Finite Group Databasebeta

← #27084175

Subgroups — #27084175

Order 12032.

2298 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689687C211yesmaximal, characteristic
2#20256591C211yesmaximal, characteristic, complemented
3#20253347C2116yesmaximal, complemented
4#20253329C218yesmaximal, complemented
5#20251978C212yesmaximal, complemented
6C2×#15689129C212yesmaximal, complemented
7C47×#2879C211yesmaximal, characteristic, complemented
8C23×D188C2211yescharacteristic
9#15689687—24no—
10#15689687C2218yes—
11C2×#13094020—21no—
12C2×#13094013—21no—
13C2×#13093998C2214yes—
14C2×#13093992—22no—
15#15689606C2218yescomplemented
16#15689601—21no—
17#15689598—22no—
18#15689598—21no—
19#41×Dic47—24no—
20C2×#13093983C2211yescharacteristic
21C2×#13093983C2212yes—
22C2×#13093983C2214yescomplemented
23#15689448C22132yescomplemented
24#15689442C22116yescomplemented
25#15689437C22116yescomplemented
26#15689165C2218yescomplemented
27#15689165C2218yescomplemented
28#15689161C2218yescomplemented
29#15689129C2218yes—
30#15689129C2218yescomplemented
31C2×#13093946C2218yescomplemented
32C22×#41×C47C2211yescharacteristic
33C22×M16×C47C2212yes—
34C47×#408C2218yescomplemented
35C47×#407C2214yescomplemented
36C24×D47—21no—
37C22×D188—28no—
38C22×D188—21no—
39C22×D188—22no—
40C22×D188C2312yes—
41C22×D188C2314yes—
42C22×C22⋊D47—24no—
43#13094020—28no—
44#13094013—28no—
45#13093998—28no—
46#13093998C23116yes—
47#13093992—216no—
48Q8×Dic47—28no—
49D4×Dic47—28no—
50D4×Dic47—216no—
51#13093983—22no—
52#13093983—22no—
53#13093983—22no—
54#13093983—24no—
55#13093983—24no—
56#13093983C2314yes—
57#13093983C2314yes—
58#13093983C2314yes—
59#13093983C2318yescomplemented
60C2×C4⋊Dic47C2314yes—
61C2×C4⋊Dic47C2312yes—
62C2×C4⋊Dic47C2318yescomplemented
63C2×C22⋊Dic47—22no—
64C2×C22⋊Dic47—21no—
65C2×C22⋊Dic47—24no—
66C2×(C2×D47)⋊C4—24no—
67C2×C4×Dic47—22no—
68C2×C4×Dic47—21no—
69C2×C4×Dic47—24no—
70#13093958C23116yescomplemented
71#13093956C23116yescomplemented
72#13093946C23132yes—
73C2×#41×C47—28no—
74C2×#41×C47D414yes—
75C2×#41×C47C2314yes—
76C22×Q8×C47D411yescharacteristic
77C22×D4×C47D412yes—
78C22×D4×C47D411yescharacteristic
79C23×C4×C47C2312yes—
80C23×C4×C47C2311yescharacteristic
81C2×M16×C47—28no—
82C2×M16×C47C2314yes—
83C2×M16×C47C2314yes—
84C22×C8×C47C2314yes—
85C22⋊C8×C47C23116yescomplemented
86C24×C47C2×D412yes—
87C24×C47C2×D411yescharacteristic
88C23×D47—22no—
89C23×D47—24no—
90C23×D47—44no—
91C23×D47—84no—
92#41×C47—216no—
93#41×C47—28no—
94#41×C47—28no—
95C2×Q8×C47—24no—
96C2×Q8×C47C2×D414yes—
97C2×D4×C47—28no—
98C2×D4×C47—24no—
99C2×D4×C47C2×D418yes—
100C2×D4×C47C2×D414yes—
101C22×C4×C47—28no—
102C22×C4×C47C2×D412yes—
103C22×C4×C47C2×D414yes—
104C22×C4×C47C2×D411yescharacteristic
105C22×C4×C47C2×D412yes—
106C22×C4×C47#4118yes—
107C22×C4×C47C2414yes—
108C22×C4×C47C2412yes—
109C22×C4×C47C2411yescharacteristic
110C2×C22⋊D47—216no—
111C2×C22⋊D47—88no—
112C22×Dic47—22no—
113C22×Dic47—24no—
114C22×Dic47—28no—
115C2×D188—28no—
116C2×D188—28no—
117C2×D188—28no—
118C2×D188—22no—
119C2×D188—24no—
120C2×D188—44no—
121C2×D188—48no—
122C2×D188C2414yes—
123M16×C47—28no—
124M16×C47—28no—
125C2×C8×C47—28no—
126C2×C8×C47C2418yes—
127C22⋊Dic47—24no—
128C22⋊Dic47—24no—
129C22⋊Dic47—24no—
130C22⋊Dic47—28no—
131(C2×D47)⋊C4—216no—
132C4⋊Dic47—24no—
133C4⋊Dic47—22no—
134C4⋊Dic47—28no—
135C4⋊Dic47C2418yes—
136C4⋊Dic47C2414yes—
137C4×Dic47—24no—
138C4×Dic47—24no—
139C4×Dic47—24no—
140C4×Dic47—28no—
141C23×C47—24no—
142C23×C47—42no—
143C23×C47—44no—
144C23×C47#12314yes—
145C23×C47C22×D414yes—
146C23×C47C22×D411yescharacteristic
147C23×C47C22×D412yes—
148C22×D47—24no—
149C22×D47—48no—
150C22×D47—44no—
151C22×D47—48no—
152C22×D47—88no—
153C22×D47—816no—
154Q8×C47—28no—
155D4×C47—216no—
156D4×C47—28no—
157C2×C4×C47—24no—
158C2×C4×C47—28no—
159C2×C4×C47—28no—
160C2×C4×C47#12314yes—
161C2×C4×C47#12318yes—
162C2×C4×C47C22×D414yes—
163C2×C4×C47C22×D411yescharacteristic
164C2×C4×C47C22×D412yes—
165C2×C4×C47C2×#4114yes—
166C2×C4×C47C2×#4114yes—
167C2×C4×C47C2×#4114yes—
168C2×C4×C47C2511yescharacteristic
169C22⋊D47—816no—
170C2×Dic47—28no—
171C2×Dic47—24no—
172C2×Dic47—44no—
173C2×Dic47—84no—
174C2×Dic47—88no—
175D188—28no—
176D188—44no—
177D188—48no—
178C8×C47—28no—
179#60518—471nomaximal, Sylow
180C22×C47—22no—
181C22×C47—24no—
182C22×C47—28no—
183C22×C47—24no—
184C22×C47—44no—
185C22×C47—84no—
186C22×C47C2×#12312yes—
187C22×C47C2×#12314yes—
188C22×C47C23×D411yescharacteristic
189C2×D47—48no—
190C2×D47—44no—
191C2×D47—88no—
192C2×D47—816no—
193C4×C47—24no—
194C4×C47C2×#12312yes—
195C4×C47C2×#12314yes—
196C4×C47C22×#4112yes—
197Dic47—44no—
198Dic47—48no—
199C2×#535—471no—
200#3428—471no—
201#3096—4716no—
202#3094—478no—
203#2911—472no—
204C2×#418—472no—
205#2879D4711yescharacteristic, complemented
206C2×C47—22no—
207C2×C47—24no—
208C2×C47—44no—
209C2×C47C22×#12311yescharacteristic
210C2×C47#358512yes—
211D47—84no—
212C22×#41C2×D4711yescharacteristic
213C23×D4—471no—
214C22×M16C2×D4712yes—
215#535—478no—
216#535—944no—
217C2×#127—941no—
218C2×#124—941no—
219C2×#121—474no—
220C2×#120—942no—
221#520—478no—
222#519—941no—
223#518—942no—
224#518—941no—
225C4×#41—944no—
226C2×#117—474no—
227C2×#117—472no—
228C2×#117—471no—
229#482—4732no—
230#481—4716no—
231#480—4716no—
232#428—478no—
233#428—478no—
234#427—478no—
235#418—478no—
236#418—478no—
237C2×#102—478no—
238#408C2×D4718yescomplemented
239#407C2×D4714yescomplemented
240C47#6051811yescharacteristic, complemented, Sylow
241C25—941no—
242C2×#41—28no—
243C2×#41D18814yes—
244C2×#41C22×D4714yes—
245C22×Q8D18811yescharacteristic
246C22×D4—472no—
247C22×D4—474no—
248C22×D4—944no—
249C22×D4—941no—
250C22×D4—942no—
251C22×D4—948no—
252C22×D4D18812yes—
253C22×D4D18811yescharacteristic
254C23×C4C22×D4712yes—
255C23×C4C22×D4711yescharacteristic
256C2×M16—28no—
257C2×M16C22×D4714yes—
258C2×M16C22×D4714yes—
259C22×C8C22×D4714yes—
260#127—948no—
261#124—948no—
262#121—4716no—
263#121—948no—
264#120—9416no—
265C4×Q8—948no—
266C4×D4—9416no—
267C4×D4—948no—
268#117—474no—
269#117—474no—
270#117—478no—
271#117—474no—
272#117—944no—
273#117—942no—
274#117—944no—
275#117—942no—
276#117—942no—
277C2×C4⋊C4—474no—
278C2×C4⋊C4—478no—
279C2×C4⋊C4—472no—
280C2×C22⋊C4—944no—
281C2×C22⋊C4—942no—
282C2×C22⋊C4—941no—
283C2×C22⋊C4—944no—
284C2×C42—944no—
285C2×C42—942no—
286C2×C42—941no—
287#107—4716no—
288#106—4716no—
289#102—4732no—
290C22⋊C8C22×D47116yescomplemented
291C24—944no—
292C24—942no—
293C24—3764no—
294C24—1884no—
295C24C2×D18812yes—
296C24C2×D18811yescharacteristic
297#41—216no—
298#41—28no—
299#41—28no—
300C2×Q8—24no—
301C2×Q8C2×D18814yes—
302C2×D4—28no—
303C2×D4—24no—
304C2×D4—474no—
305C2×D4—9416no—
306C2×D4—942no—
307C2×D4—944no—
308C2×D4—948no—
309C2×D4—948no—
310C2×D4—948no—
311C2×D4—3768no—
312C2×D4—1884no—
313C2×D4—1888no—
314C2×D4C2×D18818yes—
315C2×D4C2×D18814yes—
316C22×C4—28no—
317C22×C4—948no—
318C22×C4—944no—
319C22×C4—942no—
320C22×C4C2×D18814yes—
321C22×C4C2×D18812yes—
322C22×C4C2×D18811yescharacteristic
323C22×C4C2×D18812yes—
324C22×C4#1062535218yes—
325C22×C4C23×D4714yes—
326C22×C4C23×D4712yes—
327C22×C4C23×D4711yescharacteristic
328M16—28no—
329M16—28no—
330C2×C8—28no—
331C2×C8C23×D4718yes—
332C4⋊C4—478no—
333C4⋊C4—474no—
334C4⋊C4—948no—
335C4⋊C4—944no—
336C4⋊C4—942no—
337C22⋊C4—944no—
338C22⋊C4—948no—
339C22⋊C4—944no—
340C22⋊C4—944no—
341C22⋊C4—9416no—
342C42—944no—
343C42—948no—
344C42—944no—
345C42—944no—
346C23—24no—
347C23—42no—
348C23—44no—
349C23—944no—
350C23—3768no—
351C23—37616no—
352C23—1888no—
353C23—1888no—
354C23—1884no—
355C23#1309400814yes—
356C23C22×D18814yes—
357C23C22×D18811yescharacteristic
358C23C22×D18812yes—
359Q8—28no—
360D4—216no—
361D4—28no—
362D4—948no—
363D4—37616no—
364D4—1884no—
365D4—1888no—
366C2×C4—24no—
367C2×C4—28no—
368C2×C4—28no—
369C2×C4—948no—
370C2×C4—944no—
371C2×C4—3764no—
372C2×C4—3768no—
373C2×C4—1884no—
374C2×C4#1309400814yes—
375C2×C4#1309400818yes—
376C2×C4C22×D18814yes—
377C2×C4C22×D18811yescharacteristic
378C2×C4C22×D18812yes—
379C2×C4C2×#1062535214yes—
380C2×C4C2×#1062535214yes—
381C2×C4C2×#1062535214yes—
382C2×C4C24×D4711yescharacteristic
383C8—28no—
384C22—24no—
385C22—22no—
386C22—24no—
387C22—28no—
388C22—44no—
389C22—84no—
390C22—37616no—
391C22—3768no—
392C22—1884no—
393C22—1888no—
394C22C2×#1309400812yes—
395C22C2×#1309400814yes—
396C22C23×D18811yescharacteristic
397C4—24no—
398C4—1888no—
399C4—1884no—
400C4C2×#1309400812yes—
401C4C2×#1309400814yes—
402C4C22×#1062535212yes—
403C2—22no—
404C2—24no—
405C2—44no—
406C2—3764no—
407C2C22×#1309400811yescharacteristic
408C2#2025816912yes—
subgroupG/H
centerC22C23×D188
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC47×#2879C2
socleC22×C47C23×D4
prime radicalC2×#15689687C2
hypercenter#2879D47
hypocenterC47#60518
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×#41 ◁ #2879

Complements

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

NH ≅ G/Ntype
2. #20256591C2semidirect
3. #20253347C2semidirect
4. #20253329C2semidirect
5. #20251978C2semidirect
6. C2×#15689129C2semidirect
7. C47×#2879C2semidirect
15. #15689606C22semidirect
22. C2×#13093983C22semidirect
23. #15689448C22semidirect
24. #15689442C22semidirect
25. #15689437C22semidirect
26. #15689165C22semidirect
27. #15689165C22semidirect
28. #15689161C22semidirect
30. #15689129C22semidirect
31. C2×#13093946C22semidirect
34. C47×#408C22semidirect
35. C47×#407C22semidirect
59. #13093983C23semidirect
62. C2×C4⋊Dic47C23semidirect
70. #13093958C23semidirect
71. #13093956C23semidirect
85. C22⋊C8×C47C23semidirect
205. #2879D47semidirect
238. #408C2×D47semidirect
239. #407C2×D47semidirect
240. C47#60518semidirect
290. C22⋊C8C22×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156896872
#202565912
#202533472
#202533292
#202519782
C2×#156891292
C47×#28792
#6051847