Finite Group Databasebeta

← #26244957

Subgroups — #26244957

Order 12032.

1102 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689166C211yesmaximal, characteristic, complemented
2C2×#15689150C211yesmaximal, characteristic, complemented
3C2×#15689122C211yesmaximal, characteristic
4C2×#15689117C211yesmaximal, characteristic
5#20245976C211yesmaximal, characteristic
6#20244780C212yesmaximal, complemented
7#20244773C212yesmaximal, complemented
8#20244766C212yesmaximal, complemented
9#20244759C212yesmaximal, complemented
10C2×#15688885C211yesmaximal, characteristic, complemented
11C47×#2317C211yesmaximal, characteristic, complemented
12C22×#10625350C2211yescharacteristic
13C22×#10625329C2211yescharacteristic
14C23×C47⋊C8C2211yescharacteristic
15C22×C4⋊Dic47C2211yescharacteristic
16#15689166—24no—
17C2×#13093957C2212yes—
18C2×#13093955C2212yes—
19#15689150—24no—
20C2×C4⋊(C47⋊C8)C2212yes—
21C2×C4⋊(C47⋊C8)C2212yes—
22#15689122—24no—
23#15689117—24no—
24C2×#13093947C2211yescharacteristic
25C2×#13093947C2212yes—
26C2×#13093947C2211yescharacteristic
27C2×#13093944C2211yescharacteristic
28C2×#13093944C2212yes—
29C2×#13093944C2211yescharacteristic
30#15689029C2212yes—
31#15689025C2212yes—
32#15689000—22no—
33#15688980—21no—
34#15688980—21no—
35#15688885C2212yes—
36#15688885C2212yes—
37C2×#117×C47C2211yescharacteristic
38C2×#117×C47C2211yescharacteristic
39C22×C4⋊C4×C47C2211yescharacteristic
40C47×#389C2212yes—
41C47×#389C2212yes—
42C47×#385—22no—
43C47×#384—21no—
44C47×#382—21no—
45C2×#10625350—22no—
46C2×#10625350—22no—
47C2×#10625350—22no—
48C2×#10625350—22no—
49C2×#10625350—22no—
50C2×#10625350Q814yes—
51C22×Dic94C2311yescharacteristic
52C22×D188C2311yescharacteristic
53C22×C22⋊D47—21no—
54C23×Dic47—21no—
55C22×C4×D47—21no—
56C2×#10625329—22no—
57C2×#10625329—22no—
58C2×#10625329D414yes—
59C22×C47⋊C8—22no—
60C22×C47⋊C8—22no—
61C22×C47⋊C8—22no—
62C22×C47⋊C8C2311yescharacteristic
63C22×C47⋊C8C2311yescharacteristic
64C22×C47⋊C8C2312yes—
65C2×C4⋊Dic47—22no—
66C2×C4⋊Dic47—22no—
67C2×C4⋊Dic47—21no—
68C2×C4⋊Dic47C2312yes—
69C2×(C2×D47)⋊C4—22no—
70C2×(C2×D47)⋊C4—22no—
71#13093957—24no—
72#13093955—24no—
73C4⋊(C47⋊C8)—24no—
74C4⋊(C47⋊C8)—24no—
75#13093947—22no—
76#13093947—22no—
77#13093947—24no—
78#13093944—22no—
79#13093944—22no—
80#13093944—24no—
81C47⋊#95—22no—
82C47⋊#95—22no—
83C23×C4×C47—21no—
84C23×C4×C47C2311yescharacteristic
85#117×C47—22no—
86#117×C47—22no—
87#117×C47—22no—
88#117×C47D414yes—
89C2×C4⋊C4×C47—22no—
90C2×C4⋊C4×C47—22no—
91C2×C4⋊C4×C47—21no—
92C2×C4⋊C4×C47—22no—
93C2×C4⋊C4×C47—22no—
94C2×C4⋊C4×C47D411yescharacteristic
95C2×C4⋊C4×C47D411yescharacteristic
96C2×C4⋊C4×C47C2312yes—
97C2×C4⋊C4×C47C2311yescharacteristic
98C2×C4⋊C4×C47C2311yescharacteristic
99C2×C22⋊C4×C47—21no—
100C2×C22⋊C4×C47D412yes—
101C2×C42×C47—21no—
102C2×C42×C47C2312yes—
103#95×C47—22no—
104#95×C47—22no—
105C24×C47C2×D411yescharacteristic
106C23×D47—21no—
107C22×C4×C47—21no—
108C22×C4×C47—21no—
109C22×C4×C47—22no—
110C22×C4×C47—42no—
111C22×C4×C47—81no—
112C22×C4×C47—81no—
113C22×C4×C47C2×D412yes—
114C22×C4×C47C2×D412yes—
115C22×C4×C47C2×D411yescharacteristic
116C22×C4×C47C2×D412yes—
117C22×C4×C47C2×D412yes—
118C22×C4×C47C2×Q812yes—
119C22×C4×C47#4112yes—
120C22×C4×C47#4112yes—
121C22×C4×C47#4112yes—
122C22×C4×C47C2411yescharacteristic
123C2×C22⋊D47—24no—
124C2×C22⋊D47—42no—
125C2×C22⋊D47—42no—
126C22×Dic47—21no—
127C22×Dic47—22no—
128C22×Dic47—42no—
129C22×Dic47—81no—
130C22×Dic47—81no—
131#10625350—28no—
132#10625350—28no—
133#10625350—44no—
134#10625350—44no—
135C2×D188—21no—
136C2×D188—21no—
137C2×D188—21no—
138C2×D188—21no—
139C2×D188—21no—
140C2×D188C2×Q812yes—
141C2×C4×D47—24no—
142C2×C4×D47—42no—
143C2×C4×D47—42no—
144C2×Dic94—21no—
145C2×Dic94—21no—
146C2×Dic94—21no—
147C2×Dic94—21no—
148C2×Dic94—21no—
149C2×Dic94C2×Q812yes—
150C4⋊C4×C47—22no—
151C4⋊C4×C47—22no—
152C4⋊C4×C47—21no—
153C4⋊C4×C47—21no—
154C4⋊C4×C47—22no—
155C4⋊C4×C47—22no—
156C4⋊C4×C47—42no—
157C4⋊C4×C47—82no—
158C4⋊C4×C47—82no—
159C4⋊C4×C47C2×D412yes—
160C4⋊C4×C47C2×D412yes—
161C22⋊C4×C47—24no—
162C22⋊C4×C47—24no—
163C42×C47—24no—
164C42×C47—22no—
165C42×C47—22no—
166(C2×D47)⋊C4—82no—
167(C2×D47)⋊C4—82no—
168C4⋊Dic47—22no—
169C4⋊Dic47—22no—
170C4⋊Dic47—42no—
171#10625329—28no—
172C2×C47⋊C8—22no—
173C2×C47⋊C8—22no—
174C2×C47⋊C8—21no—
175C2×C47⋊C8—21no—
176C2×C47⋊C8—21no—
177C2×C47⋊C8—21no—
178C2×C47⋊C8—21no—
179C2×C47⋊C8—21no—
180C2×C47⋊C8—44no—
181C2×C47⋊C8C2×D414yes—
182C23×C47—21no—
183C23×C47—21no—
184C23×C47—21no—
185C23×C47—21no—
186C23×C47#12112yes—
187C23×C47#12112yes—
188C23×C47C22⋊Q812yes—
189C23×C47C22×D411yescharacteristic
190C22×D47—22no—
191C22×D47—41no—
192C22×D47—41no—
193C22×D47—41no—
194C22×D47—41no—
195C22×D47—41no—
196C22×D47—41no—
197C2×C4×C47—22no—
198C2×C4×C47—22no—
199C2×C4×C47—22no—
200C2×C4×C47—22no—
201C2×C4×C47—22no—
202C2×C4×C47—22no—
203C2×C4×C47—22no—
204C2×C4×C47—22no—
205C2×C4×C47—41no—
206C2×C4×C47—41no—
207C2×C4×C47—41no—
208C2×C4×C47—41no—
209C2×C4×C47—42no—
210C2×C4×C47—42no—
211C2×C4×C47—42no—
212C2×C4×C47—84no—
213C2×C4×C47#12112yes—
214C2×C4×C47#12112yes—
215C2×C4×C47#12114yes—
216C2×C4×C47C22⋊Q812yes—
217C2×C4×C47C22⋊Q814yes—
218C2×C4×C47C22⋊Q814yes—
219C2×C4×C47#12614yes—
220C2×C4×C47C22×D411yescharacteristic
221C2×C4×C47C22×D411yescharacteristic
222C2×C4×C47C22×Q811yescharacteristic
223C2×C4×C47C2×#4111yescharacteristic
224C2×C4×C47C2×#4111yescharacteristic
225C2×C4×C47C2×#4111yescharacteristic
226C22⋊D47—48no—
227C2×Dic47—22no—
228C2×Dic47—41no—
229C2×Dic47—41no—
230C2×Dic47—42no—
231C2×Dic47—42no—
232C2×Dic47—42no—
233C2×Dic47—84no—
234D188—22no—
235D188—22no—
236D188—41no—
237D188—41no—
238C4×D47—48no—
239Dic94—22no—
240Dic94—22no—
241Dic94—41no—
242Dic94—41no—
243C47⋊C8—24no—
244C47⋊C8—41no—
245C47⋊C8—41no—
246#23616—471nomaximal, Sylow
247C22×C47—22no—
248C22×C47—22no—
249C22×C47—22no—
250C22×C47—22no—
251C22×C47—22no—
252C22×C47—22no—
253C22×C47#39614yes—
254C22×C47C2×#12111yescharacteristic
255C22×C47C2×#12111yescharacteristic
256C22×C47C2×C22⋊Q811yescharacteristic
257C22×C47#57611yescharacteristic
258C22×C47#57711yescharacteristic
259C22×C47#57711yescharacteristic
260C22×C47#57811yescharacteristic
261C2×D47—42no—
262C2×D47—42no—
263C2×D47—42no—
264C2×D47—81no—
265C2×D47—81no—
266C2×D47—81no—
267C2×D47—81no—
268C4×C47—42no—
269C4×C47—42no—
270C4×C47—42no—
271C4×C47—42no—
272C4×C47—82no—
273C4×C47#39614yes—
274C4×C47C2×#12111yescharacteristic
275C4×C47C2×C22⋊Q811yescharacteristic
276C4×C47C2×C22⋊Q811yescharacteristic
277C4×C47C2×#12611yescharacteristic
278Dic47—42no—
279Dic47—82no—
280C2×#428—471no—
281C2×#424—471no—
282C2×#417—471no—
283C2×#416—471no—
284#2317D4711yescharacteristic, complemented
285#2311—471no—
286#2068—472no—
287#2067—472no—
288#2066—472no—
289#2065—472no—
290C2×#340—471no—
291C2×C47—24no—
292C2×C47C2×#39611yescharacteristic
293C2×C47#304511yescharacteristic
294C2×C47#304611yescharacteristic
295C2×C47#306211yescharacteristic
296C2×C47#306311yescharacteristic
297C2×C47#308311yescharacteristic
298C2×C47#308411yescharacteristic
299D47—82no—
300C22×#41—471no—
301C22×M16—471no—
302C23×C8—471no—
303C2×#117C2×D4711yescharacteristic
304C2×#117C2×D4711yescharacteristic
305C22×C4⋊C4—471no—
306C22×C4⋊C4C2×D4711yescharacteristic
307#428—944no—
308C2×#107—472no—
309C2×#106—472no—
310#424—944no—
311C2×C4⋊C8—472no—
312C2×C4⋊C8—472no—
313#417—944no—
314#416—944no—
315C2×#103—471no—
316C2×#103—472no—
317C2×#103—471no—
318C2×#102—471no—
319C2×#102—472no—
320C2×#102—471no—
321#391—472no—
322#390—472no—
323#389C2×D4712yes—
324#389C2×D4712yes—
325#386—942no—
326#385—22no—
327#384—21no—
328#382—21no—
329#380—941no—
330#380—941no—
331#340—472no—
332#340—472no—
333C47#2361611yescharacteristic, complemented, Sylow
334C2×#41—474no—
335C2×#41—942no—
336C2×#41—942no—
337C2×#41—942no—
338C2×#41—942no—
339C2×#41—942no—
340C22×Q8—471no—
341C22×D4—471no—
342C22×D4—941no—
343C23×C4—21no—
344C23×C4—941no—
345C23×C4—941no—
346C23×C4C22×D4711yescharacteristic
347C2×M16—474no—
348C2×M16—942no—
349C2×M16—942no—
350C22×C8—472no—
351C22×C8—471no—
352C22×C8—471no—
353C22×C8—942no—
354C22×C8—942no—
355C22×C8—942no—
356#117—22no—
357#117—22no—
358#117—22no—
359#117D18814yes—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—472no—
366C2×C4⋊C4—942no—
367C2×C4⋊C4—942no—
368C2×C4⋊C4—941no—
369C2×C4⋊C4C22⋊D4711yescharacteristic
370C2×C4⋊C4C22⋊D4711yescharacteristic
371C2×C4⋊C4C22×D4712yes—
372C2×C4⋊C4C22×D4711yescharacteristic
373C2×C4⋊C4C22×D4711yescharacteristic
374C2×C22⋊C4—21no—
375C2×C22⋊C4—942no—
376C2×C22⋊C4—942no—
377C2×C22⋊C4C22⋊D4712yes—
378C2×C42—21no—
379C2×C42C22×D4712yes—
380#107—944no—
381#106—944no—
382C4⋊C8—944no—
383C4⋊C8—944no—
384#103—944no—
385#103—942no—
386#103—942no—
387#102—944no—
388#102—942no—
389#102—942no—
390#95—22no—
391#95—22no—
392#95—942no—
393#95—942no—
394C24—941no—
395C24C2×C22⋊D4711yescharacteristic
396#41—948no—
397#41—948no—
398#41—1884no—
399#41—1884no—
400C2×Q8—472no—
401C2×Q8—941no—
402C2×Q8—941no—
403C2×Q8—941no—
404C2×Q8—941no—
405C2×Q8—941no—
406C2×D4—472no—
407C2×D4—944no—
408C2×D4—941no—
409C2×D4—941no—
410C2×D4—941no—
411C2×D4—941no—
412C2×D4—941no—
413C2×D4—1882no—
414C2×D4—1882no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—42no—
419C22×C4—81no—
420C22×C4—81no—
421C22×C4—941no—
422C22×C4—944no—
423C22×C4—942no—
424C22×C4—3761no—
425C22×C4—3761no—
426C22×C4—1882no—
427C22×C4—1882no—
428C22×C4—1882no—
429C22×C4C2×D18812yes—
430C22×C4#1062535012yes—
431C22×C4D4×D4712yes—
432C22×C4#1062535212yes—
433C22×C4Q8×D4712yes—
434C22×C4#1062535412yes—
435C22×C4C2×C22⋊D4712yes—
436C22×C4C2×C22⋊D4712yes—
437C22×C4C2×C22⋊D4711yescharacteristic
438C22×C4C23×D4711yescharacteristic
439M16—948no—
440C2×C8—474no—
441C2×C8—942no—
442C2×C8—942no—
443C2×C8—941no—
444C2×C8—941no—
445C2×C8—941no—
446C2×C8—941no—
447C2×C8—941no—
448C2×C8—941no—
449C2×C8—1884no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—42no—
457C4⋊C4—82no—
458C4⋊C4—82no—
459C4⋊C4—942no—
460C4⋊C4—942no—
461C4⋊C4—1882no—
462C4⋊C4C2×D18812yes—
463C4⋊C4C2×D18812yes—
464C22⋊C4—24no—
465C22⋊C4—24no—
466C22⋊C4—3762no—
467C22⋊C4—3762no—
468C42—24no—
469C42—22no—
470C42—22no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—942no—
476C23—1881no—
477C23—1881no—
478C23—1881no—
479C23—1881no—
480C23—1881no—
481C23—1881no—
482C23#1309399612yes—
483C23#1309399812yes—
484C23#1309400012yes—
485C23C22×C22⋊D4711yescharacteristic
486Q8—942no—
487Q8—942no—
488Q8—1881no—
489Q8—1881no—
490D4—942no—
491D4—942no—
492D4—1888no—
493D4—1881no—
494D4—1881no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—84no—
511C2×C4—942no—
512C2×C4—3764no—
513C2×C4—1882no—
514C2×C4—1882no—
515C2×C4—1882no—
516C2×C4—1881no—
517C2×C4—1881no—
518C2×C4—1888no—
519C2×C4#1309399612yes—
520C2×C4#1309399812yes—
521C2×C4#1309399914yes—
522C2×C4#1309400012yes—
523C2×C4#1309400214yes—
524C2×C4#1309400414yes—
525C2×C4#1309401914yes—
526C2×C4C2×D4×D4711yescharacteristic
527C2×C4C22×D18811yescharacteristic
528C2×C4C2×Q8×D4711yescharacteristic
529C2×C4C2×#1062535211yescharacteristic
530C2×C4C2×#1062535011yescharacteristic
531C2×C4C2×#1062535411yescharacteristic
532C8—944no—
533C8—1881no—
534C8—1881no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—3761no—
542C22—3761no—
543C22—3761no—
544C22—3761no—
545C22—1882no—
546C22—1882no—
547C22—1882no—
548C22#1568904514yes—
549C22C2×#1309399611yescharacteristic
550C22C2×#1309399811yescharacteristic
551C22C2×#1309400011yescharacteristic
552C22#1568992611yescharacteristic
553C22#1568993111yescharacteristic
554C22#1568993411yescharacteristic
555C22#1568993811yescharacteristic
556C4—42no—
557C4—42no—
558C4—42no—
559C4—42no—
560C4—82no—
561C4—3762no—
562C4—1882no—
563C4#1568904514yes—
564C4C2×#1309399911yescharacteristic
565C4C2×#1309400211yescharacteristic
566C4C2×#1309400411yescharacteristic
567C4C2×#1309401911yescharacteristic
568C2—24no—
569C2—3762no—
570C2C2×#1568904511yescharacteristic
571C2#2025292711yescharacteristic
572C2#2025293811yescharacteristic
573C2#2025306411yescharacteristic
574C2#2025307511yescharacteristic
575C2#2025323511yescharacteristic
576C2#2025324611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2317C2
socleC23×C47C22×D4
prime radicalC22×#10625350C22
hypercenter#2317D47
hypocenterC47#23616
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2317

Complements

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

NH ≅ G/Ntype
1. C2×#15689166C2semidirect
2. C2×#15689150C2semidirect
6. #20244780C2semidirect
7. #20244773C2semidirect
8. #20244766C2semidirect
9. #20244759C2semidirect
10. C2×#15688885C2semidirect
11. C47×#2317C2semidirect
284. #2317D47semidirect
333. C47#23616semidirect

Maximal subgroups

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

subgroupindex
C2×#156891662
C2×#156891502
C2×#156891222
C2×#156891172
#202459762
#202447802
#202447732
#202447662
#202447592
C2×#156888852
C47×#23172
#2361647