Finite Group Databasebeta

← #26190125

Subgroups — #26190125

Order 12032.

1142 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1142 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×#15689749C211yesmaximal, characteristic, complemented
2#20254145C212yesmaximal, complemented
3#20242378C214yesmaximal, complemented
4#20242310C214yesmaximal, complemented
5#20241959C212yesmaximal, complemented
6C2×#15688847C211yesmaximal, characteristic, complemented
7C47×#2617C211yesmaximal, characteristic
8#15689749—24no—
9#15689749—22no—
10#15689749C2214yes—
11C2×#13094003—21no—
12C2×#13094003C2212yes—
13C2×#13094001—22no—
14C2×#13094001—22no—
15C22×C22⋊Dic47—21no—
16#15689453C2214yescomplemented
17#15689439—22no—
18#15689350—24no—
19#15689274C2214yescomplemented
20#15689261—22no—
21C2×C22⋊(C47⋊C8)C2212yes—
22#15688859C2218yescomplemented
23#15688847C2214yes—
24C2×#121×C47C2211yescharacteristic
25C2×#120×C47—21no—
26C22×C22⋊C4×C47C2211yescharacteristic
27C47×#397C2212yes—
28C47×#394—22no—
29C47×#392C2212yes—
30C47×#386—21no—
31C47×#380C2211yescharacteristic
32C22×Dic94—21no—
33C22×Dic94D412yescomplemented
34C23×Dic47—21no—
35C23×Dic47—21no—
36C2×#10625335—24no—
37C22×C47⋊C8—22no—
38#13094003—24no—
39#13094003—22no—
40#13094003—22no—
41#13094003—24no—
42#13094003D414yescomplemented
43#13094003C2314yes—
44#13094001—28no—
45#13094001—28no—
46#13094001—28no—
47#13094001—28no—
48C2×C4⋊Dic47—21no—
49C2×C4⋊Dic47D412yescomplemented
50C2×Dic47⋊C4—22no—
51C2×Dic47⋊C4—22no—
52C2×C22⋊Dic47—24no—
53C2×C22⋊Dic47—22no—
54C2×C22⋊Dic47—24no—
55C2×C22⋊Dic47—22no—
56#13093955—24no—
57#13093947—24no—
58#13093945—24no—
59C22⋊(C47⋊C8)—22no—
60C22⋊(C47⋊C8)C2314yes—
61C25×C47D411yescharacteristic
62C22×D4×C47—21no—
63C22×D4×C47—22no—
64C22×D4×C47C2311yescharacteristic
65C23×C4×C47D411yescharacteristic
66C23×C4×C47C2311yescharacteristic
67#121×C47—22no—
68#121×C47—22no—
69#120×C47—44no—
70C2×C4⋊C4×C47—21no—
71C2×C4⋊C4×C47C2311yescharacteristic
72C2×C22⋊C4×C47—22no—
73C2×C22⋊C4×C47—22no—
74C2×C22⋊C4×C47—21no—
75C2×C22⋊C4×C47—22no—
76C2×C22⋊C4×C47—21no—
77C2×C22⋊C4×C47—22no—
78C2×C22⋊C4×C47D412yes—
79C2×C22⋊C4×C47C2312yes—
80#95×C47—22no—
81#95×C47C2312yes—
82C24×C47—21no—
83C24×C47—21no—
84C24×C47—22no—
85C24×C47—21no—
86C24×C47—81no—
87C24×C47—81no—
88C24×C47SD1614yescomplemented
89C24×C47C2×D412yes—
90C24×C47C2×D411yescharacteristic
91C2×D4×C47—22no—
92C2×D4×C47—41no—
93C2×D4×C47—41no—
94C2×D4×C47—44no—
95C2×D4×C47—42no—
96C2×D4×C47—42no—
97C2×D4×C47—81no—
98C2×D4×C47—82no—
99C2×D4×C47—82no—
100C22×C4×C47—21no—
101C22×C4×C47—21no—
102C22×C4×C47—22no—
103C22×C4×C47—21no—
104C22×C4×C47—81no—
105C22×C4×C47—81no—
106C22×C4×C47C2×D412yes—
107C22×C4×C47C2×D411yescharacteristic
108C22×C4×C47C2×D412yes—
109C22×C4×C47C2×D412yes—
110C22×C4×C47C2411yescharacteristic
111C22×Dic47—22no—
112C22×Dic47—22no—
113C22×Dic47—24no—
114C22×Dic47—24no—
115C22×Dic47—82no—
116C22×Dic47—82no—
117C2×Dic94—24no—
118C2×Dic94—22no—
119C2×Dic94—82no—
120C2×Dic94—82no—
121C2×Dic94C2×D414yes—
122C4⋊C4×C47—22no—
123C4⋊C4×C47—81no—
124C22⋊C4×C47—24no—
125C22⋊C4×C47—24no—
126C22⋊C4×C47—44no—
127C22⋊C4×C47—42no—
128C22⋊C4×C47—42no—
129C22⋊C4×C47—82no—
130C22⋊Dic47—28no—
131C22⋊Dic47—24no—
132C22⋊Dic47—24no—
133#10625335—84no—
134C4⋊Dic47—24no—
135C4⋊Dic47—22no—
136C4⋊Dic47C2×D414yes—
137Dic47⋊C4—28no—
138Dic47⋊C4—24no—
139Dic47⋊C4—24no—
140C2×C47⋊C8—24no—
141C2×C47⋊C8—82no—
142C23×C47—22no—
143C23×C47—24no—
144C23×C47—22no—
145C23×C47—24no—
146C23×C47—41no—
147C23×C47—41no—
148C23×C47—42no—
149C23×C47—42no—
150C23×C47—42no—
151C23×C47—42no—
152C23×C47—42no—
153C23×C47—42no—
154C23×C47—41no—
155C23×C47—41no—
156C23×C47—41no—
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×C47#13712yes—
166C23×C47C22×D411yescharacteristic
167D4×C47—44no—
168D4×C47—88no—
169D4×C47—82no—
170D4×C47—82no—
171C2×C4×C47—22no—
172C2×C4×C47—42no—
173C2×C4×C47—41no—
174C2×C4×C47—42no—
175C2×C4×C47—41no—
176C2×C4×C47—42no—
177C2×C4×C47—42no—
178C2×C4×C47—42no—
179C2×C4×C47—42no—
180C2×C4×C47—41no—
181C2×C4×C47—42no—
182C2×C4×C47—41no—
183C2×C4×C47—41no—
184C2×C4×C47—81no—
185C2×C4×C47#12012yes—
186C2×C4×C47C22×D412yes—
187C2×Dic47—24no—
188C2×Dic47—24no—
189C2×Dic47—44no—
190C2×Dic47—44no—
191C2×Dic47—84no—
192C2×Dic47—84no—
193Dic94—44no—
194Dic94—82no—
195Dic94—84no—
196C47⋊C8—44no—
197#18286—471nomaximal, Sylow
198C22×C47—21no—
199C22×C47—21no—
200C22×C47—21no—
201C22×C47—21no—
202C22×C47—21no—
203C22×C47—21no—
204C22×C47—24no—
205C22×C47—42no—
206C22×C47—42no—
207C22×C47—44no—
208C22×C47—48no—
209C22×C47—42no—
210C22×C47—42no—
211C22×C47—82no—
212C22×C47—84no—
213C22×C47—82no—
214C22×C47—82no—
215C22×C47—81no—
216C22×C47—81no—
217C22×C47—81no—
218C22×C47#44812yes—
219C22×C47#44812yes—
220C22×C47C2×#12011yescharacteristic
221C22×C47C22×SD1611yescharacteristic
222C22×C47C2×#13711yescharacteristic
223C4×C47—22no—
224C4×C47—22no—
225C4×C47—42no—
226C4×C47—82no—
227C4×C47—81no—
228Dic47—44no—
229Dic47—44no—
230C2×#543—471no—
231#3187—472no—
232#2617D4711yescharacteristic
233#1608—474no—
234#1596—474no—
235#1520—472no—
236C2×#323—471no—
237C2×C47—21no—
238C2×C47—22no—
239C2×C47—21no—
240C2×C47—22no—
241C2×C47—44no—
242C2×C47—161no—
243C2×C47C2×#44811yescharacteristic
244C2×C47#301911yescharacteristic
245C2×C47#302611yescharacteristic
246#543—474no—
247#543—944no—
248#543—942no—
249C2×C22⋊Q8—472no—
250C2×C22⋊Q8—942no—
251C2×C22⋊Q8—942no—
252C2×C22⋊Q8—941no—
253C2×#121C2×D4711yescharacteristic
254C2×#120—21no—
255C22×C22⋊C4—941no—
256C22×C22⋊C4C2×D4711yescharacteristic
257#483—474no—
258#481—942no—
259#465—944no—
260#450—474no—
261#448—942no—
262C2×C22⋊C8—472no—
263#397C2×D4712yes—
264#394—22no—
265#392C2×D4712yes—
266#386—21no—
267#380C2×D4711yescharacteristic
268#328—478no—
269#323—474no—
270C47#1828611yescharacteristic, complemented, Sylow
271C25C22⋊D4711yescharacteristic
272C22×Q8—472no—
273C22×Q8—941no—
274C22×D4—21no—
275C22×D4—22no—
276C22×D4C22×D4711yescharacteristic
277C23×C4—941no—
278C23×C4—941no—
279C23×C4C22⋊D4711yescharacteristic
280C23×C4C22×D4711yescharacteristic
281C2×SD16—944no—
282C22×C8—942no—
283C22⋊Q8—474no—
284C22⋊Q8—474no—
285C22⋊Q8—948no—
286C22⋊Q8—948no—
287C22⋊Q8—948no—
288C22⋊Q8—948no—
289C22⋊Q8—944no—
290C22⋊Q8—942no—
291C22⋊Q8—942no—
292C22⋊Q8—944no—
293#121—22no—
294#121—22no—
295#120—44no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—472no—
298C2×C4⋊C4—942no—
299C2×C4⋊C4—941no—
300C2×C4⋊C4—942no—
301C2×C4⋊C4C22×D4711yescharacteristic
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—942no—
309C2×C22⋊C4—944no—
310C2×C22⋊C4—942no—
311C2×C22⋊C4—944no—
312C2×C22⋊C4C22⋊D4712yes—
313C2×C22⋊C4C22×D4712yes—
314#106—944no—
315#103—944no—
316#102—944no—
317C22⋊C8—474no—
318C22⋊C8—942no—
319#95—22no—
320#95C22×D4712yes—
321C24—21no—
322C24—21no—
323C24—22no—
324C24—21no—
325C24—81no—
326C24—81no—
327C24#1062533514yescomplemented
328C24C2×C22⋊D4712yes—
329C24C2×C22⋊D4711yescharacteristic
330C2×Q8—474no—
331C2×Q8—944no—
332C2×Q8—942no—
333C2×Q8—3762no—
334C2×Q8—3762no—
335C2×D4—22no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—44no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—81no—
342C2×D4—82no—
343C2×D4—82no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—81no—
349C22×C4—81no—
350C22×C4—944no—
351C22×C4—944no—
352C22×C4—942no—
353C22×C4—942no—
354C22×C4—3762no—
355C22×C4—3762no—
356C22×C4D4×D4712yes—
357C22×C4D4×D4712yes—
358C22×C4C2×C22⋊D4712yes—
359C22×C4C2×C22⋊D4711yescharacteristic
360C22×C4C23×D4711yescharacteristic
361SD16—3764no—
362C2×C8—944no—
363C2×C8—3762no—
364C4⋊C4—22no—
365C4⋊C4—81no—
366C4⋊C4—474no—
367C4⋊C4—948no—
368C4⋊C4—944no—
369C4⋊C4—944no—
370C4⋊C4—942no—
371C4⋊C4—944no—
372C22⋊C4—24no—
373C22⋊C4—24no—
374C22⋊C4—44no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—82no—
378C22⋊C4—948no—
379C22⋊C4—944no—
380C22⋊C4—944no—
381C23—24no—
382C23—22no—
383C23—24no—
384C23—22no—
385C23—41no—
386C23—41no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—81no—
397C23—81no—
398C23—84no—
399C23—84no—
400C23#1309399211yescharacteristic
401C23#1309399211yescharacteristic
402C23C2×#1062533514yes—
403C23C2×#1062533512yes—
404C23#1309405412yes—
405C23C22×C22⋊D4711yescharacteristic
406Q8—3762no—
407Q8—3764no—
408Q8—1884no—
409D4—44no—
410D4—88no—
411D4—82no—
412D4—82no—
413C2×C4—22no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—81no—
427C2×C4—944no—
428C2×C4—944no—
429C2×C4—3764no—
430C2×C4—3764no—
431C2×C4—1884no—
432C2×C4—1884no—
433C2×C4#1309399212yes—
434C2×C4C2×D4×D4712yes—
435C8—1884no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—24no—
443C22—42no—
444C22—42no—
445C22—44no—
446C22—48no—
447C22—42no—
448C22—42no—
449C22—82no—
450C22—84no—
451C22—82no—
452C22—82no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22#1568926112yes—
457C22#1568926112yes—
458C22C2×#1309399211yescharacteristic
459C22C22×#1062533511yescharacteristic
460C22C2×#1309405411yescharacteristic
461C4—22no—
462C4—22no—
463C4—42no—
464C4—82no—
465C4—81no—
466C4—1884no—
467C4—1884no—
468C2—21no—
469C2—22no—
470C2—21no—
471C2—22no—
472C2—44no—
473C2—161no—
474C2C2×#1568926111yescharacteristic
475C2#2025274411yescharacteristic
476C2#2025278411yescharacteristic
subgroupG/H
centerC22C2×#13093992
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
Fitting, prime radicalC47×#2617C2
socleC22×C47C2×#120
hypercenter#2617D47
hypocenterC47#18286
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2617

Complements

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

NH ≅ G/Ntype
1. C2×#15689749C2semidirect
2. #20254145C2semidirect
3. #20242378C2semidirect
4. #20242310C2semidirect
5. #20241959C2semidirect
6. C2×#15688847C2semidirect
16. #15689453C22semidirect
19. #15689274C22semidirect
22. #15688859C22semidirect
33. C22×Dic94D4semidirect
42. #13094003D4semidirect
49. C2×C4⋊Dic47D4semidirect
88. C24×C47SD16semidirect
270. C47#18286semidirect
327. C24#10625335semidirect

Maximal subgroups

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

subgroupindex
C2×#156897492
#202541452
#202423782
#202423102
#202419592
C2×#156888472
C47×#26172
#1828647