Finite Group Databasebeta

← #26190118

Subgroups — #26190118

Order 12032.

1408 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1408 conjugacy classes of subgroups — too many to draw. Only its 187 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689685C211yesmaximal, characteristic
2#20254141C212yesmaximal, complemented
3#20242407C214yesmaximal, complemented
4#20242339C214yesmaximal, complemented
5#20241955C212yesmaximal, complemented
6C2×#15688847C211yesmaximal, characteristic, complemented
7C47×#2626C211yesmaximal, characteristic, complemented
8C23×C22⋊D47—21no—
9#15689685—24no—
10#15689685—22no—
11#15689685C2214yes—
12C2×#13094008—22no—
13C2×#13093998—21no—
14C2×#13093998C2212yes—
15C2×#13093994—22no—
16#15689451—22no—
17#15689441C2214yescomplemented
18#15689354—24no—
19#15689273—22no—
20#15689262C2214yescomplemented
21C2×C22⋊(C47⋊C8)C2212yescomplemented
22#15688856C2218yescomplemented
23#15688847C2214yescomplemented
24C2×C22⋊Q8×C47C2211yescharacteristic, complemented
25C22×C22⋊C4×C47C2211yescharacteristic
26C47×#399C2212yescomplemented
27C47×#396—22no—
28C47×#393C2212yescomplemented
29C47×#385—21no—
30C47×#380C2211yescharacteristic, complemented
31C47×#379—21no—
32C24×D47—21no—
33C22×D188—21no—
34C22×D188D412yes—
35C22×C22⋊D47—22no—
36C22×C22⋊D47—24no—
37C22×C22⋊D47—22no—
38C22×C22⋊D47—24no—
39C22×C22⋊D47—44no—
40C22×C22⋊D47—44no—
41C23×Dic47—21no—
42C2×#10625336—24no—
43C22×C47⋊C8—22no—
44#13094008—28no—
45#13094008—28no—
46#13093998—24no—
47#13093998—24no—
48#13093998—22no—
49#13093998—22no—
50#13093998D414yes—
51#13093998C2314yes—
52#13093994—28no—
53#13093994—28no—
54C2×C4⋊Dic47—21no—
55C2×C4⋊Dic47D412yes—
56C2×(C2×D47)⋊C4—22no—
57C2×(C2×D47)⋊C4—22no—
58#13093955—24no—
59#13093948—24no—
60#13093944—24no—
61C22⋊(C47⋊C8)—22no—
62C22⋊(C47⋊C8)C2314yescomplemented
63C25×C47D411yescharacteristic
64C22×Q8×C47C2311yescharacteristic, complemented
65C23×C4×C47D411yescharacteristic
66C23×C4×C47C2311yescharacteristic
67C22⋊Q8×C47—22no—
68C22⋊Q8×C47—22no—
69C2×C4⋊C4×C47—21no—
70C2×C4⋊C4×C47—21no—
71C2×C4⋊C4×C47—22no—
72C2×C4⋊C4×C47C2311yescharacteristic, complemented
73C2×C22⋊C4×C47—22no—
74C2×C22⋊C4×C47—22no—
75C2×C22⋊C4×C47—21no—
76C2×C22⋊C4×C47—21no—
77C2×C22⋊C4×C47—22no—
78C2×C22⋊C4×C47—22no—
79C2×C22⋊C4×C47D412yes—
80C2×C22⋊C4×C47C2312yes—
81#95×C47—22no—
82#95×C47C2312yescomplemented
83C24×C47—21no—
84C24×C47—21no—
85C24×C47—22no—
86C24×C47—81no—
87C24×C47—81no—
88C24×C47SD1614yescomplemented
89C24×C47C2×D412yes—
90C24×C47C2×D411yescharacteristic
91C23×D47—22no—
92C23×D47—24no—
93C23×D47—44no—
94C23×D47—82no—
95C23×D47—82no—
96C2×Q8×C47—22no—
97C2×Q8×C47—81no—
98C22×C4×C47—21no—
99C22×C4×C47—21no—
100C22×C4×C47—21no—
101C22×C4×C47—22no—
102C22×C4×C47—21no—
103C22×C4×C47—81no—
104C22×C4×C47—81no—
105C22×C4×C47C2×D412yes—
106C22×C4×C47C2×D411yescharacteristic
107C22×C4×C47C2×D412yes—
108C22×C4×C47C2×D412yes—
109C22×C4×C47C2411yescharacteristic
110C2×C22⋊D47—28no—
111C2×C22⋊D47—24no—
112C2×C22⋊D47—24no—
113C2×C22⋊D47—48no—
114C2×C22⋊D47—416no—
115C2×C22⋊D47—48no—
116C2×C22⋊D47—48no—
117C2×C22⋊D47—44no—
118C2×C22⋊D47—44no—
119C22×Dic47—24no—
120C22×Dic47—22no—
121C22×Dic47—44no—
122C2×D188—24no—
123C2×D188—22no—
124C2×D188—82no—
125C2×D188—82no—
126C2×D188C2×D414yes—
127C4⋊C4×C47—22no—
128C4⋊C4×C47—42no—
129C4⋊C4×C47—81no—
130C4⋊C4×C47—82no—
131C22⋊C4×C47—24no—
132C22⋊C4×C47—24no—
133C22⋊C4×C47—41no—
134C22⋊C4×C47—41no—
135C22⋊C4×C47—42no—
136C22⋊C4×C47—82no—
137C22⋊C4×C47—82no—
138#10625336—84no—
139(C2×D47)⋊C4—28no—
140(C2×D47)⋊C4—24no—
141(C2×D47)⋊C4—24no—
142C4⋊Dic47—24no—
143C4⋊Dic47—22no—
144C4⋊Dic47C2×D414yes—
145C2×C47⋊C8—24no—
146C2×C47⋊C8—82no—
147C23×C47—22no—
148C23×C47—24no—
149C23×C47—22no—
150C23×C47—24no—
151C23×C47—41no—
152C23×C47—41no—
153C23×C47—42no—
154C23×C47—42no—
155C23×C47—42no—
156C23×C47—42no—
157C23×C47—81no—
158C23×C47—81no—
159C23×C47—84no—
160C23×C47—84no—
161C23×C47#12011yescharacteristic
162C23×C47#12011yescharacteristic
163C23×C47C2×SD1614yes—
164C23×C47C2×SD1612yes—
165C23×C47AGL(1,ℤ/8)12yes—
166C23×C47C22×D411yescharacteristic
167C22×D47—24no—
168C22×D47—44no—
169C22×D47—44no—
170C22×D47—48no—
171C22×D47—44no—
172C22×D47—84no—
173C22×D47—84no—
174C22×D47—816no—
175Q8×C47—44no—
176C2×C4×C47—22no—
177C2×C4×C47—42no—
178C2×C4×C47—41no—
179C2×C4×C47—42no—
180C2×C4×C47—41no—
181C2×C4×C47—42no—
182C2×C4×C47—42no—
183C2×C4×C47—42no—
184C2×C4×C47—42no—
185C2×C4×C47—42no—
186C2×C4×C47—42no—
187C2×C4×C47—41no—
188C2×C4×C47—41no—
189C2×C4×C47—42no—
190C2×C4×C47—81no—
191C2×C4×C47#12012yes—
192C2×C4×C47C22×D412yes—
193C22⋊D47—416no—
194C22⋊D47—44no—
195C22⋊D47—44no—
196C22⋊D47—48no—
197C2×Dic47—24no—
198C2×Dic47—48no—
199C2×Dic47—44no—
200D188—44no—
201D188—82no—
202D188—84no—
203C47⋊C8—44no—
204#18285—471nomaximal, Sylow
205C22×C47—21no—
206C22×C47—21no—
207C22×C47—21no—
208C22×C47—21no—
209C22×C47—21no—
210C22×C47—21no—
211C22×C47—24no—
212C22×C47—42no—
213C22×C47—42no—
214C22×C47—44no—
215C22×C47—48no—
216C22×C47—42no—
217C22×C47—42no—
218C22×C47—82no—
219C22×C47—84no—
220C22×C47#45012yes—
221C22×C47#45012yes—
222C22×C47C2×#12011yescharacteristic
223C22×C47C22×SD1611yescharacteristic
224C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
225C2×D47—44no—
226C2×D47—44no—
227C2×D47—44no—
228C2×D47—88no—
229C2×D47—816no—
230C4×C47—22no—
231C4×C47—22no—
232C4×C47—42no—
233C4×C47—82no—
234C4×C47—81no—
235C4×C47—81no—
236Dic47—44no—
237C2×#534—471no—
238#3187—472no—
239#2626D4711yescharacteristic, complemented
240#1613—474no—
241#1601—474no—
242#1519—472no—
243C2×#323—471no—
244C2×C47—21no—
245C2×C47—22no—
246C2×C47—21no—
247C2×C47—22no—
248C2×C47—44no—
249C2×C47C2×#45011yescharacteristic
250C2×C47#301911yescharacteristic
251C2×C47#302611yescharacteristic
252D47—84no—
253C23×D4—941no—
254#534—474no—
255#534—944no—
256#534—942no—
257C2×#123—942no—
258C2×C22⋊Q8C2×D4711yescharacteristic, complemented
259C2×#121—472no—
260C2×#121—941no—
261C2×#120—942no—
262C22×C22⋊C4C2×D4711yescharacteristic
263#483—942no—
264#481—474no—
265#465—944no—
266#450—942no—
267#448—474no—
268C2×C22⋊C8—472no—
269#399C2×D4712yescomplemented
270#396—22no—
271#393C2×D4712yescomplemented
272#385—21no—
273#380C2×D4711yescharacteristic, complemented
274#379—21no—
275#327—478no—
276#323—474no—
277C47#1828511yescharacteristic, complemented, Sylow
278C25—941no—
279C25C22⋊D4711yescharacteristic
280C22×Q8C22×D4711yescharacteristic, complemented
281C22×D4—472no—
282C22×D4—941no—
283C22×D4—942no—
284C22×D4—944no—
285C22×D4—942no—
286C22×D4—944no—
287C22×D4—1884no—
288C22×D4—1884no—
289C23×C4—941no—
290C23×C4C22⋊D4711yescharacteristic
291C23×C4C22×D4711yescharacteristic
292C2×SD16—944no—
293C22×C8—942no—
294#123—948no—
295#123—948no—
296C22⋊Q8—22no—
297C22⋊Q8—22no—
298#121—474no—
299#121—474no—
300#121—944no—
301#121—942no—
302#121—942no—
303#121—944no—
304#120—948no—
305#120—948no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—472no—
310C2×C4⋊C4—941no—
311C2×C4⋊C4C22×D4711yescharacteristic, complemented
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—942no—
319C2×C22⋊C4—942no—
320C2×C22⋊C4C22⋊D4712yes—
321C2×C22⋊C4C22×D4712yes—
322#106—944no—
323#103—944no—
324#102—944no—
325C22⋊C8—474no—
326C22⋊C8—942no—
327#95—22no—
328#95C22×D4712yescomplemented
329C24—21no—
330C24—21no—
331C24—22no—
332C24—81no—
333C24—81no—
334C24—944no—
335C24—942no—
336C24—3762no—
337C24—3762no—
338C24—1884no—
339C24#1062533614yescomplemented
340C24C2×C22⋊D4712yes—
341C24C2×C22⋊D4711yescharacteristic
342C2×Q8—22no—
343C2×Q8—81no—
344C2×D4—474no—
345C2×D4—944no—
346C2×D4—948no—
347C2×D4—944no—
348C2×D4—944no—
349C2×D4—942no—
350C2×D4—3762no—
351C2×D4—3762no—
352C2×D4—18816no—
353C2×D4—1888no—
354C2×D4—1888no—
355C2×D4—1888no—
356C2×D4—1884no—
357C2×D4—1884no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—81no—
364C22×C4—81no—
365C22×C4—944no—
366C22×C4—942no—
367C22×C4—1884no—
368C22×C4D4×D4712yes—
369C22×C4D4×D4712yes—
370C22×C4C2×C22⋊D4712yes—
371C22×C4C2×C22⋊D4711yescharacteristic
372C22×C4C23×D4711yescharacteristic
373SD16—3764no—
374C2×C8—944no—
375C2×C8—3762no—
376C4⋊C4—22no—
377C4⋊C4—42no—
378C4⋊C4—81no—
379C4⋊C4—82no—
380C4⋊C4—474no—
381C4⋊C4—944no—
382C4⋊C4—942no—
383C22⋊C4—24no—
384C22⋊C4—24no—
385C22⋊C4—41no—
386C22⋊C4—41no—
387C22⋊C4—42no—
388C22⋊C4—82no—
389C22⋊C4—82no—
390C22⋊C4—948no—
391C22⋊C4—944no—
392C22⋊C4—944no—
393C23—24no—
394C23—22no—
395C23—24no—
396C23—22no—
397C23—41no—
398C23—41no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—81no—
404C23—81no—
405C23—84no—
406C23—84no—
407C23—944no—
408C23—37616no—
409C23—3764no—
410C23—3764no—
411C23—1884no—
412C23—1884no—
413C23—1888no—
414C23—1884no—
415C23#1309399211yescharacteristic
416C23#1309399211yescharacteristic
417C23C2×#1062533614yes—
418C23C2×#1062533612yes—
419C23#1309404912yes—
420C23C22×C22⋊D4711yescharacteristic
421Q8—44no—
422D4—3762no—
423D4—3764no—
424D4—1884no—
425D4—18816no—
426D4—1888no—
427D4—1884no—
428D4—1884no—
429C2×C4—22no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—81no—
444C2×C4—944no—
445C2×C4—1888no—
446C2×C4—1884no—
447C2×C4#1309399212yes—
448C2×C4C2×D4×D4712yes—
449C8—1884no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—24no—
457C22—42no—
458C22—42no—
459C22—44no—
460C22—48no—
461C22—42no—
462C22—42no—
463C22—82no—
464C22—84no—
465C22—3768no—
466C22—37616no—
467C22—1884no—
468C22—1884no—
469C22—1884no—
470C22#1568927312yes—
471C22#1568927312yes—
472C22C2×#1309399211yescharacteristic
473C22C22×#1062533611yescharacteristic
474C22C2×#1309404911yescharacteristic
475C4—22no—
476C4—22no—
477C4—42no—
478C4—82no—
479C4—81no—
480C4—81no—
481C4—1884no—
482C2—21no—
483C2—22no—
484C2—21no—
485C2—22no—
486C2—44no—
487C2—3764no—
488C2C2×#1568927311yescharacteristic
489C2#2025274411yescharacteristic
490C2#2025278411yescharacteristic
subgroupG/H
centerC22C2×#13093992
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2626C2
socleC22×C47C2×#120
prime radicalC2×#15689685C2
hypercenter#2626D47
hypocenterC47#18285
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2626

Complements

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

NH ≅ G/Ntype
2. #20254141C2semidirect
3. #20242407C2semidirect
4. #20242339C2semidirect
5. #20241955C2semidirect
6. C2×#15688847C2semidirect
7. C47×#2626C2semidirect
17. #15689441C22semidirect
20. #15689262C22semidirect
21. C2×C22⋊(C47⋊C8)C22semidirect
22. #15688856C22semidirect
23. #15688847C22semidirect
24. C2×C22⋊Q8×C47C22semidirect
26. C47×#399C22semidirect
28. C47×#393C22semidirect
30. C47×#380C22semidirect
62. C22⋊(C47⋊C8)C23semidirect
64. C22×Q8×C47C23semidirect
72. C2×C4⋊C4×C47C23semidirect
82. #95×C47C23semidirect
88. C24×C47SD16semidirect
239. #2626D47semidirect
258. C2×C22⋊Q8C2×D47semidirect
269. #399C2×D47semidirect
271. #393C2×D47semidirect
273. #380C2×D47semidirect
277. C47#18285semidirect
280. C22×Q8C22×D47semidirect
311. C2×C4⋊C4C22×D47semidirect
328. #95C22×D47semidirect
339. C24#10625336semidirect

Maximal subgroups

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

subgroupindex
C2×#156896852
#202541412
#202424072
#202423392
#202419552
C2×#156888472
C47×#26262
#1828547