Finite Group Databasebeta

← #26221331

Subgroups — #26221331

Order 12032.

1538 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689670C211yesmaximal, characteristic
2C2×#15689174C211yesmaximal, characteristic, complemented
3C4×C4⋊C4×D47C211yesmaximal, characteristic, complemented
4#20244338C212yesmaximal, complemented
5#20244313C212yesmaximal, complemented
6#20243704C212yesmaximal, complemented
7#20243702C212yesmaximal, complemented
8C4×#13093944C211yesmaximal, characteristic, complemented
9C4×#13093946C211yesmaximal, characteristic, complemented
10C4×#13093945C211yesmaximal, characteristic, complemented
11C4×#102×C47C211yesmaximal, characteristic, complemented
12C22×#10625350—21no—
13#15689670—24no—
14#15689670C418yescomplemented
15C2×#13094023C2211yescharacteristic
16C2×#13094020C2211yescharacteristic
17C2×#13094010—21no—
18C2×#13093999C2211yescharacteristic
19C2×#13093997—21no—
20C2×#13093995C2211yescharacteristic
21C2×C4×D188C2211yescharacteristic
22C2×C4×C22⋊D47—21no—
23C2×C4⋊C4×D47C2212yes—
24C2×C42×D47—21no—
25C2×C42×D47C2211yescharacteristic
26#15689174—24no—
27C2×Dic47⋊C8C2212yes—
28C2×#13093946C2212yes—
29C2×#13093945C2212yes—
30C2×#13093944C2212yes—
31C2×(C2×D47)⋊C8C2212yes—
32C2×C4×C47⋊C8C2211yescharacteristic
33C4⋊C4×Dic47C2212yes—
34C4×Dic47⋊C4—21no—
35C4×C4⋊Dic47C2211yescharacteristic
36#15688974—21no—
37#95×D47—21no—
38#15688886C2212yes—
39#15688885C2212yes—
40#15688885C2212yes—
41C2×C4×D4×C47C2211yescharacteristic
42C2×#102×C47C2212yes—
43C2×C4×C8×C47C2211yescharacteristic
44C4×C4⋊C4×C47C2211yescharacteristic
45C47×#340C2212yes—
46C2×#10625350—28no—
47C2×#10625350—44no—
48C2×#10625350—44no—
49C22×Dic94—21no—
50C22×D188—21no—
51C22×D188C2311yescharacteristic
52C22×C22⋊D47—21no—
53C22×C22⋊D47—21no—
54C22×C22⋊D47—21no—
55C22×C4×D47—22no—
56C22×C4×D47—21no—
57C22×C4×D47D411yescharacteristic
58C22×C4×D47D411yescharacteristic
59C22×C4×D47C2311yescharacteristic
60C22×C47⋊C8C2312yes—
61#13094023—22no—
62#13094023C2×C414yescomplemented
63#13094020—22no—
64#13094020C2×C414yescomplemented
65#13094010—28no—
66#13093999—22no—
67#13093999C2×C414yes—
68#13093997—28no—
69#13093995—22no—
70#13093995C2×C414yes—
71C4×D188—22no—
72C4×D188C2×C414yescomplemented
73C4×C22⋊D47—28no—
74C2×C4⋊Dic47C2311yescharacteristic
75C2×C4⋊Dic47C2312yes—
76C2×Dic47⋊C4—22no—
77C2×Dic47⋊C4—21no—
78C4⋊C4×D47—24no—
79C4⋊C4×D47—24no—
80C2×C22⋊Dic47—21no—
81C2×(C2×D47)⋊C4—21no—
82C2×(C2×D47)⋊C4—21no—
83C2×C4×Dic47—21no—
84C2×C4×Dic47—22no—
85C2×C4×Dic47D411yescharacteristic
86C2×C4×Dic47D411yescharacteristic
87C2×C4×Dic47C2311yescharacteristic
88C42×D47—44no—
89C42×D47C2×C414yes—
90C42×D47D414yes—
91Dic47⋊C8—24no—
92#13093946—24no—
93#13093945—24no—
94#13093944—24no—
95(C2×D47)⋊C8—24no—
96C4×C47⋊C8—22no—
97C47⋊#95—21no—
98C47⋊#95—22no—
99C22×D4×C47C2311yescharacteristic
100C23×C4×C47—21no—
101C22×C8×C47C2312yes—
102C4×D4×C47—22no—
103C4×D4×C47C2×C414yescomplemented
104C2×C4⋊C4×C47C2311yescharacteristic
105C2×C4⋊C4×C47C2312yes—
106C2×C22⋊C4×C47—21no—
107C2×C42×C47—21no—
108C2×C42×C47C2311yescharacteristic
109#102×C47—24no—
110C4×C8×C47—22no—
111#95×C47—21no—
112C24×C47—21no—
113C23×D47—21no—
114C23×D47C2×D411yescharacteristic
115C2×D4×C47—21no—
116C2×D4×C47—22no—
117C2×D4×C47—22no—
118C2×D4×C47C22×C412yes—
119C22×C4×C47—22no—
120C22×C4×C47—21no—
121C22×C4×C47—24no—
122C22×C4×C47—42no—
123C22×C4×C47—42no—
124C22×C4×C47C2×D411yescharacteristic
125C22×C4×C47C2×D411yescharacteristic
126C22×C4×C47C2411yescharacteristic
127C2×C22⋊D47—24no—
128C2×C22⋊D47—24no—
129C2×C22⋊D47—24no—
130C2×C22⋊D47—44no—
131C2×C22⋊D47—42no—
132C2×C22⋊D47—42no—
133C2×C22⋊D47—42no—
134C2×C22⋊D47—42no—
135C22×Dic47—22no—
136C22×Dic47—21no—
137C22×Dic47C2×D411yescharacteristic
138C22×Dic47C2×D411yescharacteristic
139C22×Dic47C2×D411yescharacteristic
140#10625350—432no—
141C2×D188—22no—
142C2×D188—22no—
143C2×D188—24no—
144C2×D188—21no—
145C2×D188—42no—
146C2×D188—42no—
147C2×D188C22×C412yes—
148C2×C4×D47—22no—
149C2×C4×D47—22no—
150C2×C4×D47—28no—
151C2×C4×D47—24no—
152C2×C4×D47—42no—
153C2×C4×D47—44no—
154C2×C4×D47—44no—
155C2×C4×D47—42no—
156C2×C4×D47—42no—
157C2×C4×D47C22⋊C414yes—
158C2×C4×D47C22⋊C414yes—
159C2×C4×D47D814yes—
160C2×C4×D47SD1614yes—
161C2×C4×D47C22×C412yes—
162C2×C4×D47C2×D412yes—
163C2×Dic94—24no—
164C2×Dic94—42no—
165C2×Dic94—42no—
166C2×C8×C47—22no—
167C2×C8×C47—22no—
168C2×C8×C47—22no—
169C4⋊C4×C47—21no—
170C4⋊C4×C47—22no—
171C4⋊C4×C47C22×C412yes—
172C4⋊C4×C47#4114yescomplemented
173C22⋊C4×C47—24no—
174C42×C47—42no—
175C42×C47C22×C412yes—
176C42×C47C2×D412yes—
177C22⋊Dic47—24no—
178(C2×D47)⋊C4—24no—
179(C2×D47)⋊C4—24no—
180C4⋊Dic47—21no—
181C4⋊Dic47—22no—
182C4⋊Dic47C22×C412yes—
183C4⋊Dic47#4114yescomplemented
184Dic47⋊C4—28no—
185Dic47⋊C4—24no—
186C4×Dic47—42no—
187C4×Dic47—44no—
188C4×Dic47C22⋊C414yes—
189C4×Dic47C22⋊C414yes—
190C4×Dic47C22×C412yes—
191C4×Dic47C2×D412yes—
192C2×C47⋊C8—22no—
193C2×C47⋊C8—22no—
194C2×C47⋊C8—22no—
195C23×C47—22no—
196C23×C47—42no—
197C23×C47—41no—
198C23×C47—41no—
199C23×C47—42no—
200C23×C47C22×D411yescharacteristic
201C22×D47—22no—
202C22×D47—22no—
203C22×D47—22no—
204C22×D47—42no—
205C22×D47—41no—
206C22×D47—41no—
207C22×D47—42no—
208C22×D47C2×C22⋊C412yes—
209C22×D47C2×D812yes—
210C22×D47C2×SD1612yes—
211D4×C47—22no—
212D4×C47—22no—
213D4×C47—42no—
214C2×C4×C47—22no—
215C2×C4×C47—22no—
216C2×C4×C47—24no—
217C2×C4×C47—42no—
218C2×C4×C47—41no—
219C2×C4×C47—41no—
220C2×C4×C47—42no—
221C2×C4×C47—48no—
222C2×C4×C47C2×C22⋊C412yes—
223C2×C4×C47C2×C22⋊C412yes—
224C2×C4×C47C2×D812yes—
225C2×C4×C47C2×SD1612yes—
226C2×C4×C47C23×C411yescharacteristic
227C2×C4×C47C22×D411yescharacteristic
228C2×C4×C47C2×#4112yes—
229C2×C4×C47C2×#4112yes—
230C22⋊D47—48no—
231C22⋊D47—48no—
232C22⋊D47—48no—
233C2×Dic47—22no—
234C2×Dic47—22no—
235C2×Dic47—22no—
236C2×Dic47—24no—
237C2×Dic47—42no—
238C2×Dic47—41no—
239C2×Dic47—41no—
240C2×Dic47—42no—
241C2×Dic47C2×C22⋊C412yes—
242C2×Dic47C2×C22⋊C412yes—
243C2×Dic47C2×C22⋊C412yes—
244C2×Dic47C2×D812yes—
245C2×Dic47C2×SD1612yes—
246D188—22no—
247D188—22no—
248D188—48no—
249D188—42no—
250C4×D47—22no—
251C4×D47—22no—
252C4×D47—24no—
253C4×D47—44no—
254C4×D47—416no—
255C4×D47—48no—
256C4×D47#102116yescomplemented
257Dic94—48no—
258C8×C47—42no—
259C47⋊C8—42no—
260#21386—471nomaximal, Sylow
261C22×C47—42no—
262C22×C47—42no—
263C22×C47—42no—
264C22×C47—82no—
265C22×C47—82no—
266C22×C47C22×C22⋊C411yescharacteristic
267C22×C47#52512yes—
268C22×C47C22×D811yescharacteristic
269C22×C47C22×SD1611yescharacteristic
270C22×C47#57512yes—
271C2×D47—24no—
272C2×D47—24no—
273C2×D47—22no—
274C2×D47—22no—
275C2×D47—42no—
276C2×D47—42no—
277C2×D47—42no—
278C2×D47—82no—
279C2×D47—82no—
280C2×D47C2×#10214yes—
281C4×C47—22no—
282C4×C47—44no—
283C4×C47—42no—
284C4×C47C2×#10214yes—
285C4×C47C4×#4111yescharacteristic
286C4×C47C4×#4111yescharacteristic
287C4×C47#53212yes—
288Dic47—22no—
289Dic47—22no—
290Dic47—44no—
291Dic47—42no—
292Dic47C2×#10214yes—
293C2×#532—471no—
294C2×#431—471no—
295C2×C4×C4⋊C4—471no—
296#1988—472no—
297#1982—472no—
298#1874—472no—
299#1874—472no—
300C4×#102—471no—
301C4×#102—471no—
302C4×#102—471no—
303C4×#102D4711yescharacteristic, complemented
304C2×C47—82no—
305C2×C47#229711yescharacteristic
306C2×C47C22×#10211yescharacteristic
307C2×C47#289611yescharacteristic
308C2×C47#310012yes—
309C2×C47#310312yes—
310D47—24no—
311D47—82no—
312C22×#41—941no—
313#532—478no—
314#532—944no—
315C2×C4⋊Q8—471no—
316C2×#127—471no—
317C2×#124—941no—
318C2×#121—471no—
319C2×#121—471no—
320C2×#121—941no—
321C2×C4×D4—471no—
322C2×C4×D4—941no—
323C2×C4×D4C2×D4711yescharacteristic
324C22×C4⋊C4—472no—
325C22×C42—471no—
326C22×C42—941no—
327#431—944no—
328C2×C4⋊C8—472no—
329C2×#102—472no—
330C2×#102—472no—
331C2×#102—472no—
332C2×#102C2×D4712yes—
333C2×C22⋊C8—472no—
334C2×C4×C8—471no—
335C2×C4×C8C2×D4711yescharacteristic
336C4×C4⋊C4—472no—
337C4×C4⋊C4—471no—
338C4×C4⋊C4—941no—
339C4×C4⋊C4—941no—
340C4×C4⋊C4C2×D4711yescharacteristic
341C2×#95—941no—
342#340—472no—
343#340—472no—
344#340—472no—
345#340C2×D4712yes—
346C47#2138611yescharacteristic, complemented, Sylow
347C2×#41—948no—
348C2×#41—1884no—
349C2×#41—1884no—
350C22×Q8—941no—
351C22×D4—471no—
352C22×D4—941no—
353C22×D4—941no—
354C22×D4—941no—
355C22×D4—941no—
356C22×D4C22×D4711yescharacteristic
357C23×C4—21no—
358C23×C4—471no—
359C23×C4—471no—
360C23×C4—471no—
361C23×C4—942no—
362C23×C4—941no—
363C22×C8—472no—
364C22×C8C22×D4712yes—
365C4⋊Q8—474no—
366C4⋊Q8—942no—
367#127—474no—
368#127—942no—
369#124—948no—
370#121—474no—
371#121—474no—
372#121—948no—
373#121—942no—
374#121—942no—
375C4×D4—22no—
376C4×D4—474no—
377C4×D4—948no—
378C4×D4—942no—
379C4×D4C4×D4714yescomplemented
380C2×C4⋊C4—472no—
381C2×C4⋊C4—471no—
382C2×C4⋊C4—944no—
383C2×C4⋊C4—942no—
384C2×C4⋊C4—941no—
385C2×C4⋊C4—944no—
386C2×C4⋊C4C22×D4711yescharacteristic
387C2×C4⋊C4C22×D4712yes—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—941no—
390C2×C22⋊C4—941no—
391C2×C22⋊C4—941no—
392C2×C42—21no—
393C2×C42—471no—
394C2×C42—471no—
395C2×C42—471no—
396C2×C42—474no—
397C2×C42—474no—
398C2×C42—942no—
399C2×C42—941no—
400C2×C42—1884no—
401C2×C42C22×D4711yescharacteristic
402C4⋊C8—944no—
403#102—24no—
404#102—944no—
405#102—944no—
406#102—944no—
407C22⋊C8—944no—
408C4×C8—22no—
409C4×C8—942no—
410#95—21no—
411#95—942no—
412#95—941no—
413C24—21no—
414C24—471no—
415C24—941no—
416#41—18832no—
417C2×Q8—944no—
418C2×Q8—1882no—
419C2×Q8—1882no—
420C2×D4—21no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—472no—
424C2×D4—944no—
425C2×D4—944no—
426C2×D4—944no—
427C2×D4—944no—
428C2×D4—941no—
429C2×D4—942no—
430C2×D4—942no—
431C2×D4—1882no—
432C2×D4—1882no—
433C2×D4—1882no—
434C2×D4—1882no—
435C2×D4—1884no—
436C2×D4—1882no—
437C2×D4—1882no—
438C2×D4C2×C4×D4712yes—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—24no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—471no—
445C22×C4—474no—
446C22×C4—474no—
447C22×C4—474no—
448C22×C4—471no—
449C22×C4—471no—
450C22×C4—472no—
451C22×C4—472no—
452C22×C4—474no—
453C22×C4—948no—
454C22×C4—942no—
455C22×C4—942no—
456C22×C4—942no—
457C22×C4—941no—
458C22×C4—944no—
459C22×C4—1882no—
460C22×C4—1884no—
461C22×C4—1884no—
462C22×C4—1882no—
463C22×C4—1882no—
464C22×C4D4×D4711yescharacteristic
465C22×C4D4×D4711yescharacteristic
466C22×C4C23×D4711yescharacteristic
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—942no—
471C2×C8—942no—
472C2×C8—942no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—472no—
476C4⋊C4—474no—
477C4⋊C4—944no—
478C4⋊C4—948no—
479C4⋊C4—941no—
480C4⋊C4—942no—
481C4⋊C4C2×C4×D4712yes—
482C4⋊C4#1062535014yescomplemented
483C22⋊C4—24no—
484C22⋊C4—944no—
485C22⋊C4—944no—
486C22⋊C4—944no—
487C42—42no—
488C42—474no—
489C42—474no—
490C42—472no—
491C42—472no—
492C42—1884no—
493C42—1882no—
494C42C2×C4×D4712yes—
495C42D4×D4712yes—
496C23—22no—
497C23—42no—
498C23—41no—
499C23—41no—
500C23—42no—
501C23—472no—
502C23—472no—
503C23—472no—
504C23—942no—
505C23—942no—
506C23—942no—
507C23—1882no—
508C23—1881no—
509C23—1881no—
510C23—1882no—
511C23C2×D4×D4711yescharacteristic
512Q8—1888no—
513D4—22no—
514D4—22no—
515D4—42no—
516D4—942no—
517D4—942no—
518D4—1888no—
519D4—1888no—
520D4—1888no—
521D4—1888no—
522D4—1882no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—48no—
531C2×C4—472no—
532C2×C4—472no—
533C2×C4—472no—
534C2×C4—4716no—
535C2×C4—472no—
536C2×C4—472no—
537C2×C4—942no—
538C2×C4—942no—
539C2×C4—942no—
540C2×C4—942no—
541C2×C4—944no—
542C2×C4—942no—
543C2×C4—944no—
544C2×C4—1884no—
545C2×C4—18816no—
546C2×C4—1882no—
547C2×C4—1881no—
548C2×C4—1881no—
549C2×C4—1882no—
550C2×C4—1888no—
551C2×C4C22⋊C4×D4712yes—
552C2×C4C22⋊C4×D4712yes—
553C2×C4D8×D4712yes—
554C2×C4SD16×D4712yes—
555C2×C4C22×C4×D4711yescharacteristic
556C2×C4C2×D4×D4711yescharacteristic
557C2×C4#41×D4712yes—
558C2×C4C2×#1062535012yes—
559C8—42no—
560C8—1882no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—82no—
565C22—82no—
566C22—474no—
567C22—942no—
568C22—944no—
569C22—944no—
570C22—942no—
571C22—3762no—
572C22—3762no—
573C22—1882no—
574C22—1882no—
575C22—1882no—
576C22C2×C22⋊C4×D4711yescharacteristic
577C22#1568963212yes—
578C22C2×D8×D4711yescharacteristic
579C22C2×SD16×D4711yescharacteristic
580C22#1568992012yes—
581C4—22no—
582C4—44no—
583C4—42no—
584C4—474no—
585C4—942no—
586C4—942no—
587C4—1884no—
588C4—1882no—
589C4#102×D4714yes—
590C4#1568959111yescharacteristic
591C4C4×#1062535011yescharacteristic
592C4#1568967012yes—
593C2—82no—
594C2—944no—
595C2—3762no—
596C2#2024587411yescharacteristic
597C2C2×#102×D4711yescharacteristic
598C2#2025190311yescharacteristic
599C2#2025338612yes—
600C2#2025341712yes—
subgroupG/H
centerC22×C4D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC4×#102×C47C2
socleC23×C47C22×D4
prime radicalC2×#15689670C2
cosocleC22×C4×C47C24
hypercenterC4×#102D47
hypocenterC47#21386
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
2. C2×#15689174C2semidirect
3. C4×C4⋊C4×D47C2semidirect
4. #20244338C2semidirect
5. #20244313C2semidirect
6. #20243704C2semidirect
7. #20243702C2semidirect
8. C4×#13093944C2semidirect
9. C4×#13093946C2semidirect
10. C4×#13093945C2semidirect
11. C4×#102×C47C2semidirect
14. #15689670C4semidirect
62. #13094023C2×C4semidirect
64. #13094020C2×C4semidirect
72. C4×D188C2×C4semidirect
103. C4×D4×C47C2×C4semidirect
172. C4⋊C4×C47#41semidirect
183. C4⋊Dic47#41semidirect
256. C4×D47#102semidirect
303. C4×#102D47semidirect
346. C47#21386semidirect
379. C4×D4C4×D47semidirect
482. C4⋊C4#10625350semidirect

Maximal subgroups

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

subgroupindex
C2×#156896702
C2×#156891742
C4×C4⋊C4×D472
#202443382
#202443132
#202437042
#202437022
C4×#130939442
C4×#130939462
C4×#130939452
C4×#102×C472
#2138647