Finite Group Databasebeta

← #26218834

Subgroups — #26218834

Order 12032.

1022 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20251905C211yesmaximal, characteristic
2C2×#15689122C211yesmaximal, characteristic
3C2×#15689105C211yesmaximal, characteristic
4#20251807C211yesmaximal, characteristic, complemented
5#20244049C214yesmaximal, complemented
6#20244042C214yesmaximal, complemented
7C2×#15688892C211yesmaximal, characteristic, complemented
8#20243114C211yesmaximal, characteristic, complemented
9C2×C47×#428C211yesmaximal, characteristic, complemented
10C22×#10625350C2211yescharacteristic
11C22×#10625329C2211yescharacteristic
12C22×#10625329C2211yescharacteristic
13C22×#10625329C2211yescharacteristic
14#15689131C2212yes—
15#15689127C2212yes—
16#15689122—22no—
17#15689122—22no—
18#15689122—21no—
19#15689122—21no—
20#15689117—22no—
21C2×#13093947C2212yes—
22C2×#13093944C2212yes—
23#15689105C418yescomplemented
24C2×#13093942C2212yes—
25C2×#13093940C2212yes—
26#15689079—22no—
27#15689079C2214yes—
28C4×#10625329—22no—
29#15688892C2214yes—
30#15688887C2214yes—
31C22×M16×C47C2211yescharacteristic
32C2×#117×C47C2211yescharacteristic
33C22×C4⋊C4×C47C2211yescharacteristic
34C47×#428—22no—
35C47×#428C2214yes—
36C2×#107×C47—21no—
37C2×#106×C47—21no—
38C2×#10625350—22no—
39C2×#10625350—21no—
40C2×#10625350—21no—
41C2×#10625350—21no—
42C2×#10625350—21no—
43C2×#10625350—22no—
44C2×#10625350—22no—
45C2×#10625350C2×C414yes—
46C22×Dic94C2311yescharacteristic
47C22×D188C2311yescharacteristic
48C22×C22⋊D47—21no—
49C22×C4×D47—21no—
50C2×#10625329—21no—
51C2×#10625329—22no—
52C2×#10625329—22no—
53C2×#10625329—22no—
54C2×#10625329—21no—
55C2×#10625329—22no—
56C2×#10625329—22no—
57C2×#10625329C2×C414yes—
58C2×#10625329D412yes—
59C2×#10625329D412yes—
60C2×#10625329C2312yes—
61C2×#10625329C2312yes—
62C22×C47⋊C8—21no—
63C22×C47⋊C8—21no—
64C22×C47⋊C8C2312yes—
65#13093947—22no—
66#13093947—22no—
67#13093947—24no—
68#13093944—22no—
69#13093944—22no—
70#13093944—24no—
71#13093942C2×C418yescomplemented
72#13093940C2×C418yescomplemented
73#13093935—24no—
74C4×C47⋊C8—24no—
75C23×C4×C47—21no—
76C23×C4×C47C2311yescharacteristic
77C2×M16×C47—21no—
78C2×M16×C47—22no—
79C2×M16×C47C2×C414yes—
80C2×M16×C47C2312yes—
81C22×C8×C47—21no—
82#117×C47—21no—
83#117×C47—21no—
84#117×C47D412yes—
85#117×C47C2312yes—
86C2×C4⋊C4×C47—21no—
87C2×C4⋊C4×C47—21no—
88C2×C4⋊C4×C47—21no—
89C2×C4⋊C4×C47—21no—
90C2×C4⋊C4×C47—21no—
91C2×C4⋊C4×C47D411yescharacteristic
92C2×C4⋊C4×C47D411yescharacteristic
93C2×C4⋊C4×C47C2312yes—
94C2×C22⋊C4×C47—21no—
95C2×C42×C47—21no—
96#107×C47—24no—
97#106×C47—24no—
98C24×C47#4111yescharacteristic
99C23×D47—21no—
100C22×C4×C47—21no—
101C22×C4×C47—21no—
102C22×C4×C47—22no—
103C22×C4×C47—41no—
104C22×C4×C47—41no—
105C22×C4×C47—82no—
106C22×C4×C47C22×C412yes—
107C22×C4×C47C2×D411yescharacteristic
108C22×C4×C47C2×D411yescharacteristic
109C22×C4×C47C2×D411yescharacteristic
110C22×C4×C47C2×D411yescharacteristic
111C22×C4×C47#4111yescharacteristic
112C22×C4×C47#4112yes—
113C22×C4×C47#4112yes—
114C22×C4×C47#4112yes—
115C22×C4×C47C2411yescharacteristic
116C2×C22⋊D47—24no—
117C2×C22⋊D47—41no—
118C2×C22⋊D47—41no—
119C2×C22⋊D47—42no—
120C22×Dic47—21no—
121#10625350—24no—
122#10625350—24no—
123#10625350—24no—
124#10625350—24no—
125#10625350—42no—
126#10625350—42no—
127#10625350—42no—
128#10625350—42no—
129C2×D188—21no—
130C2×D188—21no—
131C2×D188—21no—
132C2×D188—21no—
133C2×D188—21no—
134C2×D188C22×C412yes—
135C2×C4×D47—24no—
136C2×C4×D47—41no—
137C2×C4×D47—41no—
138C2×C4×D47—42no—
139C2×Dic94—21no—
140C2×Dic94—21no—
141C2×Dic94—21no—
142C2×Dic94—21no—
143C2×Dic94—21no—
144C2×Dic94C22×C412yes—
145M16×C47—24no—
146M16×C47—22no—
147M16×C47C22×C414yes—
148C2×C8×C47—24no—
149C2×C8×C47—22no—
150C4⋊C4×C47—21no—
151C4⋊C4×C47—21no—
152C4⋊C4×C47—22no—
153C4⋊C4×C47—22no—
154C4⋊C4×C47—41no—
155C4⋊C4×C47—41no—
156C4⋊C4×C47C2×D412yes—
157C4⋊C4×C47C2×D412yes—
158C22⋊C4×C47—22no—
159C22⋊C4×C47—22no—
160C42×C47—22no—
161C42×C47—22no—
162#10625329—24no—
163#10625329—24no—
164#10625329—24no—
165#10625329—22no—
166#10625329—22no—
167#10625329—22no—
168#10625329—42no—
169#10625329C22×C414yes—
170C2×C47⋊C8—24no—
171C2×C47⋊C8—22no—
172C2×C47⋊C8—22no—
173C2×C47⋊C8—22no—
174C2×C47⋊C8—22no—
175C2×C47⋊C8—42no—
176C2×C47⋊C8C2×D414yes—
177C23×C47—41no—
178C23×C47—41no—
179C23×C47#11712yes—
180C23×C47#12111yescharacteristic
181C23×C47#12111yescharacteristic
182C23×C47#12612yes—
183C23×C47C2×#4111yescharacteristic
184C22×D47—22no—
185C22×D47—41no—
186C22×D47—41no—
187C22×D47—81no—
188C22×D47—81no—
189C2×C4×C47—22no—
190C2×C4×C47—22no—
191C2×C4×C47—42no—
192C2×C4×C47—81no—
193C2×C4×C47—81no—
194C2×C4×C47—82no—
195C2×C4×C47—82no—
196C2×C4×C47—82no—
197C2×C4×C47#11712yes—
198C2×C4×C47C4×D412yes—
199C2×C4×C47C4×D412yes—
200C2×C4×C47C4×D412yes—
201C2×C4×C47C4×D412yes—
202C2×C4×C47#12111yescharacteristic
203C2×C4×C47#12111yescharacteristic
204C2×C4×C47#12312yes—
205C2×C4×C47#12312yes—
206C2×C4×C47#12412yes—
207C2×C4×C47#12412yes—
208C2×C4×C47#12612yes—
209C2×C4×C47C23×C411yescharacteristic
210C2×C4×C47C22×D411yescharacteristic
211C2×C4×C47C22×D411yescharacteristic
212C2×C4×C47C2×#4111yescharacteristic
213C2×C4×C47C2×#4111yescharacteristic
214C2×C4×C47C2×#4111yescharacteristic
215C22⋊D47—44no—
216C22⋊D47—44no—
217C2×Dic47—22no—
218C2×Dic47—41no—
219C2×Dic47—41no—
220D188—22no—
221D188—22no—
222D188—41no—
223D188—41no—
224C4×D47—44no—
225C4×D47—44no—
226Dic94—22no—
227Dic94—22no—
228Dic94—41no—
229Dic94—41no—
230C8×C47—24no—
231C47⋊C8—24no—
232C47⋊C8—24no—
233C47⋊C8—42no—
234#21076—471nomaximal, Sylow
235C22×C47—21no—
236C22×C47—21no—
237C22×C47—21no—
238C22×C47—21no—
239C22×C47—21no—
240C22×C47—21no—
241C22×C47—42no—
242C22×C47—42no—
243C22×C47#38812yes—
244C22×C47#38812yes—
245C22×C47C2×#11711yescharacteristic
246C22×C47C2×#12111yescharacteristic
247C22×C47C2×#12611yescharacteristic
248C2×D47—42no—
249C2×D47—82no—
250C2×D47—81no—
251C2×D47—81no—
252C2×D47—82no—
253C4×C47—42no—
254C4×C47—42no—
255C4×C47—82no—
256C4×C47#38812yes—
257C4×C47#38812yes—
258C4×C47C2×C4×D411yescharacteristic
259C4×C47C2×C4×D411yescharacteristic
260C4×C47C2×#12311yescharacteristic
261C4×C47C2×#12411yescharacteristic
262Dic47—42no—
263C2×#428D4711yescharacteristic, complemented
264#2896—471no—
265C2×#417—471no—
266C2×#413—471no—
267#2873—471no—
268#1931—474no—
269#1930—474no—
270C2×#343—471no—
271#1738—471no—
272C2×C47—21no—
273C2×C47—21no—
274C2×C47—22no—
275C2×C47—22no—
276C2×C47C2×#38811yescharacteristic
277C2×C47#306811yescharacteristic
278C2×C47#306911yescharacteristic
279D47—82no—
280C22×#41—471no—
281C22×M16—471no—
282C22×M16—471no—
283C22×M16—471no—
284C22×M16C2×D4711yescharacteristic
285C2×#117C2×D4711yescharacteristic
286C22×C4⋊C4C2×D4711yescharacteristic
287#428—22no—
288#428C2×D4714yes—
289C2×#107—21no—
290C2×#106—21no—
291#419—472no—
292#418—472no—
293#417—942no—
294#417—942no—
295#417—941no—
296#417—941no—
297#416—942no—
298C2×#103—472no—
299C2×#102—472no—
300#413—478no—
301C2×#101—472no—
302C2×#100—472no—
303#405—474no—
304#405—942no—
305C4×M16—942no—
306#343—474no—
307#341—474no—
308C47#2107611yescharacteristic, complemented, Sylow
309C2×#41—474no—
310C2×#41—942no—
311C2×#41—942no—
312C2×#41—941no—
313C2×#41—941no—
314C2×#41—942no—
315C2×#41—941no—
316C2×#41—941no—
317C22×Q8—471no—
318C22×D4—471no—
319C22×D4—941no—
320C23×C4—21no—
321C23×C4—941no—
322C23×C4C22×D4711yescharacteristic
323C2×M16—22no—
324C2×M16—21no—
325C2×M16—472no—
326C2×M16—472no—
327C2×M16—474no—
328C2×M16—472no—
329C2×M16—472no—
330C2×M16—942no—
331C2×M16—941no—
332C2×M16—942no—
333C2×M16—942no—
334C2×M16—941no—
335C2×M16—942no—
336C2×M16—942no—
337C2×M16C4×D4714yes—
338C2×M16C22×D4712yes—
339C22×C8—21no—
340C22×C8—472no—
341C22×C8—941no—
342C22×C8—941no—
343#117—21no—
344#117—21no—
345#117C22⋊D4712yes—
346#117C22×D4712yes—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4C22⋊D4711yescharacteristic
353C2×C4⋊C4C22⋊D4711yescharacteristic
354C2×C4⋊C4C22×D4712yes—
355C2×C22⋊C4—21no—
356C2×C42—21no—
357#107—24no—
358#106—24no—
359#103—944no—
360#103—942no—
361#103—942no—
362#102—944no—
363#102—942no—
364#102—942no—
365#101—478no—
366#100—478no—
367#97—944no—
368C4×C8—944no—
369C24—941no—
370C24#1062535211yescharacteristic
371#41—944no—
372#41—944no—
373#41—944no—
374#41—944no—
375#41—1882no—
376#41—1882no—
377#41—1882no—
378#41—1882no—
379C2×Q8—472no—
380C2×Q8—941no—
381C2×Q8—941no—
382C2×Q8—941no—
383C2×Q8—941no—
384C2×Q8—941no—
385C2×D4—472no—
386C2×D4—944no—
387C2×D4—941no—
388C2×D4—941no—
389C2×D4—941no—
390C2×D4—941no—
391C2×D4—941no—
392C2×D4—1881no—
393C2×D4—1881no—
394C2×D4—1882no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—82no—
401C22×C4—941no—
402C22×C4—944no—
403C22×C4—1881no—
404C22×C4—1881no—
405C22×C4—1882no—
406C22×C4C2×C4×D4712yes—
407C22×C4#1062535012yes—
408C22×C4D4×D4711yescharacteristic
409C22×C4D4×D4711yescharacteristic
410C22×C4#1062535211yescharacteristic
411C22×C4#1062535412yes—
412C22×C4#1062535412yes—
413C22×C4C2×C22⋊D4711yescharacteristic
414C22×C4C2×C22⋊D4711yescharacteristic
415C22×C4C23×D4711yescharacteristic
416M16—24no—
417M16—22no—
418M16—474no—
419M16—944no—
420M16—944no—
421M16—944no—
422M16—942no—
423M16—942no—
424M16—942no—
425M16—1882no—
426M16C2×C4×D4714yes—
427C2×C8—24no—
428C2×C8—22no—
429C2×C8—474no—
430C2×C8—944no—
431C2×C8—942no—
432C2×C8—942no—
433C2×C8—942no—
434C2×C8—942no—
435C2×C8—1882no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4C2×C22⋊D4712yes—
443C4⋊C4C2×C22⋊D4712yes—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C42—22no—
447C42—22no—
448C23—41no—
449C23—41no—
450C23—942no—
451C23—3761no—
452C23—3761no—
453C23—1881no—
454C23—1881no—
455C23#1309398212yes—
456C23#1309399611yescharacteristic
457C23#1309399611yescharacteristic
458C23#1309401912yes—
459C23C2×#1062535211yescharacteristic
460Q8—942no—
461Q8—942no—
462Q8—1881no—
463Q8—1881no—
464D4—942no—
465D4—942no—
466D4—1884no—
467D4—1884no—
468D4—1881no—
469D4—1881no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—42no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—942no—
479C2×C4—1881no—
480C2×C4—1881no—
481C2×C4—1884no—
482C2×C4—1884no—
483C2×C4#1309398212yes—
484C2×C4C4×C22⋊D4712yes—
485C2×C4C4×C22⋊D4712yes—
486C2×C4#1309398612yes—
487C2×C4#1309398612yes—
488C2×C4#1309399611yescharacteristic
489C2×C4#1309399611yescharacteristic
490C2×C4#1309400912yes—
491C2×C4#1309400912yes—
492C2×C4#1309401312yes—
493C2×C4#1309401312yes—
494C2×C4#1309401912yes—
495C2×C4C22×C4×D4711yescharacteristic
496C2×C4C2×D4×D4711yescharacteristic
497C2×C4C22×C22⋊D4711yescharacteristic
498C2×C4C2×#1062535011yescharacteristic
499C2×C4C2×#1062535411yescharacteristic
500C2×C4C2×#1062535411yescharacteristic
501C8—24no—
502C8—944no—
503C8—944no—
504C8—1882no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—42no—
512C22—42no—
513C22—3762no—
514C22—3762no—
515C22—3761no—
516C22—3761no—
517C22—1882no—
518C22#1568901312yes—
519C22#1568901312yes—
520C22C2×#1309398211yescharacteristic
521C22C2×#1309399611yescharacteristic
522C22C2×#1309401911yescharacteristic
523C4—42no—
524C4—42no—
525C4—82no—
526C4—1882no—
527C4#1568901312yes—
528C4#1568901312yes—
529C4C2×C4×C22⋊D4711yescharacteristic
530C4C2×#1309398611yescharacteristic
531C4C2×#1309400911yescharacteristic
532C4C2×#1309401311yescharacteristic
533C2—21no—
534C2—21no—
535C2—22no—
536C2—22no—
537C2—3762no—
538C2C2×#1568901311yescharacteristic
539C2#2025311311yescharacteristic
540C2#2025312211yescharacteristic
subgroupG/H
centerC22C2×#13093996
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#428C2
socleC22×C47C2×#121
prime radicalC22×#10625350C22
cosocleC22×C4×C47C24
hypercenterC2×#428D47
hypocenterC47#21076
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
4. #20251807C2semidirect
5. #20244049C2semidirect
6. #20244042C2semidirect
7. C2×#15688892C2semidirect
8. #20243114C2semidirect
9. C2×C47×#428C2semidirect
23. #15689105C4semidirect
71. #13093942C2×C4semidirect
72. #13093940C2×C4semidirect
263. C2×#428D47semidirect
308. C47#21076semidirect

Maximal subgroups

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

subgroupindex
#202519052
C2×#156891222
C2×#156891052
#202518072
#202440492
#202440422
C2×#156888922
#202431142
C2×C47×#4282
#2107647