Finite Group Databasebeta

← #25156528

Subgroups — #25156528

Order 11008.

854 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583240C211yesmaximal, characteristic, complemented
2C2×#14582940C211yesmaximal, characteristic, complemented
3#19138048C211yesmaximal, characteristic
4#19137545C211yesmaximal, characteristic
5#19136135C211yesmaximal, characteristic, complemented
6#19136075C211yesmaximal, characteristic, complemented
7#19136031C211yesmaximal, characteristic, complemented
8#19135792C211yesmaximal, characteristic, complemented
9#19135235C211yesmaximal, characteristic, complemented
10#19135185C211yesmaximal, characteristic, complemented
11#19135027C211yesmaximal, characteristic, complemented
12#19134906C211yesmaximal, characteristic, complemented
13#19134889C211yesmaximal, characteristic
14#19134786C211yesmaximal, characteristic, complemented
15C43×#1854C211yesmaximal, characteristic, complemented
16C22×#10621872C2211yescharacteristic
17C2×#13063305C2211yescharacteristic
18C2×#13063304C2211yescharacteristic
19C2×C4×C22⋊D43C2211yescharacteristic
20C2×#13063283C2211yescharacteristic
21C2×C4⋊C4×D43C2211yescharacteristic
22#14583240—24no—
23#14582940—24no—
24C2×#13063258C2211yescharacteristic
25C2×#13063257C2211yescharacteristic
26C2×#13063256C2211yescharacteristic
27C2×#13063256C2211yescharacteristic
28C2×#13063250C2211yescharacteristic
29C2×#13063250C2211yescharacteristic
30C2×#13063248C2211yescharacteristic
31C2×#13063248C2211yescharacteristic
32C2×C22⋊(C43⋊C8)C2211yescharacteristic
33C2×(C2×D43)⋊C8C2211yescharacteristic
34#14582833—21no—
35#14582822—21no—
36#14582804C2211yescharacteristic
37#14582799—21no—
38#14582795C2211yescharacteristic
39#14582790—21no—
40#14582787—21no—
41#14582775—21no—
42#14582768C2211yescharacteristic
43#14582759C2211yescharacteristic
44#14582758—21no—
45#14582756C2211yescharacteristic
46C4⋊C4×Dic43C2211yescharacteristic
47C4×Dic43⋊C4—21no—
48#14582662C2211yescharacteristic
49#14582661C2211yescharacteristic
50#14582661C2211yescharacteristic
51#14582661C2211yescharacteristic
52#14582653C2211yescharacteristic
53#14582653C2211yescharacteristic
54C2×#117×C43C2211yescharacteristic
55C2×#107×C43C2211yescharacteristic
56C2×#106×C43C2211yescharacteristic
57C2×C22⋊C8×C43C2211yescharacteristic
58C43×#380C2211yescharacteristic
59C43×#340C2211yescharacteristic
60C43×#340C2211yescharacteristic
61C22×Dic86C2311yescharacteristic
62C22×C22⋊D43D411yescharacteristic
63C22×C4×D43—21no—
64C22×C4×D43C2311yescharacteristic
65C2×#10621872—22no—
66C2×#10621872—22no—
67C2×#10621872—22no—
68C22×C43⋊C8C2311yescharacteristic
69C22×C43⋊C8C2311yescharacteristic
70#13063305—22no—
71#13063305—22no—
72#13063304—22no—
73#13063304—41no—
74C4×C22⋊D43—22no—
75C4×C22⋊D43—22no—
76#13063283—22no—
77#13063283—41no—
78C2×C4⋊Dic43—21no—
79C2×C4⋊Dic43C2311yescharacteristic
80C2×C4⋊Dic43C2311yescharacteristic
81C2×C4⋊Dic43C2311yescharacteristic
82C2×Dic43⋊C4—21no—
83C2×Dic43⋊C4—21no—
84C2×Dic43⋊C4—21no—
85C2×Dic43⋊C4—21no—
86C2×Dic43⋊C4—21no—
87C2×Dic43⋊C4D411yescharacteristic
88C4⋊C4×D43—22no—
89C4⋊C4×D43—41no—
90C2×C22⋊Dic43—21no—
91C2×C22⋊Dic43—21no—
92C2×C22⋊Dic43D411yescharacteristic
93C2×(C2×D43)⋊C4—21no—
94C2×(C2×D43)⋊C4—21no—
95C2×(C2×D43)⋊C4—21no—
96C2×(C2×D43)⋊C4D411yescharacteristic
97C2×C4×Dic43—21no—
98C2×C4×Dic43—21no—
99C2×C4×Dic43C2311yescharacteristic
100#13063258D414yescomplemented
101#13063257—22no—
102#13063256—22no—
103#13063256—22no—
104#13063250—22no—
105#13063250—41no—
106#13063248—22no—
107#13063248—22no—
108C22⋊(C43⋊C8)—22no—
109(C2×D43)⋊C8—22no—
110C43⋊#95—21no—
111C43⋊#95—21no—
112C43⋊#95—21no—
113C43⋊#95—21no—
114C43⋊#95—21no—
115C43⋊#95—21no—
116C23×C4×C43C2311yescharacteristic
117C22×C8×C43C2311yescharacteristic
118C22×C8×C43C2311yescharacteristic
119#117×C43—22no—
120#117×C43—41no—
121C2×C4⋊C4×C43C2311yescharacteristic
122C2×C4⋊C4×C43C2311yescharacteristic
123C2×C4⋊C4×C43C2311yescharacteristic
124C2×C4⋊C4×C43C2311yescharacteristic
125C2×C22⋊C4×C43—21no—
126C2×C22⋊C4×C43—21no—
127C2×C42×C43—21no—
128#107×C43—22no—
129#106×C43—41no—
130C22⋊C8×C43—22no—
131#95×C43—21no—
132C24×C43C2×D411yescharacteristic
133C23×D43C2×D411yescharacteristic
134C22×C4×C43—21no—
135C22×C4×C43—21no—
136C22×C4×C43—21no—
137C22×C4×C43—21no—
138C22×C4×C43—21no—
139C22×C4×C43—21no—
140C22×C4×C43—21no—
141C22×C4×C43—21no—
142C22×C4×C43—21no—
143C22×C4×C43—21no—
144C22×C4×C43C2×D411yescharacteristic
145C22×C4×C43C2411yescharacteristic
146C2×C22⋊D43—22no—
147C2×C22⋊D43—22no—
148C2×C22⋊D43—41no—
149C22×Dic43—21no—
150C22×Dic43—21no—
151C22×Dic43—21no—
152C22×Dic43—21no—
153C22×Dic43C2×D411yescharacteristic
154C22×Dic43C2×D411yescharacteristic
155C22×Dic43C2×D411yescharacteristic
156C2×C4×D43—21no—
157C2×C4×D43—21no—
158C2×C4×D43—21no—
159C2×C4×D43—21no—
160C2×C4×D43—42no—
161C2×C4×D43—42no—
162C2×C4×D43—81no—
163C2×C4×D43#4112yes—
164C2×C4×D43#4112yes—
165C2×Dic86—21no—
166C2×Dic86—21no—
167C2×Dic86—21no—
168C2×Dic86—21no—
169C2×Dic86—21no—
170C2×Dic86—21no—
171C2×C8×C43—21no—
172C2×C8×C43—21no—
173C2×C8×C43—21no—
174C2×C8×C43—21no—
175C2×C8×C43—21no—
176C2×C8×C43#4112yes—
177C4⋊C4×C43—21no—
178C4⋊C4×C43—21no—
179C4⋊C4×C43—21no—
180C4⋊C4×C43—21no—
181C4⋊C4×C43—21no—
182C4⋊C4×C43—21no—
183C4⋊C4×C43—21no—
184C4⋊C4×C43C2×D412yes—
185C22⋊C4×C43—42no—
186C22⋊C4×C43—81no—
187C42×C43—42no—
188C22⋊Dic43—22no—
189C22⋊Dic43—42no—
190C22⋊Dic43—81no—
191(C2×D43)⋊C4—22no—
192(C2×D43)⋊C4—42no—
193(C2×D43)⋊C4—42no—
194(C2×D43)⋊C4—81no—
195C4⋊Dic43—21no—
196C4⋊Dic43—21no—
197C4⋊Dic43—21no—
198C4⋊Dic43—21no—
199C4⋊Dic43—21no—
200C4⋊Dic43—42no—
201C4⋊Dic43C2×D412yes—
202Dic43⋊C4—22no—
203Dic43⋊C4—42no—
204Dic43⋊C4—42no—
205Dic43⋊C4—42no—
206Dic43⋊C4—81no—
207Dic43⋊C4—81no—
208C4×Dic43—21no—
209C4×Dic43—21no—
210C4×Dic43—42no—
211C4×Dic43—42no—
212C2×C43⋊C8—21no—
213C2×C43⋊C8—21no—
214C2×C43⋊C8—21no—
215C2×C43⋊C8—21no—
216C2×C43⋊C8—21no—
217C2×C43⋊C8C2×D412yes—
218#10621872—28no—
219C23×C43—21no—
220C23×C43—21no—
221C23×C43—21no—
222C23×C43—21no—
223C23×C43—21no—
224C23×C43—21no—
225C23×C43—21no—
226C23×C43C22×D411yescharacteristic
227C22×D43—21no—
228C22×D43—21no—
229C22×D43—21no—
230C22×D43—21no—
231C22×D43—21no—
232C22×D43—21no—
233C22×D43#12312yes—
234C2×C4×C43—21no—
235C2×C4×C43—22no—
236C2×C4×C43—21no—
237C2×C4×C43—21no—
238C2×C4×C43—41no—
239C2×C4×C43—41no—
240C2×C4×C43—41no—
241C2×C4×C43—41no—
242C2×C4×C43—41no—
243C2×C4×C43—41no—
244C2×C4×C43—41no—
245C2×C4×C43—41no—
246C2×C4×C43—41no—
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×C43C22×D411yescharacteristic
254C2×C4×C43C2×#4111yescharacteristic
255C2×C4×C43C2×#4111yescharacteristic
256C2×C4×C43C2×#4111yescharacteristic
257C2×C4×C43H(5,𝔽2)11yescharacteristic
258C2×C4×C432−1+411yescharacteristic
259C22⋊D43—44no—
260C2×Dic43—21no—
261C2×Dic43—21no—
262C2×Dic43—21no—
263C2×Dic43—21no—
264C2×Dic43—21no—
265C2×Dic43—22no—
266C2×Dic43—21no—
267C2×Dic43—21no—
268C2×Dic43—41no—
269C2×Dic43—41no—
270C2×Dic43—41no—
271C2×Dic43—41no—
272C2×Dic43—41no—
273C2×Dic43—41no—
274C2×Dic43—41no—
275C2×Dic43—41no—
276C2×Dic43—41no—
277C2×Dic43—41no—
278C2×Dic43—41no—
279C2×Dic43#12312yes—
280C2×Dic43#13712yes—
281C4×D43—22no—
282C4×D43—22no—
283C4×D43—22no—
284C4×D43—22no—
285C4×D43—84no—
286Dic86—41no—
287Dic86—41no—
288Dic86—41no—
289Dic86—41no—
290C8×C43—22no—
291C8×C43—41no—
292C43⋊C8—22no—
293C43⋊C8—41no—
294#26155—431nomaximal, Sylow
295C22×C43—41no—
296C22×C43—41no—
297C22×C43—41no—
298C22×C43—41no—
299C22×C43—41no—
300C22×C43—41no—
301C22×C43—41no—
302C22×C43C2×#12311yescharacteristic
303C22×C43D4211yescharacteristic
304C22×C43#54711yescharacteristic
305C22×C43C2×#13711yescharacteristic
306C22×C43#57511yescharacteristic
307C22×C43#57711yescharacteristic
308C22×C43#57711yescharacteristic
309C2×D43—22no—
310C2×D43—22no—
311C2×D43—22no—
312C2×D43—41no—
313C2×D43—41no—
314C2×D43—41no—
315C2×D43—41no—
316C4×C43—41no—
317C4×C43—42no—
318C4×C43—81no—
319C4×C43—81no—
320C4×C43—81no—
321C4×C43#52511yescharacteristic
322C4×C43#53911yescharacteristic
323C4×C43#54811yescharacteristic
324C4×C43#55311yescharacteristic
325Dic43—22no—
326Dic43—22no—
327Dic43—41no—
328Dic43—42no—
329Dic43—81no—
330Dic43—81no—
331Dic43—81no—
332C2×#485—431no—
333C2×#428—431no—
334#2417—431no—
335#2354—431no—
336#2096—431no—
337#2084—431no—
338#2077—431no—
339#2035—431no—
340#1928—431no—
341#1921—431no—
342#1894—431no—
343#1873—431no—
344#1870—431no—
345#1854D4311yescharacteristic, complemented
346#1853—431no—
347C2×C43—81no—
348C2×C43#244311yescharacteristic
349C2×C43#309511yescharacteristic
350C2×C43#309911yescharacteristic
351C2×C43#326711yescharacteristic
352C2×C43#327811yescharacteristic
353C2×C43#334811yescharacteristic
354C2×C43#335611yescharacteristic
355D43—42no—
356C22×M16—431no—
357C2×C22⋊Q8—431no—
358C2×C22⋊Q8—431no—
359C2×C4×D4—431no—
360C2×#117—431no—
361C2×#117C2×D4311yescharacteristic
362C22×C4⋊C4—431no—
363#485—864no—
364#428—864no—
365C2×#107—431no—
366C2×#107C2×D4311yescharacteristic
367C2×#106—431no—
368C2×#106—431no—
369C2×#106—431no—
370C2×#106C2×D4311yescharacteristic
371C2×#103—431no—
372C2×#103—431no—
373C2×#103—431no—
374C2×#103—431no—
375C2×C22⋊C8—431no—
376C2×C22⋊C8—431no—
377C2×C22⋊C8C2×D4311yescharacteristic
378#398—861no—
379#396—861no—
380#391—431no—
381#390—861no—
382#389—431no—
383#388—861no—
384#388—861no—
385#386—861no—
386#384—431no—
387#380—431no—
388#380—431no—
389#380—861no—
390#380C2×D4311yescharacteristic
391C4×C4⋊C4—431no—
392C4×C4⋊C4—861no—
393#340—431no—
394#340—431no—
395#340—431no—
396#340—431no—
397#340C2×D4311yescharacteristic
398#340C2×D4311yescharacteristic
399#336—431no—
400#336—431no—
401C43#2615511yescharacteristic, complemented, Sylow
402C22×Q8—431no—
403C22×D4—431no—
404C23×C4—431no—
405C23×C4—861no—
406C23×C4C22×D4311yescharacteristic
407C2×M16—862no—
408C2×M16—862no—
409C2×M16—862no—
410C22×C8—431no—
411C22×C8—431no—
412C22×C8C22×D4311yescharacteristic
413C22×C8C22×D4311yescharacteristic
414C22⋊Q8—862no—
415C22⋊Q8—862no—
416C22⋊Q8—862no—
417C22⋊Q8—1721no—
418C4×D4—862no—
419C4×D4—862no—
420#117—22no—
421#117—41no—
422#117—862no—
423#117—1721no—
424C2×C4⋊C4—431no—
425C2×C4⋊C4—431no—
426C2×C4⋊C4—431no—
427C2×C4⋊C4—431no—
428C2×C4⋊C4—862no—
429C2×C4⋊C4—861no—
430C2×C4⋊C4—861no—
431C2×C4⋊C4—861no—
432C2×C4⋊C4—861no—
433C2×C4⋊C4—861no—
434C2×C4⋊C4—861no—
435C2×C4⋊C4—1721no—
436C2×C4⋊C4C22×D4311yescharacteristic
437C2×C4⋊C4C22×D4311yescharacteristic
438C2×C4⋊C4C22×D4311yescharacteristic
439C2×C4⋊C4C22×D4311yescharacteristic
440C2×C22⋊C4—21no—
441C2×C22⋊C4—21no—
442C2×C22⋊C4—431no—
443C2×C22⋊C4—431no—
444C2×C22⋊C4—861no—
445C2×C22⋊C4—861no—
446C2×C22⋊C4—861no—
447C2×C22⋊C4—861no—
448C2×C22⋊C4—861no—
449C2×C42—21no—
450C2×C42—431no—
451C2×C42—861no—
452C2×C42—861no—
453#107—22no—
454#107—434no—
455#106—41no—
456#106—862no—
457#106—862no—
458#106—862no—
459#103—862no—
460#103—862no—
461#103—862no—
462#103—1721no—
463C22⋊C8—22no—
464C22⋊C8—862no—
465C22⋊C8—862no—
466#95—21no—
467#95—861no—
468#95—861no—
469#95—861no—
470#95—861no—
471#95—861no—
472#95—861no—
473C24—431no—
474C24D4×D4311yescharacteristic
475C2×Q8—861no—
476C2×Q8—861no—
477C2×Q8—861no—
478C2×Q8—861no—
479C2×Q8—861no—
480C2×Q8—861no—
481C2×D4—862no—
482C2×D4—862no—
483C2×D4—1721no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—431no—
495C22×C4—431no—
496C22×C4—432no—
497C22×C4—432no—
498C22×C4—431no—
499C22×C4—861no—
500C22×C4—861no—
501C22×C4—861no—
502C22×C4—861no—
503C22×C4—861no—
504C22×C4—861no—
505C22×C4—861no—
506C22×C4—861no—
507C22×C4—3441no—
508C22×C4—1722no—
509C22×C4—1722no—
510C22×C4D4×D4311yescharacteristic
511C22×C4C23×D4311yescharacteristic
512M16—868no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—432no—
519C2×C8—861no—
520C2×C8—861no—
521C2×C8—861no—
522C2×C8—861no—
523C2×C8—861no—
524C2×C8#1062189812yes—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—432no—
533C4⋊C4—862no—
534C4⋊C4—861no—
535C4⋊C4—861no—
536C4⋊C4—861no—
537C4⋊C4—861no—
538C4⋊C4—861no—
539C4⋊C4—3441no—
540C4⋊C4—3441no—
541C4⋊C4—1722no—
542C4⋊C4—1722no—
543C4⋊C4—1722no—
544C4⋊C4—1722no—
545C4⋊C4D4×D4312yes—
546C22⋊C4—42no—
547C22⋊C4—81no—
548C22⋊C4—862no—
549C22⋊C4—862no—
550C22⋊C4—3441no—
551C22⋊C4—3441no—
552C22⋊C4—1722no—
553C22⋊C4—1722no—
554C22⋊C4—1722no—
555C42—42no—
556C42—861no—
557C42—861no—
558C42—1722no—
559C42—1722no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—432no—
568C23—861no—
569C23—861no—
570C23—861no—
571C23—861no—
572C23—861no—
573C23—861no—
574C23C2×D4×D4311yescharacteristic
575Q8—1721no—
576Q8—1721no—
577Q8—1721no—
578Q8—1721no—
579D4—1724no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—432no—
600C2×C4—432no—
601C2×C4—861no—
602C2×C4—861no—
603C2×C4—861no—
604C2×C4—861no—
605C2×C4—862no—
606C2×C4—862no—
607C2×C4—862no—
608C2×C4—862no—
609C2×C4—861no—
610C2×C4—861no—
611C2×C4—861no—
612C2×C4—862no—
613C2×C4—3444no—
614C2×C4—1721no—
615C2×C4—1721no—
616C2×C4—1721no—
617C2×C4—1721no—
618C2×C4—1721no—
619C2×C4—1721no—
620C2×C4—1721no—
621C2×C4—1721no—
622C2×C4—1721no—
623C2×C4—1721no—
624C2×C4—1721no—
625C2×C4C2×D4×D4311yescharacteristic
626C2×C4#41×D4311yescharacteristic
627C2×C4#41×D4311yescharacteristic
628C2×C4C2×#1062189811yescharacteristic
629C2×C4#1306337011yescharacteristic
630C2×C4#1306337311yescharacteristic
631C8—22no—
632C8—41no—
633C8—862no—
634C8—1721no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—862no—
643C22—862no—
644C22—862no—
645C22—1721no—
646C22—1721no—
647C22—1721no—
648C22—1721no—
649C22#123×D4311yescharacteristic
650C22#1458353111yescharacteristic
651C22#1458355011yescharacteristic
652C22#137×D4311yescharacteristic
653C22#1458369611yescharacteristic
654C22#1458370611yescharacteristic
655C22#1458370911yescharacteristic
656C4—41no—
657C4—42no—
658C4—81no—
659C4—81no—
660C4—81no—
661C4—862no—
662C4—862no—
663C4—3441no—
664C4—3441no—
665C4—3441no—
666C4—1722no—
667C4—1721no—
668C4#1458340811yescharacteristic
669C4#1458349511yescharacteristic
670C4#1458355911yescharacteristic
671C4#1458359911yescharacteristic
672C2—81no—
673C2—1722no—
674C2#1913832911yescharacteristic
675C2#1914454211yescharacteristic
676C2#1914458711yescharacteristic
677C2#1914611311yescharacteristic
678C2#1914625411yescharacteristic
679C2#1914710611yescharacteristic
680C2#1914720611yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#1854C2
socleC23×C43C22×D4
prime radicalC22×C22⋊D43D4
hypercenter#1854D43
hypocenterC43#26155
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
1. C2×#14583240C2semidirect
2. C2×#14582940C2semidirect
5. #19136135C2semidirect
6. #19136075C2semidirect
7. #19136031C2semidirect
8. #19135792C2semidirect
9. #19135235C2semidirect
10. #19135185C2semidirect
11. #19135027C2semidirect
12. #19134906C2semidirect
14. #19134786C2semidirect
15. C43×#1854C2semidirect
100. #13063258D4semidirect
345. #1854D43semidirect
401. C43#26155semidirect

Maximal subgroups

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

subgroupindex
C2×#145832402
C2×#145829402
#191380482
#191375452
#191361352
#191360752
#191360312
#191357922
#191352352
#191351852
#191350272
#191349062
#191348892
#191347862
C43×#18542
#2615543