Finite Group Databasebeta

← #26182599

Subgroups — #26182599

Order 12032.

916 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689845C211yesmaximal, characteristic
2#20252477C211yesmaximal, characteristic, complemented
3#20241758C212yesmaximal, complemented
4#20241755C212yesmaximal, complemented
5#20241741C211yesmaximal, characteristic
6#20241739C211yesmaximal, characteristic
7#20241731C211yesmaximal, characteristic, complemented
8#20241720C212yesmaximal, complemented
9#20241711C211yesmaximal, characteristic, complemented
10#20241700C212yesmaximal, complemented
11C47×#2984C211yesmaximal, characteristic, complemented
12#15689845—21no—
13#15689845—22no—
14#15689845—21no—
15#15689845—21no—
16#15689845—22no—
17#15689845—21no—
18C2×#13094022C411yescharacteristic
19C2×#13094019C412yes—
20C2×#13094018C2212yes—
21C2×#13094013C411yescharacteristic
22C2×#13094012C2211yescharacteristic
23C2×#13094002C412yes—
24C2×C4×Dic94C2211yescharacteristic
25C2×#13093989C411yescharacteristic
26C2×#13093986C411yescharacteristic
27C2×#13093981C2212yes—
28#15689254—22no—
29#15689254—21no—
30Q8×C47⋊C8—21no—
31#15689180C2212yes—
32#15689180C2211yescharacteristic
33#15689176C2212yes—
34#15689176C2211yescharacteristic
35#15688848C2211yescharacteristic
36#15688848C2212yes—
37#15688848C2211yescharacteristic
38#15688848C2211yescharacteristic
39#15688848C2211yescharacteristic
40#15688848C2212yes—
41#15688846C2212yes—
42#15688846C2212yes—
43#15688846C2212yes—
44#15688846C2212yes—
45C2×C4×Q8×C47C2211yescharacteristic
46C47×#446—21no—
47C47×#446—22no—
48C8×Q8×C47—21no—
49C47×#432C2212yes—
50C47×#432C2211yescharacteristic
51C47×#431C2212yes—
52C47×#431C2211yescharacteristic
53C22×Dic94C2×C411yescharacteristic
54C22×Dic94C2×C411yescharacteristic
55C22×D188C2×C411yescharacteristic
56C22×C4×D47C2×C412yes—
57#13094022—21no—
58#13094022—21no—
59#13094022—22no—
60#13094019—22no—
61#13094019—22no—
62#13094019—22no—
63#13094019—22no—
64#13094018—22no—
65#13094018—22no—
66#13094018—22no—
67#13094018—22no—
68#13094013—21no—
69#13094013—21no—
70#13094013—22no—
71#13094012—21no—
72#13094012—21no—
73#13094012—21no—
74#13094012—21no—
75#13094002—22no—
76#13094002—22no—
77#13094002—22no—
78#13094002—22no—
79C4×Dic94—21no—
80C4×Dic94—21no—
81C4×Dic94—21no—
82C4×Dic94—21no—
83#13093989—22no—
84#13093989—21no—
85#13093989—21no—
86#13093986—22no—
87#13093986—21no—
88#13093986—21no—
89#13093981—22no—
90#13093981—22no—
91#13093981—22no—
92#13093981—22no—
93C2×C4⋊Dic47C2×C411yescharacteristic
94C2×Dic47⋊C4C2×C412yes—
95C2×Dic47⋊C4C2×C412yes—
96C2×Dic47⋊C4C2×C412yes—
97C2×Dic47⋊C4C2311yescharacteristic
98C2×Dic47⋊C4C2311yescharacteristic
99C2×(C2×D47)⋊C4C2×C412yes—
100C2×(C2×D47)⋊C4C2×C412yes—
101C2×(C2×D47)⋊C4C2×C412yes—
102C2×(C2×D47)⋊C4C2×C412yes—
103C2×(C2×D47)⋊C4C2312yes—
104C2×C4×Dic47C2×C412yes—
105C2×C4×Dic47C2×C411yescharacteristic
106C2×C4×Dic47C2×C411yescharacteristic
107C4⋊(C47⋊C8)—22no—
108C4⋊(C47⋊C8)—21no—
109C4⋊(C47⋊C8)—22no—
110C4⋊(C47⋊C8)—21no—
111C22⋊(C47⋊C8)C2311yescharacteristic
112C22⋊(C47⋊C8)C2312yes—
113C22⋊(C47⋊C8)C2311yescharacteristic
114#13093935—22no—
115#13093935—21no—
116C4×C47⋊C8—22no—
117C4×C47⋊C8—21no—
118C22×Q8×C47C2×C411yescharacteristic
119C4×Q8×C47D411yescharacteristic
120C4×Q8×C47D412yes—
121C4×Q8×C47D411yescharacteristic
122C4×Q8×C47D411yescharacteristic
123C4×Q8×C47D412yes—
124C4×Q8×C47D411yescharacteristic
125C2×C4⋊C4×C47C2×C411yescharacteristic
126C2×C4⋊C4×C47C2×C412yes—
127C2×C42×C47C2312yes—
128C2×C42×C47C2311yescharacteristic
129C4⋊C8×C47—22no—
130C4⋊C8×C47—21no—
131C4⋊C8×C47—22no—
132C4⋊C8×C47—21no—
133C22⋊C8×C47C2311yescharacteristic
134C22⋊C8×C47C2312yes—
135C22⋊C8×C47C2311yescharacteristic
136#97×C47—22no—
137#97×C47—21no—
138C4×C8×C47—22no—
139C4×C8×C47—21no—
140C23×D47C22×C412yes—
141C2×Q8×C47—42no—
142C2×Q8×C47C22⋊C411yescharacteristic
143C2×Q8×C47C22⋊C411yescharacteristic
144C2×Q8×C47C22⋊C412yes—
145C22×C4×C47C22×C412yes—
146C22×C4×C47C22×C411yescharacteristic
147C22×C4×C47C22×C411yescharacteristic
148C22×C4×C47C22×C412yes—
149C22×C4×C47C2411yescharacteristic
150C22×Dic47C22×C411yescharacteristic
151C22×Dic47C22×C411yescharacteristic
152C22×Dic47C22×C411yescharacteristic
153C22×Dic47C22×C411yescharacteristic
154C22×Dic47C22×C412yes—
155C2×D188—21no—
156C2×D188—21no—
157C2×D188—42no—
158C2×C4×D47—22no—
159C2×C4×D47—22no—
160C2×C4×D47—24no—
161C2×C4×D47—42no—
162C2×Dic94—21no—
163C2×Dic94—21no—
164C2×Dic94—21no—
165C2×Dic94—21no—
166C2×Dic94—22no—
167C2×Dic94—41no—
168C2×Dic94—42no—
169C2×C8×C47—21no—
170C2×C8×C47—22no—
171C2×C8×C47—21no—
172C4⋊C4×C47C22⋊C412yes—
173C4⋊C4×C47C22⋊C411yescharacteristic
174C4⋊C4×C47C22⋊C411yescharacteristic
175C4⋊C4×C47C22⋊C412yes—
176C4⋊C4×C47C22⋊C412yes—
177C4⋊C4×C47C22⋊C412yes—
178C4⋊C4×C47C22⋊C412yes—
179C42×C47C2×D412yes—
180C42×C47C2×D411yescharacteristic
181C42×C47C2×D412yes—
182C42×C47C2×D411yescharacteristic
183C42×C47C2×D412yes—
184C42×C47C2×D412yes—
185C42×C47C2×D411yescharacteristic
186C42×C47C2×D411yescharacteristic
187(C2×D47)⋊C4—22no—
188(C2×D47)⋊C4—22no—
189(C2×D47)⋊C4—22no—
190(C2×D47)⋊C4—22no—
191(C2×D47)⋊C4—22no—
192(C2×D47)⋊C4—22no—
193(C2×D47)⋊C4—22no—
194(C2×D47)⋊C4—22no—
195(C2×D47)⋊C4—22no—
196(C2×D47)⋊C4—22no—
197C4⋊Dic47—21no—
198C4⋊Dic47—21no—
199Dic47⋊C4—22no—
200Dic47⋊C4—22no—
201Dic47⋊C4—22no—
202Dic47⋊C4—22no—
203Dic47⋊C4—22no—
204Dic47⋊C4—22no—
205Dic47⋊C4—21no—
206Dic47⋊C4—21no—
207Dic47⋊C4—21no—
208Dic47⋊C4—21no—
209C4×Dic47—22no—
210C4×Dic47—22no—
211C4×Dic47—21no—
212C4×Dic47—21no—
213C4×Dic47—21no—
214C4×Dic47—21no—
215C2×C47⋊C8—21no—
216C2×C47⋊C8—22no—
217C2×C47⋊C8—21no—
218C23×C47C23×C411yescharacteristic
219C22×D47—22no—
220C22×D47—22no—
221C22×D47—22no—
222C22×D47—42no—
223C22×D47—42no—
224Q8×C47—22no—
225Q8×C47—22no—
226Q8×C47—42no—
227C2×C4×C47—21no—
228C2×C4×C47—22no—
229C2×C4×C47—21no—
230C2×C4×C47—22no—
231C2×C4×C47—22no—
232C2×C4×C47—42no—
233C2×C4×C47—41no—
234C2×C4×C47C2×C22⋊C411yescharacteristic
235C2×C4×C47C2×C22⋊C412yes—
236C2×C4×C47C2×C22⋊C412yes—
237C2×C4×C47C2×C22⋊C412yes—
238C2×C4×C47C2×C22⋊C412yes—
239C2×C4×C47C2×C22⋊C411yescharacteristic
240C2×C4×C47C2×C22⋊C411yescharacteristic
241C2×C4×C47C2×C22⋊C411yescharacteristic
242C2×C4×C47C22×D411yescharacteristic
243C2×C4×C47C22×D411yescharacteristic
244C2×Dic47—21no—
245C2×Dic47—21no—
246C2×Dic47—21no—
247C2×Dic47—21no—
248C2×Dic47—21no—
249C2×Dic47—22no—
250C2×Dic47—21no—
251C2×Dic47—21no—
252C2×Dic47—42no—
253C2×Dic47—41no—
254C2×Dic47—41no—
255C2×Dic47#13112yes—
256D188—42no—
257D188—42no—
258C4×D47—44no—
259C4×D47—44no—
260Dic94—42no—
261Dic94—42no—
262Dic94—42no—
263Dic94—42no—
264C8×C47—41no—
265C8×C47—42no—
266C8×C47—41no—
267C47⋊C8—41no—
268C47⋊C8—42no—
269C47⋊C8—41no—
270#17428—471nomaximal, Sylow
271C22×C47—21no—
272C22×C47—21no—
273C22×C47C22×C22⋊C411yescharacteristic
274C22×C47C2×#13111yescharacteristic
275C22×C47#56811yescharacteristic
276C2×D47—42no—
277C2×D47—42no—
278C2×D47—42no—
279C2×D47—82no—
280C2×D47—82no—
281C4×C47—21no—
282C4×C47—21no—
283C4×C47—22no—
284C4×C47—22no—
285C4×C47—21no—
286C4×C47—21no—
287C4×C47—21no—
288C4×C47—22no—
289C4×C47—22no—
290C4×C47#41312yes—
291Dic47—22no—
292Dic47—41no—
293Dic47—41no—
294Dic47—41no—
295Dic47—42no—
296C2×#556—471no—
297#2984—471no—
298#2984D4711yescharacteristic, complemented
299#1473—472no—
300#1473—472no—
301#1469—471no—
302#1469—471no—
303#1467—471no—
304#1464—472no—
305#1461—471no—
306#1458—472no—
307C2×C47—21no—
308C2×C47—21no—
309C2×C47#288011yescharacteristic
310C2×C47#288411yescharacteristic
311C2×C47C2×#41311yescharacteristic
312D47—82no—
313#556—941no—
314#556—942no—
315#556—941no—
316#556—941no—
317#556—942no—
318#556—941no—
319C2×C4⋊Q8—471no—
320C2×#126—472no—
321C2×#126—472no—
322C2×#124—471no—
323C2×#124—471no—
324C2×C22⋊Q8—472no—
325C2×C4×Q8—471no—
326C2×C4×Q8—471no—
327C2×C4×Q8C2×D4711yescharacteristic
328C2×C4×D4—471no—
329C2×#117—472no—
330#446—22no—
331#446—21no—
332#446—942no—
333#446—941no—
334C8×Q8—21no—
335C8×Q8—941no—
336#432—472no—
337#432—471no—
338#432C2×D4712yes—
339#432C2×D4711yescharacteristic
340#431—472no—
341#431—471no—
342#431C2×D4712yes—
343#431C2×D4711yescharacteristic
344#324—471no—
345#324—472no—
346#324—471no—
347#324—471no—
348#324—471no—
349#324—472no—
350#323—472no—
351#323—472no—
352#323—472no—
353#323—472no—
354C47#1742811yescharacteristic, complemented, Sylow
355C22×Q8—471no—
356C22×Q8—471no—
357C22×Q8C4×D4711yescharacteristic
358C22×D4—471no—
359C23×C4—472no—
360C4⋊Q8—941no—
361C4⋊Q8—941no—
362C4⋊Q8—942no—
363#126—942no—
364#126—942no—
365#126—942no—
366#126—942no—
367#126—942no—
368#126—942no—
369#126—942no—
370#126—942no—
371#124—941no—
372#124—941no—
373#124—942no—
374#124—941no—
375#124—941no—
376#124—941no—
377#124—941no—
378C22⋊Q8—942no—
379C22⋊Q8—942no—
380C22⋊Q8—942no—
381C22⋊Q8—942no—
382C4×Q8—942no—
383C4×Q8—941no—
384C4×Q8—941no—
385C4×Q8—941no—
386C4×Q8—941no—
387C4×Q8—941no—
388C4×Q8—941no—
389C4×Q8D18811yescharacteristic
390C4×Q8D18812yes—
391C4×Q8D18811yescharacteristic
392C4×Q8C22⋊D4711yescharacteristic
393C4×Q8C22⋊D4712yes—
394C4×Q8C22⋊D4711yescharacteristic
395C4×D4—942no—
396C4×D4—941no—
397C4×D4—941no—
398#117—942no—
399#117—942no—
400#117—942no—
401#117—942no—
402C2×C4⋊C4—472no—
403C2×C4⋊C4—472no—
404C2×C4⋊C4—471no—
405C2×C4⋊C4—472no—
406C2×C4⋊C4—471no—
407C2×C4⋊C4—471no—
408C2×C4⋊C4C4×D4711yescharacteristic
409C2×C4⋊C4C4×D4712yes—
410C2×C22⋊C4—472no—
411C2×C22⋊C4—472no—
412C2×C22⋊C4—472no—
413C2×C22⋊C4—472no—
414C2×C22⋊C4—472no—
415C2×C42—472no—
416C2×C42—471no—
417C2×C42—471no—
418C2×C42C22×D4712yes—
419C2×C42C22×D4711yescharacteristic
420C4⋊C8—22no—
421C4⋊C8—21no—
422C4⋊C8—22no—
423C4⋊C8—21no—
424C4⋊C8—942no—
425C4⋊C8—941no—
426C4⋊C8—942no—
427C4⋊C8—941no—
428C22⋊C8—471no—
429C22⋊C8—472no—
430C22⋊C8—471no—
431C22⋊C8C22×D4711yescharacteristic
432C22⋊C8C22×D4712yes—
433C22⋊C8C22×D4711yescharacteristic
434#97—22no—
435#97—21no—
436#97—942no—
437#97—941no—
438C4×C8—22no—
439C4×C8—21no—
440C4×C8—942no—
441C4×C8—941no—
442C24—472no—
443C2×Q8—42no—
444C2×Q8—942no—
445C2×Q8—941no—
446C2×Q8—941no—
447C2×Q8—941no—
448C2×Q8—941no—
449C2×Q8—1881no—
450C2×Q8—1882no—
451C2×Q8(C2×D47)⋊C411yescharacteristic
452C2×Q8(C2×D47)⋊C411yescharacteristic
453C2×Q8(C2×D47)⋊C412yes—
454C2×D4—941no—
455C2×D4—941no—
456C2×D4—1882no—
457C22×C4—471no—
458C22×C4—471no—
459C22×C4—471no—
460C22×C4—471no—
461C22×C4—472no—
462C22×C4—944no—
463C22×C4—942no—
464C22×C4—942no—
465C22×C4—1882no—
466C22×C4C2×C4×D4711yescharacteristic
467C22×C4C2×C4×D4712yes—
468C22×C4C2×C4×D4711yescharacteristic
469C22×C4C2×C4×D4712yes—
470C22×C4C23×D4711yescharacteristic
471C2×C8—21no—
472C2×C8—22no—
473C2×C8—21no—
474C2×C8—941no—
475C2×C8—942no—
476C2×C8—941no—
477C4⋊C4—942no—
478C4⋊C4—942no—
479C4⋊C4—942no—
480C4⋊C4—942no—
481C4⋊C4—942no—
482C4⋊C4—942no—
483C4⋊C4—941no—
484C4⋊C4—941no—
485C4⋊C4—941no—
486C4⋊C4—941no—
487C4⋊C4—941no—
488C4⋊C4—941no—
489C4⋊C4(C2×D47)⋊C412yes—
490C4⋊C4(C2×D47)⋊C411yescharacteristic
491C4⋊C4(C2×D47)⋊C411yescharacteristic
492C4⋊C4(C2×D47)⋊C412yes—
493C4⋊C4(C2×D47)⋊C412yes—
494C4⋊C4(C2×D47)⋊C412yes—
495C4⋊C4(C2×D47)⋊C412yes—
496C22⋊C4—942no—
497C22⋊C4—942no—
498C22⋊C4—942no—
499C22⋊C4—942no—
500C22⋊C4—942no—
501C22⋊C4—942no—
502C22⋊C4—942no—
503C22⋊C4—942no—
504C22⋊C4—942no—
505C22⋊C4—942no—
506C42—942no—
507C42—942no—
508C42—941no—
509C42—941no—
510C42—941no—
511C42—941no—
512C42C2×D18812yes—
513C42C2×D18811yescharacteristic
514C42C2×D18812yes—
515C42C2×D18811yescharacteristic
516C42C2×C22⋊D4712yes—
517C42C2×C22⋊D4711yescharacteristic
518C42C2×C22⋊D4712yes—
519C42C2×C22⋊D4711yescharacteristic
520C23—942no—
521C23—942no—
522C23—942no—
523C23—1882no—
524C23—1882no—
525C23C22×C4×D4711yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—42no—
529Q8—1882no—
530Q8—1882no—
531Q8—1882no—
532Q8—1882no—
533D4—1882no—
534D4—1882no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—472no—
543C2×C4—941no—
544C2×C4—941no—
545C2×C4—941no—
546C2×C4—941no—
547C2×C4—941no—
548C2×C4—942no—
549C2×C4—941no—
550C2×C4—941no—
551C2×C4—1884no—
552C2×C4—1884no—
553C2×C4—1881no—
554C2×C4—1881no—
555C2×C4—1882no—
556C2×C4C2×(C2×D47)⋊C411yescharacteristic
557C2×C4C2×(C2×D47)⋊C412yes—
558C2×C4C2×(C2×D47)⋊C412yes—
559C2×C4C2×(C2×D47)⋊C412yes—
560C2×C4C2×(C2×D47)⋊C412yes—
561C2×C4C2×(C2×D47)⋊C411yescharacteristic
562C2×C4C2×(C2×D47)⋊C411yescharacteristic
563C2×C4C2×(C2×D47)⋊C411yescharacteristic
564C2×C4C22×C22⋊D4711yescharacteristic
565C2×C4C22×D18811yescharacteristic
566C8—41no—
567C8—42no—
568C8—41no—
569C8—1881no—
570C8—1882no—
571C8—1881no—
572C22—21no—
573C22—21no—
574C22—3762no—
575C22—3762no—
576C22—1882no—
577C22—1882no—
578C22—1882no—
579C22C22×(C2×D47)⋊C411yescharacteristic
580C22#131×D4711yescharacteristic
581C22#1568988811yescharacteristic
582C4—22no—
583C4—22no—
584C4—21no—
585C4—21no—
586C4—21no—
587C4—21no—
588C4—21no—
589C4—22no—
590C4—22no—
591C4—942no—
592C4—1881no—
593C4—1881no—
594C4—1881no—
595C4—1882no—
596C4#1568910512yes—
597C2—21no—
598C2—21no—
599C2—3762no—
600C2#2025183511yescharacteristic
601C2#2025185111yescharacteristic
602C2C2×#1568910511yescharacteristic
subgroupG/H
centerC22C22×(C2×D47)⋊C4
commutatorC23×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2984C2
socleC22×C47C22×C22⋊C4
prime radicalC22×D188C2×C4
cosocleC22×C4×C47C24
hypercenter#2984D47
hypocenterC47#17428
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #2984

Complements

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

NH ≅ G/Ntype
2. #20252477C2semidirect
3. #20241758C2semidirect
4. #20241755C2semidirect
7. #20241731C2semidirect
8. #20241720C2semidirect
9. #20241711C2semidirect
10. #20241700C2semidirect
11. C47×#2984C2semidirect
298. #2984D47semidirect
354. C47#17428semidirect

Maximal subgroups

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

subgroupindex
C2×#156898452
#202524772
#202417582
#202417552
#202417412
#202417392
#202417312
#202417202
#202417112
#202417002
C47×#29842
#1742847