Finite Group Databasebeta

← #26279361

Subgroups — #26279361

Order 12032.

856 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689415C211yesmaximal, characteristic, complemented
2C2×C4×#10625335C211yesmaximal, characteristic
3#20246308C211yesmaximal, characteristic, complemented
4#20244909C211yesmaximal, characteristic
5#20244869C211yesmaximal, characteristic, complemented
6#20244776C211yesmaximal, characteristic
7#20244718C211yesmaximal, characteristic, complemented
8#20244118C211yesmaximal, characteristic, complemented
9#20243947C211yesmaximal, characteristic, complemented
10#20243746C211yesmaximal, characteristic
11#20243746C211yesmaximal, characteristic
12#20243655C211yesmaximal, characteristic, complemented
13#20243641C211yesmaximal, characteristic, complemented
14#20243635C211yesmaximal, characteristic
15C47×#2452C211yesmaximal, characteristic
16C22×#10625335C2211yescharacteristic
17C2×C4×Dic94C2211yescharacteristic
18C2×C4×Dic94C2211yescharacteristic
19C2×C4×Dic94C2211yescharacteristic
20#15689415—24no—
21C4×#10625335—24no—
22C2×#13093957C2211yescharacteristic
23C2×#13093957C2211yescharacteristic
24C2×#13093957C2211yescharacteristic
25C2×#13093955C2211yescharacteristic
26C2×#13093955C2211yescharacteristic
27C2×C4⋊(C47⋊C8)C2211yescharacteristic
28C2×#13093947C2211yescharacteristic
29C2×#13093947C2211yescharacteristic
30C2×#13093947C2211yescharacteristic
31C2×#13093945C2211yescharacteristic
32C2×#13093945C2211yescharacteristic
33C2×#13093945C2211yescharacteristic
34C2×C4×C47⋊C8C2211yescharacteristic
35#15689021C2211yescharacteristic
36#15689021C2211yescharacteristic
37#15689018—21no—
38#15689018—21no—
39#15688989C2211yescharacteristic
40#15688987—21no—
41#15688974—21no—
42#15688885C2211yescharacteristic
43#15688885C2211yescharacteristic
44#15688885C2211yescharacteristic
45#15688885C2211yescharacteristic
46#15688878C2211yescharacteristic
47#15688878C2211yescharacteristic
48#15688878C2211yescharacteristic
49#15688878C2211yescharacteristic
50C2×C4⋊Q8×C47C2211yescharacteristic
51C2×C22⋊Q8×C47—21no—
52C2×C4×D4×C47C2211yescharacteristic
53C47×#396—21no—
54C47×#393—21no—
55C47×#390C2211yescharacteristic
56C47×#390C2211yescharacteristic
57C47×#389C2211yescharacteristic
58C47×#389C2211yescharacteristic
59C47×#385—21no—
60C4×C4⋊C4×C47C2211yescharacteristic
61C22×Dic94C2311yescharacteristic
62C2×#10625335—22no—
63C2×#10625335—22no—
64C2×#10625335Q814yescomplemented
65C22×C47⋊C8C2311yescharacteristic
66C22×C47⋊C8C2311yescharacteristic
67C22×C47⋊C8C2311yescharacteristic
68C22×C47⋊C8C2311yescharacteristic
69C4×Dic94—22no—
70C4×Dic94—22no—
71C4×Dic94—22no—
72C4×Dic94—22no—
73C4×Dic94—41no—
74C4×Dic94—41no—
75C2×C4⋊Dic47C2311yescharacteristic
76C2×C4⋊Dic47C2311yescharacteristic
77C2×C4⋊Dic47C2311yescharacteristic
78C2×Dic47⋊C4—21no—
79C2×Dic47⋊C4—21no—
80C2×Dic47⋊C4—21no—
81C2×Dic47⋊C4—21no—
82C2×Dic47⋊C4—21no—
83C2×C4×Dic47—21no—
84C2×C4×Dic47—21no—
85C2×C4×Dic47—21no—
86#13093957—22no—
87#13093957—22no—
88#13093957—22no—
89#13093955—22no—
90#13093955—41no—
91C4⋊(C47⋊C8)—22no—
92#13093947—22no—
93#13093947—22no—
94#13093947D414yes—
95#13093945—22no—
96#13093945—22no—
97#13093945—41no—
98C4×C47⋊C8—22no—
99C47⋊#95—21no—
100C47⋊#95—21no—
101C47⋊#95—21no—
102C47⋊#95—21no—
103C22×Q8×C47—21no—
104C22×D4×C47C2311yescharacteristic
105C23×C4×C47—21no—
106C4⋊Q8×C47—22no—
107C4⋊Q8×C47—41no—
108C22⋊Q8×C47—44no—
109C4×D4×C47—22no—
110C4×D4×C47—41no—
111C2×C4⋊C4×C47—21no—
112C2×C4⋊C4×C47—21no—
113C2×C4⋊C4×C47—21no—
114C2×C4⋊C4×C47D411yescharacteristic
115C2×C4⋊C4×C47D411yescharacteristic
116C2×C4⋊C4×C47C2311yescharacteristic
117C2×C4⋊C4×C47C2311yescharacteristic
118C2×C4⋊C4×C47C2311yescharacteristic
119C2×C22⋊C4×C47—21no—
120C2×C22⋊C4×C47—21no—
121C2×C22⋊C4×C47—21no—
122C2×C22⋊C4×C47—21no—
123C2×C22⋊C4×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
131C24×C47—21no—
132C2×Q8×C47—42no—
133C2×Q8×C47—42no—
134C2×Q8×C47—81no—
135C2×D4×C47—21no—
136C2×D4×C47—21no—
137C2×D4×C47—21no—
138C2×D4×C47—21no—
139C2×D4×C47—21no—
140C2×D4×C47C2×Q812yes—
141C22×C4×C47—21no—
142C22×C4×C47—21no—
143C22×C4×C47—21no—
144C22×C4×C47—42no—
145C22×C4×C47—42no—
146C22×C4×C47—81no—
147C22×C4×C47C2×D411yescharacteristic
148C22×C4×C47C2×D411yescharacteristic
149C22×C4×C47C2×D411yescharacteristic
150C22×C4×C47C2×D411yescharacteristic
151C22×C4×C47C2×D411yescharacteristic
152C22×C4×C47C2×D411yescharacteristic
153C22×C4×C47C2411yescharacteristic
154C22×Dic47—21no—
155C22×Dic47—21no—
156C22×Dic47—21no—
157C22×Dic47—21no—
158C2×Dic94—21no—
159C2×Dic94—21no—
160C2×Dic94—21no—
161C2×Dic94C2×D412yes—
162C2×Dic94C2×Q812yes—
163C2×Dic94#4112yes—
164C4⋊C4×C47—22no—
165C4⋊C4×C47—22no—
166C4⋊C4×C47—21no—
167C4⋊C4×C47—21no—
168C4⋊C4×C47—21no—
169C4⋊C4×C47—21no—
170C4⋊C4×C47—21no—
171C4⋊C4×C47—42no—
172C4⋊C4×C47—42no—
173C4⋊C4×C47—42no—
174C4⋊C4×C47C2×D412yes—
175C22⋊C4×C47—42no—
176C22⋊C4×C47—42no—
177C22⋊C4×C47—42no—
178C22⋊C4×C47—42no—
179C22⋊C4×C47—81no—
180C42×C47—21no—
181C42×C47—21no—
182C42×C47—21no—
183C42×C47—21no—
184C42×C47—21no—
185C42×C47—21no—
186#10625335—28no—
187C4⋊Dic47—21no—
188C4⋊Dic47—21no—
189C4⋊Dic47—21no—
190C4⋊Dic47—21no—
191C4⋊Dic47—21no—
192C4⋊Dic47#4112yes—
193Dic47⋊C4—24no—
194Dic47⋊C4—42no—
195Dic47⋊C4—42no—
196Dic47⋊C4—42no—
197Dic47⋊C4—42no—
198C4×Dic47—24no—
199C4×Dic47—42no—
200C4×Dic47—42no—
201C2×C47⋊C8—21no—
202C2×C47⋊C8—21no—
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⋊C8C2×D412yes—
212C2×C47⋊C8C2×Q812yes—
213C23×C47—22no—
214C23×C47—41no—
215C23×C47—41no—
216C23×C47—41no—
217C23×C47—41no—
218C23×C47—41no—
219C23×C47—41no—
220C23×C47C22×D411yescharacteristic
221Q8×C47—84no—
222D4×C47—22no—
223D4×C47—22no—
224D4×C47—41no—
225D4×C47—41no—
226C2×C4×C47—21no—
227C2×C4×C47—21no—
228C2×C4×C47—21no—
229C2×C4×C47—21no—
230C2×C4×C47—21no—
231C2×C4×C47—21no—
232C2×C4×C47—21no—
233C2×C4×C47—21no—
234C2×C4×C47—22no—
235C2×C4×C47—21no—
236C2×C4×C47—21no—
237C2×C4×C47—21no—
238C2×C4×C47—21no—
239C2×C4×C47—21no—
240C2×C4×C47—21no—
241C2×C4×C47—21no—
242C2×C4×C47—21no—
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×C47—41no—
250C2×C4×C47—41no—
251C2×C4×C47—84no—
252C2×C4×C47#13512yes—
253C2×C4×C47AGL(1,ℤ/8)12yes—
254C2×C4×C47C22×D411yescharacteristic
255C2×C4×C47C22×Q811yescharacteristic
256C2×C4×C47C2×#4111yescharacteristic
257C2×C4×C47C2×#4111yescharacteristic
258C2×C4×C47H(5,𝔽2)11yescharacteristic
259C2×C4×C472−1+411yescharacteristic
260C2×Dic47—22no—
261C2×Dic47—22no—
262C2×Dic47—22no—
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—
272Dic94—22no—
273Dic94—22no—
274Dic94—22no—
275Dic94C22⋊Q814yescomplemented
276C47⋊C8—22no—
277C47⋊C8—22no—
278C47⋊C8—41no—
279C47⋊C8—41no—
280#26377—471nomaximal, Sylow
281C22×C47—42no—
282C22×C47—42no—
283C22×C47—42no—
284C22×C47—81no—
285C22×C47—81no—
286C22×C47—81no—
287C22×C47—81no—
288C22×C47#52511yescharacteristic
289C22×C47D4211yescharacteristic
290C22×C47D4×Q811yescharacteristic
291C22×C47C2×#13511yescharacteristic
292C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
293C22×C47#57711yescharacteristic
294C22×C47#57811yescharacteristic
295C4×C47—22no—
296C4×C47—22no—
297C4×C47—41no—
298C4×C47—41no—
299C4×C47—41no—
300C4×C47—42no—
301C4×C47—41no—
302C4×C47—81no—
303C4×C47—81no—
304C4×C47C2×C22⋊Q811yescharacteristic
305C4×C47#53911yescharacteristic
306C4×C47#54711yescharacteristic
307C4×C47#55411yescharacteristic
308Dic47—24no—
309Dic47—42no—
310Dic47—81no—
311Dic47—81no—
312C2×#477—471no—
313C2×C4×SD16—471no—
314#2452D4711yescharacteristic
315#2351—471no—
316#2092—471no—
317#2085—471no—
318#2067—471no—
319#2058—471no—
320#1942—471no—
321#1916—471no—
322#1880—471no—
323#1880—471no—
324#1865—471no—
325#1863—471no—
326#1862—471no—
327C2×C47—82no—
328C2×C47#244911yescharacteristic
329C2×C47#310611yescharacteristic
330C2×C47#312111yescharacteristic
331C2×C47#326511yescharacteristic
332C2×C47#328511yescharacteristic
333C2×C47#338111yescharacteristic
334C2×C47#338611yescharacteristic
335C22×SD16—471no—
336C2×C4⋊Q8C2×D4711yescharacteristic
337C2×C22⋊Q8—21no—
338C2×C4×Q8—471no—
339C2×C4×Q8—471no—
340C2×C4×Q8—471no—
341C2×C4×D4C2×D4711yescharacteristic
342#477—944no—
343C4×SD16—944no—
344C2×#107—471no—
345C2×#107—471no—
346C2×#107—471no—
347C2×#106—471no—
348C2×#106—471no—
349C2×C4⋊C8—471no—
350C2×#103—471no—
351C2×#103—471no—
352C2×#103—471no—
353C2×#102—471no—
354C2×#102—471no—
355C2×#102—471no—
356C2×C4×C8—471no—
357#396—21no—
358#393—21no—
359#390C2×D4711yescharacteristic
360#390C2×D4711yescharacteristic
361#389—471no—
362#389—471no—
363#389—941no—
364#389—941no—
365#389C2×D4711yescharacteristic
366#389C2×D4711yescharacteristic
367#385—21no—
368#382—471no—
369#382—941no—
370C4×C4⋊C4—941no—
371C4×C4⋊C4C2×D4711yescharacteristic
372#340—471no—
373#340—471no—
374#340—471no—
375#340—471no—
376#336—471no—
377#336—471no—
378#336—471no—
379#336—471no—
380C47#2637711yescharacteristic, complemented, Sylow
381C22×Q8—21no—
382C22×Q8—471no—
383C22×D4C22×D4711yescharacteristic
384C23×C4—21no—
385C2×SD16—474no—
386C2×SD16—942no—
387C2×SD16—942no—
388C22×C8—471no—
389C22×C8—471no—
390C22×C8—471no—
391C22×C8—471no—
392C4⋊Q8—22no—
393C4⋊Q8—41no—
394C22⋊Q8—44no—
395C4×Q8—942no—
396C4×Q8—942no—
397C4×Q8—942no—
398C4×Q8—942no—
399C4×Q8—1881no—
400C4×Q8—1881no—
401C4×D4—22no—
402C4×D4—41no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—471no—
407C2×C4⋊C4—471no—
408C2×C4⋊C4—471no—
409C2×C4⋊C4—941no—
410C2×C4⋊C4—941no—
411C2×C4⋊C4—941no—
412C2×C4⋊C4—941no—
413C2×C4⋊C4—941no—
414C2×C4⋊C4C22⋊D4711yescharacteristic
415C2×C4⋊C4C22⋊D4711yescharacteristic
416C2×C4⋊C4C22×D4711yescharacteristic
417C2×C4⋊C4C22×D4711yescharacteristic
418C2×C4⋊C4C22×D4711yescharacteristic
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C42—941no—
425C2×C42—941no—
426C2×C42—941no—
427C2×C42C22×D4711yescharacteristic
428C2×C42C22×D4711yescharacteristic
429C2×C42C22×D4711yescharacteristic
430#107—942no—
431#107—942no—
432#107—942no—
433#106—942no—
434#106—1881no—
435C4⋊C8—942no—
436#103—474no—
437#103—942no—
438#103—942no—
439#102—942no—
440#102—942no—
441#102—1881no—
442C4×C8—942no—
443#95—21no—
444#95—21no—
445#95—941no—
446#95—941no—
447#95—941no—
448#95—941no—
449#95C22⋊D4711yescharacteristic
450#95C22⋊D4711yescharacteristic
451C24—21no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—81no—
455C2×Q8—472no—
456C2×Q8—472no—
457C2×Q8—472no—
458C2×Q8—941no—
459C2×Q8—941no—
460C2×Q8—941no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4Q8×D4712yes—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—81no—
473C22×C4—941no—
474C22×C4—941no—
475C22×C4—941no—
476C22×C4—941no—
477C22×C4C2×C22⋊D4711yescharacteristic
478C22×C4C2×C22⋊D4711yescharacteristic
479C22×C4C2×C22⋊D4711yescharacteristic
480C22×C4C2×C22⋊D4711yescharacteristic
481C22×C4C2×C22⋊D4711yescharacteristic
482C22×C4C2×C22⋊D4711yescharacteristic
483C22×C4C23×D4711yescharacteristic
484SD16—948no—
485C2×C8—472no—
486C2×C8—472no—
487C2×C8—941no—
488C2×C8—941no—
489C2×C8—941no—
490C2×C8—941no—
491C2×C8—941no—
492C2×C8—941no—
493C2×C8—941no—
494C2×C8—941no—
495C2×C8—941no—
496C2×C8—941no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—472no—
508C4⋊C4—944no—
509C4⋊C4—941no—
510C4⋊C4—941no—
511C4⋊C4—941no—
512C4⋊C4—941no—
513C4⋊C4—941no—
514C4⋊C4—1882no—
515C4⋊C4—1882no—
516C4⋊C4—1882no—
517C4⋊C4—1882no—
518C4⋊C4D4×D4712yes—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—81no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—944no—
531C42—1882no—
532C42—1882no—
533C23—22no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23C22×C22⋊D4711yescharacteristic
541Q8—84no—
542Q8—474no—
543Q8—942no—
544Q8—942no—
545Q8—942no—
546D4—22no—
547D4—22no—
548D4—41no—
549D4—41no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—84no—
576C2×C4—942no—
577C2×C4—942no—
578C2×C4—942no—
579C2×C4—942no—
580C2×C4—1881no—
581C2×C4—1881no—
582C2×C4—1881no—
583C2×C4—1881no—
584C2×C4—1881no—
585C2×C4—1881no—
586C2×C4—1881no—
587C2×C4—1881no—
588C2×C4#1309404412yes—
589C2×C4#1309405012yes—
590C2×C4C2×D4×D4711yescharacteristic
591C2×C4C2×Q8×D4711yescharacteristic
592C2×C4#41×D4711yescharacteristic
593C2×C4#41×D4711yescharacteristic
594C2×C4#1309406911yescharacteristic
595C2×C4#1309407211yescharacteristic
596C8—942no—
597C8—942no—
598C8—1881no—
599C8—1881no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22#1568963011yescharacteristic
608C22D4×C22⋊D4711yescharacteristic
609C22Q8×C22⋊D4711yescharacteristic
610C22C2×#1309404411yescharacteristic
611C22C2×#1309405011yescharacteristic
612C22#1568993411yescharacteristic
613C22#1568993811yescharacteristic
614C4—22no—
615C4—22no—
616C4—41no—
617C4—41no—
618C4—41no—
619C4—42no—
620C4—41no—
621C4—81no—
622C4—81no—
623C4—944no—
624C4—3761no—
625C4—3761no—
626C4—1882no—
627C4C22⋊Q8×D4711yescharacteristic
628C4#1568971911yescharacteristic
629C4#1568977511yescharacteristic
630C4#1568982711yescharacteristic
631C2—82no—
632C2#2024719411yescharacteristic
633C2#2025344211yescharacteristic
634C2#2025361111yescharacteristic
635C2#2025487911yescharacteristic
636C2#2025512511yescharacteristic
637C2#2025626311yescharacteristic
638C2#2025630911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2452C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
hypercenter#2452D47
hypocenterC47#26377
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2452

Complements

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

NH ≅ G/Ntype
1. C2×#15689415C2semidirect
3. #20246308C2semidirect
5. #20244869C2semidirect
7. #20244718C2semidirect
8. #20244118C2semidirect
9. #20243947C2semidirect
12. #20243655C2semidirect
13. #20243641C2semidirect
64. C2×#10625335Q8semidirect
275. Dic94C22⋊Q8semidirect
380. C47#26377semidirect

Maximal subgroups

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

subgroupindex
C2×#156894152
C2×C4×#106253352
#202463082
#202449092
#202448692
#202447762
#202447182
#202441182
#202439472
#202437462
#202436552
#202436412
#202436352
C47×#24522
#2637747