Finite Group Databasebeta

← #27123142

Subgroups — #27123142

Order 12032.

7226 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20256822C2112yesmaximal, complemented
2#20256811C2112yesmaximal, complemented
3#20256800C218yesmaximal, complemented
4#20256780C2112yesmaximal, complemented
5#20256721C213yesmaximal, complemented
6#20256685C2112yesmaximal, complemented
7C2×#15689591C213yesmaximal, complemented
8C47×#3458C211yesmaximal, characteristic, complemented
9C22×#10625350C2213yes—
10#15689747C22148yescomplemented
11#15689736C22124yescomplemented
12#15689729C22124yescomplemented
13#15689722C22148yescomplemented
14#15689713C22148yescomplemented
15#15689704C22112yes—
16#15689699C22112yescomplemented
17#15689692C22112yescomplemented
18#15689685C22112yes—
19#15689678C22124yescomplemented
20#15689674C22124yescomplemented
21#15689673C22112yes—
22#15689660C22148yescomplemented
23#15689659C22148yescomplemented
24#15689658C22148yescomplemented
25#15689636C22148yescomplemented
26#15689635C22124yescomplemented
27C4×#10625354C22116yescomplemented
28#15689591C22112yescomplemented
29#15689591C22124yescomplemented
30#15689591C22112yescomplemented
31C2×#13093989C2216yescomplemented
32C2×#13093986C2216yescomplemented
33C2×(C22⋊D47)⋊C4C22112yes—
34C2×#13093981C2216yescomplemented
35C22×C4×Dic47C2211yescharacteristic
36C2×C42×D47C2216yescomplemented
37C47×#544C22112yescomplemented
38C47×#543C2214yes—
39C47×#533C22112yescomplemented
40C2×#117×C47C2213yes—
41C2×#10625354C23116yes—
42C2×#10625350—224no—
43C2×#10625350C23112yes—
44C2×#10625350C23112yes—
45C22×Dic94C2313yes—
46C22×D188C2313yes—
47C22×C22⋊D47C2316yes—
48C23×Dic47C2312yes—
49C22×C4×D47C2316yes—
50C22×C4×D47C23112yes—
51#13094024C23112yes—
52#13094021C23112yes—
53#13094019C23196yes—
54#13094015C23148yes—
55#13094013C23148yes—
56#13094012C23124yes—
57#13094009C23196yes—
58#13094008C23148yes—
59#13094004C23148yes—
60#13094000C23148yes—
61#13093999C23148yes—
62#13093999C23148yes—
63#13093998C23148yes—
64#13093994C23148yes—
65Q8×Dic47C23116yes—
66#13093989C23124yes—
67#13093989C23124yes—
68C4×D188C23148yes—
69#13093986C23124yes—
70#13093986C23124yes—
71#13093986C23148yes—
72(C22⋊D47)⋊C4C23148yes—
73(C22⋊D47)⋊C4C23148yes—
74#13093982C23148yes—
75#13093981C23124yes—
76#13093981C23124yes—
77C2×Dic47⋊C4C23112yes—
78C2×(C2×D47)⋊C4C23112yes—
79C2×C4×Dic47C2313yes—
80C2×C4×Dic47C23112yes—
81C2×C4×Dic47C2313yes—
82C2×C4×Dic47C2316yes—
83C42×D47C23112yes—
84C42×D47C23124yes—
85C42×D47C23112yes—
86C23×C4×C47C2311yescharacteristic
87C4⋊Q8×C47C23124yes—
88#125×C47C23124yes—
89C22⋊Q8×C47C23148yes—
90C4×Q8×C47C23116yes—
91#117×C47C23112yes—
92#117×C47C23112yes—
93C2×C4⋊C4×C47C2316yes—
94C2×C22⋊C4×C47C2316yes—
95C2×C42×C47C2316yes—
96C24×C47C2411yescharacteristic
97C23×D47C2416yes—
98C2×Q8×C47C24116yes—
99C22×C4×C47#4118yescomplemented
100C22×C4×C47C24112yes—
101C22×C4×C47C2413yes—
102C22×C4×C47C2413yes—
103C2×C22⋊D47—224no—
104C2×C22⋊D47C24124yes—
105C22×Dic47—44no—
106C22×Dic47C2416yes—
107C22×Dic47C2416yes—
108C22×Dic47C2416yes—
109#10625354—432no—
110#10625350—248no—
111#10625350—248no—
112C2×D188—212no—
113C2×D188C24112yes—
114C2×D188C24148yes—
115C2×C4×D47—224no—
116C2×C4×D47—424no—
117C2×C4×D47C24112yes—
118C2×C4×D47C24112yes—
119C2×C4×D47C24124yes—
120C2×C4×D47C24124yes—
121C2×Dic94—212no—
122C2×Dic94C24112yes—
123C4⋊C4×C47C24124yes—
124C4⋊C4×C47C24148yes—
125C22⋊C4×C47C24124yes—
126C42×C47C24112yes—
127C42×C47C24112yes—
128(C2×D47)⋊C4C24148yes—
129(C2×D47)⋊C4C24196yes—
130C4⋊Dic47C24148yes—
131Dic47⋊C4C24148yes—
132C4×Dic47C2411yescharacteristic
133C4×Dic47C2413yes—
134C4×Dic47C2416yes—
135C4×Dic47C24124yes—
136C4×Dic47C24124yes—
137C4×Dic47C2416yes—
138C23×C47—42no—
139C23×C47C2×#4114yes—
140C23×C47C2513yes—
141C22×D47—212no—
142C22×D47—412no—
143C22×D47C25112yes—
144Q8×C47—416no—
145C2×C4×C47—412no—
146C2×C4×C47C2×#41112yes—
147C2×C4×C47C2×#41112yes—
148C2×C4×C47C25124yes—
149C2×C4×C47C2513yes—
150C2×C4×C47C2511yescharacteristic
151C22⋊D47—248no—
152C2×Dic47—212no—
153C2×Dic47—212no—
154C2×Dic47—46no—
155C2×Dic47C2516yes—
156C2×Dic47C25112yes—
157C2×Dic47C2512yes—
158D188—224no—
159D188—448no—
160C4×D47—224no—
161C4×D47—224no—
162C4×D47—248no—
163C4×D47—424no—
164Dic94—224no—
165#62686—471nomaximal, Sylow
166C22×C47—26no—
167C22×C47—44no—
168C22×C47C22×#4116yes—
169C22×C47C2611yescharacteristic
170C2×D47—212no—
171C2×D47—224no—
172C2×D47—424no—
173C4×C47—224no—
174C4×C47C22×#4116yes—
175C4×C47C22×#4112yes—
176Dic47—212no—
177Dic47—26no—
178Dic47#58514yes—
179#3461—4712no—
180#3460—4712no—
181#3459—478no—
182#3458D4711yescharacteristic, complemented
183#3455—4712no—
184#3449—473no—
185#3444—4712no—
186C2×C4×#41—473no—
187C2×C47—26no—
188C2×C47C23×#4111yescharacteristic
189C2×C47C2×#58512yes—
190D47—412no—
191C22×#41—473no—
192#544C2×D47112yescomplemented
193#543C2×D4714yes—
194#542—4748no—
195#541—4724no—
196#540—4724no—
197#539—4748no—
198#538—4748no—
199#537—4712no—
200#536—4712no—
201#535—4712no—
202#534—4712no—
203#533—4724no—
204#533C2×D47112yescomplemented
205#532—4724no—
206#532—4712no—
207#529—4748no—
208#529—4748no—
209#529—4748no—
210#525—4748no—
211#525—4724no—
212C4×#41—4716no—
213C4×#41—4712no—
214C4×#41—4712no—
215C4×#41—4724no—
216C2×C4×Q8—476no—
217C2×C4×D4—476no—
218C2×C4×D4—4712no—
219C2×#117—476no—
220C2×#117C2×D4713yes—
221C22×C42—471no—
222C22×C42—476no—
223C47#6268611yescharacteristic, complemented, Sylow
224C2×#41—4716no—
225C2×#41—4712no—
226C2×#41—4712no—
227C2×#41—9424no—
228C22×Q8—473no—
229C22×D4—476no—
230C22×D4—473no—
231C23×C4—472no—
232C23×C4—476no—
233C23×C4—4712no—
234C23×C4C22×D4711yescharacteristic
235C4⋊Q8—4712no—
236C4⋊Q8C22×D47124yes—
237#127—4712no—
238#126—4796no—
239#125—4748no—
240#125C22×D47124yes—
241#124—4748no—
242#124—4724no—
243#123—4796no—
244#123—4748no—
245C22⋊Q8—4748no—
246C22⋊Q8—4748no—
247C22⋊Q8C22×D47148yes—
248#121—4748no—
249#121—4748no—
250#121—4748no—
251#120—4748no—
252C4×Q8—4716no—
253C4×Q8—4724no—
254C4×Q8—4724no—
255C4×Q8C22×D47116yes—
256C4×D4—4724no—
257C4×D4—4724no—
258C4×D4—4748no—
259C4×D4—4748no—
260C4×D4—4748no—
261C4×D4—4748no—
262#117—4748no—
263#117—4724no—
264#117—4724no—
265#117C22×D47112yes—
266#117C22×D47112yes—
267C2×C4⋊C4—4712no—
268C2×C4⋊C4C22×D4716yes—
269C2×C22⋊C4—4712no—
270C2×C22⋊C4C22×D4716yes—
271C2×C42—473no—
272C2×C42—476no—
273C2×C42—4712no—
274C2×C42—473no—
275C2×C42—4712no—
276C2×C42—4712no—
277C2×C42—4724no—
278C2×C42C22×D4716yes—
279C24—476no—
280C24C23×D4711yescharacteristic
281#41—9448no—
282#41—9448no—
283#41—18832no—
284C2×Q8—4712no—
285C2×Q8—9412no—
286C2×Q8C23×D47116yes—
287C2×D4—4724no—
288C2×D4—4712no—
289C2×D4—4748no—
290C2×D4—9424no—
291C2×D4—9412no—
292C22×C4—476no—
293C22×C4—476no—
294C22×C4—476no—
295C22×C4—4724no—
296C22×C4—4712no—
297C22×C4—4712no—
298C22×C4—4724no—
299C22×C4—9424no—
300C22×C4—18824no—
301C22×C4—1884no—
302C22×C4#1062535418yescomplemented
303C22×C4C23×D47112yes—
304C22×C4C23×D4713yes—
305C22×C4C23×D4713yes—
306C4⋊C4—4748no—
307C4⋊C4—4748no—
308C4⋊C4C23×D47124yes—
309C4⋊C4C23×D47148yes—
310C22⋊C4—4748no—
311C22⋊C4—4796no—
312C22⋊C4C23×D47124yes—
313C42—476no—
314C42—4724no—
315C42—476no—
316C42—4724no—
317C42—471no—
318C42—473no—
319C42C23×D47112yes—
320C42C23×D47112yes—
321C23—42no—
322C23—4712no—
323C23—9412no—
324C23—18812no—
325C23C2×#1062535414yes—
326C23C24×D4713yes—
327Q8—416no—
328Q8—9424no—
329D4—9448no—
330D4—9424no—
331D4—18848no—
332C2×C4—412no—
333C2×C4—472no—
334C2×C4—476no—
335C2×C4—4712no—
336C2×C4—9448no—
337C2×C4—9412no—
338C2×C4—9412no—
339C2×C4—9424no—
340C2×C4—9424no—
341C2×C4—18824no—
342C2×C4—1886no—
343C2×C4C2×#10625354112yes—
344C2×C4C2×#10625354112yes—
345C2×C4C24×D47124yes—
346C2×C4C24×D4713yes—
347C2×C4C24×D4711yescharacteristic
348C22—26no—
349C22—44no—
350C22—9412no—
351C22—9424no—
352C22—18824no—
353C22C22×#1062535416yes—
354C22C25×D4711yescharacteristic
355C4—224no—
356C4—474no—
357C4—9412no—
358C4—946no—
359C4C22×#1062535412yes—
360C4C22×#1062535416yes—
361C2—26no—
362C2—18812no—
363C2C23×#1062535411yescharacteristic
364C2#585×D4712yes—
subgroupG/H
centerC2×C4C24×D47
commutator, socle, cosocleC22×C47C26
FrattiniC22C25×D47
FittingC47×#3458C2
hypercenter#3458D47
hypocenterC47#62686
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ #3458

Complements

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

NH ≅ G/Ntype
1. #20256822C2semidirect
2. #20256811C2semidirect
3. #20256800C2semidirect
4. #20256780C2semidirect
5. #20256721C2semidirect
6. #20256685C2semidirect
7. C2×#15689591C2semidirect
8. C47×#3458C2semidirect
10. #15689747C22semidirect
11. #15689736C22semidirect
12. #15689729C22semidirect
13. #15689722C22semidirect
14. #15689713C22semidirect
16. #15689699C22semidirect
17. #15689692C22semidirect
19. #15689678C22semidirect
20. #15689674C22semidirect
22. #15689660C22semidirect
23. #15689659C22semidirect
24. #15689658C22semidirect
25. #15689636C22semidirect
26. #15689635C22semidirect
27. C4×#10625354C22semidirect
28. #15689591C22semidirect
29. #15689591C22semidirect
30. #15689591C22semidirect
31. C2×#13093989C22semidirect
32. C2×#13093986C22semidirect
34. C2×#13093981C22semidirect
36. C2×C42×D47C22semidirect
37. C47×#544C22semidirect
39. C47×#533C22semidirect
99. C22×C4×C47#41semidirect
182. #3458D47semidirect
192. #544C2×D47semidirect
204. #533C2×D47semidirect
223. C47#62686semidirect
302. C22×C4#10625354semidirect

Maximal subgroups

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

subgroupindex
#202568222
#202568112
#202568002
#202567802
#202567212
#202566852
C2×#156895912
C47×#34582
#6268647