Finite Group Databasebeta

← #26232911

Subgroups — #26232911

Order 12032.

1250 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689788C211yesmaximal, characteristic
2C2×#15689251C211yesmaximal, characteristic, complemented
3#20246340C211yesmaximal, characteristic, complemented
4#20244359C211yesmaximal, characteristic, complemented
5#20244347C211yesmaximal, characteristic, complemented
6#20244338C211yesmaximal, characteristic, complemented
7#20244337C211yesmaximal, characteristic, complemented
8#20244334C211yesmaximal, characteristic, complemented
9#20243397C211yesmaximal, characteristic
10#20243395C211yesmaximal, characteristic
11#20243387C211yesmaximal, characteristic
12#20243214C211yesmaximal, characteristic
13#20243212C211yesmaximal, characteristic
14#20243202C211yesmaximal, characteristic
15C47×#1993C211yesmaximal, characteristic, complemented
16C22×Q8×D47—21no—
17#15689788—24no—
18#15689788C418yescomplemented
19C2×#13094023C2211yescharacteristic
20C2×#13094023C2211yescharacteristic
21C2×#13094002—21no—
22C2×#13094002—21no—
23C2×#13094000—21no—
24C2×C4×Dic94C2211yescharacteristic
25C2×C4×D188C2211yescharacteristic
26C2×#13093986C2211yescharacteristic
27C2×#13093986C2211yescharacteristic
28#15689251—24no—
29C2×C4⋊(C47⋊C8)C2211yescharacteristic
30C2×Dic47⋊C8C2211yescharacteristic
31C2×Dic47⋊C8C2211yescharacteristic
32C2×#13093946C2211yescharacteristic
33C2×#13093946C2211yescharacteristic
34C2×#13093944C2211yescharacteristic
35C2×#13093944C2211yescharacteristic
36C2×#13093934C2211yescharacteristic
37C2×C8×Dic47C2211yescharacteristic
38#15689027—21no—
39#15689008—21no—
40#15689005—21no—
41#15688992C2211yescharacteristic
42#15688988C2211yescharacteristic
43C4⋊C4×Dic47C2211yescharacteristic
44C4⋊C4×Dic47C2211yescharacteristic
45C4×C4⋊Dic47C2211yescharacteristic
46#15688974—21no—
47#15688886C2211yescharacteristic
48#15688886C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688885C2211yescharacteristic
53#15688885C2211yescharacteristic
54#15688885C2211yescharacteristic
55C2×C4⋊Q8×C47C2211yescharacteristic
56C2×#97×C47C2211yescharacteristic
57C47×#382C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#340C2211yescharacteristic
60C47×#340C2211yescharacteristic
61C47×#340C2211yescharacteristic
62C22×Dic94—21no—
63C22×Dic94—21no—
64C22×Dic94D411yescharacteristic
65C2×Q8×D47—28no—
66C2×Q8×D47—44no—
67C2×Q8×D47—44no—
68C22×D188C2311yescharacteristic
69C22×C4×D47—21no—
70C22×C4×D47—21no—
71C22×C4×D47—21no—
72C22×C47⋊C8C2311yescharacteristic
73C22×C47⋊C8C2311yescharacteristic
74#13094023—22no—
75#13094023—22no—
76#13094023C2×C414yes—
77#13094023C2×C414yes—
78#13094002—28no—
79#13094002—28no—
80#13094000—28no—
81C4×Dic94C2×C414yes—
82C4×Dic94D414yes—
83C4×D188—22no—
84C4×D188C2×C414yes—
85#13093986—22no—
86#13093986—22no—
87#13093986C2×C414yes—
88#13093986C2×C414yes—
89C2×C4⋊Dic47D411yescharacteristic
90C2×C4⋊Dic47C2311yescharacteristic
91C2×C4⋊Dic47C2311yescharacteristic
92C2×C4⋊Dic47C2311yescharacteristic
93C2×Dic47⋊C4—21no—
94C2×Dic47⋊C4—21no—
95C2×Dic47⋊C4—21no—
96C2×Dic47⋊C4—21no—
97C2×Dic47⋊C4D411yescharacteristic
98C2×Dic47⋊C4D411yescharacteristic
99C2×(C2×D47)⋊C4—21no—
100C2×(C2×D47)⋊C4—21no—
101C2×(C2×D47)⋊C4—21no—
102C2×C4×Dic47—21no—
103C2×C4×Dic47—21no—
104C2×C4×Dic47—21no—
105C2×C4×Dic47C2311yescharacteristic
106C2×C4×Dic47C2311yescharacteristic
107C4⋊(C47⋊C8)—22no—
108Dic47⋊C8—22no—
109Dic47⋊C8—22no—
110#13093946—22no—
111#13093946Q814yescomplemented
112#13093944—22no—
113#13093944—22no—
114#13093934—22no—
115C8×Dic47—22no—
116C47⋊#95—21no—
117C47⋊#95—21no—
118C47⋊#95—21no—
119C47⋊#95—21no—
120C47⋊#95—21no—
121C22×Q8×C47—21no—
122C22×C8×C47C2311yescharacteristic
123C22×C8×C47C2311yescharacteristic
124C4⋊Q8×C47—22no—
125C4⋊Q8×C47C2×C414yes—
126C2×C4⋊C4×C47—21no—
127C2×C4⋊C4×C47C2311yescharacteristic
128C2×C4⋊C4×C47C2311yescharacteristic
129C2×C4⋊C4×C47C2311yescharacteristic
130C2×C4⋊C4×C47C2311yescharacteristic
131C2×C42×C47C2311yescharacteristic
132#97×C47—22no—
133#95×C47—21no—
134#95×C47—21no—
135C23×D47—21no—
136C2×Q8×C47—24no—
137C2×Q8×C47—42no—
138C2×Q8×C47—42no—
139C22×C4×C47—21no—
140C22×C4×C47—21no—
141C22×C4×C47—21no—
142C22×C4×C47—21no—
143C22×C4×C47C2×D411yescharacteristic
144C22×C4×C47C2×D411yescharacteristic
145C22×C4×C47C2411yescharacteristic
146C22×Dic47—21no—
147C22×Dic47—21no—
148C22×Dic47—21no—
149C22×Dic47C2×D411yescharacteristic
150C22×Dic47C2×D411yescharacteristic
151C22×Dic47C2×D411yescharacteristic
152C22×Dic47C2×D411yescharacteristic
153Q8×D47—432no—
154C2×D188—21no—
155C2×D188—21no—
156C2×D188—21no—
157C2×D188C22×C412yes—
158C2×D188C2×Q812yes—
159C2×D188#4112yes—
160C2×C4×D47—24no—
161C2×C4×D47—24no—
162C2×C4×D47—24no—
163C2×C4×D47—42no—
164C2×C4×D47—42no—
165C2×C4×D47—42no—
166C2×C4×D47—42no—
167C2×C4×D47—42no—
168C2×C4×D47—42no—
169C2×Dic94—22no—
170C2×Dic94—22no—
171C2×Dic94—24no—
172C2×Dic94—24no—
173C2×Dic94—42no—
174C2×Dic94—42no—
175C2×Dic94—42no—
176C2×Dic94—42no—
177C2×Dic94C22⋊C414yes—
178C2×C8×C47—21no—
179C2×C8×C47—21no—
180C2×C8×C47—21no—
181C2×C8×C47—21no—
182C2×C8×C47—21no—
183C2×C8×C47C2×Q812yescomplemented
184C4⋊C4×C47—24no—
185C4⋊C4×C47—21no—
186C4⋊C4×C47—21no—
187C4⋊C4×C47—21no—
188C4⋊C4×C47—21no—
189C4⋊C4×C47—21no—
190C4⋊C4×C47—21no—
191C4⋊C4×C47C22×C412yes—
192C4⋊C4×C47C22×C412yes—
193C42×C47C22×C412yes—
194C42×C47C2×D412yes—
195(C2×D47)⋊C4—24no—
196(C2×D47)⋊C4—24no—
197(C2×D47)⋊C4—24no—
198C4⋊Dic47—21no—
199C4⋊Dic47—21no—
200C4⋊Dic47—21no—
201C4⋊Dic47—21no—
202C4⋊Dic47C22⋊C414yes—
203C4⋊Dic47C22×C412yes—
204C4⋊Dic47C2×Q812yescomplemented
205Dic47⋊C4—24no—
206Dic47⋊C4—24no—
207Dic47⋊C4—42no—
208Dic47⋊C4—81no—
209Dic47⋊C4C22⋊C414yes—
210Dic47⋊C4C22⋊C414yes—
211C4×Dic47—42no—
212C4×Dic47—81no—
213C4×Dic47—81no—
214C4×Dic47C22×C412yes—
215C4×Dic47C22×C412yes—
216C4×Dic47C2×D412yes—
217C4×Dic47C2×D412yes—
218C2×C47⋊C8—21no—
219C2×C47⋊C8—21no—
220C2×C47⋊C8—21no—
221C2×C47⋊C8—21no—
222C2×C47⋊C8—21no—
223C2×C47⋊C8—21no—
224C23×C47C22×D411yescharacteristic
225C22×D47—22no—
226C22×D47—22no—
227C22×D47—22no—
228C22×D47—41no—
229C22×D47—41no—
230C22×D47—41no—
231C22×D47—41no—
232Q8×C47—48no—
233C2×C4×C47—21no—
234C2×C4×C47—21no—
235C2×C4×C47—21no—
236C2×C4×C47—21no—
237C2×C4×C47—22no—
238C2×C4×C47—22no—
239C2×C4×C47—41no—
240C2×C4×C47—41no—
241C2×C4×C47—41no—
242C2×C4×C47—41no—
243C2×C4×C47—41no—
244C2×C4×C47—41no—
245C2×C4×C47—41no—
246C2×C4×C47—41no—
247C2×C4×C47—41no—
248C2×C4×C47—41no—
249C2×C4×C47C2×C22⋊C412yes—
250C2×C4×C47C2×C22⋊C412yes—
251C2×C4×C47C23×C411yescharacteristic
252C2×C4×C47C22×D411yescharacteristic
253C2×C4×C47C22×Q811yescharacteristic
254C2×C4×C47C2×#4111yescharacteristic
255C2×C4×C472−1+411yescharacteristic
256C2×C4×C472−1+411yescharacteristic
257C2×Dic47—21no—
258C2×Dic47—21no—
259C2×Dic47—21no—
260C2×Dic47—21no—
261C2×Dic47—21no—
262C2×Dic47—21no—
263C2×Dic47—22no—
264C2×Dic47—22no—
265C2×Dic47—41no—
266C2×Dic47—41no—
267C2×Dic47—41no—
268C2×Dic47—41no—
269C2×Dic47—41no—
270C2×Dic47—41no—
271C2×Dic47—41no—
272C2×Dic47C2×C22⋊C412yes—
273C2×Dic47C2×C22⋊C412yes—
274C2×Dic47C2×C22⋊C412yes—
275C2×Dic47C2×C22⋊C412yes—
276C2×Dic47AGL(1,ℤ/8)12yes—
277C2×Dic47AGL(1,ℤ/8)12yes—
278D188—22no—
279D188—22no—
280D188—22no—
281D188C4×Q814yes—
282C4×D47—48no—
283C4×D47—48no—
284C4×D47—48no—
285Dic94—28no—
286Dic94—48no—
287Dic94—48no—
288C8×C47—22no—
289C8×C47—41no—
290C47⋊C8—41no—
291C47⋊C8—41no—
292#22523—471nomaximal, Sylow
293C22×C47C22×C22⋊C411yescharacteristic
294C22×C47#54811yescharacteristic
295C22×C47D4×Q811yescharacteristic
296C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
297C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
298C22×C47#57711yescharacteristic
299C22×C47#57811yescharacteristic
300C2×D47—24no—
301C2×D47—42no—
302C2×D47—42no—
303C2×D47—42no—
304C2×D47—42no—
305C2×D47—42no—
306C2×D47—42no—
307C4×C47—22no—
308C4×C47—22no—
309C4×C47—42no—
310C4×C47—42no—
311C4×C47—81no—
312C4×C47—81no—
313C4×C47C2×C4×Q811yescharacteristic
314C4×C47#51911yescharacteristic
315C4×C47#54811yescharacteristic
316C4×C47D4×Q811yescharacteristic
317Dic47—22no—
318Dic47—22no—
319Dic47—22no—
320Dic47—42no—
321Dic47—42no—
322Dic47—81no—
323Dic47#41814yes—
324C2×D4×Q8—471no—
325C2×#446—471no—
326#2355—471no—
327#1993—471no—
328#1993D4711yescharacteristic, complemented
329#1990—471no—
330#1988—471no—
331#1987—471no—
332#1987—471no—
333#1804—471no—
334#1804—471no—
335#1802—471no—
336#1762—471no—
337#1762—471no—
338C4×#102—471no—
339C2×C47Q8×C22⋊C411yescharacteristic
340C2×C47C2×#41811yescharacteristic
341C2×C47#289811yescharacteristic
342C2×C47#336711yescharacteristic
343C2×C47#337311yescharacteristic
344C2×C47#338411yescharacteristic
345C2×C47#338611yescharacteristic
346D47—44no—
347C23×Q8—941no—
348D4×Q8—478no—
349D4×Q8—944no—
350C2×C4⋊Q8—471no—
351C2×C4⋊Q8—471no—
352C2×C4⋊Q8C2×D4711yescharacteristic
353C2×C22⋊Q8—941no—
354C2×C22⋊Q8—941no—
355C2×C22⋊Q8—941no—
356C2×C4×Q8—471no—
357C2×C4×D4—471no—
358C2×C4×D4—471no—
359C2×C4×D4—471no—
360#446—944no—
361C2×C4⋊C8—471no—
362C2×C4⋊C8—471no—
363C2×C4⋊C8—471no—
364C2×#102—471no—
365C2×#102—471no—
366C2×#102—471no—
367C2×#102—471no—
368C2×#97—471no—
369C2×#97C2×D4711yescharacteristic
370C2×C4×C8—471no—
371#391—941no—
372#387—941no—
373#387—941no—
374#384—471no—
375#382—471no—
376#382C2×D4711yescharacteristic
377C4×C4⋊C4—471no—
378C4×C4⋊C4—471no—
379C4×C4⋊C4—471no—
380C4×C4⋊C4—941no—
381#340—471no—
382#340—471no—
383#340—471no—
384#340—471no—
385#340—471no—
386#340—471no—
387#340—471no—
388#340—471no—
389#340C2×D4711yescharacteristic
390#340C2×D4711yescharacteristic
391#340C2×D4711yescharacteristic
392#340C2×D4711yescharacteristic
393C47#2252311yescharacteristic, complemented, Sylow
394C22×Q8—21no—
395C22×Q8—471no—
396C22×Q8—941no—
397C22×Q8—941no—
398C22×Q8—948no—
399C22×Q8—1884no—
400C22×Q8—1884no—
401C22×D4—471no—
402C23×C4—941no—
403C23×C4—941no—
404C23×C4—941no—
405C22×C8—471no—
406C22×C8—471no—
407C22×C8C22×D4711yescharacteristic
408C22×C8C22×D4711yescharacteristic
409C4⋊Q8—22no—
410C4⋊Q8—474no—
411C4⋊Q8—474no—
412C4⋊Q8—942no—
413C4⋊Q8—942no—
414C4⋊Q8C4×D4714yes—
415C22⋊Q8—948no—
416C22⋊Q8—948no—
417C22⋊Q8—948no—
418C4×Q8—474no—
419C4×Q8—474no—
420C4×D4—474no—
421C4×D4—474no—
422C4×D4—474no—
423C4×D4—942no—
424C4×D4—942no—
425C4×D4—942no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—471no—
428C2×C4⋊C4—471no—
429C2×C4⋊C4—471no—
430C2×C4⋊C4—471no—
431C2×C4⋊C4—471no—
432C2×C4⋊C4—471no—
433C2×C4⋊C4—941no—
434C2×C4⋊C4—941no—
435C2×C4⋊C4—941no—
436C2×C4⋊C4—941no—
437C2×C4⋊C4C22×D4711yescharacteristic
438C2×C4⋊C4C22×D4711yescharacteristic
439C2×C4⋊C4C22×D4711yescharacteristic
440C2×C4⋊C4C22×D4711yescharacteristic
441C2×C22⋊C4—941no—
442C2×C22⋊C4—941no—
443C2×C22⋊C4—941no—
444C2×C42—471no—
445C2×C42—471no—
446C2×C42—941no—
447C2×C42—941no—
448C2×C42—941no—
449C2×C42C22×D4711yescharacteristic
450C4⋊C8—942no—
451C4⋊C8—942no—
452C4⋊C8—942no—
453#102—474no—
454#102—942no—
455#102—942no—
456#102—942no—
457#97—22no—
458#97—942no—
459C4×C8—942no—
460#95—21no—
461#95—21no—
462#95—941no—
463#95—941no—
464#95—941no—
465#95—941no—
466#95—941no—
467C24—941no—
468C2×Q8—24no—
469C2×Q8—42no—
470C2×Q8—42no—
471C2×Q8—474no—
472C2×Q8—942no—
473C2×Q8—942no—
474C2×Q8—944no—
475C2×Q8—944no—
476C2×Q8—1882no—
477C2×Q8—1882no—
478C2×Q8—1882no—
479C2×Q8—1882no—
480C2×Q8—18832no—
481C2×D4—472no—
482C2×D4—472no—
483C2×D4—472no—
484C2×D4—941no—
485C2×D4—941no—
486C2×D4—941no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—471no—
492C22×C4—471no—
493C22×C4—471no—
494C22×C4—471no—
495C22×C4—944no—
496C22×C4—944no—
497C22×C4—941no—
498C22×C4—941no—
499C22×C4—944no—
500C22×C4—941no—
501C22×C4—1882no—
502C22×C4—1882no—
503C22×C4—1882no—
504C22×C4—1882no—
505C22×C4—1882no—
506C22×C4—1882no—
507C22×C4D4×D4711yescharacteristic
508C22×C4D4×D4711yescharacteristic
509C22×C4C23×D4711yescharacteristic
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—941no—
516C2×C8—941no—
517C2×C8—941no—
518C2×C8—941no—
519C2×C8—941no—
520C2×C8—941no—
521C2×C8Q8×D4712yescomplemented
522C4⋊C4—24no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—474no—
530C4⋊C4—474no—
531C4⋊C4—474no—
532C4⋊C4—472no—
533C4⋊C4—472no—
534C4⋊C4—944no—
535C4⋊C4—944no—
536C4⋊C4—941no—
537C4⋊C4—941no—
538C4⋊C4—941no—
539C4⋊C4—941no—
540C4⋊C4—3761no—
541C4⋊C4—1882no—
542C4⋊C4C2×C4×D4712yes—
543C4⋊C4C2×C4×D4712yes—
544C22⋊C4—944no—
545C22⋊C4—944no—
546C22⋊C4—944no—
547C42—472no—
548C42—472no—
549C42—472no—
550C42—472no—
551C42—3761no—
552C42—3761no—
553C42—1882no—
554C42C2×C4×D4712yes—
555C42D4×D4712yes—
556C23—942no—
557C23—942no—
558C23—942no—
559C23—1881no—
560C23—1881no—
561C23—1881no—
562C23—1881no—
563C23C2×D4×D4711yescharacteristic
564Q8—48no—
565Q8—948no—
566Q8—1888no—
567Q8—1888no—
568D4—474no—
569D4—942no—
570D4—942no—
571D4—942no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—472no—
589C2×C4—472no—
590C2×C4—472no—
591C2×C4—472no—
592C2×C4—472no—
593C2×C4—472no—
594C2×C4—941no—
595C2×C4—941no—
596C2×C4—941no—
597C2×C4—941no—
598C2×C4—941no—
599C2×C4—941no—
600C2×C4—942no—
601C2×C4—942no—
602C2×C4—1888no—
603C2×C4—1888no—
604C2×C4—1888no—
605C2×C4—1881no—
606C2×C4—1881no—
607C2×C4—1881no—
608C2×C4—1881no—
609C2×C4—1881no—
610C2×C4—1881no—
611C2×C4—1881no—
612C2×C4C22⋊C4×D4712yes—
613C2×C4C22⋊C4×D4712yes—
614C2×C4C22×C4×D4711yescharacteristic
615C2×C4C2×D4×D4711yescharacteristic
616C2×C4C2×Q8×D4711yescharacteristic
617C2×C4#41×D4711yescharacteristic
618C2×C4#1309407111yescharacteristic
619C2×C4#1309407211yescharacteristic
620C8—22no—
621C8—41no—
622C8—1881no—
623C8—1881no—
624C22—944no—
625C22—1882no—
626C22—1882no—
627C22—1882no—
628C22—1882no—
629C22—1882no—
630C22—1882no—
631C22C2×C22⋊C4×D4711yescharacteristic
632C22#1568978111yescharacteristic
633C22#1568978811yescharacteristic
634C22AGL(1,ℤ/8)×D4711yescharacteristic
635C22AGL(1,ℤ/8)×D4711yescharacteristic
636C22#1568993211yescharacteristic
637C22#1568993911yescharacteristic
638C4—22no—
639C4—22no—
640C4—42no—
641C4—42no—
642C4—81no—
643C4—81no—
644C4—474no—
645C4—942no—
646C4—942no—
647C4—942no—
648C4—3761no—
649C4—1882no—
650C4—1882no—
651C4C4×Q8×D4711yescharacteristic
652C4#1568959911yescharacteristic
653C4#1568978311yescharacteristic
654C4#1568978611yescharacteristic
655C2—1884no—
656C2#2024622211yescharacteristic
657C2#418×D4711yescharacteristic
658C2#2025192011yescharacteristic
659C2#2025610311yescharacteristic
660C2#2025617711yescharacteristic
661C2#2025628411yescharacteristic
662C2#2025630611yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1993C2
socleC23×C47C22×D4
prime radicalC22×D188C23
cosocleC22×C4×C47C24
hypercenter#1993D47
hypocenterC47#22523
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1993

Complements

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

NH ≅ G/Ntype
2. C2×#15689251C2semidirect
3. #20246340C2semidirect
4. #20244359C2semidirect
5. #20244347C2semidirect
6. #20244338C2semidirect
7. #20244337C2semidirect
8. #20244334C2semidirect
15. C47×#1993C2semidirect
18. #15689788C4semidirect
111. #13093946Q8semidirect
183. C2×C8×C47C2×Q8semidirect
204. C4⋊Dic47C2×Q8semidirect
328. #1993D47semidirect
393. C47#22523semidirect
521. C2×C8Q8×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156897882
C2×#156892512
#202463402
#202443592
#202443472
#202443382
#202443372
#202443342
#202433972
#202433952
#202433872
#202432142
#202432122
#202432022
C47×#19932
#2252347