Finite Group Databasebeta

← #26220318

Subgroups — #26220318

Order 12032.

1562 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20256721C211yesmaximal, characteristic
2C2×#15689141C212yesmaximal, complemented
3C22×#13093952C212yesmaximal, complemented
4#20251807C211yesmaximal, characteristic, complemented
5#20244306C218yesmaximal, complemented
6C47×#2296C211yesmaximal, characteristic, complemented
7C22×#10625350C2212yes—
8C22×#10625329C2212yes—
9#15689704—22no—
10#15689699C414yescomplemented
11#15689692C414yescomplemented
12#15689685—22no—
13#15689673C2214yes—
14#15689635—24no—
15#15689141—28no—
16C2×#13093952—28no—
17C2×#13093952C2218yescomplemented
18C2×#13093952C2218yescomplemented
19#15689079—24no—
20C4×#10625329C2214yescomplemented
21C2×#117×C47C2211yescharacteristic
22C2×#117×C47C2211yescharacteristic
23C22×C42×C47C2211yescharacteristic
24C47×#389—22no—
25C47×#385—22no—
26C4×C4⋊C4×C47C2214yescomplemented
27C2×#10625350—22no—
28C2×#10625350—22no—
29C2×#10625350—24no—
30C2×#10625350—24no—
31C2×#10625350—24no—
32C2×#10625350—24no—
33C2×#10625350C2×C414yes—
34C2×#10625350C2314yes—
35C22×Dic94C2×C412yes—
36C22×D188C2×C412yes—
37C22×C22⋊D47—22no—
38C22×C4×D47—22no—
39C2×#10625329—22no—
40C2×#10625329—24no—
41C2×#10625329D418yescomplemented
42C2×#10625329C2314yes—
43C22×C47⋊C8—22no—
44#13094024C2×C414yescomplemented
45#13094021C2×C414yescomplemented
46#13094012C2×C418yescomplemented
47#13094008—28no—
48#13094004—28no—
49#13093999—28no—
50#13093998—28no—
51#13093994—28no—
52C4×D188—28no—
53#13093952—216no—
54#13093952—216no—
55#13093935—24no—
56C4×C47⋊C8—24no—
57C23×C4×C47—21no—
58C23×C4×C47C2311yescharacteristic
59#117×C47—24no—
60#117×C47C2×C414yes—
61#117×C47D414yes—
62C2×C4⋊C4×C47—21no—
63C2×C4⋊C4×C47—24no—
64C2×C4⋊C4×C47D412yes—
65C2×C22⋊C4×C47—21no—
66C2×C22⋊C4×C47D412yes—
67C2×C42×C47—24no—
68C2×C42×C47—21no—
69C2×C42×C47C2312yes—
70C2×C42×C47C2312yes—
71C2×C42×C47C2312yes—
72#95×C47—24no—
73C24×C47C2×D411yescharacteristic
74C23×D47—22no—
75C22×C4×C47—22no—
76C22×C4×C47—22no—
77C22×C4×C47—44no—
78C22×C4×C47—42no—
79C22×C4×C47C22×C412yes—
80C22×C4×C47C2×D414yes—
81C22×C4×C47C2×D411yescharacteristic
82C22×C4×C47C2×D412yes—
83C22×C4×C47C2×D414yes—
84C22×C4×C47#4114yes—
85C22×C4×C47C2411yescharacteristic
86C2×C22⋊D47—24no—
87C2×C22⋊D47—24no—
88C2×C22⋊D47—44no—
89C2×C22⋊D47—44no—
90C22×Dic47—22no—
91#10625350—28no—
92#10625350—28no—
93#10625350—24no—
94#10625350—24no—
95#10625350—24no—
96#10625350—24no—
97#10625350—44no—
98#10625350—44no—
99#10625350—44no—
100#10625350—44no—
101C2×D188—24no—
102C2×D188—24no—
103C2×D188—22no—
104C2×D188—28no—
105C2×D188C22×C414yes—
106C2×C4×D47—24no—
107C2×C4×D47—24no—
108C2×C4×D47—44no—
109C2×C4×D47—44no—
110C2×Dic94—24no—
111C2×Dic94—24no—
112C2×Dic94—22no—
113C2×Dic94C22×C414yes—
114C4⋊C4×C47—48no—
115C4⋊C4×C47—42no—
116C4⋊C4×C47C22⋊C418yes—
117C22⋊C4×C47—42no—
118C22⋊C4×C47C22⋊C418yes—
119C42×C47—24no—
120C42×C47—24no—
121C42×C47—42no—
122C42×C47C22×C414yes—
123C42×C47C2×D414yes—
124(C2×D47)⋊C4—216no—
125C4⋊Dic47—28no—
126#10625329—28no—
127#10625329—24no—
128#10625329C2×D418yes—
129C2×C47⋊C8—28no—
130C2×C47⋊C8—24no—
131C23×C47—42no—
132C23×C47C2×C22⋊C412yes—
133C23×C47#12412yes—
134C23×C47#12712yes—
135C23×C47C22×D411yescharacteristic
136C22×D47—24no—
137C22×D47—44no—
138C22×D47—84no—
139C2×C4×C47—24no—
140C2×C4×C47—44no—
141C2×C4×C47—42no—
142C2×C4×C47—48no—
143C2×C4×C47—42no—
144C2×C4×C47—44no—
145C2×C4×C47—42no—
146C2×C4×C47C2×C22⋊C418yes—
147C2×C4×C47C2×C22⋊C412yes—
148C2×C4×C47C4×D414yes—
149C2×C4×C47C4×D414yes—
150C2×C4×C47#12114yes—
151C2×C4×C47#12114yes—
152C2×C4×C47#12412yes—
153C2×C4×C47#12712yes—
154C2×C4×C47C23×C411yescharacteristic
155C2×C4×C47C22×D412yes—
156C2×C4×C47C22×D411yescharacteristic
157C2×C4×C47C2×#4112yes—
158C22⋊D47—44no—
159C22⋊D47—44no—
160C22⋊D47—48no—
161C2×Dic47—24no—
162C2×Dic47—44no—
163D188—24no—
164D188—24no—
165D188—44no—
166D188—88no—
167C4×D47—44no—
168C4×D47—44no—
169C4×D47—48no—
170Dic94—24no—
171Dic94—24no—
172Dic94—44no—
173C47⋊C8—28no—
174#21253—471nomaximal, Sylow
175C22×C47—22no—
176C22×C47—24no—
177C22×C47—42no—
178C22×C47—42no—
179C22×C47#39014yes—
180C22×C47C22×C22⋊C411yescharacteristic
181C22×C47C2×#12411yescharacteristic
182C22×C47C2×#12711yescharacteristic
183C2×D47—48no—
184C2×D47—44no—
185C2×D47—88no—
186C4×C47—28no—
187C4×C47—44no—
188C4×C47—82no—
189C4×C47#39014yes—
190C4×C47C2×C4×D412yes—
191C4×C47C2×#12112yes—
192Dic47—44no—
193#3449—471no—
194C2×#421—472no—
195C22×#104—472no—
196#2873—471no—
197#2296D4711yescharacteristic, complemented
198#1980—478no—
199C2×C47—21no—
200C2×C47—24no—
201C2×C47—21no—
202C2×C47C2×#39011yescharacteristic
203C2×C47#290112yes—
204D47—84no—
205C22×#41—472no—
206C22×M16—472no—
207#537—942no—
208#536—474no—
209#535—474no—
210#534—942no—
211#532—474no—
212#525—944no—
213C2×#117C2×D4711yescharacteristic
214C2×#117C2×D4711yescharacteristic
215C22×C42C2×D4711yescharacteristic
216#421—948no—
217C2×#104—478no—
218C2×#104—478no—
219C2×#104—948no—
220#405—944no—
221C4×M16—474no—
222#389—22no—
223#385—22no—
224C4×C4⋊C4C2×D4714yescomplemented
225C47#2125311yescharacteristic, complemented, Sylow
226C2×#41—474no—
227C2×#41—474no—
228C2×#41—942no—
229C2×#41—942no—
230C2×#41—944no—
231C2×#41—944no—
232C2×#41—944no—
233C2×#41—944no—
234C22×Q8—472no—
235C22×D4—472no—
236C22×D4—942no—
237C23×C4—21no—
238C23×C4—942no—
239C23×C4C22×D4711yescharacteristic
240C2×M16—478no—
241C2×M16—474no—
242C2×M16—942no—
243C2×M16—944no—
244C22×C8—942no—
245C4⋊Q8—474no—
246#127—474no—
247#124—478no—
248#123—948no—
249C22⋊Q8—948no—
250#121—948no—
251#121—948no—
252#120—948no—
253C4×D4—948no—
254#117—24no—
255#117C4×D4714yes—
256#117D18814yes—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4C22⋊D4712yes—
260C2×C22⋊C4—21no—
261C2×C22⋊C4C22⋊D4712yes—
262C2×C42—24no—
263C2×C42—21no—
264C2×C42C22×D4712yes—
265C2×C42C22×D4712yes—
266C2×C42C22×D4712yes—
267#104—9416no—
268#104—9416no—
269#97—944no—
270C4×C8—944no—
271#95—24no—
272C24—942no—
273C24C2×C22⋊D4711yescharacteristic
274#41—948no—
275#41—948no—
276#41—944no—
277#41—944no—
278#41—944no—
279#41—944no—
280#41—1884no—
281#41—1884no—
282#41—1884no—
283#41—1884no—
284C2×Q8—474no—
285C2×Q8—942no—
286C2×Q8—944no—
287C2×Q8—944no—
288C2×D4—474no—
289C2×D4—944no—
290C2×D4—944no—
291C2×D4—948no—
292C2×D4—942no—
293C2×D4—944no—
294C2×D4—944no—
295C2×D4—1884no—
296C2×D4—1884no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—44no—
300C22×C4—42no—
301C22×C4—942no—
302C22×C4—944no—
303C22×C4—944no—
304C22×C4—1884no—
305C22×C4—1884no—
306C22×C4C2×C4×D4712yes—
307C22×C4C2×D18812yes—
308C22×C4D4×D4714yes—
309C22×C4#1062535414yes—
310C22×C4C2×C22⋊D4714yes—
311C22×C4C2×C22⋊D4711yescharacteristic
312C22×C4C23×D4711yescharacteristic
313M16—478no—
314M16—948no—
315M16—944no—
316C2×C8—948no—
317C2×C8—944no—
318C4⋊C4—48no—
319C4⋊C4—42no—
320C4⋊C4—948no—
321C4⋊C4(C2×D47)⋊C418yes—
322C22⋊C4—42no—
323C22⋊C4—9416no—
324C22⋊C4(C2×D47)⋊C418yes—
325C42—24no—
326C42—24no—
327C42—42no—
328C42C2×C4×D4714yes—
329C42C2×D18814yes—
330C23—42no—
331C23—944no—
332C23—3764no—
333C23—1884no—
334C23C2×(C2×D47)⋊C412yes—
335C23#1309401312yes—
336C23#1309402012yes—
337C23C22×C22⋊D4711yescharacteristic
338Q8—944no—
339Q8—944no—
340Q8—1884no—
341D4—944no—
342D4—944no—
343D4—3768no—
344D4—1888no—
345D4—1884no—
346D4—1884no—
347D4—1884no—
348C2×C4—24no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—48no—
352C2×C4—42no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—944no—
356C2×C4—1884no—
357C2×C4—1888no—
358C2×C4—1884no—
359C2×C4—1884no—
360C2×C4C2×(C2×D47)⋊C418yes—
361C2×C4C2×(C2×D47)⋊C412yes—
362C2×C4#1309398614yes—
363C2×C4#1309398614yes—
364C2×C4#1309399914yes—
365C2×C4#1309399914yes—
366C2×C4#1309401312yes—
367C2×C4#1309402012yes—
368C2×C4C22×C4×D4711yescharacteristic
369C2×C4C2×D4×D4712yes—
370C2×C4C22×D18811yescharacteristic
371C2×C4C2×#1062535412yes—
372C8—948no—
373C22—22no—
374C22—24no—
375C22—42no—
376C22—42no—
377C22—3768no—
378C22—1888no—
379C22—1884no—
380C22#1568902414yes—
381C22C22×(C2×D47)⋊C411yescharacteristic
382C22C2×#1309401311yescharacteristic
383C22C2×#1309402011yescharacteristic
384C4—28no—
385C4—44no—
386C4—82no—
387C4—1884no—
388C4#1568902414yes—
389C4C2×#1309398612yes—
390C4C2×#1309399912yes—
391C2—21no—
392C2—24no—
393C2—21no—
394C2—3764no—
395C2C2×#1568902411yescharacteristic
396C2#2025193612yes—
subgroupG/H
centerC2×C4C22×D188
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2296C2
socleC22×C47C22×C22⋊C4
prime radical#20256721C2
cosocleC22×C4×C47C24
hypercenter#2296D47
hypocenterC47#21253
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×#117 ◁ #2296

Complements

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

NH ≅ G/Ntype
2. C2×#15689141C2semidirect
3. C22×#13093952C2semidirect
4. #20251807C2semidirect
5. #20244306C2semidirect
6. C47×#2296C2semidirect
10. #15689699C4semidirect
11. #15689692C4semidirect
17. C2×#13093952C22semidirect
18. C2×#13093952C22semidirect
20. C4×#10625329C22semidirect
26. C4×C4⋊C4×C47C22semidirect
41. C2×#10625329D4semidirect
44. #13094024C2×C4semidirect
45. #13094021C2×C4semidirect
46. #13094012C2×C4semidirect
197. #2296D47semidirect
224. C4×C4⋊C4C2×D47semidirect
225. C47#21253semidirect

Maximal subgroups

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

subgroupindex
#202567212
C2×#156891412
C22×#130939522
#202518072
#202443062
C47×#22962
#2125347