Finite Group Databasebeta

← #26244853

Subgroups — #26244853

Order 12032.

1002 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689172C211yesmaximal, characteristic, complemented
2C2×#15689166C211yesmaximal, characteristic, complemented
3#20251905C211yesmaximal, characteristic
4C2×#15689125C211yesmaximal, characteristic
5C2×#15689105C211yesmaximal, characteristic
6#20244754C214yesmaximal, complemented
7#20244747C214yesmaximal, complemented
8C2×#15688891C211yesmaximal, characteristic, complemented
9C47×#2905C211yesmaximal, characteristic, complemented
10C22×#10625350C2211yescharacteristic
11C22×#10625329C2211yescharacteristic
12C22×#10625329C2211yescharacteristic
13C22×C4⋊Dic47C2211yescharacteristic
14#15689172—24no—
15C2×#13093960C2214yes—
16#15689166—22no—
17#15689166C2214yes—
18C2×#13093957—21no—
19C2×#13093955—21no—
20#15689131C2212yes—
21#15689127C2212yes—
22#15689125—22no—
23#15689125—22no—
24#15689122—21no—
25#15689122—21no—
26#15689117—22no—
27C2×#13093949C2212yes—
28C2×#13093946C2212yes—
29#15689105—24no—
30C2×#13093942C2212yes—
31C2×#13093940C2212yes—
32#15688891C2214yes—
33C22×M16×C47C2211yescharacteristic
34C22×M16×C47C2211yescharacteristic
35C2×#117×C47C2211yescharacteristic
36C47×#424—22no—
37C47×#424C2214yes—
38C47×#423—22no—
39C2×#10625350—22no—
40C2×#10625350—21no—
41C2×#10625350—21no—
42C2×#10625350—21no—
43C2×#10625350—21no—
44C2×#10625350—22no—
45C2×#10625350—22no—
46C2×#10625350Q814yes—
47C22×Dic94C2311yescharacteristic
48C22×D188C2311yescharacteristic
49C22×C22⋊D47—21no—
50C23×Dic47—21no—
51C22×C4×D47—21no—
52C2×#10625329—21no—
53C2×#10625329—22no—
54C2×#10625329—21no—
55C2×#10625329—22no—
56C2×#10625329—22no—
57C2×#10625329D414yes—
58C2×#10625329C2312yes—
59C2×#10625329C2312yes—
60C22×C47⋊C8—21no—
61C22×C47⋊C8—21no—
62C2×C4⋊Dic47—21no—
63C2×C4⋊Dic47—21no—
64C2×C4⋊Dic47—21no—
65C2×C4⋊Dic47—21no—
66C2×C4⋊Dic47—21no—
67C2×C4⋊Dic47C2312yes—
68#13093960—28no—
69#13093957—24no—
70#13093955—24no—
71#13093949—24no—
72#13093947—22no—
73#13093947—22no—
74#13093946—24no—
75#13093944—22no—
76#13093944—22no—
77#13093942—24no—
78#13093940—24no—
79C23×C4×C47C2311yescharacteristic
80C2×M16×C47—22no—
81C2×M16×C47—22no—
82C2×M16×C47—21no—
83C2×M16×C47—22no—
84C2×M16×C47—22no—
85C2×M16×C47D412yes—
86C2×M16×C47D412yes—
87C2×M16×C47C2312yes—
88C22×C8×C47—21no—
89C22×C8×C47C2312yes—
90#117×C47—21no—
91#117×C47—21no—
92#117×C47D412yes—
93#117×C47C2312yes—
94C2×C4⋊C4×C47D411yescharacteristic
95C2×C4⋊C4×C47D411yescharacteristic
96C2×C22⋊C4×C47—21no—
97C2×C42×C47—21no—
98C4⋊C8×C47—28no—
99C24×C47#4111yescharacteristic
100C23×D47—21no—
101C22×C4×C47—21no—
102C22×C4×C47—21no—
103C22×C4×C47C2×D412yes—
104C22×C4×C47C2×D411yescharacteristic
105C22×C4×C47C2×D411yescharacteristic
106C22×C4×C47C2×D411yescharacteristic
107C22×C4×C47C2×D411yescharacteristic
108C22×C4×C47C2×Q812yes—
109C22×C4×C47#4111yescharacteristic
110C22×C4×C47#4112yes—
111C22×C4×C47#4112yes—
112C22×C4×C47C2411yescharacteristic
113C2×C22⋊D47—24no—
114C2×C22⋊D47—41no—
115C2×C22⋊D47—41no—
116C2×C22⋊D47—42no—
117C22×Dic47—21no—
118C22×Dic47—22no—
119C22×Dic47—41no—
120C22×Dic47—41no—
121C22×Dic47—82no—
122#10625350—24no—
123#10625350—24no—
124#10625350—24no—
125#10625350—24no—
126#10625350—42no—
127#10625350—42no—
128#10625350—42no—
129#10625350—42no—
130C2×D188—21no—
131C2×D188—21no—
132C2×D188—21no—
133C2×D188—21no—
134C2×D188—21no—
135C2×D188C2×Q812yes—
136C2×C4×D47—24no—
137C2×C4×D47—41no—
138C2×C4×D47—41no—
139C2×C4×D47—42no—
140C2×Dic94—21no—
141C2×Dic94—21no—
142C2×Dic94—21no—
143C2×Dic94—21no—
144C2×Dic94—21no—
145C2×Dic94C2×Q812yes—
146M16×C47—24no—
147M16×C47—24no—
148M16×C47—22no—
149M16×C47—22no—
150M16×C47—42no—
151C2×C8×C47—22no—
152C2×C8×C47—22no—
153C2×C8×C47—22no—
154C2×C8×C47—42no—
155C2×C8×C47C2×D414yes—
156C4⋊C4×C47—21no—
157C4⋊C4×C47—21no—
158C4⋊C4×C47C2×D412yes—
159C4⋊C4×C47C2×D412yes—
160C22⋊C4×C47—22no—
161C22⋊C4×C47—22no—
162C42×C47—22no—
163C42×C47—22no—
164C4⋊Dic47—22no—
165C4⋊Dic47—22no—
166C4⋊Dic47—41no—
167C4⋊Dic47—41no—
168#10625329—24no—
169#10625329—22no—
170#10625329—22no—
171#10625329—22no—
172#10625329—42no—
173#10625329C2×D414yes—
174C2×C47⋊C8—24no—
175C2×C47⋊C8—22no—
176C2×C47⋊C8—22no—
177C2×C47⋊C8—42no—
178C23×C47—41no—
179C23×C47—41no—
180C23×C47#12111yescharacteristic
181C23×C47#12111yescharacteristic
182C23×C47C22⋊Q812yes—
183C23×C47#12612yes—
184C23×C47C2×#4111yescharacteristic
185C22×D47—22no—
186C22×D47—41no—
187C22×D47—41no—
188C22×D47—81no—
189C22×D47—81no—
190C2×C4×C47—22no—
191C2×C4×C47—22no—
192C2×C4×C47—81no—
193C2×C4×C47—81no—
194C2×C4×C47#12112yes—
195C2×C4×C47#12112yes—
196C2×C4×C47#12111yescharacteristic
197C2×C4×C47#12111yescharacteristic
198C2×C4×C47#12112yes—
199C2×C4×C47#12112yes—
200C2×C4×C47C22⋊Q812yes—
201C2×C4×C47C22⋊Q812yes—
202C2×C4×C47C22⋊Q812yes—
203C2×C4×C47C22⋊Q812yes—
204C2×C4×C47C22⋊Q812yes—
205C2×C4×C47#12612yes—
206C2×C4×C47C22×D411yescharacteristic
207C2×C4×C47C22×D411yescharacteristic
208C2×C4×C47C22×D411yescharacteristic
209C2×C4×C47C22×Q811yescharacteristic
210C2×C4×C47C2×#4111yescharacteristic
211C2×C4×C47C2×#4111yescharacteristic
212C22⋊D47—44no—
213C22⋊D47—44no—
214C2×Dic47—22no—
215C2×Dic47—41no—
216C2×Dic47—41no—
217C2×Dic47—42no—
218C2×Dic47—82no—
219C2×Dic47—82no—
220C2×Dic47—82no—
221D188—22no—
222D188—22no—
223D188—41no—
224D188—41no—
225C4×D47—44no—
226C4×D47—44no—
227Dic94—22no—
228Dic94—22no—
229Dic94—41no—
230Dic94—41no—
231C8×C47—24no—
232C8×C47—42no—
233C47⋊C8—24no—
234C47⋊C8—42no—
235#23604—471nomaximal, Sylow
236C22×C47—21no—
237C22×C47—21no—
238C22×C47—21no—
239C22×C47—21no—
240C22×C47—21no—
241C22×C47—21no—
242C22×C47—42no—
243C22×C47—42no—
244C22×C47#39612yes—
245C22×C47#39612yes—
246C22×C47C2×#12111yescharacteristic
247C22×C47C2×C22⋊Q811yescharacteristic
248C22×C47C2×#12611yescharacteristic
249C2×D47—42no—
250C2×D47—82no—
251C2×D47—81no—
252C2×D47—81no—
253C2×D47—82no—
254C4×C47—42no—
255C4×C47—42no—
256C4×C47#39612yes—
257C4×C47#39612yes—
258C4×C47C2×#12111yescharacteristic
259C4×C47C2×#12111yescharacteristic
260C4×C47C2×C22⋊Q811yescharacteristic
261C4×C47C2×C22⋊Q811yescharacteristic
262Dic47—42no—
263Dic47—82no—
264C2×#430—471no—
265C2×#428—471no—
266#2905D4711yescharacteristic, complemented
267#2896—471no—
268C2×#417—471no—
269C2×#413—471no—
270#2064—474no—
271#2063—474no—
272C2×#343—471no—
273C2×C47—21no—
274C2×C47—21no—
275C2×C47—22no—
276C2×C47—22no—
277C2×C47C2×#39611yescharacteristic
278C2×C47#306811yescharacteristic
279C2×C47#306911yescharacteristic
280D47—82no—
281C22×#41—471no—
282C22×M16—471no—
283C22×M16—471no—
284C22×M16C2×D4711yescharacteristic
285C22×M16C2×D4711yescharacteristic
286C2×#117C2×D4711yescharacteristic
287C22×C4⋊C4—471no—
288#430—944no—
289C2×#108—474no—
290#428—474no—
291#428—942no—
292C2×#107—941no—
293C2×#106—941no—
294#424—22no—
295#424C2×D4714yes—
296#423—22no—
297#419—472no—
298#418—472no—
299#417—942no—
300#417—942no—
301#417—941no—
302#417—941no—
303#416—942no—
304C2×#103—472no—
305C2×#102—472no—
306#413—944no—
307C2×#101—472no—
308C2×#100—472no—
309#343—474no—
310C47#2360411yescharacteristic, complemented, Sylow
311C2×#41—474no—
312C2×#41—942no—
313C2×#41—941no—
314C2×#41—941no—
315C2×#41—942no—
316C2×#41—942no—
317C2×#41—941no—
318C2×#41—941no—
319C22×Q8—471no—
320C22×D4—471no—
321C22×D4—941no—
322C23×C4—941no—
323C23×C4—941no—
324C23×C4C22×D4711yescharacteristic
325C2×M16—22no—
326C2×M16—22no—
327C2×M16—21no—
328C2×M16—22no—
329C2×M16—22no—
330C2×M16—474no—
331C2×M16—472no—
332C2×M16—472no—
333C2×M16—941no—
334C2×M16—942no—
335C2×M16—941no—
336C2×M16—942no—
337C2×M16—942no—
338C2×M16C22⋊D4712yes—
339C2×M16C22⋊D4712yes—
340C2×M16C22×D4712yes—
341C22×C8—21no—
342C22×C8—941no—
343C22×C8—941no—
344C22×C8C22×D4712yes—
345#117—21no—
346#117—21no—
347#117D18812yes—
348#117C22×D4712yes—
349C2×C4⋊C4—472no—
350C2×C4⋊C4—941no—
351C2×C4⋊C4—941no—
352C2×C4⋊C4—941no—
353C2×C4⋊C4—941no—
354C2×C4⋊C4—941no—
355C2×C4⋊C4D18811yescharacteristic
356C2×C4⋊C4D18811yescharacteristic
357C2×C22⋊C4—21no—
358C2×C42—21no—
359#108—948no—
360#107—944no—
361#106—944no—
362C4⋊C8—28no—
363#103—944no—
364#103—942no—
365#103—942no—
366#102—944no—
367#102—942no—
368#102—942no—
369#101—944no—
370#100—944no—
371C24—941no—
372C24#1062535011yescharacteristic
373#41—944no—
374#41—944no—
375#41—944no—
376#41—944no—
377#41—1882no—
378#41—1882no—
379#41—1882no—
380#41—1882no—
381C2×Q8—472no—
382C2×Q8—941no—
383C2×Q8—941no—
384C2×Q8—941no—
385C2×Q8—941no—
386C2×Q8—941no—
387C2×D4—472no—
388C2×D4—944no—
389C2×D4—941no—
390C2×D4—941no—
391C2×D4—941no—
392C2×D4—941no—
393C2×D4—941no—
394C2×D4—1881no—
395C2×D4—1881no—
396C2×D4—1882no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—941no—
400C22×C4—944no—
401C22×C4—942no—
402C22×C4—3762no—
403C22×C4—1881no—
404C22×C4—1881no—
405C22×C4—1881no—
406C22×C4—1881no—
407C22×C4—1882no—
408C22×C4C2×D18811yescharacteristic
409C22×C4C2×D18811yescharacteristic
410C22×C4#1062535011yescharacteristic
411C22×C4D4×D4712yes—
412C22×C4#1062535212yes—
413C22×C4Q8×D4712yes—
414C22×C4#1062535412yes—
415C22×C4C2×C22⋊D4711yescharacteristic
416C22×C4C2×C22⋊D4711yescharacteristic
417C22×C4C23×D4711yescharacteristic
418M16—24no—
419M16—24no—
420M16—22no—
421M16—22no—
422M16—42no—
423M16—474no—
424M16—944no—
425M16—942no—
426M16—942no—
427M16—942no—
428M16—1882no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—42no—
433C2×C8—944no—
434C2×C8—942no—
435C2×C8—942no—
436C2×C8—1882no—
437C2×C8C2×C22⋊D4714yes—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—942no—
441C4⋊C4—942no—
442C4⋊C4—1881no—
443C4⋊C4—1881no—
444C4⋊C4C2×D18812yes—
445C4⋊C4C2×D18812yes—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C42—22no—
449C42—22no—
450C23—41no—
451C23—41no—
452C23—942no—
453C23—3761no—
454C23—3761no—
455C23—1881no—
456C23—1881no—
457C23#1309399811yescharacteristic
458C23#1309399811yescharacteristic
459C23#1309400212yes—
460C23#1309401912yes—
461C23C2×#1062535011yescharacteristic
462Q8—942no—
463Q8—942no—
464Q8—1881no—
465Q8—1881no—
466D4—942no—
467D4—942no—
468D4—1884no—
469D4—1884no—
470D4—1881no—
471D4—1881no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—942no—
477C2×C4—3762no—
478C2×C4—3762no—
479C2×C4—3762no—
480C2×C4—1882no—
481C2×C4—1881no—
482C2×C4—1881no—
483C2×C4—1884no—
484C2×C4—1884no—
485C2×C4#1309399612yes—
486C2×C4#1309399612yes—
487C2×C4#1309399811yescharacteristic
488C2×C4#1309399811yescharacteristic
489C2×C4#1309399912yes—
490C2×C4#1309399912yes—
491C2×C4#1309400012yes—
492C2×C4#1309400012yes—
493C2×C4#1309400212yes—
494C2×C4#1309400412yes—
495C2×C4#1309400412yes—
496C2×C4#1309401912yes—
497C2×C4C2×D4×D4711yescharacteristic
498C2×C4C22×C22⋊D4711yescharacteristic
499C2×C4C22×D18811yescharacteristic
500C2×C4C2×Q8×D4711yescharacteristic
501C2×C4C2×#1062535211yescharacteristic
502C2×C4C2×#1062535411yescharacteristic
503C8—24no—
504C8—42no—
505C8—944no—
506C8—1882no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—42no—
514C22—42no—
515C22—3762no—
516C22—3762no—
517C22—3761no—
518C22—3761no—
519C22—1882no—
520C22#1568904512yes—
521C22#1568904512yes—
522C22C2×#1309399811yescharacteristic
523C22C2×#1309400211yescharacteristic
524C22C2×#1309401911yescharacteristic
525C4—42no—
526C4—42no—
527C4—3762no—
528C4—1882no—
529C4#1568904512yes—
530C4#1568904512yes—
531C4C2×#1309399611yescharacteristic
532C4C2×#1309399911yescharacteristic
533C4C2×#1309400011yescharacteristic
534C4C2×#1309400411yescharacteristic
535C2—21no—
536C2—21no—
537C2—22no—
538C2—22no—
539C2—3762no—
540C2C2×#1568904511yescharacteristic
541C2#2025311611yescharacteristic
542C2#2025312511yescharacteristic
subgroupG/H
centerC22C2×#13093998
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2905C2
socleC22×C47C2×#121
prime radicalC22×#10625350C22
hypercenter#2905D47
hypocenterC47#23604
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2905

Complements

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

NH ≅ G/Ntype
1. C2×#15689172C2semidirect
2. C2×#15689166C2semidirect
6. #20244754C2semidirect
7. #20244747C2semidirect
8. C2×#15688891C2semidirect
9. C47×#2905C2semidirect
266. #2905D47semidirect
310. C47#23604semidirect

Maximal subgroups

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

subgroupindex
C2×#156891722
C2×#156891662
#202519052
C2×#156891252
C2×#156891052
#202447542
#202447472
C2×#156888912
C47×#29052
#2360447