Finite Group Databasebeta

← #27079843

Subgroups — #27079843

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
1C22×#13094031C211yesmaximal, characteristic, complemented
2#20256591C211yesmaximal, characteristic, complemented
3#20251921C218yesmaximal, complemented
4#20251914C218yesmaximal, complemented
5#20251907C216yesmaximal, complemented
6C2×#15689119C216yesmaximal, complemented
7C2×C47×#585C211yesmaximal, characteristic
8C2×#13094031—28no—
9C2×#13094031C2218yescomplemented
10C22×#10625329C2213yescomplemented
11C23×C47⋊C8C2213yescomplemented
12#15689606C2218yescomplemented
13#15689601—21no—
14#15689598—23no—
15#41×Dic47—24no—
16C2×#13093983C2216yescomplemented
17#15689132C22112yescomplemented
18#15689128C22112yescomplemented
19#15689124C22124yescomplemented
20#15689119C22112yescomplemented
21#15689119C22124yescomplemented
22#15689119C2214yescomplemented
23C2×#13093948C22112yescomplemented
24C2×#13093945C22112yescomplemented
25C47×#585—28no—
26C47×#585C4116yescomplemented
27C2×2−1+4×C47—21no—
28C2×2−1+4×C47C2214yes—
29C2×H(5,𝔽2)×C47—23no—
30C2×H(5,𝔽2)×C47C2214yes—
31C22×#41×C47—24no—
32C22×#41×C47C2216yes—
33C22×#41×C47C2211yescharacteristic
34#13094031—232no—
35C2×#10625329—212no—
36C2×#10625329C23112yes—
37C22×C47⋊C8—212no—
38C22×C47⋊C8C23112yes—
39C22×C47⋊C8C2311yescharacteristic
40C22×C47⋊C8C2313yes—
41Q8×Dic47—28no—
42D4×Dic47—224no—
43#13093983—26no—
44#13093983—26no—
45#13093983C2314yes—
46#13093983C23112yes—
47C2×C4⋊Dic47C23112yes—
48C2×C22⋊Dic47—26no—
49C2×C4×Dic47—26no—
50#13093948C2318yes—
51#13093948C23124yes—
52#13093945C2318yes—
53#13093945C23124yes—
542−1+4×C47—216no—
552−1+4×C47—216no—
562−1+4×C47C2×C4132yescomplemented
57H(5,𝔽2)×C47—216no—
58H(5,𝔽2)×C47—248no—
59H(5,𝔽2)×C47C2×C4132yescomplemented
60C2×#41×C47—264no—
61C2×#41×C47—24no—
62C2×#41×C47—212no—
63C2×#41×C47—26no—
64C2×#41×C47—26no—
65C2×#41×C47—224no—
66C2×#41×C47C2×C418yes—
67C2×#41×C47C2×C4148yescomplemented
68C2×#41×C47D418yes—
69C2×#41×C47D414yes—
70C2×#41×C47C2314yes—
71C2×#41×C47C23112yes—
72C2×#41×C47C2314yes—
73C22×Q8×C47—24no—
74C22×Q8×C47D411yescharacteristic
75C22×Q8×C47C2316yes—
76C22×D4×C47—26no—
77C22×D4×C47—212no—
78C22×D4×C47D413yes—
79C22×D4×C47C2316yes—
80C23×C4×C47—26no—
81C23×C4×C47C2313yes—
82C24×C47—26no—
83C24×C47C2×D413yes—
84#41×C47—232no—
85#41×C47—296no—
86#41×C47—248no—
87#41×C47—248no—
88#41×C47—28no—
89#41×C47—224no—
90#41×C47C22⋊C4132yes—
91#41×C47C22×C4116yes—
92#41×C47C22×C4148yes—
93#41×C47C22×C4116yes—
94#41×C47C2×D4116yes—
95C2×Q8×C47—232no—
96C2×Q8×C47—26no—
97C2×Q8×C47—212no—
98C2×Q8×C47C22⋊C418yes—
99C2×Q8×C47C22×C4124yes—
100C2×Q8×C47C2×D414yes—
101C2×Q8×C47C2414yes—
102C2×D4×C47—248no—
103C2×D4×C47—24no—
104C2×D4×C47—212no—
105C2×D4×C47—296no—
106C2×D4×C47—26no—
107C2×D4×C47—212no—
108C2×D4×C47C22⋊C4124yes—
109C2×D4×C47C22×C4124yes—
110C2×D4×C47C2×D4112yes—
111C2×D4×C47C2414yes—
112C22×C4×C47—26no—
113C22×C4×C47—248no—
114C22×C4×C47—24no—
115C22×C4×C47—212no—
116C22×C4×C47C22×C4112yes—
117C22×C4×C47C2×D416yes—
118C22×C4×C47C2×D413yes—
119C22×C4×C47C2×D4112yes—
120C22×C4×C47C2411yescharacteristic
121C22×C4×C47C2416yes—
122C22×Dic47—212no—
123C22⋊Dic47—24no—
124C22⋊Dic47—212no—
125C4⋊Dic47—212no—
126C4⋊Dic47C2418yes—
127C4×Dic47—24no—
128C4×Dic47—212no—
129#10625329—224no—
130C2×C47⋊C8—22no—
131C2×C47⋊C8—26no—
132C2×C47⋊C8—224no—
133C2×C47⋊C8C2412yes—
134C2×C47⋊C8C2416yes—
135C23×C47—24no—
136C23×C47—224no—
137C23×C47—46no—
138C23×C47—812no—
139C23×C47C2×C22⋊C4112yes—
140C23×C47C22×D411yescharacteristic
141C23×C47C22×D416yes—
142Q8×C47—224no—
143Q8×C47—24no—
144Q8×C47C2×C22⋊C4116yes—
145Q8×C47C23×C418yes—
146D4×C47—248no—
147D4×C47—216no—
148D4×C47—224no—
149D4×C47—24no—
150D4×C47C2×C22⋊C4148yes—
151D4×C47C23×C418yes—
152C2×C4×C47—224no—
153C2×C4×C47—24no—
154C2×C4×C47—216no—
155C2×C4×C47—248no—
156C2×C4×C47C2×C22⋊C4124yes—
157C2×C4×C47C2×C22⋊C4112yes—
158C2×C4×C47C23×C412yes—
159C2×C4×C47C23×C4112yes—
160C2×C4×C47C22×D416yes—
161C2×C4×C47C22×D412yes—
162C2×C4×C47C22×D4112yes—
163C2×C4×C47C22×D411yescharacteristic
164C2×C4×C47C2511yescharacteristic
165C2×Dic47—28no—
166C2×Dic47—812no—
167C47⋊C8—22no—
168C47⋊C8—26no—
169#60086—471nomaximal, Sylow
170C22×C47—212no—
171C22×C47—22no—
172C22×C47—28no—
173C22×C47—412no—
174C22×C47—48no—
175C22×C47—824no—
176C22×C47C22×C22⋊C4112yes—
177C22×C47C22×C22⋊C412yes—
178C22×C47C23×D411yescharacteristic
179C4×C47—28no—
180C4×C47C22×C22⋊C412yes—
181C4×C47C22×C22⋊C4112yes—
182C4×C47C24×C411yescharacteristic
183C4×C47C23×D411yescharacteristic
184Dic47—48no—
185C2×#585D4711yescharacteristic
186C22×#131—471no—
187#3428—471no—
188#2898—478no—
189#2897—478no—
190#2896—476no—
191C2×#416—476no—
192C2×C47—22no—
193C2×C47—212no—
194C2×C47—48no—
195C2×C47C23×C22⋊C411yescharacteristic
196C2×C47#358412yes—
197#585—28no—
198#585Dic47116yescomplemented
199C2×2−1+4—21no—
200C2×2−1+4C2×D4714yes—
201C2×H(5,𝔽2)—23no—
202C2×H(5,𝔽2)C2×D4714yes—
203C22×#41—24no—
204C22×#41C2×D4716yes—
205C22×#41C2×D4711yescharacteristic
206C2×#131—478no—
207C2×#131—948no—
208C22×M16—473no—
209C23×C8—473no—
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—216no—
2252−1+4—216no—
2262−1+4C2×Dic47132yescomplemented
227H(5,𝔽2)—216no—
228H(5,𝔽2)—248no—
229H(5,𝔽2)C2×Dic47132yescomplemented
230C2×#41—264no—
231C2×#41—24no—
232C2×#41—212no—
233C2×#41—26no—
234C2×#41—26no—
235C2×#41—224no—
236C2×#41C2×Dic4718yes—
237C2×#41C2×Dic47148yescomplemented
238C2×#41C22⋊D4718yes—
239C2×#41C22⋊D4714yes—
240C2×#41C22×D4714yes—
241C2×#41C22×D47112yes—
242C2×#41C22×D4714yes—
243C22×Q8—24no—
244C22×Q8C22⋊D4711yescharacteristic
245C22×Q8C22×D4716yes—
246C22×D4—26no—
247C22×D4—212no—
248C22×D4C22⋊D4713yes—
249C22×D4C22×D4716yes—
250C23×C4—26no—
251C23×C4C22×D4713yes—
252#131—9432no—
253C2×M16—4712no—
254C2×M16—9412no—
255C22×C8—471no—
256C22×C8—473no—
257C22×C8—4712no—
258C22×C8—9412no—
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—26no—
273C24C2×C22⋊D4713yes—
274#41—232no—
275#41—296no—
276#41—248no—
277#41—248no—
278#41—28no—
279#41—224no—
280#41C22⋊Dic47132yes—
281#41C22×Dic47116yes—
282#41C22×Dic47148yes—
283#41C22×Dic47116yes—
284#41C2×C22⋊D47116yes—
285C2×Q8—232no—
286C2×Q8—26no—
287C2×Q8—212no—
288C2×Q8C22⋊Dic4718yes—
289C2×Q8C22×Dic47124yes—
290C2×Q8C2×C22⋊D4714yes—
291C2×Q8C23×D4714yes—
292C2×D4—248no—
293C2×D4—24no—
294C2×D4—212no—
295C2×D4—296no—
296C2×D4—26no—
297C2×D4—212no—
298C2×D4C22⋊Dic47124yes—
299C2×D4C22×Dic47124yes—
300C2×D4C2×C22⋊D47112yes—
301C2×D4C23×D4714yes—
302C22×C4—26no—
303C22×C4—248no—
304C22×C4—212no—
305C22×C4—24no—
306C22×C4—9412no—
307C22×C4C22×Dic47112yes—
308C22×C4C2×C22⋊D4713yes—
309C22×C4C2×C22⋊D4716yes—
310C22×C4C2×C22⋊D47112yes—
311C22×C4C23×D4716yes—
312C22×C4C23×D4711yescharacteristic
313M16—9424no—
314C2×C8—472no—
315C2×C8—476no—
316C2×C8—942no—
317C2×C8—946no—
318C2×C8—9424no—
319C4⋊C4—478no—
320C4⋊C4—9412no—
321C22⋊C4—944no—
322C22⋊C4—9412no—
323C42—9412no—
324C42—944no—
325C23—24no—
326C23—224no—
327C23—46no—
328C23—812no—
329C23C2×C22⋊Dic47112yes—
330C23C22×C22⋊D4711yescharacteristic
331C23C22×C22⋊D4716yes—
332Q8—224no—
333Q8—24no—
334Q8C2×C22⋊Dic47116yes—
335Q8C23×Dic4718yes—
336D4—248no—
337D4—216no—
338D4—224no—
339D4—24no—
340D4C2×C22⋊Dic47148yes—
341D4C23×Dic4718yes—
342C2×C4—224no—
343C2×C4—24no—
344C2×C4—216no—
345C2×C4—248no—
346C2×C4—948no—
347C2×C4—37612no—
348C2×C4C2×C22⋊Dic47124yes—
349C2×C4C2×C22⋊Dic47112yes—
350C2×C4C23×Dic4712yes—
351C2×C4C23×Dic47112yes—
352C2×C4C22×C22⋊D4716yes—
353C2×C4C22×C22⋊D4712yes—
354C2×C4C22×C22⋊D47112yes—
355C2×C4C22×C22⋊D4711yescharacteristic
356C2×C4C24×D4711yescharacteristic
357C8—942no—
358C8—946no—
359C22—212no—
360C22—22no—
361C22—28no—
362C22—412no—
363C22—48no—
364C22—824no—
365C22C22×C22⋊Dic47112yes—
366C22C22×C22⋊Dic4712yes—
367C22C23×C22⋊D4711yescharacteristic
368C4—28no—
369C4—1888no—
370C4C22×C22⋊Dic4712yes—
371C4C22×C22⋊Dic47112yes—
372C4C24×Dic4711yescharacteristic
373C4C23×C22⋊D4711yescharacteristic
374C2—22no—
375C2—212no—
376C2—48no—
377C2C23×C22⋊Dic4711yescharacteristic
378C2#2025815912yes—
subgroupG/H
centerC2×C4C22×C22⋊D47
commutatorC4×C47C24×C4
FrattiniC2×C4C24×D47
Fitting, prime radicalC2×C47×#585C2
socleC22×C47C23×D4
cosocleC2×C4×C47C25
hypercenterC2×#585D47
hypocenterC47#60086
LCSG ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C22×#41 ◁ C2×#585

Complements

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

NH ≅ G/Ntype
1. C22×#13094031C2semidirect
2. #20256591C2semidirect
3. #20251921C2semidirect
4. #20251914C2semidirect
5. #20251907C2semidirect
6. C2×#15689119C2semidirect
9. C2×#13094031C22semidirect
10. C22×#10625329C22semidirect
11. C23×C47⋊C8C22semidirect
12. #15689606C22semidirect
16. C2×#13093983C22semidirect
17. #15689132C22semidirect
18. #15689128C22semidirect
19. #15689124C22semidirect
20. #15689119C22semidirect
21. #15689119C22semidirect
22. #15689119C22semidirect
23. C2×#13093948C22semidirect
24. C2×#13093945C22semidirect
26. C47×#585C4semidirect
56. 2−1+4×C47C2×C4semidirect
59. H(5,𝔽2)×C47C2×C4semidirect
67. C2×#41×C47C2×C4semidirect
198. #585Dic47semidirect
223. C47#60086semidirect
226. 2−1+4C2×Dic47semidirect
229. H(5,𝔽2)C2×Dic47semidirect
237. C2×#41C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
C22×#130940312
#202565912
#202519212
#202519142
#202519072
C2×#156891192
C2×C47×#5852
#6008647