Finite Group Databasebeta

← #27080100

Subgroups — #27080100

Order 12032.

2038 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20256587C211yesmaximal, characteristic
2#20252056C213yesmaximal, complemented
3#20252053C213yesmaximal, complemented
4#20252045C213yesmaximal, complemented
5#20252045C213yesmaximal, complemented
6#20252038C213yesmaximal, complemented
7#20252031C213yesmaximal, complemented
8#20252025C213yesmaximal, complemented
9C2×#15689183C213yesmaximal, complemented
10#20251956C211yesmaximal, characteristic, complemented
11#20251835C211yesmaximal, characteristic
12#20251835C211yesmaximal, characteristic
13#20251814C211yesmaximal, characteristic, complemented
14#131×Dic47C211yesmaximal, characteristic, complemented
15C47×#2906C211yesmaximal, characteristic, complemented
16C2×#41×D47C2211yescharacteristic
17C2×#13094031C2211yescharacteristic
18C2×#13094031C2211yescharacteristic
19#15689605C411yescharacteristic
20#15689605C411yescharacteristic
21#15689602C413yes—
22#15689602C413yes—
23#15689599C411yescharacteristic
24#15689599C411yescharacteristic
25#15689595C413yes—
26#15689595C413yes—
27#41×Dic47C2211yescharacteristic
28C4×#10625350C2213yes—
29#15689591C2213yes—
30C2×#13093981C2213yes—
31#117×D47C2213yes—
32#15689183C2211yescharacteristic
33#15689183C2213yes—
34#15689183C2213yes—
35#15689183C2213yes—
36#15689183C2216yes—
37#15689183C2213yescomplemented
38#15689183C2213yescomplemented
39#15689183C2213yescomplemented
40#15689183C2213yescomplemented
41#15689183C2216yescomplemented
42#15689183C2216yescomplemented
43#15689179C2216yescomplemented
44#15689179C2216yescomplemented
45#15689175C2216yescomplemented
46#15689175C2216yescomplemented
47#15689150C2213yescomplemented
48#15689150C2213yescomplemented
49#15689147C2213yescomplemented
50C2×C4⋊(C47⋊C8)C2213yescomplemented
51#15689091C2213yes—
52#15689091C2213yes—
53#15689089C2213yes—
54#15689089C2213yes—
55#15689086C2213yes—
56#15689086C2213yes—
57C2×(C2×D47)⋊C8C2213yes—
58C2×(C2×D47)⋊C8C2213yes—
59#15689081C2213yescomplemented
60#15689081C2213yescomplemented
61#15689078C2213yescomplemented
62#15689078C2213yescomplemented
63M16×Dic47C2213yescomplemented
64M16×Dic47C2213yescomplemented
65C2×#13093934C2213yescomplemented
66C2×C8×Dic47C2213yescomplemented
67C2×#131×C47C2211yescharacteristic
68C2×#131×C47C2211yescharacteristic
69C4×#41×C47C2211yescharacteristic
70C47×#424C2213yescomplemented
71C47×#424C2213yescomplemented
72C47×#423C2213yescomplemented
73C2×C4⋊C8×C47C2213yescomplemented
74C2×#10625354C2×C411yescharacteristic
75C2×#10625350C2313yes—
76C2×#10625352C2×C413yes—
77#41×D47—24no—
78#41×D47—24no—
79C2×Q8×D47C2×C411yescharacteristic
80C2×D4×D47C2×C413yes—
81C22×C4×D47C2313yes—
82#13094031—24no—
83#13094031—24no—
84C2×#10625329C2313yes—
85C2×#10625329C2313yes—
86C22×C47⋊C8C2313yes—
87C22×C47⋊C8C2313yes—
88Q8×Dic47C2×C411yescharacteristic
89Q8×Dic47C2×C411yescharacteristic
90C4×Dic94C2×C413yes—
91C4×Dic94C2×C413yes—
92#13093989C2×C413yes—
93#13093989C2×C413yes—
94C4×D188C2×C413yes—
95C4×D188C2×C413yes—
96D4×Dic47C2×C413yes—
97D4×Dic47C2×C413yes—
98#13093986C2×C413yes—
99#13093986C2×C413yes—
100C4×C22⋊D47C2×C416yes—
101C4×C22⋊D47C2×C416yes—
102(C22⋊D47)⋊C4C2×C416yes—
103(C22⋊D47)⋊C4C2×C416yes—
104#13093983C2313yes—
105#13093982C2×C413yes—
106#13093982C2×C413yes—
107#13093981C2×C413yes—
108#13093981C2×C413yes—
109#13093981C2×C413yes—
110#13093981C2×C413yes—
111#13093981C2313yes—
112#13093981C2311yescharacteristic
113#13093981C2313yes—
114#13093981C2313yes—
115#13093980C2×C416yes—
116C2×Dic47⋊C4C2×C413yes—
117C2×Dic47⋊C4C2×C413yes—
118C4⋊C4×D47C2×C413yes—
119C4⋊C4×D47C2×C413yes—
120C2×(C2×D47)⋊C4C2×C413yes—
121C2×(C2×D47)⋊C4C2×C413yes—
122C22⋊C4×D47C2×C416yes—
123C2×C4×Dic47C2313yes—
124C42×D47C2316yes—
125C4⋊(C47⋊C8)C2311yescharacteristic
126C4⋊(C47⋊C8)C2313yes—
127C4⋊(C47⋊C8)C2313yes—
128C4⋊(C47⋊C8)C2313yes—
129C4⋊(C47⋊C8)C2316yes—
130(C2×D47)⋊C8C2311yescharacteristic
131(C2×D47)⋊C8C2311yescharacteristic
132(C2×D47)⋊C8C2313yes—
133(C2×D47)⋊C8C2313yes—
134(C2×D47)⋊C8C2313yes—
135(C2×D47)⋊C8C2313yes—
136(C2×D47)⋊C8C2313yes—
137(C2×D47)⋊C8C2313yes—
138(C2×D47)⋊C8C2316yes—
139(C2×D47)⋊C8C2316yes—
140#13093934C2311yescharacteristic
141#13093934C2313yes—
142#13093934C2313yes—
143#13093934C2313yes—
144#13093934C2316yes—
145C8×Dic47C2311yescharacteristic
146C8×Dic47C2313yes—
147C8×Dic47C2313yes—
148C8×Dic47C2313yes—
149C8×Dic47C2316yes—
150C2×#41×C47C2311yescharacteristic
151#131×C47—24no—
152#131×C47—24no—
153C2×M16×C47C2313yes—
154C2×M16×C47C2313yes—
155C22×C8×C47C2313yes—
156C22×C8×C47C2313yes—
157C4×Q8×C47C2×C411yescharacteristic
158C4×Q8×C47C2×C411yescharacteristic
159C4×D4×C47C2×C413yes—
160C4×D4×C47C2×C413yes—
161#117×C47C2313yes—
162C2×C42×C47C2313yes—
163C4⋊C8×C47C2313yes—
164C4⋊C8×C47C2316yes—
165C4⋊C8×C47C2313yes—
166C4⋊C8×C47C2311yescharacteristic
167C4⋊C8×C47C2313yes—
168C23×D47C22×C413yes—
169#41×C47#4114yes—
170#41×C47#4114yes—
171C2×Q8×C47C22×C411yescharacteristic
172C2×D4×C47C22×C413yes—
173C22×C4×C47C22×C413yes—
174C22×C4×C47C22×C413yes—
175C22×C4×C47C2413yes—
176C2×C22⋊D47C22×C416yes—
177C22×Dic47C22×C413yes—
178C22×Dic47C22×C413yes—
179C22×Dic47C22×C413yes—
180#10625354—24no—
181Q8×D47—24no—
182#10625352—212no—
183D4×D47—212no—
184#10625350—26no—
185#10625350—26no—
186C2×D188C22×C413yes—
187C2×C4×D47—26no—
188C2×C4×D47—26no—
189C2×C4×D47C22×C413yes—
190C2×C4×D47C22×C413yes—
191C2×C4×D47C22×C416yes—
192C2×C4×D47C22×C416yes—
193C2×C4×D47C2413yes—
194C2×C4×D47C2411yescharacteristic
195C2×Dic94C22×C413yes—
196M16×C47—26no—
197M16×C47—26no—
198C2×C8×C47—26no—
199C2×C8×C47—26no—
200C2×C8×C47C2411yescharacteristic
201C2×C8×C47C2411yescharacteristic
202C2×C8×C47C2413yes—
203C2×C8×C47C2413yes—
204C4⋊C4×C47C22×C413yes—
205C4⋊C4×C47C22×C413yes—
206C22⋊C4×C47C22×C416yes—
207C42×C47C22×C413yes—
208C42×C47C22×C413yes—
209C42×C47C2411yescharacteristic
210C42×C47C2413yes—
211C22⋊Dic47C22×C416yes—
212(C2×D47)⋊C4C22×C413yes—
213(C2×D47)⋊C4C22×C413yes—
214(C2×D47)⋊C4C22×C411yescharacteristic
215(C2×D47)⋊C4C22×C411yescharacteristic
216(C2×D47)⋊C4C22×C413yes—
217(C2×D47)⋊C4C22×C413yes—
218(C2×D47)⋊C4C22×C413yes—
219(C2×D47)⋊C4C22×C413yes—
220C4⋊Dic47C22×C413yes—
221C4⋊Dic47C22×C413yes—
222Dic47⋊C4C22×C413yes—
223Dic47⋊C4C22×C413yes—
224Dic47⋊C4C22×C411yescharacteristic
225Dic47⋊C4C22×C411yescharacteristic
226Dic47⋊C4C22×C413yes—
227Dic47⋊C4C22×C413yes—
228Dic47⋊C4C22×C413yes—
229Dic47⋊C4C22×C413yes—
230C4×Dic47C22×C416yes—
231C4×Dic47C22×C416yes—
232C4×Dic47C22×C413yes—
233C4×Dic47C22×C413yes—
234C4×Dic47C2413yes—
235C4×Dic47C2411yescharacteristic
236#10625329—26no—
237#10625329—26no—
238C2×C47⋊C8—26no—
239C2×C47⋊C8—26no—
240C2×C47⋊C8C2411yescharacteristic
241C2×C47⋊C8C2413yes—
242C2×C47⋊C8C2413yes—
243C2×C47⋊C8C2411yescharacteristic
244C23×C47C23×C413yes—
245C22×D47—26no—
246C22×D47—43no—
247C22×D47—43no—
248C22×D47C23×C411yescharacteristic
249C22×D47C23×C413yes—
250Q8×C47#11714yes—
251D4×C47#117112yes—
252C2×C4×C47—43no—
253C2×C4×C47—43no—
254C2×C4×C47C23×C413yes—
255C2×C4×C47C23×C413yes—
256C2×C4×C47C23×C411yescharacteristic
257C2×C4×C47C23×C411yescharacteristic
258C2×C4×C47C23×C413yes—
259C2×C4×C47C2×#4116yes—
260C2×C4×C47C2×#4116yes—
261C2×C4×C47C2511yescharacteristic
262C22⋊D47—212no—
263C2×Dic47—26no—
264C2×Dic47—43no—
265C2×Dic47—43no—
266C2×Dic47C23×C413yes—
267C2×Dic47C23×C411yescharacteristic
268C2×Dic47C23×C411yescharacteristic
269C2×Dic47C23×C413yes—
270C2×Dic47C23×C413yes—
271C2×Dic47C23×C411yescharacteristic
272D188—26no—
273C4×D47—26no—
274C4×D47—26no—
275C4×D47—21no—
276C4×D47—21no—
277C4×D47—23no—
278C4×D47—23no—
279C4×D47—46no—
280C4×D47—46no—
281Dic94—26no—
282C8×C47—21no—
283C8×C47—21no—
284C8×C47—23no—
285C8×C47—23no—
286C47⋊C8—21no—
287C47⋊C8—21no—
288C47⋊C8—23no—
289C47⋊C8—23no—
290#60129—471nomaximal, Sylow
291C22×C47—23no—
292C22×C47—23no—
293C22×C47C2×#11716yes—
294C22×C47C24×C411yescharacteristic
295C2×D47—21no—
296C2×D47—23no—
297C2×D47—23no—
298C2×D47—23no—
299C2×D47—21no—
300C2×D47—21no—
301C2×D47—86no—
302C4×C47—23no—
303C4×C47—23no—
304C4×C47—21no—
305C4×C47—21no—
306C4×C47C2×#11716yes—
307C4×C47C22×#4111yescharacteristic
308C4×C47C22×#4111yescharacteristic
309Dic47—23no—
310Dic47—21no—
311Dic47—23no—
312Dic47—23no—
313Dic47—21no—
314Dic47—21no—
315#3428—471no—
316#2925—473no—
317#2925—473no—
318#2924—473no—
319#2924—473no—
320#2923—473no—
321#2922—473no—
322#2921—473no—
323C2×#433—473no—
324#2906—471no—
325#2906D4711yescharacteristic, complemented
326#2880—471no—
327#2880—471no—
328#2875—471no—
329C4×#131—471no—
330C2×C47—26no—
331C2×C47C22×#11711yescharacteristic
332C2×C47#357311yescharacteristic
333C2×C47#357311yescharacteristic
334D47—43no—
335D47—41no—
336C22×#41—471no—
337C2×#131—471no—
338C2×#131—471no—
339C2×#131C2×D4711yescharacteristic
340C2×#131C2×D4711yescharacteristic
341#520—471no—
342#520—471no—
343#520—473no—
344#520—473no—
345#519—471no—
346#519—471no—
347#518—473no—
348#518—473no—
349C4×#41—471no—
350C4×#41—473no—
351C4×#41—473no—
352C4×#41C2×D4711yescharacteristic
353C2×#117—473no—
354C2×#117—473no—
355#433—471no—
356#433—473no—
357#433—473no—
358#433—473no—
359#433—473no—
360#433—473no—
361#433—473no—
362#433—473no—
363#433—476no—
364#433—476no—
365#433—476no—
366#432—476no—
367#432—476no—
368#431—476no—
369#431—476no—
370#424—473no—
371#424—473no—
372#424C2×D4713yescomplemented
373#424C2×D4713yescomplemented
374#423—473no—
375#423C2×D4713yescomplemented
376C2×C4⋊C8—473no—
377C2×C4⋊C8C2×D4713yescomplemented
378#408—473no—
379#408—473no—
380#408—473no—
381#408—473no—
382#407—473no—
383#407—473no—
384C2×C22⋊C8—473no—
385C2×C22⋊C8—473no—
386#405—473no—
387#405—473no—
388#405—473no—
389#405—473no—
390C4×M16—473no—
391C4×M16—473no—
392C2×#97—473no—
393C2×C4×C8—473no—
394C47#6012911yescharacteristic, complemented, Sylow
395C2×#41—471no—
396C2×#41—473no—
397C2×#41—473no—
398C2×#41—944no—
399C2×#41—944no—
400C2×#41C22×D4711yescharacteristic
401C22×Q8—471no—
402C22×D4—473no—
403C23×C4—473no—
404#131—24no—
405#131—24no—
406#131—944no—
407#131—944no—
408C2×M16—473no—
409C2×M16—473no—
410C2×M16C22×D4713yes—
411C2×M16C22×D4713yes—
412C22×C8—473no—
413C22×C8—473no—
414C22×C8C22×D4713yes—
415C22×C8C22×D4713yes—
416C4×Q8—471no—
417C4×Q8—471no—
418C4×Q8—473no—
419C4×Q8—473no—
420C4×Q8—473no—
421C4×Q8—473no—
422C4×Q8C4×D4711yescharacteristic
423C4×Q8C4×D4711yescharacteristic
424C4×D4—473no—
425C4×D4—473no—
426C4×D4—476no—
427C4×D4—476no—
428C4×D4—473no—
429C4×D4—473no—
430C4×D4—476no—
431C4×D4—476no—
432C4×D4—473no—
433C4×D4—473no—
434C4×D4C4×D4713yes—
435C4×D4C4×D4713yes—
436#117—473no—
437#117—473no—
438#117—473no—
439#117—476no—
440#117—473no—
441#117—473no—
442#117—473no—
443#117—473no—
444#117—473no—
445#117—471no—
446#117—473no—
447#117—473no—
448#117C22×D4713yes—
449C2×C4⋊C4—473no—
450C2×C4⋊C4—473no—
451C2×C4⋊C4—473no—
452C2×C4⋊C4—473no—
453C2×C22⋊C4—473no—
454C2×C22⋊C4—473no—
455C2×C22⋊C4—476no—
456C2×C42—473no—
457C2×C42—476no—
458C2×C42C22×D4713yes—
459C4⋊C8—471no—
460C4⋊C8—473no—
461C4⋊C8—473no—
462C4⋊C8—473no—
463C4⋊C8—476no—
464C4⋊C8C22×D4711yescharacteristic
465C4⋊C8C22×D4713yes—
466C4⋊C8C22×D4713yes—
467C4⋊C8C22×D4713yes—
468C4⋊C8C22×D4716yes—
469C22⋊C8—471no—
470C22⋊C8—471no—
471C22⋊C8—473no—
472C22⋊C8—473no—
473C22⋊C8—473no—
474C22⋊C8—473no—
475C22⋊C8—473no—
476C22⋊C8—473no—
477C22⋊C8—476no—
478C22⋊C8—476no—
479#97—471no—
480#97—473no—
481#97—473no—
482#97—473no—
483#97—476no—
484C4×C8—471no—
485C4×C8—473no—
486C4×C8—473no—
487C4×C8—473no—
488C4×C8—476no—
489C24—473no—
490#41—944no—
491#41—9412no—
492#41—946no—
493#41—946no—
494#41#1062535214yes—
495#41#1062535414yes—
496C2×Q8—473no—
497C2×Q8—944no—
498C2×Q8C2×C4×D4711yescharacteristic
499C2×D4—476no—
500C2×D4—473no—
501C2×D4—9412no—
502C2×D4C2×C4×D4713yes—
503C22×C4—473no—
504C22×C4—473no—
505C22×C4—473no—
506C22×C4—473no—
507C22×C4—473no—
508C22×C4—473no—
509C22×C4—471no—
510C22×C4—476no—
511C22×C4—476no—
512C22×C4—946no—
513C22×C4—946no—
514C22×C4C2×C4×D4713yes—
515C22×C4C2×C4×D4713yes—
516C22×C4C23×D4713yes—
517M16—26no—
518M16—26no—
519M16—946no—
520M16—946no—
521C2×C8—26no—
522C2×C8—26no—
523C2×C8—471no—
524C2×C8—473no—
525C2×C8—473no—
526C2×C8—471no—
527C2×C8—946no—
528C2×C8—946no—
529C2×C8C23×D4711yescharacteristic
530C2×C8C23×D4713yes—
531C2×C8C23×D4713yes—
532C2×C8C23×D4711yescharacteristic
533C4⋊C4—473no—
534C4⋊C4—473no—
535C4⋊C4—471no—
536C4⋊C4—471no—
537C4⋊C4—473no—
538C4⋊C4—473no—
539C4⋊C4—473no—
540C4⋊C4—473no—
541C4⋊C4—473no—
542C4⋊C4—473no—
543C4⋊C4C2×C4×D4713yes—
544C4⋊C4C2×C4×D4713yes—
545C22⋊C4—476no—
546C22⋊C4—473no—
547C22⋊C4—473no—
548C22⋊C4—471no—
549C22⋊C4—471no—
550C22⋊C4—473no—
551C22⋊C4—473no—
552C22⋊C4—473no—
553C22⋊C4—473no—
554C22⋊C4C2×C4×D4716yes—
555C42—473no—
556C42—473no—
557C42—476no—
558C42—476no—
559C42—473no—
560C42—471no—
561C42C2×C4×D4713yes—
562C42C2×C4×D4713yes—
563C42C23×D4713yes—
564C42C23×D4711yescharacteristic
565C23—471no—
566C23—473no—
567C23—946no—
568C23—1883no—
569C23—1883no—
570C23C22×C4×D4713yes—
571Q8—946no—
572Q8#1309398214yes—
573D4—9412no—
574D4—946no—
575D4#13093982112yes—
576C2×C4—43no—
577C2×C4—43no—
578C2×C4—471no—
579C2×C4—473no—
580C2×C4—471no—
581C2×C4—473no—
582C2×C4—473no—
583C2×C4—471no—
584C2×C4—946no—
585C2×C4—946no—
586C2×C4—946no—
587C2×C4—941no—
588C2×C4—943no—
589C2×C4—941no—
590C2×C4—943no—
591C2×C4—1886no—
592C2×C4—1886no—
593C2×C4—1883no—
594C2×C4—1883no—
595C2×C4C22×C4×D4713yes—
596C2×C4C22×C4×D4713yes—
597C2×C4C22×C4×D4711yescharacteristic
598C2×C4C22×C4×D4711yescharacteristic
599C2×C4C22×C4×D4713yes—
600C2×C4C2×#1062535216yes—
601C2×C4C2×#1062535416yes—
602C2×C4C24×D4711yescharacteristic
603C8—21no—
604C8—21no—
605C8—23no—
606C8—23no—
607C8—941no—
608C8—941no—
609C8—943no—
610C8—943no—
611C22—23no—
612C22—23no—
613C22—941no—
614C22—943no—
615C22—943no—
616C22—941no—
617C22—943no—
618C22—941no—
619C22—3766no—
620C22C2×#1309398216yes—
621C22C23×C4×D4711yescharacteristic
622C4—21no—
623C4—23no—
624C4—21no—
625C4—23no—
626C4—941no—
627C4—943no—
628C4—943no—
629C4—941no—
630C4—943no—
631C4—941no—
632C4C2×#1309398216yes—
633C4C22×#1062535211yescharacteristic
634C4C22×#1062535411yescharacteristic
635C2—26no—
636C2—1881no—
637C2—1883no—
638C2C22×#1309398211yescharacteristic
639C2#2025809911yescharacteristic
640C2#2025810111yescharacteristic
subgroupG/H
center, FrattiniC2×C4C24×D47
commutator, socleC22×C47C24×C4
FittingC47×#2906C2
prime radicalC2×#41×D47C22
cosocleC2×C4×C47C25
hypercenter#2906D47
hypocenterC47#60129
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ #2906

Complements

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

NH ≅ G/Ntype
2. #20252056C2semidirect
3. #20252053C2semidirect
4. #20252045C2semidirect
5. #20252045C2semidirect
6. #20252038C2semidirect
7. #20252031C2semidirect
8. #20252025C2semidirect
9. C2×#15689183C2semidirect
10. #20251956C2semidirect
13. #20251814C2semidirect
14. #131×Dic47C2semidirect
15. C47×#2906C2semidirect
37. #15689183C22semidirect
38. #15689183C22semidirect
39. #15689183C22semidirect
40. #15689183C22semidirect
41. #15689183C22semidirect
42. #15689183C22semidirect
43. #15689179C22semidirect
44. #15689179C22semidirect
45. #15689175C22semidirect
46. #15689175C22semidirect
47. #15689150C22semidirect
48. #15689150C22semidirect
49. #15689147C22semidirect
50. C2×C4⋊(C47⋊C8)C22semidirect
59. #15689081C22semidirect
60. #15689081C22semidirect
61. #15689078C22semidirect
62. #15689078C22semidirect
63. M16×Dic47C22semidirect
64. M16×Dic47C22semidirect
65. C2×#13093934C22semidirect
66. C2×C8×Dic47C22semidirect
70. C47×#424C22semidirect
71. C47×#424C22semidirect
72. C47×#423C22semidirect
73. C2×C4⋊C8×C47C22semidirect
325. #2906D47semidirect
372. #424C2×D47semidirect
373. #424C2×D47semidirect
375. #423C2×D47semidirect
377. C2×C4⋊C8C2×D47semidirect
394. C47#60129semidirect

Maximal subgroups

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

subgroupindex
#202565872
#202520562
#202520532
#202520452
#202520382
#202520312
#202520252
C2×#156891832
#202519562
#202518352
#202518142
#131×Dic472
C47×#29062
#6012947