Finite Group Databasebeta

← #14671263

Subgroups — #14671263

Order 2816.

1088 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#66897C211yesmaximal, characteristic, complemented
2C2×#10622174C212yesmaximal, complemented
3#13068531C211yesmaximal, characteristic, complemented
4#13066734C212yesmaximal, complemented
5C11×#2403C211yesmaximal, characteristic
6C23×D4×C11C411yescharacteristic, complemented
7C23×D4×C11C2211yescharacteristic
8C2×C11×#120—21no—
9C2×C11×#120C411yescharacteristic, complemented
10C2×C11×#120C2211yescharacteristic
11C22×#4318—21no—
12C11×#392—21no—
13C11×#392—21no—
14C11×#386—21no—
15C11×#386C411yescharacteristic, complemented
16C11×#386C2211yescharacteristic
17C11×#379C411yescharacteristic, complemented
18#10622860—21no—
19C2×#66898—21no—
20C2×#66898C2212yes—
21C2×#66897—21no—
22C2×#66897—24no—
23C2×#66897C2212yes—
24#10622174—24no—
25C25×C11C2×C411yescharacteristic
26C25×C11C2×C411yescharacteristic
27C25×C11D411yescharacteristic
28C22×D4×C11—21no—
29C22×D4×C11—22no—
30C22×D4×C11—22no—
31C22×D4×C11—21no—
32C22×D4×C11—21no—
33C22×D4×C11—21no—
34C22×D4×C11—42no—
35C22×D4×C11—41no—
36C22×D4×C11—41no—
37C22×D4×C11—44no—
38C22×D4×C11—41no—
39C22×D4×C11—41no—
40C22×D4×C11C2×C411yescharacteristic
41C22×D4×C11D411yescharacteristic
42C22×D4×C11D411yescharacteristic
43C22×D4×C11C2311yescharacteristic
44C23×C4×C11C2×C411yescharacteristic
45C23×C4×C11D411yescharacteristic
46C11×#120—22no—
47C11×#120—22no—
48C11×#120—44no—
49C11×#120—42no—
50C2×C11×C22⋊C4—21no—
51C2×C11×C22⋊C4—21no—
52C2×C11×C22⋊C4—21no—
53C2×C11×C22⋊C4—21no—
54C2×C11×C22⋊C4—21no—
55C2×C11×C22⋊C4—21no—
56C2×C11×C22⋊C4C2×C411yescharacteristic
57C2×C11×C22⋊C4C2×C411yescharacteristic
58C2×C22⋊Dic11—22no—
59C2×C22⋊Dic11—41no—
60C2×C22⋊Dic11—41no—
61C2×#4318—24no—
62C2×#4318—22no—
63C2×#4318—41no—
64C2×#4318—42no—
65C22×C11⋊C8—41no—
66C11×#95—21no—
67C11×#95—21no—
68#66898—22no—
69#66898—22no—
70#66898—42no—
71#66897—24no—
72#66897—22no—
73#66897D414yescomplemented
74#13082—111nomaximal, Sylow
75C24×C11—21no—
76C24×C11—21no—
77C24×C11—21no—
78C24×C11—21no—
79C24×C11—21no—
80C24×C11—22no—
81C24×C11—22no—
82C24×C11—22no—
83C24×C11—41no—
84C24×C11—41no—
85C24×C11—42no—
86C24×C11—41no—
87C24×C11—42no—
88C24×C11—42no—
89C24×C11—41no—
90C24×C11—81no—
91C24×C11—81no—
92C24×C11—81no—
93C24×C11C22⋊C411yescharacteristic
94C24×C11C22⋊C411yescharacteristic
95C24×C11C22⋊C411yescharacteristic
96C24×C11C22×C411yescharacteristic
97C24×C11C2×D411yescharacteristic
98C2×D4×C11—24no—
99C2×D4×C11—22no—
100C2×D4×C11—22no—
101C2×D4×C11—22no—
102C2×D4×C11—22no—
103C2×D4×C11—21no—
104C2×D4×C11—21no—
105C2×D4×C11—41no—
106C2×D4×C11—41no—
107C2×D4×C11—41no—
108C2×D4×C11—41no—
109C2×D4×C11—41no—
110C2×D4×C11—41no—
111C2×D4×C11—44no—
112C2×D4×C11—44no—
113C2×D4×C11—44no—
114C2×D4×C11—44no—
115C2×D4×C11—42no—
116C2×D4×C11—42no—
117C2×D4×C11—42no—
118C2×D4×C11—42no—
119C2×D4×C11—44no—
120C2×D4×C11—44no—
121C2×D4×C11—44no—
122C2×D4×C11—41no—
123C2×D4×C11—41no—
124C2×D4×C11—44no—
125C2×D4×C11—42no—
126C2×D4×C11—42no—
127C2×D4×C11—42no—
128C2×D4×C11—42no—
129C2×D4×C11—81no—
130C2×D4×C11—81no—
131C2×D4×C11C2×D412yes—
132C2×D4×C11#4112yes—
133C22×C4×C11—21no—
134C22×C4×C11—21no—
135C22×C4×C11—22no—
136C22×C4×C11—21no—
137C22×C4×C11—21no—
138C22×C4×C11—21no—
139C22×C4×C11—21no—
140C22×C4×C11—21no—
141C22×C4×C11—42no—
142C22×C4×C11—41no—
143C22×C4×C11C22⋊C411yescharacteristic
144C22×C4×C11C2×D411yescharacteristic
145C22×Dic11—42no—
146C11×C22⋊C4—22no—
147C11×C22⋊C4—42no—
148C11×C22⋊C4—42no—
149C11×C22⋊C4—42no—
150C11×C22⋊C4—42no—
151C11×C22⋊C4—41no—
152C11×C22⋊C4—41no—
153C11×C22⋊C4—41no—
154C11×C22⋊C4—81no—
155C22⋊Dic11—42no—
156C22⋊Dic11—42no—
157C22⋊Dic11—82no—
158C22⋊Dic11—82no—
159#4318—24no—
160#4318—44no—
161#4318—42no—
162C2×C11⋊C8—44no—
163C2×C11⋊C8—42no—
164C22×#100—111no—
165#2403D1111yescharacteristic
166C2×AΣL(1,𝔽16)—112no—
167#1781—111no—
168#1390—112no—
169C23×C11—22no—
170C23×C11—21no—
171C23×C11—21no—
172C23×C11—21no—
173C23×C11—21no—
174C23×C11—21no—
175C23×C11—21no—
176C23×C11—22no—
177C23×C11—21no—
178C23×C11—21no—
179C23×C11—21no—
180C23×C11—21no—
181C23×C11—21no—
182C23×C11—21no—
183C23×C11—41no—
184C23×C11—42no—
185C23×C11—42no—
186C23×C11—42no—
187C23×C11—42no—
188C23×C11—41no—
189C23×C11—41no—
190C23×C11—41no—
191C23×C11—41no—
192C23×C11—41no—
193C23×C11—41no—
194C23×C11—42no—
195C23×C11—41no—
196C23×C11—42no—
197C23×C11—41no—
198C23×C11—41no—
199C23×C11—41no—
200C23×C11—41no—
201C23×C11—42no—
202C23×C11—42no—
203C23×C11—42no—
204C23×C11—42no—
205C23×C11—41no—
206C23×C11—41no—
207C23×C11—41no—
208C23×C11—81no—
209C23×C11—81no—
210C23×C11—81no—
211C23×C11—81no—
212C23×C11—82no—
213C23×C11—84no—
214C23×C11—84no—
215C23×C11—84no—
216C23×C11—84no—
217C23×C11—84no—
218C23×C11—84no—
219C23×C11—82no—
220C23×C11#9911yescharacteristic
221C23×C11#9911yescharacteristic
222C23×C11C2×C22⋊C411yescharacteristic
223C23×C11C4×D412yes—
224C23×C11#12112yes—
225D4×C11—42no—
226D4×C11—42no—
227D4×C11—44no—
228D4×C11—44no—
229D4×C11—42no—
230D4×C11—42no—
231D4×C11—44no—
232D4×C11—48no—
233D4×C11—44no—
234D4×C11—84no—
235D4×C11—84no—
236C2×C4×C11—22no—
237C2×C4×C11—22no—
238C2×C4×C11—22no—
239C2×C4×C11—22no—
240C2×C4×C11—21no—
241C2×C4×C11—21no—
242C2×C4×C11—21no—
243C2×C4×C11—21no—
244C2×C4×C11—22no—
245C2×C4×C11—41no—
246C2×C4×C11—42no—
247C2×C4×C11—41no—
248C2×C4×C11—42no—
249C2×C4×C11—41no—
250C2×C4×C11—41no—
251C2×C4×C11—41no—
252C2×C4×C11—41no—
253C2×C4×C11—41no—
254C2×C4×C11—41no—
255C2×C4×C11#9912yes—
256C2×C4×C11#12011yescharacteristic
257C2×C4×C11#12311yescharacteristic
258C2×Dic11—82no—
259C2×Dic11—82no—
260C2×Dic11—82no—
261C11⋊C8—44no—
262C23×D4Dic1111yescharacteristic, complemented
263C23×D4C2×D1111yescharacteristic
264C22×M16—221no—
265C2×#120—21no—
266C2×#120Dic1111yescharacteristic, complemented
267C2×#120C2×D1111yescharacteristic
268C2×#100—112no—
269C2×#100—224no—
270C2×#100—221no—
271C2×#99—112no—
272C2×#99—221no—
273#392—21no—
274#392—21no—
275#386—21no—
276#386Dic1111yescharacteristic, complemented
277#386C2×D1111yescharacteristic
278#379—221no—
279#379Dic1111yescharacteristic, complemented
280AΣL(1,𝔽16)—224no—
281C22×C11—21no—
282C22×C11—21no—
283C22×C11—21no—
284C22×C11—21no—
285C22×C11—21no—
286C22×C11—21no—
287C22×C11—21no—
288C22×C11—22no—
289C22×C11—22no—
290C22×C11—42no—
291C22×C11—42no—
292C22×C11—42no—
293C22×C11—42no—
294C22×C11—42no—
295C22×C11—42no—
296C22×C11—42no—
297C22×C11—42no—
298C22×C11—41no—
299C22×C11—41no—
300C22×C11—41no—
301C22×C11—41no—
302C22×C11—42no—
303C22×C11—41no—
304C22×C11—41no—
305C22×C11—41no—
306C22×C11—41no—
307C22×C11—41no—
308C22×C11—41no—
309C22×C11—81no—
310C22×C11—84no—
311C22×C11—84no—
312C22×C11—81no—
313C22×C11—81no—
314C22×C11—81no—
315C22×C11—81no—
316C22×C11—81no—
317C22×C11—84no—
318C22×C11—84no—
319C22×C11—84no—
320C22×C11—84no—
321C22×C11—81no—
322C22×C11—81no—
323C22×C11AΣL(1,𝔽16)12yescomplemented
324C22×C11#38611yescharacteristic
325C22×C11C2×#9911yescharacteristic
326C22×C11C2×#9911yescharacteristic
327C4×C11—22no—
328C4×C11—21no—
329C4×C11—21no—
330C4×C11—42no—
331C4×C11—81no—
332C4×C11—81no—
333Dic11—162no—
334C25C2×Dic1111yescharacteristic
335C25C2×Dic1111yescharacteristic
336C25C22⋊D1111yescharacteristic
337C22×D4—21no—
338C22×D4—22no—
339C22×D4—22no—
340C22×D4—21no—
341C22×D4—21no—
342C22×D4—21no—
343C22×D4—42no—
344C22×D4—41no—
345C22×D4—41no—
346C22×D4—44no—
347C22×D4—41no—
348C22×D4—41no—
349C22×D4C2×Dic1111yescharacteristic
350C22×D4C22⋊D1111yescharacteristic
351C22×D4C22⋊D1111yescharacteristic
352C22×D4C22×D1111yescharacteristic
353C23×C4C2×Dic1111yescharacteristic
354C23×C4C22⋊D1111yescharacteristic
355C2×M16—224no—
356C2×M16—222no—
357C2×M16—441no—
358C2×M16—442no—
359C22×C8—441no—
360#120—22no—
361#120—22no—
362#120—44no—
363#120—42no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—222no—
371C2×C22⋊C4—441no—
372C2×C22⋊C4—441no—
373C2×C22⋊C4C2×Dic1111yescharacteristic
374C2×C22⋊C4C2×Dic1111yescharacteristic
375#100—114no—
376#100—224no—
377#100—222no—
378#99—222no—
379#99—222no—
380#99—442no—
381#95—21no—
382#95—21no—
383C2×C11—21no—
384C2×C11—21no—
385C2×C11—22no—
386C2×C11—42no—
387C2×C11—41no—
388C2×C11—42no—
389C2×C11—81no—
390C2×C11—82no—
391C2×C11—81no—
392C2×C11#189811yescharacteristic
393C2×C11C2×AΣL(1,𝔽16)11yescharacteristic
394C2×C11#212111yescharacteristic
395C24—21no—
396C24—21no—
397C24—21no—
398C24—21no—
399C24—21no—
400C24—22no—
401C24—22no—
402C24—22no—
403C24—41no—
404C24—41no—
405C24—41no—
406C24—42no—
407C24—42no—
408C24—42no—
409C24—41no—
410C24—81no—
411C24—81no—
412C24—81no—
413C24C22⋊Dic1111yescharacteristic
414C24C22⋊Dic1111yescharacteristic
415C24C22⋊Dic1111yescharacteristic
416C24C22×Dic1111yescharacteristic
417C24C2×C22⋊D1111yescharacteristic
418C2×D4—24no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—41no—
426C2×D4—41no—
427C2×D4—41no—
428C2×D4—41no—
429C2×D4—41no—
430C2×D4—41no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—44no—
440C2×D4—44no—
441C2×D4—44no—
442C2×D4—41no—
443C2×D4—41no—
444C2×D4—44no—
445C2×D4—42no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—81no—
450C2×D4—81no—
451C2×D4D4×D1112yes—
452C2×D4#434112yes—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—42no—
462C22×C4—41no—
463C22×C4—442no—
464C22×C4C22⋊Dic1111yescharacteristic
465C22×C4C2×C22⋊D1111yescharacteristic
466M16—224no—
467M16—444no—
468M16—442no—
469C2×C8—444no—
470C2×C8—442no—
471C22⋊C4—22no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—41no—
477C22⋊C4—41no—
478C22⋊C4—41no—
479C22⋊C4—81no—
480C22⋊C4—442no—
481C22⋊C4—442no—
482C22⋊C4—882no—
483C22⋊C4—882no—
484C11#1308211yescharacteristic, complemented, Sylow
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—22no—
491C23—21no—
492C23—22no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—41no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—42no—
511C23—41no—
512C23—42no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—42no—
518C23—42no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—42no—
523C23—42no—
524C23—81no—
525C23—81no—
526C23—81no—
527C23—81no—
528C23—82no—
529C23—84no—
530C23—84no—
531C23—84no—
532C23—84no—
533C23—84no—
534C23—84no—
535C23—82no—
536C23#6689811yescharacteristic
537C23#6689811yescharacteristic
538C23D4×Dic1112yes—
539C23#6699212yes—
540C23C2×C22⋊Dic1111yescharacteristic
541D4—42no—
542D4—42no—
543D4—44no—
544D4—44no—
545D4—42no—
546D4—42no—
547D4—44no—
548D4—48no—
549D4—44no—
550D4—84no—
551D4—84no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—882no—
572C2×C4—882no—
573C2×C4—882no—
574C2×C4#6689812yes—
575C2×C4#6698811yescharacteristic
576C2×C4#6699011yescharacteristic
577C8—444no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—22no—
584C22—22no—
585C22—21no—
586C22—21no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—42no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—81no—
607C22—84no—
608C22—84no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—84no—
615C22—84no—
616C22—84no—
617C22—84no—
618C22—81no—
619C22—81no—
620C22#1062217412yescomplemented
621C22C2×#6689811yescharacteristic
622C22C2×#6689811yescharacteristic
623C22#1062283311yescharacteristic
624C4—21no—
625C4—22no—
626C4—21no—
627C4—42no—
628C4—81no—
629C4—81no—
630C4—1762no—
631C2—21no—
632C2—21no—
633C2—22no—
634C2—42no—
635C2—41no—
636C2—42no—
637C2—81no—
638C2—82no—
639C2—81no—
640C2#1306907011yescharacteristic
641C2C2×#1062217411yescharacteristic
642C2#1307026211yescharacteristic
subgroupG/H
centerC22C2×#66898
commutatorC24×C11C22×C4
FrattiniC22×D4C22×D11
Fitting, prime radicalC11×#2403C2
socleC22×C11C2×#99
cosocleC22×D4×C11C23
hypercenter#2403D11
hypocenterC11#13082
LCSG ▷ C24×C11 ▷ C22×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #2403

Complements

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

NH ≅ G/Ntype
1. C22×#66897C2semidirect
2. C2×#10622174C2semidirect
3. #13068531C2semidirect
4. #13066734C2semidirect
6. C23×D4×C11C4semidirect
9. C2×C11×#120C4semidirect
15. C11×#386C4semidirect
17. C11×#379C4semidirect
73. #66897D4semidirect
262. C23×D4Dic11semidirect
266. C2×#120Dic11semidirect
276. #386Dic11semidirect
279. #379Dic11semidirect
323. C22×C11AΣL(1,𝔽16)semidirect
484. C11#13082semidirect
620. C22#10622174semidirect

Maximal subgroups

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

subgroupindex
C22×#668972
C2×#106221742
#130685312
#130667342
C11×#24032
#1308211