Finite Group Databasebeta

← #25062709

Subgroups — #25062709

Order 11008.

990 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×#14583428C211yesmaximal, characteristic
2#19143225C211yesmaximal, characteristic, complemented
3#19143223C211yesmaximal, characteristic, complemented
4#19132961C214yesmaximal, complemented
5#19132952C211yesmaximal, characteristic, complemented
6#19132950C211yesmaximal, characteristic, complemented
7#19132947C211yesmaximal, characteristic, complemented
8#19132930C214yesmaximal, complemented
9C43×#1469C211yesmaximal, characteristic, complemented
10#14583428—21no—
11#14583428—21no—
12#14583428—21no—
13#14583428—21no—
14#14583428—21no—
15#14583428—21no—
16#14583428—21no—
17#14583428—21no—
18C2×#13063318C2212yes—
19C2×#13063317C2211yescharacteristic
20C2×#13063307C412yes—
21C2×#13063301C411yescharacteristic
22C2×#13063296C411yescharacteristic
23C2×Q8×Dic43C411yescharacteristic
24C2×D4×Dic43C411yescharacteristic
25C2×(C22⋊D43)⋊C4C2212yes—
26C2×#13063281C412yes—
27C2×C42×D43C2211yescharacteristic
28#14582956C2211yescharacteristic
29#14582955—21no—
30#14582954—21no—
31#14582953C2212yes—
32#14582952C2211yescharacteristic
33#14582951—21no—
34#14582950—21no—
35#14582949C2212yes—
36#14582921—21no—
37#14582921—21no—
38#14582861C2212yes—
39#14582861C2212yes—
40#14582851—21no—
41C4×#10621872—21no—
42#14582624C2214yes—
43#14582624C2214yes—
44#14582623C2214yes—
45C2×#124×C43C2211yescharacteristic
46C43×#432C2211yescharacteristic
47C43×#431C2211yescharacteristic
48C43×#323C2214yes—
49C22×C22⋊D43C2×C412yes—
50C23×Dic43C2×C412yes—
51C22×C4×D43C2×C411yescharacteristic
52C22×C4×D43C2×C411yescharacteristic
53C22×C4×D43C2311yescharacteristic
54C2×#10621872—22no—
55C2×#10621872—22no—
56#13063318—22no—
57#13063318—22no—
58#13063318—22no—
59#13063318—22no—
60#13063317—21no—
61#13063317—21no—
62#13063317—21no—
63#13063317—21no—
64#13063307—22no—
65#13063307—22no—
66#13063307—22no—
67#13063307—22no—
68#13063301—21no—
69#13063301—21no—
70#13063301—21no—
71#13063301—21no—
72#13063296—21no—
73#13063296—21no—
74#13063296—21no—
75#13063296—21no—
76Q8×Dic43—21no—
77Q8×Dic43—21no—
78Q8×Dic43—21no—
79Q8×Dic43—21no—
80D4×Dic43—21no—
81D4×Dic43—21no—
82D4×Dic43—21no—
83D4×Dic43—21no—
84(C22⋊D43)⋊C4—22no—
85(C22⋊D43)⋊C4—22no—
86(C22⋊D43)⋊C4—22no—
87(C22⋊D43)⋊C4—22no—
88#13063281—22no—
89#13063281—22no—
90#13063281—22no—
91#13063281—22no—
92C2×C4⋊Dic43C2×C411yescharacteristic
93C2×C4⋊Dic43C2×C411yescharacteristic
94C2×C4⋊Dic43C2×C412yes—
95C2×Dic43⋊C4C2×C412yes—
96C2×Dic43⋊C4C2×C412yes—
97C2×Dic43⋊C4C2312yes—
98C2×C22⋊Dic43C2×C412yes—
99C2×C22⋊Dic43C2×C412yes—
100C2×(C2×D43)⋊C4C2×C412yes—
101C2×C4×Dic43C2×C412yes—
102C2×C4×Dic43C2×C411yescharacteristic
103C2×C4×Dic43C2×C411yescharacteristic
104C2×C4×Dic43C2311yescharacteristic
105C42×D43D411yescharacteristic
106C42×D43D411yescharacteristic
107C42×D43D411yescharacteristic
108C42×D43D411yescharacteristic
109C42×D43D411yescharacteristic
110C42×D43D411yescharacteristic
111C42×D43D411yescharacteristic
112C42×D43D411yescharacteristic
113C4⋊(C43⋊C8)—21no—
114C4⋊(C43⋊C8)—21no—
115Dic43⋊C8—22no—
116Dic43⋊C8—22no—
117C22⋊(C43⋊C8)C2312yes—
118C22⋊(C43⋊C8)C2312yes—
119(C2×D43)⋊C8—22no—
120(C2×D43)⋊C8—22no—
121#13063236—21no—
122#13063235—22no—
123C4×C43⋊C8—21no—
124C8×Dic43—22no—
125C22×Q8×C43C2×C411yescharacteristic
126C22×D4×C43C2×C411yescharacteristic
127#124×C43—21no—
128#124×C43—21no—
129#124×C43—21no—
130#124×C43—21no—
131C2×C22⋊C4×C43C2×C412yes—
132C2×C22⋊C4×C43C2312yes—
133C2×C42×C43C2311yescharacteristic
134C4⋊C8×C43—21no—
135C4⋊C8×C43—21no—
136C22⋊C8×C43C2312yes—
137C22⋊C8×C43C2312yes—
138#97×C43—21no—
139C4×C8×C43—21no—
140C24×C43C22×C412yes—
141C23×D43C22×C411yescharacteristic
142C2×Q8×C43—21no—
143C2×Q8×C43—21no—
144C2×Q8×C43—42no—
145C2×D4×C43—21no—
146C2×D4×C43—21no—
147C2×D4×C43—42no—
148C22×C4×C43C22×C411yescharacteristic
149C22×C4×C43C22×C411yescharacteristic
150C22×C4×C43C22×C412yes—
151C22×C4×C43C2411yescharacteristic
152C2×C22⋊D43—22no—
153C2×C22⋊D43—22no—
154C2×C22⋊D43—44no—
155C22×Dic43—22no—
156C22×Dic43—22no—
157C22×Dic43—44no—
158C22×Dic43C22×C412yes—
159C22×Dic43C22×C411yescharacteristic
160C22×Dic43C22×C411yescharacteristic
161C22×Dic43C22×C412yes—
162C22×Dic43C22×C411yescharacteristic
163C2×C4×D43—42no—
164C2×C4×D43—42no—
165C2×C4×D43—42no—
166C2×C4×D43C22⋊C411yescharacteristic
167C2×C4×D43C22⋊C411yescharacteristic
168C2×C4×D43C22⋊C411yescharacteristic
169C2×C4×D43C22⋊C411yescharacteristic
170C2×C4×D43C22⋊C411yescharacteristic
171C2×C4×D43C22⋊C411yescharacteristic
172C2×C4×D43C22⋊C411yescharacteristic
173C2×C4×D43C22⋊C411yescharacteristic
174C2×C4×D43C2×D411yescharacteristic
175C2×C4×D43C2×D411yescharacteristic
176C2×C4×D43C2×D411yescharacteristic
177C2×C4×D43C2×D411yescharacteristic
178C2×C8×C43—22no—
179C2×C8×C43—22no—
180C22⋊C4×C43—22no—
181C22⋊C4×C43—22no—
182C22⋊C4×C43—22no—
183C22⋊C4×C43—22no—
184C42×C43C2×D411yescharacteristic
185C42×C43C2×D411yescharacteristic
186C42×C43C2×D411yescharacteristic
187C42×C43C2×D411yescharacteristic
188C22⋊Dic43—22no—
189C22⋊Dic43—22no—
190C22⋊Dic43—22no—
191C22⋊Dic43—22no—
192(C2×D43)⋊C4—22no—
193(C2×D43)⋊C4—22no—
194C4⋊Dic43—21no—
195C4⋊Dic43—21no—
196C4⋊Dic43—21no—
197C4⋊Dic43—21no—
198C4⋊Dic43—22no—
199C4⋊Dic43—22no—
200Dic43⋊C4—22no—
201Dic43⋊C4—22no—
202Dic43⋊C4—22no—
203Dic43⋊C4—22no—
204Dic43⋊C4—22no—
205Dic43⋊C4—22no—
206C4×Dic43—22no—
207C4×Dic43—22no—
208C4×Dic43C22⋊C411yescharacteristic
209C4×Dic43C22⋊C411yescharacteristic
210C4×Dic43C22⋊C411yescharacteristic
211C4×Dic43C22⋊C411yescharacteristic
212C4×Dic43C22⋊C411yescharacteristic
213C4×Dic43C22⋊C411yescharacteristic
214C4×Dic43C22⋊C411yescharacteristic
215C4×Dic43C22⋊C411yescharacteristic
216C4×Dic43C2×D411yescharacteristic
217C4×Dic43C2×D411yescharacteristic
218C4×Dic43C2×D411yescharacteristic
219C4×Dic43C2×D411yescharacteristic
220C2×C43⋊C8—22no—
221C2×C43⋊C8—22no—
222#10621872—44no—
223#10621872—44no—
224C23×C43—22no—
225C23×C43—24no—
226C23×C43—42no—
227C23×C43—42no—
228C23×C43C23×C411yescharacteristic
229C22×D43—22no—
230C22×D43—41no—
231C22×D43—41no—
232C22×D43C2×C22⋊C411yescharacteristic
233C22×D43C2×C22⋊C411yescharacteristic
234Q8×C43—42no—
235Q8×C43—42no—
236D4×C43—42no—
237D4×C43—42no—
238C2×C4×C43—22no—
239C2×C4×C43—22no—
240C2×C4×C43—22no—
241C2×C4×C43—41no—
242C2×C4×C43—42no—
243C2×C4×C43C2×C22⋊C411yescharacteristic
244C2×C4×C43C2×C22⋊C411yescharacteristic
245C2×C4×C43C2×C22⋊C411yescharacteristic
246C2×C4×C43C2×C22⋊C411yescharacteristic
247C2×C4×C43C22×D411yescharacteristic
248C2×C4×C43C22×D411yescharacteristic
249C22⋊D43—44no—
250C22⋊D43—44no—
251C2×Dic43—22no—
252C2×Dic43—22no—
253C2×Dic43—22no—
254C2×Dic43—22no—
255C2×Dic43—24no—
256C2×Dic43—41no—
257C2×Dic43—44no—
258C2×Dic43—44no—
259C2×Dic43—42no—
260C2×Dic43C2×C22⋊C411yescharacteristic
261C2×Dic43C2×C22⋊C411yescharacteristic
262C2×Dic43C2×C22⋊C411yescharacteristic
263C2×Dic43C2×C22⋊C411yescharacteristic
264C2×Dic43C2×C22⋊C411yescharacteristic
265C2×Dic43C2×C22⋊C411yescharacteristic
266C4×D43—22no—
267C4×D43—22no—
268C4×D43—22no—
269C4×D43—22no—
270C4×D43—41no—
271C4×D43—41no—
272C4×D43—41no—
273C4×D43—41no—
274C4×D43—41no—
275C4×D43—41no—
276C4×D43—41no—
277C4×D43—41no—
278C8×C43—42no—
279C8×C43—42no—
280C43⋊C8—42no—
281C43⋊C8—42no—
282#17401—431nomaximal, Sylow
283C22×C43—21no—
284C22×C43—21no—
285C22×C43—42no—
286C22×C43—42no—
287C22×C43—42no—
288C22×C43—84no—
289C22×C43C22×C22⋊C411yescharacteristic
290C22×C43#56812yes—
291C2×D43—21no—
292C2×D43—21no—
293C2×D43—21no—
294C2×D43—21no—
295C2×D43—21no—
296C2×D43—21no—
297C2×D43—44no—
298C4×C43—21no—
299C4×C43—21no—
300C4×C43—21no—
301C4×C43—21no—
302C4×C43—21no—
303C4×C43—42no—
304C4×C43#41312yes—
305Dic43—21no—
306Dic43—21no—
307Dic43—21no—
308Dic43—21no—
309Dic43—21no—
310Dic43—42no—
311Dic43—42no—
312Dic43#41012yes—
313C2×#529—431no—
314#2920—431no—
315#2920—431no—
316#1473—434no—
317#1470—431no—
318#1470—431no—
319#1469—431no—
320#1469D4311yescharacteristic, complemented
321#1465—434no—
322C2×C43—22no—
323C2×C43—82no—
324C2×C43#287911yescharacteristic
325C2×C43C2×#41011yescharacteristic
326C2×C43C2×#41311yescharacteristic
327D43—41no—
328D43—41no—
329#529—861no—
330#529—861no—
331#529—861no—
332#529—861no—
333#529—861no—
334#529—861no—
335#529—861no—
336#529—861no—
337C2×#126—432no—
338C2×#125—431no—
339C2×#124C2×D4311yescharacteristic
340C2×#123—432no—
341C2×C22⋊Q8—431no—
342C2×#121—431no—
343C2×C4×Q8—431no—
344C2×C4×D4—431no—
345C2×C4×D4—432no—
346C2×#117—432no—
347C22×C42—431no—
348#432—431no—
349#432—432no—
350#432—861no—
351#432—861no—
352#432C2×D4311yescharacteristic
353#431—431no—
354#431—432no—
355#431—861no—
356#431—861no—
357#431C2×D4311yescharacteristic
358#423—861no—
359#423—861no—
360#407—432no—
361#407—432no—
362C4×M16—861no—
363C4×M16—861no—
364#324—434no—
365#324—434no—
366#323—434no—
367#323C2×D4314yes—
368C43#1740111yescharacteristic, complemented, Sylow
369C22×Q8C4×D4311yescharacteristic
370C22×D4—432no—
371C22×D4C4×D4311yescharacteristic
372C23×C4—431no—
373C23×C4—431no—
374C23×C4—431no—
375C23×C4—432no—
376C2×M16—862no—
377C2×M16—862no—
378#126—862no—
379#126—862no—
380#126—862no—
381#126—862no—
382#125—861no—
383#125—861no—
384#125—861no—
385#125—861no—
386#124—21no—
387#124—21no—
388#124—21no—
389#124—21no—
390#123—862no—
391#123—862no—
392#123—862no—
393#123—862no—
394C22⋊Q8—861no—
395C22⋊Q8—861no—
396C22⋊Q8—861no—
397C22⋊Q8—861no—
398#121—861no—
399#121—861no—
400#121—861no—
401#121—861no—
402C4×Q8—861no—
403C4×Q8—861no—
404C4×Q8—861no—
405C4×Q8—861no—
406C4×D4—861no—
407C4×D4—861no—
408C4×D4—861no—
409C4×D4—861no—
410C4×D4—862no—
411C4×D4—862no—
412C4×D4—862no—
413C4×D4—862no—
414#117—862no—
415#117—862no—
416#117—862no—
417#117—862no—
418C2×C4⋊C4—432no—
419C2×C4⋊C4—432no—
420C2×C4⋊C4—431no—
421C2×C4⋊C4—431no—
422C2×C4⋊C4—432no—
423C2×C4⋊C4—432no—
424C2×C22⋊C4—432no—
425C2×C22⋊C4—432no—
426C2×C22⋊C4—432no—
427C2×C22⋊C4C4×D4312yes—
428C2×C22⋊C4C22×D4312yes—
429C2×C42—431no—
430C2×C42—431no—
431C2×C42—431no—
432C2×C42—431no—
433C2×C42—431no—
434C2×C42—431no—
435C2×C42—431no—
436C2×C42—431no—
437C2×C42—431no—
438C2×C42—431no—
439C2×C42—431no—
440C2×C42—432no—
441C2×C42C22×D4311yescharacteristic
442C4⋊C8—21no—
443C4⋊C8—21no—
444C4⋊C8—862no—
445C4⋊C8—862no—
446C4⋊C8—861no—
447C4⋊C8—861no—
448C22⋊C8—432no—
449C22⋊C8—432no—
450C22⋊C8—862no—
451C22⋊C8—862no—
452C22⋊C8C22×D4312yes—
453C22⋊C8C22×D4312yes—
454#97—21no—
455#97—862no—
456#97—861no—
457C4×C8—21no—
458C4×C8—862no—
459C4×C8—861no—
460C24—431no—
461C24C2×C4×D4312yes—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—42no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—42no—
468C2×D4—862no—
469C2×D4—862no—
470C2×D4—1724no—
471C22×C4—431no—
472C22×C4—431no—
473C22×C4—431no—
474C22×C4—431no—
475C22×C4—431no—
476C22×C4—431no—
477C22×C4—431no—
478C22×C4—431no—
479C22×C4—431no—
480C22×C4—431no—
481C22×C4—431no—
482C22×C4—431no—
483C22×C4—431no—
484C22×C4—431no—
485C22×C4—431no—
486C22×C4—432no—
487C22×C4—432no—
488C22×C4—862no—
489C22×C4—862no—
490C22×C4—1722no—
491C22×C4—1722no—
492C22×C4—1722no—
493C22×C4—1724no—
494C22×C4C2×C4×D4311yescharacteristic
495C22×C4C2×C4×D4311yescharacteristic
496C22×C4C2×C4×D4312yes—
497C22×C4C23×D4311yescharacteristic
498M16—1724no—
499M16—1724no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—862no—
503C2×C8—862no—
504C4⋊C4—862no—
505C4⋊C4—862no—
506C4⋊C4—862no—
507C4⋊C4—862no—
508C4⋊C4—862no—
509C4⋊C4—862no—
510C4⋊C4—861no—
511C4⋊C4—861no—
512C4⋊C4—861no—
513C4⋊C4—861no—
514C4⋊C4—862no—
515C4⋊C4—862no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—862no—
521C22⋊C4—862no—
522C22⋊C4—862no—
523C22⋊C4—862no—
524C22⋊C4—862no—
525C22⋊C4—862no—
526C42—431no—
527C42—431no—
528C42—431no—
529C42—431no—
530C42—431no—
531C42—431no—
532C42—431no—
533C42—431no—
534C42—431no—
535C42—431no—
536C42—431no—
537C42—431no—
538C42—862no—
539C42—862no—
540C42D4×D4311yescharacteristic
541C42D4×D4311yescharacteristic
542C42D4×D4311yescharacteristic
543C42D4×D4311yescharacteristic
544C23—22no—
545C23—24no—
546C23—42no—
547C23—42no—
548C23—431no—
549C23—431no—
550C23—862no—
551C23—1721no—
552C23—1721no—
553C23C22×C4×D4311yescharacteristic
554Q8—42no—
555Q8—42no—
556D4—42no—
557D4—42no—
558D4—1724no—
559D4—1724no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—431no—
566C2×C4—431no—
567C2×C4—431no—
568C2×C4—431no—
569C2×C4—431no—
570C2×C4—431no—
571C2×C4—862no—
572C2×C4—862no—
573C2×C4—862no—
574C2×C4—862no—
575C2×C4—862no—
576C2×C4—862no—
577C2×C4—862no—
578C2×C4—862no—
579C2×C4—864no—
580C2×C4—1721no—
581C2×C4—1721no—
582C2×C4—1721no—
583C2×C4—1721no—
584C2×C4—1721no—
585C2×C4—1721no—
586C2×C4—1721no—
587C2×C4—1721no—
588C2×C4—1724no—
589C2×C4—1721no—
590C2×C4—1724no—
591C2×C4—1722no—
592C2×C4C22⋊C4×D4311yescharacteristic
593C2×C4C22⋊C4×D4311yescharacteristic
594C2×C4C22⋊C4×D4311yescharacteristic
595C2×C4C22⋊C4×D4311yescharacteristic
596C2×C4C2×D4×D4311yescharacteristic
597C2×C4C2×D4×D4311yescharacteristic
598C8—42no—
599C8—42no—
600C8—1722no—
601C8—1722no—
602C22—21no—
603C22—21no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—84no—
608C22—861no—
609C22—861no—
610C22—861no—
611C22—861no—
612C22—861no—
613C22—861no—
614C22—1724no—
615C22C2×C22⋊C4×D4311yescharacteristic
616C22#1458366412yes—
617C4—21no—
618C4—21no—
619C4—21no—
620C4—21no—
621C4—21no—
622C4—42no—
623C4—432no—
624C4—861no—
625C4—861no—
626C4—861no—
627C4—861no—
628C4—861no—
629C4—1722no—
630C4—1722no—
631C4#1458288012yes—
632C2—22no—
633C2—82no—
634C2—1721no—
635C2—1721no—
636C2#1914303711yescharacteristic
637C2#410×D4311yescharacteristic
638C2C2×#1458288011yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D43
commutatorC23×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1469C2
socleC22×C43C22×C22⋊C4
prime radicalC2×#13063296C4
cosocleC22×C4×C43C24
hypercenter#1469D43
hypocenterC43#17401
LCSG ▷ C23×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1469

Complements

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

NH ≅ G/Ntype
2. #19143225C2semidirect
3. #19143223C2semidirect
4. #19132961C2semidirect
5. #19132952C2semidirect
6. #19132950C2semidirect
7. #19132947C2semidirect
8. #19132930C2semidirect
9. C43×#1469C2semidirect
320. #1469D43semidirect
368. C43#17401semidirect

Maximal subgroups

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

subgroupindex
C2×#145834282
#191432252
#191432232
#191329612
#191329522
#191329502
#191329472
#191329302
C43×#14692
#1740143