Finite Group Databasebeta

← #25133166

Subgroups — #25133166

Order 11008.

1064 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583605C211yesmaximal, characteristic
2C2×#14583183C211yesmaximal, characteristic, complemented
3C2×#14582998C211yesmaximal, characteristic
4#19139963C211yesmaximal, characteristic, complemented
5#19135931C211yesmaximal, characteristic, complemented
6#19135745C211yesmaximal, characteristic, complemented
7#19135743C211yesmaximal, characteristic, complemented
8#19135566C211yesmaximal, characteristic, complemented
9#19135533C211yesmaximal, characteristic, complemented
10#19134930C211yesmaximal, characteristic
11#19134928C211yesmaximal, characteristic
12#19134855C211yesmaximal, characteristic
13#19134853C211yesmaximal, characteristic
14#19134604C211yesmaximal, characteristic
15C43×#2059C211yesmaximal, characteristic, complemented
16C22×#10621873C2211yescharacteristic
17#14583605—24no—
18#14583605—24no—
19C2×#13063324C2211yescharacteristic
20C2×#13063316C2211yescharacteristic
21C2×#13063305—21no—
22C2×#13063303—21no—
23C2×#13063303—21no—
24C2×C4×Dic86C2211yescharacteristic
25C2×#13063290C2211yescharacteristic
26C2×C4×D172C2211yescharacteristic
27C2×#13063287C2211yescharacteristic
28C2×#13063287C2211yescharacteristic
29C2×C4⋊C4×D43—21no—
30#14583183—24no—
31#14582998—24no—
32C2×#13063256C2211yescharacteristic
33C2×Dic43⋊C8C2211yescharacteristic
34C2×#13063250C2211yescharacteristic
35C2×#13063248C2211yescharacteristic
36C2×#13063248C2211yescharacteristic
37C2×#13063247C2211yescharacteristic
38C2×#13063245C2211yescharacteristic
39C2×#13063245C2211yescharacteristic
40C2×#13063235C2211yescharacteristic
41#14582840—21no—
42#14582819—21no—
43#14582805C2211yescharacteristic
44#14582804C2211yescharacteristic
45#14582795C2211yescharacteristic
46#14582785—21no—
47#14582781—21no—
48#14582764C2211yescharacteristic
49#14582662C2211yescharacteristic
50#14582661C2211yescharacteristic
51#14582661C2211yescharacteristic
52#14582661C2211yescharacteristic
53#14582654C2211yescharacteristic
54#14582653C2211yescharacteristic
55#14582653C2211yescharacteristic
56C2×#125×C43C2211yescharacteristic
57C2×#107×C43C2211yescharacteristic
58C2×#106×C43C2211yescharacteristic
59C43×#389C2211yescharacteristic
60C43×#340C2211yescharacteristic
61C43×#340C2211yescharacteristic
62C43×#336C2211yescharacteristic
63C22×Dic86—21no—
64C22×Dic86D411yescharacteristic
65C22×Dic86C2311yescharacteristic
66C22×D172C2311yescharacteristic
67C22×C4×D43—21no—
68C22×C4×D43—21no—
69C22×C4×D43—21no—
70C2×#10621873—22no—
71C2×#10621873—22no—
72C2×#10621873Q814yescomplemented
73C22×C43⋊C8C2311yescharacteristic
74C22×C43⋊C8C2311yescharacteristic
75#13063324—22no—
76#13063324D414yes—
77#13063316—22no—
78#13063316—22no—
79#13063305—28no—
80#13063303—28no—
81#13063303—28no—
82C4×Dic86—22no—
83C4×Dic86—22no—
84#13063290—22no—
85#13063290—22no—
86C4×D172—22no—
87C4×D172—22no—
88#13063287—22no—
89#13063287—22no—
90#13063287—22no—
91#13063287D414yes—
92C2×C4⋊Dic43—21no—
93C2×C4⋊Dic43D411yescharacteristic
94C2×C4⋊Dic43C2311yescharacteristic
95C2×C4⋊Dic43C2311yescharacteristic
96C2×Dic43⋊C4—21no—
97C2×Dic43⋊C4—21no—
98C2×Dic43⋊C4—21no—
99C2×Dic43⋊C4—21no—
100C2×Dic43⋊C4D411yescharacteristic
101C2×Dic43⋊C4D411yescharacteristic
102C4⋊C4×D43—28no—
103C2×(C2×D43)⋊C4—21no—
104C2×(C2×D43)⋊C4—21no—
105C2×(C2×D43)⋊C4—21no—
106C2×C4×Dic43—21no—
107C2×C4×Dic43C2311yescharacteristic
108C2×C4×Dic43C2311yescharacteristic
109#13063256—22no—
110Dic43⋊C8—22no—
111#13063250—22no—
112#13063248—22no—
113#13063248—22no—
114#13063247—22no—
115#13063245—22no—
116#13063245—22no—
117#13063235—22no—
118C43⋊#95—21no—
119C43⋊#95—21no—
120C43⋊#95—21no—
121C43⋊#95—21no—
122C43⋊#95—21no—
123C43⋊#95—21no—
124C43⋊#95—21no—
125C22×C8×C43C2311yescharacteristic
126C22×C8×C43C2311yescharacteristic
127#125×C43—22no—
128#125×C43—22no—
129C2×C4⋊C4×C43—21no—
130C2×C4⋊C4×C43—21no—
131C2×C4⋊C4×C43—21no—
132C2×C4⋊C4×C43C2311yescharacteristic
133C2×C4⋊C4×C43C2311yescharacteristic
134C2×C4⋊C4×C43C2311yescharacteristic
135C2×C4⋊C4×C43C2311yescharacteristic
136C2×C42×C43C2311yescharacteristic
137#107×C43—22no—
138#106×C43—22no—
139#95×C43—21no—
140C23×D43—21no—
141C22×C4×C43—21no—
142C22×C4×C43—21no—
143C22×C4×C43—21no—
144C22×C4×C43—21no—
145C22×C4×C43C2×D411yescharacteristic
146C22×C4×C43C2×D411yescharacteristic
147C22×C4×C43C2411yescharacteristic
148C22×Dic43—21no—
149C22×Dic43—21no—
150C22×Dic43—21no—
151C22×Dic43C2×D411yescharacteristic
152C22×Dic43C2×D411yescharacteristic
153C22×Dic43C2×D411yescharacteristic
154C22×Dic43C2×D411yescharacteristic
155C2×D172—21no—
156C2×D172—21no—
157C2×D172—21no—
158C2×D172C2×D412yes—
159C2×D172C2×Q812yes—
160C2×D172#4112yes—
161C2×C4×D43—24no—
162C2×C4×D43—24no—
163C2×C4×D43—24no—
164C2×C4×D43—42no—
165C2×C4×D43—42no—
166C2×C4×D43—81no—
167C2×C4×D43—81no—
168C2×C4×D43—81no—
169C2×C4×D43—81no—
170C2×Dic86—21no—
171C2×Dic86—21no—
172C2×Dic86—21no—
173C2×Dic86—21no—
174C2×Dic86—21no—
175C2×Dic86—22no—
176C2×Dic86—24no—
177C2×Dic86—41no—
178C2×Dic86—41no—
179C2×Dic86—81no—
180C2×Dic86—81no—
181C2×Dic86C2×Q812yes—
182C2×C8×C43—21no—
183C2×C8×C43—21no—
184C2×C8×C43—21no—
185C2×C8×C43—21no—
186C2×C8×C43—21no—
187C2×C8×C43C2×Q812yes—
188C4⋊C4×C43—24no—
189C4⋊C4×C43—24no—
190C4⋊C4×C43—21no—
191C4⋊C4×C43—21no—
192C4⋊C4×C43—21no—
193C4⋊C4×C43—21no—
194C4⋊C4×C43—21no—
195C4⋊C4×C43—21no—
196C4⋊C4×C43—42no—
197C4⋊C4×C43C2×D412yes—
198C4⋊C4×C43C2×D412yes—
199C42×C43—21no—
200C42×C43—21no—
201(C2×D43)⋊C4—24no—
202(C2×D43)⋊C4—24no—
203(C2×D43)⋊C4—24no—
204C4⋊Dic43—24no—
205C4⋊Dic43—21no—
206C4⋊Dic43—22no—
207C4⋊Dic43—21no—
208C4⋊Dic43—21no—
209C4⋊Dic43C2×D412yes—
210Dic43⋊C4—24no—
211Dic43⋊C4—22no—
212Dic43⋊C4—24no—
213Dic43⋊C4—22no—
214Dic43⋊C4—42no—
215Dic43⋊C4—42no—
216C4×Dic43—21no—
217C4×Dic43—21no—
218C4×Dic43—42no—
219C4×Dic43C2×D412yes—
220C4×Dic43C2×D412yes—
221C2×C43⋊C8—21no—
222C2×C43⋊C8—21no—
223C2×C43⋊C8—21no—
224C2×C43⋊C8—21no—
225C2×C43⋊C8—21no—
226C2×C43⋊C8—21no—
227#10621873—28no—
228C23×C43C22×D411yescharacteristic
229C22×D43—22no—
230C22×D43—22no—
231C22×D43—22no—
232C22×D43—41no—
233C22×D43—41no—
234C22×D43—41no—
235C22×D43—41no—
236C2×C4×C43—21no—
237C2×C4×C43—22no—
238C2×C4×C43—22no—
239C2×C4×C43—21no—
240C2×C4×C43—21no—
241C2×C4×C43—21no—
242C2×C4×C43—21no—
243C2×C4×C43—21no—
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×C43—41no—
254C2×C4×C43C22×D411yescharacteristic
255C2×C4×C43C22×D411yescharacteristic
256C2×C4×C43C22×Q811yescharacteristic
257C2×C4×C43C2×#4111yescharacteristic
258C2×C4×C432−1+411yescharacteristic
259C2×C4×C432−1+411yescharacteristic
260C2×Dic43—21no—
261C2×Dic43—21no—
262C2×Dic43—21no—
263C2×Dic43—21no—
264C2×Dic43—21no—
265C2×Dic43—21no—
266C2×Dic43—22no—
267C2×Dic43—22no—
268C2×Dic43—21no—
269C2×Dic43—21no—
270C2×Dic43—21no—
271C2×Dic43—21no—
272C2×Dic43—41no—
273C2×Dic43—41no—
274C2×Dic43—41no—
275C2×Dic43—41no—
276C2×Dic43—41no—
277C2×Dic43—41no—
278C2×Dic43—41no—
279C2×Dic43#12012yes—
280C2×Dic43#12012yes—
281D172—22no—
282D172—22no—
283D172—22no—
284D172C22⋊Q814yes—
285C4×D43—48no—
286C4×D43—84no—
287C4×D43—84no—
288Dic86—22no—
289Dic86—22no—
290Dic86—44no—
291Dic86—41no—
292Dic86—41no—
293Dic86—84no—
294C8×C43—22no—
295C8×C43—41no—
296C43⋊C8—41no—
297C43⋊C8—41no—
298#24343—431nomaximal, Sylow
299C22×C43C2×#12011yescharacteristic
300C22×C43#54811yescharacteristic
301C22×C43D4×Q811yescharacteristic
302C22×C43#57511yescharacteristic
303C22×C43#57511yescharacteristic
304C22×C43#57711yescharacteristic
305C22×C43#57711yescharacteristic
306C2×D43—24no—
307C2×D43—42no—
308C2×D43—42no—
309C2×D43—42no—
310C2×D43—42no—
311C2×D43—42no—
312C2×D43—42no—
313C4×C43—41no—
314C4×C43—42no—
315C4×C43—42no—
316C4×C43—41no—
317C4×C43—81no—
318C4×C43—81no—
319C4×C43C2×C22⋊Q811yescharacteristic
320C4×C43#53611yescharacteristic
321C4×C43#54811yescharacteristic
322C4×C43D4×Q811yescharacteristic
323Dic43—22no—
324Dic43—22no—
325Dic43—42no—
326Dic43—41no—
327Dic43—42no—
328Dic43—41no—
329Dic43—81no—
330C2×#554—431no—
331C2×#475—431no—
332C2×#440—431no—
333#2581—431no—
334#2059—431no—
335#2059D4311yescharacteristic, complemented
336#2026—431no—
337#2026—431no—
338#1993—431no—
339#1985—431no—
340#1877—431no—
341#1877—431no—
342#1864—431no—
343#1864—431no—
344#1804—431no—
345C2×C43#240911yescharacteristic
346C2×C43#300511yescharacteristic
347C2×C43#300911yescharacteristic
348C2×C43#337611yescharacteristic
349C2×C43#337611yescharacteristic
350C2×C43#338811yescharacteristic
351C2×C43#338811yescharacteristic
352D43—44no—
353C22×SD16—431no—
354#554—864no—
355#554—864no—
356C2×C4⋊Q8—431no—
357C2×#125—431no—
358C2×#125C2×D4311yescharacteristic
359C2×C22⋊Q8—861no—
360C2×C22⋊Q8—861no—
361C2×C22⋊Q8—861no—
362C2×C4×Q8—431no—
363C2×C4×Q8—431no—
364C2×C4×D4—431no—
365C2×C4×D4—431no—
366C2×C4×D4—431no—
367C22×C4⋊C4—861no—
368#475—864no—
369#440—864no—
370C2×#107C2×D4311yescharacteristic
371C2×#106—431no—
372C2×#106C2×D4311yescharacteristic
373C2×C4⋊C8—431no—
374C2×#103—431no—
375C2×#103—431no—
376C2×#103—431no—
377C2×#102—431no—
378C2×#102—431no—
379C2×#102—431no—
380C2×#97—431no—
381#400—861no—
382#395—861no—
383#391—431no—
384#391—431no—
385#389—431no—
386#389C2×D4311yescharacteristic
387#387—861no—
388#387—861no—
389#382—431no—
390#340—431no—
391#340—431no—
392#340—431no—
393#340—431no—
394#340C2×D4311yescharacteristic
395#340C2×D4311yescharacteristic
396#336—431no—
397#336—431no—
398#336—431no—
399#336C2×D4311yescharacteristic
400C43#2434311yescharacteristic, complemented, Sylow
401C22×Q8—431no—
402C22×Q8—431no—
403C22×Q8—861no—
404C22×D4—431no—
405C23×C4—861no—
406C23×C4—861no—
407C23×C4—861no—
408C2×SD16—434no—
409C2×SD16—862no—
410C2×SD16—862no—
411C22×C8—431no—
412C22×C8—431no—
413C22×C8C22×D4311yescharacteristic
414C22×C8C22×D4311yescharacteristic
415C4⋊Q8—434no—
416C4⋊Q8—862no—
417#125—22no—
418#125—22no—
419#125—862no—
420#125—862no—
421C22⋊Q8—868no—
422C22⋊Q8—868no—
423C22⋊Q8—868no—
424C4×Q8—862no—
425C4×Q8—862no—
426C4×Q8—862no—
427C4×Q8—862no—
428C4×D4—434no—
429C4×D4—862no—
430C4×D4—862no—
431C4×D4—862no—
432C4×D4—862no—
433C4×D4—862no—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—21no—
437C2×C4⋊C4—431no—
438C2×C4⋊C4—431no—
439C2×C4⋊C4—431no—
440C2×C4⋊C4—431no—
441C2×C4⋊C4—431no—
442C2×C4⋊C4—868no—
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×C4⋊C4C22×D4311yescharacteristic
451C2×C4⋊C4C22×D4311yescharacteristic
452C2×C22⋊C4—861no—
453C2×C22⋊C4—861no—
454C2×C22⋊C4—861no—
455C2×C42—431no—
456C2×C42—431no—
457C2×C42—861no—
458C2×C42C22×D4311yescharacteristic
459#107—22no—
460#106—22no—
461#106—862no—
462C4⋊C8—862no—
463#103—862no—
464#103—862no—
465#103—862no—
466#102—862no—
467#102—862no—
468#102—862no—
469#97—862no—
470#95—21no—
471#95—861no—
472#95—861no—
473#95—861no—
474#95—861no—
475#95—861no—
476#95—861no—
477#95—861no—
478C24—861no—
479C2×Q8—432no—
480C2×Q8—864no—
481C2×Q8—862no—
482C2×Q8—861no—
483C2×Q8—861no—
484C2×Q8—861no—
485C2×Q8—861no—
486C2×Q8—861no—
487C2×Q8—3441no—
488C2×Q8—3441no—
489C2×Q8—1721no—
490C2×Q8—1721no—
491C2×D4—432no—
492C2×D4—432no—
493C2×D4—432no—
494C2×D4—861no—
495C2×D4—861no—
496C2×D4—861no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—431no—
502C22×C4—431no—
503C22×C4—431no—
504C22×C4—431no—
505C22×C4—864no—
506C22×C4—864no—
507C22×C4—864no—
508C22×C4—861no—
509C22×C4—861no—
510C22×C4—861no—
511C22×C4—3441no—
512C22×C4—3441no—
513C22×C4—3441no—
514C22×C4—3441no—
515C22×C4—1722no—
516C22×C4—1722no—
517C22×C4D4×D4311yescharacteristic
518C22×C4D4×D4311yescharacteristic
519C22×C4C23×D4311yescharacteristic
520SD16—868no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—861no—
527C2×C8—861no—
528C2×C8—861no—
529C2×C8—861no—
530C2×C8—861no—
531C2×C8—861no—
532C2×C8Q8×D4312yes—
533C4⋊C4—24no—
534C4⋊C4—24no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—42no—
542C4⋊C4—432no—
543C4⋊C4—862no—
544C4⋊C4—862no—
545C4⋊C4—862no—
546C4⋊C4—864no—
547C4⋊C4—864no—
548C4⋊C4—864no—
549C4⋊C4—861no—
550C4⋊C4—861no—
551C4⋊C4—861no—
552C4⋊C4—1722no—
553C4⋊C4—1722no—
554C4⋊C4D4×D4312yes—
555C4⋊C4D4×D4312yes—
556C22⋊C4—864no—
557C22⋊C4—864no—
558C22⋊C4—864no—
559C42—21no—
560C42—21no—
561C42—432no—
562C42—432no—
563C42—861no—
564C42—861no—
565C42—1722no—
566C23—862no—
567C23—862no—
568C23—862no—
569C23—1721no—
570C23—1721no—
571C23—1721no—
572C23—1721no—
573C23C2×D4×D4311yescharacteristic
574Q8—862no—
575Q8—862no—
576Q8—3444no—
577Q8—1724no—
578Q8—1721no—
579Q8—1721no—
580D4—434no—
581D4—862no—
582D4—862no—
583D4—862no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—432no—
603C2×C4—432no—
604C2×C4—861no—
605C2×C4—861no—
606C2×C4—861no—
607C2×C4—861no—
608C2×C4—862no—
609C2×C4—861no—
610C2×C4—861no—
611C2×C4—862no—
612C2×C4—861no—
613C2×C4—861no—
614C2×C4—861no—
615C2×C4—861no—
616C2×C4—3444no—
617C2×C4—3444no—
618C2×C4—1728no—
619C2×C4—1721no—
620C2×C4—1721no—
621C2×C4—1721no—
622C2×C4—1721no—
623C2×C4—1721no—
624C2×C4—1721no—
625C2×C4—1721no—
626C2×C4C2×D4×D4311yescharacteristic
627C2×C4C2×D4×D4311yescharacteristic
628C2×C4C2×Q8×D4311yescharacteristic
629C2×C4#41×D4311yescharacteristic
630C2×C4#1306337311yescharacteristic
631C2×C4#1306337311yescharacteristic
632C8—22no—
633C8—41no—
634C8—1721no—
635C8—1721no—
636C22—864no—
637C22—1722no—
638C22—1722no—
639C22—1722no—
640C22—1722no—
641C22—1722no—
642C22—1722no—
643C22#120×D4311yescharacteristic
644C22#1458355911yescharacteristic
645C22#1458356411yescharacteristic
646C22#1458369511yescharacteristic
647C22#1458369611yescharacteristic
648C22#1458370611yescharacteristic
649C22#1458370811yescharacteristic
650C4—41no—
651C4—42no—
652C4—41no—
653C4—42no—
654C4—81no—
655C4—81no—
656C4—862no—
657C4—862no—
658C4—3441no—
659C4—1721no—
660C4—1722no—
661C4—1721no—
662C4—1722no—
663C4C22⋊Q8×D4311yescharacteristic
664C4#1458347411yescharacteristic
665C4#1458355911yescharacteristic
666C4#1458356411yescharacteristic
667C2—1724no—
668C2#1913795511yescharacteristic
669C2#1914383311yescharacteristic
670C2#1914386911yescharacteristic
671C2#1914742011yescharacteristic
672C2#1914742511yescharacteristic
673C2#1914753511yescharacteristic
674C2#1914754611yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#2059C2
socleC23×C43C22×D4
prime radicalC22×D172C23
hypercenter#2059D43
hypocenterC43#24343
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2059

Complements

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

NH ≅ G/Ntype
2. C2×#14583183C2semidirect
4. #19139963C2semidirect
5. #19135931C2semidirect
6. #19135745C2semidirect
7. #19135743C2semidirect
8. #19135566C2semidirect
9. #19135533C2semidirect
15. C43×#2059C2semidirect
72. C2×#10621873Q8semidirect
335. #2059D43semidirect
400. C43#24343semidirect

Maximal subgroups

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

subgroupindex
C2×#145836052
C2×#145831832
C2×#145829982
#191399632
#191359312
#191357452
#191357432
#191355662
#191355332
#191349302
#191349282
#191348552
#191348532
#191346042
C43×#20592
#2434343