Finite Group Databasebeta

← #25107913

Subgroups — #25107913

Order 11008.

920 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583056C211yesmaximal, characteristic
2#19137278C211yesmaximal, characteristic
3#19137178C211yesmaximal, characteristic
4#19135463C211yesmaximal, characteristic, complemented
5#19135454C211yesmaximal, characteristic, complemented
6#19135286C211yesmaximal, characteristic, complemented
7#19135273C211yesmaximal, characteristic, complemented
8#19135229C211yesmaximal, characteristic, complemented
9#19134939C211yesmaximal, characteristic, complemented
10#19134935C211yesmaximal, characteristic, complemented
11#19134906C211yesmaximal, characteristic
12#19134791C211yesmaximal, characteristic
13#19134528C211yesmaximal, characteristic
14#19134360C211yesmaximal, characteristic
15C43×#1928C211yesmaximal, characteristic, complemented
16C22×Q8×D43C2211yescharacteristic
17C22×Dic172C2211yescharacteristic
18C22×#10621885C2211yescharacteristic
19C2×#13063305C2211yescharacteristic
20C2×C4×Dic86C2211yescharacteristic
21C2×#13063283C2211yescharacteristic
22C2×#13063282C2211yescharacteristic
23#14583056C418yescomplemented
24C2×#13063256C2211yescharacteristic
25C2×#13063250C2211yescharacteristic
26C2×#13063250C2211yescharacteristic
27C2×#13063249C2211yescharacteristic
28C2×#13063248C2211yescharacteristic
29C2×#13063248C2211yescharacteristic
30C2×#13063248C2211yescharacteristic
31C2×(C2×D43)⋊C8C2211yescharacteristic
32C2×(C2×D43)⋊C8C2211yescharacteristic
33C2×#13063235C2211yescharacteristic
34C2×C8×Dic43C2211yescharacteristic
35#14582806C2211yescharacteristic
36#14582805C2211yescharacteristic
37#14582804C2211yescharacteristic
38#14582804C2211yescharacteristic
39#14582803—21no—
40#14582803—21no—
41#14582798C2211yescharacteristic
42#14582797—21no—
43#14582791—21no—
44#14582790—21no—
45#14582784—21no—
46#14582761—21no—
47#14582761—21no—
48#14582756C2211yescharacteristic
49#14582662C2211yescharacteristic
50#14582661C2211yescharacteristic
51#14582661C2211yescharacteristic
52#14582654C2211yescharacteristic
53C2×#107×C43C2211yescharacteristic
54C2×#103×C43C2211yescharacteristic
55C2×#103×C43C2211yescharacteristic
56C43×#391C2211yescharacteristic
57C4×C4⋊C4×C43C2211yescharacteristic
58C43×#340C2211yescharacteristic
59C43×#336C2211yescharacteristic
60C22×Dic86—21no—
61C22×Dic86C2311yescharacteristic
62C22×Dic86C2311yescharacteristic
63C2×Q8×D43—22no—
64C2×Q8×D43—41no—
65C2×Q8×D43—41no—
66C2×Q8×D43—41no—
67C2×Q8×D43—41no—
68C2×Q8×D43C2×C414yes—
69C22×C4×D43—21no—
70C22×C4×D43C2311yescharacteristic
71C2×Dic172—41no—
72C2×Dic172—41no—
73C2×Dic172C2×C414yes—
74C2×#10621885—41no—
75C2×#10621885—41no—
76C2×#10621885C2×C414yes—
77C22×C43⋊C8C2311yescharacteristic
78C22×C43⋊C8C2311yescharacteristic
79#13063305—22no—
80#13063305C2×C414yes—
81C4×Dic86—41no—
82C4×Dic86—41no—
83#13063283—41no—
84#13063283—41no—
85#13063282—41no—
86#13063282—41no—
87C2×C4⋊Dic43—21no—
88C2×C4⋊Dic43—21no—
89C2×C4⋊Dic43C2311yescharacteristic
90C2×C4⋊Dic43C2311yescharacteristic
91C2×Dic43⋊C4—21no—
92C2×Dic43⋊C4D411yescharacteristic
93C2×Dic43⋊C4D411yescharacteristic
94C2×(C2×D43)⋊C4—21no—
95C2×(C2×D43)⋊C4—21no—
96C2×(C2×D43)⋊C4—21no—
97C2×(C2×D43)⋊C4D411yescharacteristic
98C2×(C2×D43)⋊C4D411yescharacteristic
99C2×C4×Dic43—21no—
100C2×C4×Dic43—21no—
101C2×C4×Dic43C2311yescharacteristic
102#13063256—41no—
103#13063250—41no—
104#13063250—41no—
105#13063249—41no—
106#13063248—41no—
107#13063248—41no—
108#13063248C2×C414yes—
109(C2×D43)⋊C8—22no—
110(C2×D43)⋊C8C2×C414yes—
111#13063235—41no—
112C8×Dic43—41no—
113C43⋊#95—21no—
114C43⋊#95—21no—
115C43⋊#95—21no—
116C43⋊#95—21no—
117C43⋊#95—21no—
118C43⋊#95—21no—
119C43⋊#95—21no—
120C43⋊#95—21no—
121C43⋊#95—21no—
122C43⋊#95—21no—
123C22×Q8×C43C2311yescharacteristic
124C22×C8×C43C2311yescharacteristic
125C22×C8×C43C2311yescharacteristic
126C2×C4⋊C4×C43C2311yescharacteristic
127C2×C4⋊C4×C43C2311yescharacteristic
128C2×C4⋊C4×C43C2311yescharacteristic
129C2×C42×C43—21no—
130C2×C42×C43C2311yescharacteristic
131#107×C43—41no—
132#103×C43—41no—
133#103×C43C2×C414yes—
134#95×C43—21no—
135#95×C43—21no—
136#95×C43—21no—
137C23×D43C2×D411yescharacteristic
138C2×Q8×C43—21no—
139C2×Q8×C43—21no—
140C2×Q8×C43—21no—
141C2×Q8×C43—21no—
142C2×Q8×C43—21no—
143C2×Q8×C43C22×C412yes—
144C22×C4×C43—21no—
145C22×C4×C43—21no—
146C22×C4×C43—21no—
147C22×C4×C43—21no—
148C22×C4×C43C2×D411yescharacteristic
149C22×C4×C43C2×D411yescharacteristic
150C22×C4×C43C2411yescharacteristic
151C22×Dic43—21no—
152C22×Dic43—21no—
153C22×Dic43—21no—
154C22×Dic43—21no—
155C22×Dic43C2×D411yescharacteristic
156C22×Dic43C2×D411yescharacteristic
157C22×Dic43C2×D411yescharacteristic
158Q8×D43—44no—
159Q8×D43—44no—
160Q8×D43—44no—
161Q8×D43—44no—
162C2×C4×D43—21no—
163C2×C4×D43—21no—
164C2×C4×D43—21no—
165C2×C4×D43—21no—
166C2×C4×D43—24no—
167C2×C4×D43—42no—
168C2×C4×D43—42no—
169C2×C4×D43C22×C412yes—
170C2×C4×D43#4112yes—
171C2×Dic86—21no—
172C2×Dic86—21no—
173C2×Dic86—21no—
174C2×Dic86—21no—
175C2×Dic86—21no—
176C2×Dic86—21no—
177C2×Dic86—21no—
178C2×Dic86—21no—
179C2×Dic86—21no—
180C2×Dic86—21no—
181C2×Dic86—24no—
182C2×Dic86—42no—
183C2×Dic86—42no—
184C2×Dic86C22×C412yes—
185C2×Dic86C22×C412yes—
186C2×C8×C43—21no—
187C2×C8×C43—21no—
188C2×C8×C43—21no—
189C2×C8×C43—21no—
190C2×C8×C43—21no—
191C2×C8×C43C22×C412yes—
192C4⋊C4×C43—21no—
193C4⋊C4×C43—21no—
194C4⋊C4×C43—21no—
195C4⋊C4×C43—21no—
196C4⋊C4×C43—21no—
197C4⋊C4×C43C22×C412yes—
198C42×C43—21no—
199C42×C43—21no—
200C42×C43—81no—
201#10621885—44no—
202(C2×D43)⋊C4—24no—
203(C2×D43)⋊C4—41no—
204(C2×D43)⋊C4—41no—
205(C2×D43)⋊C4—42no—
206(C2×D43)⋊C4—81no—
207C4⋊Dic43—24no—
208C4⋊Dic43—21no—
209C4⋊Dic43—21no—
210C4⋊Dic43—21no—
211C4⋊Dic43—21no—
212C4⋊Dic43—81no—
213Dic43⋊C4—41no—
214Dic43⋊C4—41no—
215Dic43⋊C4—42no—
216C4×Dic43—21no—
217C4×Dic43—21no—
218C4×Dic43—42no—
219C4×Dic43—42no—
220C2×C43⋊C8—21no—
221C2×C43⋊C8—21no—
222C2×C43⋊C8—21no—
223C2×C43⋊C8—21no—
224C2×C43⋊C8—21no—
225C2×C43⋊C8C22×C412yes—
226Dic172—44no—
227C23×C43C22×D411yescharacteristic
228C22×D43—21no—
229C22×D43—21no—
230C22×D43—21no—
231C22×D43—21no—
232C22×D43—21no—
233C22×D43—21no—
234C22×D43C4×D412yes—
235Q8×C43—22no—
236Q8×C43—22no—
237Q8×C43—41no—
238Q8×C43—41no—
239C2×C4×C43—22no—
240C2×C4×C43—22no—
241C2×C4×C43—21no—
242C2×C4×C43—21no—
243C2×C4×C43—21no—
244C2×C4×C43—21no—
245C2×C4×C43—21no—
246C2×C4×C43—21no—
247C2×C4×C43—41no—
248C2×C4×C43—41no—
249C2×C4×C43—41no—
250C2×C4×C43—41no—
251C2×C4×C43—41no—
252C2×C4×C43—41no—
253C2×C4×C43—41no—
254C2×C4×C43—41no—
255C2×C4×C43—41no—
256C2×C4×C43—41no—
257C2×C4×C43C23×C411yescharacteristic
258C2×C4×C43C2×#4111yescharacteristic
259C2×C4×C43H(5,𝔽2)11yescharacteristic
260C2×C4×C43H(5,𝔽2)11yescharacteristic
261C2×C4×C43H(5,𝔽2)11yescharacteristic
262C2×C4×C43H(5,𝔽2)11yescharacteristic
263C2×Dic43—21no—
264C2×Dic43—21no—
265C2×Dic43—22no—
266C2×Dic43—22no—
267C2×Dic43—21no—
268C2×Dic43—21no—
269C2×Dic43—21no—
270C2×Dic43—21no—
271C2×Dic43—21no—
272C2×Dic43—21no—
273C2×Dic43—41no—
274C2×Dic43—41no—
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×Dic43C4×D412yes—
284C4×D43—22no—
285C4×D43—22no—
286C4×D43—22no—
287C4×D43—22no—
288C4×D43—48no—
289Dic86—22no—
290Dic86—22no—
291Dic86—22no—
292Dic86—22no—
293Dic86—48no—
294Dic86—41no—
295Dic86—41no—
296Dic86—41no—
297Dic86—41no—
298C8×C43—22no—
299C8×C43—41no—
300C43⋊C8—22no—
301C43⋊C8—41no—
302#22054—431nomaximal, Sylow
303C22×C43C2×C4×D411yescharacteristic
304C22×C43#53411yescharacteristic
305C22×C43#53511yescharacteristic
306C22×C43#57711yescharacteristic
307C22×C43#57811yescharacteristic
308C22×C43#57811yescharacteristic
309C22×C43#57811yescharacteristic
310C2×D43—22no—
311C2×D43—22no—
312C2×D43—22no—
313C2×D43—41no—
314C2×D43—41no—
315C2×D43—41no—
316C2×D43—41no—
317C4×C43—42no—
318C4×C43—41no—
319C4×C43—41no—
320C4×C43—42no—
321C4×C43—81no—
322C4×C43—81no—
323C4×C43#51811yescharacteristic
324C4×C43#51811yescharacteristic
325C4×C43#53811yescharacteristic
326C4×C43#53811yescharacteristic
327Dic43—22no—
328Dic43—42no—
329Dic43—42no—
330Dic43—41no—
331Dic43—41no—
332Dic43—81no—
333Dic43—81no—
334C2×#451—431no—
335#2323—431no—
336#2310—431no—
337#1971—431no—
338#1969—431no—
339#1935—431no—
340#1933—431no—
341#1928D4311yescharacteristic, complemented
342#1927—431no—
343#1878—431no—
344#1878—431no—
345#1873—431no—
346#1854—431no—
347#1787—431no—
348#1750—431no—
349C2×C43#232111yescharacteristic
350C2×C43#294711yescharacteristic
351C2×C43#295011yescharacteristic
352C2×C43#320111yescharacteristic
353C2×C43#320211yescharacteristic
354C2×C43#322311yescharacteristic
355C2×C43#322511yescharacteristic
356D43—42no—
357C23×Q8—431no—
358C22×Q16—431no—
359C22×Q16—431no—
360C2×C22⋊Q8—431no—
361C2×C4×Q8—431no—
362C2×#117—431no—
363C2×#117—431no—
364#451—438no—
365C2×#107C2×D4311yescharacteristic
366C2×#106—431no—
367C2×#103—431no—
368C2×#103—431no—
369C2×#103—431no—
370C2×#103—431no—
371C2×#103—431no—
372C2×#103—431no—
373C2×#103C2×D4311yescharacteristic
374C2×#103C2×D4311yescharacteristic
375C2×C22⋊C8—431no—
376C2×C22⋊C8—431no—
377C2×#97—431no—
378C2×C4×C8—431no—
379#391—431no—
380#391—431no—
381#391—431no—
382#391—431no—
383#391—861no—
384#391—861no—
385#391C2×D4311yescharacteristic
386#389—431no—
387#389—861no—
388#388—861no—
389#388—861no—
390#387—861no—
391#381—861no—
392#381—861no—
393#380—431no—
394C4×C4⋊C4C2×D4311yescharacteristic
395#340—431no—
396#340—431no—
397#340—431no—
398#340C2×D4311yescharacteristic
399#336—431no—
400#336C2×D4311yescharacteristic
401C43#2205411yescharacteristic, complemented, Sylow
402C22×Q8—434no—
403C22×Q8—431no—
404C22×Q8—431no—
405C22×Q8—861no—
406C22×Q8—862no—
407C22×Q8—1721no—
408C22×Q8—1721no—
409C22×Q8—1721no—
410C22×Q8—1721no—
411C22×Q8C22×D4311yescharacteristic
412C23×C4—431no—
413C23×C4—861no—
414C2×Q16—434no—
415C2×Q16—434no—
416C2×Q16—1721no—
417C2×Q16—1721no—
418C2×Q16—1721no—
419C2×Q16—1721no—
420C22×C8—431no—
421C22×C8—431no—
422C22×C8C22×D4311yescharacteristic
423C22×C8C22×D4311yescharacteristic
424C22⋊Q8—434no—
425C22⋊Q8—862no—
426C4×Q8—1721no—
427C4×Q8—1721no—
428#117—1721no—
429#117—1721no—
430#117—1721no—
431#117—1721no—
432C2×C4⋊C4—431no—
433C2×C4⋊C4—431no—
434C2×C4⋊C4—431no—
435C2×C4⋊C4—431no—
436C2×C4⋊C4—861no—
437C2×C4⋊C4—861no—
438C2×C4⋊C4—861no—
439C2×C4⋊C4C22×D4311yescharacteristic
440C2×C4⋊C4C22×D4311yescharacteristic
441C2×C4⋊C4C22×D4311yescharacteristic
442C2×C22⋊C4—431no—
443C2×C22⋊C4—431no—
444C2×C22⋊C4—861no—
445C2×C22⋊C4—861no—
446C2×C22⋊C4—861no—
447C2×C42—21no—
448C2×C42—431no—
449C2×C42—861no—
450C2×C42—861no—
451C2×C42C22×D4311yescharacteristic
452#107—41no—
453#106—1721no—
454#103—41no—
455#103—434no—
456#103—1721no—
457#103—1721no—
458#103—1721no—
459#103—1721no—
460#103—1721no—
461#103C4×D4314yes—
462C22⋊C8—434no—
463C22⋊C8—862no—
464#97—1721no—
465C4×C8—1721no—
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#95—861no—
475#95—861no—
476#95—861no—
477#95—861no—
478#95—861no—
479C24—431no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—432no—
486C2×Q8—432no—
487C2×Q8—864no—
488C2×Q8—861no—
489C2×Q8—861no—
490C2×Q8—861no—
491C2×Q8—861no—
492C2×Q8—861no—
493C2×Q8—861no—
494C2×Q8—861no—
495C2×Q8—861no—
496C2×Q8—861no—
497C2×Q8—861no—
498C2×Q8—1722no—
499C2×Q8—1722no—
500C2×Q8—1724no—
501C2×Q8—1724no—
502C2×Q8—1724no—
503C2×Q8—1724no—
504C2×Q8C2×C4×D4312yes—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—431no—
510C22×C4—432no—
511C22×C4—432no—
512C22×C4—431no—
513C22×C4—431no—
514C22×C4—864no—
515C22×C4—861no—
516C22×C4—861no—
517C22×C4—861no—
518C22×C4—861no—
519C22×C4—861no—
520C22×C4—861no—
521C22×C4—861no—
522C22×C4—861no—
523C22×C4—1722no—
524C22×C4—1722no—
525C22×C4D4×D4311yescharacteristic
526C22×C4D4×D4311yescharacteristic
527C22×C4C23×D4311yescharacteristic
528Q16—1724no—
529Q16—1724no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—432no—
536C2×C8—861no—
537C2×C8—861no—
538C2×C8—861no—
539C2×C8—861no—
540C2×C8—861no—
541C2×C8C2×C4×D4312yes—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—864no—
548C4⋊C4—861no—
549C4⋊C4—861no—
550C4⋊C4—861no—
551C4⋊C4—861no—
552C4⋊C4—3441no—
553C4⋊C4—1721no—
554C4⋊C4—1721no—
555C4⋊C4—1722no—
556C4⋊C4C2×C4×D4312yes—
557C22⋊C4—864no—
558C22⋊C4—3441no—
559C22⋊C4—1721no—
560C22⋊C4—1721no—
561C22⋊C4—1722no—
562C42—21no—
563C42—21no—
564C42—81no—
565C42—861no—
566C42—861no—
567C42—1722no—
568C42—1722no—
569C23—432no—
570C23—861no—
571C23—861no—
572C23—861no—
573C23—861no—
574C23—861no—
575C23—861no—
576C23C2×D4×D4311yescharacteristic
577Q8—22no—
578Q8—22no—
579Q8—41no—
580Q8—41no—
581Q8—862no—
582Q8—862no—
583Q8—862no—
584Q8—862no—
585Q8—1728no—
586Q8—1721no—
587Q8—1721no—
588Q8—1721no—
589Q8—1721no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—432no—
609C2×C4—861no—
610C2×C4—861no—
611C2×C4—862no—
612C2×C4—862no—
613C2×C4—862no—
614C2×C4—862no—
615C2×C4—862no—
616C2×C4—862no—
617C2×C4—861no—
618C2×C4—861no—
619C2×C4—861no—
620C2×C4—861no—
621C2×C4—861no—
622C2×C4—861no—
623C2×C4—1728no—
624C2×C4—1721no—
625C2×C4—1721no—
626C2×C4—1721no—
627C2×C4—1721no—
628C2×C4—1721no—
629C2×C4—1721no—
630C2×C4—1721no—
631C2×C4—1721no—
632C2×C4—1721no—
633C2×C4—1721no—
634C2×C4C22×C4×D4311yescharacteristic
635C2×C4#41×D4311yescharacteristic
636C2×C4#1306337011yescharacteristic
637C2×C4#1306337111yescharacteristic
638C2×C4#1306337111yescharacteristic
639C2×C4#1306337111yescharacteristic
640C8—22no—
641C8—41no—
642C8—862no—
643C8—1721no—
644C22—862no—
645C22—862no—
646C22—862no—
647C22—1721no—
648C22—1721no—
649C22—1721no—
650C22—1721no—
651C22C4×D4×D4311yescharacteristic
652C22#1458346011yescharacteristic
653C22#1458346511yescharacteristic
654C22#1458370911yescharacteristic
655C22#1458371211yescharacteristic
656C22#1458371311yescharacteristic
657C22#1458371511yescharacteristic
658C4—41no—
659C4—42no—
660C4—41no—
661C4—42no—
662C4—81no—
663C4—81no—
664C4—862no—
665C4—3441no—
666C4—3441no—
667C4—1721no—
668C4—1721no—
669C4—1722no—
670C4—1722no—
671C4#1458337111yescharacteristic
672C4#1458337311yescharacteristic
673C4#1458348311yescharacteristic
674C4#1458348811yescharacteristic
675C2—1722no—
676C2#1913725711yescharacteristic
677C2#1914343511yescharacteristic
678C2#1914345811yescharacteristic
679C2#1914548211yescharacteristic
680C2#1914548911yescharacteristic
681C2#1914567711yescharacteristic
682C2#1914570111yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1928C2
socleC23×C43C22×D4
prime radicalC23×D43C2×D4
cosocleC22×C4×C43C24
hypercenter#1928D43
hypocenterC43#22054
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1928

Complements

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

NH ≅ G/Ntype
4. #19135463C2semidirect
5. #19135454C2semidirect
6. #19135286C2semidirect
7. #19135273C2semidirect
8. #19135229C2semidirect
9. #19134939C2semidirect
10. #19134935C2semidirect
15. C43×#1928C2semidirect
23. #14583056C4semidirect
341. #1928D43semidirect
401. C43#22054semidirect

Maximal subgroups

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

subgroupindex
C2×#145830562
#191372782
#191371782
#191354632
#191354542
#191352862
#191352732
#191352292
#191349392
#191349352
#191349062
#191347912
#191345282
#191343602
C43×#19282
#2205443