11⋊C8—22no—
90C22×C11⋊C8—22no—
91C22×C11⋊C8C2311yescharacteristic
92C22×C11⋊C8C2311yescharacteristic
93#66943—24no—
94#66943—24no—
95#66941—24no—
96#66941—24no—
97#66931—216no—
98#66923—28no—
99#66922—28no—
100C11⋊#95—22no—
101C11⋊#95—22no—
102#66885—24no—
103#66880—24no—
104#66874—22no—
105#66874—22no—
106#66873—22no—
107#66873—22no—
108#23458—111nomaximal, Sylow
109C24×C11C2×D411yescharacteristic
110C23×D11—21no—
111C22×C4×C11—21no—
112C22×C4×C11—21no—
113C22×C4×C11—22no—
114C22×C4×C11—42no—
115C22×C4×C11—81no—
116C22×C4×C11—81no—
117C22×C4×C11C2×D411yescharacteristic
118C22×C4×C11C2×D412yes—
119C22×C4×C11C2×D412yes—
120C22×C4×C11C2×Q812yes—
121C22×C4×C11#4112yes—
122C22×C4×C11#4112yes—
123C22×C4×C11#4112yes—
124C22×C4×C11C2411yescharacteristic
125C2×C22⋊D11—24no—
126C2×C22⋊D11—42no—
127C2×C22⋊D11—42no—
128C22×Dic11—24no—
129C22×Dic11—21no—
130C22×Dic11—22no—
131C22×Dic11—81no—
132C22×Dic11—81no—
133C22×Dic11C2×D412yes—
134C22×Dic11C2×D412yes—
135#4339—28no—
136#4339—28no—
137#4339—44no—
138#4339—44no—
139C2×D44—21no—
140C2×D44—21no—
141C2×D44—21no—
142C2×D44—21no—
143C2×D44—21no—
144C2×D44C2×Q812yes—
145C2×C4×D11—24no—
146C2×C4×D11—42no—
147C2×C4×D11—42no—
148C2×Dic22—28no—
149C2×Dic22—21no—
150C2×Dic22—21no—
151C2×Dic22—21no—
152C2×Dic22—21no—
153C2×Dic22—21no—
154C2×Dic22—82no—
155C2×Dic22—82no—
156C2×Dic22C2×Q812yes—
157C11×M16—44no—
158C2×C8×C11—22no—
159C2×C8×C11—22no—
160C2×C8×C11—22no—
161C11×C4⋊C4—21no—
162C11×C4⋊C4—21no—
163C11×C4⋊C4—22no—
164C11×C4⋊C4—22no—
165C11×C4⋊C4—42no—
166C11×C4⋊C4C2×D412yes—
167C11×C4⋊C4C2×D412yes—
168C11×C22⋊C4—24no—
169C42×C11—24no—
170C22⋊Dic11—24no—
171(C2×D11)⋊C4—82no—
172(C2×D11)⋊C4—82no—
173C4⋊Dic11—28no—
174C4⋊Dic11—22no—
175C4⋊Dic11—24no—
176C4⋊Dic11—22no—
177C4⋊Dic11—22no—
178C4⋊Dic11C2×D414yes—
179Dic11⋊C4—28no—
180Dic11⋊C4—28no—
181C4×Dic11—22no—
182C4×Dic11—22no—
183C2×C11⋊C8—21no—
184C2×C11⋊C8—21no—
185C2×C11⋊C8—21no—
186C2×C11⋊C8—21no—
187C2×C11⋊C8—21no—
188C2×C11⋊C8—21no—
189C2×C11⋊C8—44no—
190C2×#544—111no—
191C2×#417—111no—
192C2×#416—111no—
193#2311—111no—
194#2026—112no—
195#2025—112no—
196#2024—112no—
197#2023—112no—
198#1736D1111yescharacteristic, complemented
199C2×#340—111no—
200#1729—111no—
201C23×C11—21no—
202C23×C11—21no—
203C23×C11—21no—
204C23×C11—21no—
205C23×C11#12012yes—
206C23×C11C22⋊Q812yes—
207C23×C11#12412yes—
208C23×C11C22×D411yescharacteristic
209C22×D11—22no—
210C22×D11—41no—
211C22×D11—41no—
212C22×D11—41no—
213C22×D11—41no—
214C22×D11—41no—
215C22×D11—41no—
216C2×C4×C11—22no—
217C2×C4×C11—22no—
218C2×C4×C11—41no—
219C2×C4×C11—41no—
220C2×C4×C11—41no—
221C2×C4×C11—41no—
222C2×C4×C11—42no—
223C2×C4×C11—42no—
224C2×C4×C11—42no—
225C2×C4×C11—84no—
226C2×C4×C11#12012yes—
227C2×C4×C11C22⋊Q814yes—
228C2×C4×C11C22⋊Q812yes—
229C2×C4×C11C22⋊Q814yes—
230C2×C4×C11#12414yes—
231C2×C4×C11#12414yes—
232C2×C4×C11#12412yes—
233C2×C4×C11C22×D411yescharacteristic
234C2×C4×C11C22×D411yescharacteristic
235C2×C4×C11C22×Q811yescharacteristic
236C2×C4×C11C2×#4111yescharacteristic
237C2×C4×C11C2×#4111yescharacteristic
238C2×C4×C11C2×#4111yescharacteristic
239C22⋊D11—48no—
240C2×Dic11—22no—
241C2×Dic11—22no—
242C2×Dic11—24no—
243C2×Dic11—22no—
244C2×Dic11—22no—
245C2×Dic11—22no—
246C2×Dic11—22no—
247C2×Dic11—22no—
248C2×Dic11—41no—
249C2×Dic11—41no—
250C2×Dic11—42no—
251C2×Dic11—42no—
252C2×Dic11—84no—
253D44—22no—
254D44—22no—
255D44—41no—
256D44—41no—
257C4×D11—48no—
258Dic22—22no—
259Dic22—22no—
260Dic22—41no—
261Dic22—41no—
262Dic22—88no—
263C8×C11—42no—
264C11⋊C8—41no—
265C11⋊C8—41no—
266C22×#41—111no—
267C22×M16C2×D1111yescharacteristic
268C23×C8—111no—
269#544—224no—
270#544—224no—
271C2×C4⋊Q8—112no—
272C2×C4⋊Q8—221no—
273C2×#125—112no—
274C2×#125—221no—
275C2×C22⋊Q8—222no—
276C2×#117—111no—
277C2×#117C2×D1111yescharacteristic
278C22×C4⋊C4—111no—
279C22×C4⋊C4C2×D1111yescharacteristic
280#417—224no—
281#416—224no—
282C2×#103—112no—
283C2×#103—111no—
284C2×#103—111no—
285C2×#102—112no—
286C2×#102—111no—
287C2×#102—111no—
288#391—112no—
289#390—112no—
290#386—222no—
291#380—221no—
292#380—221no—
293#340—112no—
294#340—112no—
295#340C2×D1112yes—
296#340C2×D1112yes—
297#336—112no—
298#336—112no—
299C22×C11—22no—
300C22×C11—22no—
301C22×C11—22no—
302C22×C11—22no—
303C22×C11—22no—
304C22×C11—22no—
305C22×C11#39314yes—
306C22×C11C2×#12011yescharacteristic
307C22×C11C2×C22⋊Q811yescharacteristic
308C22×C11C2×#12411yescharacteristic
309C22×C11#57611yescharacteristic
310C22×C11#57711yescharacteristic
311C22×C11#57711yescharacteristic
312C22×C11#57811yescharacteristic
313C2×D11—42no—
314C2×D11—42no—
315C2×D11—42no—
316C2×D11—81no—
317C2×D11—81no—
318C2×D11—81no—
319C2×D11—81no—
320C4×C11—42no—
321C4×C11—42no—
322C4×C11—82no—
323C4×C11#39314yes—
324C4×C11C2×C22⋊Q811yescharacteristic
325C4×C11C2×C22⋊Q811yescharacteristic
326C4×C11C2×#12411yescharacteristic
327C4×C11C2×#12411yescharacteristic
328Dic11—42no—
329Dic11—42no—
330Dic11—44no—
331Dic11—42no—
332C2×#41—114no—
333C2×#41—222no—
334C2×#41—222no—
335C2×#41—222no—
336C2×#41—222no—
337C2×#41—222no—
338C22×Q8—111no—
339C22×Q8—222no—
340C22×D4—111no—
341C22×D4—221no—
342C23×C4—21no—
343C23×C4—221no—
344C23×C4—221no—
345C23×C4C22×D1111yescharacteristic
346C2×M16—22no—
347C2×M16—22no—
348C2×M16—22no—
349C22×C8—111no—
350C22×C8—111no—
351C22×C8—222no—
352C22×C8—222no—
353C22×C8—222no—
354C22×C8C22×D1112yes—
355C4⋊Q8—228no—
356C4⋊Q8—224no—
357C4⋊Q8—224no—
358#125—228no—
359#125—224no—
360#125—224no—
361C22⋊Q8—2216no—
362#117—22no—
363#117—222no—
364#117—222no—
365#117D4414yes—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—114no—
370C2×C4⋊C4—112no—
371C2×C4⋊C4—111no—
372C2×C4⋊C4—111no—
373C2×C4⋊C4—222no—
374C2×C4⋊C4—222no—
375C2×C4⋊C4—222no—
376C2×C4⋊C4—222no—
377C2×C4⋊C4—221no—
378C2×C4⋊C4C22×D1112yes—
379C2×C4⋊C4C22×D1111yescharacteristic
380C2×C4⋊C4C22×D1111yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—112no—
383C2×C22⋊C4—222no—
384C2×C22⋊C4—222no—
385C2×C42—21no—
386C2×C42—112no—
387#103—224no—
388#103—222no—
389#103—222no—
390#102—224no—
391#102—222no—
392#102—222no—
393#95—222no—
394#95—222no—
395C2×C11—24no—
396C2×C11C2×#39311yescharacteristic
397C2×C11#301211yescharacteristic
398C2×C11#301311yescharacteristic
399C2×C11#308311yescharacteristic
400C2×C11#308411yescharacteristic
401C2×C11#313611yescharacteristic
402C2×C11#313711yescharacteristic
403D11—82no—
404C24—221no—
405C24D4×D1111yescharacteristic
406#41—228no—
407#41—228no—
408#41—444no—
409#41—444no—
410C2×Q8—112no—
411C2×Q8—228no—
412C2×Q8—221no—
413C2×Q8—221no—
414C2×Q8—221no—
415C2×Q8—221no—
416C2×Q8—221no—
417C2×Q8—882no—
418C2×Q8—882no—
419C2×D4—112no—
420C2×D4—224no—
421C2×D4—221no—
422C2×D4—221no—
423C2×D4—221no—
424C2×D4—221no—
425C2×D4—221no—
426C2×D4—442no—
427C2×D4—442no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—42no—
432C22×C4—81no—
433C22×C4—81no—
434C22×C4—112no—
435C22×C4—112no—
436C22×C4—221no—
437C22×C4—222no—
438C22×C4—224no—
439C22×C4—224no—
440C22×C4—442no—
441C22×C4—442no—
442C22×C4—881no—
443C22×C4—881no—
444C22×C4C2×D4412yes—
445C22×C4#433912yes—
446C22×C4#433912yes—
447C22×C4D4×D1111yescharacteristic
448C22×C4D4×D1112yes—
449C22×C4#434112yes—
450C22×C4Q8×D1112yes—
451C22×C4C23×D1111yescharacteristic
452M16—44no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—221no—
457C2×C8—221no—
458C2×C8—221no—
459C2×C8—221no—
460C2×C8—221no—
461C2×C8—221no—
462C2×C8—444no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—42no—
468C4⋊C4—114no—
469C4⋊C4—228no—
470C4⋊C4—228no—
471C4⋊C4—228no—
472C4⋊C4—224no—
473C4⋊C4—222no—
474C4⋊C4—222no—
475C4⋊C4—222no—
476C4⋊C4C2×D4412yes—
477C4⋊C4C2×D4412yes—
478C22⋊C4—24no—
479C22⋊C4—224no—
480C22⋊C4—882no—
481C22⋊C4—882no—
482C42—24no—
483C42—222no—
484C42—222no—
485C11#2345811yescharacteristic, complemented, Sylow
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—222no—
491C23—441no—
492C23—441no—
493C23—441no—
494C23—441no—
495C23—441no—
496C23—441no—
497C23#6693512yes—
498C23#6693812yes—
499C23#6694912yes—
500C23C2×D4×D1111yescharacteristic
501Q8—222no—
502Q8—222no—
503Q8—441no—
504Q8—441no—
505Q8—888no—
506D4—222no—
507D4—222no—
508D4—448no—
509D4—441no—
510D4—441no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—84no—
521C2×C4—222no—
522C2×C4—222no—
523C2×C4—224no—
524C2×C4—222no—
525C2×C4—222no—
526C2×C4—222no—
527C2×C4—222no—
528C2×C4—222no—
529C2×C4—441no—
530C2×C4—441no—
531C2×C4—442no—
532C2×C4—442no—
533C2×C4—448no—
534C2×C4—884no—
535C2×C4#6692814yes—
536C2×C4#6693512yes—
537C2×C4#6693814yes—
538C2×C4#6693812yes—
539C2×C4#6694914yes—
540C2×C4#6694912yes—
541C2×C4#6695014yes—
542C2×C4C22×D4411yescharacteristic
543C2×C4C2×#433911yescharacteristic
544C2×C4C2×#433911yescharacteristic
545C2×C4C2×D4×D1111yescharacteristic
546C2×C4C2×#434111yescharacteristic
547C2×C4C2×Q8×D1111yescharacteristic
548C8—42no—
549C8—441no—
550C8—441no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—442no—
558C22—442no—
559C22—442no—
560C22—881no—
561C22—881no—
562C22—881no—
563C22—881no—
564C22#1062228614yes—
565C22C2×#6693511yescharacteristic
566C22C2×#6693811yescharacteristic
567C22C2×#6694911yescharacteristic
568C22#1062330511yescharacteristic
569C22#1062330611yescharacteristic
570C22#1062330911yescharacteristic
571C22#1062331011yescharacteristic
572C4—42no—
573C4—42no—
574C4—82no—
575C4—442no—
576C4—444no—
577C4—442no—
578C4—442no—
579C4#1062228614yes—
580C4C2×#6692811yescharacteristic
581C4C2×#6693811yescharacteristic
582C4C2×#6694911yescharacteristic
583C4C2×#6695011yescharacteristic
584C2—24no—
585C2—882no—
586C2C2×#1062228611yescharacteristic
587C2#1307792211yescharacteristic
588C2#1307793311yescharacteristic
589C2#1307846611yescharacteristic
590C2#1307847711yescharacteristic
591C2#1307896611yescharacteristic
592C2#1307897711yescharacteristic
subgroupG/H
centerC23C2×D4×D11
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
FittingC11×#1736C2
socleC23×C11C22×D4
prime radicalC22×#4339C22
hypercenter#1736D11
hypocenterC11#23458
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
1. C2×#10623050C2semidirect
5. #13069766C2semidirect
6. #13069761C2semidirect
7. #13069756C2semidirect
8. #13069751C2semidirect
9. C2×#10622167C2semidirect
10. #13068314C2semidirect
11. C11×#1736C2semidirect
54. C2×#4339Q8semidirect
198. #1736D11semidirect
485. C11#23458semidirect

Maximal subgroups

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

subgroupindex
C2×#106230502
C2×#106227462
C2×#106226152
#130712092
#130697662
#130697612
#130697562
#130697512
C2×#106221672
#130683142
C11×#17362
#2345811