Finite Group Databasebeta

← #26221158

Subgroups — #26221158

Order 12032.

894 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689178C211yesmaximal, characteristic
2#20245792C211yesmaximal, characteristic
3C4×#13093982C211yesmaximal, characteristic
4#20244150C211yesmaximal, characteristic, complemented
5#20244148C211yesmaximal, characteristic, complemented
6#20244145C211yesmaximal, characteristic, complemented
7#20244143C211yesmaximal, characteristic, complemented
8#20243590C211yesmaximal, characteristic
9#20243588C211yesmaximal, characteristic
10#20243586C211yesmaximal, characteristic
11#20243584C211yesmaximal, characteristic
12#20243286C211yesmaximal, characteristic, complemented
13#20243286C211yesmaximal, characteristic, complemented
14#20243285C211yesmaximal, characteristic, complemented
15C47×#1776C211yesmaximal, characteristic, complemented
16C2×#13093982C2211yescharacteristic
17C2×#13093982C2211yescharacteristic
18C2×#13093981—21no—
19C2×C42×D47C2211yescharacteristic
20#15689178C418yescomplemented
21C2×#13093958C2211yescharacteristic
22C2×#13093957C2211yescharacteristic
23C2×#13093957C2211yescharacteristic
24C2×#13093956C2211yescharacteristic
25C2×#13093955C2211yescharacteristic
26C2×#13093955C2211yescharacteristic
27C2×Dic47⋊C8C2211yescharacteristic
28C2×Dic47⋊C8C2211yescharacteristic
29C2×(C2×D47)⋊C8C2211yescharacteristic
30C2×(C2×D47)⋊C8C2211yescharacteristic
31C2×#13093935C2211yescharacteristic
32#15688992C2211yescharacteristic
33#15688992C2211yescharacteristic
34#15688987—21no—
35#15688982—21no—
36#15688980C2211yescharacteristic
37#15688980C2211yescharacteristic
38C4⋊C4×Dic47C2211yescharacteristic
39C4⋊C4×Dic47C2211yescharacteristic
40C4×Dic47⋊C4—21no—
41C4×C4⋊Dic47C2211yescharacteristic
42C4×C4⋊Dic47C2211yescharacteristic
43C4×(C2×D47)⋊C4—21no—
44#15688970—21no—
45#15688968—21no—
46C42×Dic47—21no—
47#15688886C2211yescharacteristic
48#15688886C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688885C2211yescharacteristic
53C2×#107×C47C2211yescharacteristic
54C2×#106×C47C2211yescharacteristic
55C2×#97×C47C2211yescharacteristic
56C4×C4⋊C4×C47C2211yescharacteristic
57C4×C4⋊C4×C47C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#340C2211yescharacteristic
60C22×C4×D47D411yescharacteristic
61C22×C4×D47D411yescharacteristic
62C22×C4×D47C2311yescharacteristic
63C22×C47⋊C8C2311yescharacteristic
64C22×C47⋊C8C2311yescharacteristic
65#13093982—22no—
66#13093982—22no—
67#13093982—22no—
68#13093982—22no—
69#13093981—44no—
70C2×C4⋊Dic47C2311yescharacteristic
71C2×C4⋊Dic47C2311yescharacteristic
72C2×C4⋊Dic47C2311yescharacteristic
73C2×C4⋊Dic47C2311yescharacteristic
74C2×Dic47⋊C4—21no—
75C2×Dic47⋊C4—21no—
76C2×Dic47⋊C4—21no—
77C2×Dic47⋊C4—21no—
78C2×(C2×D47)⋊C4—21no—
79C2×(C2×D47)⋊C4—21no—
80C2×(C2×D47)⋊C4—21no—
81C2×(C2×D47)⋊C4—21no—
82C2×C4×Dic47—21no—
83C2×C4×Dic47—21no—
84C2×C4×Dic47—21no—
85C2×C4×Dic47—21no—
86C2×C4×Dic47—21no—
87C2×C4×Dic47—21no—
88C2×C4×Dic47D411yescharacteristic
89C2×C4×Dic47D411yescharacteristic
90C2×C4×Dic47C2311yescharacteristic
91C42×D47C2×C414yes—
92C42×D47Q814yes—
93#13093958—22no—
94#13093957—22no—
95#13093957—22no—
96#13093956—22no—
97#13093955—22no—
98#13093955—22no—
99Dic47⋊C8C2×C414yes—
100Dic47⋊C8C2×C414yes—
101(C2×D47)⋊C8C2×C414yes—
102(C2×D47)⋊C8C2×C414yes—
103#13093935C2×C414yes—
104C47⋊#95—21no—
105C47⋊#95—21no—
106C47⋊#95—21no—
107C47⋊#95—21no—
108C47⋊#95—21no—
109C47⋊#95—21no—
110C22×C8×C47C2311yescharacteristic
111C22×C8×C47C2311yescharacteristic
112C2×C4⋊C4×C47C2311yescharacteristic
113C2×C4⋊C4×C47C2311yescharacteristic
114C2×C4⋊C4×C47C2311yescharacteristic
115C2×C4⋊C4×C47C2311yescharacteristic
116C2×C42×C47—21no—
117C2×C42×C47—21no—
118C2×C42×C47C2311yescharacteristic
119#107×C47—22no—
120#106×C47—22no—
121#97×C47C2×C414yes—
122#95×C47—21no—
123#95×C47—21no—
124C23×D47C2×D411yescharacteristic
125C22×C4×C47—21no—
126C22×C4×C47—21no—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47C2×D411yescharacteristic
130C22×C4×C47C2×D411yescharacteristic
131C22×C4×C47C2411yescharacteristic
132C22×Dic47—21no—
133C22×Dic47—21no—
134C22×Dic47—21no—
135C22×Dic47—21no—
136C22×Dic47C2×D411yescharacteristic
137C22×Dic47C2×D411yescharacteristic
138C22×Dic47C2×D411yescharacteristic
139C2×C4×D47—21no—
140C2×C4×D47—21no—
141C2×C4×D47—21no—
142C2×C4×D47—21no—
143C2×C4×D47—22no—
144C2×C4×D47—22no—
145C2×C4×D47—41no—
146C2×C4×D47—41no—
147C2×C4×D47C4⋊C414yes—
148C2×C4×D47C4⋊C414yes—
149C2×C4×D47C22×C412yes—
150C2×C4×D47C2×Q812yes—
151C2×C8×C47—21no—
152C2×C8×C47—21no—
153C2×C8×C47—21no—
154C2×C8×C47—21no—
155C2×C8×C47C22×C412yes—
156C2×C8×C47C22×C412yes—
157C4⋊C4×C47—21no—
158C4⋊C4×C47—21no—
159C4⋊C4×C47—21no—
160C4⋊C4×C47—21no—
161C4⋊C4×C47—21no—
162C4⋊C4×C47—21no—
163C4⋊C4×C47#4112yes—
164C4⋊C4×C47#4112yes—
165C42×C47—42no—
166C42×C47—42no—
167C42×C47C22×C412yes—
168C42×C47C2×Q812yes—
169(C2×D47)⋊C4—24no—
170(C2×D47)⋊C4—24no—
171(C2×D47)⋊C4—42no—
172(C2×D47)⋊C4—42no—
173C4⋊Dic47—21no—
174C4⋊Dic47—21no—
175C4⋊Dic47—21no—
176C4⋊Dic47—21no—
177C4⋊Dic47—21no—
178C4⋊Dic47—21no—
179C4⋊Dic47#4112yes—
180C4⋊Dic47#4112yes—
181Dic47⋊C4—42no—
182Dic47⋊C4—42no—
183Dic47⋊C4—42no—
184Dic47⋊C4—42no—
185C4×Dic47—24no—
186C4×Dic47—24no—
187C4×Dic47—42no—
188C4×Dic47—42no—
189C4×Dic47—42no—
190C4×Dic47—42no—
191C4×Dic47C4⋊C414yes—
192C4×Dic47C4⋊C414yes—
193C4×Dic47C22×C412yes—
194C4×Dic47C2×Q812yes—
195C2×C47⋊C8—21no—
196C2×C47⋊C8—21no—
197C2×C47⋊C8—21no—
198C2×C47⋊C8—21no—
199C2×C47⋊C8C22×C412yes—
200C2×C47⋊C8C22×C412yes—
201C23×C47C22×D411yescharacteristic
202C22×D47—21no—
203C22×D47—21no—
204C22×D47—21no—
205C22×D47—21no—
206C22×D47—21no—
207C22×D47—21no—
208C22×D47C2×C4⋊C412yes—
209C2×C4×C47—21no—
210C2×C4×C47—21no—
211C2×C4×C47—21no—
212C2×C4×C47—21no—
213C2×C4×C47—22no—
214C2×C4×C47—22no—
215C2×C4×C47—41no—
216C2×C4×C47—41no—
217C2×C4×C47—41no—
218C2×C4×C47—41no—
219C2×C4×C47—41no—
220C2×C4×C47—41no—
221C2×C4×C47—41no—
222C2×C4×C47—41no—
223C2×C4×C47—41no—
224C2×C4×C47—41no—
225C2×C4×C47C2×C4⋊C412yes—
226C2×C4×C47C2×C4⋊C412yes—
227C2×C4×C47C23×C411yescharacteristic
228C2×C4×C47C22×Q811yescharacteristic
229C2×C4×C47C2×#4111yescharacteristic
230C2×C4×C47C2×#4111yescharacteristic
231C2×C4×C47C2×#4111yescharacteristic
232C2×C4×C47C2×#4111yescharacteristic
233C2×Dic47—21no—
234C2×Dic47—21no—
235C2×Dic47—21no—
236C2×Dic47—21no—
237C2×Dic47—22no—
238C2×Dic47—22no—
239C2×Dic47—41no—
240C2×Dic47—41no—
241C2×Dic47—41no—
242C2×Dic47—41no—
243C2×Dic47—41no—
244C2×Dic47—41no—
245C2×Dic47—41no—
246C2×Dic47—41no—
247C2×Dic47—41no—
248C2×Dic47—41no—
249C2×Dic47C2×C4⋊C412yes—
250C2×Dic47C2×C4⋊C412yes—
251C2×Dic47C2×C4⋊C412yes—
252C2×Dic47AGL(1,ℤ/8)12yes—
253C2×Dic47#13712yes—
254C4×D47—22no—
255C4×D47—22no—
256C4×D47—22no—
257C4×D47—22no—
258C4×D47—28no—
259C4×D47—44no—
260C8×C47—22no—
261C8×C47—22no—
262C47⋊C8—22no—
263C47⋊C8—22no—
264#21364—471nomaximal, Sylow
265C22×C47C22×C4⋊C411yescharacteristic
266C22×C47#52511yescharacteristic
267C22×C47#52511yescharacteristic
268C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
269C22×C47C2×#13711yescharacteristic
270C22×C47#57511yescharacteristic
271C22×C47#57511yescharacteristic
272C2×D47—22no—
273C2×D47—22no—
274C2×D47—22no—
275C2×D47—41no—
276C2×D47—41no—
277C2×D47—41no—
278C2×D47—41no—
279C4×C47—22no—
280C4×C47—22no—
281C4×C47—42no—
282C4×C47—42no—
283C4×C47—81no—
284C4×C47—81no—
285C4×C47C4×#4111yescharacteristic
286C4×C47C4×#4111yescharacteristic
287C4×C47#53311yescharacteristic
288C4×C47#53311yescharacteristic
289Dic47—22no—
290Dic47—22no—
291Dic47—42no—
292Dic47—42no—
293Dic47—81no—
294Dic47—81no—
295Dic47#42814yes—
296C2×#432—471no—
297#2285—471no—
298C4×#117—471no—
299#1948—471no—
300#1948—471no—
301#1947—471no—
302#1947—471no—
303#1854—471no—
304#1854—471no—
305#1854—471no—
306#1854—471no—
307#1776—471no—
308#1776—471no—
309#1776—471no—
310#1776D4711yescharacteristic, complemented
311C2×C47#229611yescharacteristic
312C2×C47C2×#42811yescharacteristic
313C2×C47#291111yescharacteristic
314C2×C47#311511yescharacteristic
315C2×C47#311511yescharacteristic
316C2×C47#311611yescharacteristic
317C2×C47#311611yescharacteristic
318D47—42no—
319C2×#117—471no—
320C2×#117—471no—
321C2×#117—941no—
322C22×C42—471no—
323#432—478no—
324C2×#107—471no—
325C2×#107—471no—
326C2×#107—471no—
327C2×#107C2×D4711yescharacteristic
328C2×#106—471no—
329C2×#106—471no—
330C2×#106—471no—
331C2×#106C2×D4711yescharacteristic
332C2×C4⋊C8—471no—
333C2×C4⋊C8—471no—
334C2×C22⋊C8—471no—
335C2×C22⋊C8—471no—
336C2×#97—471no—
337C2×#97C2×D4711yescharacteristic
338#384—471no—
339#384—471no—
340#382—941no—
341#380—471no—
342#380—471no—
343#380—941no—
344C4×C4⋊C4—471no—
345C4×C4⋊C4—471no—
346C4×C4⋊C4—471no—
347C4×C4⋊C4—471no—
348C4×C4⋊C4—941no—
349C4×C4⋊C4C2×D4711yescharacteristic
350C4×C4⋊C4C2×D4711yescharacteristic
351C4×C22⋊C4—941no—
352C4×C22⋊C4—941no—
353#376—941no—
354C43—941no—
355#340—471no—
356#340—471no—
357#340—471no—
358#340—471no—
359#340—471no—
360#340—471no—
361#340C2×D4711yescharacteristic
362#340C2×D4711yescharacteristic
363C47#2136411yescharacteristic, complemented, Sylow
364C23×C4—471no—
365C23×C4—471no—
366C23×C4—471no—
367C22×C8—471no—
368C22×C8—471no—
369C22×C8C22×D4711yescharacteristic
370C22×C8C22×D4711yescharacteristic
371#117—942no—
372#117—942no—
373#117—942no—
374#117—942no—
375#117—1884no—
376C2×C4⋊C4—471no—
377C2×C4⋊C4—471no—
378C2×C4⋊C4—471no—
379C2×C4⋊C4—471no—
380C2×C4⋊C4—941no—
381C2×C4⋊C4—941no—
382C2×C4⋊C4—941no—
383C2×C4⋊C4—941no—
384C2×C4⋊C4C22×D4711yescharacteristic
385C2×C4⋊C4C22×D4711yescharacteristic
386C2×C4⋊C4C22×D4711yescharacteristic
387C2×C4⋊C4C22×D4711yescharacteristic
388C2×C22⋊C4—941no—
389C2×C22⋊C4—941no—
390C2×C22⋊C4—941no—
391C2×C22⋊C4—941no—
392C2×C42—21no—
393C2×C42—21no—
394C2×C42—471no—
395C2×C42—471no—
396C2×C42—471no—
397C2×C42—474no—
398C2×C42—474no—
399C2×C42—941no—
400C2×C42—941no—
401C2×C42—941no—
402C2×C42—941no—
403C2×C42—941no—
404C2×C42—941no—
405C2×C42C22×D4711yescharacteristic
406#107—22no—
407#107—942no—
408#107—942no—
409#107—942no—
410#106—22no—
411#106—942no—
412#106—942no—
413#106—942no—
414C4⋊C8—474no—
415C4⋊C8—474no—
416C22⋊C8—474no—
417C22⋊C8—474no—
418#97—474no—
419#97C4×D4714yes—
420#95—21no—
421#95—21no—
422#95—941no—
423#95—941no—
424#95—941no—
425#95—941no—
426#95—941no—
427#95—941no—
428C24—471no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—471no—
434C22×C4—471no—
435C22×C4—471no—
436C22×C4—474no—
437C22×C4—472no—
438C22×C4—472no—
439C22×C4—474no—
440C22×C4—942no—
441C22×C4—942no—
442C22×C4—941no—
443C22×C4—941no—
444C22×C4—941no—
445C22×C4—941no—
446C22×C4—941no—
447C22×C4—941no—
448C22×C4—941no—
449C22×C4—941no—
450C22×C4—1881no—
451C22×C4—1881no—
452C22×C4D4×D4711yescharacteristic
453C22×C4D4×D4711yescharacteristic
454C22×C4C23×D4711yescharacteristic
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—472no—
460C2×C8—472no—
461C2×C8—941no—
462C2×C8—941no—
463C2×C8—941no—
464C2×C8—941no—
465C2×C8C2×C4×D4712yes—
466C2×C8C2×C4×D4712yes—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—472no—
474C4⋊C4—472no—
475C4⋊C4—941no—
476C4⋊C4—941no—
477C4⋊C4—941no—
478C4⋊C4—941no—
479C4⋊C4—941no—
480C4⋊C4—941no—
481C4⋊C4—1882no—
482C4⋊C4—1882no—
483C4⋊C4—1882no—
484C4⋊C4—1882no—
485C4⋊C4#1062535012yes—
486C4⋊C4#1062535012yes—
487C22⋊C4—944no—
488C22⋊C4—944no—
489C22⋊C4—1882no—
490C22⋊C4—1882no—
491C42—42no—
492C42—42no—
493C42—474no—
494C42—474no—
495C42—472no—
496C42—472no—
497C42—944no—
498C42—944no—
499C42—1882no—
500C42—1882no—
501C42—1882no—
502C42—1882no—
503C42C2×C4×D4712yes—
504C42Q8×D4712yes—
505C23—472no—
506C23—941no—
507C23—941no—
508C23—941no—
509C23—941no—
510C23—941no—
511C23—941no—
512C23C2×D4×D4711yescharacteristic
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—472no—
530C2×C4—472no—
531C2×C4—472no—
532C2×C4—472no—
533C2×C4—472no—
534C2×C4—948no—
535C2×C4—941no—
536C2×C4—941no—
537C2×C4—941no—
538C2×C4—941no—
539C2×C4—942no—
540C2×C4—942no—
541C2×C4—942no—
542C2×C4—942no—
543C2×C4—942no—
544C2×C4—942no—
545C2×C4—1884no—
546C2×C4—1881no—
547C2×C4—1881no—
548C2×C4—1881no—
549C2×C4—1881no—
550C2×C4—1881no—
551C2×C4—1881no—
552C2×C4—1881no—
553C2×C4—1881no—
554C2×C4—1881no—
555C2×C4—1881no—
556C2×C4C4⋊C4×D4712yes—
557C2×C4C4⋊C4×D4712yes—
558C2×C4C22×C4×D4711yescharacteristic
559C2×C4C2×Q8×D4711yescharacteristic
560C2×C4#41×D4711yescharacteristic
561C2×C4#41×D4711yescharacteristic
562C2×C4C2×#1062535011yescharacteristic
563C2×C4C2×#1062535011yescharacteristic
564C8—22no—
565C8—22no—
566C8—942no—
567C8—942no—
568C22—942no—
569C22—942no—
570C22—942no—
571C22—1881no—
572C22—1881no—
573C22—1881no—
574C22—1881no—
575C22C2×C4⋊C4×D4711yescharacteristic
576C22#1568963211yescharacteristic
577C22#1568963211yescharacteristic
578C22AGL(1,ℤ/8)×D4711yescharacteristic
579C22#137×D4711yescharacteristic
580C22#1568991911yescharacteristic
581C22#1568992211yescharacteristic
582C4—22no—
583C4—22no—
584C4—42no—
585C4—42no—
586C4—81no—
587C4—81no—
588C4—474no—
589C4—942no—
590C4—942no—
591C4—3761no—
592C4—3761no—
593C4—1882no—
594C4—1882no—
595C4#1568959111yescharacteristic
596C4C4×#1062535011yescharacteristic
597C4#1568967711yescharacteristic
598C4#1568967711yescharacteristic
599C2—1882no—
600C2#2024586811yescharacteristic
601C2#428×D4711yescharacteristic
602C2#2025197511yescharacteristic
603C2#2025353511yescharacteristic
604C2#2025353711yescharacteristic
605C2#2025354611yescharacteristic
606C2#2025354811yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1776C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
cosocleC22×C4×C47C24
hypercenter#1776D47
hypocenterC47#21364
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1776

Complements

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

NH ≅ G/Ntype
4. #20244150C2semidirect
5. #20244148C2semidirect
6. #20244145C2semidirect
7. #20244143C2semidirect
12. #20243286C2semidirect
13. #20243286C2semidirect
14. #20243285C2semidirect
15. C47×#1776C2semidirect
20. #15689178C4semidirect
310. #1776D47semidirect
363. C47#21364semidirect

Maximal subgroups

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

subgroupindex
C2×#156891782
#202457922
C4×#130939822
#202441502
#202441482
#202441452
#202441432
#202435902
#202435882
#202435862
#202435842
#202432862
#202432852
C47×#17762
#2136447