Finite Group Databasebeta

← #14775367

Subgroups — #14775367

Order 2816.

846 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622335C211yesmaximal, characteristic
2#13071641C211yesmaximal, characteristic
3#13071483C211yesmaximal, characteristic
4#13069354C211yesmaximal, characteristic, complemented
5#13069292C211yesmaximal, characteristic, complemented
6#13069251C211yesmaximal, characteristic, complemented
7#13069209C211yesmaximal, characteristic, complemented
8#13069188C211yesmaximal, characteristic, complemented
9#13069049C211yesmaximal, characteristic
10#13068873C211yesmaximal, characteristic, complemented
11#13068817C211yesmaximal, characteristic
12C4×#66882C211yesmaximal, characteristic, complemented
13#13068447C211yesmaximal, characteristic, complemented
14#13068336C211yesmaximal, characteristic
15C11×#1753C211yesmaximal, characteristic
16C22×C11×M16C2211yescharacteristic
17C2×#66979C2211yescharacteristic
18C22×C4⋊Dic11C2211yescharacteristic
19C2×#66977C2211yescharacteristic
20C2×#66930—21no—
21C2×C4×Dic22C2211yescharacteristic
22C2×C11×C22⋊C8C2211yescharacteristic
23C2×C11×#97C2211yescharacteristic
24C2×C4×C8×C11C2211yescharacteristic
25C4×C11×C22⋊C4C2211yescharacteristic
26#10622825C2211yescharacteristic
27C2×#66882C2211yescharacteristic
28C2×#66882C2211yescharacteristic
29C2×#66882C2211yescharacteristic
30C2×#66881C2211yescharacteristic
31C2×#66880C2211yescharacteristic
32C2×#66880C2211yescharacteristic
33C2×#66880C2211yescharacteristic
34C2×#66880C2211yescharacteristic
35#10622558—21no—
36#10622557—21no—
37#10622556—21no—
38C4×C4⋊Dic11C2211yescharacteristic
39C4×C4⋊Dic11C2211yescharacteristic
40C4×C4⋊Dic11C2211yescharacteristic
41#10622542C2211yescharacteristic
42#10622541C2211yescharacteristic
43#10622541C2211yescharacteristic
44#10622541C2211yescharacteristic
45#10622335C418yescomplemented
46#10622266—21no—
47#10622266—21no—
48#10622265—21no—
49#10622262—21no—
50C11×#336C2211yescharacteristic
51C11×#336C2211yescharacteristic
52#10622189C2211yescharacteristic
53#10622189C2211yescharacteristic
54#10622189C2211yescharacteristic
55#10622189C2211yescharacteristic
56#10622189C2211yescharacteristic
57#10622189C2211yescharacteristic
58#10622189C2211yescharacteristic
59#10622189C2211yescharacteristic
60C23×C4×C11C2311yescharacteristic
61C23×Dic11—21no—
62C22×Dic22C2311yescharacteristic
63C2×C11×M16—22no—
64C2×C11×M16—22no—
65C2×C11×M16—41no—
66C22×C8×C11C2311yescharacteristic
67C22×C8×C11C2311yescharacteristic
68C22×C8×C11C2311yescharacteristic
69C22×C8×C11C2311yescharacteristic
70C2×C11×C22⋊C4D411yescharacteristic
71C2×C11×C22⋊C4D411yescharacteristic
72C2×C42×C11C2311yescharacteristic
73C2×C42×C11C2311yescharacteristic
74C2×C22⋊Dic11—21no—
75C2×C22⋊Dic11—21no—
76C2×C22⋊Dic11—21no—
77#66979—22no—
78#66979—41no—
79C2×C4⋊Dic11—21no—
80C2×C4⋊Dic11—22no—
81C2×C4⋊Dic11C2×C414yes—
82C2×C4⋊Dic11C2311yescharacteristic
83C2×C4⋊Dic11C2311yescharacteristic
84C2×C4⋊Dic11C2311yescharacteristic
85C2×C4⋊Dic11C2311yescharacteristic
86C2×C4⋊Dic11C2311yescharacteristic
87C2×C4⋊Dic11C2311yescharacteristic
88C2×C4⋊Dic11C2311yescharacteristic
89#66977—22no—
90#66977C2×C414yes—
91C2×Dic11⋊C4—21no—
92C2×Dic11⋊C4—21no—
93C2×Dic11⋊C4—21no—
94C2×Dic11⋊C4—21no—
95C2×Dic11⋊C4—21no—
96C2×C4×Dic11—21no—
97C2×C4×Dic11—21no—
98C2×C4×Dic11—21no—
99C2×C4×Dic11—21no—
100C2×C4×Dic11—21no—
101#66930—44no—
102C4×Dic22—22no—
103C4×Dic22—41no—
104C11×C22⋊C8C2×C414yes—
105C11×#97—41no—
106C4×C8×C11—22no—
107C11×#95D411yescharacteristic
108C11×#95D411yescharacteristic
109C11⋊#95—21no—
110C11⋊#95—21no—
111C11⋊#95—21no—
112C11⋊#95—21no—
113C11⋊#95—21no—
114C11⋊#95—21no—
115C11⋊#95—21no—
116C11⋊#95—21no—
117C11⋊#95—21no—
118#66882—22no—
119#66882C2×C414yes—
120#66882C2×C414yes—
121#66881—41no—
122#66880—22no—
123#66880—41no—
124#66880C2×C414yes—
125#66880C2×C414yes—
126#22947—111nomaximal, Sylow
127C24×C11C2×D411yescharacteristic
128C22×C4×C11—21no—
129C22×C4×C11—21no—
130C22×C4×C11—21no—
131C22×C4×C11—21no—
132C22×C4×C11C22×C412yes—
133C22×C4×C11C2×D411yescharacteristic
134C22×C4×C11C2×D411yescharacteristic
135C22×C4×C11C2×D411yescharacteristic
136C22×C4×C11C2×D411yescharacteristic
137C22×C4×C11C2×D411yescharacteristic
138C22×C4×C11#4112yes—
139C22×C4×C11C2411yescharacteristic
140C22×Dic11—24no—
141C22×Dic11—21no—
142C22×Dic11—21no—
143C22×Dic11—21no—
144C22×Dic11—21no—
145C22×Dic11—21no—
146C22×Dic11—21no—
147C22×Dic11—21no—
148C22×Dic11—21no—
149C22×Dic11—42no—
150C22×Dic11—42no—
151C2×Dic22—21no—
152C2×Dic22—21no—
153C2×Dic22—21no—
154C2×Dic22—21no—
155C2×Dic22—21no—
156C2×Dic22C22×C412yes—
157C11×M16—44no—
158C2×C8×C11—21no—
159C2×C8×C11—21no—
160C2×C8×C11—21no—
161C2×C8×C11—21no—
162C2×C8×C11—21no—
163C2×C8×C11—21no—
164C2×C8×C11—21no—
165C2×C8×C11—21no—
166C2×C8×C11—21no—
167C2×C8×C11C22×C412yes—
168C2×C8×C11C22×C412yes—
169C2×C8×C11#4112yes—
170C11×C22⋊C4—22no—
171C11×C22⋊C4—41no—
172C42×C11—21no—
173C42×C11—21no—
174C42×C11—21no—
175C42×C11—21no—
176C22⋊Dic11—24no—
177C22⋊Dic11—42no—
178C22⋊Dic11—42no—
179C4⋊Dic11—24no—
180C4⋊Dic11—21no—
181C4⋊Dic11—21no—
182C4⋊Dic11—21no—
183C4⋊Dic11—21no—
184C4⋊Dic11—21no—
185C4⋊Dic11—21no—
186C4⋊Dic11—21no—
187C4⋊Dic11—21no—
188C4⋊Dic11—21no—
189C4⋊Dic11—21no—
190C4⋊Dic11C22×C412yes—
191C4⋊Dic11C22×C412yes—
192C4⋊Dic11C22×C412yes—
193C4⋊Dic11#4112yes—
194Dic11⋊C4—24no—
195Dic11⋊C4—42no—
196Dic11⋊C4—42no—
197Dic11⋊C4—42no—
198Dic11⋊C4—42no—
199C4×Dic11—42no—
200C4×Dic11—42no—
201C4×Dic11—42no—
202C4×Dic11—42no—
203C4×Dic11—42no—
204C2×#484—111no—
205#2362—111no—
206#2344—111no—
207#1943—111no—
208#1933—111no—
209#1927—111no—
210#1921—111no—
211#1918—111no—
212#1894—111no—
213#1863—111no—
214#1854—111no—
215C4×#107—111no—
216#1763—111no—
217#1753D1111yescharacteristic
218#1736—111no—
219C23×C11—21no—
220C23×C11—21no—
221C23×C11—21no—
222C23×C11—21no—
223C23×C11—21no—
224C23×C11—21no—
225C23×C11C4×D412yes—
226C23×C11C22×D411yescharacteristic
227C2×C4×C11—21no—
228C2×C4×C11—21no—
229C2×C4×C11—21no—
230C2×C4×C11—21no—
231C2×C4×C11—22no—
232C2×C4×C11—22no—
233C2×C4×C11—22no—
234C2×C4×C11—22no—
235C2×C4×C11—21no—
236C2×C4×C11—21no—
237C2×C4×C11—21no—
238C2×C4×C11—21no—
239C2×C4×C11—21no—
240C2×C4×C11—21no—
241C2×C4×C11—21no—
242C2×C4×C11—21no—
243C2×C4×C11—21no—
244C2×C4×C11C4×D412yes—
245C2×C4×C11AGL(1,ℤ/8)12yes—
246C2×C4×C11C23×C411yescharacteristic
247C2×C4×C11C2×#4111yescharacteristic
248C2×C4×C11C2×#4111yescharacteristic
249C2×C4×C11C2×#4111yescharacteristic
250C2×C4×C11H(5,𝔽2)11yescharacteristic
251C2×C4×C112−1+411yescharacteristic
252C2×Dic11—22no—
253C2×Dic11—22no—
254C2×Dic11—22no—
255C2×Dic11—22no—
256C2×Dic11—22no—
257C2×Dic11—48no—
258C2×Dic11—41no—
259C2×Dic11—41no—
260C2×Dic11—41no—
261C2×Dic11—41no—
262C2×Dic11—41no—
263C2×Dic11—41no—
264C2×Dic11—41no—
265C2×Dic11—41no—
266C2×Dic11—41no—
267C2×Dic11—41no—
268C2×Dic11—41no—
269C2×Dic11—41no—
270C2×Dic11—41no—
271C2×Dic11—41no—
272C2×Dic11—41no—
273C2×Dic11—41no—
274C2×Dic11—41no—
275C2×Dic11—41no—
276C2×Dic11—41no—
277Dic22—22no—
278Dic22—22no—
279Dic22—41no—
280Dic22—41no—
281C8×C11—22no—
282C8×C11—22no—
283C8×C11—22no—
284C8×C11—41no—
285C22×M16C2×D1111yescharacteristic
286C2×C22⋊Q8—111no—
287C2×C4×Q8—111no—
288C2×#117—111no—
289C2×#117—221no—
290C22×C4⋊C4—111no—
291#484—118no—
292C2×#107—111no—
293C2×#107—111no—
294C2×#107—111no—
295C2×#106—111no—
296C2×#103—111no—
297C2×#103—111no—
298C2×#103—111no—
299C2×#103—111no—
300C2×C22⋊C8C2×D1111yescharacteristic
301C2×#97C2×D1111yescharacteristic
302C2×C4×C8C2×D1111yescharacteristic
303#391—111no—
304#389—111no—
305#389—111no—
306#389—111no—
307#388—221no—
308#388—221no—
309#387—221no—
310#387—221no—
311#387—221no—
312#381—221no—
313#380—111no—
314C4×C4⋊C4—111no—
315C4×C4⋊C4—111no—
316C4×C4⋊C4—111no—
317C4×C4⋊C4—221no—
318C4×C22⋊C4C2×D1111yescharacteristic
319#340—111no—
320#340—111no—
321#340—111no—
322#340—111no—
323#340—111no—
324#340—111no—
325#340—111no—
326#340—111no—
327#336C2×D1111yescharacteristic
328#336C2×D1111yescharacteristic
329C22×C11—22no—
330C22×C11—22no—
331C22×C11—22no—
332C22×C11—41no—
333C22×C11—41no—
334C22×C11—41no—
335C22×C11—41no—
336C22×C11C2×C4×D411yescharacteristic
337C22×C11#54611yescharacteristic
338C22×C11#54711yescharacteristic
339C22×C11C2×AGL(1,ℤ/8)11yescharacteristic
340C22×C11#57511yescharacteristic
341C22×C11#57811yescharacteristic
342C22×C11#57811yescharacteristic
343C4×C11—22no—
344C4×C11—22no—
345C4×C11—41no—
346C4×C11—41no—
347C4×C11—41no—
348C4×C11C4×#4111yescharacteristic
349C4×C11#52011yescharacteristic
350C4×C11#55111yescharacteristic
351C4×C11#55211yescharacteristic
352Dic11—42no—
353Dic11—42no—
354Dic11—42no—
355Dic11—42no—
356Dic11—42no—
357Dic11—81no—
358Dic11—81no—
359Dic11—81no—
360C22×Q8—111no—
361C23×C4—221no—
362C23×C4C22×D1111yescharacteristic
363C2×M16—22no—
364C2×M16—22no—
365C2×M16—41no—
366C22×C8C22×D1111yescharacteristic
367C22×C8C22×D1111yescharacteristic
368C22×C8C22×D1111yescharacteristic
369C22×C8C22×D1111yescharacteristic
370C22⋊Q8—114no—
371C22⋊Q8—222no—
372C4×Q8—222no—
373C4×Q8—441no—
374#117—222no—
375#117—444no—
376#117—441no—
377C2×C4⋊C4—114no—
378C2×C4⋊C4—111no—
379C2×C4⋊C4—111no—
380C2×C4⋊C4—111no—
381C2×C4⋊C4—111no—
382C2×C4⋊C4—111no—
383C2×C4⋊C4—111no—
384C2×C4⋊C4—111no—
385C2×C4⋊C4—221no—
386C2×C4⋊C4—221no—
387C2×C4⋊C4—222no—
388C2×C4⋊C4—221no—
389C2×C4⋊C4—221no—
390C2×C4⋊C4—221no—
391C2×C4⋊C4—221no—
392C2×C22⋊C4—221no—
393C2×C22⋊C4—221no—
394C2×C22⋊C4—221no—
395C2×C22⋊C4D4411yescharacteristic
396C2×C22⋊C4D4411yescharacteristic
397C2×C42—221no—
398C2×C42—221no—
399C2×C42—221no—
400C2×C42—221no—
401C2×C42—221no—
402C2×C42C22×D1111yescharacteristic
403C2×C42C22×D1111yescharacteristic
404#107—114no—
405#107—114no—
406#107—222no—
407#106—441no—
408#103—114no—
409#103—114no—
410#103—222no—
411#103—441no—
412C22⋊C8C4×D1114yes—
413#97—41no—
414C4×C8—22no—
415#95—221no—
416#95—221no—
417#95—221no—
418#95—221no—
419#95—221no—
420#95—221no—
421#95—221no—
422#95—221no—
423#95—221no—
424#95D4411yescharacteristic
425#95D4411yescharacteristic
426C2×C11—42no—
427C2×C11#235911yescharacteristic
428C2×C11C4×AGL(1,ℤ/8)11yescharacteristic
429C2×C11#295011yescharacteristic
430C2×C11#332711yescharacteristic
431C2×C11#333011yescharacteristic
432C2×C11#334311yescharacteristic
433C2×C11#334911yescharacteristic
434C24C2×D4411yescharacteristic
435C2×Q8—112no—
436C2×Q8—221no—
437C2×Q8—221no—
438C2×Q8—221no—
439C2×Q8—221no—
440C2×Q8—221no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—224no—
446C22×C4—221no—
447C22×C4—221no—
448C22×C4—221no—
449C22×C4—221no—
450C22×C4—221no—
451C22×C4—221no—
452C22×C4—221no—
453C22×C4—221no—
454C22×C4—442no—
455C22×C4—442no—
456C22×C4C2×C4×D1112yes—
457C22×C4C2×D4411yescharacteristic
458C22×C4C2×D4411yescharacteristic
459C22×C4C2×D4411yescharacteristic
460C22×C4C2×D4411yescharacteristic
461C22×C4C2×D4411yescharacteristic
462C22×C4#433912yes—
463C22×C4C23×D1111yescharacteristic
464M16—44no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8C2×C4×D1112yes—
475C2×C8C2×C4×D1112yes—
476C2×C8#434112yes—
477C4⋊C4—112no—
478C4⋊C4—112no—
479C4⋊C4—112no—
480C4⋊C4—112no—
481C4⋊C4—224no—
482C4⋊C4—224no—
483C4⋊C4—221no—
484C4⋊C4—221no—
485C4⋊C4—221no—
486C4⋊C4—221no—
487C4⋊C4—221no—
488C4⋊C4—221no—
489C4⋊C4—221no—
490C4⋊C4—221no—
491C4⋊C4—221no—
492C4⋊C4—221no—
493C4⋊C4—442no—
494C4⋊C4—442no—
495C4⋊C4—442no—
496C4⋊C4—442no—
497C22⋊C4—22no—
498C22⋊C4—41no—
499C22⋊C4—224no—
500C22⋊C4—442no—
501C22⋊C4—442no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—442no—
507C42—442no—
508C42—442no—
509C42—442no—
510C42—442no—
511C11#2294711yescharacteristic, complemented, Sylow
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23C4×D4412yes—
519C23C22×D4411yescharacteristic
520Q8—222no—
521Q8—222no—
522Q8—441no—
523Q8—441no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—222no—
542C2×C4—222no—
543C2×C4—222no—
544C2×C4—222no—
545C2×C4—222no—
546C2×C4—448no—
547C2×C4—441no—
548C2×C4—441no—
549C2×C4—441no—
550C2×C4—441no—
551C2×C4—441no—
552C2×C4—441no—
553C2×C4—441no—
554C2×C4—441no—
555C2×C4—441no—
556C2×C4—441no—
557C2×C4—441no—
558C2×C4—441no—
559C2×C4—441no—
560C2×C4—441no—
561C2×C4—441no—
562C2×C4—441no—
563C2×C4—441no—
564C2×C4—441no—
565C2×C4—441no—
566C2×C4C4×D4412yes—
567C2×C4#6696112yes—
568C2×C4C22×C4×D1111yescharacteristic
569C2×C4C2×#433911yescharacteristic
570C2×C4C2×#434111yescharacteristic
571C2×C4#6703711yescharacteristic
572C2×C4#41×D1111yescharacteristic
573C2×C4#6704311yescharacteristic
574C8—22no—
575C8—22no—
576C8—22no—
577C8—41no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22C2×C4×D4411yescharacteristic
586C22#1062308511yescharacteristic
587C22#1062308611yescharacteristic
588C22C2×#6696111yescharacteristic
589C22#1062327911yescharacteristic
590C22#1062328411yescharacteristic
591C22#1062328411yescharacteristic
592C4—22no—
593C4—22no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—442no—
598C4—442no—
599C4—442no—
600C4—442no—
601C4—442no—
602C4—881no—
603C4—881no—
604C4—881no—
605C4#1062302611yescharacteristic
606C4#1062306211yescharacteristic
607C4C4×#434111yescharacteristic
608C4#1062307711yescharacteristic
609C2—42no—
610C2#1307161011yescharacteristic
611C2C4×#6696111yescharacteristic
612C2#1307748511yescharacteristic
613C2#1308088611yescharacteristic
614C2#1308092211yescharacteristic
615C2#1308105811yescharacteristic
616C2#1308114811yescharacteristic
subgroupG/H
centerC23C22×D44
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingC11×#1753C2
socleC23×C11C22×D4
prime radicalC24×C11C2×D4
cosocleC22×C4×C11C24
hypercenter#1753D11
hypocenterC11#22947
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1753

Complements

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

NH ≅ G/Ntype
4. #13069354C2semidirect
5. #13069292C2semidirect
6. #13069251C2semidirect
7. #13069209C2semidirect
8. #13069188C2semidirect
10. #13068873C2semidirect
12. C4×#66882C2semidirect
13. #13068447C2semidirect
45. #10622335C4semidirect
511. C11#22947semidirect

Maximal subgroups

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

subgroupindex
C2×#106223352
#130716412
#130714832
#130693542
#130692922
#130692512
#130692092
#130691882
#130690492
#130688732
#130688172
C4×#668822
#130684472
#130683362
C11×#17532
#2294711