Finite Group Databasebeta

← #26279304

Subgroups — #26279304

Order 12032.

864 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689408C211yesmaximal, characteristic, complemented
2C2×C4×#10625336C211yesmaximal, characteristic
3#20247238C211yesmaximal, characteristic
4#20244922C211yesmaximal, characteristic
5#20244865C211yesmaximal, characteristic, complemented
6#20244759C211yesmaximal, characteristic
7#20244721C211yesmaximal, characteristic, complemented
8#20244113C211yesmaximal, characteristic, complemented
9#20243947C211yesmaximal, characteristic, complemented
10#20243743C211yesmaximal, characteristic
11#20243743C211yesmaximal, characteristic
12#20243656C211yesmaximal, characteristic, complemented
13#20243656C211yesmaximal, characteristic, complemented
14#20243648C211yesmaximal, characteristic
15Q8×C4⋊C4×C47C211yesmaximal, characteristic, complemented
16C22×#10625336C2211yescharacteristic
17C2×#13094024C2211yescharacteristic
18C2×#13094002—21no—
19C2×C4×D188C2211yescharacteristic
20#15689408—24no—
21C4×#10625336—24no—
22C2×#13093955C2211yescharacteristic
23C2×#13093955C2211yescharacteristic
24C2×#13093955C2211yescharacteristic
25C2×#13093955C2211yescharacteristic
26C2×#13093955C2211yescharacteristic
27C2×C4⋊(C47⋊C8)C2211yescharacteristic
28C2×#13093948C2211yescharacteristic
29C2×#13093948C2211yescharacteristic
30C2×#13093948C2211yescharacteristic
31C2×#13093944C2211yescharacteristic
32C2×#13093944C2211yescharacteristic
33C2×#13093944C2211yescharacteristic
34C2×C4×C47⋊C8C2211yescharacteristic
35#15689045—21no—
36#15689036—21no—
37#15689025C2211yescharacteristic
38#15689025C2211yescharacteristic
39#15689021C2211yescharacteristic
40#15689021C2211yescharacteristic
41#15688997—21no—
42#15688885C2211yescharacteristic
43#15688885C2211yescharacteristic
44#15688885C2211yescharacteristic
45#15688885C2211yescharacteristic
46#15688878C2211yescharacteristic
47#15688878C2211yescharacteristic
48#15688878C2211yescharacteristic
49#15688878C2211yescharacteristic
50C2×C4×Q8×C47C2211yescharacteristic
51C2×C4×Q8×C47C2211yescharacteristic
52C2×C4×Q8×C47C2211yescharacteristic
53C47×#389—21no—
54C47×#389—21no—
55C47×#389C2211yescharacteristic
56C47×#389C2211yescharacteristic
57C47×#384—21no—
58C47×#384C2211yescharacteristic
59C4×C4⋊C4×C47—21no—
60C4×C4⋊C4×C47C2211yescharacteristic
61C22×Dic94—21no—
62C22×D188C2311yescharacteristic
63C22×C4×D47—21no—
64C2×#10625336—22no—
65C2×#10625336—22no—
66C2×#10625336Q814yescomplemented
67C22×C47⋊C8C2311yescharacteristic
68C22×C47⋊C8C2311yescharacteristic
69C22×C47⋊C8C2311yescharacteristic
70C22×C47⋊C8C2311yescharacteristic
71#13094024—22no—
72#13094024—41no—
73#13094002—44no—
74C4×D188—22no—
75C4×D188—41no—
76C2×C4⋊Dic47—21no—
77C2×C4⋊Dic47C2311yescharacteristic
78C2×C4⋊Dic47C2311yescharacteristic
79C2×C4⋊Dic47C2311yescharacteristic
80C2×Dic47⋊C4—21no—
81C2×Dic47⋊C4—21no—
82C2×(C2×D47)⋊C4—21no—
83C2×(C2×D47)⋊C4—21no—
84C2×(C2×D47)⋊C4—21no—
85C2×(C2×D47)⋊C4—21no—
86C2×(C2×D47)⋊C4—21no—
87#13093955—22no—
88#13093955—22no—
89#13093955—22no—
90#13093955—22no—
91#13093955—41no—
92C4⋊(C47⋊C8)—22no—
93#13093948—22no—
94#13093948—22no—
95#13093948D414yes—
96#13093944—22no—
97#13093944—22no—
98#13093944—41no—
99C4×C47⋊C8—22no—
100C47⋊#95—21no—
101C47⋊#95—21no—
102C22×Q8×C47C2311yescharacteristic
103C4×Q8×C47—22no—
104C4×Q8×C47—22no—
105C4×Q8×C47—22no—
106C4×Q8×C47—22no—
107C4×Q8×C47—41no—
108C4×Q8×C47—41no—
109C2×C4⋊C4×C47—21no—
110C2×C4⋊C4×C47—21no—
111C2×C4⋊C4×C47—21no—
112C2×C4⋊C4×C47—21no—
113C2×C4⋊C4×C47—21no—
114C2×C4⋊C4×C47—21no—
115C2×C4⋊C4×C47—21no—
116C2×C4⋊C4×C47D411yescharacteristic
117C2×C4⋊C4×C47D411yescharacteristic
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×C47C2311yescharacteristic
125C2×C42×C47C2311yescharacteristic
126C2×C42×C47C2311yescharacteristic
127#95×C47—21no—
128#95×C47—21no—
129#95×C47D411yescharacteristic
130#95×C47D411yescharacteristic
131C23×D47—21no—
132C2×Q8×C47—21no—
133C2×Q8×C47—21no—
134C2×Q8×C47—21no—
135C2×Q8×C47C2×D412yes—
136C2×Q8×C47C2×Q812yes—
137C2×Q8×C47#4112yes—
138C22×C4×C47—21no—
139C22×C4×C47—21no—
140C22×C4×C47—21no—
141C22×C4×C47—21no—
142C22×C4×C47C2×D411yescharacteristic
143C22×C4×C47C2×D411yescharacteristic
144C22×C4×C47C2×D411yescharacteristic
145C22×C4×C47C2×D411yescharacteristic
146C22×C4×C47C2×D411yescharacteristic
147C22×C4×C47C2×D411yescharacteristic
148C22×C4×C47C2411yescharacteristic
149C22×Dic47—21no—
150C22×Dic47—21no—
151C22×Dic47—21no—
152C2×D188—21no—
153C2×D188—21no—
154C2×D188—21no—
155C2×D188—21no—
156C2×D188—21no—
157C2×D188C2×Q812yes—
158C2×C4×D47—42no—
159C2×C4×D47—42no—
160C2×C4×D47—81no—
161C2×Dic94—42no—
162C2×Dic94—42no—
163C2×Dic94—81no—
164C4⋊C4×C47—24no—
165C4⋊C4×C47—21no—
166C4⋊C4×C47—21no—
167C4⋊C4×C47—21no—
168C4⋊C4×C47—21no—
169C4⋊C4×C47—21no—
170C4⋊C4×C47—41no—
171C4⋊C4×C47—42no—
172C4⋊C4×C47—42no—
173C4⋊C4×C47—42no—
174C4⋊C4×C47—81no—
175C4⋊C4×C47—81no—
176C4⋊C4×C47—81no—
177C4⋊C4×C47SD1614yescomplemented
178C4⋊C4×C47#4112yes—
179C42×C47—24no—
180C42×C47—21no—
181C42×C47—21no—
182C42×C47—21no—
183C42×C47—21no—
184C42×C47—21no—
185C42×C47—21no—
186C42×C47—42no—
187C42×C47—42no—
188#10625336—28no—
189(C2×D47)⋊C4—42no—
190(C2×D47)⋊C4—42no—
191(C2×D47)⋊C4—42no—
192(C2×D47)⋊C4—42no—
193(C2×D47)⋊C4—81no—
194C4⋊Dic47—21no—
195C4⋊Dic47—21no—
196C4⋊Dic47—21no—
197C4⋊Dic47—21no—
198C4⋊Dic47—21no—
199C4⋊Dic47—42no—
200C4⋊Dic47C2×D412yes—
201Dic47⋊C4—42no—
202Dic47⋊C4—42no—
203C2×C47⋊C8—21no—
204C2×C47⋊C8—21no—
205C2×C47⋊C8—21no—
206C2×C47⋊C8—21no—
207C2×C47⋊C8—21no—
208C2×C47⋊C8—21no—
209C2×C47⋊C8—21no—
210C2×C47⋊C8—21no—
211C2×C47⋊C8—21no—
212C2×C47⋊C8—21no—
213C2×C47⋊C8C2×D412yes—
214C2×C47⋊C8C2×Q812yes—
215C23×C47C22×D411yescharacteristic
216C22×D47—22no—
217C22×D47—41no—
218C22×D47—41no—
219C22×D47—41no—
220C22×D47—41no—
221C22×D47—41no—
222C22×D47—41no—
223Q8×C47—22no—
224Q8×C47—22no—
225Q8×C47—22no—
226Q8×C47C22⋊Q814yescomplemented
227C2×C4×C47—21no—
228C2×C4×C47—22no—
229C2×C4×C47—21no—
230C2×C4×C47—21no—
231C2×C4×C47—22no—
232C2×C4×C47—22no—
233C2×C4×C47—21no—
234C2×C4×C47—21no—
235C2×C4×C47—21no—
236C2×C4×C47—21no—
237C2×C4×C47—22no—
238C2×C4×C47—21no—
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—41no—
247C2×C4×C47—41no—
248C2×C4×C47—41no—
249C2×C4×C47—41no—
250C2×C4×C47—41no—
251C2×C4×C47—41no—
252C2×C4×C47—41no—
253C2×C4×C47—41no—
254C2×C4×C47C2×SD1612yes—
255C2×C4×C47C2×SD1612yes—
256C2×C4×C47C2×SD1612yes—
257C2×C4×C47C22×D411yescharacteristic
258C2×C4×C47C22×Q811yescharacteristic
259C2×C4×C47C2×#4111yescharacteristic
260C2×C4×C47C2×#4111yescharacteristic
261C2×C4×C47H(5,𝔽2)11yescharacteristic
262C2×C4×C472−1+411yescharacteristic
263C2×Dic47—22no—
264C2×Dic47—41no—
265C2×Dic47—41no—
266C2×Dic47—41no—
267C2×Dic47—41no—
268C2×Dic47—41no—
269C2×Dic47—41no—
270C2×Dic47—41no—
271C2×Dic47—41no—
272D188—22no—
273D188—22no—
274D188—41no—
275D188—41no—
276C4×D47—84no—
277Dic94—84no—
278C47⋊C8—22no—
279C47⋊C8—22no—
280C47⋊C8—41no—
281C47⋊C8—41no—
282#26373—471nomaximal, Sylow
283C22×C47#52511yescharacteristic
284C22×C47D4211yescharacteristic
285C22×C47D4×Q811yescharacteristic
286C22×C47C22×SD1611yescharacteristic
287C22×C47#57511yescharacteristic
288C22×C47#57711yescharacteristic
289C22×C47#57711yescharacteristic
290C2×D47—42no—
291C2×D47—42no—
292C2×D47—42no—
293C2×D47—81no—
294C2×D47—81no—
295C2×D47—81no—
296C2×D47—81no—
297C4×C47—22no—
298C4×C47—24no—
299C4×C47—22no—
300C4×C47—22no—
301C4×C47—41no—
302C4×C47—42no—
303C4×C47—41no—
304C4×C47—41no—
305C4×C47—81no—
306C4×C47—81no—
307C4×C47C2×C22⋊Q811yescharacteristic
308C4×C47#53911yescharacteristic
309C4×C47#54711yescharacteristic
310C4×C47#55411yescharacteristic
311Dic47—42no—
312Dic47—81no—
313Dic47—81no—
314C2×#475—471no—
315C2×C4×SD16—471no—
316#2452—471no—
317Q8×C4⋊C4D4711yescharacteristic, complemented
318#2094—471no—
319#2084—471no—
320#2065—471no—
321#2058—471no—
322#1941—471no—
323#1916—471no—
324#1880—471no—
325#1880—471no—
326#1865—471no—
327#1865—471no—
328#1864—471no—
329C2×C47#244911yescharacteristic
330C2×C47#310311yescharacteristic
331C2×C47#312511yescharacteristic
332C2×C47#327111yescharacteristic
333C2×C47D4×SD1611yescharacteristic
334C2×C47Q8×SD1611yescharacteristic
335C2×C47#338811yescharacteristic
336D47—82no—
337C22×SD16—471no—
338C2×C4⋊Q8—471no—
339C2×C22⋊Q8—941no—
340C2×C4×Q8C2×D4711yescharacteristic
341C2×C4×Q8C2×D4711yescharacteristic
342C2×C4×Q8C2×D4711yescharacteristic
343C2×C4×D4—471no—
344#475—944no—
345C4×SD16—944no—
346C2×#106—471no—
347C2×#106—471no—
348C2×#106—471no—
349C2×#106—471no—
350C2×#106—471no—
351C2×C4⋊C8—471no—
352C2×#103—471no—
353C2×#103—471no—
354C2×#103—471no—
355C2×#102—471no—
356C2×#102—471no—
357C2×#102—471no—
358C2×C4×C8—471no—
359#396—941no—
360#393—941no—
361#390—471no—
362#390—471no—
363#389—21no—
364#389—21no—
365#389—471no—
366#389—471no—
367#389C2×D4711yescharacteristic
368#389C2×D4711yescharacteristic
369#385—941no—
370#384—21no—
371#384C2×D4711yescharacteristic
372C4×C4⋊C4—21no—
373C4×C4⋊C4C2×D4711yescharacteristic
374#340—471no—
375#340—471no—
376#340—471no—
377#340—471no—
378#336—471no—
379#336—471no—
380#336—471no—
381#336—471no—
382C47#2637311yescharacteristic, complemented, Sylow
383C22×Q8—941no—
384C22×Q8C22×D4711yescharacteristic
385C22×D4—471no—
386C23×C4—941no—
387C2×SD16—474no—
388C2×SD16—942no—
389C2×SD16—942no—
390C22×C8—471no—
391C22×C8—471no—
392C22×C8—471no—
393C22×C8—471no—
394C4⋊Q8—942no—
395C4⋊Q8—1881no—
396C22⋊Q8—1884no—
397C4×Q8—22no—
398C4×Q8—22no—
399C4×Q8—22no—
400C4×Q8—22no—
401C4×Q8—41no—
402C4×Q8—41no—
403C4×D4—942no—
404C4×D4—1881no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—471no—
413C2×C4⋊C4—471no—
414C2×C4⋊C4—471no—
415C2×C4⋊C4—941no—
416C2×C4⋊C4—941no—
417C2×C4⋊C4—941no—
418C2×C4⋊C4C22⋊D4711yescharacteristic
419C2×C4⋊C4C22⋊D4711yescharacteristic
420C2×C4⋊C4C22×D4711yescharacteristic
421C2×C4⋊C4C22×D4711yescharacteristic
422C2×C4⋊C4C22×D4711yescharacteristic
423C2×C22⋊C4—941no—
424C2×C22⋊C4—941no—
425C2×C22⋊C4—941no—
426C2×C22⋊C4—941no—
427C2×C22⋊C4—941no—
428C2×C42—21no—
429C2×C42—21no—
430C2×C42—21no—
431C2×C42C22×D4711yescharacteristic
432C2×C42C22×D4711yescharacteristic
433C2×C42C22×D4711yescharacteristic
434#106—942no—
435#106—942no—
436#106—942no—
437#106—942no—
438#106—1881no—
439C4⋊C8—942no—
440#103—474no—
441#103—942no—
442#103—942no—
443#102—942no—
444#102—942no—
445#102—1881no—
446C4×C8—942no—
447#95—21no—
448#95—21no—
449#95—941no—
450#95—941no—
451#95C22⋊D4711yescharacteristic
452#95C22⋊D4711yescharacteristic
453C24—941no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—3761no—
458C2×Q8—1882no—
459C2×Q8—1882no—
460C2×Q8#1062535012yes—
461C2×Q8D4×D4712yes—
462C2×Q8Q8×D4712yes—
463C2×D4—472no—
464C2×D4—941no—
465C2×D4—941no—
466C2×D4—941no—
467C2×D4—941no—
468C2×D4—941no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—941no—
474C22×C4—941no—
475C22×C4—941no—
476C22×C4—3761no—
477C22×C4—1882no—
478C22×C4—1882no—
479C22×C4C2×C22⋊D4711yescharacteristic
480C22×C4C2×C22⋊D4711yescharacteristic
481C22×C4C2×C22⋊D4711yescharacteristic
482C22×C4C2×C22⋊D4711yescharacteristic
483C22×C4C2×C22⋊D4711yescharacteristic
484C22×C4C2×C22⋊D4711yescharacteristic
485C22×C4C23×D4711yescharacteristic
486SD16—948no—
487C2×C8—472no—
488C2×C8—472no—
489C2×C8—941no—
490C2×C8—941no—
491C2×C8—941no—
492C2×C8—941no—
493C2×C8—941no—
494C2×C8—941no—
495C2×C8—941no—
496C2×C8—941no—
497C2×C8—941no—
498C2×C8—941no—
499C4⋊C4—24no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—41no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—81no—
510C4⋊C4—81no—
511C4⋊C4—81no—
512C4⋊C4—472no—
513C4⋊C4—941no—
514C4⋊C4—941no—
515C4⋊C4—941no—
516C4⋊C4—941no—
517C4⋊C4—941no—
518C4⋊C4—1882no—
519C4⋊C4—1882no—
520C4⋊C4—1882no—
521C4⋊C4#1062533614yescomplemented
522C4⋊C4#1062535012yes—
523C22⋊C4—3761no—
524C22⋊C4—1882no—
525C22⋊C4—1882no—
526C22⋊C4—1882no—
527C22⋊C4—1882no—
528C42—24no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42—42no—
536C42—42no—
537C23—942no—
538C23—1881no—
539C23—1881no—
540C23—1881no—
541C23—1881no—
542C23—1881no—
543C23—1881no—
544C23C22×C22⋊D4711yescharacteristic
545Q8—22no—
546Q8—22no—
547Q8—22no—
548Q8—3764no—
549Q8#1309400214yescomplemented
550D4—942no—
551D4—942no—
552D4—1881no—
553D4—1881no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—942no—
582C2×C4—3764no—
583C2×C4—1881no—
584C2×C4—1881no—
585C2×C4—1881no—
586C2×C4—1881no—
587C2×C4—1881no—
588C2×C4—1881no—
589C2×C4—1881no—
590C2×C4—1881no—
591C2×C4C2×#1062533612yes—
592C2×C4C2×#1062533612yes—
593C2×C4C2×#1062533612yes—
594C2×C4C2×D4×D4711yescharacteristic
595C2×C4C2×Q8×D4711yescharacteristic
596C2×C4C2×#1062535011yescharacteristic
597C2×C4C2×#1062535011yescharacteristic
598C2×C4#1309406911yescharacteristic
599C2×C4#1309407211yescharacteristic
600C8—942no—
601C8—942no—
602C8—1881no—
603C8—1881no—
604C22—3761no—
605C22—3761no—
606C22—3761no—
607C22—3761no—
608C22—1882no—
609C22—1882no—
610C22—1882no—
611C22#1568963411yescharacteristic
612C22D4×C22⋊D4711yescharacteristic
613C22Q8×C22⋊D4711yescharacteristic
614C22C22×#1062533611yescharacteristic
615C22#1568992111yescharacteristic
616C22#1568993111yescharacteristic
617C22#1568993411yescharacteristic
618C4—22no—
619C4—22no—
620C4—24no—
621C4—22no—
622C4—41no—
623C4—42no—
624C4—41no—
625C4—41no—
626C4—81no—
627C4—81no—
628C4—3761no—
629C4—3761no—
630C4—1882no—
631C4C2×#1309400211yescharacteristic
632C4#1568972011yescharacteristic
633C4#1568977411yescharacteristic
634C4#1568982811yescharacteristic
635C2—3762no—
636C2#2024719711yescharacteristic
637C2#2025341911yescharacteristic
638C2#2025366511yescharacteristic
639C2#2025496911yescharacteristic
640C2D4×#1062533611yescharacteristic
641C2Q8×#1062533611yescharacteristic
642C2#2025632711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingQ8×C4⋊C4×C47C2
socleC23×C47C22×D4
prime radicalC22×D188C23
hypercenterQ8×C4⋊C4D47
hypocenterC47#26373
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#15689408C2semidirect
5. #20244865C2semidirect
7. #20244721C2semidirect
8. #20244113C2semidirect
9. #20243947C2semidirect
12. #20243656C2semidirect
13. #20243656C2semidirect
15. Q8×C4⋊C4×C47C2semidirect
66. C2×#10625336Q8semidirect
177. C4⋊C4×C47SD16semidirect
226. Q8×C47C22⋊Q8semidirect
317. Q8×C4⋊C4D47semidirect
382. C47#26373semidirect
521. C4⋊C4#10625336semidirect
549. Q8#13094002semidirect

Maximal subgroups

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

subgroupindex
C2×#156894082
C2×C4×#106253362
#202472382
#202449222
#202448652
#202447592
#202447212
#202441132
#202439472
#202437432
#202436562
#202436482
Q8×C4⋊C4×C472
#2637347