Finite Group Databasebeta

← #25065161

Subgroups — #25065161

Order 11008.

1018 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19147892C211yesmaximal, characteristic
2#19143227C211yesmaximal, characteristic, complemented
3#423×D43C211yesmaximal, characteristic, complemented
4#19133333C211yesmaximal, characteristic, complemented
5#19133328C211yesmaximal, characteristic, complemented
6#19133328C211yesmaximal, characteristic, complemented
7#19133324C212yesmaximal, complemented
8#19133316C212yesmaximal, complemented
9#19133251C212yesmaximal, complemented
10#19133243C212yesmaximal, complemented
11C43×#1561C211yesmaximal, characteristic, complemented
12C2×M16×D43—21no—
13#14583454C2211yescharacteristic
14#14583454C2211yescharacteristic
15#14583450C2211yescharacteristic
16#14583450C2211yescharacteristic
17#14583449C2211yescharacteristic
18#14583436—21no—
19#14583436—21no—
20#14583435—21no—
21#14583435—21no—
22#14583434C412yes—
23#14583434C412yes—
24#14583412—21no—
25#14583412—21no—
26#14583411C412yes—
27C4×#10621902—21no—
28C4×#10621902—21no—
29#14583367C412yes—
30C2×C42×D43C2211yescharacteristic
31#14582955—21no—
32#14582955C2212yes—
33#14582951—21no—
34#14582951C2212yes—
35#14582923C2211yescharacteristic
36#14582923C2211yescharacteristic
37#14582922—21no—
38#14582921—21no—
39#14582921C2212yes—
40C4⋊C8×D43—21no—
41C4⋊C8×D43C2212yes—
42#14582862—21no—
43M16×Dic43—21no—
44#14582647C2212yescomplemented
45#14582647C2212yescomplemented
46#14582645C2212yescomplemented
47#14582645C2212yescomplemented
48#14582642C2212yescomplemented
49#14582642C2212yescomplemented
50C43×#533C2211yescharacteristic
51C43×#423C2211yescharacteristic
52C43×#423C2211yescharacteristic
53C43×#332C2212yescomplemented
54C43×#332C2212yescomplemented
55C2×#10621902—21no—
56C2×#10621902—21no—
57C2×#10621898C2×C412yes—
58C22×C4×D43C2×C411yescharacteristic
59C22×C4×D43D411yescharacteristic
60C22×C4×D43D411yescharacteristic
61C2×#10621877—21no—
62C2×#10621877—21no—
63C2×#10621872—22no—
64M16×D43—44no—
65C2×C8×D43—22no—
66#13063325C2311yescharacteristic
67#13063322C2311yescharacteristic
68#13063320—22no—
69#13063320—22no—
70#13063316—21no—
71#13063316—21no—
72#13063314—21no—
73#13063314—21no—
74#13063313C2×C412yes—
75#13063310—22no—
76#13063310—22no—
77#13063309—22no—
78#13063305C2×C412yes—
79#13063305C2×C412yes—
80#13063301—21no—
81#13063301—21no—
82#13063300—21no—
83#13063300—21no—
84#13063300C2×C412yes—
85#13063300C2×C412yes—
86#13063299—21no—
87#13063299—21no—
88#13063295—22no—
89Q8×Dic43—21no—
90Q8×Dic43—21no—
91#13063290C2×C412yes—
92#13063290C2×C412yes—
93C4×D172—22no—
94C4×D172—22no—
95C4×D172—21no—
96C4×D172—21no—
97#13063287—22no—
98#13063287—22no—
99#13063287—21no—
100#13063287—21no—
101#13063287C2×C412yes—
102#13063287C2×C412yes—
103(C22⋊D43)⋊C4—22no—
104(C22⋊D43)⋊C4—22no—
105#13063283—21no—
106#13063283—21no—
107#13063283—22no—
108#13063283—22no—
109#13063282—22no—
110C2×C4×Dic43C2×C411yescharacteristic
111C2×C4×Dic43D411yescharacteristic
112C2×C4×Dic43D411yescharacteristic
113C42×D43—22no—
114C42×D43C2×C411yescharacteristic
115C42×D43C2×C411yescharacteristic
116C42×D43D411yescharacteristic
117C42×D43D411yescharacteristic
118C42×D43D411yescharacteristic
119C42×D43D411yescharacteristic
120C42×D43C2311yescharacteristic
121C42×D43C2311yescharacteristic
122C4⋊(C43⋊C8)—21no—
123C4⋊(C43⋊C8)—21no—
124C4⋊(C43⋊C8)C2312yes—
125C4⋊(C43⋊C8)C2312yes—
126Dic43⋊C8—22no—
127Dic43⋊C8—22no—
128(C2×D43)⋊C8—22no—
129(C2×D43)⋊C8—22no—
130#13063235—22no—
131C8×Dic43—22no—
132C2×M16×C43—21no—
133C2×M16×C43—21no—
134C4⋊Q8×C43C2311yescharacteristic
135C4⋊Q8×C43C2311yescharacteristic
136#125×C43C2×C412yes—
137C22⋊Q8×C43—21no—
138C22⋊Q8×C43—21no—
139C4×Q8×C43—21no—
140C4×Q8×C43—21no—
141#117×C43C2×C412yes—
142C2×C42×C43C2311yescharacteristic
143C4⋊C8×C43—21no—
144C4⋊C8×C43—21no—
145C4⋊C8×C43C2312yes—
146C4⋊C8×C43C2312yes—
147C23×D43C22⋊C411yescharacteristic
148C2×Q8×C43—21no—
149C2×Q8×C43—21no—
150C22×C4×C43C22×C411yescharacteristic
151C22×C4×C43C2×D411yescharacteristic
152C22×C4×C43C2×D411yescharacteristic
153C2×C22⋊D43—22no—
154C22×Dic43C22⋊C411yescharacteristic
155C22×Dic43C22⋊C411yescharacteristic
156C22×Dic43C22⋊C411yescharacteristic
157#10621902—42no—
158#10621902—42no—
159#10621902—42no—
160#10621902—42no—
161#10621898—44no—
162C2×D172—21no—
163C2×D172—21no—
164C2×D172—22no—
165C2×D172—22no—
166C2×D172C22×C412yes—
167C2×C4×D43—22no—
168C2×C4×D43—22no—
169C2×C4×D43—22no—
170C2×C4×D43—22no—
171C2×C4×D43—22no—
172C2×C4×D43—42no—
173C2×C4×D43—41no—
174C2×C4×D43—41no—
175C2×C4×D43C22⋊C411yescharacteristic
176C2×C4×D43C22⋊C411yescharacteristic
177C2×C4×D43C22⋊C411yescharacteristic
178C2×C4×D43C22⋊C411yescharacteristic
179C2×C4×D43C22⋊C411yescharacteristic
180C2×C4×D43C22⋊C411yescharacteristic
181C2×C4×D43C22×C411yescharacteristic
182C2×C4×D43C22×C411yescharacteristic
183C2×C4×D43C2×D411yescharacteristic
184C2×C4×D43C2×D411yescharacteristic
185C2×C4×D43C2×D411yescharacteristic
186C2×C4×D43C2×D411yescharacteristic
187C2×Dic86C22×C412yes—
188M16×C43—42no—
189M16×C43—42no—
190C2×C8×C43—22no—
191C2×C8×C43—22no—
192C4⋊C4×C43—22no—
193C4⋊C4×C43—22no—
194C4⋊C4×C43—21no—
195C4⋊C4×C43—21no—
196C4⋊C4×C43C22×C412yes—
197C4⋊C4×C43C22×C412yes—
198C22⋊C4×C43—22no—
199C42×C43—22no—
200C42×C43C22×C411yescharacteristic
201C42×C43C2×D411yescharacteristic
202C42×C43C2×D411yescharacteristic
203C42×C43C2411yescharacteristic
204(C2×D43)⋊C4—22no—
205(C2×D43)⋊C4—22no—
206(C2×D43)⋊C4—22no—
207(C2×D43)⋊C4—22no—
208(C2×D43)⋊C4—22no—
209(C2×D43)⋊C4—22no—
210(C2×D43)⋊C4—22no—
211(C2×D43)⋊C4—22no—
212C4⋊Dic43—21no—
213C4⋊Dic43—21no—
214C4⋊Dic43—22no—
215C4⋊Dic43—22no—
216Dic43⋊C4—22no—
217Dic43⋊C4—22no—
218C4×Dic43—22no—
219C4×Dic43—22no—
220C4×Dic43C22⋊C411yescharacteristic
221C4×Dic43C22⋊C411yescharacteristic
222C4×Dic43C22⋊C411yescharacteristic
223C4×Dic43C22⋊C411yescharacteristic
224C4×Dic43C22⋊C411yescharacteristic
225C4×Dic43C22⋊C411yescharacteristic
226C4×Dic43C22×C411yescharacteristic
227C4×Dic43C22×C411yescharacteristic
228C4×Dic43C2×D411yescharacteristic
229C4×Dic43C2×D411yescharacteristic
230C4×Dic43C2×D411yescharacteristic
231C4×Dic43C2×D411yescharacteristic
232#10621877—42no—
233#10621877—42no—
234C2×C43⋊C8—22no—
235C2×C43⋊C8—22no—
236#10621872—44no—
237C8×D43—44no—
238C23×C43C2×C22⋊C411yescharacteristic
239C22×D43—22no—
240C22×D43—22no—
241C22×D43—41no—
242C22×D43—41no—
243C22×D43C2×C22⋊C411yescharacteristic
244C22×D43C2×C22⋊C411yescharacteristic
245Q8×C43—41no—
246Q8×C43—41no—
247Q8×C43—41no—
248Q8×C43—41no—
249C2×C4×C43—21no—
250C2×C4×C43—21no—
251C2×C4×C43—21no—
252C2×C4×C43—21no—
253C2×C4×C43—22no—
254C2×C4×C43—22no—
255C2×C4×C43—22no—
256C2×C4×C43C2×C22⋊C411yescharacteristic
257C2×C4×C43C2×C22⋊C411yescharacteristic
258C2×C4×C43C2×C22⋊C411yescharacteristic
259C2×C4×C43C23×C411yescharacteristic
260C2×C4×C43C22×D411yescharacteristic
261C2×C4×C43C22×D411yescharacteristic
262C22⋊D43—44no—
263C2×Dic43—22no—
264C2×Dic43—22no—
265C2×Dic43—41no—
266C2×Dic43—41no—
267C2×Dic43C2×C22⋊C411yescharacteristic
268C2×Dic43C2×C22⋊C411yescharacteristic
269C2×Dic43C2×C22⋊C411yescharacteristic
270C2×Dic43C2×C22⋊C411yescharacteristic
271C2×Dic43C2×C22⋊C411yescharacteristic
272C2×Dic43C2×C22⋊C411yescharacteristic
273D172—42no—
274D172—44no—
275D172—44no—
276D172—41no—
277D172—41no—
278D172—41no—
279D172—41no—
280C4×D43—22no—
281C4×D43—21no—
282C4×D43—21no—
283C4×D43—21no—
284C4×D43—21no—
285C4×D43—21no—
286C4×D43—21no—
287C4×D43—22no—
288C4×D43—22no—
289C4×D43—22no—
290C4×D43—41no—
291C4×D43—41no—
292C4×D43—41no—
293C4×D43—41no—
294C4×D43—44no—
295C4×D43—44no—
296C4×D43—44no—
297C4×D43AGL(1,ℤ/8)12yes—
298C4×D43#13712yes—
299Dic86—42no—
300C8×C43—42no—
301C8×C43—42no—
302C43⋊C8—42no—
303C43⋊C8—42no—
304#17737—431nomaximal, Sylow
305C22×C43—21no—
306C22×C43—21no—
307C22×C43—21no—
308C22×C43C22×C22⋊C411yescharacteristic
309C2×D43—21no—
310C2×D43—21no—
311C2×D43—21no—
312C2×D43—21no—
313C2×D43—21no—
314C2×D43—44no—
315C2×D43—42no—
316C2×D43—42no—
317C2×D43—42no—
318C2×D43#41712yes—
319C4×C43—21no—
320C4×C43—21no—
321C4×C43—21no—
322C4×C43—42no—
323C4×C43—42no—
324C4×C43#41312yes—
325C4×C43C2×AGL(1,ℤ/8)11yescharacteristic
326C4×C43C2×#13711yescharacteristic
327C4×C43#57511yescharacteristic
328C4×C43#57511yescharacteristic
329Dic43—21no—
330Dic43—21no—
331Dic43—21no—
332Dic43—21no—
333Dic43—21no—
334Dic43—42no—
335Dic43#41712yes—
336#3444—431no—
337#2920—431no—
338C2×#423—431no—
339#1561—431no—
340#1561D4311yescharacteristic, complemented
341#1560—431no—
342#1560—431no—
343#1559—432no—
344#1557—432no—
345#1542—432no—
346#1540—432no—
347C2×C43—41no—
348C2×C43C2×#41311yescharacteristic
349C2×C43C2×#41711yescharacteristic
350C2×C43#289611yescharacteristic
351D43—22no—
352D43—41no—
353D43—82no—
354C22×M16—861no—
355#533—431no—
356#533—431no—
357#533C2×D4311yescharacteristic
358#532—431no—
359#532—431no—
360#532—431no—
361#529—432no—
362#529—432no—
363#529—861no—
364#529—861no—
365#529—861no—
366#529—861no—
367#525—432no—
368#525—861no—
369#525—861no—
370C4×#41—432no—
371C4×#41—861no—
372C4×#41—861no—
373C22×C42—431no—
374#432—432no—
375#432—861no—
376#431—432no—
377#431—861no—
378#423—431no—
379#423—431no—
380#423—432no—
381#423—861no—
382#423—861no—
383#423C2×D4311yescharacteristic
384#423C2×D4311yescharacteristic
385C2×C4⋊C8—432no—
386C2×C4⋊C8—861no—
387#407—861no—
388C4×M16—861no—
389#333—432no—
390#333—432no—
391#332—432no—
392#332—432no—
393#332C2×D4312yescomplemented
394#332C2×D4312yescomplemented
395#331—432no—
396#331—432no—
397C43#1773711yescharacteristic, complemented, Sylow
398C2×#41—432no—
399C2×#41—861no—
400C2×#41—861no—
401C23×C4—431no—
402C23×C4—431no—
403C23×C4—431no—
404C2×M16—21no—
405C2×M16—21no—
406C2×M16—862no—
407C2×M16—861no—
408C2×M16—861no—
409C2×M16—1724no—
410C22×C8—862no—
411C4⋊Q8—431no—
412C4⋊Q8C22×D4311yescharacteristic
413C4⋊Q8C22×D4311yescharacteristic
414#127—431no—
415#126—862no—
416#126—862no—
417#125—861no—
418#125—861no—
419#125C4×D4312yes—
420#124—432no—
421#124—861no—
422#124—861no—
423#123—862no—
424#123—862no—
425#123—862no—
426C22⋊Q8—21no—
427C22⋊Q8—21no—
428C22⋊Q8—432no—
429C22⋊Q8—432no—
430C22⋊Q8—861no—
431C22⋊Q8—861no—
432#121—432no—
433#121—432no—
434#121—861no—
435#121—861no—
436#121—861no—
437#121—861no—
438#120—862no—
439C4×Q8—21no—
440C4×Q8—21no—
441C4×Q8—432no—
442C4×Q8—432no—
443C4×Q8—861no—
444C4×Q8—861no—
445C4×D4—432no—
446C4×D4—432no—
447C4×D4—862no—
448C4×D4—862no—
449C4×D4—861no—
450C4×D4—861no—
451C4×D4—862no—
452C4×D4—862no—
453C4×D4—862no—
454C4×D4—862no—
455C4×D4—861no—
456C4×D4—861no—
457#117—862no—
458#117—862no—
459#117—862no—
460#117—861no—
461#117—861no—
462#117C4×D4312yes—
463C2×C42—431no—
464C2×C42—431no—
465C2×C42—431no—
466C2×C42—431no—
467C2×C42—431no—
468C2×C42—431no—
469C2×C42—431no—
470C2×C42—431no—
471C2×C42—431no—
472C2×C42—431no—
473C2×C42—431no—
474C2×C42—862no—
475C2×C42C22×D4311yescharacteristic
476C4⋊C8—21no—
477C4⋊C8—21no—
478C4⋊C8—432no—
479C4⋊C8—432no—
480C4⋊C8—862no—
481C4⋊C8—862no—
482C4⋊C8—861no—
483C4⋊C8—861no—
484C4⋊C8C22×D4312yes—
485C4⋊C8C22×D4312yes—
486C22⋊C8—862no—
487C22⋊C8—862no—
488#97—862no—
489C4×C8—862no—
490C24—431no—
491#41—1724no—
492#41—1722no—
493#41—1722no—
494#41—1722no—
495#41—1722no—
496C2×Q8—21no—
497C2×Q8—21no—
498C2×Q8—432no—
499C2×D4—432no—
500C2×D4—862no—
501C2×D4—862no—
502C2×D4—862no—
503C2×D4—861no—
504C2×D4—861no—
505C22×C4—431no—
506C22×C4—431no—
507C22×C4—431no—
508C22×C4—431no—
509C22×C4—431no—
510C22×C4—431no—
511C22×C4—431no—
512C22×C4—431no—
513C22×C4—431no—
514C22×C4—431no—
515C22×C4—431no—
516C22×C4—431no—
517C22×C4—431no—
518C22×C4—431no—
519C22×C4—431no—
520C22×C4—862no—
521C22×C4—862no—
522C22×C4—862no—
523C22×C4—862no—
524C22×C4—862no—
525C22×C4—1722no—
526C22×C4—1721no—
527C22×C4—1721no—
528C22×C4C2×C4×D4311yescharacteristic
529C22×C4D4×D4311yescharacteristic
530C22×C4D4×D4311yescharacteristic
531M16—42no—
532M16—42no—
533M16—1724no—
534M16—1722no—
535M16—1722no—
536C2×C8—22no—
537C2×C8—22no—
538C2×C8—862no—
539C2×C8—862no—
540C2×C8—1724no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—862no—
546C4⋊C4—862no—
547C4⋊C4—862no—
548C4⋊C4—862no—
549C4⋊C4—861no—
550C4⋊C4—861no—
551C4⋊C4C2×C4×D4312yes—
552C4⋊C4C2×C4×D4312yes—
553C22⋊C4—22no—
554C22⋊C4—862no—
555C22⋊C4—862no—
556C22⋊C4—862no—
557C22⋊C4—862no—
558C22⋊C4—862no—
559C22⋊C4—862no—
560C22⋊C4—862no—
561C22⋊C4—862no—
562C42—22no—
563C42—431no—
564C42—431no—
565C42—431no—
566C42—431no—
567C42—431no—
568C42—431no—
569C42—431no—
570C42—431no—
571C42—431no—
572C42—431no—
573C42—431no—
574C42—431no—
575C42—862no—
576C42—862no—
577C42C2×C4×D4311yescharacteristic
578C42D4×D4311yescharacteristic
579C42D4×D4311yescharacteristic
580C42C23×D4311yescharacteristic
581C23—431no—
582C23—431no—
583C23—862no—
584C23—862no—
585C23—1721no—
586C23—1721no—
587C23C22⋊C4×D4311yescharacteristic
588Q8—41no—
589Q8—41no—
590Q8—41no—
591Q8—41no—
592Q8—1722no—
593D4—1722no—
594D4—1724no—
595D4—1724no—
596D4—1724no—
597D4—1721no—
598D4—1721no—
599D4—1721no—
600D4—1721no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—431no—
609C2×C4—432no—
610C2×C4—432no—
611C2×C4—431no—
612C2×C4—431no—
613C2×C4—431no—
614C2×C4—431no—
615C2×C4—431no—
616C2×C4—862no—
617C2×C4—862no—
618C2×C4—862no—
619C2×C4—862no—
620C2×C4—862no—
621C2×C4—861no—
622C2×C4—861no—
623C2×C4—861no—
624C2×C4—861no—
625C2×C4—861no—
626C2×C4—861no—
627C2×C4—862no—
628C2×C4—1721no—
629C2×C4—1721no—
630C2×C4—1721no—
631C2×C4—1721no—
632C2×C4—1724no—
633C2×C4—1724no—
634C2×C4—1721no—
635C2×C4—1721no—
636C2×C4—1724no—
637C2×C4C22⋊C4×D4311yescharacteristic
638C2×C4C22⋊C4×D4311yescharacteristic
639C2×C4C22⋊C4×D4311yescharacteristic
640C2×C4C22×C4×D4311yescharacteristic
641C2×C4C2×D4×D4311yescharacteristic
642C2×C4C2×D4×D4311yescharacteristic
643C8—42no—
644C8—42no—
645C8—1722no—
646C8—1722no—
647C22—21no—
648C22—21no—
649C22—21no—
650C22—432no—
651C22—861no—
652C22—861no—
653C22—861no—
654C22—861no—
655C22—861no—
656C22—1724no—
657C22—1722no—
658C22—1722no—
659C22—1722no—
660C22C2×C22⋊C4×D4311yescharacteristic
661C4—21no—
662C4—21no—
663C4—21no—
664C4—42no—
665C4—42no—
666C4—432no—
667C4—861no—
668C4—861no—
669C4—861no—
670C4—861no—
671C4—861no—
672C4—1722no—
673C4#1458288212yes—
674C4AGL(1,ℤ/8)×D4311yescharacteristic
675C4#137×D4311yescharacteristic
676C4#1458369511yescharacteristic
677C4#1458369811yescharacteristic
678C2—41no—
679C2—862no—
680C2—3442no—
681C2—1721no—
682C2C2×#1458288211yescharacteristic
683C2#417×D4311yescharacteristic
684C2#1914311111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D43
commutatorC2×C4×C43C23×C4
FrattiniC42C23×D43
FittingC43×#1561C2
socleC22×C43C22×C22⋊C4
prime radical#19147892C2
cosocleC42×C43C24
hypercenter#1561D43
hypocenterC43#17737
LCSG ▷ C2×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1561

Complements

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

NH ≅ G/Ntype
2. #19143227C2semidirect
3. #423×D43C2semidirect
4. #19133333C2semidirect
5. #19133328C2semidirect
6. #19133328C2semidirect
7. #19133324C2semidirect
8. #19133316C2semidirect
9. #19133251C2semidirect
10. #19133243C2semidirect
11. C43×#1561C2semidirect
44. #14582647C22semidirect
45. #14582647C22semidirect
46. #14582645C22semidirect
47. #14582645C22semidirect
48. #14582642C22semidirect
49. #14582642C22semidirect
53. C43×#332C22semidirect
54. C43×#332C22semidirect
340. #1561D43semidirect
393. #332C2×D43semidirect
394. #332C2×D43semidirect
397. C43#17737semidirect

Maximal subgroups

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

subgroupindex
#191478922
#191432272
#423×D432
#191333332
#191333282
#191333242
#191333162
#191332512
#191332432
C43×#15612
#1773743