Finite Group Databasebeta

← #26222587

Subgroups — #26222587

Order 12032.

1004 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689345C211yesmaximal, characteristic
2C2×#15689228C211yesmaximal, characteristic, complemented
3C2×#15689221C211yesmaximal, characteristic
4C4×#13094012C211yesmaximal, characteristic
5#20244320C211yesmaximal, characteristic
6#20244156C211yesmaximal, characteristic, complemented
7#20244029C211yesmaximal, characteristic, complemented
8#20244015C211yesmaximal, characteristic
9#20243743C211yesmaximal, characteristic
10#20243729C211yesmaximal, characteristic, complemented
11#20243403C211yesmaximal, characteristic, complemented
12#20243241C211yesmaximal, characteristic, complemented
13#20243221C211yesmaximal, characteristic, complemented
14#20243216C211yesmaximal, characteristic
15C42×Q8×C47C211yesmaximal, characteristic, complemented
16C22×#10625337C2211yescharacteristic
17C22×#10625336C2211yescharacteristic
18C2×#13094012C2211yescharacteristic
19C2×C4×Dic94C2211yescharacteristic
20C2×C4×D188C2211yescharacteristic
21#15689345C418yescomplemented
22#15689228—24no—
23#15689221—24no—
24C2×#13093957C2211yescharacteristic
25C2×#13093955C2211yescharacteristic
26C2×C4⋊(C47⋊C8)C2211yescharacteristic
27C2×C4⋊(C47⋊C8)C2211yescharacteristic
28C2×#13093948C2211yescharacteristic
29C2×#13093948C2211yescharacteristic
30C2×#13093947C2211yescharacteristic
31C2×#13093947C2211yescharacteristic
32C2×#13093947C2211yescharacteristic
33C2×#13093944C2211yescharacteristic
34C2×#13093944C2211yescharacteristic
35C2×#13093944C2211yescharacteristic
36C2×#13093935C2211yescharacteristic
37C2×#13093935C2211yescharacteristic
38#15689029C2211yescharacteristic
39#15689025C2211yescharacteristic
40#15689014—21no—
41#15689014—21no—
42#15688989C2211yescharacteristic
43C4×(C2×D47)⋊C4—21no—
44C4×(C2×D47)⋊C4—21no—
45#15688885C2211yescharacteristic
46#15688885C2211yescharacteristic
47#15688885C2211yescharacteristic
48#15688885C2211yescharacteristic
49#15688878C2211yescharacteristic
50#15688878C2211yescharacteristic
51C2×C4×Q8×C47C2211yescharacteristic
52C2×C4×Q8×C47C2211yescharacteristic
53C2×C4×Q8×C47C2211yescharacteristic
54C4×C4⋊C4×C47—21no—
55C4×C4⋊C4×C47—21no—
56C4×C4⋊C4×C47—21no—
57C4×C4⋊C4×C47C2211yescharacteristic
58C4×C4⋊C4×C47C2211yescharacteristic
59C4×C4⋊C4×C47C2211yescharacteristic
60C43×C47—21no—
61C43×C47C2211yescharacteristic
62C22×Dic94C2311yescharacteristic
63C22×D188C2311yescharacteristic
64C22×C4×D47—21no—
65C2×#10625337—22no—
66C2×#10625337—22no—
67C2×#10625337C2×C414yescomplemented
68C2×#10625336—22no—
69C2×#10625336—22no—
70C2×#10625336C2×C414yes—
71C22×C47⋊C8C2311yescharacteristic
72C22×C47⋊C8C2311yescharacteristic
73C22×C47⋊C8C2311yescharacteristic
74C22×C47⋊C8C2311yescharacteristic
75#13094012—22no—
76#13094012C2×C414yes—
77C4×Dic94—22no—
78C4×Dic94—22no—
79C4×D188—22no—
80C4×D188—22no—
81C2×C4⋊Dic47C2311yescharacteristic
82C2×C4⋊Dic47C2311yescharacteristic
83C2×Dic47⋊C4—21no—
84C2×(C2×D47)⋊C4—21no—
85C2×(C2×D47)⋊C4—21no—
86C2×(C2×D47)⋊C4—21no—
87C2×(C2×D47)⋊C4—21no—
88C2×(C2×D47)⋊C4—21no—
89C2×C4×Dic47—21no—
90#13093957—22no—
91#13093955—22no—
92C4⋊(C47⋊C8)—22no—
93C4⋊(C47⋊C8)C2×C414yescomplemented
94#13093948—22no—
95#13093948—22no—
96#13093947—22no—
97#13093947—22no—
98#13093947C2×C414yescomplemented
99#13093944—22no—
100#13093944—22no—
101#13093944C2×C414yes—
102#13093935—22no—
103#13093935—22no—
104C47⋊#95—21no—
105C47⋊#95—21no—
106C47⋊#95—21no—
107C47⋊#95—21no—
108C22×Q8×C47C2311yescharacteristic
109C4×Q8×C47—22no—
110C4×Q8×C47—22no—
111C4×Q8×C47—22no—
112C4×Q8×C47—22no—
113C4×Q8×C47C2×C414yescomplemented
114C4×Q8×C47D414yes—
115C2×C4⋊C4×C47—21no—
116C2×C4⋊C4×C47—21no—
117C2×C4⋊C4×C47—21no—
118C2×C4⋊C4×C47C2311yescharacteristic
119C2×C4⋊C4×C47C2311yescharacteristic
120C2×C4⋊C4×C47C2311yescharacteristic
121C2×C42×C47—21no—
122C2×C42×C47—21no—
123C2×C42×C47—21no—
124C2×C42×C47—21no—
125C2×C42×C47—21no—
126C2×C42×C47—21no—
127C2×C42×C47D411yescharacteristic
128C2×C42×C47D411yescharacteristic
129C2×C42×C47D411yescharacteristic
130C2×C42×C47D411yescharacteristic
131C2×C42×C47C2311yescharacteristic
132C2×C42×C47C2311yescharacteristic
133C2×C42×C47C2311yescharacteristic
134#95×C47—21no—
135#95×C47—21no—
136#95×C47—21no—
137C23×D47—21no—
138C2×Q8×C47—21no—
139C2×Q8×C47—21no—
140C2×Q8×C47—21no—
141C2×Q8×C47C22×C412yes—
142C2×Q8×C47C2×D412yes—
143C2×Q8×C47#4112yes—
144C22×C4×C47—21no—
145C22×C4×C47—21no—
146C22×C4×C47—21no—
147C22×C4×C47—21no—
148C22×C4×C47C2×D411yescharacteristic
149C22×C4×C47C2×D411yescharacteristic
150C22×C4×C47C2×D411yescharacteristic
151C22×C4×C47C2×D411yescharacteristic
152C22×C4×C47C2×D411yescharacteristic
153C22×C4×C47C2×D411yescharacteristic
154C22×C4×C47C2411yescharacteristic
155C22×Dic47—21no—
156C22×Dic47—21no—
157C22×Dic47—21no—
158C2×D188—21no—
159C2×D188—21no—
160C2×D188—21no—
161C2×D188—21no—
162C2×D188—21no—
163C2×D188C22×C412yes—
164C2×C4×D47—42no—
165C2×C4×D47—42no—
166C2×C4×D47—42no—
167C2×Dic94—21no—
168C2×Dic94—21no—
169C2×Dic94—21no—
170C2×Dic94—21no—
171C2×Dic94—21no—
172C2×Dic94C22×C412yes—
173C4⋊C4×C47—24no—
174C4⋊C4×C47—24no—
175C4⋊C4×C47—24no—
176C4⋊C4×C47—21no—
177C4⋊C4×C47—21no—
178C4⋊C4×C47—21no—
179C4⋊C4×C47C22×C412yes—
180C4⋊C4×C47C2×D412yes—
181C4⋊C4×C47#4112yes—
182C42×C47—24no—
183C42×C47—24no—
184C42×C47—24no—
185C42×C47—22no—
186C42×C47—21no—
187C42×C47—21no—
188C42×C47—41no—
189C42×C47—41no—
190C42×C47—41no—
191C42×C47—42no—
192C42×C47—42no—
193C42×C47—42no—
194C42×C47C22×C412yes—
195C42×C47C2×D412yes—
196C42×C47#4112yes—
197C42×C47#4112yes—
198#10625337—28no—
199#10625336—28no—
200(C2×D47)⋊C4—24no—
201(C2×D47)⋊C4—24no—
202(C2×D47)⋊C4—42no—
203(C2×D47)⋊C4—42no—
204(C2×D47)⋊C4—42no—
205C4⋊Dic47—21no—
206C4⋊Dic47—21no—
207C4⋊Dic47—21no—
208C4⋊Dic47—21no—
209Dic47⋊C4—42no—
210C4×Dic47—42no—
211C2×C47⋊C8—21no—
212C2×C47⋊C8—21no—
213C2×C47⋊C8—21no—
214C2×C47⋊C8—21no—
215C2×C47⋊C8—21no—
216C2×C47⋊C8—21no—
217C2×C47⋊C8—21no—
218C2×C47⋊C8—21no—
219C2×C47⋊C8—21no—
220C2×C47⋊C8—21no—
221C2×C47⋊C8C22×C412yes—
222C2×C47⋊C8C22×C412yes—
223C23×C47C22×D411yescharacteristic
224C22×D47—22no—
225C22×D47—41no—
226C22×D47—41no—
227C22×D47—41no—
228C22×D47—41no—
229C22×D47—41no—
230C22×D47—41no—
231Q8×C47—22no—
232Q8×C47—22no—
233Q8×C47—22no—
234Q8×C47C4×D414yes—
235C2×C4×C47—22no—
236C2×C4×C47—22no—
237C2×C4×C47—21no—
238C2×C4×C47—22no—
239C2×C4×C47—21no—
240C2×C4×C47—21no—
241C2×C4×C47—21no—
242C2×C4×C47—21no—
243C2×C4×C47—21no—
244C2×C4×C47—21no—
245C2×C4×C47—21no—
246C2×C4×C47—21no—
247C2×C4×C47—21no—
248C2×C4×C47—22no—
249C2×C4×C47—22no—
250C2×C4×C47—21no—
251C2×C4×C47—22no—
252C2×C4×C47—21no—
253C2×C4×C47—21no—
254C2×C4×C47—41no—
255C2×C4×C47—41no—
256C2×C4×C47—41no—
257C2×C4×C47—41no—
258C2×C4×C47—41no—
259C2×C4×C47—41no—
260C2×C4×C47C4×D412yes—
261C2×C4×C47C4×D412yes—
262C2×C4×C47#12112yes—
263C2×C4×C47#12112yes—
264C2×C4×C47#13712yes—
265C2×C4×C47C23×C411yescharacteristic
266C2×C4×C47C22×D411yescharacteristic
267C2×C4×C47C2×#4111yescharacteristic
268C2×C4×C47C2×#4111yescharacteristic
269C2×C4×C47C2×#4111yescharacteristic
270C2×C4×C47C2×#4111yescharacteristic
271C2×Dic47—22no—
272C2×Dic47—41no—
273C2×Dic47—41no—
274C2×Dic47—41no—
275C2×Dic47—41no—
276C2×Dic47—41no—
277C2×Dic47—41no—
278C2×Dic47—41no—
279C2×Dic47—41no—
280D188—22no—
281D188—22no—
282D188—41no—
283D188—41no—
284C4×D47—84no—
285Dic94—22no—
286Dic94—22no—
287Dic94—41no—
288Dic94—41no—
289C47⋊C8—22no—
290C47⋊C8—22no—
291C47⋊C8—41no—
292C47⋊C8—41no—
293#21508—471nomaximal, Sylow
294C22×C47C2×C4×D411yescharacteristic
295C22×C47C2×#12111yescharacteristic
296C22×C47#52511yescharacteristic
297C22×C47C2×#13711yescharacteristic
298C22×C47#57511yescharacteristic
299C22×C47#57511yescharacteristic
300C22×C47#57511yescharacteristic
301C2×D47—42no—
302C2×D47—42no—
303C2×D47—42no—
304C2×D47—81no—
305C2×D47—81no—
306C2×D47—81no—
307C2×D47—81no—
308C4×C47—22no—
309C4×C47—22no—
310C4×C47—22no—
311C4×C47—22no—
312C4×C47—22no—
313C4×C47—24no—
314C4×C47—41no—
315C4×C47—42no—
316C4×C47—42no—
317C4×C47—42no—
318C4×C47C2×C4×D411yescharacteristic
319C4×C47C4×#4111yescharacteristic
320C4×C47#52911yescharacteristic
321C4×C47#53211yescharacteristic
322Dic47—42no—
323Dic47—81no—
324Dic47—81no—
325C2×#464—471no—
326C2×#441—471no—
327C2×#440—471no—
328C4×#124—471no—
329C42×Q8D4711yescharacteristic, complemented
330#1983—471no—
331#1949—471no—
332#1928—471no—
333#1926—471no—
334#1880—471no—
335#1878—471no—
336#1805—471no—
337#1767—471no—
338#1763—471no—
339#1762—471no—
340C2×C47C4×#12111yescharacteristic
341C2×C47#294311yescharacteristic
342C2×C47C4×#13711yescharacteristic
343C2×C47#303711yescharacteristic
344C2×C47#304011yescharacteristic
345C2×C47#311111yescharacteristic
346C2×C47#311711yescharacteristic
347D47—82no—
348C22×Q16—471no—
349C22×SD16—471no—
350C2×#124—471no—
351C2×C4×Q8—471no—
352C2×C4×Q8C2×D4711yescharacteristic
353C2×C4×Q8C2×D4711yescharacteristic
354C2×C4×Q8C2×D4711yescharacteristic
355C2×C4×D4—471no—
356#464—478no—
357#441—944no—
358#440—944no—
359C2×#107—471no—
360C2×#106—471no—
361C2×C4⋊C8—471no—
362C2×C4⋊C8—471no—
363C2×#103—471no—
364C2×#103—471no—
365C2×#103—471no—
366C2×#103—471no—
367C2×#103—471no—
368C2×#102—471no—
369C2×#102—471no—
370C2×#102—471no—
371C2×#97—471no—
372C2×#97—471no—
373#391—471no—
374#390—471no—
375#388—941no—
376#388—941no—
377#382—471no—
378C4×C4⋊C4—21no—
379C4×C4⋊C4—21no—
380C4×C4⋊C4—21no—
381C4×C4⋊C4C2×D4711yescharacteristic
382C4×C4⋊C4C2×D4711yescharacteristic
383C4×C4⋊C4C2×D4711yescharacteristic
384C4×C22⋊C4—941no—
385C4×C22⋊C4—941no—
386C43—21no—
387C43C2×D4711yescharacteristic
388#340—471no—
389#340—471no—
390#340—471no—
391#340—471no—
392#336—471no—
393#336—471no—
394C47#2150811yescharacteristic, complemented, Sylow
395C22×Q8—471no—
396C22×Q8C22×D4711yescharacteristic
397C22×D4—471no—
398C23×C4—941no—
399C2×Q16—474no—
400C2×Q16—942no—
401C2×Q16—942no—
402C2×SD16—474no—
403C2×SD16—942no—
404C2×SD16—942no—
405C22×C8—471no—
406C22×C8—471no—
407C22×C8—471no—
408C22×C8—471no—
409#124—474no—
410#124—942no—
411C4×Q8—22no—
412C4×Q8—22no—
413C4×Q8—22no—
414C4×Q8—22no—
415C4×Q8—942no—
416C4×Q8—942no—
417C4×Q8C4×D4714yescomplemented
418C4×Q8D18814yes—
419C4×D4—942no—
420C4×D4—942no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—471no—
425C2×C4⋊C4—471no—
426C2×C4⋊C4—941no—
427C2×C4⋊C4C22×D4711yescharacteristic
428C2×C4⋊C4C22×D4711yescharacteristic
429C2×C4⋊C4C22×D4711yescharacteristic
430C2×C22⋊C4—941no—
431C2×C22⋊C4—941no—
432C2×C22⋊C4—941no—
433C2×C22⋊C4—941no—
434C2×C22⋊C4—941no—
435C2×C42—21no—
436C2×C42—21no—
437C2×C42—21no—
438C2×C42—21no—
439C2×C42—21no—
440C2×C42—21no—
441C2×C42—941no—
442C2×C42C22⋊D4711yescharacteristic
443C2×C42C22⋊D4711yescharacteristic
444C2×C42C22⋊D4711yescharacteristic
445C2×C42C22⋊D4711yescharacteristic
446C2×C42C22×D4711yescharacteristic
447C2×C42C22×D4711yescharacteristic
448C2×C42C22×D4711yescharacteristic
449#107—942no—
450#106—942no—
451C4⋊C8—474no—
452C4⋊C8—942no—
453#103—474no—
454#103—942no—
455#103—942no—
456#103—942no—
457#103—942no—
458#102—474no—
459#102—942no—
460#102—942no—
461#97—942no—
462#97—942no—
463#95—21no—
464#95—21no—
465#95—21no—
466#95—941no—
467#95—941no—
468#95—941no—
469#95—941no—
470C24—941no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—472no—
475C2×Q8—941no—
476C2×Q8—941no—
477C2×Q8—941no—
478C2×Q8—941no—
479C2×Q8—941no—
480C2×Q8C2×C4×D4712yes—
481C2×Q8C2×D18812yes—
482C2×Q8#1062535012yes—
483C2×D4—472no—
484C2×D4—941no—
485C2×D4—941no—
486C2×D4—941no—
487C2×D4—941no—
488C2×D4—941no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—941no—
494C22×C4—941no—
495C22×C4—941no—
496C22×C4—1882no—
497C22×C4—1882no—
498C22×C4—1882no—
499C22×C4C2×C22⋊D4711yescharacteristic
500C22×C4C2×C22⋊D4711yescharacteristic
501C22×C4C2×C22⋊D4711yescharacteristic
502C22×C4C2×C22⋊D4711yescharacteristic
503C22×C4C2×C22⋊D4711yescharacteristic
504C22×C4C2×C22⋊D4711yescharacteristic
505C22×C4C23×D4711yescharacteristic
506Q16—948no—
507SD16—948no—
508C2×C8—472no—
509C2×C8—472no—
510C2×C8—941no—
511C2×C8—941no—
512C2×C8—941no—
513C2×C8—941no—
514C2×C8—941no—
515C2×C8—941no—
516C2×C8—941no—
517C2×C8—941no—
518C2×C8—941no—
519C2×C8—941no—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—24no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—941no—
527C4⋊C4—941no—
528C4⋊C4—941no—
529C4⋊C4—941no—
530C4⋊C4—1882no—
531C4⋊C4C2×C4×D4712yes—
532C4⋊C4C2×D18812yes—
533C4⋊C4#1062535012yes—
534C22⋊C4—944no—
535C22⋊C4—944no—
536C22⋊C4—1882no—
537C22⋊C4—1882no—
538C22⋊C4—1882no—
539C42—24no—
540C42—24no—
541C42—24no—
542C42—22no—
543C42—21no—
544C42—21no—
545C42—41no—
546C42—41no—
547C42—41no—
548C42—42no—
549C42—42no—
550C42—42no—
551C42—1882no—
552C42C2×C4×D4712yes—
553C42C2×D18812yes—
554C42#1062535012yes—
555C42#1062535012yes—
556C23—942no—
557C23—1881no—
558C23—1881no—
559C23—1881no—
560C23—1881no—
561C23—1881no—
562C23—1881no—
563C23C22×C22⋊D4711yescharacteristic
564Q8—22no—
565Q8—22no—
566Q8—22no—
567Q8—942no—
568Q8—942no—
569Q8—1881no—
570Q8—1881no—
571Q8C4×D18814yes—
572D4—942no—
573D4—942no—
574D4—1881no—
575D4—1881no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—942no—
602C2×C4—3764no—
603C2×C4—1881no—
604C2×C4—1881no—
605C2×C4—1881no—
606C2×C4—1881no—
607C2×C4—1881no—
608C2×C4—1881no—
609C2×C4—1881no—
610C2×C4—1881no—
611C2×C4C4×C22⋊D4712yes—
612C2×C4C4×C22⋊D4712yes—
613C2×C4#1309399812yes—
614C2×C4#1309399812yes—
615C2×C4#1309405512yes—
616C2×C4C22×C4×D4711yescharacteristic
617C2×C4C22×D18811yescharacteristic
618C2×C4C2×#1062535011yescharacteristic
619C2×C4C2×#1062535011yescharacteristic
620C2×C4C2×#1062535011yescharacteristic
621C2×C4C2×#1062535011yescharacteristic
622C8—942no—
623C8—942no—
624C8—1881no—
625C8—1881no—
626C22—3761no—
627C22—3761no—
628C22—3761no—
629C22—3761no—
630C22—1882no—
631C22—1882no—
632C22—1882no—
633C22C2×C4×C22⋊D4711yescharacteristic
634C22C2×#1309399811yescharacteristic
635C22#1568963411yescharacteristic
636C22C2×#1309405511yescharacteristic
637C22#1568992111yescharacteristic
638C22#1568992111yescharacteristic
639C22#1568992111yescharacteristic
640C4—22no—
641C4—22no—
642C4—22no—
643C4—24no—
644C4—22no—
645C4—22no—
646C4—42no—
647C4—41no—
648C4—42no—
649C4—42no—
650C4—3761no—
651C4—3761no—
652C4—1882no—
653C4C2×C4×D18811yescharacteristic
654C4C4×#1062535011yescharacteristic
655C4#1568965711yescharacteristic
656C4#1568967311yescharacteristic
657C2—3762no—
658C2C4×#1309399811yescharacteristic
659C2#2025218611yescharacteristic
660C2C4×#1309405511yescharacteristic
661C2#2025286011yescharacteristic
662C2#2025288211yescharacteristic
663C2#2025349711yescharacteristic
664C2#2025355911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC42×Q8×C47C2
socleC23×C47C22×D4
prime radicalC22×D188C23
cosocleC22×C4×C47C24
hypercenterC42×Q8D47
hypocenterC47#21508
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C43 ◁ C42×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#15689228C2semidirect
6. #20244156C2semidirect
7. #20244029C2semidirect
10. #20243729C2semidirect
11. #20243403C2semidirect
12. #20243241C2semidirect
13. #20243221C2semidirect
15. C42×Q8×C47C2semidirect
21. #15689345C4semidirect
67. C2×#10625337C2×C4semidirect
93. C4⋊(C47⋊C8)C2×C4semidirect
98. #13093947C2×C4semidirect
113. C4×Q8×C47C2×C4semidirect
329. C42×Q8D47semidirect
394. C47#21508semidirect
417. C4×Q8C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156893452
C2×#156892282
C2×#156892212
C4×#130940122
#202443202
#202441562
#202440292
#202440152
#202437432
#202437292
#202434032
#202432412
#202432212
#202432162
C42×Q8×C472
#2150847