Finite Group Databasebeta

← #14914362

Subgroups — #14914362

Order 2816.

2194 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10623109C211yesmaximal, characteristic
2#13071715C214yesmaximal, complemented
3#13071642C212yesmaximal
4#13071639C212yesmaximal
5#13071583C211yesmaximal, characteristic, complemented
6#13071581C211yesmaximal, characteristic, complemented
7#13071536C211yesmaximal, characteristic, complemented
8#13071510C212yesmaximal
9#13071496C212yesmaximal
10#13071293C214yesmaximal
11#13071264C212yesmaximal, complemented
12#13071187C212yesmaximal, complemented
13C4×#66997C212yesmaximal, complemented
14#13071022C211yesmaximal, characteristic
15#13070980C212yesmaximal, complemented
16C4×#66945C211yesmaximal, characteristic
17C11×#2351C211yesmaximal, characteristic, complemented
18C2×C4×Q8×C11C2212yes—
19C2×C4×Q8×C11C2211yescharacteristic
20C2×#66999C2211yescharacteristic
21C2×#66997C2211yescharacteristic
22#10623109C4116yescomplemented
23C2×#66950C2211yescharacteristic
24C2×#66949C2212yes—
25C2×#66945C2212yes—
26C2×#66945C2212yes—
27C2×#66942C2212yes—
28C2×#66941C2211yescharacteristic
29C2×#66940C2214yes—
30C2×#66925C2212yes—
31C2×C42×D11C2211yescharacteristic
32C2×C4×Dic22C2212yes—
33C2×C4×Dic22C2212yes—
34C2×C4×Dic22C2211yescharacteristic
35C11×#389C2212yes—
36C11×#389C2214yes—
37C11×#382C2212yes—
38C11×#382C2211yescharacteristic
39C4×C11×C4⋊C4C2211yescharacteristic
40C4×C11×C4⋊C4C2212yes—
41#10622843C2212yes—
42#10622603C2214yes—
43#10622603C2214yes—
44#10622594C2214yes—
45#10622591C2214yes—
46#10622589C2212yes—
47C4⋊C4×Dic11C2212yes—
48C4⋊C4×Dic11C2212yes—
49C4⋊C4×Dic11C2214yes—
50C4⋊C4×Dic11C2212yes—
51#10622587C2212yes—
52#10622585C2214yes—
53#10622585C2214yes—
54#10622585C2214yes—
55#10622585C2214yes—
56#10622553C2214yes—
57C4×(C2×D11)⋊C4C2212yes—
58#10622550C2212yes—
59#10622549C2211yescharacteristic
60C4×C4⋊Dic11C2211yescharacteristic
61#10622545C2212yes—
62C4×Dic11⋊C4C2214yes—
63C4×Dic11⋊C4C2214yes—
64C4×Dic11⋊C4C2214yes—
65C42×Dic11C2212yes—
66#10622541C2212yes—
67#10622282C2214yes—
68#10622280C2212yes—
69#10622280C2212yes—
70#10622279C2211yescharacteristic
71#10622279C2211yescharacteristic
72#10622268C2214yes—
73#10622267C2214yes—
74#10622264C2212yes—
75#10622263C2212yes—
76#10622262C2214yes—
77#10622262C2212yes—
78#10622262C2212yes—
79#10622262C2212yes—
80#10622261C2211yescharacteristic
81#10622261C2211yescharacteristic
82#10622261C2212yes—
83#10622259C2214yes—
84C22×Q8×C11C2311yescharacteristic
85C22×C4×D11C2311yescharacteristic
86C22×C4×D11C2311yescharacteristic
87C22×C4×D11C2311yescharacteristic
88C22×Dic22C2311yescharacteristic
89C22×Dic22C2312yes—
90C4×Q8×C11—44no—
91C4×Q8×C11C2×C418yes—
92C2×C11×C4⋊C4C2314yes—
93C2×C11×C4⋊C4C2312yes—
94C2×C11×C4⋊C4C2312yes—
95C2×C11×C4⋊C4C2314yes—
96C2×C11×C4⋊C4C2311yescharacteristic
97C2×C11×C4⋊C4C2312yes—
98C2×C42×C11C2312yes—
99C2×C42×C11C2314yes—
100C2×C42×C11C2311yescharacteristic
101C2×C42×C11C2312yes—
102#66999C2×C418yes—
103#66997C2×C418yes—
104C2×(C2×D11)⋊C4C2312yes—
105C2×(C2×D11)⋊C4C2312yes—
106C2×(C2×D11)⋊C4C2314yes—
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⋊C4C2314yes—
117C2×Dic11⋊C4C2312yes—
118C2×Dic11⋊C4C2312yes—
119C2×Dic11⋊C4C2312yes—
120C2×Dic11⋊C4C2312yes—
121C2×Dic11⋊C4C2312yes—
122C2×C4×Dic11C2312yes—
123C2×C4×Dic11C2314yes—
124C2×C4×Dic11C2314yes—
125C2×C4×Dic11C2312yes—
126C2×C4×Dic11C2312yes—
127C2×C4×Dic11C2312yes—
128C2×C4×Dic11C2314yes—
129C2×C4×Dic11C2314yes—
130C2×C4×Dic11C2311yescharacteristic
131C2×C4×Dic11C2312yes—
132C2×C4×Dic11C2311yescharacteristic
133C2×C4×Dic11C2311yescharacteristic
134C2×C4×Dic11C2312yes—
135#66950C2×C418yes—
136#66949C2×C4116yes—
137#66945—28no—
138#66945—44no—
139#66942C2×C4116yes—
140#66941C2×C418yes—
141#66940—216no—
142#66925C2×C4116yes—
143C42×D11C2×C418yes—
144C4×Dic22—44no—
145C4×Dic22C2×C4116yes—
146C4×Dic22C2×C418yes—
147C11×#95C2314yes—
148C11×#95C2311yescharacteristic
149C11×#95C2312yes—
150C11×#95C2311yescharacteristic
151C11×#95C2312yes—
152C11⋊#95C2314yes—
153C11⋊#95C2314yes—
154C11⋊#95C2314yes—
155C11⋊#95C2311yescharacteristic
156C11⋊#95C2311yescharacteristic
157C11⋊#95C2312yes—
158C11⋊#95C2314yes—
159C11⋊#95C2312yes—
160C11⋊#95C2312yes—
161C11⋊#95C2312yes—
162C11⋊#95C2314yes—
163C11⋊#95C2314yes—
164C11⋊#95C2314yes—
165#34225—111nomaximal, Sylow
166C23×D11C2411yescharacteristic
167C2×Q8×C11—22no—
168C2×Q8×C11—22no—
169C2×Q8×C11C22×C414yes—
170C22×C4×C11C2414yes—
171C22×C4×C11C2412yes—
172C22×C4×C11C2411yescharacteristic
173C22×C4×C11C2412yes—
174C22×C4×C11C2412yes—
175C22×C4×C11C2411yescharacteristic
176C22×C4×C11C2412yes—
177C22×C4×C11C2411yescharacteristic
178C22×Dic11C2412yes—
179C22×Dic11C2414yes—
180C22×Dic11C2411yescharacteristic
181C22×Dic11C2412yes—
182C22×Dic11C2412yes—
183C22×Dic11C2412yes—
184C22×Dic11C2411yescharacteristic
185C22×Dic11C2411yescharacteristic
186C2×C4×D11—24no—
187C2×C4×D11—24no—
188C2×C4×D11—22no—
189C2×C4×D11—22no—
190C2×C4×D11C22×C414yes—
191C2×C4×D11C22×C414yes—
192C2×C4×D11C22×C414yes—
193C2×Dic22—28no—
194C2×Dic22—22no—
195C2×Dic22—22no—
196C2×Dic22C22×C418yes—
197C2×Dic22C22×C414yes—
198C11×C4⋊C4—28no—
199C11×C4⋊C4—24no—
200C11×C4⋊C4—24no—
201C11×C4⋊C4—44no—
202C11×C4⋊C4C22×C414yes—
203C11×C4⋊C4C22×C418yes—
204C42×C11—24no—
205C42×C11—44no—
206C42×C11C22×C414yes—
207C42×C11C22×C418yes—
208(C2×D11)⋊C4—24no—
209(C2×D11)⋊C4—24no—
210(C2×D11)⋊C4—44no—
211(C2×D11)⋊C4C22×C418yes—
212(C2×D11)⋊C4C22×C418yes—
213C4⋊Dic11—24no—
214C4⋊Dic11—24no—
215C4⋊Dic11C22×C414yes—
216C4⋊Dic11C22×C418yes—
217Dic11⋊C4—28no—
218Dic11⋊C4—28no—
219Dic11⋊C4—28no—
220Dic11⋊C4—44no—
221Dic11⋊C4C22×C418yes—
222Dic11⋊C4C22×C418yes—
223Dic11⋊C4C22×C418yes—
224Dic11⋊C4C22×C418yes—
225Dic11⋊C4C22×C418yes—
226C4×Dic11—28no—
227C4×Dic11—28no—
228C4×Dic11—24no—
229C4×Dic11—24no—
230C4×Dic11—24no—
231C4×Dic11—24no—
232C4×Dic11—44no—
233C4×Dic11—44no—
234C4×Dic11C22×C414yes—
235C4×Dic11C22×C418yes—
236C4×Dic11C22×C414yes—
237C4×Dic11C22×C414yes—
238C4×Dic11#4118yes—
239C2×#533—111no—
240#2370—114no—
241#2362—112no—
242#2362—112no—
243#2356—111no—
244#2356—111no—
245#2351—111no—
246#2351D1111yescharacteristic, complemented
247#2347—112no—
248#2346—112no—
249#2323—114no—
250#2320—112no—
251C4×C4⋊Q8—112no—
252C4×C4⋊Q8—112no—
253#2285—111no—
254#2276—112no—
255C4×#117—111no—
256C23×C11C2511yescharacteristic
257C22×D11—22no—
258C22×D11—21no—
259C22×D11—21no—
260C22×D11—22no—
261C22×D11C23×C412yes—
262Q8×C11—28no—
263C2×C4×C11—24no—
264C2×C4×C11—22no—
265C2×C4×C11—22no—
266C2×C4×C11—22no—
267C2×C4×C11—22no—
268C2×C4×C11—22no—
269C2×C4×C11—22no—
270C2×C4×C11—24no—
271C2×C4×C11—22no—
272C2×C4×C11—24no—
273C2×C4×C11—22no—
274C2×C4×C11—22no—
275C2×C4×C11—24no—
276C2×C4×C11C23×C412yes—
277C2×C4×C11C23×C412yes—
278C2×C4×C11C23×C414yes—
279C2×C4×C11C23×C414yes—
280C2×C4×C11C23×C412yes—
281C2×C4×C11C2×#4114yes—
282C2×C4×C11H(5,𝔽2)12yes—
283C2×C4×C112−1+412yes—
284C2×Dic11—24no—
285C2×Dic11—21no—
286C2×Dic11—22no—
287C2×Dic11—22no—
288C2×Dic11—22no—
289C2×Dic11—22no—
290C2×Dic11—21no—
291C2×Dic11—24no—
292C2×Dic11—22no—
293C2×Dic11—24no—
294C2×Dic11—22no—
295C2×Dic11—22no—
296C2×Dic11—24no—
297C2×Dic11C23×C412yes—
298C2×Dic11C23×C414yes—
299C2×Dic11C23×C414yes—
300C2×Dic11C23×C412yes—
301C2×Dic11C23×C412yes—
302C2×Dic11C2×#4114yes—
303C2×Dic11C2×#4114yes—
304C2×Dic112−1+414yes—
305C4×D11—28no—
306C4×D11—28no—
307C4×D11—28no—
308Dic22—28no—
309Dic22—216no—
310#533—1116no—
311C2×C4⋊Q8—111no—
312C2×C4⋊Q8—111no—
313C2×#125—112no—
314C2×C22⋊Q8—111no—
315C2×C22⋊Q8—112no—
316C2×C22⋊Q8—111no—
317C2×C4×Q8—114no—
318C2×C4×Q8—112no—
319C2×C4×Q8—111no—
320C2×C4×Q8—112no—
321C2×C4×Q8C2×D1112yes—
322C2×C4×Q8C2×D1111yescharacteristic
323C2×#117—112no—
324C2×#117—112no—
325C2×#117—112no—
326C22×C42—111no—
327#391—112no—
328#391—112no—
329#391—114no—
330#389—114no—
331#389—114no—
332#389—114no—
333#389—114no—
334#389—112no—
335#389C2×D1112yes—
336#389C2×D1114yes—
337#388—114no—
338#388—114no—
339#388—114no—
340#388—114no—
341#387—114no—
342#387—114no—
343#387—114no—
344#387—114no—
345#384—112no—
346#383—112no—
347#383—112no—
348#382—111no—
349#382—112no—
350#382C2×D1112yes—
351#382C2×D1111yescharacteristic
352#380—112no—
353#380—112no—
354C4×C4⋊C4—112no—
355C4×C4⋊C4—112no—
356C4×C4⋊C4—114no—
357C4×C4⋊C4—112no—
358C4×C4⋊C4—114no—
359C4×C4⋊C4—114no—
360C4×C4⋊C4—111no—
361C4×C4⋊C4—114no—
362C4×C4⋊C4—114no—
363C4×C4⋊C4—112no—
364C4×C4⋊C4—112no—
365C4×C4⋊C4—112no—
366C4×C4⋊C4C2×D1112yes—
367C4×C4⋊C4C2×D1111yescharacteristic
368C4×C22⋊C4—112no—
369C4×C22⋊C4—111no—
370C4×C22⋊C4—111no—
371#376—112no—
372#376—111no—
373#376—112no—
374#376—111no—
375C43—112no—
376C22×C11C24×C411yescharacteristic
377C22×C11C22×#4112yes—
378C22×C11C2×H(5,𝔽2)11yescharacteristic
379C22×C11C2×2−1+411yescharacteristic
380C22×C11C2×2−1+412yes—
381C2×D11—22no—
382C2×D11—22no—
383C2×D11—22no—
384C2×D11—42no—
385C2×D11—42no—
386C4×C11—24no—
387C4×C11—22no—
388C4×C11—22no—
389C4×C11—24no—
390C4×C11—24no—
391C4×C11—44no—
392C4×C11—42no—
393C4×C11#52014yes—
394Dic11—24no—
395Dic11—24no—
396Dic11—22no—
397Dic11—24no—
398Dic11—42no—
399Dic11—44no—
400Dic11C4×#4114yes—
401Dic11#51914yes—
402C22×Q8—112no—
403C22×Q8—111no—
404C22×Q8C22×D1111yescharacteristic
405C23×C4—111no—
406C23×C4—111no—
407C23×C4—111no—
408C4⋊Q8—118no—
409C4⋊Q8—118no—
410#125—1116no—
411C22⋊Q8—118no—
412C22⋊Q8—1116no—
413C22⋊Q8—118no—
414C4×Q8—44no—
415C4×Q8—1116no—
416C4×Q8—118no—
417C4×Q8—2216no—
418C4×Q8—444no—
419C4×Q8C4×D1118yes—
420#117—1116no—
421#117—228no—
422#117—444no—
423C2×C4⋊C4—112no—
424C2×C4⋊C4—112no—
425C2×C4⋊C4—112no—
426C2×C4⋊C4—112no—
427C2×C4⋊C4—112no—
428C2×C4⋊C4—112no—
429C2×C4⋊C4—111no—
430C2×C4⋊C4—112no—
431C2×C4⋊C4—112no—
432C2×C4⋊C4—114no—
433C2×C4⋊C4—114no—
434C2×C4⋊C4—114no—
435C2×C4⋊C4—114no—
436C2×C4⋊C4C22×D1112yes—
437C2×C4⋊C4C22×D1112yes—
438C2×C4⋊C4C22×D1114yes—
439C2×C4⋊C4C22×D1114yes—
440C2×C4⋊C4C22×D1112yes—
441C2×C4⋊C4C22×D1111yescharacteristic
442C2×C22⋊C4—112no—
443C2×C22⋊C4—112no—
444C2×C22⋊C4—112no—
445C2×C22⋊C4—112no—
446C2×C22⋊C4—114no—
447C2×C42—112no—
448C2×C42—112no—
449C2×C42—111no—
450C2×C42—111no—
451C2×C42—111no—
452C2×C42—118no—
453C2×C42—112no—
454C2×C42—114no—
455C2×C42—114no—
456C2×C42—112no—
457C2×C42—112no—
458C2×C42—112no—
459C2×C42—114no—
460C2×C42—114no—
461C2×C42C22×D1112yes—
462C2×C42C22×D1114yes—
463C2×C42C22×D1112yes—
464C2×C42C22×D1111yescharacteristic
465#95—111no—
466#95—112no—
467#95—111no—
468#95—112no—
469#95—112no—
470#95—114no—
471#95—114no—
472#95—112no—
473#95—114no—
474#95—114no—
475#95—114no—
476#95—114no—
477#95—114no—
478#95C22×D1111yescharacteristic
479#95C22×D1111yescharacteristic
480#95C22×D1112yes—
481#95C22×D1112yes—
482#95C22×D1114yes—
483C2×C11C2×C4×#4111yescharacteristic
484C2×C11C2×#51911yescharacteristic
485C2×C11C2×#52011yescharacteristic
486C2×C11#350912yes—
487C2×C11#351112yes—
488D11—42no—
489C24—111no—
490C2×Q8—22no—
491C2×Q8—22no—
492C2×Q8—118no—
493C2×Q8—114no—
494C2×Q8—228no—
495C2×Q8—222no—
496C2×Q8—222no—
497C2×Q8C2×C4×D1114yes—
498C22×C4—111no—
499C22×C4—111no—
500C22×C4—112no—
501C22×C4—111no—
502C22×C4—112no—
503C22×C4—112no—
504C22×C4—114no—
505C22×C4—114no—
506C22×C4—114no—
507C22×C4—112no—
508C22×C4—114no—
509C22×C4—224no—
510C22×C4—224no—
511C22×C4—222no—
512C22×C4—222no—
513C22×C4C23×D1112yes—
514C22×C4C23×D1114yes—
515C22×C4C23×D1111yescharacteristic
516C22×C4C23×D1112yes—
517C22×C4C23×D1112yes—
518C22×C4C23×D1111yescharacteristic
519C22×C4C23×D1112yes—
520C22×C4C23×D1111yescharacteristic
521C4⋊C4—24no—
522C4⋊C4—24no—
523C4⋊C4—28no—
524C4⋊C4—44no—
525C4⋊C4—118no—
526C4⋊C4—118no—
527C4⋊C4—118no—
528C4⋊C4—118no—
529C4⋊C4—118no—
530C4⋊C4—118no—
531C4⋊C4—114no—
532C4⋊C4—224no—
533C4⋊C4—228no—
534C4⋊C4—228no—
535C4⋊C4—224no—
536C4⋊C4—228no—
537C4⋊C4—444no—
538C4⋊C4C2×C4×D1118yes—
539C4⋊C4C2×C4×D1114yes—
540C22⋊C4—118no—
541C22⋊C4—118no—
542C22⋊C4—224no—
543C22⋊C4—224no—
544C22⋊C4—444no—
545C42—24no—
546C42—44no—
547C42—114no—
548C42—114no—
549C42—114no—
550C42—118no—
551C42—118no—
552C42—224no—
553C42—228no—
554C42—228no—
555C42—224no—
556C42—224no—
557C42—224no—
558C42—444no—
559C42—444no—
560C42C2×C4×D1114yes—
561C42C2×C4×D1118yes—
562C11#3422511yescharacteristic, complemented, Sylow
563C23—112no—
564C23—221no—
565C23—221no—
566C23—222no—
567C23—222no—
568C23C24×D1111yescharacteristic
569Q8—28no—
570Q8—2216no—
571Q8—228no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—112no—
586C2×C4—112no—
587C2×C4—114no—
588C2×C4—114no—
589C2×C4—112no—
590C2×C4—114no—
591C2×C4—114no—
592C2×C4—114no—
593C2×C4—228no—
594C2×C4—228no—
595C2×C4—221no—
596C2×C4—222no—
597C2×C4—222no—
598C2×C4—221no—
599C2×C4—222no—
600C2×C4—224no—
601C2×C4—222no—
602C2×C4—228no—
603C2×C4—224no—
604C2×C4—222no—
605C2×C4—222no—
606C2×C4—224no—
607C2×C4—222no—
608C2×C4—224no—
609C2×C4C22×C4×D1112yes—
610C2×C4C22×C4×D1112yes—
611C2×C4C22×C4×D1114yes—
612C2×C4C22×C4×D1114yes—
613C2×C4C22×C4×D1112yes—
614C2×C4#6703712yes—
615C2×C4#6704012yes—
616C2×C4#41×D1114yes—
617C22—222no—
618C22—222no—
619C22—222no—
620C22—442no—
621C22—442no—
622C22C23×C4×D1111yescharacteristic
623C22C2×#6703711yescharacteristic
624C22C2×#6704011yescharacteristic
625C22C2×#41×D1112yes—
626C22D11×2−1+412yes—
627C4—24no—
628C4—22no—
629C4—24no—
630C4—22no—
631C4—24no—
632C4—42no—
633C4—44no—
634C4—114no—
635C4—114no—
636C4—224no—
637C4—224no—
638C4—222no—
639C4—224no—
640C4—442no—
641C4—444no—
642C4#1062303514yes—
643C2—442no—
644C2C4×#41×D1111yescharacteristic
645C2D11×#51911yescharacteristic
646C2C2×#1062303511yescharacteristic
647C2#1308249912yes—
648C2#1308252212yes—
subgroupG/H
center, FrattiniC23C24×D11
commutatorC22×C11C24×C4
FittingC11×#2351C2
socle, cosocleC23×C11C25
prime radicalC23×D11C24
hypercenter#2351D11
hypocenterC11#34225
LCSG ▷ C22×C11 ▷ C11
UCSC1 ◁ C23 ◁ #2351

Complements

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

NH ≅ G/Ntype
2. #13071715C2semidirect
5. #13071583C2semidirect
6. #13071581C2semidirect
7. #13071536C2semidirect
11. #13071264C2semidirect
12. #13071187C2semidirect
13. C4×#66997C2semidirect
15. #13070980C2semidirect
17. C11×#2351C2semidirect
22. #10623109C4semidirect
246. #2351D11semidirect
562. C11#34225semidirect

Maximal subgroups

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

subgroupindex
C2×#106231092
#130717152
#130716422
#130716392
#130715832
#130715812
#130715362
#130715102
#130714962
#130712932
#130712642
#130711872
C4×#669972
#130710222
#130709802
C4×#669452
C11×#23512
#3422511