Finite Group Databasebeta

← #26108321

Subgroups — #26108321

Order 12032.

760 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20243408C211yesmaximal, characteristic
2C4×#13093948C211yesmaximal, characteristic, complemented
3#20241887C211yesmaximal, characteristic, complemented
4#20241843C211yesmaximal, characteristic, complemented
5#20241817C211yesmaximal, characteristic, complemented
6#20241770C211yesmaximal, characteristic
7Q8×#41×C47C211yesmaximal, characteristic
8#15689176C2211yescharacteristic
9#15689147C2211yescharacteristic
10C2×#13093951—21no—
11C2×#13093948—21no—
12C2×C4×C47⋊C8C2211yescharacteristic
13C4×C4⋊Dic47C2211yescharacteristic
14#15688885—21no—
15#15688882—21no—
16#15688860—22no—
17Q8⋊(C47⋊C8)—21no—
18Q8⋊(C47⋊C8)—21no—
19Q8⋊(C47⋊C8)—21no—
20Q8⋊(C47⋊C8)—22no—
21D4⋊(C47⋊C8)—21no—
22Q82×C47C411yescharacteristic, complemented
23C47×#557—21no—
24C47×#557C411yescharacteristic, complemented
25C47×#554—22no—
26C47×#554—21no—
27C47×#554—21no—
28C47×#554C411yescharacteristic, complemented
29D4×Q8×C47—21no—
30D4×Q8×C47C411yescharacteristic, complemented
31C47×#533—22no—
32C47×#533—21no—
33C47×#533—21no—
34C47×#533C2211yescharacteristic
35C4×#41×C47—21no—
36C4×#41×C47C2211yescharacteristic
37C2×C4×Q8×C47—21no—
38C2×C4×Q8×C47C2211yescharacteristic
39C2×#10625329—21no—
40C22×C47⋊C8—21no—
41C2×C4⋊Dic47—21no—
42C2×C4×Dic47—21no—
43C4⋊(C47⋊C8)—21no—
44C4⋊(C47⋊C8)—21no—
45C4⋊(C47⋊C8)—21no—
46#13093951—42no—
47#13093948—24no—
48C22⋊(C47⋊C8)—21no—
49C4×C47⋊C8—21no—
50C4×C47⋊C8—21no—
51C4×C47⋊C8—21no—
52C47⋊#95—21no—
53C2×#41×C47D411yescharacteristic
54C22×Q8×C47—21no—
55C22×Q8×C47D411yescharacteristic
56C4⋊Q8×C47—22no—
57C4⋊Q8×C47—21no—
58C4⋊Q8×C47—22no—
59C4⋊Q8×C47—21no—
60C4⋊Q8×C47—21no—
61C4⋊Q8×C47—21no—
62C4⋊Q8×C47C2×C411yescharacteristic
63C4⋊Q8×C47C2×C411yescharacteristic
64#125×C47—22no—
65#125×C47—22no—
66#125×C47—22no—
67#125×C47—21no—
68#125×C47—21no—
69#125×C47—21no—
70C22⋊Q8×C47—22no—
71C22⋊Q8×C47—22no—
72C22⋊Q8×C47—22no—
73C22⋊Q8×C47—22no—
74C22⋊Q8×C47—22no—
75C22⋊Q8×C47—21no—
76C22⋊Q8×C47—22no—
77C22⋊Q8×C47—21no—
78C22⋊Q8×C47—21no—
79C22⋊Q8×C47—22no—
80C22⋊Q8×C47D412yes—
81C4×Q8×C47—22no—
82C4×Q8×C47—22no—
83C4×Q8×C47—22no—
84C4×Q8×C47—21no—
85C4×Q8×C47—21no—
86C4×Q8×C47—21no—
87C4×Q8×C47—21no—
88C4×Q8×C47—21no—
89C4×Q8×C47—21no—
90C4×Q8×C47—21no—
91C4×Q8×C47C2×C411yescharacteristic
92C4×Q8×C47C2×C411yescharacteristic
93C4×Q8×C47C2×C411yescharacteristic
94C4×Q8×C47D411yescharacteristic
95C4×Q8×C47D411yescharacteristic
96C4×Q8×C47D411yescharacteristic
97C4×Q8×C47D412yes—
98C4×D4×C47—22no—
99C4×D4×C47—22no—
100C4×D4×C47—21no—
101C4×D4×C47—21no—
102C4×D4×C47—21no—
103C4×D4×C47—21no—
104C4×D4×C47C2×C411yescharacteristic
105C4×D4×C47D411yescharacteristic
106#117×C47—22no—
107#117×C47—21no—
108#117×C47—21no—
109#117×C47D411yescharacteristic
110C2×C4⋊C4×C47—22no—
111C2×C4⋊C4×C47—21no—
112C2×C4⋊C4×C47—21no—
113C2×C4⋊C4×C47D411yescharacteristic
114C2×C42×C47—22no—
115C2×C42×C47—21no—
116C2×C42×C47—21no—
117C2×C42×C47C2311yescharacteristic
118#41×C47—21no—
119#41×C47—21no—
120#41×C47SD1612yes—
121#41×C47Q1612yes—
122C2×Q8×C47—22no—
123C2×Q8×C47—22no—
124C2×Q8×C47—21no—
125C2×Q8×C47—21no—
126C2×Q8×C47—22no—
127C2×Q8×C47—41no—
128C2×Q8×C47—42no—
129C2×Q8×C47—42no—
130C2×Q8×C47C22⋊C411yescharacteristic
131C2×Q8×C47C22⋊C411yescharacteristic
132C2×Q8×C47C22⋊C411yescharacteristic
133C2×Q8×C47C22⋊C412yes—
134C2×D4×C47—21no—
135C2×D4×C47C22⋊C411yescharacteristic
136C22×C4×C47—21no—
137C22×C4×C47—22no—
138C22×C4×C47—21no—
139C22×C4×C47—22no—
140C22×C4×C47—21no—
141C22×C4×C47—21no—
142C22×C4×C47—21no—
143C22×C4×C47C2×D411yescharacteristic
144C22×C4×C47C2×D411yescharacteristic
145C22×C4×C47C2×D411yescharacteristic
146C22×Dic47—41no—
147C4⋊C4×C47—22no—
148C4⋊C4×C47—22no—
149C4⋊C4×C47—22no—
150C4⋊C4×C47—22no—
151C4⋊C4×C47—22no—
152C4⋊C4×C47—22no—
153C4⋊C4×C47—22no—
154C4⋊C4×C47—22no—
155C4⋊C4×C47—21no—
156C4⋊C4×C47—21no—
157C4⋊C4×C47—21no—
158C4⋊C4×C47—21no—
159C4⋊C4×C47—21no—
160C4⋊C4×C47—21no—
161C4⋊C4×C47—21no—
162C4⋊C4×C47—22no—
163C4⋊C4×C47—21no—
164C4⋊C4×C47—21no—
165C4⋊C4×C47—21no—
166C4⋊C4×C47—22no—
167C4⋊C4×C47C22⋊C411yescharacteristic
168C4⋊C4×C47C22⋊C411yescharacteristic
169C4⋊C4×C47C22⋊C411yescharacteristic
170C4⋊C4×C47C22⋊C411yescharacteristic
171C4⋊C4×C47C22⋊C412yes—
172C22⋊C4×C47—22no—
173C22⋊C4×C47—22no—
174C22⋊C4×C47—22no—
175C22⋊C4×C47—21no—
176C22⋊C4×C47—21no—
177C22⋊C4×C47—21no—
178C42×C47—22no—
179C42×C47—22no—
180C42×C47—21no—
181C42×C47—21no—
182C42×C47—21no—
183C42×C47—21no—
184C42×C47—22no—
185C42×C47—21no—
186C42×C47—21no—
187C42×C47—21no—
188C42×C47C22×C411yescharacteristic
189C42×C47C2×D411yescharacteristic
190C42×C47C2×D411yescharacteristic
191C42×C47C2×D411yescharacteristic
192C4⋊Dic47—22no—
193C4⋊Dic47—41no—
194C4×Dic47—41no—
195C4×Dic47—41no—
196#10625329—41no—
197#10625329—41no—
198C2×C47⋊C8—22no—
199C2×C47⋊C8—41no—
200C2×C47⋊C8—41no—
201C2×C47⋊C8—41no—
202C2×C47⋊C8—41no—
203C23×C47—21no—
204C23×C47#12011yescharacteristic
205Q8×C47—21no—
206Q8×C47—21no—
207Q8×C47—21no—
208Q8×C47—21no—
209Q8×C47—42no—
210Q8×C47—42no—
211Q8×C47—44no—
212Q8×C47—41no—
213Q8×C47—41no—
214Q8×C47—82no—
215Q8×C47#10312yescomplemented
216Q8×C47#10412yescomplemented
217D4×C47—22no—
218D4×C47—22no—
219D4×C47—21no—
220D4×C47#10312yescomplemented
221C2×C4×C47—21no—
222C2×C4×C47—21no—
223C2×C4×C47—21no—
224C2×C4×C47—21no—
225C2×C4×C47—22no—
226C2×C4×C47—22no—
227C2×C4×C47—21no—
228C2×C4×C47—22no—
229C2×C4×C47—21no—
230C2×C4×C47—21no—
231C2×C4×C47—21no—
232C2×C4×C47—22no—
233C2×C4×C47—21no—
234C2×C4×C47—41no—
235C2×C4×C47—42no—
236C2×C4×C47—44no—
237C2×C4×C47—41no—
238C2×C4×C47—41no—
239C2×C4×C47—81no—
240C2×C4×C47—81no—
241C2×C4×C47—82no—
242C2×C4×C47C2×C22⋊C411yescharacteristic
243C2×C4×C47C2×C22⋊C411yescharacteristic
244C2×C4×C47C2×C22⋊C411yescharacteristic
245C2×C4×C47#12011yescharacteristic
246C2×C4×C47#12011yescharacteristic
247C2×C4×C47#12011yescharacteristic
248C2×C4×C47C2×SD1611yescharacteristic
249C2×C4×C47C2×Q1611yescharacteristic
250C2×C4×C47#13711yescharacteristic
251C2×C4×C47#13711yescharacteristic
252C2×Dic47—42no—
253C2×Dic47—81no—
254C2×Dic47—81no—
255C47⋊C8—42no—
256C47⋊C8—81no—
257C47⋊C8—81no—
258#8439—471nomaximal, Sylow
259C22×C47—21no—
260C22×C47—21no—
261C22×C47—22no—
262C22×C47—41no—
263C22×C47—41no—
264C22×C47#37911yescharacteristic
265C22×C47#44811yescharacteristic
266C22×C47#45111yescharacteristic
267C4×C47—21no—
268C4×C47—21no—
269C4×C47—22no—
270C4×C47—21no—
271C4×C47—22no—
272C4×C47—41no—
273C4×C47—42no—
274C4×C47—41no—
275C4×C47—41no—
276C4×C47—41no—
277C4×C47—42no—
278C4×C47—41no—
279C4×C47C2×#10311yescharacteristic
280C4×C47#41911yescharacteristic
281C4×C47C2×#10411yescharacteristic
282C4×C47#42111yescharacteristic
283C4×C47#44811yescharacteristic
284C4×C47#45111yescharacteristic
285Dic47—82no—
286Q8×#41D4711yescharacteristic
287#1806—471no—
288C4×#103—471no—
289#1503—471no—
290#1493—471no—
291#1487—471no—
292#1476—471no—
293C2×C47—21no—
294C2×C47—21no—
295C2×C47—42no—
296C2×C47#178711yescharacteristic
297C2×C47#178911yescharacteristic
298C2×C47#179411yescharacteristic
299Q82Dic4711yescharacteristic, complemented
300#557—21no—
301#557Dic4711yescharacteristic, complemented
302#554—22no—
303#554—21no—
304#554—21no—
305#554Dic4711yescharacteristic, complemented
306D4×Q8—21no—
307D4×Q8Dic4711yescharacteristic, complemented
308#533—22no—
309#533—21no—
310#533—21no—
311#533C2×D4711yescharacteristic
312C4×#41—21no—
313C4×#41C2×D4711yescharacteristic
314C2×C4×Q8—21no—
315C2×C4×Q8C2×D4711yescharacteristic
316#431—471no—
317#423—471no—
318C2×#104—941no—
319C2×#103—941no—
320C2×C4×C8—471no—
321C4×C4⋊C4—471no—
322#340—941no—
323#339—941no—
324#328—942no—
325Q8⋊C8—941no—
326Q8⋊C8—941no—
327Q8⋊C8—941no—
328Q8⋊C8—942no—
329D4⋊C8—941no—
330C47#843911yescharacteristic, complemented, Sylow
331C2×#41C22⋊D4711yescharacteristic
332C22×Q8—21no—
333C22×Q8C22⋊D4711yescharacteristic
334C2×M16—941no—
335C22×C8—941no—
336C4⋊Q8—22no—
337C4⋊Q8—21no—
338C4⋊Q8—22no—
339C4⋊Q8—21no—
340C4⋊Q8—21no—
341C4⋊Q8—21no—
342C4⋊Q8C2×Dic4711yescharacteristic
343C4⋊Q8C2×Dic4711yescharacteristic
344#125—22no—
345#125—22no—
346#125—22no—
347#125—21no—
348#125—21no—
349#125—21no—
350C22⋊Q8—22no—
351C22⋊Q8—22no—
352C22⋊Q8—22no—
353C22⋊Q8—22no—
354C22⋊Q8—22no—
355C22⋊Q8—21no—
356C22⋊Q8—22no—
357C22⋊Q8—21no—
358C22⋊Q8—21no—
359C22⋊Q8—22no—
360C22⋊Q8C22⋊D4712yes—
361C4×Q8—22no—
362C4×Q8—22no—
363C4×Q8—22no—
364C4×Q8—21no—
365C4×Q8—21no—
366C4×Q8—21no—
367C4×Q8—21no—
368C4×Q8—21no—
369C4×Q8—21no—
370C4×Q8—21no—
371C4×Q8C2×Dic4711yescharacteristic
372C4×Q8C2×Dic4711yescharacteristic
373C4×Q8C2×Dic4711yescharacteristic
374C4×Q8C22⋊D4711yescharacteristic
375C4×Q8C22⋊D4711yescharacteristic
376C4×Q8C22⋊D4711yescharacteristic
377C4×Q8C22⋊D4712yes—
378C4×D4—22no—
379C4×D4—22no—
380C4×D4—21no—
381C4×D4—21no—
382C4×D4—21no—
383C4×D4—21no—
384C4×D4C2×Dic4711yescharacteristic
385C4×D4C22⋊D4711yescharacteristic
386#117—22no—
387#117—21no—
388#117—21no—
389#117C22⋊D4711yescharacteristic
390C2×C4⋊C4—22no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—941no—
394C2×C4⋊C4C22⋊D4711yescharacteristic
395C2×C42—22no—
396C2×C42—21no—
397C2×C42—21no—
398C2×C42—941no—
399C2×C42C22×D4711yescharacteristic
400C4⋊C8—941no—
401C4⋊C8—941no—
402C4⋊C8—941no—
403#104—1882no—
404#103—944no—
405C22⋊C8—941no—
406C4×C8—941no—
407C4×C8—941no—
408C4×C8—941no—
409#95—941no—
410#41—21no—
411#41—21no—
412#41#1062533612yes—
413#41#1062533712yes—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—22no—
419C2×Q8—41no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8C22⋊Dic4711yescharacteristic
423C2×Q8C22⋊Dic4711yescharacteristic
424C2×Q8C22⋊Dic4711yescharacteristic
425C2×Q8C22⋊Dic4712yes—
426C2×D4—21no—
427C2×D4C22⋊Dic4711yescharacteristic
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—1881no—
436C22×C4C2×C22⋊D4711yescharacteristic
437C22×C4C2×C22⋊D4711yescharacteristic
438C22×C4C2×C22⋊D4711yescharacteristic
439M16—1881no—
440M16—1881no—
441C2×C8—942no—
442C2×C8—1881no—
443C2×C8—1881no—
444C2×C8—1881no—
445C2×C8—1881no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—942no—
467C4⋊C4—1881no—
468C4⋊C4C22⋊Dic4711yescharacteristic
469C4⋊C4C22⋊Dic4711yescharacteristic
470C4⋊C4C22⋊Dic4711yescharacteristic
471C4⋊C4C22⋊Dic4711yescharacteristic
472C4⋊C4C22⋊Dic4712yes—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C42—22no—
480C42—22no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—22no—
487C42—21no—
488C42—21no—
489C42—1881no—
490C42—1881no—
491C42C22×Dic4711yescharacteristic
492C42C2×C22⋊D4711yescharacteristic
493C42C2×C22⋊D4711yescharacteristic
494C42C2×C22⋊D4711yescharacteristic
495C23—21no—
496C23#1309399311yescharacteristic
497Q8—21no—
498Q8—21no—
499Q8—21no—
500Q8—21no—
501Q8—42no—
502Q8—42no—
503Q8—44no—
504Q8—41no—
505Q8—41no—
506Q8—82no—
507Q8#1309394812yescomplemented
508Q8#1309395112yescomplemented
509D4—22no—
510D4—22no—
511D4—21no—
512D4#1309394812yescomplemented
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—82no—
534C2×C4—3761no—
535C2×C4—3761no—
536C2×C4—1882no—
537C2×C4C2×C22⋊Dic4711yescharacteristic
538C2×C4C2×C22⋊Dic4711yescharacteristic
539C2×C4C2×C22⋊Dic4711yescharacteristic
540C2×C4#1309399311yescharacteristic
541C2×C4#1309399311yescharacteristic
542C2×C4#1309399311yescharacteristic
543C2×C4C2×#1062533611yescharacteristic
544C2×C4C2×#1062533711yescharacteristic
545C2×C4#1309405511yescharacteristic
546C2×C4#1309405511yescharacteristic
547C8—3761no—
548C8—3761no—
549C8—1882no—
550C22—21no—
551C22—21no—
552C22—22no—
553C22—41no—
554C22—41no—
555C22#1568897911yescharacteristic
556C22#1568926211yescharacteristic
557C22#1568927911yescharacteristic
558C4—22no—
559C4—21no—
560C4—22no—
561C4—21no—
562C4—21no—
563C4—41no—
564C4—42no—
565C4—41no—
566C4—41no—
567C4—42no—
568C4—41no—
569C4—41no—
570C4—3762no—
571C4C2×#1309394811yescharacteristic
572C4#1568913211yescharacteristic
573C4C2×#1309395111yescharacteristic
574C4#1568914011yescharacteristic
575C4#1568926211yescharacteristic
576C4#1568927911yescharacteristic
577C2—21no—
578C2—21no—
579C2—42no—
580C2#2024332311yescharacteristic
581C2#2024333211yescharacteristic
582C2#2024335211yescharacteristic
subgroupG/H
centerC2×C4#13093993
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingQ8×#41×C47C2
socleC22×C47#379
prime radicalC2×#41×C47D4
cosocleC2×C42×C47C23
hypercenterQ8×#41D47
hypocenterC47#8439
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ Q8×#41

Complements

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

NH ≅ G/Ntype
2. C4×#13093948C2semidirect
3. #20241887C2semidirect
4. #20241843C2semidirect
5. #20241817C2semidirect
22. Q82×C47C4semidirect
24. C47×#557C4semidirect
28. C47×#554C4semidirect
30. D4×Q8×C47C4semidirect
215. Q8×C47#103semidirect
216. Q8×C47#104semidirect
220. D4×C47#103semidirect
299. Q82Dic47semidirect
301. #557Dic47semidirect
305. #554Dic47semidirect
307. D4×Q8Dic47semidirect
330. C47#8439semidirect
507. Q8#13093948semidirect
508. Q8#13093951semidirect
512. D4#13093948semidirect

Maximal subgroups

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

subgroupindex
#202434082
C4×#130939482
#202418872
#202418432
#202418172
#202417702
Q8×#41×C472
#843947