Finite Group Databasebeta

← #25112054

Subgroups — #25112054

Order 11008.

1490 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583552C211yesmaximal, characteristic
2C2×C8×D172C211yesmaximal, characteristic, complemented
3C2×#14582894C212yesmaximal, complemented
4#19137546C211yesmaximal, characteristic, complemented
5#19135519C211yesmaximal, characteristic, complemented
6#19135015C212yesmaximal, complemented
7#19134904C212yesmaximal, complemented
8#19134592C211yesmaximal, characteristic, complemented
9#19134534C212yesmaximal, complemented
10C4×#13063247C211yesmaximal, characteristic, complemented
11C43×#1956C211yesmaximal, characteristic, complemented
12C22×#10621902C2212yes—
13C22×C8×D43C2212yes—
14#14583552—24no—
15#14583552C418yescomplemented
16C2×#13063310—22no—
17C2×#13063301C2212yes—
18C2×#13063300—21no—
19C2×C4×D172C2211yescharacteristic
20C2×C4×D172C2211yescharacteristic
21C2×#13063283C2212yes—
22C2×C4⋊C4×D43—21no—
23C8×D172—24no—
24C2×C4⋊(C43⋊C8)C2211yescharacteristic
25#14582894—28no—
26C2×#13063249C2212yes—
27C2×#13063249C2212yes—
28C2×#13063247C2211yescharacteristic
29C2×#13063247C2211yescharacteristic
30C2×#13063245C2212yes—
31C2×(C2×D43)⋊C8C2212yes—
32#14582800—21no—
33#14582789—22no—
34#14582756C2212yes—
35C4⋊C4×Dic43—21no—
36C4×C4⋊Dic43C2211yescharacteristic
37#14582662C2211yescharacteristic
38#14582662C2211yescharacteristic
39#14582661C2212yes—
40C2×C4⋊Q8×C43C2211yescharacteristic
41C2×#103×C43C2212yes—
42C2×#103×C43C2212yes—
43C2×C4×C8×C43C2211yescharacteristic
44C43×#384C2211yescharacteristic
45C2×#10621902—24no—
46C2×#10621902—24no—
47C2×#10621902—24no—
48C2×#10621902—42no—
49C2×#10621902—42no—
50C2×#10621902C2×C418yes—
51C22×D172—22no—
52C22×D172D411yescharacteristic
53C22×D172C2311yescharacteristic
54C22×C4×D43—21no—
55C22×C4×D43—22no—
56C22×C4×D43C2312yes—
57C22×C43⋊C8C2312yes—
58C2×C8×D43—24no—
59C2×C8×D43—24no—
60C2×C8×D43—42no—
61#13063310—216no—
62#13063301—24no—
63#13063301C2×C418yes—
64#13063300—28no—
65C4×D172—22no—
66C4×D172C2×C414yes—
67C4×D172C2×C414yes—
68C4×D172D414yes—
69#13063283—24no—
70#13063283—42no—
71C2×C4⋊Dic43D411yescharacteristic
72C2×C4⋊Dic43C2311yescharacteristic
73C2×C4⋊Dic43C2311yescharacteristic
74C2×C4⋊Dic43C2311yescharacteristic
75C2×Dic43⋊C4—22no—
76C4⋊C4×D43—28no—
77C2×(C2×D43)⋊C4—22no—
78C2×(C2×D43)⋊C4—21no—
79C2×(C2×D43)⋊C4—22no—
80C2×(C2×D43)⋊C4—22no—
81C2×(C2×D43)⋊C4D412yes—
82C2×C4×Dic43—21no—
83C2×C4×Dic43—22no—
84C4⋊(C43⋊C8)—22no—
85#13063249—24no—
86#13063249—42no—
87#13063247—22no—
88#13063247D414yescomplemented
89#13063245—24no—
90(C2×D43)⋊C8—24no—
91C43⋊#95—21no—
92C43⋊#95—22no—
93C22×Q8×C43C2312yes—
94C22×C8×C43C2311yescharacteristic
95C22×C8×C43C2311yescharacteristic
96C4⋊Q8×C43—22no—
97C4⋊Q8×C43C2×C414yes—
98C2×C4⋊C4×C43—21no—
99C2×C4⋊C4×C43—21no—
100C2×C4⋊C4×C43—21no—
101C2×C4⋊C4×C43—21no—
102C2×C4⋊C4×C43C2312yes—
103C2×C42×C43C2311yescharacteristic
104#103×C43—24no—
105#103×C43—42no—
106C4×C8×C43—22no—
107C23×D43—21no—
108C23×D43C2×D412yes—
109C2×Q8×C43—22no—
110C2×Q8×C43—22no—
111C2×Q8×C43—22no—
112C2×Q8×C43—22no—
113C2×Q8×C43—22no—
114C2×Q8×C43C22×C414yes—
115C22×C4×C43—22no—
116C22×C4×C43—22no—
117C22×C4×C43C2×D411yescharacteristic
118C22×C4×C43C2×D411yescharacteristic
119C22×C4×C43C2411yescharacteristic
120C22×Dic43—21no—
121C22×Dic43—21no—
122C22×Dic43—21no—
123C22×Dic43C2×D412yes—
124#10621902—216no—
125#10621902—48no—
126#10621902—48no—
127#10621902—48no—
128C2×D172—22no—
129C2×D172—22no—
130C2×D172—21no—
131C2×D172—21no—
132C2×D172—21no—
133C2×D172—28no—
134C2×D172—44no—
135C2×D172—44no—
136C2×D172C22⋊C414yes—
137C2×D172C22×C412yes—
138C2×D172C2×D412yes—
139C2×D172#4112yes—
140C2×C4×D43—22no—
141C2×C4×D43—22no—
142C2×C4×D43—22no—
143C2×C4×D43—22no—
144C2×C4×D43—24no—
145C2×C4×D43—28no—
146C2×C4×D43—44no—
147C2×C4×D43—44no—
148C2×C4×D43—42no—
149C2×C4×D43—42no—
150C2×C4×D43C22×C414yes—
151C2×C4×D43C2×D414yes—
152C2×C8×C43—21no—
153C2×C8×C43—21no—
154C2×C8×C43—21no—
155C2×C8×C43—21no—
156C2×C8×C43—21no—
157C2×C8×C43C2×D412yes—
158C4⋊C4×C43—24no—
159C4⋊C4×C43—24no—
160C4⋊C4×C43—22no—
161C4⋊C4×C43—22no—
162C4⋊C4×C43—42no—
163C4⋊C4×C43—81no—
164C42×C43C22×C412yes—
165C42×C43C2×D412yes—
166(C2×D43)⋊C4—24no—
167(C2×D43)⋊C4—28no—
168(C2×D43)⋊C4—44no—
169(C2×D43)⋊C4—82no—
170(C2×D43)⋊C4C22⋊C418yes—
171C4⋊Dic43—21no—
172C4⋊Dic43—21no—
173C4⋊Dic43—21no—
174C4⋊Dic43—21no—
175C4⋊Dic43C22⋊C414yes—
176C4⋊Dic43C22×C412yes—
177C4⋊Dic43C2×D412yes—
178Dic43⋊C4—28no—
179C4×Dic43—44no—
180C4×Dic43—42no—
181C2×C43⋊C8—22no—
182C2×C43⋊C8—22no—
183C2×C43⋊C8—22no—
184C8×D43—48no—
185C23×C43C22×D411yescharacteristic
186C22×D43—22no—
187C22×D43—22no—
188C22×D43—22no—
189C22×D43—22no—
190C22×D43—22no—
191C22×D43—22no—
192C22×D43—22no—
193C22×D43—22no—
194C22×D43—22no—
195C22×D43—41no—
196C22×D43—41no—
197C22×D43—41no—
198C22×D43—41no—
199C22×D43C2×C22⋊C414yes—
200Q8×C43—24no—
201Q8×C43—24no—
202Q8×C43—42no—
203Q8×C43—42no—
204C2×C4×C43—21no—
205C2×C4×C43—21no—
206C2×C4×C43—24no—
207C2×C4×C43—42no—
208C2×C4×C43—42no—
209C2×C4×C43—42no—
210C2×C4×C43—42no—
211C2×C4×C43—42no—
212C2×C4×C43C2×C22⋊C412yes—
213C2×C4×C43C2×C22⋊C412yes—
214C2×C4×C43#13512yes—
215C2×C4×C43#13512yes—
216C2×C4×C43C23×C411yescharacteristic
217C2×C4×C43C22×D411yescharacteristic
218C2×C4×C43C22×D411yescharacteristic
219C2×C4×C43C2×#4111yescharacteristic
220C2×C4×C43H(5,𝔽2)11yescharacteristic
221C2×C4×C43H(5,𝔽2)11yescharacteristic
222C2×Dic43—22no—
223C2×Dic43—22no—
224C2×Dic43—22no—
225C2×Dic43—22no—
226C2×Dic43—41no—
227C2×Dic43—41no—
228C2×Dic43—41no—
229C2×Dic43—41no—
230C2×Dic43—41no—
231C2×Dic43—41no—
232C2×Dic43—41no—
233C2×Dic43C2×C22⋊C414yes—
234D172—22no—
235D172—22no—
236D172—22no—
237D172—28no—
238D172—416no—
239D172C4×D414yes—
240C4×D43—24no—
241C4×D43—24no—
242C4×D43—24no—
243C4×D43—24no—
244C4×D43—416no—
245C4×D43—48no—
246C8×C43—22no—
247C8×C43—41no—
248C43⋊C8—42no—
249#22417—431nomaximal, Sylow
250C22×C43C22×C22⋊C411yescharacteristic
251C22×C43D4211yescharacteristic
252C22×C43#54611yescharacteristic
253C22×C43C2×#13511yescharacteristic
254C22×C43C2×#13511yescharacteristic
255C22×C43#57711yescharacteristic
256C22×C43#57811yescharacteristic
257C2×D43—24no—
258C2×D43—24no—
259C2×D43—24no—
260C2×D43—24no—
261C2×D43—42no—
262C2×D43—42no—
263C2×D43—42no—
264C2×D43—42no—
265C2×D43—42no—
266C2×D43—42no—
267C2×D43—42no—
268C2×D43—42no—
269C2×D43—42no—
270C2×D43—42no—
271C4×C43—22no—
272C4×C43—44no—
273C4×C43—82no—
274C4×C43#41614yescomplemented
275C4×C43C2×C4×D411yescharacteristic
276C4×C43#51811yescharacteristic
277C4×C43D4211yescharacteristic
278C4×C43#54611yescharacteristic
279Dic43—24no—
280Dic43—42no—
281Dic43—42no—
282Dic43—81no—
283C2×#547—431no—
284C2×C8×D4—431no—
285C2×#416—432no—
286#2354—431no—
287#1981—431no—
288#1956D4311yescharacteristic, complemented
289#1892—432no—
290#1873—432no—
291#1802—431no—
292#1788—432no—
293C4×#102—431no—
294C2×C43D4×C22⋊C411yescharacteristic
295C2×C43C2×#41611yescharacteristic
296C2×C43#289811yescharacteristic
297C2×C43#328311yescharacteristic
298C2×C43#328411yescharacteristic
299C2×C43#330211yescharacteristic
300C2×C43#330311yescharacteristic
301D43—44no—
302D43—44no—
303C22×#41—432no—
304C23×C8—432no—
305#547—438no—
306#547—864no—
307C2×C4⋊Q8C2×D4311yescharacteristic
308C2×#123—862no—
309C2×C22⋊Q8—432no—
310C2×#121—861no—
311C2×C4×D4—431no—
312C2×C4×D4—431no—
313C2×#117—432no—
314C22×C4⋊C4—861no—
315C8×D4—864no—
316C2×C4⋊C8—431no—
317#416—868no—
318C2×#103—432no—
319C2×#103—432no—
320C2×#103C2×D4312yes—
321C2×#103C2×D4312yes—
322C2×#102—431no—
323C2×#102—431no—
324C2×#102—432no—
325C2×C22⋊C8—432no—
326C2×C4×C8C2×D4311yescharacteristic
327#390—861no—
328#388—862no—
329#384C2×D4311yescharacteristic
330#380—432no—
331C4×C4⋊C4—431no—
332C4×C4⋊C4—861no—
333#340—431no—
334#340—431no—
335#340—432no—
336C43#2241711yescharacteristic, complemented, Sylow
337C2×#41—438no—
338C2×#41—864no—
339C2×#41—864no—
340C2×#41—864no—
341C2×#41—1722no—
342C2×#41—1722no—
343C22×Q8C22×D4312yes—
344C22×D4—431no—
345C22×D4—431no—
346C22×D4—862no—
347C23×C4—432no—
348C23×C4—862no—
349C23×C4—861no—
350C22×C8—432no—
351C22×C8—864no—
352C22×C8—864no—
353C22×C8—1722no—
354C22×C8C22×D4311yescharacteristic
355C22×C8C22×D4311yescharacteristic
356C4⋊Q8—22no—
357C4⋊Q8C4×D4314yes—
358#123—8616no—
359C22⋊Q8—438no—
360C22⋊Q8—864no—
361#121—868no—
362C4×D4—434no—
363C4×D4—434no—
364C4×D4—434no—
365C4×D4—862no—
366#117—864no—
367#117—1722no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—431no—
373C2×C4⋊C4—431no—
374C2×C4⋊C4—431no—
375C2×C4⋊C4—431no—
376C2×C4⋊C4—868no—
377C2×C4⋊C4—862no—
378C2×C4⋊C4C22×D4312yes—
379C2×C22⋊C4—432no—
380C2×C22⋊C4—862no—
381C2×C22⋊C4—861no—
382C2×C22⋊C4—862no—
383C2×C22⋊C4—862no—
384C2×C42—862no—
385C2×C42—861no—
386C2×C42C22×D4311yescharacteristic
387C4⋊C8—862no—
388#103—24no—
389#103—42no—
390#103—864no—
391#103—1722no—
392#102—434no—
393#102—864no—
394#102—862no—
395C22⋊C8—864no—
396C4×C8—22no—
397#95—862no—
398#95—861no—
399C24—432no—
400C24—861no—
401#41—8616no—
402#41—1728no—
403#41—1728no—
404#41—1728no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8C2×C4×D4314yes—
411C2×D4—434no—
412C2×D4—432no—
413C2×D4—432no—
414C2×D4—432no—
415C2×D4—868no—
416C2×D4—862no—
417C2×D4—862no—
418C2×D4—861no—
419C2×D4—861no—
420C2×D4—861no—
421C2×D4—1724no—
422C2×D4—1724no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—432no—
426C22×C4—434no—
427C22×C4—434no—
428C22×C4—868no—
429C22×C4—862no—
430C22×C4—862no—
431C22×C4—862no—
432C22×C4—862no—
433C22×C4—861no—
434C22×C4—861no—
435C22×C4—864no—
436C22×C4—861no—
437C22×C4—1724no—
438C22×C4—1724no—
439C22×C4—1722no—
440C22×C4—1722no—
441C22×C4D4×D4311yescharacteristic
442C22×C4D4×D4311yescharacteristic
443C22×C4C23×D4311yescharacteristic
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—862no—
450C2×C8—862no—
451C2×C8—862no—
452C2×C8—1728no—
453C2×C8D4×D4312yes—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—42no—
459C4⋊C4—81no—
460C4⋊C4—434no—
461C4⋊C4—432no—
462C4⋊C4—432no—
463C4⋊C4—868no—
464C4⋊C4—861no—
465C4⋊C4—861no—
466C4⋊C4—861no—
467C4⋊C4—861no—
468C22⋊C4—438no—
469C22⋊C4—868no—
470C22⋊C4—864no—
471C22⋊C4—3442no—
472C22⋊C4—1724no—
473C42—1724no—
474C42—1722no—
475C42C2×C4×D4312yes—
476C42D4×D4312yes—
477C23—434no—
478C23—862no—
479C23—862no—
480C23—862no—
481C23—862no—
482C23—862no—
483C23—862no—
484C23—862no—
485C23—862no—
486C23—862no—
487C23—1721no—
488C23—1721no—
489C23—1721no—
490C23—1721no—
491C23C2×D4×D4311yescharacteristic
492Q8—24no—
493Q8—24no—
494Q8—42no—
495Q8—42no—
496D4—434no—
497D4—862no—
498D4—868no—
499D4—862no—
500D4—862no—
501D4—17216no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—24no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—434no—
511C2×C4—862no—
512C2×C4—862no—
513C2×C4—864no—
514C2×C4—864no—
515C2×C4—864no—
516C2×C4—864no—
517C2×C4—862no—
518C2×C4—862no—
519C2×C4—17216no—
520C2×C4—1721no—
521C2×C4—1721no—
522C2×C4—1721no—
523C2×C4—1721no—
524C2×C4—1721no—
525C2×C4—1721no—
526C2×C4—1721no—
527C2×C4—1728no—
528C2×C4C22⋊C4×D4312yes—
529C2×C4C22⋊C4×D4312yes—
530C2×C4#1306334612yes—
531C2×C4#1306334812yes—
532C2×C4C22×C4×D4311yescharacteristic
533C2×C4C2×D4×D4311yescharacteristic
534C2×C4C2×D4×D4311yescharacteristic
535C2×C4#41×D4311yescharacteristic
536C2×C4#1306337011yescharacteristic
537C2×C4#1306337111yescharacteristic
538C8—22no—
539C8—41no—
540C8—1722no—
541C22—864no—
542C22—864no—
543C22—864no—
544C22—864no—
545C22—1722no—
546C22—1722no—
547C22—1722no—
548C22—1722no—
549C22—1722no—
550C22—1722no—
551C22—1722no—
552C22—1722no—
553C22—1722no—
554C22—1722no—
555C22C2×C22⋊C4×D4311yescharacteristic
556C22#1458353411yescharacteristic
557C22#1458354211yescharacteristic
558C22C2×#1306334611yescharacteristic
559C22C2×#1306334811yescharacteristic
560C22#1458370511yescharacteristic
561C22#1458371211yescharacteristic
562C4—22no—
563C4—44no—
564C4—82no—
565C4—864no—
566C4—3441no—
567C4—1722no—
568C4—1722no—
569C4#1458289414yescomplemented
570C4C4×D4×D4311yescharacteristic
571C4#1458337111yescharacteristic
572C4#1458353311yescharacteristic
573C4#1458354611yescharacteristic
574C2—1724no—
575C2—1724no—
576C2#1913741611yescharacteristic
577C2C2×#1458289411yescharacteristic
578C2#1914312311yescharacteristic
579C2#1914631511yescharacteristic
580C2#1914632411yescharacteristic
581C2#1914651111yescharacteristic
582C2#1914652211yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1956C2
socleC23×C43C22×D4
prime radicalC2×#14583552C2
cosocleC22×C4×C43C24
hypercenter#1956D43
hypocenterC43#22417
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1956

Complements

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

NH ≅ G/Ntype
2. C2×C8×D172C2semidirect
3. C2×#14582894C2semidirect
4. #19137546C2semidirect
5. #19135519C2semidirect
6. #19135015C2semidirect
7. #19134904C2semidirect
8. #19134592C2semidirect
9. #19134534C2semidirect
10. C4×#13063247C2semidirect
11. C43×#1956C2semidirect
15. #14583552C4semidirect
88. #13063247D4semidirect
274. C4×C43#416semidirect
288. #1956D43semidirect
336. C43#22417semidirect
569. C4#14582894semidirect

Maximal subgroups

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

subgroupindex
C2×#145835522
C2×C8×D1722
C2×#145828942
#191375462
#191355192
#191350152
#191349042
#191345922
#191345342
C4×#130632472
C43×#19562
#2241743