Finite Group Databasebeta

← #27085816

Subgroups — #27085816

Order 12032.

3250 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689931C214yesmaximal, complemented
2#20257166C211yesmaximal, characteristic, complemented
3#20253866C2112yesmaximal, complemented
4#20253845C213yesmaximal, complemented
5#20253841C213yesmaximal, complemented
6#20253814C213yesmaximal, complemented
7C2×#15689497C213yesmaximal, complemented
8C4×#13094031C211yesmaximal, characteristic, complemented
9C47×#3499C211yesmaximal, characteristic, complemented
10C2×#13094070C2212yes—
11#15689931—232no—
12C2×#13094055C22112yescomplemented
13C2×#13094049C22112yescomplemented
14C2×#13094044C22112yescomplemented
15C22×#10625336C22112yescomplemented
16C2×#13094031C2212yes—
17#15689795—22no—
18D4×D188—26no—
19#15689692C2216yes—
20#15689673C2213yes—
21C2×#13094021C2213yes—
22#15689515C22112yescomplemented
23#15689513C22112yescomplemented
24#15689505C22112yescomplemented
25#15689502C2213yescomplemented
26#15689499C2213yescomplemented
27#15689497C2213yescomplemented
28#15689497C2216yescomplemented
29#15689497C22112yescomplemented
30#15689495C2211yescharacteristic, complemented
31#15689079C2216yescomplemented
32C4×#10625329C2213yescomplemented
33C2×C4×C47⋊C8C2213yescomplemented
34C2×2−1+4×C47C2212yes—
35D4×Q8×C47—22no—
36C47×#547—26no—
37C47×#536C2216yescomplemented
38C47×#532C2213yescomplemented
39C2×C4⋊Q8×C47C2213yescomplemented
40C4×#41×C47C2211yescharacteristic
41#13094070—216no—
42C2×#10625354—22no—
43C2×#10625350C2316yes—
44C22×D188—26no—
45C22×D188C2316yes—
46C2×D4×D47—26no—
47#13094055—248no—
48#13094049—248no—
49#13094044—248no—
50C2×#10625337C23112yes—
51C2×#10625335C2314yes—
52C2×#10625336—248no—
53C2×#10625336C23112yes—
54C2×#10625336C23124yes—
55C2×#10625334C23112yes—
56#13094031D4116yescomplemented
57C2×#10625329C2316yes—
58C22×C47⋊C8C2316yes—
59#13094024C2311yescharacteristic
60#13094021—26no—
61#13094021C2313yes—
62#13094021C2316yes—
63#13094012C2316yes—
64#13093999—212no—
65#13093998—26no—
66#13093994—212no—
67C4×D188—26no—
68#13093935C2316yes—
69C4×C47⋊C8C2311yescharacteristic
70C4×C47⋊C8C2313yes—
71C4×C47⋊C8C2316yes—
722−1+4×C47—216no—
73C2×#41×C47—26no—
74C2×#41×C47C2316yes—
75C2×#41×C47C2311yescharacteristic
76C22×Q8×C47—22no—
77C22×Q8×C47C2316yes—
78C4⋊Q8×C47—26no—
79C4⋊Q8×C47C2313yes—
80C4⋊Q8×C47C2316yes—
81#127×C47C2311yescharacteristic
82#124×C47C2316yes—
83#123×C47—212no—
84C22⋊Q8×C47—212no—
85#121×C47—26no—
86C4×Q8×C47D412yes—
87C4×D4×C47—26no—
88C4×D4×C47D416yes—
89#117×C47C2313yes—
90C2×C4⋊C4×C47—26no—
91C2×C42×C47C2313yes—
92C23×D47—26no—
93#41×C47—224no—
94#41×C47—424no—
95#41×C47C2×D418yes—
96C2×Q8×C47—26no—
97C2×Q8×C47—224no—
98C2×Q8×C47—48no—
99C2×Q8×C47C2×D411yescharacteristic
100C2×Q8×C47C2416yes—
101C2×D4×C47—26no—
102C2×D4×C47—22no—
103C2×D4×C47C2×D413yes—
104C2×D4×C47C2412yes—
105C22×C4×C47—26no—
106C22×C4×C47—26no—
107C22×C4×C47C2×D416yes—
108C22×C4×C47C2413yes—
109C2×C22⋊D47—26no—
110#10625354—48no—
111D4×D47—424no—
112#10625350—224no—
113C2×D188—224no—
114C2×D188—26no—
115C2×D188—26no—
116C2×D188—212no—
117C2×D188—812no—
118C2×D188C2416yes—
119C2×C4×D47—26no—
120C2×Dic94C2412yes—
121C4⋊C4×C47—26no—
122C4⋊C4×C47—212no—
123C4⋊C4×C47C2×D416yes—
124C22⋊C4×C47—212no—
125C22⋊C4×C47C2×D416yes—
126C42×C47C2×D416yes—
127C42×C47C2411yescharacteristic
128C42×C47C2413yes—
129#10625337—224no—
130#10625336—224no—
131#10625336—248no—
132#10625335—28no—
133#10625334—224no—
134(C2×D47)⋊C4—212no—
135C4⋊Dic47—22no—
136#10625329C2×D4124yescomplemented
137C2×C47⋊C8C2×D4124yescomplemented
138C2×C47⋊C8C2412yes—
139C2×C47⋊C8C2416yes—
140C23×C47—22no—
141C23×C47C22×D413yes—
142C22×D47—26no—
143C22×D47—412no—
144C22×D47—812no—
145Q8×C47—212no—
146Q8×C47—412no—
147Q8×C47#12714yes—
148D4×C47—24no—
149D4×C47—412no—
150D4×C47—44no—
151D4×C47#127112yes—
152C2×C4×C47—212no—
153C2×C4×C47—26no—
154C2×C4×C47—412no—
155C2×C4×C47—412no—
156C2×C4×C47C22×D4112yes—
157C2×C4×C47C22×D416yes—
158C2×C4×C47C22×D413yes—
159C2×C4×C47C22×D412yes—
160C2×C4×C47C2511yescharacteristic
161C22⋊D47—412no—
162C2×Dic47—22no—
163D188—212no—
164D188—412no—
165D188—412no—
166D188—812no—
167C4×D47—412no—
168Dic94—24no—
169C47⋊C8C22×D414yes—
170C47⋊C8C22×D4112yes—
171#60646—471nomaximal, Sylow
172C22×C47—26no—
173C22×C47—42no—
174C22×C47—44no—
175C22×C47C2×#12716yes—
176C22×C47C23×D411yescharacteristic
177C2×D47—46no—
178C2×D47—412no—
179C2×D47—1612no—
180C4×C47—26no—
181C4×C47—46no—
182C4×C47C2×#12716yes—
183C4×C47#57714yescomplemented
184C4×C47C23×D412yes—
185Dic47—42no—
186C2×#577—474no—
187#3499D4711yescharacteristic, complemented
188#3498—471no—
189#3154—4712no—
190#3152—473no—
191#3151—473no—
192#3147—473no—
193C2×#492—473no—
194C4×#131—471no—
195C2×C47—26no—
196C2×C47—82no—
197C2×C47C22×#12711yescharacteristic
198C2×C47C2×#57712yes—
199D47—86no—
200C2×2−1+4C2×D4712yes—
201C2×H(5,𝔽2)—472no—
202#577—9432no—
203C2×#137—4712no—
204C2×AGL(1,ℤ/8)—4712no—
205C2×#135—4712no—
206C22×SD16—4712no—
207C2×#131—472no—
208#550—942no—
209D4×Q8—22no—
210#547—26no—
211D42—946no—
212#536C2×D4716yescomplemented
213#535—476no—
214#532—473no—
215#532C2×D4713yescomplemented
216C2×C4⋊Q8C2×D4713yescomplemented
217C2×#127—473no—
218C4×#41C2×D4711yescharacteristic
219#497—4712no—
220#496—4712no—
221#495—4712no—
222#494—473no—
223#493—473no—
224#492—473no—
225#492—476no—
226#492—4712no—
227#492—471no—
228#405—476no—
229C4×M16—473no—
230C2×C4×C8—473no—
231C47#6064611yescharacteristic, complemented, Sylow
2322−1+4—216no—
233H(5,𝔽2)—9416no—
234C2×#41—26no—
235C2×#41—476no—
236C2×#41—942no—
237C2×#41C22×D4716yes—
238C2×#41C22×D4711yescharacteristic
239C22×Q8—22no—
240C22×Q8C22×D4716yes—
241C22×D4—476no—
242C22×D4—946no—
243C22×D4—946no—
244#137—9448no—
245AGL(1,ℤ/8)—9448no—
246#135—9448no—
247C2×Q16—4712no—
248C2×SD16—4712no—
249C2×SD16—4724no—
250C2×SD16—474no—
251C2×SD16—9448no—
252C2×D8—4712no—
253#131—4716no—
254C2×M16—476no—
255C22×C8—476no—
256C4⋊Q8—26no—
257C4⋊Q8—471no—
258C4⋊Q8C22×D4713yes—
259C4⋊Q8C22×D4716yes—
260#127—473no—
261#127—476no—
262#127—946no—
263#127C22×D4711yescharacteristic
264#124—476no—
265#124C22×D4716yes—
266#123—212no—
267C22⋊Q8—212no—
268#121—26no—
269#121—946no—
270#121—9412no—
271#120—9412no—
272C4×Q8C22⋊D4712yes—
273C4×D4—26no—
274C4×D4—946no—
275C4×D4C22⋊D4716yes—
276#117C22×D4713yes—
277C2×C4⋊C4—26no—
278C2×C42C22×D4713yes—
279#97—476no—
280C4×C8—471no—
281C4×C8—473no—
282C4×C8—476no—
283C24—946no—
284#41—224no—
285#41—424no—
286#41—9424no—
287#41—1888no—
288#41D4×D4718yes—
289C2×Q8—26no—
290C2×Q8—224no—
291C2×Q8—48no—
292C2×Q8—472no—
293C2×Q8C2×C22⋊D4711yescharacteristic
294C2×Q8C23×D4716yes—
295C2×D4—26no—
296C2×D4—22no—
297C2×D4—476no—
298C2×D4—946no—
299C2×D4—9424no—
300C2×D4—9412no—
301C2×D4—946no—
302C2×D4—946no—
303C2×D4—37612no—
304C2×D4—18824no—
305C2×D4C2×C22⋊D4713yes—
306C2×D4C23×D4712yes—
307C22×C4—26no—
308C22×C4—26no—
309C22×C4—946no—
310C22×C4C2×C22⋊D4716yes—
311C22×C4C23×D4713yes—
312Q16—9424no—
313SD16—948no—
314SD16—9424no—
315SD16—9448no—
316D8—9424no—
317M16—4724no—
318C2×C8—4724no—
319C2×C8—472no—
320C2×C8—476no—
321C4⋊C4—26no—
322C4⋊C4—212no—
323C4⋊C4—942no—
324C4⋊C4C2×C22⋊D4716yes—
325C22⋊C4—212no—
326C22⋊C4—9412no—
327C22⋊C4C2×C22⋊D4716yes—
328C42C2×C22⋊D4716yes—
329C42C23×D4711yescharacteristic
330C42C23×D4713yes—
331C23—22no—
332C23—946no—
333C23—37612no—
334C23—18812no—
335C23C22×C22⋊D4713yes—
336Q8—212no—
337Q8—412no—
338Q8—944no—
339Q8#1309402014yes—
340D4—24no—
341D4—412no—
342D4—44no—
343D4—9412no—
344D4—37612no—
345D4—18812no—
346D4—18812no—
347D4—18812no—
348D4#13094020112yes—
349C2×C4—212no—
350C2×C4—26no—
351C2×C4—412no—
352C2×C4—412no—
353C2×C4—942no—
354C2×C4—18812no—
355C2×C4C2×D4×D47112yes—
356C2×C4C22×C22⋊D4716yes—
357C2×C4C22×C22⋊D4713yes—
358C2×C4C22×C22⋊D4712yes—
359C2×C4C24×D4711yescharacteristic
360C8—474no—
361C8—4712no—
362C22—26no—
363C22—44no—
364C22—42no—
365C22—18812no—
366C22—1886no—
367C22—75212no—
368C22C2×#1309402016yes—
369C22C23×C22⋊D4711yescharacteristic
370C4—26no—
371C4—46no—
372C4—1882no—
373C4C2×#1309402016yes—
374C4#1568993114yescomplemented
375C4C22×D4×D4712yes—
376C2—26no—
377C2—82no—
378C2—3766no—
379C2C22×#1309402011yescharacteristic
380C2C2×#1568993112yes—
subgroupG/H
centerC22C23×C22⋊D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC47×#3499C2
socleC22×C47C23×D4
hypercenter#3499D47
hypocenterC47#60646
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C4×#41 ◁ #3499

Complements

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

NH ≅ G/Ntype
1. C2×#15689931C2semidirect
2. #20257166C2semidirect
3. #20253866C2semidirect
4. #20253845C2semidirect
5. #20253841C2semidirect
6. #20253814C2semidirect
7. C2×#15689497C2semidirect
8. C4×#13094031C2semidirect
9. C47×#3499C2semidirect
12. C2×#13094055C22semidirect
13. C2×#13094049C22semidirect
14. C2×#13094044C22semidirect
15. C22×#10625336C22semidirect
22. #15689515C22semidirect
23. #15689513C22semidirect
24. #15689505C22semidirect
25. #15689502C22semidirect
26. #15689499C22semidirect
27. #15689497C22semidirect
28. #15689497C22semidirect
29. #15689497C22semidirect
30. #15689495C22semidirect
31. #15689079C22semidirect
32. C4×#10625329C22semidirect
33. C2×C4×C47⋊C8C22semidirect
37. C47×#536C22semidirect
38. C47×#532C22semidirect
39. C2×C4⋊Q8×C47C22semidirect
56. #13094031D4semidirect
136. #10625329C2×D4semidirect
137. C2×C47⋊C8C2×D4semidirect
183. C4×C47#577semidirect
187. #3499D47semidirect
212. #536C2×D47semidirect
215. #532C2×D47semidirect
216. C2×C4⋊Q8C2×D47semidirect
231. C47#60646semidirect
374. C4#15689931semidirect

Maximal subgroups

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

subgroupindex
C2×#156899312
#202571662
#202538662
#202538452
#202538412
#202538142
C2×#156894972
C4×#130940312
C47×#34992
#6064647