Finite Group Databasebeta

← #27079844

Subgroups — #27079844

Order 12032.

4874 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689959C211yesmaximal, characteristic
2#20256591C211yesmaximal, characteristic, complemented
3#20251922C218yesmaximal, complemented
4#20251915C218yesmaximal, complemented
5#20251908C216yesmaximal, complemented
6C2×#15689120C216yesmaximal, complemented
7C22×#131×C47C211yesmaximal, characteristic, complemented
8#15689959—28no—
9#15689959C4116yescomplemented
10C2×#13094072—21no—
11C2×#13094071C2214yes—
12C2×#13094070C2214yes—
13C2×#13094069—23no—
14C22×#10625350C2216yes—
15C2×#41×D47—24no—
16#15689606C2218yescomplemented
17#15689601—21no—
18#15689598—23no—
19#41×Dic47—24no—
20C2×#13093983C2216yescomplemented
21#15689133C22112yescomplemented
22#15689129C22112yescomplemented
23#15689125C22124yescomplemented
24#15689120C22112yescomplemented
25#15689120C22124yescomplemented
26#15689120C2214yescomplemented
27C2×#13093949C22112yescomplemented
28C2×#13093946C22112yescomplemented
29C22×#41×C47C2211yescharacteristic
30C2×#131×C47—28no—
31C2×#131×C47C2218yescomplemented
32C22×M16×C47C2213yescomplemented
33C23×C8×C47C2213yescomplemented
34#13094072—216no—
35#13094071—216no—
36#13094071C2×C4132yescomplemented
37#13094070—216no—
38#13094070C2×C4132yescomplemented
39#13094069—248no—
40C2×#10625354—24no—
41C2×#10625350—26no—
42C2×#10625350—26no—
43C2×#10625350—224no—
44C2×#10625350C2×C4148yescomplemented
45C2×#10625350C2314yes—
46C2×#10625350C23112yes—
47C2×#10625352—212no—
48#41×D47—264no—
49C22×Dic94C2316yes—
50C2×Q8×D47—24no—
51C22×D188C2316yes—
52C22×C22⋊D47—26no—
53C2×D4×D47—212no—
54C22×C4×D47—26no—
55Q8×Dic47—28no—
56D4×Dic47—224no—
57#13093983—26no—
58#13093983—26no—
59#13093983C2314yes—
60#13093983C23112yes—
61C2×C4⋊Dic47C23112yes—
62C2×C22⋊Dic47—26no—
63C2×C4×Dic47—26no—
64#13093949C2318yes—
65#13093949C23124yes—
66#13093946C2318yes—
67#13093946C23124yes—
68C2×#41×C47C2×C418yes—
69C2×#41×C47D418yes—
70C2×#41×C47D414yes—
71C2×#41×C47C2314yes—
72C22×Q8×C47D411yescharacteristic
73C22×D4×C47D413yes—
74C23×C4×C47C2313yes—
75#131×C47—232no—
76C2×M16×C47—212no—
77C2×M16×C47C23112yes—
78C22×C8×C47—212no—
79C22×C8×C47C2311yescharacteristic
80C22×C8×C47C2313yes—
81C22×C8×C47C23112yes—
82C24×C47C2×D413yes—
83C23×D47—26no—
84#41×C47C22⋊C4132yes—
85#41×C47C22×C4116yes—
86#41×C47C2×D4116yes—
87C2×Q8×C47C22⋊C418yes—
88C2×Q8×C47C2×D414yes—
89C2×D4×C47C22⋊C4124yes—
90C2×D4×C47C2×D4112yes—
91C22×C4×C47C22×C4112yes—
92C22×C4×C47C2×D413yes—
93C22×C4×C47C2×D416yes—
94C22×C4×C47C2×D4112yes—
95C22×C4×C47C2411yescharacteristic
96C22×C4×C47C2416yes—
97C2×C22⋊D47—248no—
98C2×C22⋊D47—212no—
99C2×C22⋊D47—24no—
100C22×Dic47—26no—
101C22×Dic47—212no—
102#10625354—232no—
103Q8×D47—232no—
104#10625352—296no—
105D4×D47—296no—
106#10625350—248no—
107#10625350—248no—
108#10625350—28no—
109#10625350—224no—
110#10625350C22×C4116yes—
111#10625350C22×C4148yes—
112C2×D188—26no—
113C2×D188—212no—
114C2×D188C22×C4124yes—
115C2×D188C2414yes—
116C2×C4×D47—248no—
117C2×C4×D47—24no—
118C2×C4×D47—212no—
119C2×Dic94—26no—
120C2×Dic94—212no—
121C2×Dic94C22×C4124yes—
122C2×Dic94C2414yes—
123M16×C47—224no—
124C2×C8×C47—22no—
125C2×C8×C47—26no—
126C2×C8×C47—224no—
127C2×C8×C47C2412yes—
128C2×C8×C47C2416yes—
129C22⋊Dic47—24no—
130C22⋊Dic47—212no—
131C4⋊Dic47—212no—
132C4⋊Dic47C2418yes—
133C4×Dic47—24no—
134C4×Dic47—212no—
135C23×C47—46no—
136C23×C47C2×C22⋊C4112yes—
137C23×C47C22×D411yescharacteristic
138C23×C47C22×D416yes—
139C22×D47—24no—
140C22×D47—224no—
141C22×D47—812no—
142Q8×C47C2×C22⋊C4116yes—
143D4×C47C2×C22⋊C4148yes—
144C2×C4×C47C2×C22⋊C4124yes—
145C2×C4×C47C2×C22⋊C4112yes—
146C2×C4×C47C23×C4112yes—
147C2×C4×C47C23×C412yes—
148C2×C4×C47C22×D412yes—
149C2×C4×C47C22×D4112yes—
150C2×C4×C47C22×D416yes—
151C2×C4×C47C22×D411yescharacteristic
152C2×C4×C47C2511yescharacteristic
153C22⋊D47—216no—
154C22⋊D47—248no—
155C2×Dic47—224no—
156C2×Dic47—24no—
157C2×Dic47—28no—
158C2×Dic47—812no—
159D188—224no—
160D188—24no—
161D188C23×C418yes—
162C4×D47—216no—
163C4×D47—248no—
164Dic94—224no—
165Dic94—24no—
166Dic94C23×C418yes—
167C8×C47—22no—
168C8×C47—26no—
169#60086—471nomaximal, Sylow
170C22×C47—212no—
171C22×C47—22no—
172C22×C47—412no—
173C22×C47C22×C22⋊C4112yes—
174C22×C47C22×C22⋊C412yes—
175C22×C47C23×D411yescharacteristic
176C2×D47—28no—
177C2×D47—48no—
178C2×D47—824no—
179C4×C47C22×C22⋊C4112yes—
180C4×C47C22×C22⋊C412yes—
181C4×C47C24×C411yescharacteristic
182C4×C47C23×D411yescharacteristic
183Dic47—28no—
184Dic47—48no—
185C2×#585—471no—
186C22×#131D4711yescharacteristic, complemented
187#3428—471no—
188#2898—478no—
189#2897—478no—
190#2896—476no—
191C2×#416—476no—
192C2×C47—212no—
193C2×C47—22no—
194C2×C47C23×C22⋊C411yescharacteristic
195C2×C47#358412yes—
196D47—48no—
197#585—4716no—
198#585—948no—
199C2×2−1+4—474no—
200C2×2−1+4—941no—
201C2×H(5,𝔽2)—474no—
202C2×H(5,𝔽2)—943no—
203C22×#41—476no—
204C22×#41—944no—
205C22×#41C2×D4711yescharacteristic
206C2×#131—28no—
207C2×#131C2×D4718yescomplemented
208C22×M16C2×D4713yescomplemented
209C23×C8C2×D4713yescomplemented
210#520—478no—
211#519—941no—
212#518—943no—
213C4×#41—944no—
214C2×#117—476no—
215#419—4712no—
216#418—4712no—
217#417—4724no—
218#416—474no—
219#416—4712no—
220#416—4724no—
221C2×#103—4712no—
222C2×#102—4712no—
223C47#6008611yescharacteristic, complemented, Sylow
2242−1+4—4732no—
2252−1+4—9416no—
2262−1+4—9416no—
227H(5,𝔽2)—4732no—
228H(5,𝔽2)—9416no—
229H(5,𝔽2)—9448no—
230C2×#41—4748no—
231C2×#41—474no—
232C2×#41—4712no—
233C2×#41—9464no—
234C2×#41—944no—
235C2×#41—9412no—
236C2×#41—946no—
237C2×#41—946no—
238C2×#41—9424no—
239C2×#41C4×D4718yes—
240C2×#41D18814yes—
241C2×#41C22⋊D4718yes—
242C2×#41C22×D4714yes—
243C22×Q8—476no—
244C22×Q8—944no—
245C22×Q8D18811yescharacteristic
246C22×D4—476no—
247C22×D4—946no—
248C22×D4—9412no—
249C22×D4D18813yes—
250C23×C4—946no—
251C23×C4C22×D4713yes—
252#131—232no—
253C2×M16—212no—
254C2×M16C22×D47112yes—
255C22×C8—212no—
256C22×C8C22×D4711yescharacteristic
257C22×C8C22×D4713yes—
258C22×C8C22×D47112yes—
259C4×Q8—948no—
260C4×D4—9424no—
261#117—474no—
262#117—4712no—
263#117—946no—
264#117—946no—
265C2×C4⋊C4—4712no—
266C2×C22⋊C4—946no—
267C2×C42—946no—
268#103—478no—
269#103—4724no—
270#102—478no—
271#102—4724no—
272C24—946no—
273C24C2×D18813yes—
274#41—4716no—
275#41—4748no—
276#41—9432no—
277#41—9496no—
278#41—9448no—
279#41—9448no—
280#41—948no—
281#41—9424no—
282#41(C2×D47)⋊C4132yes—
283#41C2×C4×D47116yes—
284#41C2×C22⋊D47116yes—
285C2×Q8—4724no—
286C2×Q8—474no—
287C2×Q8—9432no—
288C2×Q8—9412no—
289C2×Q8—946no—
290C2×Q8(C2×D47)⋊C418yes—
291C2×Q8C2×D18814yes—
292C2×D4—4724no—
293C2×D4—474no—
294C2×D4—9448no—
295C2×D4—944no—
296C2×D4—9412no—
297C2×D4—9496no—
298C2×D4—9412no—
299C2×D4—946no—
300C2×D4(C2×D47)⋊C4124yes—
301C2×D4C2×D188112yes—
302C22×C4—946no—
303C22×C4—9448no—
304C22×C4—9412no—
305C22×C4—944no—
306C22×C4—9412no—
307C22×C4C2×C4×D47112yes—
308C22×C4C2×D18813yes—
309C22×C4C2×D18816yes—
310C22×C4C2×C22⋊D47112yes—
311C22×C4C23×D4716yes—
312C22×C4C23×D4711yescharacteristic
313M16—224no—
314C2×C8—22no—
315C2×C8—26no—
316C2×C8—224no—
317C2×C8C23×D4712yes—
318C2×C8C23×D4716yes—
319C4⋊C4—478no—
320C4⋊C4—9412no—
321C22⋊C4—944no—
322C22⋊C4—9412no—
323C42—9412no—
324C42—944no—
325C23—46no—
326C23—9424no—
327C23—944no—
328C23—37612no—
329C23C2×(C2×D47)⋊C4112yes—
330C23C22×D18811yescharacteristic
331C23C22×D18816yes—
332Q8—478no—
333Q8—9424no—
334Q8—944no—
335Q8C2×(C2×D47)⋊C4116yes—
336D4—478no—
337D4—9448no—
338D4—9416no—
339D4—9424no—
340D4—944no—
341D4C2×(C2×D47)⋊C4148yes—
342C2×C4—9424no—
343C2×C4—944no—
344C2×C4—9416no—
345C2×C4—9448no—
346C2×C4—948no—
347C2×C4—37612no—
348C2×C4C2×(C2×D47)⋊C4124yes—
349C2×C4C2×(C2×D47)⋊C4112yes—
350C2×C4C22×C4×D4712yes—
351C2×C4C22×C4×D47112yes—
352C2×C4C22×C22⋊D4712yes—
353C2×C4C22×C22⋊D47112yes—
354C2×C4C22×D18816yes—
355C2×C4C22×D18811yescharacteristic
356C2×C4C24×D4711yescharacteristic
357C8—22no—
358C8—26no—
359C22—212no—
360C22—22no—
361C22—412no—
362C22—948no—
363C22—37624no—
364C22—1888no—
365C22C22×(C2×D47)⋊C4112yes—
366C22C22×(C2×D47)⋊C412yes—
367C22C23×D18811yescharacteristic
368C4—948no—
369C4—1888no—
370C4C22×(C2×D47)⋊C412yes—
371C4C22×(C2×D47)⋊C4112yes—
372C4C23×C4×D4711yescharacteristic
373C4C23×C22⋊D4711yescharacteristic
374C2—22no—
375C2—212no—
376C2—1888no—
377C2C23×(C2×D47)⋊C411yescharacteristic
378C2#2025816212yes—
subgroupG/H
centerC2×C4C22×D188
commutatorC4×C47C24×C4
FrattiniC2×C4C24×D47
FittingC22×#131×C47C2
socleC22×C47C23×D4
prime radicalC2×#15689959C2
cosocleC2×C4×C47C25
hypercenterC22×#131D47
hypocenterC47#60086
LCSG ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C22×#41 ◁ C22×#131

Complements

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

NH ≅ G/Ntype
2. #20256591C2semidirect
3. #20251922C2semidirect
4. #20251915C2semidirect
5. #20251908C2semidirect
6. C2×#15689120C2semidirect
7. C22×#131×C47C2semidirect
9. #15689959C4semidirect
16. #15689606C22semidirect
20. C2×#13093983C22semidirect
21. #15689133C22semidirect
22. #15689129C22semidirect
23. #15689125C22semidirect
24. #15689120C22semidirect
25. #15689120C22semidirect
26. #15689120C22semidirect
27. C2×#13093949C22semidirect
28. C2×#13093946C22semidirect
31. C2×#131×C47C22semidirect
32. C22×M16×C47C22semidirect
33. C23×C8×C47C22semidirect
36. #13094071C2×C4semidirect
38. #13094070C2×C4semidirect
44. C2×#10625350C2×C4semidirect
186. C22×#131D47semidirect
207. C2×#131C2×D47semidirect
208. C22×M16C2×D47semidirect
209. C23×C8C2×D47semidirect
223. C47#60086semidirect

Maximal subgroups

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

subgroupindex
C2×#156899592
#202565912
#202519222
#202519152
#202519082
C2×#156891202
C22×#131×C472
#6008647