Finite Group Databasebeta

← #26214796

Subgroups — #26214796

Order 12032.

1130 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689225C211yesmaximal, characteristic
2C2×#15689222C211yesmaximal, characteristic
3C2×#15689210C211yesmaximal, characteristic
4#20245752C211yesmaximal, characteristic, complemented
5#20245748C211yesmaximal, characteristic
6#20243288C211yesmaximal, characteristic, complemented
7#107×Dic47C211yesmaximal, characteristic, complemented
8#20243217C211yesmaximal, characteristic, complemented
9#20243212C211yesmaximal, characteristic
10C4×#13093947C211yesmaximal, characteristic, complemented
11#20243207C211yesmaximal, characteristic, complemented
12C4×#13093944C211yesmaximal, characteristic
13#20243202C211yesmaximal, characteristic
14#20243076C211yesmaximal, characteristic, complemented
15C47×#1776C211yesmaximal, characteristic, complemented
16C22×#10625325C2211yescharacteristic
17C2×C4×Dic94C2211yescharacteristic
18C2×#13093989C2211yescharacteristic
19C2×#13093989C2211yescharacteristic
20C2×C4×D188C2211yescharacteristic
21C2×#13093986C2211yescharacteristic
22C2×#13093986C2211yescharacteristic
23#15689225C418yescomplemented
24#15689222C418yescomplemented
25#15689210C418yescomplemented
26C2×#13093958C2211yescharacteristic
27C2×#13093949C2211yescharacteristic
28C2×#13093947C2211yescharacteristic
29C2×#13093947C2211yescharacteristic
30C2×#13093946C2211yescharacteristic
31C2×#13093944C2211yescharacteristic
32C2×#13093944C2211yescharacteristic
33C2×#13093934C2211yescharacteristic
34C2×#13093934C2211yescharacteristic
35C2×C4×C47⋊C8C2211yescharacteristic
36C2×C8×Dic47C2211yescharacteristic
37C2×C8×Dic47C2211yescharacteristic
38C4⋊C4×Dic47C2211yescharacteristic
39C4⋊C4×Dic47C2211yescharacteristic
40C4⋊C4×Dic47C2211yescharacteristic
41C4⋊C4×Dic47C2211yescharacteristic
42C4×Dic47⋊C4—21no—
43C4×Dic47⋊C4—21no—
44#15688974—21no—
45C4×(C2×D47)⋊C4—21no—
46C4×(C2×D47)⋊C4—21no—
47#15688970—21no—
48#15688969C2211yescharacteristic
49#15688968—21no—
50#95×D47—21no—
51#15688885C2211yescharacteristic
52#15688885C2211yescharacteristic
53#15688885C2211yescharacteristic
54#15688885C2211yescharacteristic
55C2×#107×C47C2211yescharacteristic
56C2×#106×C47C2211yescharacteristic
57C2×#97×C47C2211yescharacteristic
58C4×C4⋊C4×C47C2211yescharacteristic
59C4×C4⋊C4×C47C2211yescharacteristic
60C47×#340C2211yescharacteristic
61C47×#340C2211yescharacteristic
62C22×Dic94C2311yescharacteristic
63C22×D188C2311yescharacteristic
64C22×C4×D47—21no—
65C22×C4×D47—21no—
66C22×C4×D47—21no—
67C2×#10625325C2×C414yes—
68C2×#10625325C2×C414yes—
69C2×#10625325C2×C414yes—
70C22×C47⋊C8C2311yescharacteristic
71C22×C47⋊C8C2311yescharacteristic
72C4×Dic94—22no—
73C4×Dic94C2×C414yescomplemented
74#13093989—22no—
75#13093989—22no—
76#13093989C2×C414yescomplemented
77#13093989C2×C414yescomplemented
78C4×D188—22no—
79C4×D188C2×C414yes—
80#13093986—22no—
81#13093986—22no—
82#13093986C2×C414yes—
83#13093986C2×C414yes—
84C2×C4⋊Dic47C2311yescharacteristic
85C2×C4⋊Dic47C2311yescharacteristic
86C2×Dic47⋊C4—21no—
87C2×Dic47⋊C4—21no—
88C2×Dic47⋊C4—21no—
89C2×(C2×D47)⋊C4—21no—
90C2×(C2×D47)⋊C4—21no—
91C2×(C2×D47)⋊C4—21no—
92C2×C4×Dic47—21no—
93C2×C4×Dic47—21no—
94C2×C4×Dic47—21no—
95C2×C4×Dic47—21no—
96C2×C4×Dic47—21no—
97C2×C4×Dic47—21no—
98C2×C4×Dic47—21no—
99C2×C4×Dic47C2311yescharacteristic
100C2×C4×Dic47C2311yescharacteristic
101#13093958C2×C414yescomplemented
102#13093949C2×C414yescomplemented
103#13093947C2×C414yescomplemented
104#13093947C2×C414yescomplemented
105#13093946C2×C414yes—
106#13093944C2×C414yes—
107#13093944C2×C414yes—
108#13093934—22no—
109#13093934C2×C414yescomplemented
110C4×C47⋊C8—22no—
111C8×Dic47—22no—
112C8×Dic47C2×C414yescomplemented
113C47⋊#95—21no—
114C47⋊#95—21no—
115C47⋊#95—21no—
116C47⋊#95—21no—
117C47⋊#95D411yescharacteristic
118C47⋊#95D411yescharacteristic
119C47⋊#95D411yescharacteristic
120C47⋊#95D411yescharacteristic
121C22×C8×C47C2311yescharacteristic
122C22×C8×C47C2311yescharacteristic
123C2×C4⋊C4×C47C2311yescharacteristic
124C2×C4⋊C4×C47C2311yescharacteristic
125C2×C4⋊C4×C47C2311yescharacteristic
126C2×C4⋊C4×C47C2311yescharacteristic
127C2×C42×C47—21no—
128C2×C42×C47—21no—
129C2×C42×C47C2311yescharacteristic
130#107×C47C2×C414yescomplemented
131#106×C47C2×C414yescomplemented
132#97×C47C2×C414yescomplemented
133#95×C47—21no—
134#95×C47—21no—
135C23×D47—21no—
136C22×C4×C47—21no—
137C22×C4×C47—21no—
138C22×C4×C47—21no—
139C22×C4×C47—21no—
140C22×C4×C47C2×D411yescharacteristic
141C22×C4×C47C2×D411yescharacteristic
142C22×C4×C47C2411yescharacteristic
143C22×Dic47—21no—
144C22×Dic47—21no—
145C22×Dic47—21no—
146C22×Dic47C2×D411yescharacteristic
147C22×Dic47C2×D411yescharacteristic
148C22×Dic47C2×D411yescharacteristic
149C22×Dic47C2×D411yescharacteristic
150C2×D188—21no—
151C2×D188—21no—
152C2×D188—21no—
153C2×D188C22×C412yes—
154C2×D188C22×C412yes—
155C2×D188C22×C412yes—
156C2×C4×D47—24no—
157C2×C4×D47—24no—
158C2×C4×D47—24no—
159C2×C4×D47—81no—
160C2×C4×D47—81no—
161C2×C4×D47—81no—
162C2×C4×D47—81no—
163C2×C4×D47—81no—
164C2×C4×D47—81no—
165C2×Dic94—21no—
166C2×Dic94—21no—
167C2×Dic94—21no—
168C2×Dic94C22×C412yes—
169C2×Dic94C22×C412yes—
170C2×Dic94C22×C412yes—
171C2×C8×C47—21no—
172C2×C8×C47—21no—
173C2×C8×C47C22×C412yes—
174C2×C8×C47C22×C412yes—
175C2×C8×C47C22×C412yes—
176C2×C8×C47C22×C412yes—
177C4⋊C4×C47—21no—
178C4⋊C4×C47—21no—
179C4⋊C4×C47—21no—
180C4⋊C4×C47—21no—
181C4⋊C4×C47C22×C412yes—
182C4⋊C4×C47C22×C412yes—
183C4⋊C4×C47C22×C412yes—
184C4⋊C4×C47C22×C412yes—
185C42×C47—81no—
186C42×C47—81no—
187C42×C47C22×C412yes—
188C42×C47#4112yes—
189(C2×D47)⋊C4—24no—
190(C2×D47)⋊C4—24no—
191(C2×D47)⋊C4—24no—
192C4⋊Dic47—21no—
193C4⋊Dic47—21no—
194C4⋊Dic47C22×C412yes—
195C4⋊Dic47C22×C412yes—
196Dic47⋊C4—24no—
197Dic47⋊C4—24no—
198Dic47⋊C4—24no—
199C4×Dic47—24no—
200C4×Dic47—24no—
201C4×Dic47—24no—
202C4×Dic47—81no—
203C4×Dic47—81no—
204C4×Dic47—81no—
205C4×Dic47—81no—
206C4×Dic47C22×C412yes—
207C4×Dic47C22×C412yes—
208C4×Dic47#4112yes—
209C4×Dic47#4112yes—
210C2×C47⋊C8—21no—
211C2×C47⋊C8—21no—
212C2×C47⋊C8—21no—
213C2×C47⋊C8—21no—
214C2×C47⋊C8C22×C412yes—
215C2×C47⋊C8C22×C412yes—
216#10625325C42116yescomplemented
217C23×C47C22×D411yescharacteristic
218C22×D47—22no—
219C22×D47—22no—
220C22×D47—22no—
221C22×D47—41no—
222C22×D47—41no—
223C22×D47—41no—
224C22×D47—41no—
225C2×C4×C47—21no—
226C2×C4×C47—21no—
227C2×C4×C47—21no—
228C2×C4×C47—21no—
229C2×C4×C47—22no—
230C2×C4×C47—22no—
231C2×C4×C47—22no—
232C2×C4×C47—22no—
233C2×C4×C47—41no—
234C2×C4×C47—41no—
235C2×C4×C47—41no—
236C2×C4×C47—41no—
237C2×C4×C47—41no—
238C2×C4×C47—41no—
239C2×C4×C47—41no—
240C2×C4×C47—41no—
241C2×C4×C47C4×D412yes—
242C2×C4×C47C4×D412yes—
243C2×C4×C47C23×C411yescharacteristic
244C2×C4×C47C23×C411yescharacteristic
245C2×C4×C47C23×C411yescharacteristic
246C2×C4×C47C2×#4111yescharacteristic
247C2×C4×C47C2×#4111yescharacteristic
248C2×C4×C47C2×#4111yescharacteristic
249C2×Dic47—22no—
250C2×Dic47—22no—
251C2×Dic47—22no—
252C2×Dic47—21no—
253C2×Dic47—21no—
254C2×Dic47—21no—
255C2×Dic47—21no—
256C2×Dic47—21no—
257C2×Dic47—21no—
258C2×Dic47—21no—
259C2×Dic47—21no—
260C2×Dic47—22no—
261C2×Dic47—22no—
262C2×Dic47—41no—
263C2×Dic47—41no—
264C2×Dic47—41no—
265C2×Dic47—41no—
266C2×Dic47C4×D412yes—
267C2×Dic47C4×D412yes—
268C2×Dic47C4×D412yes—
269C2×Dic47C4×D412yes—
270D188—22no—
271D188—22no—
272D188—22no—
273D188C2×C4214yes—
274C4×D47—84no—
275C4×D47—84no—
276C4×D47—84no—
277Dic94—22no—
278Dic94—22no—
279Dic94—22no—
280Dic94C2×C4214yes—
281C8×C47—22no—
282C8×C47C2×C4214yes—
283C47⋊C8—22no—
284C47⋊C8—22no—
285#20504—471nomaximal, Sylow
286C22×C47C2×C4×D411yescharacteristic
287C22×C47C2×C4×D411yescharacteristic
288C22×C47C2×C4×D411yescharacteristic
289C22×C47#57611yescharacteristic
290C22×C47#57711yescharacteristic
291C22×C47#57711yescharacteristic
292C22×C47#57811yescharacteristic
293C2×D47—24no—
294C2×D47—42no—
295C2×D47—42no—
296C2×D47—42no—
297C2×D47—42no—
298C2×D47—42no—
299C2×D47—42no—
300C4×C47—22no—
301C4×C47—22no—
302C4×C47—42no—
303C4×C47—42no—
304C4×C47—42no—
305C4×C47—42no—
306C4×C47C22×C4211yescharacteristic
307C4×C47C4×#4111yescharacteristic
308C4×C47C4×#4111yescharacteristic
309C4×C47C4×#4111yescharacteristic
310Dic47—22no—
311Dic47—22no—
312Dic47—22no—
313Dic47—22no—
314Dic47—24no—
315Dic47—42no—
316Dic47—42no—
317C2×#440—471no—
318C2×#440—471no—
319C2×C4×SD16—471no—
320#2276—471no—
321#2275—471no—
322#1776—471no—
323#1776D4711yescharacteristic, complemented
324C4×#107—471no—
325#1763—471no—
326#1762—471no—
327C4×#103—471no—
328C4×#103—471no—
329C4×#102—471no—
330C4×#102—471no—
331#1726—471no—
332C2×C47C42×D411yescharacteristic
333C2×C47#294611yescharacteristic
334C2×C47#294611yescharacteristic
335C2×C47#294711yescharacteristic
336C2×C47#294711yescharacteristic
337C2×C47#294911yescharacteristic
338C2×C47#295111yescharacteristic
339D47—44no—
340C22×SD16—471no—
341C2×C4×Q8—471no—
342C2×C4×Q8—471no—
343C2×C4×Q8—471no—
344C2×C4×D4—471no—
345C2×C4×D4—471no—
346C2×C4×D4—471no—
347#440—478no—
348#440—478no—
349C4×SD16—478no—
350C2×#107—471no—
351C2×#107C2×D4711yescharacteristic
352C2×#106C2×D4711yescharacteristic
353C2×#103—471no—
354C2×#103—471no—
355C2×#103—471no—
356C2×#102—471no—
357C2×#102—471no—
358C2×#102—471no—
359C2×#97—471no—
360C2×#97—471no—
361C2×#97C2×D4711yescharacteristic
362C2×C4×C8—471no—
363C2×C4×C8—471no—
364C2×C4×C8—471no—
365C4×C4⋊C4—471no—
366C4×C4⋊C4—471no—
367C4×C4⋊C4—471no—
368C4×C4⋊C4—471no—
369C4×C4⋊C4—941no—
370C4×C4⋊C4—941no—
371C4×C4⋊C4—941no—
372C4×C4⋊C4C2×D4711yescharacteristic
373C4×C4⋊C4C2×D4711yescharacteristic
374C4×C22⋊C4—941no—
375C4×C22⋊C4—941no—
376C4×C22⋊C4—941no—
377#376—471no—
378#376—941no—
379C2×#95—941no—
380#340—471no—
381#340—471no—
382#340—471no—
383#340—471no—
384#340C2×D4711yescharacteristic
385#340C2×D4711yescharacteristic
386C47#2050411yescharacteristic, complemented, Sylow
387C22×Q8—471no—
388C22×D4—471no—
389C23×C4—941no—
390C23×C4—941no—
391C23×C4—941no—
392C2×SD16—474no—
393C2×SD16—474no—
394C2×SD16—474no—
395C22×C8—471no—
396C22×C8—471no—
397C22×C8C22×D4711yescharacteristic
398C22×C8C22×D4711yescharacteristic
399C4×Q8—474no—
400C4×Q8—474no—
401C4×Q8—474no—
402C4×Q8—942no—
403C4×Q8—942no—
404C4×Q8—942no—
405C4×D4—474no—
406C4×D4—474no—
407C4×D4—474no—
408C4×D4—942no—
409C4×D4—942no—
410C4×D4—942no—
411C2×C4⋊C4—471no—
412C2×C4⋊C4—471no—
413C2×C4⋊C4—941no—
414C2×C4⋊C4—941no—
415C2×C4⋊C4—941no—
416C2×C4⋊C4C22×D4711yescharacteristic
417C2×C4⋊C4C22×D4711yescharacteristic
418C2×C4⋊C4C22×D4711yescharacteristic
419C2×C4⋊C4C22×D4711yescharacteristic
420C2×C22⋊C4—941no—
421C2×C22⋊C4—941no—
422C2×C22⋊C4—941no—
423C2×C42—21no—
424C2×C42—21no—
425C2×C42—471no—
426C2×C42—471no—
427C2×C42—941no—
428C2×C42—941no—
429C2×C42—941no—
430C2×C42—941no—
431C2×C42—941no—
432C2×C42—941no—
433C2×C42—941no—
434C2×C42C22×D4711yescharacteristic
435#107—474no—
436#107C4×D4714yescomplemented
437#106C4×D4714yescomplemented
438#103—474no—
439#103—474no—
440#103—474no—
441#102—474no—
442#102—474no—
443#102—474no—
444#97—474no—
445#97—942no—
446#97C4×D4714yescomplemented
447C4×C8—474no—
448C4×C8—942no—
449C4×C8—942no—
450#95—21no—
451#95—21no—
452#95—471no—
453#95—471no—
454#95—471no—
455#95—471no—
456#95—941no—
457#95—941no—
458#95—941no—
459#95—941no—
460C24—941no—
461C2×Q8—472no—
462C2×Q8—472no—
463C2×Q8—472no—
464C2×Q8—941no—
465C2×Q8—941no—
466C2×Q8—941no—
467C2×D4—472no—
468C2×D4—472no—
469C2×D4—472no—
470C2×D4—941no—
471C2×D4—941no—
472C2×D4—941no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—471no—
478C22×C4—471no—
479C22×C4—471no—
480C22×C4—471no—
481C22×C4—944no—
482C22×C4—944no—
483C22×C4—941no—
484C22×C4—941no—
485C22×C4—941no—
486C22×C4—944no—
487C22×C4—3761no—
488C22×C4—3761no—
489C22×C4—3761no—
490C22×C4—3761no—
491C22×C4—3761no—
492C22×C4—3761no—
493C22×C4D4×D4711yescharacteristic
494C22×C4D4×D4711yescharacteristic
495C22×C4C23×D4711yescharacteristic
496SD16—4716no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—472no—
500C2×C8—472no—
501C2×C8—941no—
502C2×C8—941no—
503C2×C8—941no—
504C2×C8—941no—
505C2×C8C2×C4×D4712yes—
506C2×C8C2×C4×D4712yes—
507C2×C8C2×C4×D4712yes—
508C2×C8C2×C4×D4712yes—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—472no—
514C4⋊C4—472no—
515C4⋊C4—944no—
516C4⋊C4—944no—
517C4⋊C4—944no—
518C4⋊C4—941no—
519C4⋊C4—941no—
520C4⋊C4C2×C4×D4712yes—
521C4⋊C4C2×C4×D4712yes—
522C4⋊C4C2×C4×D4712yes—
523C4⋊C4C2×C4×D4712yes—
524C22⋊C4—944no—
525C22⋊C4—944no—
526C22⋊C4—944no—
527C42—81no—
528C42—81no—
529C42—472no—
530C42—472no—
531C42—472no—
532C42—472no—
533C42—944no—
534C42—944no—
535C42—944no—
536C42—3761no—
537C42—3761no—
538C42—3761no—
539C42—3761no—
540C42C2×C4×D4712yes—
541C42#1062535412yes—
542C23—942no—
543C23—942no—
544C23—942no—
545C23—1881no—
546C23—1881no—
547C23—1881no—
548C23—1881no—
549C23C2×D4×D4711yescharacteristic
550Q8—474no—
551Q8—942no—
552Q8—942no—
553Q8—942no—
554D4—474no—
555D4—942no—
556D4—942no—
557D4—942no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—472no—
575C2×C4—472no—
576C2×C4—472no—
577C2×C4—472no—
578C2×C4—942no—
579C2×C4—942no—
580C2×C4—942no—
581C2×C4—941no—
582C2×C4—941no—
583C2×C4—941no—
584C2×C4—941no—
585C2×C4—941no—
586C2×C4—941no—
587C2×C4—941no—
588C2×C4—941no—
589C2×C4—942no—
590C2×C4—942no—
591C2×C4—3764no—
592C2×C4—3764no—
593C2×C4—3764no—
594C2×C4—1881no—
595C2×C4—1881no—
596C2×C4—1881no—
597C2×C4—1881no—
598C2×C4#1309398612yes—
599C2×C4#1309398612yes—
600C2×C4C22×C4×D4711yescharacteristic
601C2×C4C22×C4×D4711yescharacteristic
602C2×C4C22×C4×D4711yescharacteristic
603C2×C4#41×D4711yescharacteristic
604C2×C4#41×D4711yescharacteristic
605C2×C4C2×#1062535411yescharacteristic
606C8—22no—
607C8—942no—
608C8—942no—
609C8C42×D4714yes—
610C22—944no—
611C22—1882no—
612C22—1882no—
613C22—1882no—
614C22—1882no—
615C22—1882no—
616C22—1882no—
617C22C4×D4×D4711yescharacteristic
618C22C4×D4×D4711yescharacteristic
619C22C2×#1309398611yescharacteristic
620C22#1568992711yescharacteristic
621C22#1568993211yescharacteristic
622C22#1568993311yescharacteristic
623C22#1568993911yescharacteristic
624C4—22no—
625C4—22no—
626C4—42no—
627C4—42no—
628C4—42no—
629C4—42no—
630C4—942no—
631C4—942no—
632C4—942no—
633C4—942no—
634C4—944no—
635C4—1882no—
636C4—1882no—
637C4C2×C42×D4711yescharacteristic
638C4#1568959111yescharacteristic
639C4C4×#1062535411yescharacteristic
640C4C4×#1062535411yescharacteristic
641C2—1884no—
642C2C4×#1309398611yescharacteristic
643C2#2025221711yescharacteristic
644C2#2025221811yescharacteristic
645C2#2025222911yescharacteristic
646C2#2025223011yescharacteristic
647C2#2025224411yescharacteristic
648C2#2025226011yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC4×C47C22×C42
FrattiniC22×C4C23×D47
FittingC47×#1776C2
socleC23×C47C22×D4
prime radicalC22×D188C23
cosocleC22×C4×C47C24
hypercenter#1776D47
hypocenterC47#20504
LCSG ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1776

Complements

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

NH ≅ G/Ntype
4. #20245752C2semidirect
6. #20243288C2semidirect
7. #107×Dic47C2semidirect
8. #20243217C2semidirect
10. C4×#13093947C2semidirect
11. #20243207C2semidirect
14. #20243076C2semidirect
15. C47×#1776C2semidirect
23. #15689225C4semidirect
24. #15689222C4semidirect
25. #15689210C4semidirect
73. C4×Dic94C2×C4semidirect
76. #13093989C2×C4semidirect
77. #13093989C2×C4semidirect
101. #13093958C2×C4semidirect
102. #13093949C2×C4semidirect
103. #13093947C2×C4semidirect
104. #13093947C2×C4semidirect
109. #13093934C2×C4semidirect
112. C8×Dic47C2×C4semidirect
130. #107×C47C2×C4semidirect
131. #106×C47C2×C4semidirect
132. #97×C47C2×C4semidirect
216. #10625325C42semidirect
323. #1776D47semidirect
386. C47#20504semidirect
436. #107C4×D47semidirect
437. #106C4×D47semidirect
446. #97C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156892252
C2×#156892222
C2×#156892102
#202457522
#202457482
#202432882
#107×Dic472
#202432172
#202432122
C4×#130939472
#202432072
C4×#130939442
#202432022
#202430762
C47×#17762
#2050447