Finite Group Databasebeta

← #25133061

Subgroups — #25133061

Order 11008.

1042 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583628C211yesmaximal, characteristic, complemented
2C2×#14583201C211yesmaximal, characteristic
3C2×SD16×Dic43C211yesmaximal, characteristic
4#19139951C211yesmaximal, characteristic
5#19135932C211yesmaximal, characteristic, complemented
6#19135929C211yesmaximal, characteristic, complemented
7#19135739C211yesmaximal, characteristic
8#19135731C211yesmaximal, characteristic
9#19135558C211yesmaximal, characteristic, complemented
10#19135529C211yesmaximal, characteristic, complemented
11#19134952C211yesmaximal, characteristic
12#19134865C211yesmaximal, characteristic, complemented
13#19134849C211yesmaximal, characteristic, complemented
14#19134608C211yesmaximal, characteristic, complemented
15C43×#1880C211yesmaximal, characteristic
16#14583628—24no—
17#14583628—24no—
18C2×#13063325—21no—
19C2×#13063325C2211yescharacteristic
20C2×#13063316—21no—
21C2×#13063316—21no—
22C2×#13063316C2211yescharacteristic
23C2×#13063316C2211yescharacteristic
24C2×Q8×Dic43C2211yescharacteristic
25C2×Q8×Dic43C2211yescharacteristic
26C2×C4×Dic86—21no—
27C2×C4×Dic86C2211yescharacteristic
28C2×D4×Dic43C2211yescharacteristic
29#14583201—24no—
30SD16×Dic43—24no—
31C2×#13063258C2211yescharacteristic
32C2×#13063257C2211yescharacteristic
33C2×#13063257C2211yescharacteristic
34C2×Dic43⋊C8C2211yescharacteristic
35C2×#13063249C2211yescharacteristic
36C2×#13063249C2211yescharacteristic
37C2×#13063246C2211yescharacteristic
38C2×#13063246C2211yescharacteristic
39C2×C8×Dic43C2211yescharacteristic
40#14582835—21no—
41#14582811—21no—
42#14582802C2211yescharacteristic
43#14582798C2211yescharacteristic
44#14582797C2211yescharacteristic
45#14582774—21no—
46#14582771—21no—
47#14582764C2211yescharacteristic
48#14582662C2211yescharacteristic
49#14582662C2211yescharacteristic
50#14582661C2211yescharacteristic
51#14582661C2211yescharacteristic
52#14582661C2211yescharacteristic
53#14582654C2211yescharacteristic
54#14582653C2211yescharacteristic
55#14582653C2211yescharacteristic
56C22×SD16×C43C2211yescharacteristic
57C2×C4×Q8×C43C2211yescharacteristic
58C2×#103×C43C2211yescharacteristic
59C2×#102×C43C2211yescharacteristic
60C43×#390C2211yescharacteristic
61C43×#340C2211yescharacteristic
62C43×#336C2211yescharacteristic
63C22×Dic86—21no—
64C22×Dic86D411yescharacteristic
65C23×Dic43—21no—
66C22×C43⋊C8C2311yescharacteristic
67C22×C43⋊C8C2311yescharacteristic
68#13063325—28no—
69#13063325—22no—
70#13063325—22no—
71#13063316—28no—
72#13063316—28no—
73#13063316—22no—
74#13063316—22no—
75#13063316—22no—
76#13063316D414yes—
77Q8×Dic43—22no—
78Q8×Dic43—22no—
79Q8×Dic43—22no—
80Q8×Dic43D414yes—
81C4×Dic86—28no—
82C4×Dic86—22no—
83C4×Dic86—22no—
84D4×Dic43—22no—
85D4×Dic43—22no—
86C2×C4⋊Dic43—21no—
87C2×C4⋊Dic43—21no—
88C2×C4⋊Dic43—21no—
89C2×C4⋊Dic43—21no—
90C2×C4⋊Dic43—21no—
91C2×C4⋊Dic43—21no—
92C2×C4⋊Dic43D411yescharacteristic
93C2×C4⋊Dic43C2311yescharacteristic
94C2×C4⋊Dic43C2311yescharacteristic
95C2×C4⋊Dic43C2311yescharacteristic
96C2×C4⋊Dic43C2311yescharacteristic
97C2×Dic43⋊C4—21no—
98C2×Dic43⋊C4—21no—
99C2×Dic43⋊C4—21no—
100C2×Dic43⋊C4D411yescharacteristic
101C2×Dic43⋊C4D411yescharacteristic
102C2×C22⋊Dic43—21no—
103C2×C22⋊Dic43—21no—
104C2×C22⋊Dic43—21no—
105C2×C4×Dic43—21no—
106C2×C4×Dic43—21no—
107C2×C4×Dic43C2311yescharacteristic
108C2×C4×Dic43C2311yescharacteristic
109#13063258—22no—
110#13063257—22no—
111#13063257—22no—
112Dic43⋊C8—22no—
113#13063249—22no—
114#13063249—22no—
115#13063246—22no—
116#13063246—22no—
117C8×Dic43—22no—
118C43⋊#95—21no—
119C43⋊#95—21no—
120C43⋊#95—21no—
121C43⋊#95—21no—
122C22×Q8×C43C2311yescharacteristic
123C22×D4×C43C2311yescharacteristic
124C2×SD16×C43—22no—
125C2×SD16×C43—22no—
126C2×SD16×C43Q814yescomplemented
127C22×C8×C43C2311yescharacteristic
128C22×C8×C43C2311yescharacteristic
129C4×Q8×C43—22no—
130C4×Q8×C43—22no—
131C2×C4⋊C4×C43—21no—
132C2×C4⋊C4×C43C2311yescharacteristic
133C2×C4⋊C4×C43C2311yescharacteristic
134C2×C22⋊C4×C43—21no—
135C2×C22⋊C4×C43—21no—
136C2×C42×C43—21no—
137C2×C42×C43C2311yescharacteristic
138#103×C43—22no—
139#102×C43—22no—
140C24×C43—21no—
141C2×Q8×C43—21no—
142C2×Q8×C43—21no—
143C2×Q8×C43—21no—
144C2×Q8×C43C2×D412yes—
145C2×Q8×C43C2×Q812yes—
146C2×Q8×C43#4112yes—
147C2×D4×C43—21no—
148C2×D4×C43—21no—
149C2×D4×C43—21no—
150C2×D4×C43—21no—
151C2×D4×C43—21no—
152C2×D4×C43C2×Q812yes—
153C22×C4×C43—21no—
154C22×C4×C43—21no—
155C22×C4×C43—21no—
156C22×C4×C43C2×D411yescharacteristic
157C22×C4×C43C2×D411yescharacteristic
158C22×C4×C43C2411yescharacteristic
159C22×Dic43—21no—
160C22×Dic43—21no—
161C22×Dic43—21no—
162C22×Dic43—21no—
163C22×Dic43—42no—
164C22×Dic43—81no—
165C22×Dic43—81no—
166C22×Dic43C2×D411yescharacteristic
167C22×Dic43C2×D411yescharacteristic
168C22×Dic43C2×D411yescharacteristic
169C22×Dic43C2×D411yescharacteristic
170C2×Dic86—24no—
171C2×Dic86—22no—
172C2×Dic86—22no—
173C2×Dic86—41no—
174C2×Dic86—42no—
175C2×Dic86—42no—
176SD16×C43—28no—
177C2×C8×C43—21no—
178C2×C8×C43—21no—
179C2×C8×C43—21no—
180C2×C8×C43—21no—
181C2×C8×C43—21no—
182C2×C8×C43C2×Q812yes—
183C4⋊C4×C43—24no—
184C4⋊C4×C43—21no—
185C4⋊C4×C43—21no—
186C4⋊C4×C43—21no—
187C4⋊C4×C43C2×D412yes—
188C22⋊C4×C43—42no—
189C22⋊C4×C43—81no—
190C42×C43—24no—
191C42×C43—21no—
192C42×C43—21no—
193C22⋊Dic43—42no—
194C22⋊Dic43—42no—
195C22⋊Dic43—81no—
196C4⋊Dic43—24no—
197C4⋊Dic43—24no—
198C4⋊Dic43—24no—
199C4⋊Dic43—21no—
200C4⋊Dic43—21no—
201C4⋊Dic43—24no—
202C4⋊Dic43—24no—
203C4⋊Dic43—22no—
204C4⋊Dic43—21no—
205C4⋊Dic43—21no—
206C4⋊Dic43—21no—
207C4⋊Dic43—21no—
208C4⋊Dic43—81no—
209C4⋊Dic43C2×D412yes—
210C4⋊Dic43C2×D412yes—
211Dic43⋊C4—24no—
212Dic43⋊C4—24no—
213Dic43⋊C4—22no—
214Dic43⋊C4—22no—
215Dic43⋊C4—81no—
216C4×Dic43—24no—
217C4×Dic43—21no—
218C4×Dic43—21no—
219C4×Dic43—24no—
220C4×Dic43C2×D412yes—
221C4×Dic43C2×D412yes—
222C2×C43⋊C8—21no—
223C2×C43⋊C8—21no—
224C2×C43⋊C8—21no—
225C2×C43⋊C8—21no—
226C2×C43⋊C8—21no—
227C2×C43⋊C8—21no—
228C23×C43—22no—
229C23×C43—41no—
230C23×C43—41no—
231C23×C43—41no—
232C23×C43—41no—
233C23×C43—41no—
234C23×C43—41no—
235C23×C43C22×D411yescharacteristic
236Q8×C43—22no—
237Q8×C43—22no—
238Q8×C43—22no—
239Q8×C43C22⋊Q814yes—
240D4×C43—22no—
241D4×C43—22no—
242D4×C43—41no—
243D4×C43—41no—
244C2×C4×C43—22no—
245C2×C4×C43—21no—
246C2×C4×C43—22no—
247C2×C4×C43—22no—
248C2×C4×C43—21no—
249C2×C4×C43—21no—
250C2×C4×C43—22no—
251C2×C4×C43—21no—
252C2×C4×C43—21no—
253C2×C4×C43—41no—
254C2×C4×C43—41no—
255C2×C4×C43—41no—
256C2×C4×C43—41no—
257C2×C4×C43—41no—
258C2×C4×C43#13512yes—
259C2×C4×C43C22×D411yescharacteristic
260C2×C4×C43C22×D411yescharacteristic
261C2×C4×C43C22×Q811yescharacteristic
262C2×C4×C43C2×#4111yescharacteristic
263C2×C4×C432−1+411yescharacteristic
264C2×C4×C432−1+411yescharacteristic
265C2×Dic43—21no—
266C2×Dic43—21no—
267C2×Dic43—21no—
268C2×Dic43—21no—
269C2×Dic43—21no—
270C2×Dic43—21no—
271C2×Dic43—22no—
272C2×Dic43—22no—
273C2×Dic43—21no—
274C2×Dic43—21no—
275C2×Dic43—21no—
276C2×Dic43—21no—
277C2×Dic43—41no—
278C2×Dic43—41no—
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—84no—
288C2×Dic43#12012yes—
289C2×Dic43#12012yes—
290Dic86—44no—
291Dic86—48no—
292C8×C43—22no—
293C8×C43—41no—
294C43⋊C8—41no—
295C43⋊C8—41no—
296#24334—431nomaximal, Sylow
297C22×C43—42no—
298C22×C43—42no—
299C22×C43—42no—
300C22×C43—81no—
301C22×C43—81no—
302C22×C43—81no—
303C22×C43—81no—
304C22×C43C2×#12011yescharacteristic
305C22×C43#54811yescharacteristic
306C22×C43D4×Q811yescharacteristic
307C22×C43C2×#13511yescharacteristic
308C22×C43#57511yescharacteristic
309C22×C43#57711yescharacteristic
310C22×C43#57711yescharacteristic
311C4×C43—24no—
312C4×C43—22no—
313C4×C43—41no—
314C4×C43—42no—
315C4×C43—81no—
316C4×C43C2×C22⋊Q811yescharacteristic
317C4×C43#53611yescharacteristic
318C4×C43#54811yescharacteristic
319C4×C43D4×Q811yescharacteristic
320Dic43—22no—
321Dic43—22no—
322Dic43—41no—
323Dic43—42no—
324Dic43—42no—
325Dic43—41no—
326Dic43—81no—
327Dic43—81no—
328C2×#557—431no—
329C2×#478—431no—
330C2×C4×SD16—431no—
331#2580—431no—
332#2059—431no—
333#2058—431no—
334#2025—431no—
335#2023—431no—
336#1991—431no—
337#1984—431no—
338#1880—431no—
339#1880D4311yescharacteristic
340#1865—431no—
341#1863—431no—
342#1805—431no—
343C2×C43—82no—
344C2×C43#240911yescharacteristic
345C2×C43#300211yescharacteristic
346C2×C43#301011yescharacteristic
347C2×C43#336511yescharacteristic
348C2×C43#337111yescharacteristic
349C2×C43#338111yescharacteristic
350C2×C43#338811yescharacteristic
351C22×SD16C2×D4311yescharacteristic
352#557—864no—
353#557—864no—
354C2×C4⋊Q8—431no—
355C2×C4⋊Q8—861no—
356C2×#125—431no—
357C2×#125—431no—
358C2×#125—861no—
359C2×#125—861no—
360C2×C4×Q8—431no—
361C2×C4×Q8—431no—
362C2×C4×Q8—431no—
363C2×C4×Q8—861no—
364C2×C4×Q8C2×D4311yescharacteristic
365C2×C4×D4—431no—
366#478—864no—
367C4×SD16—864no—
368C2×#107—431no—
369C2×#106—431no—
370C2×#106—431no—
371C2×C4⋊C8—431no—
372C2×#103—431no—
373C2×#103—431no—
374C2×#103C2×D4311yescharacteristic
375C2×#102—431no—
376C2×#102—431no—
377C2×#102C2×D4311yescharacteristic
378C2×C4×C8—431no—
379#399—861no—
380#393—861no—
381#390—431no—
382#390C2×D4311yescharacteristic
383#389—431no—
384#389—431no—
385#385—861no—
386#385—861no—
387#382—431no—
388#340—431no—
389#340—431no—
390#340—431no—
391#340—431no—
392#340—431no—
393#340C2×D4311yescharacteristic
394#336—431no—
395#336—431no—
396#336—431no—
397#336C2×D4311yescharacteristic
398C43#2433411yescharacteristic, complemented, Sylow
399C22×Q8—431no—
400C22×Q8—861no—
401C22×Q8C22×D4311yescharacteristic
402C22×D4C22×D4311yescharacteristic
403C23×C4—861no—
404C2×SD16—22no—
405C2×SD16—22no—
406C2×SD16Dic8614yescomplemented
407C22×C8—431no—
408C22×C8—431no—
409C22×C8C22×D4311yescharacteristic
410C22×C8C22×D4311yescharacteristic
411C4⋊Q8—862no—
412C4⋊Q8—862no—
413C4⋊Q8—868no—
414#125—434no—
415#125—862no—
416#125—862no—
417#125—868no—
418#125—868no—
419#125—862no—
420C4×Q8—22no—
421C4×Q8—22no—
422C4×Q8—434no—
423C4×Q8—862no—
424C4×Q8—862no—
425C4×Q8—868no—
426C4×Q8—862no—
427C4×Q8—862no—
428C4×Q8—862no—
429C4×D4—862no—
430C4×D4—862no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—431no—
433C2×C4⋊C4—431no—
434C2×C4⋊C4—431no—
435C2×C4⋊C4—431no—
436C2×C4⋊C4—431no—
437C2×C4⋊C4—431no—
438C2×C4⋊C4—431no—
439C2×C4⋊C4—861no—
440C2×C4⋊C4—861no—
441C2×C4⋊C4—861no—
442C2×C4⋊C4—861no—
443C2×C4⋊C4—861no—
444C2×C4⋊C4—861no—
445C2×C4⋊C4—861no—
446C2×C4⋊C4—861no—
447C2×C4⋊C4—861no—
448C2×C4⋊C4C22×D4311yescharacteristic
449C2×C4⋊C4C22×D4311yescharacteristic
450C2×C22⋊C4—21no—
451C2×C22⋊C4—21no—
452C2×C22⋊C4—861no—
453C2×C22⋊C4—861no—
454C2×C22⋊C4—861no—
455C2×C42—21no—
456C2×C42—431no—
457C2×C42—431no—
458C2×C42—861no—
459C2×C42—861no—
460C2×C42C22×D4311yescharacteristic
461#107—862no—
462#106—862no—
463#106—862no—
464C4⋊C8—862no—
465#103—22no—
466#103—862no—
467#103—862no—
468#102—22no—
469#102—862no—
470#102—862no—
471C4×C8—862no—
472#95—861no—
473#95—861no—
474#95—861no—
475#95—861no—
476C24—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—862no—
481C2×Q8—864no—
482C2×Q8—862no—
483C2×Q8—1721no—
484C2×Q8—1722no—
485C2×Q8—1722no—
486C2×Q8C2×Dic8612yes—
487C2×Q8D4×D4312yes—
488C2×Q8#1062190012yes—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4C2×Dic8612yes—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—431no—
499C22×C4—431no—
500C22×C4—431no—
501C22×C4—431no—
502C22×C4—861no—
503C22×C4—861no—
504C22×C4—861no—
505C22×C4—861no—
506C22×C4—3441no—
507C22×C4—3441no—
508C22×C4—1722no—
509C22×C4D4×D4311yescharacteristic
510C22×C4D4×D4311yescharacteristic
511C22×C4C23×D4311yescharacteristic
512SD16—28no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—861no—
519C2×C8—861no—
520C2×C8—861no—
521C2×C8—861no—
522C2×C8—861no—
523C2×C8—861no—
524C2×C8C2×Dic8612yes—
525C4⋊C4—24no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—432no—
530C4⋊C4—432no—
531C4⋊C4—862no—
532C4⋊C4—862no—
533C4⋊C4—862no—
534C4⋊C4—864no—
535C4⋊C4—864no—
536C4⋊C4—864no—
537C4⋊C4—864no—
538C4⋊C4—864no—
539C4⋊C4—864no—
540C4⋊C4—864no—
541C4⋊C4—861no—
542C4⋊C4—861no—
543C4⋊C4—861no—
544C4⋊C4—861no—
545C4⋊C4—861no—
546C4⋊C4—861no—
547C4⋊C4—3441no—
548C4⋊C4—3441no—
549C4⋊C4D4×D4312yes—
550C22⋊C4—42no—
551C22⋊C4—81no—
552C22⋊C4—3441no—
553C22⋊C4—1722no—
554C22⋊C4—1722no—
555C42—24no—
556C42—21no—
557C42—21no—
558C42—432no—
559C42—432no—
560C42—864no—
561C42—864no—
562C42—861no—
563C42—861no—
564C23—22no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23C2×D4×D4311yescharacteristic
572Q8—22no—
573Q8—22no—
574Q8—22no—
575Q8—1724no—
576Q8—1728no—
577Q8#1306330214yes—
578D4—22no—
579D4—22no—
580D4—41no—
581D4—41no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—432no—
597C2×C4—432no—
598C2×C4—861no—
599C2×C4—861no—
600C2×C4—861no—
601C2×C4—861no—
602C2×C4—861no—
603C2×C4—862no—
604C2×C4—862no—
605C2×C4—861no—
606C2×C4—861no—
607C2×C4—861no—
608C2×C4—861no—
609C2×C4—861no—
610C2×C4—3444no—
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—1721no—
620C2×C4—1721no—
621C2×C4#1306334612yes—
622C2×C4C2×D4×D4311yescharacteristic
623C2×C4C2×D4×D4311yescharacteristic
624C2×C4C22×Dic8611yescharacteristic
625C2×C4C2×#1062190011yescharacteristic
626C2×C4#1306337211yescharacteristic
627C2×C4#1306337211yescharacteristic
628C8—22no—
629C8—41no—
630C8—1721no—
631C8—1721no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22#120×D4311yescharacteristic
640C22#1458355511yescharacteristic
641C22D4×Dic8611yescharacteristic
642C22C2×#1306334611yescharacteristic
643C22#1458369811yescharacteristic
644C22#1458370611yescharacteristic
645C22#1458370911yescharacteristic
646C4—22no—
647C4—24no—
648C4—41no—
649C4—42no—
650C4—81no—
651C4—862no—
652C4—862no—
653C4—3441no—
654C4—3441no—
655C4—1721no—
656C4—1722no—
657C4—1722no—
658C4—1721no—
659C4C2×#1306330211yescharacteristic
660C4#1458347111yescharacteristic
661C4#1458355511yescharacteristic
662C4D4×Dic8611yescharacteristic
663C2—82no—
664C2#1913795711yescharacteristic
665C2#1914381211yescharacteristic
666C2#1914387811yescharacteristic
667C2#1914729511yescharacteristic
668C2#1914736011yescharacteristic
669C2#1914746811yescharacteristic
670C2#1914754111yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#1880C2
socleC23×C43C22×D4
prime radicalC22×D4×C43C23
hypercenter#1880D43
hypocenterC43#24334
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1880

Complements

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

NH ≅ G/Ntype
1. C2×#14583628C2semidirect
5. #19135932C2semidirect
6. #19135929C2semidirect
9. #19135558C2semidirect
10. #19135529C2semidirect
12. #19134865C2semidirect
13. #19134849C2semidirect
14. #19134608C2semidirect
126. C2×SD16×C43Q8semidirect
398. C43#24334semidirect
406. C2×SD16Dic86semidirect

Maximal subgroups

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

subgroupindex
C2×#145836282
C2×#145832012
C2×SD16×Dic432
#191399512
#191359322
#191359292
#191357392
#191357312
#191355582
#191355292
#191349522
#191348652
#191348492
#191346082
C43×#18802
#2433443