Finite Group Databasebeta

← #26215975

Subgroups — #26215975

Order 12032.

1330 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689889C211yesmaximal, characteristic
2C2×#15689171C211yesmaximal, characteristic, complemented
3C2×#15689163C211yesmaximal, characteristic, complemented
4#20251908C211yesmaximal, characteristic
5C2×#15689122C211yesmaximal, characteristic
6#20243451C214yesmaximal, complemented
7#20243446C214yesmaximal, complemented
8#20243126C211yesmaximal, characteristic, complemented
9C2×C47×#343C211yesmaximal, characteristic, complemented
10C22×#10625350C2211yescharacteristic
11#15689889—24no—
12#15689889C418yescomplemented
13C2×#13094032C2214yes—
14C2×#13094030—22no—
15C22×#10625329C2211yescharacteristic
16C22×#10625329C2211yescharacteristic
17C2×M16×D47—22no—
18C2×#13093983C2211yescharacteristic
19#15689171—24no—
20C2×#13093959C2214yescomplemented
21#15689163—22no—
22#15689163—22no—
23C2×#13093957C2212yescomplemented
24C2×#13093955C2212yescomplemented
25#15689133—21no—
26#15689129—21no—
27#15689125C2212yes—
28#15689125C2212yes—
29#15689122—22no—
30#15689122C2214yes—
31#15689120—22no—
32C2×#13093947—21no—
33C2×#13093944—21no—
34#15688891C2212yescomplemented
35#15688891C2212yescomplemented
36C22×M16×C47C2211yescharacteristic
37C22×M16×C47C2211yescharacteristic
38C22×C4⋊C4×C47C2211yescharacteristic
39C47×#343C2214yescomplemented
40C2×#10625350—21no—
41C2×#10625350—22no—
42C2×#10625350—21no—
43C2×#10625350—22no—
44C2×#10625350C2×C414yes—
45C2×#10625350D412yes—
46C2×#10625350Q814yes—
47C2×#10625350C2312yes—
48C22×Dic94D411yescharacteristic
49C22×D188D411yescharacteristic
50C22×C22⋊D47—21no—
51C22×C4×D47—21no—
52#13094032—28no—
53#13094032C2×C4116yescomplemented
54#13094030—216no—
55C2×#10625329—21no—
56C2×#10625329—22no—
57C2×#10625329—22no—
58C2×#10625329C2×C414yes—
59C2×#10625329D412yes—
60C2×#10625329D412yes—
61C2×#10625329D414yes—
62C2×#10625329C2312yes—
63C2×#10625324—24no—
64M16×D47—216no—
65C22×C47⋊C8—21no—
66C22×C47⋊C8C2312yes—
67C2×C8×D47—24no—
68#13093983—21no—
69#13093983—21no—
70#13093983—21no—
71#13093983—21no—
72C2×C4⋊Dic47C2311yescharacteristic
73C2×C4⋊Dic47C2311yescharacteristic
74C2×C22⋊Dic47—21no—
75C2×C4×Dic47—21no—
76#13093959—28no—
77#13093957—24no—
78#13093955—24no—
79#13093949—22no—
80#13093949—22no—
81#13093947—24no—
82#13093946—22no—
83#13093946—22no—
84#13093944—24no—
85C23×C4×C47—21no—
86C23×C4×C47C2311yescharacteristic
87C2×M16×C47—21no—
88C2×M16×C47—22no—
89C2×M16×C47—21no—
90C2×M16×C47—22no—
91C2×M16×C47—22no—
92C2×M16×C47C2×C414yes—
93C2×M16×C47C2312yes—
94C2×M16×C47C2312yes—
95C22×C8×C47—21no—
96C22×C8×C47—21no—
97C2×C4⋊C4×C47—21no—
98C2×C4⋊C4×C47—21no—
99C2×C4⋊C4×C47—21no—
100C2×C4⋊C4×C47—21no—
101C2×C4⋊C4×C47—21no—
102C2×C4⋊C4×C47C2312yes—
103C24×C47#4111yescharacteristic
104C23×D47—21no—
105C22×C4×C47—22no—
106C22×C4×C47—41no—
107C22×C4×C47—41no—
108C22×C4×C47—82no—
109C22×C4×C47C22×C412yes—
110C22×C4×C47C2×D412yes—
111C22×C4×C47C2×D411yescharacteristic
112C22×C4×C47C2×D411yescharacteristic
113C22×C4×C47C2×D411yescharacteristic
114C22×C4×C47C2×D411yescharacteristic
115C22×C4×C47C2×Q812yes—
116C22×C4×C47#4111yescharacteristic
117C22×C4×C47#4112yes—
118C22×C4×C47C2411yescharacteristic
119C2×C22⋊D47—24no—
120C2×C22⋊D47—22no—
121C2×C22⋊D47—24no—
122C2×C22⋊D47—22no—
123C22×Dic47—21no—
124C22×Dic47—21no—
125C22×Dic47—21no—
126#10625350—24no—
127#10625350—24no—
128#10625350—24no—
129#10625350—24no—
130#10625350—24no—
131#10625350—22no—
132#10625350—22no—
133#10625350C4⋊C418yes—
134#10625350C22×C414yes—
135#10625350C2×Q814yes—
136C2×D188—22no—
137C2×D188—21no—
138C2×D188C4⋊C414yes—
139C2×D188C2×D412yes—
140C2×C4×D47—24no—
141C2×C4×D47—22no—
142C2×C4×D47—24no—
143C2×C4×D47—22no—
144C2×Dic94—22no—
145C2×Dic94—21no—
146C2×Dic94C4⋊C414yes—
147C2×Dic94C2×D412yes—
148M16×C47—24no—
149M16×C47—22no—
150M16×C47—22no—
151M16×C47—22no—
152M16×C47—42no—
153M16×C47C22×C414yes—
154C2×C8×C47—24no—
155C2×C8×C47—22no—
156C2×C8×C47—22no—
157C2×C8×C47—42no—
158C4⋊C4×C47—22no—
159C4⋊C4×C47—22no—
160C4⋊C4×C47—41no—
161C4⋊C4×C47—41no—
162C22⋊Dic47—41no—
163C22⋊Dic47—41no—
164C4⋊Dic47—21no—
165C4⋊Dic47—21no—
166C4⋊Dic47—21no—
167C4⋊Dic47—21no—
168C4×Dic47—41no—
169C4×Dic47—41no—
170#10625329—22no—
171#10625329—22no—
172#10625329—42no—
173#10625329C22⋊C418yes—
174#10625329C22⋊C418yes—
175C2×C47⋊C8—22no—
176C2×C47⋊C8—42no—
177C2×C47⋊C8C22×C414yes—
178C2×C47⋊C8C2×D414yes—
179C2×C47⋊C8C2×D414yes—
180#10625324—216no—
181C8×D47—216no—
182C23×C47—41no—
183C23×C47—41no—
184C23×C47#11712yes—
185C23×C47#12111yescharacteristic
186C23×C47#12111yescharacteristic
187C23×C47C22⋊Q812yes—
188C23×C47C2×#4111yescharacteristic
189C22×D47—22no—
190C22×D47—24no—
191C22×D47—81no—
192C22×D47—81no—
193C2×C4×C47—42no—
194C2×C4×C47—82no—
195C2×C4×C47—82no—
196C2×C4×C47—82no—
197C2×C4×C47C2×C22⋊C412yes—
198C2×C4×C47C2×C22⋊C412yes—
199C2×C4×C47C2×C4⋊C412yes—
200C2×C4×C47C2×C4⋊C412yes—
201C2×C4×C47#11712yes—
202C2×C4×C47#12112yes—
203C2×C4×C47#12112yes—
204C2×C4×C47#12111yescharacteristic
205C2×C4×C47#12111yescharacteristic
206C2×C4×C47C22⋊Q812yes—
207C2×C4×C47C22⋊Q812yes—
208C2×C4×C47C22⋊Q812yes—
209C2×C4×C47C23×C411yescharacteristic
210C2×C4×C47C22×D411yescharacteristic
211C2×C4×C47C22×D411yescharacteristic
212C2×C4×C47C22×D411yescharacteristic
213C2×C4×C47C22×Q811yescharacteristic
214C2×C4×C47C2×#4111yescharacteristic
215C22⋊D47—24no—
216C22⋊D47—24no—
217C22⋊D47—28no—
218C2×Dic47—22no—
219C2×Dic47—24no—
220C2×Dic47—41no—
221C2×Dic47—41no—
222C2×Dic47—81no—
223C2×Dic47—81no—
224D188—22no—
225D188—24no—
226D188C2×C4⋊C414yes—
227C4×D47—24no—
228C4×D47—24no—
229C4×D47—28no—
230Dic94—22no—
231Dic94—24no—
232Dic94C2×C4⋊C414yes—
233C8×C47—24no—
234C8×C47—42no—
235C47⋊C8—42no—
236C47⋊C8C2×C22⋊C418yes—
237#20679—471nomaximal, Sylow
238C22×C47—21no—
239C22×C47—21no—
240C22×C47—21no—
241C22×C47—21no—
242C22×C47—21no—
243C22×C47—21no—
244C22×C47—42no—
245C22×C47—42no—
246C22×C47#38012yes—
247C22×C47#38012yes—
248C22×C47C2×#11711yescharacteristic
249C22×C47C2×#12111yescharacteristic
250C22×C47C2×C22⋊Q811yescharacteristic
251C2×D47—24no—
252C2×D47—42no—
253C2×D47—42no—
254C2×D47—84no—
255C4×C47—82no—
256C4×C47#38012yes—
257C4×C47#38012yes—
258C4×C47C22×C22⋊C411yescharacteristic
259C4×C47C22×C4⋊C411yescharacteristic
260C4×C47C2×#12111yescharacteristic
261C4×C47C2×C22⋊Q811yescharacteristic
262Dic47—24no—
263Dic47—81no—
264Dic47—81no—
265C2×#568—471no—
266C2×#430—471no—
267C2×#428—471no—
268#2896—471no—
269C2×#417—471no—
270#1816—474no—
271#1815—474no—
272#1742—471no—
273C2×#343D4711yescharacteristic, complemented
274C2×C47—22no—
275C2×C47—21no—
276C2×C47—21no—
277C2×C47—22no—
278C2×C47C2×#38011yescharacteristic
279C2×C47#306811yescharacteristic
280C2×C47#306911yescharacteristic
281D47—44no—
282C22×#41—471no—
283#568—478no—
284#568—944no—
285C2×#131—474no—
286C2×#131—942no—
287C22×M16—471no—
288C22×M16—471no—
289C22×M16—942no—
290C22×M16C2×D4711yescharacteristic
291C22×M16C2×D4711yescharacteristic
292C2×#117—471no—
293C22×C4⋊C4C2×D4711yescharacteristic
294#430—944no—
295C2×#108—474no—
296#428—942no—
297#428—942no—
298C2×#107—472no—
299C2×#106—472no—
300#419—941no—
301#418—941no—
302#417—472no—
303#417—472no—
304#417—474no—
305#417—942no—
306#416—942no—
307C2×#103—941no—
308C2×#102—941no—
309#343—472no—
310#343—472no—
311#343C2×D4714yescomplemented
312C47#2067911yescharacteristic, complemented, Sylow
313C2×#41—474no—
314C2×#41—472no—
315C2×#41—474no—
316C2×#41—472no—
317C2×#41—941no—
318C2×#41—942no—
319C2×#41—941no—
320C2×#41—942no—
321C22×Q8—471no—
322C22×D4—471no—
323C22×D4—941no—
324C23×C4—21no—
325C23×C4—941no—
326C23×C4C22×D4711yescharacteristic
327#131—4716no—
328#131—9416no—
329#131—948no—
330C2×M16—21no—
331C2×M16—22no—
332C2×M16—21no—
333C2×M16—22no—
334C2×M16—22no—
335C2×M16—474no—
336C2×M16—472no—
337C2×M16—472no—
338C2×M16—474no—
339C2×M16—472no—
340C2×M16—944no—
341C2×M16—9416no—
342C2×M16—941no—
343C2×M16—942no—
344C2×M16—942no—
345C2×M16C4×D4714yes—
346C2×M16C22×D4712yes—
347C2×M16C22×D4712yes—
348C22×C8—21no—
349C22×C8—21no—
350C22×C8—472no—
351C22×C8—944no—
352C22×C8—941no—
353#117—941no—
354#117—941no—
355#117—941no—
356#117—941no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—471no—
363C2×C4⋊C4—471no—
364C2×C4⋊C4C22×D4712yes—
365C2×C22⋊C4—941no—
366C2×C42—941no—
367#108—948no—
368#107—944no—
369#106—944no—
370#103—944no—
371#103—942no—
372#103—942no—
373#102—944no—
374#102—942no—
375#102—942no—
376C24—941no—
377C24#1062535011yescharacteristic
378#41—478no—
379#41—474no—
380#41—474no—
381#41—944no—
382#41—944no—
383#41—944no—
384#41—944no—
385#41—944no—
386#41—942no—
387#41—942no—
388C2×Q8—474no—
389C2×Q8—472no—
390C2×Q8—942no—
391C2×Q8—941no—
392C2×D4—474no—
393C2×D4—472no—
394C2×D4—944no—
395C2×D4—942no—
396C2×D4—944no—
397C2×D4—942no—
398C2×D4—941no—
399C2×D4—942no—
400C22×C4—22no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—82no—
404C22×C4—941no—
405C22×C4—944no—
406C22×C4—942no—
407C22×C4—944no—
408C22×C4—942no—
409C22×C4—941no—
410C22×C4—941no—
411C22×C4C2×C4×D4712yes—
412C22×C4#1062535011yescharacteristic
413C22×C4#1062535012yes—
414C22×C4D4×D4712yes—
415C22×C4D4×D4711yescharacteristic
416C22×C4D4×D4711yescharacteristic
417C22×C4D4×D4711yescharacteristic
418C22×C4D4×D4711yescharacteristic
419C22×C4Q8×D4712yes—
420C22×C4C23×D4711yescharacteristic
421M16—24no—
422M16—22no—
423M16—22no—
424M16—22no—
425M16—42no—
426M16—478no—
427M16—478no—
428M16—9416no—
429M16—942no—
430M16—942no—
431M16—1882no—
432M16C2×C4×D4714yes—
433C2×C8—24no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—42no—
437C2×C8—474no—
438C2×C8—474no—
439C2×C8—474no—
440C2×C8—9416no—
441C2×C8—942no—
442C2×C8—1882no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—941no—
448C4⋊C4—941no—
449C4⋊C4—941no—
450C4⋊C4—941no—
451C22⋊C4—1881no—
452C22⋊C4—1881no—
453C42—1881no—
454C42—1881no—
455C23—41no—
456C23—41no—
457C23—944no—
458C23—942no—
459C23—3761no—
460C23—3761no—
461C23#1309398112yes—
462C23#1309399711yescharacteristic
463C23#1309399711yescharacteristic
464C23#1309400212yes—
465C23C2×#1062535011yescharacteristic
466Q8—474no—
467Q8—944no—
468Q8—942no—
469D4—474no—
470D4—948no—
471D4—944no—
472D4—944no—
473D4—944no—
474D4—942no—
475C2×C4—42no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—944no—
480C2×C4—942no—
481C2×C4—948no—
482C2×C4—944no—
483C2×C4—944no—
484C2×C4—3761no—
485C2×C4—3761no—
486C2×C4—1881no—
487C2×C4—1881no—
488C2×C4C22⋊C4×D4712yes—
489C2×C4C22⋊C4×D4712yes—
490C2×C4C4⋊C4×D4712yes—
491C2×C4C4⋊C4×D4712yes—
492C2×C4#1309398112yes—
493C2×C4#1309399711yescharacteristic
494C2×C4#1309399711yescharacteristic
495C2×C4#1309399712yes—
496C2×C4#1309399712yes—
497C2×C4#1309400212yes—
498C2×C4#1309400212yes—
499C2×C4#1309400212yes—
500C2×C4C22×C4×D4711yescharacteristic
501C2×C4C2×D4×D4711yescharacteristic
502C2×C4C2×D4×D4711yescharacteristic
503C2×C4C2×D4×D4711yescharacteristic
504C2×C4C2×Q8×D4711yescharacteristic
505C2×C4C2×#1062535011yescharacteristic
506C8—24no—
507C8—42no—
508C8—478no—
509C8—1882no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—42no—
517C22—42no—
518C22—944no—
519C22—3764no—
520C22—1882no—
521C22—1882no—
522C22#1568898212yes—
523C22#1568898212yes—
524C22C2×#1309398111yescharacteristic
525C22C2×#1309399711yescharacteristic
526C22C2×#1309400211yescharacteristic
527C4—82no—
528C4—944no—
529C4—3761no—
530C4—3761no—
531C4#1568898212yes—
532C4#1568898212yes—
533C4C2×C22⋊C4×D4711yescharacteristic
534C4C2×C4⋊C4×D4711yescharacteristic
535C4C2×#1309399711yescharacteristic
536C4C2×#1309400211yescharacteristic
537C2—22no—
538C2—22no—
539C2—21no—
540C2—21no—
541C2—1884no—
542C2C2×#1568898211yescharacteristic
543C2#2025311511yescharacteristic
544C2#2025312411yescharacteristic
subgroupG/H
centerC22C2×#13093997
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#343C2
socleC22×C47C2×#121
prime radicalC22×#10625350C22
cosocleC22×C4×C47C24
hypercenterC2×#343D47
hypocenterC47#20679
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#343

Complements

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

NH ≅ G/Ntype
2. C2×#15689171C2semidirect
3. C2×#15689163C2semidirect
6. #20243451C2semidirect
7. #20243446C2semidirect
8. #20243126C2semidirect
9. C2×C47×#343C2semidirect
12. #15689889C4semidirect
20. C2×#13093959C22semidirect
23. C2×#13093957C22semidirect
24. C2×#13093955C22semidirect
34. #15688891C22semidirect
35. #15688891C22semidirect
39. C47×#343C22semidirect
53. #13094032C2×C4semidirect
273. C2×#343D47semidirect
311. #343C2×D47semidirect
312. C47#20679semidirect

Maximal subgroups

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

subgroupindex
C2×#156898892
C2×#156891712
C2×#156891632
#202519082
C2×#156891222
#202434512
#202434462
#202431262
C2×C47×#3432
#2067947