Finite Group Databasebeta

← #25156604

Subgroups — #25156604

Order 11008.

908 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583114C211yesmaximal, characteristic
2C2×C4×#10621882C211yesmaximal, characteristic
3#19138382C211yesmaximal, characteristic, complemented
4#19136146C211yesmaximal, characteristic, complemented
5#19136086C211yesmaximal, characteristic, complemented
6#19136009C211yesmaximal, characteristic, complemented
7#19135836C211yesmaximal, characteristic, complemented
8#19135270C211yesmaximal, characteristic, complemented
9#19135154C211yesmaximal, characteristic, complemented
10#19134943C211yesmaximal, characteristic
11#19134929C211yesmaximal, characteristic, complemented
12#19134929C211yesmaximal, characteristic, complemented
13#19134856C211yesmaximal, characteristic, complemented
14#19134855C211yesmaximal, characteristic, complemented
15C43×#2349C211yesmaximal, characteristic, complemented
16C22×#10621883C2211yescharacteristic
17C22×#10621882C2211yescharacteristic
18C2×#13063313C2211yescharacteristic
19C2×C4×D172C2211yescharacteristic
20#14583114—24no—
21C4×#10621882—24no—
22C2×#13063258C2211yescharacteristic
23C2×#13063256C2211yescharacteristic
24C2×C4⋊(C43⋊C8)C2211yescharacteristic
25C2×#13063248C2211yescharacteristic
26C2×#13063248C2211yescharacteristic
27C2×#13063248C2211yescharacteristic
28C2×#13063246C2211yescharacteristic
29C2×#13063246C2211yescharacteristic
30C2×#13063245C2211yescharacteristic
31C2×#13063245C2211yescharacteristic
32C2×#13063245C2211yescharacteristic
33C2×C4×C43⋊C8C2211yescharacteristic
34#14582828—21no—
35#14582825—21no—
36#14582805C2211yescharacteristic
37#14582805C2211yescharacteristic
38#14582801C2211yescharacteristic
39#14582797C2211yescharacteristic
40#14582770—21no—
41#14582761—21no—
42#14582661C2211yescharacteristic
43#14582661C2211yescharacteristic
44#14582661C2211yescharacteristic
45#14582661C2211yescharacteristic
46#14582654C2211yescharacteristic
47#14582654C2211yescharacteristic
48#14582654C2211yescharacteristic
49#14582654C2211yescharacteristic
50C2×C4×D4×C43C2211yescharacteristic
51C2×C4×D4×C43C2211yescharacteristic
52C2×C4×D4×C43C2211yescharacteristic
53C43×#389C2211yescharacteristic
54C43×#389C2211yescharacteristic
55C43×#385—21no—
56C43×#385—21no—
57C43×#382C2211yescharacteristic
58C43×#380—21no—
59C4×C4⋊C4×C43C2211yescharacteristic
60C2×#95×C43—21no—
61C22×Dic86C2311yescharacteristic
62C22×D172C2311yescharacteristic
63C22×C4×D43—21no—
64C2×#10621883—22no—
65C2×#10621883—22no—
66C2×#10621883—22no—
67C2×#10621882—22no—
68C2×#10621882—22no—
69C2×#10621882D414yes—
70C22×C43⋊C8C2311yescharacteristic
71C22×C43⋊C8C2311yescharacteristic
72C22×C43⋊C8C2311yescharacteristic
73C22×C43⋊C8C2311yescharacteristic
74#13063313—22no—
75#13063313—41no—
76C4×D172—22no—
77C4×D172—41no—
78C2×C4⋊Dic43C2311yescharacteristic
79C2×C4⋊Dic43C2311yescharacteristic
80C2×Dic43⋊C4—21no—
81C2×(C2×D43)⋊C4—21no—
82C2×(C2×D43)⋊C4—21no—
83C2×(C2×D43)⋊C4—21no—
84C2×(C2×D43)⋊C4—21no—
85C2×(C2×D43)⋊C4—21no—
86#13063258—22no—
87#13063256—41no—
88C4⋊(C43⋊C8)—22no—
89#13063248—22no—
90#13063248—22no—
91#13063248—41no—
92#13063246—22no—
93#13063246—22no—
94#13063245—22no—
95#13063245—22no—
96#13063245—22no—
97C4×C43⋊C8—22no—
98C43⋊#95—21no—
99C43⋊#95—21no—
100C43⋊#95—21no—
101C43⋊#95—21no—
102C43⋊#95—21no—
103C22×D4×C43C2311yescharacteristic
104C23×C4×C43—21no—
105C23×C4×C43—21no—
106C23×C4×C43—21no—
107C4×D4×C43—22no—
108C4×D4×C43—22no—
109C4×D4×C43—22no—
110C4×D4×C43—22no—
111C4×D4×C43—41no—
112C4×D4×C43—41no—
113C2×C4⋊C4×C43—21no—
114C2×C4⋊C4×C43—21no—
115C2×C4⋊C4×C43D411yescharacteristic
116C2×C4⋊C4×C43D411yescharacteristic
117C2×C4⋊C4×C43C2311yescharacteristic
118C2×C4⋊C4×C43C2311yescharacteristic
119C2×C4⋊C4×C43C2311yescharacteristic
120C2×C22⋊C4×C43—21no—
121C2×C22⋊C4×C43—21no—
122C2×C22⋊C4×C43—21no—
123C2×C42×C43C2311yescharacteristic
124C2×C42×C43C2311yescharacteristic
125C2×C42×C43C2311yescharacteristic
126#95×C43—21no—
127#95×C43—21no—
128#95×C43—21no—
129#95×C43—21no—
130#95×C43D411yescharacteristic
131#95×C43D411yescharacteristic
132C24×C43—21no—
133C23×D43—21no—
134C2×D4×C43—21no—
135C2×D4×C43—21no—
136C2×D4×C43—21no—
137C2×D4×C43C2×D412yes—
138C2×D4×C43#4112yes—
139C2×D4×C43#4112yes—
140C22×C4×C43—24no—
141C22×C4×C43—21no—
142C22×C4×C43—21no—
143C22×C4×C43—21no—
144C22×C4×C43—42no—
145C22×C4×C43—42no—
146C22×C4×C43—42no—
147C22×C4×C43—42no—
148C22×C4×C43—81no—
149C22×C4×C43—81no—
150C22×C4×C43—81no—
151C22×C4×C43—81no—
152C22×C4×C43C2×D411yescharacteristic
153C22×C4×C43C2×D411yescharacteristic
154C22×C4×C43C2×D411yescharacteristic
155C22×C4×C43C2×D411yescharacteristic
156C22×C4×C43C2×D411yescharacteristic
157C22×C4×C43C2×D411yescharacteristic
158C22×C4×C43C2411yescharacteristic
159C22×Dic43—21no—
160C22×Dic43—21no—
161C22×Dic43—21no—
162C2×D172—21no—
163C2×D172—21no—
164C2×D172—21no—
165C2×D172—21no—
166C2×D172—21no—
167C2×D172C2×D412yes—
168C2×C4×D43—42no—
169C2×C4×D43—42no—
170C2×C4×D43—81no—
171C2×Dic86—21no—
172C2×Dic86—21no—
173C2×Dic86—21no—
174C2×Dic86—21no—
175C2×Dic86—21no—
176C2×Dic86—21no—
177C4⋊C4×C43—22no—
178C4⋊C4×C43—21no—
179C4⋊C4×C43—21no—
180C4⋊C4×C43—21no—
181C4⋊C4×C43—21no—
182C4⋊C4×C43—21no—
183C4⋊C4×C43—41no—
184C4⋊C4×C43—81no—
185C4⋊C4×C43—81no—
186C4⋊C4×C43#4112yes—
187C22⋊C4×C43—24no—
188C22⋊C4×C43—42no—
189C22⋊C4×C43—42no—
190C42×C43—21no—
191C42×C43—21no—
192C42×C43—21no—
193C42×C43—21no—
194C42×C43—21no—
195C42×C43—21no—
196#10621883—44no—
197#10621882—28no—
198(C2×D43)⋊C4—42no—
199(C2×D43)⋊C4—42no—
200(C2×D43)⋊C4—42no—
201(C2×D43)⋊C4—42no—
202(C2×D43)⋊C4—81no—
203C4⋊Dic43—21no—
204C4⋊Dic43—21no—
205C4⋊Dic43—21no—
206C4⋊Dic43—21no—
207Dic43⋊C4—81no—
208C2×C43⋊C8—21no—
209C2×C43⋊C8—21no—
210C2×C43⋊C8—21no—
211C2×C43⋊C8—21no—
212C2×C43⋊C8—21no—
213C2×C43⋊C8—21no—
214C2×C43⋊C8—21no—
215C2×C43⋊C8—21no—
216C2×C43⋊C8—21no—
217C2×C43⋊C8—21no—
218C2×C43⋊C8—21no—
219C2×C43⋊C8C2×D412yes—
220C23×C43—22no—
221C23×C43—22no—
222C23×C43—22no—
223C23×C43—41no—
224C23×C43—41no—
225C23×C43—41no—
226C23×C43—41no—
227C23×C43C22×D411yescharacteristic
228C22×D43—22no—
229C22×D43—41no—
230C22×D43—41no—
231C22×D43—41no—
232C22×D43—41no—
233C22×D43—41no—
234C22×D43—41no—
235D4×C43—22no—
236D4×C43—22no—
237D4×C43—22no—
238D4×C43#12314yes—
239C2×C4×C43—21no—
240C2×C4×C43—21no—
241C2×C4×C43—21no—
242C2×C4×C43—21no—
243C2×C4×C43—22no—
244C2×C4×C43—21no—
245C2×C4×C43—21no—
246C2×C4×C43—21no—
247C2×C4×C43—21no—
248C2×C4×C43—21no—
249C2×C4×C43—21no—
250C2×C4×C43—21no—
251C2×C4×C43—21no—
252C2×C4×C43—21no—
253C2×C4×C43—21no—
254C2×C4×C43—21no—
255C2×C4×C43—21no—
256C2×C4×C43—21no—
257C2×C4×C43—41no—
258C2×C4×C43—41no—
259C2×C4×C43—41no—
260C2×C4×C43—41no—
261C2×C4×C43—41no—
262C2×C4×C43—41no—
263C2×C4×C43—41no—
264C2×C4×C43—41no—
265C2×C4×C43—84no—
266C2×C4×C43—84no—
267C2×C4×C43—84no—
268C2×C4×C43#13512yes—
269C2×C4×C43C22×D411yescharacteristic
270C2×C4×C43C2×#4111yescharacteristic
271C2×C4×C43C2×#4111yescharacteristic
272C2×C4×C43C2×#4111yescharacteristic
273C2×C4×C43H(5,𝔽2)11yescharacteristic
274C2×C4×C432−1+411yescharacteristic
275C2×Dic43—41no—
276C2×Dic43—41no—
277C2×Dic43—41no—
278C2×Dic43—41no—
279C2×Dic43—41no—
280C2×Dic43—41no—
281C2×Dic43—41no—
282C2×Dic43—41no—
283C2×Dic43—41no—
284D172—22no—
285D172—22no—
286D172—41no—
287D172—41no—
288C4×D43—84no—
289Dic86—41no—
290Dic86—41no—
291Dic86—41no—
292Dic86—41no—
293C43⋊C8—22no—
294C43⋊C8—41no—
295C43⋊C8—41no—
296C43⋊C8—41no—
297#26160—431nomaximal, Sylow
298C22×C43—24no—
299C22×C43—42no—
300C22×C43—42no—
301C22×C43—42no—
302C22×C43—42no—
303C22×C43—42no—
304C22×C43—42no—
305C22×C43#52511yescharacteristic
306C22×C43D4211yescharacteristic
307C22×C43#54811yescharacteristic
308C22×C43C2×#13511yescharacteristic
309C22×C43#57511yescharacteristic
310C22×C43#57611yescharacteristic
311C22×C43#57711yescharacteristic
312C2×D43—42no—
313C2×D43—42no—
314C2×D43—42no—
315C2×D43—81no—
316C2×D43—81no—
317C2×D43—81no—
318C2×D43—81no—
319C4×C43—22no—
320C4×C43—41no—
321C4×C43—41no—
322C4×C43—42no—
323C4×C43—41no—
324C4×C43—41no—
325C4×C43—41no—
326C4×C43—81no—
327C4×C43—81no—
328C4×C43C2×#12311yescharacteristic
329C4×C43#53911yescharacteristic
330C4×C43#54711yescharacteristic
331C4×C43#55311yescharacteristic
332Dic43—81no—
333Dic43—81no—
334Dic43—81no—
335C2×#463—431no—
336C2×C4×D8—431no—
337#2447—431no—
338#2349D4311yescharacteristic, complemented
339#2097—431no—
340#2086—431no—
341#2073—431no—
342#2042—431no—
343#1933—431no—
344#1916—431no—
345#1879—431no—
346#1877—431no—
347#1877—431no—
348#1864—431no—
349#1864—431no—
350C2×C43—44no—
351C2×C43#244311yescharacteristic
352C2×C43#310211yescharacteristic
353C2×C43#311811yescharacteristic
354C2×C43#326711yescharacteristic
355C2×C43#328011yescharacteristic
356C2×C43#336011yescharacteristic
357C2×C43#337011yescharacteristic
358D43—82no—
359C22×SD16—431no—
360C22×D8—431no—
361C2×#124—431no—
362C2×C4×D4—431no—
363C2×C4×D4C2×D4311yescharacteristic
364C2×C4×D4C2×D4311yescharacteristic
365C2×C4×D4C2×D4311yescharacteristic
366#463—864no—
367C4×D8—864no—
368C2×#107—431no—
369C2×#106—431no—
370C2×C4⋊C8—431no—
371C2×#103—431no—
372C2×#103—431no—
373C2×#103—431no—
374C2×#102—431no—
375C2×#102—431no—
376C2×#102—431no—
377C2×#102—431no—
378C2×#102—431no—
379C2×C4×C8—431no—
380#397—861no—
381#397—861no—
382#391—431no—
383#391—431no—
384#390—431no—
385#389—431no—
386#389C2×D4311yescharacteristic
387#389C2×D4311yescharacteristic
388#385—21no—
389#385—21no—
390#385—861no—
391#382C2×D4311yescharacteristic
392#381—861no—
393#380—21no—
394C4×C4⋊C4C2×D4311yescharacteristic
395C2×#95—21no—
396#340—431no—
397#340—431no—
398#340—431no—
399#340—431no—
400#336—431no—
401#336—431no—
402#336—431no—
403#336—431no—
404C43#2616011yescharacteristic, complemented, Sylow
405C22×Q8—431no—
406C22×D4—431no—
407C22×D4C22×D4311yescharacteristic
408C23×C4—21no—
409C23×C4—21no—
410C23×C4—21no—
411C23×C4—861no—
412C2×SD16—862no—
413C2×SD16—862no—
414C2×SD16—862no—
415C2×D8—434no—
416C2×D8—862no—
417C2×D8—862no—
418C22×C8—431no—
419C22×C8—431no—
420C22×C8—431no—
421C22×C8—431no—
422#124—862no—
423#124—1721no—
424C4×D4—22no—
425C4×D4—22no—
426C4×D4—22no—
427C4×D4—22no—
428C4×D4—41no—
429C4×D4—41no—
430C4×D4—862no—
431C4×D4—1721no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—21no—
434C2×C4⋊C4—431no—
435C2×C4⋊C4—431no—
436C2×C4⋊C4—861no—
437C2×C4⋊C4C22⋊D4311yescharacteristic
438C2×C4⋊C4C22⋊D4311yescharacteristic
439C2×C4⋊C4C22×D4311yescharacteristic
440C2×C4⋊C4C22×D4311yescharacteristic
441C2×C4⋊C4C22×D4311yescharacteristic
442C2×C22⋊C4—21no—
443C2×C22⋊C4—21no—
444C2×C22⋊C4—21no—
445C2×C22⋊C4—861no—
446C2×C22⋊C4—861no—
447C2×C22⋊C4—861no—
448C2×C22⋊C4—861no—
449C2×C22⋊C4—861no—
450C2×C42C22×D4311yescharacteristic
451C2×C42C22×D4311yescharacteristic
452C2×C42C22×D4311yescharacteristic
453#107—862no—
454#106—1721no—
455C4⋊C8—862no—
456#103—862no—
457#103—862no—
458#103—1721no—
459#102—862no—
460#102—862no—
461#102—862no—
462#102—862no—
463#102—862no—
464C4×C8—862no—
465#95—21no—
466#95—21no—
467#95—21no—
468#95—21no—
469#95—861no—
470#95—861no—
471#95—861no—
472#95—861no—
473#95—861no—
474#95C22⋊D4311yescharacteristic
475#95C22⋊D4311yescharacteristic
476C24—21no—
477C24—861no—
478C2×Q8—861no—
479C2×Q8—861no—
480C2×Q8—861no—
481C2×Q8—861no—
482C2×Q8—861no—
483C2×Q8—861no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—432no—
488C2×D4—861no—
489C2×D4—861no—
490C2×D4—861no—
491C2×D4—861no—
492C2×D4—861no—
493C2×D4#1062189812yes—
494C2×D4D4×D4312yes—
495C2×D4#1062190212yes—
496C22×C4—21no—
497C22×C4—24no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—42no—
501C22×C4—42no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4—81no—
505C22×C4—81no—
506C22×C4—81no—
507C22×C4—81no—
508C22×C4—861no—
509C22×C4—861no—
510C22×C4—861no—
511C22×C4—3441no—
512C22×C4—1722no—
513C22×C4—1722no—
514C22×C4C2×C22⋊D4311yescharacteristic
515C22×C4C2×C22⋊D4311yescharacteristic
516C22×C4C2×C22⋊D4311yescharacteristic
517C22×C4C2×C22⋊D4311yescharacteristic
518C22×C4C2×C22⋊D4311yescharacteristic
519C22×C4C2×C22⋊D4311yescharacteristic
520C22×C4C23×D4311yescharacteristic
521SD16—1724no—
522D8—868no—
523C2×C8—432no—
524C2×C8—861no—
525C2×C8—861no—
526C2×C8—861no—
527C2×C8—861no—
528C2×C8—861no—
529C2×C8—861no—
530C2×C8—861no—
531C2×C8—861no—
532C2×C8—861no—
533C2×C8—861no—
534C2×C8—861no—
535C4⋊C4—22no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—41no—
542C4⋊C4—81no—
543C4⋊C4—81no—
544C4⋊C4—861no—
545C4⋊C4—861no—
546C4⋊C4—861no—
547C4⋊C4—861no—
548C4⋊C4—3441no—
549C4⋊C4#1062189812yes—
550C22⋊C4—24no—
551C22⋊C4—42no—
552C22⋊C4—42no—
553C22⋊C4—3441no—
554C22⋊C4—1722no—
555C22⋊C4—1722no—
556C22⋊C4—1722no—
557C22⋊C4—1722no—
558C42—21no—
559C42—21no—
560C42—21no—
561C42—21no—
562C42—21no—
563C42—21no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—862no—
572C23—1721no—
573C23—1721no—
574C23—1721no—
575C23—1721no—
576C23—1721no—
577C23—1721no—
578C23C22×C22⋊D4311yescharacteristic
579Q8—1721no—
580Q8—1721no—
581Q8—1721no—
582Q8—1721no—
583D4—22no—
584D4—22no—
585D4—22no—
586D4—862no—
587D4—862no—
588D4—1721no—
589D4—1721no—
590D4#1306331014yes—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—84no—
618C2×C4—84no—
619C2×C4—84no—
620C2×C4—3444no—
621C2×C4—1721no—
622C2×C4—1721no—
623C2×C4—1721no—
624C2×C4—1721no—
625C2×C4—1721no—
626C2×C4—1721no—
627C2×C4—1721no—
628C2×C4—1721no—
629C2×C4—1721no—
630C2×C4#1306334512yes—
631C2×C4C2×D4×D4311yescharacteristic
632C2×C4C2×#1062189811yescharacteristic
633C2×C4C2×#1062189811yescharacteristic
634C2×C4C2×#1062190211yescharacteristic
635C2×C4#1306337011yescharacteristic
636C2×C4#1306337311yescharacteristic
637C8—862no—
638C8—1721no—
639C8—1721no—
640C8—1721no—
641C22—24no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—3441no—
649C22—3441no—
650C22—3441no—
651C22—3441no—
652C22—1722no—
653C22—1722no—
654C22—1722no—
655C22#1458341011yescharacteristic
656C22D4×C22⋊D4311yescharacteristic
657C22#1458356011yescharacteristic
658C22C2×#1306334511yescharacteristic
659C22#1458369711yescharacteristic
660C22#1458370211yescharacteristic
661C22#1458371011yescharacteristic
662C4—22no—
663C4—41no—
664C4—41no—
665C4—42no—
666C4—41no—
667C4—41no—
668C4—41no—
669C4—81no—
670C4—81no—
671C4—3441no—
672C4—3441no—
673C4—3441no—
674C4C2×#1306331011yescharacteristic
675C4#1458349611yescharacteristic
676C4#1458355011yescharacteristic
677C4#1458359711yescharacteristic
678C2—44no—
679C2—3442no—
680C2#1913833011yescharacteristic
681C2#1914461311yescharacteristic
682C2#1914477911yescharacteristic
683C2#1914611611yescharacteristic
684C2#1914627611yescharacteristic
685C2#1914724411yescharacteristic
686C2#1914734811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#2349C2
socleC23×C43C22×D4
prime radicalC22×#10621882C22
hypercenter#2349D43
hypocenterC43#26160
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2349

Complements

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

NH ≅ G/Ntype
3. #19138382C2semidirect
4. #19136146C2semidirect
5. #19136086C2semidirect
6. #19136009C2semidirect
7. #19135836C2semidirect
8. #19135270C2semidirect
9. #19135154C2semidirect
11. #19134929C2semidirect
12. #19134929C2semidirect
13. #19134856C2semidirect
14. #19134855C2semidirect
15. C43×#2349C2semidirect
338. #2349D43semidirect
404. C43#26160semidirect

Maximal subgroups

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

subgroupindex
C2×#145831142
C2×C4×#106218822
#191383822
#191361462
#191360862
#191360092
#191358362
#191352702
#191351542
#191349432
#191349292
#191348562
#191348552
C43×#23492
#2616043