Finite Group Databasebeta

← #14785968

Subgroups — #14785968

Order 2816.

1304 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622653C211yesmaximal, characteristic, complemented
2C22×#66874C211yesmaximal, characteristic, complemented
3#13071981C211yesmaximal, characteristic, complemented
4#13069764C212yesmaximal, complemented
5#13069619C212yesmaximal, complemented
6#13069045C211yesmaximal, characteristic, complemented
7#13068950C212yesmaximal, complemented
8#13068928C212yesmaximal, complemented
9#13068909C211yesmaximal, characteristic, complemented
10#13068706C211yesmaximal, characteristic, complemented
11C2×C11×#553C211yesmaximal, characteristic
12C23×Dic22C2211yescharacteristic
13C11×#553—24no—
14C11×#553—24no—
15C2×C11×#126—21no—
16C2×C11×#125C2211yescharacteristic
17C2×C11×#123—21no—
18C2×C11×#121—21no—
19C2×C11×#121C2211yescharacteristic
20C2×C11×#121C2211yescharacteristic
21C2×C4×D4×C11—21no—
22C2×C4×D4×C11C2211yescharacteristic
23C2×C4×D4×C11C2211yescharacteristic
24C2×C11×#117C2211yescharacteristic
25C22×C11×C4⋊C4C2211yescharacteristic
26C22×#4324C2212yes—
27C2×#66977C2212yes—
28C23×C11⋊C8C2211yescharacteristic
29#10622833—21no—
30C2×#66896C2212yes—
31#10622825C2211yescharacteristic
32C2×C22⋊(C11⋊C8)C2211yescharacteristic
33C2×C22⋊(C11⋊C8)C2212yes—
34#10622653—24no—
35#10622592—21no—
36C2×#66874—24no—
37C2×#66874C2212yes—
38C2×#66874C2212yes—
39C2×#66874C2212yes—
40#10622560—21no—
41#10622559—21no—
42#10622542C2212yes—
43#10622167C2212yes—
44#10622167C2212yes—
45#10622167C2212yes—
46#10622162C2212yes—
47C22×D4×C11—21no—
48C22×D4×C11—21no—
49C22×D4×C11D411yescharacteristic
50C22×D4×C11C2311yescharacteristic
51C23×C4×C11—21no—
52C23×C4×C11—21no—
53C23×C4×C11C2311yescharacteristic
54C23×C4×C11C2311yescharacteristic
55C23×Dic11—21no—
56C22×Dic22—22no—
57C22×Dic22—24no—
58C22×Dic22—22no—
59C22×Dic22—21no—
60C22×Dic22—42no—
61C22×Dic22C2312yes—
62C11×#126—28no—
63C11×#125—22no—
64C11×#125—22no—
65C11×#123—28no—
66C11×#121—28no—
67C11×#121—22no—
68C11×#121—22no—
69C11×#121—22no—
70C11×#121D414yes—
71C4×D4×C11—28no—
72C4×D4×C11—22no—
73C4×D4×C11—22no—
74C4×D4×C11—22no—
75C4×D4×C11—22no—
76C11×#117—22no—
77C11×#117—22no—
78C2×C11×C4⋊C4—21no—
79C2×C11×C4⋊C4—22no—
80C2×C11×C4⋊C4—21no—
81C2×C11×C4⋊C4—21no—
82C2×C11×C4⋊C4D411yescharacteristic
83C2×C11×C4⋊C4D414yes—
84C2×C11×C4⋊C4C2311yescharacteristic
85C2×C11×C4⋊C4C2311yescharacteristic
86C2×C11×C4⋊C4C2311yescharacteristic
87C2×C11×C22⋊C4—21no—
88C2×C11×C22⋊C4—21no—
89C2×C11×C22⋊C4—21no—
90C2×C11×C22⋊C4—21no—
91C2×C11×C22⋊C4D411yescharacteristic
92C2×C11×C22⋊C4D411yescharacteristic
93C2×C42×C11—21no—
94C2×C42×C11C2311yescharacteristic
95C2×C22⋊Dic11—22no—
96C2×C22⋊Dic11—21no—
97C2×#4324—24no—
98C2×#4324—24no—
99C2×#4324—42no—
100C2×C4⋊Dic11C2312yes—
101#66977—24no—
102#66977—42no—
103C2×Dic11⋊C4—22no—
104C22×C11⋊C8—22no—
105C22×C11⋊C8—42no—
106C22×C11⋊C8C2312yes—
107C22×C11⋊C8C2312yes—
108#66896—42no—
109C11⋊#95—22no—
110C11⋊#95—22no—
111C11⋊#95—21no—
112C22⋊(C11⋊C8)—22no—
113C22⋊(C11⋊C8)—42no—
114#66874—24no—
115#66874—24no—
116#66874—24no—
117#23964—111nomaximal, Sylow
118C24×C11—21no—
119C24×C11C2×D411yescharacteristic
120C24×C11C2×D411yescharacteristic
121C2×D4×C11—22no—
122C2×D4×C11—24no—
123C2×D4×C11—22no—
124C2×D4×C11—22no—
125C2×D4×C11—24no—
126C2×D4×C11—21no—
127C2×D4×C11—22no—
128C2×D4×C11—22no—
129C2×D4×C11—42no—
130C2×D4×C11—42no—
131C2×D4×C11—82no—
132C2×D4×C11C2×D412yes—
133C22×C4×C11—22no—
134C22×C4×C11—22no—
135C22×C4×C11—21no—
136C22×C4×C11—21no—
137C22×C4×C11—21no—
138C22×C4×C11—24no—
139C22×C4×C11—24no—
140C22×C4×C11—21no—
141C22×C4×C11—21no—
142C22×C4×C11—22no—
143C22×C4×C11—22no—
144C22×C4×C11—82no—
145C22×C4×C11—82no—
146C22×C4×C11C2×D411yescharacteristic
147C22×C4×C11C2×D411yescharacteristic
148C22×C4×C11C2×D411yescharacteristic
149C22×C4×C11C2×D411yescharacteristic
150C22×C4×C11C2×D412yes—
151C22×C4×C11C2×D412yes—
152C22×C4×C11C2411yescharacteristic
153C22×Dic11—22no—
154C22×Dic11—22no—
155C22×Dic11—44no—
156C22×Dic11—42no—
157C2×Dic22—28no—
158C2×Dic22—22no—
159C2×Dic22—22no—
160C2×Dic22—22no—
161C2×Dic22—22no—
162C2×Dic22—22no—
163C2×Dic22—48no—
164C2×Dic22—44no—
165C2×Dic22—44no—
166C2×Dic22—42no—
167C2×Dic22#4114yes—
168C11×C4⋊C4—24no—
169C11×C4⋊C4—24no—
170C11×C4⋊C4—24no—
171C11×C4⋊C4—22no—
172C11×C4⋊C4—21no—
173C11×C4⋊C4—21no—
174C11×C4⋊C4—21no—
175C11×C4⋊C4C2×D412yes—
176C11×C4⋊C4C2×D412yes—
177C11×C4⋊C4C2×D412yes—
178C11×C22⋊C4—24no—
179C11×C22⋊C4—24no—
180C11×C22⋊C4—24no—
181C11×C22⋊C4—22no—
182C11×C22⋊C4—24no—
183C11×C22⋊C4—22no—
184C42×C11—24no—
185C42×C11—21no—
186C42×C11—21no—
187C22⋊Dic11—44no—
188C22⋊Dic11—42no—
189#4324—48no—
190C4⋊Dic11—22no—
191C4⋊Dic11—22no—
192Dic11⋊C4—44no—
193C2×C11⋊C8—22no—
194C2×C11⋊C8—22no—
195C2×C11⋊C8—22no—
196C2×C11⋊C8—22no—
197C2×C11⋊C8—22no—
198C2×C11⋊C8—22no—
199C2×C11⋊C8—44no—
200C2×#553D1111yescharacteristic
201C2×#448—111no—
202C22×#103—111no—
203#2409—111no—
204#2026—112no—
205#2000—112no—
206#1893—111no—
207#1877—112no—
208#1873—112no—
209#1869—111no—
210#1824—111no—
211C23×C11—22no—
212C23×C11—22no—
213C23×C11—21no—
214C23×C11—22no—
215C23×C11—22no—
216C23×C11—21no—
217C23×C11—21no—
218C23×C11—22no—
219C23×C11—42no—
220C23×C11—41no—
221C23×C11—41no—
222C23×C11—42no—
223C23×C11#12012yes—
224C23×C11#13512yes—
225C23×C11#13712yes—
226C23×C11C22×D411yescharacteristic
227D4×C11—24no—
228D4×C11—44no—
229D4×C11—48no—
230D4×C11—41no—
231D4×C11—41no—
232D4×C11—84no—
233C2×C4×C11—22no—
234C2×C4×C11—22no—
235C2×C4×C11—22no—
236C2×C4×C11—24no—
237C2×C4×C11—21no—
238C2×C4×C11—21no—
239C2×C4×C11—22no—
240C2×C4×C11—21no—
241C2×C4×C11—22no—
242C2×C4×C11—22no—
243C2×C4×C11—22no—
244C2×C4×C11—21no—
245C2×C4×C11—21no—
246C2×C4×C11—22no—
247C2×C4×C11—41no—
248C2×C4×C11—41no—
249C2×C4×C11—42no—
250C2×C4×C11—42no—
251C2×C4×C11—41no—
252C2×C4×C11—41no—
253C2×C4×C11—42no—
254C2×C4×C11—84no—
255C2×C4×C11—84no—
256C2×C4×C11#12012yes—
257C2×C4×C11C22×D411yescharacteristic
258C2×C4×C11C22×D411yescharacteristic
259C2×C4×C11C2×#4112yes—
260C2×C4×C11H(5,𝔽2)12yes—
261C2×Dic11—24no—
262C2×Dic11—48no—
263C2×Dic11—42no—
264C2×Dic11—42no—
265C2×Dic11—42no—
266C2×Dic11—42no—
267C2×Dic11—42no—
268Dic22—24no—
269Dic22—24no—
270Dic22—48no—
271Dic22—42no—
272Dic22—42no—
273C11⋊C8—42no—
274C11⋊C8—42no—
275C23×Q8—111no—
276C22×SD16—112no—
277C23×C8—111no—
278#553—24no—
279#553—24no—
280C2×#126—21no—
281C2×#125C2×D1111yescharacteristic
282C2×#123—21no—
283C2×C22⋊Q8—112no—
284C2×#121—21no—
285C2×#121C2×D1111yescharacteristic
286C2×#121C2×D1111yescharacteristic
287C2×C4×D4—21no—
288C2×C4×D4C2×D1111yescharacteristic
289C2×C4×D4C2×D1111yescharacteristic
290C2×#117C2×D1111yescharacteristic
291C22×C4⋊C4C2×D1111yescharacteristic
292#448—224no—
293C2×#103—112no—
294C2×#103—112no—
295C2×#103—112no—
296C2×#103—224no—
297C2×#102—112no—
298C2×C22⋊C8—111no—
299C2×C22⋊C8—112no—
300#395—221no—
301#393—221no—
302#391—112no—
303#386—221no—
304#385—221no—
305#380—111no—
306#340—112no—
307#340—112no—
308#340—112no—
309#336—112no—
310C22×C11—22no—
311C22×C11—22no—
312C22×C11—24no—
313C22×C11—24no—
314C22×C11—41no—
315C22×C11—42no—
316C22×C11—41no—
317C22×C11—42no—
318C22×C11—42no—
319C22×C11—41no—
320C22×C11—42no—
321C22×C11—42no—
322C22×C11—82no—
323C22×C11—82no—
324C22×C11#45214yes—
325C22×C11C2×#12011yescharacteristic
326C22×C11#54612yes—
327C22×C11C2×#13511yescharacteristic
328C22×C11C2×#13711yescharacteristic
329C22×C11#57712yes—
330C4×C11—22no—
331C4×C11—41no—
332C4×C11—41no—
333C4×C11—42no—
334C4×C11—41no—
335C4×C11—42no—
336C4×C11—42no—
337C4×C11#52511yescharacteristic
338C4×C11#53511yescharacteristic
339C4×C11#54612yes—
340Dic11—44no—
341Dic11—82no—
342C22×Q8—112no—
343C22×Q8—224no—
344C22×Q8—221no—
345C22×Q8—222no—
346C22×Q8—222no—
347C22×Q8—442no—
348C22×D4—21no—
349C22×D4—21no—
350C22×D4C22⋊D1111yescharacteristic
351C22×D4C22×D1111yescharacteristic
352C23×C4—21no—
353C23×C4—21no—
354C23×C4—221no—
355C23×C4C22×D1111yescharacteristic
356C23×C4C22×D1111yescharacteristic
357C2×SD16—224no—
358C2×SD16—224no—
359C2×SD16—442no—
360C22×C8—112no—
361C22×C8—112no—
362C22×C8—222no—
363C22×C8—442no—
364#126—28no—
365#125—22no—
366#125—22no—
367#123—28no—
368C22⋊Q8—224no—
369C22⋊Q8—442no—
370#121—28no—
371#121—22no—
372#121—22no—
373#121—22no—
374#121C22⋊D1114yes—
375C4×D4—28no—
376C4×D4—22no—
377C4×D4—22no—
378C4×D4—22no—
379C4×D4—22no—
380#117—22no—
381#117—22no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—112no—
387C2×C4⋊C4—222no—
388C2×C4⋊C4C22⋊D1111yescharacteristic
389C2×C4⋊C4C22⋊D1114yes—
390C2×C4⋊C4C22×D1111yescharacteristic
391C2×C4⋊C4C22×D1111yescharacteristic
392C2×C4⋊C4C22×D1111yescharacteristic
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—222no—
398C2×C22⋊C4—221no—
399C2×C22⋊C4C22⋊D1111yescharacteristic
400C2×C22⋊C4C22⋊D1111yescharacteristic
401C2×C42—21no—
402C2×C42C22×D1111yescharacteristic
403#103—224no—
404#103—224no—
405#103—224no—
406#102—442no—
407C22⋊C8—222no—
408C22⋊C8—442no—
409#95—221no—
410#95—222no—
411#95—222no—
412C2×C11—24no—
413C2×C11—81no—
414C2×C11—82no—
415C2×C11#240411yescharacteristic
416C2×C11C2×#45211yescharacteristic
417C2×C11#300911yescharacteristic
418C2×C11#330112yes—
419C2×C11#332012yes—
420C24—21no—
421C24C2×C22⋊D1111yescharacteristic
422C24C2×C22⋊D1111yescharacteristic
423C2×Q8—114no—
424C2×Q8—228no—
425C2×Q8—222no—
426C2×Q8—222no—
427C2×Q8—222no—
428C2×Q8—222no—
429C2×Q8—222no—
430C2×Q8—444no—
431C2×Q8—448no—
432C2×Q8—444no—
433C2×Q8—442no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—24no—
438C2×D4—24no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—21no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4—82no—
445C2×D4C2×C22⋊D1112yes—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—82no—
458C22×C4—82no—
459C22×C4—222no—
460C22×C4—222no—
461C22×C4—444no—
462C22×C4—442no—
463C22×C4C2×C22⋊D1111yescharacteristic
464C22×C4C2×C22⋊D1111yescharacteristic
465C22×C4C2×C22⋊D1111yescharacteristic
466C22×C4C2×C22⋊D1111yescharacteristic
467C22×C4C2×C22⋊D1112yes—
468C22×C4C2×C22⋊D1112yes—
469C22×C4C23×D1111yescharacteristic
470SD16—448no—
471C2×C8—222no—
472C2×C8—222no—
473C2×C8—222no—
474C2×C8—222no—
475C2×C8—222no—
476C2×C8—222no—
477C2×C8—444no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—222no—
486C4⋊C4—222no—
487C4⋊C4—444no—
488C4⋊C4C2×C22⋊D1112yes—
489C4⋊C4C2×C22⋊D1112yes—
490C4⋊C4C2×C22⋊D1112yes—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—22no—
495C22⋊C4—24no—
496C22⋊C4—22no—
497C22⋊C4—444no—
498C22⋊C4—442no—
499C42—24no—
500C42—21no—
501C42—21no—
502C11#2396411yescharacteristic, complemented, Sylow
503C23—22no—
504C23—22no—
505C23—21no—
506C23—21no—
507C23—22no—
508C23—22no—
509C23—21no—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—42no—
514C23—42no—
515C23#6700312yes—
516C23#6700412yes—
517C23#6700612yes—
518C23C22×C22⋊D1111yescharacteristic
519Q8—224no—
520Q8—224no—
521Q8—448no—
522Q8—442no—
523Q8—442no—
524D4—24no—
525D4—44no—
526D4—48no—
527D4—41no—
528D4—41no—
529D4—84no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—84no—
552C2×C4—84no—
553C2×C4—224no—
554C2×C4—448no—
555C2×C4—442no—
556C2×C4—442no—
557C2×C4—442no—
558C2×C4—442no—
559C2×C4—442no—
560C2×C4#6700612yes—
561C2×C4#41×D1112yes—
562C2×C4#6704212yes—
563C2×C4C22×C22⋊D1111yescharacteristic
564C2×C4C22×C22⋊D1111yescharacteristic
565C8—442no—
566C8—442no—
567C22—22no—
568C22—22no—
569C22—24no—
570C22—24no—
571C22—41no—
572C22—42no—
573C22—41no—
574C22—42no—
575C22—42no—
576C22—41no—
577C22—42no—
578C22—42no—
579C22—82no—
580C22—82no—
581C22#1062285214yes—
582C22C2×#6700311yescharacteristic
583C22C2×#6700411yescharacteristic
584C22#1062336012yes—
585C22#1062336712yes—
586C22C2×#6700611yescharacteristic
587C4—22no—
588C4—41no—
589C4—41no—
590C4—42no—
591C4—41no—
592C4—42no—
593C4—42no—
594C4—444no—
595C4—882no—
596C4#1062335911yescharacteristic
597C4#1062336012yes—
598C4#1062336111yescharacteristic
599C2—24no—
600C2—81no—
601C2—82no—
602C2#1307193111yescharacteristic
603C2C2×#1062285211yescharacteristic
604C2#1307789311yescharacteristic
605C2#1308051212yes—
606C2#1308078112yes—
subgroupG/H
centerC23C22×C22⋊D11
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
Fitting, prime radicalC2×C11×#553C2
socleC23×C11C22×D4
hypercenterC2×#553D11
hypocenterC11#23964
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#553

Complements

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

NH ≅ G/Ntype
1. C2×#10622653C2semidirect
2. C22×#66874C2semidirect
3. #13071981C2semidirect
4. #13069764C2semidirect
5. #13069619C2semidirect
6. #13069045C2semidirect
7. #13068950C2semidirect
8. #13068928C2semidirect
9. #13068909C2semidirect
10. #13068706C2semidirect
502. C11#23964semidirect

Maximal subgroups

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

subgroupindex
C2×#106226532
C22×#668742
#130719812
#130697642
#130696192
#130690452
#130689502
#130689282
#130689092
#130687062
C2×C11×#5532
#2396411