Finite Group Databasebeta

← #14917025

Subgroups — #14917025

Order 2816.

2026 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10623109C211yesmaximal, characteristic
2#13071798C214yesmaximal, complemented
3#13071620C212yesmaximal
4#13071617C212yesmaximal
5#13071583C211yesmaximal, characteristic, complemented
6#13071581C211yesmaximal, characteristic, complemented
7#13071536C211yesmaximal, characteristic, complemented
8#13071479C212yesmaximal
9#13071469C212yesmaximal
10#13071363C212yesmaximal, complemented
11#13071350C214yesmaximal
12#13071332C212yesmaximal, complemented
13#13071264C212yesmaximal, complemented
14#13071262C212yesmaximal, complemented
15#13071012C211yesmaximal, characteristic
16D11×#382C211yesmaximal, characteristic
17C11×#2351C211yesmaximal, characteristic, complemented
18C2×C4×Q8×C11C2212yes—
19C2×C4×Q8×C11C2211yescharacteristic
20C2×#66999C2211yescharacteristic
21C2×#66997C2211yescharacteristic
22#10623109C4116yescomplemented
23C2×#66950C2211yescharacteristic
24C2×#66949C2212yes—
25C2×C4⋊C4×D11C2212yes—
26C2×#66942C2212yes—
27C2×#66941C2211yescharacteristic
28C2×#66925C2212yes—
29C2×C42×D11C2211yescharacteristic
30C2×C4×Dic22C2212yes—
31C2×C4×Dic22C2212yes—
32C2×C4×Dic22C2211yescharacteristic
33C11×#389C2212yes—
34C11×#389C2214yes—
35C11×#382C2212yes—
36C11×#382C2211yescharacteristic
37C4×C11×C4⋊C4C2212yes—
38C4×C11×C4⋊C4C2211yescharacteristic
39#10622843C2212yes—
40#10622602C2214yes—
41#10622602C2214yes—
42#10622600C2212yes—
43#10622594C2214yes—
44#10622591C2214yes—
45#10622591C2214yes—
46#10622591C2214yes—
47#10622590C2212yes—
48#10622590C2212yes—
49C4⋊C4×Dic11C2212yes—
50C4⋊C4×Dic11C2212yes—
51#10622587C2214yes—
52#10622587C2212yes—
53#10622586C2214yes—
54#10622585C2214yes—
55#10622585C2214yes—
56C4×(C2×D11)⋊C4C2212yes—
57C4×(C2×D11)⋊C4C2214yes—
58#10622550C2212yes—
59#10622549C2211yescharacteristic
60C4×C4⋊Dic11C2211yescharacteristic
61#10622546C2214yes—
62#10622546C2214yes—
63#10622546C2214yes—
64#10622545C2212yes—
65#10622545C2212yes—
66#10622541C2212yes—
67#10622279C2214yes—
68#10622277C2211yescharacteristic
69#10622277C2211yescharacteristic
70D11×#95C2212yes—
71D11×#95C2212yes—
72#10622268C2214yes—
73#10622267C2214yes—
74#10622265C2214yes—
75#10622264C2212yes—
76#10622264C2211yescharacteristic
77#10622264C2211yescharacteristic
78#10622263C2212yes—
79#10622263C2212yes—
80#10622263C2212yes—
81#10622262C2212yes—
82#10622261C2212yes—
83#10622259C2214yes—
84C22×Q8×C11C2311yescharacteristic
85C22×C4×D11C2311yescharacteristic
86C22×C4×D11C2311yescharacteristic
87C22×C4×D11C2314yes—
88C22×C4×D11C2311yescharacteristic
89C22×Dic22C2311yescharacteristic
90C22×Dic22C2312yes—
91C4×Q8×C11—44no—
92C4×Q8×C11C2×C418yescomplemented
93C2×C11×C4⋊C4C2312yes—
94C2×C11×C4⋊C4C2312yes—
95C2×C11×C4⋊C4C2314yes—
96C2×C11×C4⋊C4C2314yes—
97C2×C11×C4⋊C4C2311yescharacteristic
98C2×C11×C4⋊C4C2312yes—
99C2×C42×C11C2312yes—
100C2×C42×C11C2314yes—
101C2×C42×C11C2311yescharacteristic
102C2×C42×C11C2312yes—
103#66999C2×C418yes—
104#66997C2×C418yescomplemented
105C2×(C2×D11)⋊C4C2312yes—
106C2×(C2×D11)⋊C4C2312yes—
107C2×(C2×D11)⋊C4C2312yes—
108C2×(C2×D11)⋊C4C2312yes—
109C2×C4⋊Dic11C2312yes—
110C2×C4⋊Dic11C2312yes—
111C2×C4⋊Dic11C2311yescharacteristic
112C2×C4⋊Dic11C2312yes—
113C2×Dic11⋊C4C2314yes—
114C2×Dic11⋊C4C2314yes—
115C2×Dic11⋊C4C2314yes—
116C2×Dic11⋊C4C2312yes—
117C2×Dic11⋊C4C2312yes—
118C2×Dic11⋊C4C2312yes—
119C2×Dic11⋊C4C2312yes—
120C2×Dic11⋊C4C2312yes—
121C2×C4×Dic11C2312yes—
122C2×C4×Dic11C2314yes—
123C2×C4×Dic11C2314yes—
124C2×C4×Dic11C2314yes—
125C2×C4×Dic11C2311yescharacteristic
126C2×C4×Dic11C2312yes—
127C2×C4×Dic11C2311yescharacteristic
128C2×C4×Dic11C2311yescharacteristic
129#66950C2×C418yes—
130#66949C2×C4116yes—
131C4⋊C4×D11—44no—
132#66942C2×C4116yescomplemented
133#66941C2×C418yescomplemented
134#66925C2×C4116yes—
135C42×D11C2×C418yes—
136C4×Dic22—44no—
137C4×Dic22C2×C4116yescomplemented
138C4×Dic22C2×C418yescomplemented
139C11×#95C2314yes—
140C11×#95C2312yes—
141C11×#95C2311yescharacteristic
142C11×#95C2311yescharacteristic
143C11×#95C2312yes—
144C11⋊#95C2312yes—
145C11⋊#95C2314yes—
146C11⋊#95C2314yes—
147C11⋊#95C2311yescharacteristic
148C11⋊#95C2314yes—
149C11⋊#95C2312yes—
150C11⋊#95C2314yes—
151C11⋊#95C2312yes—
152C11⋊#95C2314yes—
153C11⋊#95C2311yescharacteristic
154C11⋊#95C2314yes—
155C11⋊#95C2312yes—
156C11⋊#95C2314yes—
157C11⋊#95C2312yes—
158C11⋊#95C2312yes—
159C11⋊#95C2312yes—
160C11⋊#95C2312yes—
161C11⋊#95C2314yes—
162C11⋊#95C2314yes—
163#34399—111nomaximal, Sylow
164C23×D11C2411yescharacteristic
165C2×Q8×C11—22no—
166C2×Q8×C11—22no—
167C2×Q8×C11C22×C414yes—
168C22×C4×C11C2414yes—
169C22×C4×C11C2412yes—
170C22×C4×C11C2412yes—
171C22×C4×C11C2411yescharacteristic
172C22×C4×C11C2412yes—
173C22×C4×C11C2412yes—
174C22×C4×C11C2411yescharacteristic
175C22×C4×C11C2411yescharacteristic
176C22×Dic11C2412yes—
177C22×Dic11C2412yes—
178C22×Dic11C2414yes—
179C22×Dic11C2411yescharacteristic
180C22×Dic11C2411yescharacteristic
181C22×Dic11C2412yes—
182C22×Dic11C2412yes—
183C22×Dic11C2411yescharacteristic
184C2×C4×D11—24no—
185C2×C4×D11—28no—
186C2×C4×D11—22no—
187C2×C4×D11—22no—
188C2×C4×D11—42no—
189C2×C4×D11—44no—
190C2×C4×D11—44no—
191C2×C4×D11C22×C414yes—
192C2×C4×D11C22×C414yes—
193C2×C4×D11C22×C414yes—
194C2×Dic22—22no—
195C2×Dic22—22no—
196C2×Dic22—44no—
197C2×Dic22C22×C414yes—
198C2×Dic22C22×C418yes—
199C11×C4⋊C4—24no—
200C11×C4⋊C4—44no—
201C11×C4⋊C4—42no—
202C11×C4⋊C4—44no—
203C11×C4⋊C4C22×C414yes—
204C11×C4⋊C4C22×C418yes—
205C42×C11—24no—
206C42×C11—44no—
207C42×C11C22×C414yes—
208C42×C11C22×C418yes—
209(C2×D11)⋊C4—24no—
210(C2×D11)⋊C4—24no—
211(C2×D11)⋊C4C22×C418yes—
212(C2×D11)⋊C4C22×C418yes—
213C4⋊Dic11—24no—
214C4⋊Dic11—42no—
215C4⋊Dic11C22×C414yes—
216C4⋊Dic11C22×C418yes—
217Dic11⋊C4—44no—
218Dic11⋊C4—44no—
219Dic11⋊C4—44no—
220Dic11⋊C4C22×C418yes—
221Dic11⋊C4C22×C418yes—
222Dic11⋊C4C22×C418yes—
223Dic11⋊C4C22×C418yes—
224Dic11⋊C4C22×C418yes—
225C4×Dic11—24no—
226C4×Dic11—44no—
227C4×Dic11—44no—
228C4×Dic11—44no—
229C4×Dic11C22×C414yes—
230C4×Dic11C22×C418yes—
231C4×Dic11C22×C414yes—
232C4×Dic11C22×C414yes—
233C2×#533—111no—
234#2382—114no—
235#2360—112no—
236#2360—112no—
237#2356—111no—
238#2356—111no—
239#2351—111no—
240#2351D1111yescharacteristic, complemented
241#2343—112no—
242#2343—112no—
243#2331—112no—
244#2329—114no—
245#2327—112no—
246#2320—112no—
247#2320—112no—
248#2283—111no—
249C2×#382—111no—
250C23×C11C2511yescharacteristic
251C22×D11—21no—
252C22×D11—21no—
253C22×D11—22no—
254C22×D11C23×C412yes—
255C22×D112−1+414yes—
256Q8×C11—28no—
257C2×C4×C11—22no—
258C2×C4×C11—22no—
259C2×C4×C11—22no—
260C2×C4×C11—24no—
261C2×C4×C11—22no—
262C2×C4×C11—22no—
263C2×C4×C11—22no—
264C2×C4×C11—24no—
265C2×C4×C11—22no—
266C2×C4×C11—22no—
267C2×C4×C11—24no—
268C2×C4×C11—24no—
269C2×C4×C11—22no—
270C2×C4×C11—22no—
271C2×C4×C11C23×C412yes—
272C2×C4×C11C23×C412yes—
273C2×C4×C11C23×C414yes—
274C2×C4×C11C23×C414yes—
275C2×C4×C11C23×C412yes—
276C2×C4×C11H(5,𝔽2)12yes—
277C2×C4×C112−1+412yes—
278C2×Dic11—22no—
279C2×Dic11—21no—
280C2×Dic11—22no—
281C2×Dic11—21no—
282C2×Dic11—22no—
283C2×Dic11—24no—
284C2×Dic11—22no—
285C2×Dic11—24no—
286C2×Dic11—22no—
287C2×Dic11—22no—
288C2×Dic11—22no—
289C2×Dic11—22no—
290C2×Dic11—24no—
291C2×Dic11—22no—
292C2×Dic11—22no—
293C2×Dic11—24no—
294C2×Dic11C23×C412yes—
295C2×Dic11C23×C412yes—
296C2×Dic11C23×C412yes—
297C2×Dic11C23×C414yes—
298C2×Dic11C23×C414yes—
299C4×D11—28no—
300C4×D11—28no—
301C4×D11—44no—
302C4×D11—88no—
303Dic22—28no—
304Dic22—48no—
305#533—1116no—
306C2×C4⋊Q8—111no—
307C2×C4⋊Q8—111no—
308C2×#125—112no—
309C2×C22⋊Q8—111no—
310C2×C22⋊Q8—112no—
311C2×C22⋊Q8—111no—
312C2×C4×Q8—112no—
313C2×C4×Q8—111no—
314C2×C4×Q8—112no—
315C2×C4×Q8C2×D1112yes—
316C2×C4×Q8C2×D1111yescharacteristic
317C2×#117—112no—
318C22×C4⋊C4—112no—
319C22×C42—111no—
320#391—112no—
321#391—114no—
322#391—112no—
323#391—114no—
324#389—114no—
325#389—114no—
326#389—114no—
327#389—112no—
328#389C2×D1114yes—
329#389C2×D1112yes—
330#387—114no—
331#387—114no—
332#387—114no—
333#387—114no—
334#387—114no—
335#387—114no—
336#387—114no—
337#387—114no—
338#387—114no—
339#387—114no—
340#385—114no—
341#385—114no—
342#383—112no—
343#383—111no—
344#383—111no—
345#383—112no—
346#382—112no—
347#382—112no—
348#382—111no—
349#382—112no—
350#382—112no—
351#382—112no—
352#382C2×D1112yes—
353#382C2×D1111yescharacteristic
354#381—111no—
355#381—111no—
356#380—112no—
357C4×C4⋊C4—112no—
358C4×C4⋊C4—112no—
359C4×C4⋊C4—111no—
360C4×C4⋊C4—112no—
361C4×C4⋊C4C2×D1112yes—
362C4×C4⋊C4C2×D1111yescharacteristic
363C4×C22⋊C4—114no—
364C4×C22⋊C4—112no—
365C4×C22⋊C4—114no—
366#376—112no—
367#376—112no—
368#376—112no—
369C2×#95—112no—
370C2×#95—112no—
371C22×C11C24×C411yescharacteristic
372C22×C11C2×H(5,𝔽2)11yescharacteristic
373C22×C11C2×2−1+411yescharacteristic
374C22×C11C2×2−1+412yes—
375C22×C11#58512yes—
376C2×D11—22no—
377C2×D11—22no—
378C2×D11—24no—
379C2×D11—24no—
380C2×D11#51914yes—
381C4×C11—22no—
382C4×C11—24no—
383C4×C11—24no—
384C4×C11—22no—
385C4×C11—42no—
386C4×C11—42no—
387C4×C11—44no—
388C4×C11#52014yes—
389Dic11—22no—
390Dic11—24no—
391Dic11—22no—
392Dic11—22no—
393Dic11—24no—
394Dic11—44no—
395Dic11—42no—
396Dic11—42no—
397C22×Q8—111no—
398C22×Q8—112no—
399C22×Q8C22×D1111yescharacteristic
400C23×C4—111no—
401C23×C4—111no—
402C23×C4—111no—
403C23×C4—114no—
404C4⋊Q8—118no—
405C4⋊Q8—118no—
406#125—1116no—
407C22⋊Q8—118no—
408C22⋊Q8—1116no—
409C22⋊Q8—118no—
410C4×Q8—44no—
411C4×Q8—1116no—
412C4×Q8—118no—
413C4×Q8—444no—
414C4×Q8C4×D1118yescomplemented
415#117—1116no—
416C2×C4⋊C4—112no—
417C2×C4⋊C4—112no—
418C2×C4⋊C4—112no—
419C2×C4⋊C4—112no—
420C2×C4⋊C4—112no—
421C2×C4⋊C4—111no—
422C2×C4⋊C4—112no—
423C2×C4⋊C4—112no—
424C2×C4⋊C4—114no—
425C2×C4⋊C4—114no—
426C2×C4⋊C4—112no—
427C2×C4⋊C4—114no—
428C2×C4⋊C4—444no—
429C2×C4⋊C4C22×D1112yes—
430C2×C4⋊C4C22×D1114yes—
431C2×C4⋊C4C22×D1112yes—
432C2×C4⋊C4C22×D1114yes—
433C2×C4⋊C4C22×D1112yes—
434C2×C4⋊C4C22×D1111yescharacteristic
435C2×C22⋊C4—112no—
436C2×C22⋊C4—112no—
437C2×C22⋊C4—112no—
438C2×C22⋊C4—112no—
439C2×C42—112no—
440C2×C42—111no—
441C2×C42—111no—
442C2×C42—111no—
443C2×C42—118no—
444C2×C42—112no—
445C2×C42—114no—
446C2×C42—114no—
447C2×C42—114no—
448C2×C42C22×D1112yes—
449C2×C42C22×D1114yes—
450C2×C42C22×D1112yes—
451C2×C42C22×D1111yescharacteristic
452#95—112no—
453#95—114no—
454#95—112no—
455#95—114no—
456#95—111no—
457#95—111no—
458#95—112no—
459#95—112no—
460#95—112no—
461#95—112no—
462#95—114no—
463#95—114no—
464#95—112no—
465#95—112no—
466#95—114no—
467#95—114no—
468#95—114no—
469#95—114no—
470#95—114no—
471#95C22×D1112yes—
472#95C22×D1112yes—
473#95C22×D1114yes—
474#95C22×D1111yescharacteristic
475#95C22×D1111yescharacteristic
476C2×C11C2×#51911yescharacteristic
477C2×C11C2×#52011yescharacteristic
478C2×C11#342811yescharacteristic
479C2×C11#356112yes—
480C2×C11#356312yes—
481D11—24no—
482C24—111no—
483C2×Q8—22no—
484C2×Q8—22no—
485C2×Q8—114no—
486C2×Q8—118no—
487C2×Q8—222no—
488C2×Q8—222no—
489C2×Q8—444no—
490C2×Q8C2×C4×D1114yes—
491C22×C4—111no—
492C22×C4—111no—
493C22×C4—112no—
494C22×C4—112no—
495C22×C4—111no—
496C22×C4—114no—
497C22×C4—114no—
498C22×C4—114no—
499C22×C4—112no—
500C22×C4—112no—
501C22×C4—114no—
502C22×C4—224no—
503C22×C4—228no—
504C22×C4—222no—
505C22×C4—222no—
506C22×C4—442no—
507C22×C4—444no—
508C22×C4—444no—
509C22×C4C23×D1114yes—
510C22×C4C23×D1112yes—
511C22×C4C23×D1112yes—
512C22×C4C23×D1111yescharacteristic
513C22×C4C23×D1112yes—
514C22×C4C23×D1112yes—
515C22×C4C23×D1111yescharacteristic
516C22×C4C23×D1111yescharacteristic
517C4⋊C4—24no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—42no—
521C4⋊C4—118no—
522C4⋊C4—118no—
523C4⋊C4—118no—
524C4⋊C4—118no—
525C4⋊C4—118no—
526C4⋊C4—118no—
527C4⋊C4—114no—
528C4⋊C4—224no—
529C4⋊C4—444no—
530C4⋊C4—444no—
531C4⋊C4—444no—
532C4⋊C4—442no—
533C4⋊C4C2×C4×D1118yes—
534C4⋊C4C2×C4×D1114yes—
535C22⋊C4—118no—
536C22⋊C4—118no—
537C22⋊C4—224no—
538C22⋊C4—224no—
539C42—24no—
540C42—44no—
541C42—114no—
542C42—114no—
543C42—114no—
544C42—118no—
545C42—224no—
546C42—444no—
547C42—444no—
548C42—444no—
549C42C2×C4×D1114yes—
550C42C2×C4×D1118yes—
551C11#3439911yescharacteristic, complemented, Sylow
552C23—112no—
553C23—114no—
554C23—221no—
555C23—221no—
556C23—222no—
557C23C24×D1111yescharacteristic
558Q8—28no—
559Q8—228no—
560Q8—448no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—112no—
576C2×C4—112no—
577C2×C4—112no—
578C2×C4—114no—
579C2×C4—114no—
580C2×C4—228no—
581C2×C4—222no—
582C2×C4—221no—
583C2×C4—221no—
584C2×C4—222no—
585C2×C4—224no—
586C2×C4—222no—
587C2×C4—222no—
588C2×C4—228no—
589C2×C4—224no—
590C2×C4—222no—
591C2×C4—222no—
592C2×C4—222no—
593C2×C4—222no—
594C2×C4—224no—
595C2×C4—222no—
596C2×C4—222no—
597C2×C4—224no—
598C2×C4—444no—
599C2×C4—888no—
600C2×C4C22×C4×D1112yes—
601C2×C4C22×C4×D1112yes—
602C2×C4C22×C4×D1114yes—
603C2×C4C22×C4×D1114yes—
604C2×C4C22×C4×D1112yes—
605C2×C4#6703712yes—
606C2×C4#6704012yes—
607C22—114no—
608C22—224no—
609C22—224no—
610C22—222no—
611C22—222no—
612C22C23×C4×D1111yescharacteristic
613C22C2×#6703711yescharacteristic
614C22C2×#6704011yescharacteristic
615C22#1062345312yes—
616C22D11×2−1+412yes—
617C4—22no—
618C4—24no—
619C4—22no—
620C4—24no—
621C4—44no—
622C4—42no—
623C4—42no—
624C4—222no—
625C4—222no—
626C4—224no—
627C4—222no—
628C4—224no—
629C4—444no—
630C4—442no—
631C4—442no—
632C4#1062303514yes—
633C2—224no—
634C2D11×#51911yescharacteristic
635C2C2×#1062303511yescharacteristic
636C2#1308181511yescharacteristic
637C2#1308317912yes—
638C2#1308321212yes—
subgroupG/H
center, FrattiniC23C24×D11
commutatorC22×C11C24×C4
FittingC11×#2351C2
socle, cosocleC23×C11C25
prime radicalC23×D11C24
hypercenter#2351D11
hypocenterC11#34399
LCSG ▷ C22×C11 ▷ C11
UCSC1 ◁ C23 ◁ #2351

Complements

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

NH ≅ G/Ntype
2. #13071798C2semidirect
5. #13071583C2semidirect
6. #13071581C2semidirect
7. #13071536C2semidirect
10. #13071363C2semidirect
12. #13071332C2semidirect
13. #13071264C2semidirect
14. #13071262C2semidirect
17. C11×#2351C2semidirect
22. #10623109C4semidirect
92. C4×Q8×C11C2×C4semidirect
104. #66997C2×C4semidirect
132. #66942C2×C4semidirect
133. #66941C2×C4semidirect
137. C4×Dic22C2×C4semidirect
138. C4×Dic22C2×C4semidirect
240. #2351D11semidirect
414. C4×Q8C4×D11semidirect
551. C11#34399semidirect

Maximal subgroups

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

subgroupindex
C2×#106231092
#130717982
#130716202
#130716172
#130715832
#130715812
#130715362
#130714792
#130714692
#130713632
#130713502
#130713322
#130712642
#130712622
#130710122
D11×#3822
C11×#23512
#3439911