Finite Group Databasebeta

← #25107902

Subgroups — #25107902

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×#14583054C211yesmaximal, characteristic
2#19137179C211yesmaximal, characteristic
3#19135465C211yesmaximal, characteristic, complemented
4#19135455C211yesmaximal, characteristic, complemented
5#19135284C211yesmaximal, characteristic, complemented
6#19135270C211yesmaximal, characteristic, complemented
7#19135236C211yesmaximal, characteristic, complemented
8#19135229C211yesmaximal, characteristic, complemented
9#19134936C211yesmaximal, characteristic, complemented
10#19134936C211yesmaximal, characteristic, complemented
11#19134908C211yesmaximal, characteristic
12#19134792C211yesmaximal, characteristic
13#19134529C211yesmaximal, characteristic
14#19134361C211yesmaximal, characteristic
15C43×#2323C211yesmaximal, characteristic
16C23×Dic86C2211yescharacteristic
17C22×#10621885C2211yescharacteristic
18C22×#10621885C2211yescharacteristic
19C2×#13063284C2211yescharacteristic
20#14583054C418yescomplemented
21C2×#13063258C2211yescharacteristic
22C2×#13063256C2211yescharacteristic
23C2×#13063249C2211yescharacteristic
24C2×#13063249C2211yescharacteristic
25C2×#13063248C2211yescharacteristic
26C2×#13063248C2211yescharacteristic
27C2×#13063248C2211yescharacteristic
28C2×#13063248C2211yescharacteristic
29C2×#13063248C2211yescharacteristic
30C2×#13063248C2211yescharacteristic
31C2×C22⋊(C43⋊C8)C2211yescharacteristic
32C2×C22⋊(C43⋊C8)C2211yescharacteristic
33C2×#13063236C2211yescharacteristic
34C2×C4×C43⋊C8C2211yescharacteristic
35#14582805C2211yescharacteristic
36#14582805C2211yescharacteristic
37#14582805C2211yescharacteristic
38#14582805C2211yescharacteristic
39#14582804—21no—
40#14582804—21no—
41#14582760—21no—
42#14582760—21no—
43#14582661C2211yescharacteristic
44#14582661C2211yescharacteristic
45#14582661C2211yescharacteristic
46#14582661C2211yescharacteristic
47#14582654C2211yescharacteristic
48#14582654C2211yescharacteristic
49C2×C22⋊Q8×C43C2211yescharacteristic
50C2×C4×Q8×C43C2211yescharacteristic
51C2×#117×C43C2211yescharacteristic
52C43×#391C2211yescharacteristic
53C43×#389—21no—
54C43×#389C2211yescharacteristic
55C43×#388—21no—
56C43×#388—21no—
57C43×#387—21no—
58C43×#380C2211yescharacteristic
59C4×C4⋊C4×C43C2211yescharacteristic
60C22×Dic86—21no—
61C22×Dic86—22no—
62C22×Dic86—41no—
63C22×Dic86—41no—
64C22×Dic86—41no—
65C22×Dic86—41no—
66C22×Dic86C2×C414yes—
67C22×Dic86C2311yescharacteristic
68C22×Dic86C2311yescharacteristic
69C23×Dic43—21no—
70C2×#10621885—41no—
71C2×#10621885—41no—
72C2×#10621885—41no—
73C2×#10621885—41no—
74C2×#10621885C2×C414yes—
75C2×#10621885C2×C414yes—
76C22×C43⋊C8C2311yescharacteristic
77C22×C43⋊C8C2311yescharacteristic
78C22×C43⋊C8C2311yescharacteristic
79C22×C43⋊C8C2311yescharacteristic
80#13063284—41no—
81#13063284—41no—
82C2×C4⋊Dic43C2311yescharacteristic
83C2×C4⋊Dic43C2311yescharacteristic
84C2×C22⋊Dic43—21no—
85C2×C4×Dic43—21no—
86#13063258—41no—
87#13063256—41no—
88#13063249—41no—
89#13063249—41no—
90#13063248—41no—
91#13063248—41no—
92#13063248—41no—
93#13063248—41no—
94#13063248C2×C414yes—
95#13063248C2×C414yes—
96C22⋊(C43⋊C8)—22no—
97C22⋊(C43⋊C8)C2×C414yes—
98#13063236—41no—
99C4×C43⋊C8—41no—
100C43⋊#95—21no—
101C43⋊#95—21no—
102C43⋊#95—21no—
103C43⋊#95—21no—
104C43⋊#95—21no—
105C43⋊#95—21no—
106C43⋊#95—21no—
107C43⋊#95—21no—
108C22×Q8×C43C2311yescharacteristic
109C23×C4×C43C2311yescharacteristic
110C22⋊Q8×C43—22no—
111C22⋊Q8×C43C2×C414yes—
112C4×Q8×C43—41no—
113C4×Q8×C43—41no—
114#117×C43—41no—
115#117×C43—41no—
116C2×C4⋊C4×C43—21no—
117C2×C4⋊C4×C43—21no—
118C2×C4⋊C4×C43—21no—
119C2×C4⋊C4×C43D411yescharacteristic
120C2×C4⋊C4×C43D411yescharacteristic
121C2×C4⋊C4×C43C2311yescharacteristic
122C2×C4⋊C4×C43C2311yescharacteristic
123C2×C4⋊C4×C43C2311yescharacteristic
124C2×C22⋊C4×C43—21no—
125C2×C22⋊C4×C43—21no—
126C2×C22⋊C4×C43D411yescharacteristic
127C2×C22⋊C4×C43D411yescharacteristic
128C2×C42×C43—21no—
129C2×C42×C43—21no—
130C2×C42×C43C2311yescharacteristic
131C2×C42×C43C2311yescharacteristic
132#95×C43—21no—
133#95×C43—21no—
134#95×C43—21no—
135#95×C43—21no—
136#95×C43—21no—
137C24×C43C2×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×C43—21no—
149C22×C4×C43—21no—
150C22×C4×C43—21no—
151C22×C4×C43—21no—
152C22×C4×C43C22×C412yes—
153C22×C4×C43C2×D411yescharacteristic
154C22×C4×C43C2×D411yescharacteristic
155C22×C4×C43C2×D411yescharacteristic
156C22×C4×C43C2×D411yescharacteristic
157C22×C4×C43C2×D411yescharacteristic
158C22×C4×C43#4112yes—
159C22×C4×C43C2411yescharacteristic
160C22×Dic43—24no—
161C22×Dic43—21no—
162C22×Dic43—21no—
163C22×Dic43—21no—
164C22×Dic43—21no—
165C22×Dic43—42no—
166C22×Dic43—42no—
167C2×Dic86—21no—
168C2×Dic86—21no—
169C2×Dic86—21no—
170C2×Dic86—21no—
171C2×Dic86—21no—
172C2×Dic86—21no—
173C2×Dic86—21no—
174C2×Dic86—21no—
175C2×Dic86—24no—
176C2×Dic86—21no—
177C2×Dic86—21no—
178C2×Dic86—44no—
179C2×Dic86—44no—
180C2×Dic86—42no—
181C2×Dic86—42no—
182C2×Dic86—44no—
183C2×Dic86—44no—
184C2×Dic86C22×C412yes—
185C2×Dic86C22×C412yes—
186C4⋊C4×C43—24no—
187C4⋊C4×C43—21no—
188C4⋊C4×C43—21no—
189C4⋊C4×C43—21no—
190C4⋊C4×C43—21no—
191C4⋊C4×C43—21no—
192C4⋊C4×C43—41no—
193C4⋊C4×C43—41no—
194C4⋊C4×C43—42no—
195C4⋊C4×C43—81no—
196C4⋊C4×C43C22×C412yes—
197C22⋊C4×C43—24no—
198C22⋊C4×C43—41no—
199C22⋊C4×C43—41no—
200C22⋊C4×C43—81no—
201C42×C43—21no—
202C42×C43—21no—
203C42×C43—21no—
204C42×C43—21no—
205C42×C43—42no—
206C42×C43—81no—
207C22⋊Dic43—42no—
208#10621885—44no—
209#10621885—44no—
210C4⋊Dic43—21no—
211C4⋊Dic43—21no—
212C4⋊Dic43—21no—
213C4⋊Dic43—21no—
214C4×Dic43—42no—
215C2×C43⋊C8—21no—
216C2×C43⋊C8—21no—
217C2×C43⋊C8—21no—
218C2×C43⋊C8—21no—
219C2×C43⋊C8—21no—
220C2×C43⋊C8—21no—
221C2×C43⋊C8—21no—
222C2×C43⋊C8—21no—
223C2×C43⋊C8—21no—
224C2×C43⋊C8—21no—
225C2×C43⋊C8C22×C412yes—
226C2×C43⋊C8C22×C412yes—
227C23×C43—21no—
228C23×C43—21no—
229C23×C43—21no—
230C23×C43—21no—
231C23×C43—21no—
232C23×C43—21no—
233C23×C43C4×D412yes—
234C23×C43C22×D411yescharacteristic
235Q8×C43—22no—
236Q8×C43—22no—
237Q8×C43—41no—
238Q8×C43—41no—
239C2×C4×C43—21no—
240C2×C4×C43—21no—
241C2×C4×C43—22no—
242C2×C4×C43—22no—
243C2×C4×C43—22no—
244C2×C4×C43—22no—
245C2×C4×C43—22no—
246C2×C4×C43—22no—
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—21no—
258C2×C4×C43—21no—
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—41no—
266C2×C4×C43—41no—
267C2×C4×C43—41no—
268C2×C4×C43—41no—
269C2×C4×C43C4×D412yes—
270C2×C4×C43C23×C411yescharacteristic
271C2×C4×C43C2×#4111yescharacteristic
272C2×C4×C43H(5,𝔽2)11yescharacteristic
273C2×C4×C43H(5,𝔽2)11yescharacteristic
274C2×C4×C43H(5,𝔽2)11yescharacteristic
275C2×C4×C43H(5,𝔽2)11yescharacteristic
276C2×Dic43—22no—
277C2×Dic43—22no—
278C2×Dic43—48no—
279C2×Dic43—41no—
280C2×Dic43—41no—
281C2×Dic43—41no—
282C2×Dic43—41no—
283C2×Dic43—41no—
284C2×Dic43—41no—
285C2×Dic43—41no—
286C2×Dic43—41no—
287C2×Dic43—41no—
288C2×Dic43—41no—
289Dic86—22no—
290Dic86—22no—
291Dic86—22no—
292Dic86—22no—
293Dic86—48no—
294Dic86—41no—
295Dic86—41no—
296Dic86—41no—
297Dic86—41no—
298C43⋊C8—22no—
299C43⋊C8—22no—
300C43⋊C8—41no—
301C43⋊C8—41no—
302#22054—431nomaximal, Sylow
303C22×C43—22no—
304C22×C43—22no—
305C22×C43—22no—
306C22×C43—41no—
307C22×C43—41no—
308C22×C43—41no—
309C22×C43—41no—
310C22×C43C2×C4×D411yescharacteristic
311C22×C43#53411yescharacteristic
312C22×C43#53511yescharacteristic
313C22×C43#57711yescharacteristic
314C22×C43#57811yescharacteristic
315C22×C43#57811yescharacteristic
316C22×C43#57811yescharacteristic
317C4×C43—22no—
318C4×C43—42no—
319C4×C43—41no—
320C4×C43—41no—
321C4×C43—41no—
322C4×C43—41no—
323C4×C43—42no—
324C4×C43—81no—
325C4×C43—81no—
326C4×C43#51811yescharacteristic
327C4×C43#51811yescharacteristic
328C4×C43#53811yescharacteristic
329C4×C43#53811yescharacteristic
330Dic43—42no—
331Dic43—42no—
332Dic43—81no—
333Dic43—81no—
334C2×#451—431no—
335#2323D4311yescharacteristic
336#2310—431no—
337#1971—431no—
338#1969—431no—
339#1935—431no—
340#1933—431no—
341#1928—431no—
342#1927—431no—
343#1878—431no—
344#1878—431no—
345#1873—431no—
346#1854—431no—
347#1787—431no—
348#1750—431no—
349C2×C43—42no—
350C2×C43#232111yescharacteristic
351C2×C43#294711yescharacteristic
352C2×C43#295011yescharacteristic
353C2×C43#320111yescharacteristic
354C2×C43#320211yescharacteristic
355C2×C43#322311yescharacteristic
356C2×C43#322511yescharacteristic
357C23×Q8—431no—
358C22×Q16—431no—
359C22×Q16—431no—
360C2×C22⋊Q8C2×D4311yescharacteristic
361C2×C4×Q8C2×D4311yescharacteristic
362C2×#117—431no—
363C2×#117C2×D4311yescharacteristic
364#451—438no—
365C2×#107—431no—
366C2×#106—431no—
367C2×#103—431no—
368C2×#103—431no—
369C2×#103—431no—
370C2×#103—431no—
371C2×#103—431no—
372C2×#103—431no—
373C2×#103—431no—
374C2×#103—431no—
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—21no—
387#389C2×D4311yescharacteristic
388#388—21no—
389#388—21no—
390#387—21no—
391#381—861no—
392#381—861no—
393#380C2×D4311yescharacteristic
394C4×C4⋊C4C2×D4311yescharacteristic
395#340—431no—
396#340—431no—
397#340—431no—
398#340—431no—
399#336—431no—
400#336—431no—
401C43#2205411yescharacteristic, complemented, Sylow
402C22×Q8—434no—
403C22×Q8—431no—
404C22×Q8—431no—
405C22×Q8—862no—
406C22×Q8—861no—
407C22×Q8—1721no—
408C22×Q8—1721no—
409C22×Q8—1721no—
410C22×Q8—1721no—
411C22×Q8C22×D4311yescharacteristic
412C23×C4—861no—
413C23×C4C22×D4311yescharacteristic
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×C8—431no—
423C22×C8—431no—
424C22⋊Q8—22no—
425C22⋊Q8C4×D4314yes—
426C4×Q8—41no—
427C4×Q8—41no—
428#117—41no—
429#117—41no—
430#117—1721no—
431#117—1721no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—21no—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—431no—
436C2×C4⋊C4—431no—
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—861no—
445C2×C22⋊C4C22⋊D4311yescharacteristic
446C2×C22⋊C4C22⋊D4311yescharacteristic
447C2×C42—21no—
448C2×C42—21no—
449C2×C42—861no—
450C2×C42C22×D4311yescharacteristic
451C2×C42C22×D4311yescharacteristic
452#107—1721no—
453#106—1721no—
454#103—434no—
455#103—434no—
456#103—1721no—
457#103—1721no—
458#103—1721no—
459#103—1721no—
460#103—1721no—
461#103—1721no—
462C22⋊C8—434no—
463C22⋊C8—862no—
464#97—1721no—
465C4×C8—1721no—
466#95—21no—
467#95—21no—
468#95—21no—
469#95—21no—
470#95—21no—
471#95—861no—
472#95—861no—
473#95—861no—
474#95—861no—
475#95—861no—
476#95—861no—
477#95—861no—
478#95—861no—
479C24C2×C22⋊D4311yescharacteristic
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—1724no—
499C2×Q8—1724no—
500C2×Q8—1722no—
501C2×Q8—1722no—
502C2×Q8—1724no—
503C2×Q8—1724no—
504C2×Q8C2×C4×D4312yes—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—864no—
514C22×C4—861no—
515C22×C4—861no—
516C22×C4—861no—
517C22×C4—861no—
518C22×C4—1722no—
519C22×C4—1722no—
520C22×C4C2×C4×D4312yes—
521C22×C4#1062189812yes—
522C22×C4C2×C22⋊D4311yescharacteristic
523C22×C4C2×C22⋊D4311yescharacteristic
524C22×C4C2×C22⋊D4311yescharacteristic
525C22×C4C2×C22⋊D4311yescharacteristic
526C22×C4C2×C22⋊D4311yescharacteristic
527C22×C4C23×D4311yescharacteristic
528Q16—1724no—
529Q16—1724no—
530C2×C8—432no—
531C2×C8—432no—
532C2×C8—861no—
533C2×C8—861no—
534C2×C8—861no—
535C2×C8—861no—
536C2×C8—861no—
537C2×C8—861no—
538C2×C8—861no—
539C2×C8—861no—
540C2×C8—861no—
541C2×C8—861no—
542C4⋊C4—24no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—41no—
549C4⋊C4—41no—
550C4⋊C4—42no—
551C4⋊C4—81no—
552C4⋊C4—861no—
553C4⋊C4—861no—
554C4⋊C4—861no—
555C4⋊C4—861no—
556C4⋊C4C2×C4×D4312yes—
557C22⋊C4—24no—
558C22⋊C4—41no—
559C22⋊C4—41no—
560C22⋊C4—81no—
561C22⋊C4—1722no—
562C42—21no—
563C42—21no—
564C42—21no—
565C42—21no—
566C42—42no—
567C42—81no—
568C42—1722no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23C4×C22⋊D4312yes—
576C23C22×C22⋊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—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
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—21no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—862no—
621C2×C4—862no—
622C2×C4—1721no—
623C2×C4—1721no—
624C2×C4—1721no—
625C2×C4—1721no—
626C2×C4—1721no—
627C2×C4—1721no—
628C2×C4—1728no—
629C2×C4—1721no—
630C2×C4—1721no—
631C2×C4—1721no—
632C2×C4—1721no—
633C2×C4C4×C22⋊D4312yes—
634C2×C4C22×C4×D4311yescharacteristic
635C2×C4C2×#1062189811yescharacteristic
636C2×C4#1306337011yescharacteristic
637C2×C4#1306337011yescharacteristic
638C2×C4#1306337111yescharacteristic
639C2×C4#1306337111yescharacteristic
640C8—862no—
641C8—862no—
642C8—1721no—
643C8—1721no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22C2×C4×C22⋊D4311yescharacteristic
652C22#1458345811yescharacteristic
653C22#1458346311yescharacteristic
654C22#1458371011yescharacteristic
655C22#1458371411yescharacteristic
656C22#1458371411yescharacteristic
657C22#1458371411yescharacteristic
658C4—22no—
659C4—42no—
660C4—41no—
661C4—41no—
662C4—42no—
663C4—41no—
664C4—41no—
665C4—81no—
666C4—81no—
667C4—3441no—
668C4—3441no—
669C4—1722no—
670C4—1722no—
671C4#1458337111yescharacteristic
672C4#1458337111yescharacteristic
673C4#1458348211yescharacteristic
674C4#1458348211yescharacteristic
675C2—42no—
676C2#1913724911yescharacteristic
677C2#1914343411yescharacteristic
678C2#1914346111yescharacteristic
679C2#1914547111yescharacteristic
680C2#1914548611yescharacteristic
681C2#1914567211yescharacteristic
682C2#1914569811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#2323C2
socleC23×C43C22×D4
prime radicalC24×C43C2×D4
cosocleC22×C4×C43C24
hypercenter#2323D43
hypocenterC43#22054
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2323

Complements

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

NH ≅ G/Ntype
3. #19135465C2semidirect
4. #19135455C2semidirect
5. #19135284C2semidirect
6. #19135270C2semidirect
7. #19135236C2semidirect
8. #19135229C2semidirect
9. #19134936C2semidirect
10. #19134936C2semidirect
20. #14583054C4semidirect
401. C43#22054semidirect

Maximal subgroups

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

subgroupindex
C2×#145830542
#191371792
#191354652
#191354552
#191352842
#191352702
#191352362
#191352292
#191349362
#191349082
#191347922
#191345292
#191343612
C43×#23232
#2205443