Finite Group Databasebeta

← #26234780

Subgroups — #26234780

Order 12032.

850 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689245C211yesmaximal, characteristic
2#20252264C211yesmaximal, characteristic
3#20251986C211yesmaximal, characteristic, complemented
4#20251981C211yesmaximal, characteristic, complemented
5#20244173C211yesmaximal, characteristic, complemented
6#20244166C211yesmaximal, characteristic, complemented
7#20243621C211yesmaximal, characteristic, complemented
8#20243619C211yesmaximal, characteristic, complemented
9#20243614C211yesmaximal, characteristic, complemented
10#20243612C211yesmaximal, characteristic, complemented
11#20243553C211yesmaximal, characteristic, complemented
12#20243450C211yesmaximal, characteristic, complemented
13#20243445C211yesmaximal, characteristic, complemented
14#20243273C211yesmaximal, characteristic, complemented
15C47×#2952C211yesmaximal, characteristic
16C2×#13094031C2211yescharacteristic
17C2×#13094031C2211yescharacteristic
18#15689606C2211yescharacteristic
19#15689606C2211yescharacteristic
20#15689245C418yescomplemented
21#15689224—21no—
22#15689224—21no—
23Q16×Dic47—21no—
24D8×Dic47—21no—
25C2×#13093960C2211yescharacteristic
26C2×#13093959C2211yescharacteristic
27#15689166—21no—
28#15689166—21no—
29#15689166C2211yescharacteristic
30#15689166C2211yescharacteristic
31#15689163—21no—
32#15689163—21no—
33#15689163C2211yescharacteristic
34#15689163C2211yescharacteristic
35#15689159—21no—
36#15689159—21no—
37#15689159C2211yescharacteristic
38C2×#13093958C2211yescharacteristic
39C2×#13093957—21no—
40C2×#13093957C2211yescharacteristic
41C2×#13093955—21no—
42C2×#13093955C2211yescharacteristic
43C2×#13093951C2211yescharacteristic
44C2×#13093951C2211yescharacteristic
45#15689124C2211yescharacteristic
46#15689124C2211yescharacteristic
47#15689078C2211yescharacteristic
48C2×#13093934C2211yescharacteristic
49#15688992C2211yescharacteristic
50#15688988C2211yescharacteristic
51#15688892C2211yescharacteristic
52#15688892C2211yescharacteristic
53#15688891C2211yescharacteristic
54#15688891C2211yescharacteristic
55#15688884C2211yescharacteristic
56#15688882C2211yescharacteristic
57C2×#135×C47C2211yescharacteristic
58C47×#520C2211yescharacteristic
59C47×#520C2211yescharacteristic
60C47×#442—21no—
61C47×#441—21no—
62C4×SD16×C47—21no—
63C4×SD16×C47—21no—
64C2×#106×C47C2211yescharacteristic
65C47×#417C2211yescharacteristic
66C47×#417C2211yescharacteristic
67C47×#405C2211yescharacteristic
68#13094031—22no—
69#13094031—22no—
70#13094031C2×C414yescomplemented
71#13094031C2×C414yescomplemented
72C2×#10625329—21no—
73C2×#10625329—21no—
74C2×#10625329C2311yescharacteristic
75C2×#10625329C2311yescharacteristic
76C2×#10625329C2311yescharacteristic
77C22×C47⋊C8—21no—
78C22×C47⋊C8—21no—
79C22×C47⋊C8C2311yescharacteristic
80Q8×Dic47—21no—
81Q8×Dic47—21no—
82D4×Dic47—21no—
83D4×Dic47—21no—
84D4×Dic47—41no—
85D4×Dic47—41no—
86#13093983—21no—
87#13093983—21no—
88#13093983C2311yescharacteristic
89C2×C4⋊Dic47—21no—
90C2×C4⋊Dic47—21no—
91C2×C4⋊Dic47C2311yescharacteristic
92C2×C4⋊Dic47C2311yescharacteristic
93C2×Dic47⋊C4—41no—
94C2×C4×Dic47C2311yescharacteristic
95#13093960C2×C414yescomplemented
96#13093959—22no—
97#13093958—21no—
98#13093958—21no—
99#13093957—21no—
100#13093957—21no—
101#13093957—21no—
102#13093957—21no—
103#13093957—21no—
104#13093957—21no—
105#13093957D411yescharacteristic
106#13093957D411yescharacteristic
107#13093955—21no—
108#13093955—21no—
109#13093955—21no—
110#13093955—21no—
111#13093955—21no—
112#13093955—21no—
113#13093955D411yescharacteristic
114#13093955D411yescharacteristic
115#13093951C2×C414yescomplemented
116#13093951C2×C414yescomplemented
117#13093948—21no—
118#13093948—21no—
119#13093945—21no—
120#13093945—21no—
121#13093934—21no—
122#13093934C2×C414yescomplemented
123C8×Dic47—21no—
124C47⋊#95—41no—
125C2×#41×C47C2311yescharacteristic
126C2×#41×C47C2311yescharacteristic
127#135×C47C2×C414yes—
128#135×C47Q814yes—
129C2×Q16×C47D411yescharacteristic
130C2×SD16×C47D411yescharacteristic
131C2×SD16×C47D411yescharacteristic
132C2×D8×C47D411yescharacteristic
133C2×M16×C47C2311yescharacteristic
134C22×C8×C47C2311yescharacteristic
135C4×Q8×C47—21no—
136C4×Q8×C47—21no—
137C4×D4×C47—21no—
138C4×D4×C47—21no—
139C4×D4×C47—41no—
140C4×D4×C47—41no—
141#117×C47—21no—
142#117×C47—21no—
143#117×C47C2311yescharacteristic
144C2×C4⋊C4×C47—21no—
145C2×C4⋊C4×C47—21no—
146C2×C4⋊C4×C47C2311yescharacteristic
147C2×C4⋊C4×C47C2311yescharacteristic
148#106×C47—21no—
149#106×C47—21no—
150#103×C47—21no—
151#103×C47—21no—
152#102×C47—21no—
153#102×C47—21no—
154#97×C47—21no—
155C4×C8×C47—21no—
156#41×C47—21no—
157#41×C47—21no—
158#41×C47—21no—
159#41×C47—21no—
160#41×C47C22×C412yes—
161#41×C47C22×C412yes—
162#41×C47C2×Q812yes—
163#41×C47C2×Q812yes—
164C2×Q8×C47C2×D411yescharacteristic
165C2×Q8×C47C2×D411yescharacteristic
166C2×D4×C47—21no—
167C2×D4×C47—21no—
168C2×D4×C47C2×D411yescharacteristic
169C2×D4×C47C2×D411yescharacteristic
170C22×C4×C47—21no—
171C22×C4×C47—21no—
172C22×C4×C47—21no—
173C22×C4×C47—21no—
174C22×C4×C47—41no—
175C22×C4×C47—41no—
176C22×C4×C47C2411yescharacteristic
177C22×Dic47—21no—
178C22×Dic47—21no—
179C22×Dic47—21no—
180C22×Dic47—41no—
181C22×Dic47—41no—
182Q16×C47C4⋊C414yescomplemented
183SD16×C47C4⋊C414yescomplemented
184SD16×C47C4⋊C414yescomplemented
185D8×C47C4⋊C414yescomplemented
186M16×C47—21no—
187M16×C47—21no—
188C2×C8×C47—21no—
189C2×C8×C47C22×C412yes—
190C2×C8×C47C2×D411yescharacteristic
191C2×C8×C47C2×D411yescharacteristic
192C2×C8×C47C2×Q812yes—
193C4⋊C4×C47—21no—
194C4⋊C4×C47—21no—
195C4⋊C4×C47—21no—
196C4⋊C4×C47—21no—
197C4⋊C4×C47—41no—
198C4⋊C4×C47—41no—
199C4⋊C4×C47C2×D411yescharacteristic
200C4⋊C4×C47C2×D411yescharacteristic
201C22⋊C4×C47—21no—
202C22⋊C4×C47—41no—
203C22⋊C4×C47—41no—
204C42×C47—21no—
205C42×C47—41no—
206C42×C47—41no—
207C22⋊Dic47—21no—
208C22⋊Dic47—41no—
209C22⋊Dic47—41no—
210C4⋊Dic47—21no—
211C4⋊Dic47—21no—
212C4⋊Dic47—21no—
213C4⋊Dic47—21no—
214C4⋊Dic47—41no—
215C4⋊Dic47—41no—
216C4⋊Dic47C2×D411yescharacteristic
217C4⋊Dic47C2×D411yescharacteristic
218Dic47⋊C4—82no—
219C4×Dic47—21no—
220C4×Dic47—21no—
221C4×Dic47—41no—
222C4×Dic47—41no—
223C4×Dic47C22×C412yes—
224#10625329—21no—
225#10625329—21no—
226#10625329—24no—
227#10625329—24no—
228#10625329—21no—
229#10625329—21no—
230#10625329C22×C412yes—
231#10625329C22×C412yes—
232C2×C47⋊C8—21no—
233C2×C47⋊C8—21no—
234C2×C47⋊C8—24no—
235C2×C47⋊C8—24no—
236C2×C47⋊C8—21no—
237C2×C47⋊C8C22×C412yes—
238C2×C47⋊C8C2×D411yescharacteristic
239C2×C47⋊C8C2×D411yescharacteristic
240C2×C47⋊C8#4112yes—
241C23×C47—21no—
242C23×C47—21no—
243C23×C47H(5,𝔽2)11yescharacteristic
244Q8×C47—21no—
245Q8×C47—21no—
246Q8×C47C2×C4⋊C412yes—
247Q8×C47C2×C4⋊C412yes—
248D4×C47—22no—
249D4×C47—22no—
250D4×C47—22no—
251D4×C47—22no—
252D4×C47—21no—
253D4×C47—21no—
254D4×C47C2×C4⋊C412yes—
255D4×C47C2×C4⋊C412yes—
256C2×C4×C47—21no—
257C2×C4×C47—21no—
258C2×C4×C47—22no—
259C2×C4×C47—22no—
260C2×C4×C47—22no—
261C2×C4×C47—22no—
262C2×C4×C47—21no—
263C2×C4×C47—21no—
264C2×C4×C47—41no—
265C2×C4×C47—41no—
266C2×C4×C47—81no—
267C2×C4×C47—81no—
268C2×C4×C47—161no—
269C2×C4×C47—161no—
270C2×C4×C47C23×C411yescharacteristic
271C2×C4×C47C22×D411yescharacteristic
272C2×C4×C47C22×D411yescharacteristic
273C2×C4×C47C22×Q811yescharacteristic
274C2×C4×C47C2×#4111yescharacteristic
275C2×C4×C472−1+411yescharacteristic
276C2×Dic47—22no—
277C2×Dic47—21no—
278C2×Dic47—21no—
279C2×Dic47—41no—
280C2×Dic47—41no—
281C2×Dic47—41no—
282C2×Dic47—41no—
283C2×Dic47—81no—
284C2×Dic47—81no—
285C2×Dic47—161no—
286C2×Dic47—161no—
287C8×C47—41no—
288C8×C47C2×C4⋊C412yes—
289C8×C47C2×C4⋊C412yes—
290C47⋊C8—22no—
291C47⋊C8—22no—
292C47⋊C8—41no—
293C47⋊C8C4×D412yes—
294C47⋊C8C4×D412yes—
295#22693—471nomaximal, Sylow
296C22×C47—21no—
297C22×C47—21no—
298C22×C47—22no—
299C22×C47—22no—
300C22×C47—41no—
301C22×C47—41no—
302C22×C47—41no—
303C22×C47—41no—
304C22×C47#52011yescharacteristic
305C22×C47D4211yescharacteristic
306C22×C47#55411yescharacteristic
307C4×C47—22no—
308C4×C47—22no—
309C4×C47—41no—
310C4×C47—41no—
311C4×C47—81no—
312C4×C47—81no—
313C4×C47C22×C4⋊C411yescharacteristic
314C4×C47C2×C4×D411yescharacteristic
315C4×C47#54711yescharacteristic
316C4×C47D4×Q811yescharacteristic
317Dic47—42no—
318Dic47—41no—
319Dic47—41no—
320Dic47—81no—
321Dic47—81no—
322C2×#444—471no—
323#2952D4711yescharacteristic
324#2951—471no—
325#2913—471no—
326#2912—471no—
327#1952—471no—
328#1951—471no—
329#1860—471no—
330#1860—471no—
331#1859—471no—
332#1859—471no—
333#1847—471no—
334#1816—471no—
335#1815—471no—
336#1773—471no—
337C2×C47—21no—
338C2×C47—21no—
339C2×C47—42no—
340C2×C47—42no—
341C2×C47D4×C4⋊C411yescharacteristic
342C2×C47#328811yescharacteristic
343C2×C47#328911yescharacteristic
344C2×#135C2×D4711yescharacteristic
345C2×#131—471no—
346C2×#131—471no—
347#520—471no—
348#520—471no—
349#520C2×D4711yescharacteristic
350#520C2×D4711yescharacteristic
351#444—478no—
352#442—21no—
353#441—21no—
354#440—941no—
355#440—941no—
356C4×Q16—941no—
357C4×SD16—21no—
358C4×SD16—21no—
359C4×D8—941no—
360C2×#108—471no—
361C2×#108—471no—
362#428—471no—
363#428—471no—
364#428—471no—
365#428—471no—
366#428—941no—
367#428—941no—
368#428—941no—
369#428—941no—
370#427—471no—
371#427—941no—
372#427—941no—
373C2×#107—471no—
374C2×#107—471no—
375C2×#107—941no—
376C2×#106—471no—
377C2×#106—941no—
378C2×#106C2×D4711yescharacteristic
379C2×#104—471no—
380C2×#104—471no—
381#417—471no—
382#417—471no—
383#417C2×D4711yescharacteristic
384#417C2×D4711yescharacteristic
385#405—471no—
386#405C2×D4711yescharacteristic
387C2×#97—471no—
388#384—471no—
389#382—471no—
390#343—471no—
391#343—471no—
392#343—471no—
393#343—471no—
394#339—471no—
395#339—471no—
396C47#2269311yescharacteristic, complemented, Sylow
397C2×#41C22×D4711yescharacteristic
398C2×#41C22×D4711yescharacteristic
399#135Dic9414yes—
400#135C4×D4714yes—
401C2×Q16C22⋊D4711yescharacteristic
402C2×SD16C22⋊D4711yescharacteristic
403C2×SD16C22⋊D4711yescharacteristic
404C2×D8C22⋊D4711yescharacteristic
405#131—474no—
406#131—474no—
407#131—942no—
408#131—942no—
409C2×M16—471no—
410C2×M16—471no—
411C2×M16—471no—
412C2×M16—941no—
413C2×M16—941no—
414C2×M16C22×D4711yescharacteristic
415C22×C8—471no—
416C22×C8—941no—
417C22×C8—941no—
418C22×C8C22×D4711yescharacteristic
419C4×Q8—21no—
420C4×Q8—21no—
421C4×Q8—941no—
422C4×Q8—941no—
423C4×D4—21no—
424C4×D4—21no—
425C4×D4—41no—
426C4×D4—41no—
427C4×D4—941no—
428C4×D4—941no—
429C4×D4—1881no—
430C4×D4—1881no—
431#117—21no—
432#117—21no—
433#117—471no—
434#117—941no—
435#117—941no—
436#117C22×D4711yescharacteristic
437C2×C4⋊C4—21no—
438C2×C4⋊C4—21no—
439C2×C4⋊C4—471no—
440C2×C4⋊C4—471no—
441C2×C4⋊C4—941no—
442C2×C4⋊C4—941no—
443C2×C4⋊C4—1881no—
444C2×C4⋊C4C22×D4711yescharacteristic
445C2×C4⋊C4C22×D4711yescharacteristic
446C2×C42—471no—
447#108—474no—
448#108—942no—
449#107—471no—
450#107—471no—
451#107—941no—
452#107—941no—
453#107—941no—
454#107—941no—
455#107—941no—
456#107—941no—
457#107—941no—
458#107—941no—
459#106—21no—
460#106—21no—
461#106—471no—
462#106—471no—
463#106—941no—
464#106—941no—
465#106—941no—
466#106—941no—
467#106—941no—
468#106—941no—
469#104—474no—
470#104—474no—
471#103—21no—
472#103—21no—
473#103—941no—
474#103—941no—
475#102—21no—
476#102—21no—
477#102—941no—
478#102—941no—
479#97—21no—
480#97—474no—
481#97—941no—
482C4×C8—21no—
483C4×C8—941no—
484#95—1881no—
485#41—21no—
486#41—21no—
487#41—21no—
488#41—21no—
489#41C2×Dic9412yes—
490#41C2×Dic9412yes—
491#41C2×C4×D4712yes—
492#41C2×C4×D4712yes—
493C2×Q8C2×C22⋊D4711yescharacteristic
494C2×Q8C2×C22⋊D4711yescharacteristic
495C2×D4—21no—
496C2×D4—21no—
497C2×D4C2×C22⋊D4711yescharacteristic
498C2×D4C2×C22⋊D4711yescharacteristic
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—41no—
504C22×C4—41no—
505C22×C4—941no—
506C22×C4—941no—
507C22×C4—941no—
508C22×C4—1881no—
509C22×C4—1881no—
510C22×C4C23×D4711yescharacteristic
511Q16Dic47⋊C414yescomplemented
512SD16Dic47⋊C414yescomplemented
513SD16Dic47⋊C414yescomplemented
514D8Dic47⋊C414yescomplemented
515M16—21no—
516M16—21no—
517M16—472no—
518M16—472no—
519M16—944no—
520M16—944no—
521M16—941no—
522M16—941no—
523M16—941no—
524M16—941no—
525C2×C8—21no—
526C2×C8—472no—
527C2×C8—471no—
528C2×C8—471no—
529C2×C8—472no—
530C2×C8—944no—
531C2×C8—944no—
532C2×C8—941no—
533C2×C8—941no—
534C2×C8—941no—
535C2×C8C2×Dic9412yes—
536C2×C8C2×C4×D4712yes—
537C2×C8C2×C22⋊D4711yescharacteristic
538C2×C8C2×C22⋊D4711yescharacteristic
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—41no—
544C4⋊C4—41no—
545C4⋊C4—471no—
546C4⋊C4—471no—
547C4⋊C4—941no—
548C4⋊C4—941no—
549C4⋊C4—941no—
550C4⋊C4—941no—
551C4⋊C4—3762no—
552C4⋊C4—1881no—
553C4⋊C4—1881no—
554C4⋊C4D4×D4711yescharacteristic
555C4⋊C4D4×D4711yescharacteristic
556C22⋊C4—21no—
557C22⋊C4—41no—
558C22⋊C4—41no—
559C22⋊C4—941no—
560C22⋊C4—1881no—
561C22⋊C4—1881no—
562C42—21no—
563C42—41no—
564C42—41no—
565C42—472no—
566C42—941no—
567C42—941no—
568C42—1881no—
569C42—1881no—
570C23—21no—
571C23—21no—
572C23#1309406911yescharacteristic
573Q8—21no—
574Q8—21no—
575Q8C2×Dic47⋊C412yes—
576Q8C2×Dic47⋊C412yes—
577D4—22no—
578D4—22no—
579D4—22no—
580D4—22no—
581D4—21no—
582D4—21no—
583D4C2×Dic47⋊C412yes—
584D4C2×Dic47⋊C412yes—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—161no—
598C2×C4—161no—
599C2×C4—941no—
600C2×C4—941no—
601C2×C4—942no—
602C2×C4—3761no—
603C2×C4—3761no—
604C2×C4—1881no—
605C2×C4—1881no—
606C2×C4—1881no—
607C2×C4—1881no—
608C2×C4—7521no—
609C2×C4—7521no—
610C2×C4C22×C4×D4711yescharacteristic
611C2×C4C2×D4×D4711yescharacteristic
612C2×C4C22×C22⋊D4711yescharacteristic
613C2×C4C22×Dic9411yescharacteristic
614C2×C4#41×D4711yescharacteristic
615C2×C4#1309407111yescharacteristic
616C8—41no—
617C8—472no—
618C8—472no—
619C8—942no—
620C8—942no—
621C8—1881no—
622C8C2×Dic47⋊C412yes—
623C8C2×Dic47⋊C412yes—
624C22—21no—
625C22—21no—
626C22—22no—
627C22—22no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22#1568960211yescharacteristic
633C22D4×C22⋊D4711yescharacteristic
634C22#1568983211yescharacteristic
635C4—22no—
636C4—22no—
637C4—41no—
638C4—41no—
639C4—81no—
640C4—81no—
641C4—3761no—
642C4—3761no—
643C4—1882no—
644C4—1881no—
645C4—1881no—
646C4C22×Dic47⋊C411yescharacteristic
647C4C4×D4×D4711yescharacteristic
648C4#1568977211yescharacteristic
649C4D4×Dic9411yescharacteristic
650C2—21no—
651C2—21no—
652C2—42no—
653C2—42no—
654C2D4×Dic47⋊C411yescharacteristic
655C2#2025515011yescharacteristic
656C2#2025516111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2952C2
socleC22×C47D42
prime radicalC2×#135×C47C22
cosocleC22×C4×C47C24
hypercenter#2952D47
hypocenterC47#22693
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2952

Complements

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

NH ≅ G/Ntype
3. #20251986C2semidirect
4. #20251981C2semidirect
5. #20244173C2semidirect
6. #20244166C2semidirect
7. #20243621C2semidirect
8. #20243619C2semidirect
9. #20243614C2semidirect
10. #20243612C2semidirect
11. #20243553C2semidirect
12. #20243450C2semidirect
13. #20243445C2semidirect
14. #20243273C2semidirect
20. #15689245C4semidirect
70. #13094031C2×C4semidirect
71. #13094031C2×C4semidirect
95. #13093960C2×C4semidirect
115. #13093951C2×C4semidirect
116. #13093951C2×C4semidirect
122. #13093934C2×C4semidirect
182. Q16×C47C4⋊C4semidirect
183. SD16×C47C4⋊C4semidirect
184. SD16×C47C4⋊C4semidirect
185. D8×C47C4⋊C4semidirect
396. C47#22693semidirect
511. Q16Dic47⋊C4semidirect
512. SD16Dic47⋊C4semidirect
513. SD16Dic47⋊C4semidirect
514. D8Dic47⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#156892452
#202522642
#202519862
#202519812
#202441732
#202441662
#202436212
#202436192
#202436142
#202436122
#202435532
#202434502
#202434452
#202432732
C47×#29522
#2269347