Finite Group Databasebeta

← #25146136

Subgroups — #25146136

Order 11008.

848 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583115C211yesmaximal, characteristic
2#19140083C211yesmaximal, characteristic, complemented
3#19139953C211yesmaximal, characteristic, complemented
4#19135989C211yesmaximal, characteristic, complemented
5#19135968C211yesmaximal, characteristic, complemented
6#19135913C211yesmaximal, characteristic, complemented
7#19135906C211yesmaximal, characteristic
8#19135461C211yesmaximal, characteristic
9#19135449C211yesmaximal, characteristic, complemented
10#19135406C211yesmaximal, characteristic, complemented
11#19135365C211yesmaximal, characteristic, complemented
12#19135278C211yesmaximal, characteristic, complemented
13#19135271C211yesmaximal, characteristic, complemented
14C4×#13063247C211yesmaximal, characteristic, complemented
15C2×C8×Q8×C43C211yesmaximal, characteristic, complemented
16C22×#10621873C2211yescharacteristic
17C22×D344C2211yescharacteristic
18C2×#13063313C2211yescharacteristic
19C2×C4×D172C2211yescharacteristic
20#14583115—24no—
21C2×#13063259C2211yescharacteristic
22C2×#13063257C2211yescharacteristic
23C2×#13063250C2211yescharacteristic
24C2×#13063250C2211yescharacteristic
25C2×#13063250C2211yescharacteristic
26C2×#13063247C2211yescharacteristic
27C2×#13063247C2211yescharacteristic
28C2×#13063247C2211yescharacteristic
29C2×#13063247C2211yescharacteristic
30C2×#13063247C2211yescharacteristic
31#14582834—21no—
32#14582825—21no—
33#14582810—21no—
34#14582810—21no—
35#14582805C2211yescharacteristic
36#14582805C2211yescharacteristic
37#14582801C2211yescharacteristic
38#14582801C2211yescharacteristic
39#14582801C2211yescharacteristic
40#14582801C2211yescharacteristic
41#14582797C2211yescharacteristic
42#14582797C2211yescharacteristic
43#14582789—21no—
44#14582789—21no—
45#14582770—21no—
46#14582770—21no—
47#14582765C2211yescharacteristic
48C4×C4⋊Dic43C2211yescharacteristic
49#14582662C2211yescharacteristic
50#14582662C2211yescharacteristic
51#14582662C2211yescharacteristic
52#14582662C2211yescharacteristic
53C2×C4×Q8×C43C2211yescharacteristic
54C8×Q8×C43—24no—
55C2×C4⋊C8×C43C2211yescharacteristic
56C2×C4⋊C8×C43C2211yescharacteristic
57C2×C4⋊C8×C43C2211yescharacteristic
58C2×C4×C8×C43C2211yescharacteristic
59C2×C4×C8×C43C2211yescharacteristic
60C2×C4×C8×C43C2211yescharacteristic
61C22×Dic86C2311yescharacteristic
62C22×D172C2311yescharacteristic
63C22×D172C2311yescharacteristic
64C22×C4×D43—21no—
65C2×#10621873—22no—
66C2×#10621873—22no—
67C2×#10621873—41no—
68C2×D344—22no—
69C2×D344—22no—
70C2×D344—41no—
71#13063313—22no—
72#13063313—22no—
73C4×D172—22no—
74C4×D172—22no—
75C2×C4⋊Dic43C2311yescharacteristic
76C2×C4⋊Dic43C2311yescharacteristic
77C2×C4⋊Dic43C2311yescharacteristic
78C2×C4⋊Dic43C2311yescharacteristic
79C2×C4⋊Dic43C2311yescharacteristic
80C2×Dic43⋊C4—21no—
81C2×Dic43⋊C4—21no—
82C2×(C2×D43)⋊C4—21no—
83C2×(C2×D43)⋊C4—21no—
84C2×(C2×D43)⋊C4—21no—
85C2×(C2×D43)⋊C4—21no—
86C2×(C2×D43)⋊C4—21no—
87C2×(C2×D43)⋊C4—21no—
88C2×(C2×D43)⋊C4—21no—
89C2×(C2×D43)⋊C4—21no—
90C2×(C2×D43)⋊C4—21no—
91C2×(C2×D43)⋊C4—21no—
92C2×(C2×D43)⋊C4—21no—
93C2×C4×Dic43—21no—
94#13063259—41no—
95#13063257—41no—
96#13063250—22no—
97#13063250—22no—
98#13063250—41no—
99#13063247—22no—
100#13063247—22no—
101#13063247—22no—
102#13063247—22no—
103#13063247—41no—
104C43⋊#95—21no—
105C43⋊#95—21no—
106C43⋊#95—21no—
107C43⋊#95—21no—
108C43⋊#95—21no—
109C43⋊#95—21no—
110C43⋊#95—21no—
111C43⋊#95—21no—
112C43⋊#95—21no—
113C22×Q8×C43D411yescharacteristic
114C22×C8×C43C2311yescharacteristic
115C22×C8×C43C2311yescharacteristic
116C22×C8×C43C2311yescharacteristic
117C22×C8×C43C2311yescharacteristic
118C4×Q8×C43—22no—
119C4×Q8×C43—22no—
120C2×C4⋊C4×C43D411yescharacteristic
121C2×C4⋊C4×C43D411yescharacteristic
122C2×C4⋊C4×C43D411yescharacteristic
123C2×C42×C43C2311yescharacteristic
124C2×C42×C43C2311yescharacteristic
125C2×C42×C43C2311yescharacteristic
126C4⋊C8×C43—22no—
127C4⋊C8×C43—22no—
128C4⋊C8×C43D414yes—
129C4×C8×C43—22no—
130C4×C8×C43—22no—
131C4×C8×C43—22no—
132C23×D43—21no—
133C23×D43—21no—
134C2×Q8×C43—22no—
135C2×Q8×C43—22no—
136C2×Q8×C43—22no—
137C22×C4×C43C2×D411yescharacteristic
138C22×C4×C43C2×D411yescharacteristic
139C22×C4×C43C2×D411yescharacteristic
140C22×C4×C43C2×D411yescharacteristic
141C22×C4×C43C2×D411yescharacteristic
142C22×C4×C43C2×D411yescharacteristic
143C22×C4×C43C2411yescharacteristic
144C22×Dic43—21no—
145C22×Dic43—21no—
146C22×Dic43—21no—
147C22×Dic43—21no—
148C22×Dic43—21no—
149C22×Dic43—21no—
150C2×D172—21no—
151C2×D172—21no—
152C2×D172—21no—
153C2×D172—21no—
154C2×D172—21no—
155C2×D172—21no—
156C2×D172—21no—
157C2×D172—21no—
158C2×D172—21no—
159C2×D172—21no—
160C2×D172—21no—
161C2×D172#4112yes—
162C2×C4×D43—42no—
163C2×C4×D43—81no—
164C2×C4×D43—81no—
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—21no—
177C2×C8×C43—21no—
178C2×C8×C43—21no—
179C2×C8×C43—21no—
180C2×C8×C43C2×D412yes—
181C2×C8×C43C2×D412yes—
182C2×C8×C43#4112yes—
183C4⋊C4×C43—22no—
184C4⋊C4×C43—22no—
185C4⋊C4×C43—22no—
186C42×C43—21no—
187C42×C43—21no—
188C42×C43—21no—
189C42×C43—21no—
190C42×C43C2×D412yes—
191C42×C43#4112yes—
192(C2×D43)⋊C4—42no—
193(C2×D43)⋊C4—42no—
194(C2×D43)⋊C4—42no—
195(C2×D43)⋊C4—42no—
196(C2×D43)⋊C4—42no—
197(C2×D43)⋊C4—42no—
198(C2×D43)⋊C4—42no—
199(C2×D43)⋊C4—81no—
200(C2×D43)⋊C4—81no—
201(C2×D43)⋊C4—81no—
202(C2×D43)⋊C4—81no—
203C4⋊Dic43—21no—
204C4⋊Dic43—21no—
205C4⋊Dic43—21no—
206C4⋊Dic43—21no—
207C4⋊Dic43—21no—
208C4⋊Dic43—21no—
209C4⋊Dic43—21no—
210C4⋊Dic43—21no—
211C4⋊Dic43—21no—
212C4⋊Dic43—21no—
213Dic43⋊C4—81no—
214Dic43⋊C4—81no—
215C4×Dic43—81no—
216D344—44no—
217#10621873—44no—
218C23×C43C22×D411yescharacteristic
219C22×D43—22no—
220C22×D43—41no—
221C22×D43—41no—
222C22×D43—41no—
223C22×D43—41no—
224C22×D43—41no—
225C22×D43—41no—
226C22×D43—41no—
227C22×D43—41no—
228C22×D43—41no—
229C22×D43—41no—
230C22×D43—41no—
231C22×D43—41no—
232C22×D43—41no—
233Q8×C43—28no—
234C2×C4×C43—21no—
235C2×C4×C43—21no—
236C2×C4×C43—21no—
237C2×C4×C43—21no—
238C2×C4×C43—21no—
239C2×C4×C43—21no—
240C2×C4×C43—21no—
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—21no—
248C2×C4×C43#12112yes—
249C2×C4×C43#12112yes—
250C2×C4×C43#13512yes—
251C2×C4×C43#13512yes—
252C2×C4×C43C22×D411yescharacteristic
253C2×C4×C43C2×#4111yescharacteristic
254C2×C4×C43C2×#4111yescharacteristic
255C2×C4×C43C2×#4111yescharacteristic
256C2×C4×C43H(5,𝔽2)11yescharacteristic
257C2×C4×C432−1+411yescharacteristic
258C2×Dic43—41no—
259C2×Dic43—41no—
260C2×Dic43—41no—
261C2×Dic43—41no—
262C2×Dic43—41no—
263C2×Dic43—41no—
264C2×Dic43—41no—
265C2×Dic43—41no—
266C2×Dic43—41no—
267C2×Dic43—41no—
268C2×Dic43—41no—
269C2×Dic43—41no—
270C2×Dic43—41no—
271C2×Dic43—41no—
272C2×Dic43—41no—
273C2×Dic43—41no—
274C2×Dic43—41no—
275C2×Dic43—41no—
276D172—22no—
277D172—22no—
278D172—41no—
279D172—41no—
280D172—41no—
281D172—41no—
282D172—41no—
283D172—41no—
284C4×D43—84no—
285Dic86—41no—
286Dic86—41no—
287Dic86—41no—
288Dic86—41no—
289C8×C43—22no—
290C8×C43—22no—
291C8×C43—22no—
292C8×C43—41no—
293#25296—431nomaximal, Sylow
294C22×C43C2×#12111yescharacteristic
295C22×C43#54811yescharacteristic
296C22×C43#55011yescharacteristic
297C22×C43C2×#13511yescharacteristic
298C22×C43C2×#13511yescharacteristic
299C22×C43#57611yescharacteristic
300C22×C43#57711yescharacteristic
301C2×D43—42no—
302C2×D43—42no—
303C2×D43—42no—
304C2×D43—81no—
305C2×D43—81no—
306C2×D43—81no—
307C2×D43—81no—
308C2×D43—81no—
309C2×D43—81no—
310C2×D43—81no—
311C2×D43—81no—
312C2×D43—81no—
313C2×D43—81no—
314C2×D43—81no—
315C4×C43—22no—
316C4×C43—22no—
317C4×C43—22no—
318C4×C43—22no—
319C4×C43—41no—
320C4×C43—41no—
321C4×C43#52911yescharacteristic
322C4×C43#54211yescharacteristic
323C4×C43#54611yescharacteristic
324C4×C43#54811yescharacteristic
325Dic43—81no—
326Dic43—81no—
327Dic43—81no—
328Dic43—81no—
329Dic43—81no—
330Dic43—81no—
331C2×#463—431no—
332C2×C8×Q8D4311yescharacteristic, complemented
333#2592—431no—
334#2580—431no—
335#2068—431no—
336#2065—431no—
337#2055—431no—
338#2054—431no—
339#1970—431no—
340#1968—431no—
341#1958—431no—
342#1949—431no—
343#1934—431no—
344#1933—431no—
345C4×#102—431no—
346C2×C43#242611yescharacteristic
347C2×C43#305011yescharacteristic
348C2×C43#306211yescharacteristic
349C2×C43#336011yescharacteristic
350C2×C43#336111yescharacteristic
351C2×C43#339411yescharacteristic
352C2×C43#339611yescharacteristic
353D43—82no—
354D43—161no—
355C22×SD16—431no—
356C22×D8—431no—
357C2×#124—431no—
358C2×C4×Q8C2×D4311yescharacteristic
359C2×C4×D4—431no—
360#463—864no—
361C8×Q8—24no—
362C2×#107—431no—
363C2×#106—431no—
364C2×C4⋊C8C2×D4311yescharacteristic
365C2×C4⋊C8C2×D4311yescharacteristic
366C2×C4⋊C8C2×D4311yescharacteristic
367C2×#103—431no—
368C2×#103—431no—
369C2×#103—431no—
370C2×#102—431no—
371C2×#102—431no—
372C2×#102—431no—
373C2×#102—431no—
374C2×#102—431no—
375C2×C4×C8C2×D4311yescharacteristic
376C2×C4×C8C2×D4311yescharacteristic
377C2×C4×C8C2×D4311yescharacteristic
378#399—861no—
379#397—861no—
380#393—861no—
381#393—861no—
382#391—431no—
383#391—431no—
384#390—431no—
385#390—431no—
386#390—431no—
387#390—431no—
388#389—431no—
389#389—431no—
390#388—861no—
391#388—861no—
392#385—861no—
393#385—861no—
394#382—431no—
395C4×C4⋊C4—431no—
396#340—431no—
397#340—431no—
398#340—431no—
399#340—431no—
400C43#2529611yescharacteristic, complemented, Sylow
401C22×Q8—431no—
402C22×Q8D17211yescharacteristic
403C22×D4—431no—
404C22×D4—431no—
405C23×C4—861no—
406C2×SD16—862no—
407C2×SD16—862no—
408C2×SD16—1721no—
409C2×D8—862no—
410C2×D8—862no—
411C2×D8—1721no—
412C22×C8C22×D4311yescharacteristic
413C22×C8C22×D4311yescharacteristic
414C22×C8C22×D4311yescharacteristic
415C22×C8C22×D4311yescharacteristic
416#124—862no—
417#124—862no—
418C4×Q8—22no—
419C4×Q8—22no—
420C4×D4—862no—
421C4×D4—862no—
422C2×C4⋊C4—431no—
423C2×C4⋊C4—431no—
424C2×C4⋊C4—431no—
425C2×C4⋊C4—431no—
426C2×C4⋊C4—431no—
427C2×C4⋊C4—861no—
428C2×C4⋊C4—861no—
429C2×C4⋊C4D17211yescharacteristic
430C2×C4⋊C4D17211yescharacteristic
431C2×C4⋊C4D17211yescharacteristic
432C2×C22⋊C4—861no—
433C2×C22⋊C4—861no—
434C2×C22⋊C4—861no—
435C2×C22⋊C4—861no—
436C2×C22⋊C4—861no—
437C2×C22⋊C4—861no—
438C2×C22⋊C4—861no—
439C2×C22⋊C4—861no—
440C2×C22⋊C4—861no—
441C2×C22⋊C4—861no—
442C2×C22⋊C4—861no—
443C2×C42—861no—
444C2×C42C22×D4311yescharacteristic
445C2×C42C22×D4311yescharacteristic
446C2×C42C22×D4311yescharacteristic
447#107—1721no—
448#106—1721no—
449C4⋊C8—22no—
450C4⋊C8—22no—
451C4⋊C8C22⋊D4314yes—
452#103—862no—
453#103—862no—
454#103—1721no—
455#102—862no—
456#102—862no—
457#102—862no—
458#102—862no—
459#102—1721no—
460C4×C8—22no—
461C4×C8—22no—
462C4×C8—22no—
463#95—861no—
464#95—861no—
465#95—861no—
466#95—861no—
467#95—861no—
468#95—861no—
469#95—861no—
470#95—861no—
471#95—861no—
472C24—861no—
473C24—861no—
474C2×Q8—22no—
475C2×Q8—22no—
476C2×Q8—22no—
477C2×Q8—861no—
478C2×Q8—861no—
479C2×Q8—861no—
480C2×Q8—861no—
481C2×Q8—861no—
482C2×Q8—861no—
483C2×D4—432no—
484C2×D4—861no—
485C2×D4—861no—
486C2×D4—861no—
487C2×D4—861no—
488C2×D4—861no—
489C2×D4—861no—
490C2×D4—861no—
491C2×D4—861no—
492C2×D4—861no—
493C2×D4—861no—
494C2×D4—861no—
495C22×C4—861no—
496C22×C4—861no—
497C22×C4—861no—
498C22×C4—861no—
499C22×C4—861no—
500C22×C4—861no—
501C22×C4—3441no—
502C22×C4—3441no—
503C22×C4—1722no—
504C22×C4C2×D17211yescharacteristic
505C22×C4C2×D17211yescharacteristic
506C22×C4C2×D17211yescharacteristic
507C22×C4C2×D17211yescharacteristic
508C22×C4C2×D17211yescharacteristic
509C22×C4C2×D17211yescharacteristic
510C22×C4C23×D4311yescharacteristic
511SD16—1724no—
512D8—1724no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8#1062190212yes—
523C2×C8C2×C22⋊D4312yes—
524C2×C8C2×C22⋊D4312yes—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—861no—
529C4⋊C4—861no—
530C4⋊C4—861no—
531C4⋊C4—861no—
532C4⋊C4—861no—
533C4⋊C4—861no—
534C4⋊C4—861no—
535C4⋊C4—861no—
536C4⋊C4—861no—
537C4⋊C4—861no—
538C4⋊C4—3441no—
539C4⋊C4—3441no—
540C22⋊C4—3441no—
541C22⋊C4—3441no—
542C22⋊C4—3441no—
543C22⋊C4—3441no—
544C22⋊C4—1722no—
545C22⋊C4—1722no—
546C22⋊C4—1722no—
547C22⋊C4—1722no—
548C22⋊C4—1722no—
549C22⋊C4—1722no—
550C22⋊C4—1722no—
551C42—21no—
552C42—21no—
553C42—21no—
554C42—21no—
555C42—3441no—
556C42#1062189812yes—
557C42C2×C22⋊D4312yes—
558C23—862no—
559C23—1721no—
560C23—1721no—
561C23—1721no—
562C23—1721no—
563C23—1721no—
564C23—1721no—
565C23—1721no—
566C23—1721no—
567C23—1721no—
568C23—1721no—
569C23—1721no—
570C23—1721no—
571C23—1721no—
572C23C22×D17211yescharacteristic
573Q8—28no—
574Q8—1721no—
575Q8—1721no—
576Q8—1721no—
577Q8—1721no—
578D4—862no—
579D4—862no—
580D4—1721no—
581D4—1721no—
582D4—1721no—
583D4—1721no—
584D4—1721no—
585D4—1721no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—3444no—
601C2×C4—1721no—
602C2×C4—1721no—
603C2×C4—1721no—
604C2×C4—1721no—
605C2×C4—1721no—
606C2×C4—1721no—
607C2×C4—1721no—
608C2×C4—1721no—
609C2×C4—1721no—
610C2×C4—1721no—
611C2×C4—1721no—
612C2×C4—1721no—
613C2×C4—1721no—
614C2×C4—1721no—
615C2×C4—1721no—
616C2×C4—1721no—
617C2×C4—1721no—
618C2×C4—1721no—
619C2×C4#1306329912yes—
620C2×C4#1306329912yes—
621C2×C4#1306334712yes—
622C2×C4#1306334712yes—
623C2×C4C22×C22⋊D4311yescharacteristic
624C2×C4#41×D4311yescharacteristic
625C2×C4C2×#1062189811yescharacteristic
626C2×C4C2×#1062190211yescharacteristic
627C2×C4#1306337111yescharacteristic
628C2×C4#1306337311yescharacteristic
629C8—22no—
630C8—22no—
631C8—22no—
632C8—41no—
633C22—3441no—
634C22—3441no—
635C22—3441no—
636C22—3441no—
637C22—3441no—
638C22—3441no—
639C22—3441no—
640C22—3441no—
641C22—3441no—
642C22—3441no—
643C22—3441no—
644C22—1722no—
645C22—1722no—
646C22—1722no—
647C22C2×#1306329911yescharacteristic
648C22#1458356111yescharacteristic
649C22#1458357111yescharacteristic
650C22C2×#1306334711yescharacteristic
651C22C2×#1306334711yescharacteristic
652C22#1458370411yescharacteristic
653C22#1458371111yescharacteristic
654C4—22no—
655C4—22no—
656C4—22no—
657C4—22no—
658C4—41no—
659C4—41no—
660C4—3441no—
661C4—3441no—
662C4—3441no—
663C4—3441no—
664C4—3441no—
665C4—3441no—
666C4#1458343611yescharacteristic
667C4#1458352111yescharacteristic
668C4#1458353711yescharacteristic
669C4#1458356011yescharacteristic
670C2—3442no—
671C2—6881no—
672C2#1913815511yescharacteristic
673C2#1914417611yescharacteristic
674C2#1914427511yescharacteristic
675C2#1914724611yescharacteristic
676C2#1914725711yescharacteristic
677C2#1914759311yescharacteristic
678C2#1914760811yescharacteristic
subgroupG/H
centerC23C22×D172
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC2×C8×Q8×C43C2
socleC23×C43C22×D4
prime radicalC22×D344C22
hypercenterC2×C8×Q8D43
hypocenterC43#25296
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8

Complements

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

NH ≅ G/Ntype
2. #19140083C2semidirect
3. #19139953C2semidirect
4. #19135989C2semidirect
5. #19135968C2semidirect
6. #19135913C2semidirect
9. #19135449C2semidirect
10. #19135406C2semidirect
11. #19135365C2semidirect
12. #19135278C2semidirect
13. #19135271C2semidirect
14. C4×#13063247C2semidirect
15. C2×C8×Q8×C43C2semidirect
332. C2×C8×Q8D43semidirect
400. C43#25296semidirect

Maximal subgroups

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

subgroupindex
C2×#145831152
#191400832
#191399532
#191359892
#191359682
#191359132
#191359062
#191354612
#191354492
#191354062
#191353652
#191352782
#191352712
C4×#130632472
C2×C8×Q8×C432
#2529643