Finite Group Databasebeta

← #26218116

Subgroups — #26218116

Order 12032.

1866 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689687C211yesmaximal, characteristic
2C22×#13093940C212yesmaximal, complemented
3#20251833C211yesmaximal, characteristic, complemented
4#20243851C218yesmaximal, complemented
5#20243126C212yesmaximal, complemented
6C2×C47×#568C211yesmaximal, characteristic, complemented
7C23×D188C2211yescharacteristic
8C22×#10625329C2212yes—
9#15689687—24no—
10#15689687—24no—
11C2×#13094020C412yes—
12C2×#13094013C412yes—
13C2×#13093998C2214yes—
14C2×#13093992C414yes—
15C2×#13093983C2211yescharacteristic
16C2×#13093940—28no—
17C2×#13093940C2218yes—
18#15689090—24no—
19#15689087C2214yescomplemented
20#15688891C2218yescomplemented
21C22×#41×C47C2211yescharacteristic
22C47×#568—28no—
23C2×#131×C47—24no—
24C22×M16×C47C2212yes—
25C22×M16×C47C2214yescomplemented
26C24×D47C2×C412yes—
27C22×D188—24no—
28C22×D188—22no—
29C22×D188—22no—
30C22×D188—24no—
31C22×D188C2×C412yes—
32C22×D188C2312yes—
33C22×C22⋊D47C2×C418yes—
34C2×#10625329—22no—
35C2×#10625329—24no—
36C2×#10625329—24no—
37C2×#10625329C2314yes—
38C22×C47⋊C8—22no—
39#13094020—28no—
40#13094013—28no—
41#13093998—28no—
42#13093998—28no—
43#13093992—216no—
44#13093983—22no—
45#13093983—22no—
46C2×C4⋊Dic47C2312yes—
47C2×C22⋊Dic47C2×C412yes—
48C2×(C2×D47)⋊C4C2×C418yes—
49C2×C4×Dic47C2×C412yes—
50#13093940—216no—
51C22⋊(C47⋊C8)—28no—
52C2×#41×C47—24no—
53C2×#41×C47—24no—
54C2×#41×C47D414yes—
55C2×#41×C47D414yes—
56C22×Q8×C47C2×C411yescharacteristic
57C22×D4×C47C2×C412yes—
58C22×D4×C47C2×C411yescharacteristic
59C23×C4×C47C2312yes—
60C23×C4×C47C2311yescharacteristic
61#131×C47—232no—
62C2×M16×C47—22no—
63C2×M16×C47—24no—
64C2×M16×C47—24no—
65C2×M16×C47—216no—
66C2×M16×C47D418yescomplemented
67C2×M16×C47C2314yes—
68C22×C8×C47—22no—
69C22×C8×C47—24no—
70C24×C47C22×C412yes—
71C24×C47C22×C411yescharacteristic
72C23×D47—24no—
73C23×D47—28no—
74C23×D47—84no—
75C23×D47C22×C414yes—
76#41×C47—28no—
77#41×C47—28no—
78#41×C47—28no—
79#41×C47—28no—
80C2×Q8×C47—24no—
81C2×Q8×C47C22⋊C414yes—
82C2×D4×C47—28no—
83C2×D4×C47—24no—
84C2×D4×C47C22⋊C418yes—
85C2×D4×C47C22⋊C414yes—
86C22×C4×C47—24no—
87C22×C4×C47—24no—
88C22×C4×C47C22×C412yes—
89C22×C4×C47C22×C411yescharacteristic
90C22×C4×C47C2×D414yes—
91C22×C4×C47C2×D414yes—
92C22×C4×C47C2×D412yes—
93C22×C4×C47C2×D412yes—
94C22×C4×C47C2×D414yes—
95C22×C4×C47#4114yes—
96C22×C4×C47C2411yescharacteristic
97C2×C22⋊D47—216no—
98C2×C22⋊D47—88no—
99C22×Dic47C22×C414yes—
100C2×D188—24no—
101C2×D188—28no—
102C2×D188—24no—
103C2×D188—22no—
104C2×D188—22no—
105C2×D188—24no—
106C2×D188—44no—
107C2×D188—44no—
108C2×D188—44no—
109C2×D188—44no—
110M16×C47—216no—
111M16×C47—28no—
112M16×C47—24no—
113M16×C47C2×D418yes—
114C2×C8×C47—216no—
115C2×C8×C47—28no—
116C2×C8×C47—24no—
117C22⋊Dic47—24no—
118(C2×D47)⋊C4—216no—
119C4⋊Dic47—22no—
120C4⋊Dic47—22no—
121C4×Dic47—24no—
122#10625329—24no—
123#10625329—24no—
124#10625329—44no—
125C2×C47⋊C8—24no—
126C2×C47⋊C8—44no—
127C23×C47—24no—
128C23×C47—42no—
129C23×C47—44no—
130C23×C47C2×C22⋊C414yes—
131C23×C47C2×C22⋊C412yes—
132C23×C47C4×D414yes—
133C23×C47C23×C411yescharacteristic
134C22×D47—24no—
135C22×D47—28no—
136C22×D47—48no—
137C22×D47—48no—
138C22×D47—88no—
139C22×D47—816no—
140Q8×C47—28no—
141D4×C47—216no—
142D4×C47—28no—
143C2×C4×C47—24no—
144C2×C4×C47—24no—
145C2×C4×C47—28no—
146C2×C4×C47—24no—
147C2×C4×C47C2×C22⋊C414yes—
148C2×C4×C47C2×C22⋊C412yes—
149C2×C4×C47C4×D414yes—
150C2×C4×C47#12014yes—
151C2×C4×C47#12114yes—
152C2×C4×C47#12114yes—
153C2×C4×C47#12314yes—
154C2×C4×C47C22×D412yes—
155C2×C4×C47C22×D411yescharacteristic
156C2×C4×C47C22×D411yescharacteristic
157C2×C4×C47C2×#4112yes—
158C22⋊D47—816no—
159C2×Dic47—24no—
160C2×Dic47—44no—
161D188—42no—
162D188—44no—
163D188—42no—
164D188—48no—
165C8×C47—28no—
166C47⋊C8—44no—
167#20990—471nomaximal, Sylow
168C22×C47—22no—
169C22×C47—24no—
170C22×C47—28no—
171C22×C47—24no—
172C22×C47—44no—
173C22×C47—84no—
174C22×C47#38614yes—
175C22×C47C22×C22⋊C411yescharacteristic
176C22×C47C2×C4×D412yes—
177C2×D47—48no—
178C2×D47—44no—
179C2×D47—816no—
180C2×D47—88no—
181C4×C47—24no—
182C4×C47#38614yes—
183C4×C47C2×#12011yescharacteristic
184C4×C47C2×#12112yes—
185C4×C47C2×#12311yescharacteristic
186Dic47—44no—
187C2×#568D4711yescharacteristic, complemented
188C2×#535—471no—
189C22×#100—472no—
190#2879—471no—
191#1900—478no—
192#1742—472no—
193C2×C47—22no—
194C2×C47—24no—
195C2×C47—44no—
196C2×C47C2×#38611yescharacteristic
197C2×C47#288812yes—
198D47—84no—
199C22×#41C2×D4711yescharacteristic
200C23×D4—471no—
201#568—28no—
202C2×#131—24no—
203C22×M16—472no—
204C22×M16C2×D4712yes—
205C22×M16C2×D4714yescomplemented
206#535—944no—
207#535—944no—
208C2×#127—472no—
209C2×#124—472no—
210C2×#121—474no—
211C2×#120—474no—
212C2×#117—471no—
213C2×#100—478no—
214C2×#100—948no—
215#408—944no—
216#407—474no—
217#343—478no—
218C47#2099011yescharacteristic, complemented, Sylow
219C25—472no—
220C2×#41—24no—
221C2×#41—24no—
222C2×#41D18814yes—
223C2×#41C22⋊D4714yes—
224C22×Q8C4×D4711yescharacteristic
225C22×D4—478no—
226C22×D4—472no—
227C22×D4—472no—
228C22×D4—942no—
229C22×D4—944no—
230C22×D4—942no—
231C22×D4—944no—
232C22×D4C4×D4712yes—
233C22×D4C4×D4711yescharacteristic
234C23×C4C22×D4712yes—
235C23×C4C22×D4711yescharacteristic
236#131—232no—
237C2×M16—22no—
238C2×M16—24no—
239C2×M16—24no—
240C2×M16—216no—
241C2×M16—474no—
242C2×M16—942no—
243C2×M16—944no—
244C2×M16—944no—
245C2×M16C22⋊D4718yescomplemented
246C2×M16C22×D4714yes—
247C22×C8—22no—
248C22×C8—24no—
249C22×C8—942no—
250#127—948no—
251#124—948no—
252#121—948no—
253#121—948no—
254#120—9416no—
255#117—942no—
256#117—942no—
257C2×C4⋊C4—472no—
258C2×C22⋊C4—472no—
259C2×C22⋊C4—478no—
260C2×C42—472no—
261#100—9416no—
262C22⋊C8—948no—
263C24—474no—
264C24—944no—
265C24—948no—
266C24—3764no—
267C24C2×C4×D4712yes—
268C24C2×C4×D4711yescharacteristic
269#41—28no—
270#41—28no—
271#41—28no—
272#41—28no—
273C2×Q8—24no—
274C2×Q8(C2×D47)⋊C414yes—
275C2×D4—28no—
276C2×D4—24no—
277C2×D4—9416no—
278C2×D4—944no—
279C2×D4—948no—
280C2×D4—944no—
281C2×D4—942no—
282C2×D4—942no—
283C2×D4—944no—
284C2×D4—3768no—
285C2×D4—1884no—
286C2×D4—1884no—
287C2×D4—1884no—
288C2×D4—1884no—
289C2×D4(C2×D47)⋊C418yes—
290C2×D4(C2×D47)⋊C414yes—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—474no—
294C22×C4C2×C4×D4712yes—
295C22×C4C2×C4×D4711yescharacteristic
296C22×C4C2×D18814yes—
297C22×C4C2×D18812yes—
298C22×C4#1062535014yes—
299C22×C4C2×C22⋊D4714yes—
300C22×C4C2×C22⋊D4712yes—
301C22×C4C2×C22⋊D4714yes—
302C22×C4C23×D4711yescharacteristic
303M16—216no—
304M16—28no—
305M16—24no—
306M16—944no—
307M16—944no—
308M16—1884no—
309M16C2×C22⋊D4718yes—
310C2×C8—216no—
311C2×C8—28no—
312C2×C8—24no—
313C2×C8—944no—
314C2×C8—1884no—
315C4⋊C4—942no—
316C4⋊C4—942no—
317C22⋊C4—944no—
318C22⋊C4—9416no—
319C42—944no—
320C23—24no—
321C23—42no—
322C23—44no—
323C23—948no—
324C23—944no—
325C23—3768no—
326C23—37616no—
327C23—1888no—
328C23—1888no—
329C23C2×(C2×D47)⋊C414yes—
330C23C2×(C2×D47)⋊C412yes—
331C23C4×C22⋊D4714yes—
332C23C22×C4×D4711yescharacteristic
333Q8—28no—
334D4—216no—
335D4—28no—
336D4—37616no—
337D4—1888no—
338D4—1882no—
339D4—1884no—
340D4—1882no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—28no—
344C2×C4—24no—
345C2×C4—944no—
346C2×C4—1884no—
347C2×C4C2×(C2×D47)⋊C414yes—
348C2×C4C2×(C2×D47)⋊C412yes—
349C2×C4C4×C22⋊D4714yes—
350C2×C4#1309399314yes—
351C2×C4#1309399814yes—
352C2×C4#1309399814yes—
353C2×C4#1309400714yes—
354C2×C4C22×C22⋊D4712yes—
355C2×C4C22×C22⋊D4711yescharacteristic
356C2×C4C22×D18811yescharacteristic
357C2×C4C2×#1062535012yes—
358C8—28no—
359C8—1884no—
360C22—22no—
361C22—24no—
362C22—24no—
363C22—28no—
364C22—44no—
365C22—84no—
366C22—3768no—
367C22—37616no—
368C22—1884no—
369C22—1888no—
370C22#1568900214yes—
371C22C22×(C2×D47)⋊C411yescharacteristic
372C22C2×C4×C22⋊D4712yes—
373C4—24no—
374C4—1884no—
375C4#1568900214yes—
376C4C2×#1309399311yescharacteristic
377C4C2×#1309399812yes—
378C4C2×#1309400711yescharacteristic
379C2—22no—
380C2—24no—
381C2—44no—
382C2—3764no—
383C2C2×#1568900211yescharacteristic
384C2#2025186612yes—
subgroupG/H
centerC22C22×(C2×D47)⋊C4
commutatorC23×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#568C2
socleC22×C47C22×C22⋊C4
prime radicalC2×#15689687C2
cosocleC22×C4×C47C24
hypercenterC2×#568D47
hypocenterC47#20990
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×#41 ◁ C2×#568

Complements

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

NH ≅ G/Ntype
2. C22×#13093940C2semidirect
3. #20251833C2semidirect
4. #20243851C2semidirect
5. #20243126C2semidirect
6. C2×C47×#568C2semidirect
19. #15689087C22semidirect
20. #15688891C22semidirect
25. C22×M16×C47C22semidirect
66. C2×M16×C47D4semidirect
187. C2×#568D47semidirect
205. C22×M16C2×D47semidirect
218. C47#20990semidirect
245. C2×M16C22⋊D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156896872
C22×#130939402
#202518332
#202438512
#202431262
C2×C47×#5682
#2099047