Finite Group Databasebeta

← #26182661

Subgroups — #26182661

Order 12032.

1290 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13094070C211yesmaximal, characteristic
2#20252499C211yesmaximal, characteristic, complemented
3#20241780C213yesmaximal, complemented
4#20241776C213yesmaximal, complemented
5#20241771C213yesmaximal, complemented
6C2×#15688852C213yesmaximal, complemented
7C8×#41×C47C211yesmaximal, characteristic, complemented
8C2×#13094070C411yescharacteristic
9#15689606C411yescharacteristic
10#15689605—21no—
11#15689603C413yes—
12#15689597C413yes—
13#15689595—23no—
14C4×#10625354—21no—
15C4×#10625350C2213yes—
16C2×C4×D188C2213yes—
17C4×D4×D47—23no—
18Q8×C47⋊C8—21no—
19D4×C47⋊C8—23no—
20#15689176C2213yescomplemented
21C2×C4⋊(C47⋊C8)C2213yescomplemented
22#15688855C2211yescharacteristic, complemented
23#15688855C2213yescomplemented
24#15688852C2213yescomplemented
25#15688852C2216yescomplemented
26#15688852C2213yescomplemented
27C4×#41×C47C2211yescharacteristic
28C8×Q8×C47—21no—
29C8×D4×C47—23no—
30C47×#431C2213yescomplemented
31C2×C4×C8×C47C2213yescomplemented
32#13094070—24no—
33#13094070C818yescomplemented
34C2×#10625354—21no—
35C2×#10625350C2×C413yes—
36C22×D188C2×C413yes—
37C2×D4×D47—23no—
38C22×C4×D47—23no—
39C22×C47⋊C8—23no—
40Q8×Dic47—21no—
41C4×Dic94C2×C413yes—
42C4×Dic94C2311yescharacteristic
43C4×D188—23no—
44C4×D188—23no—
45C4×D188C2×C416yes—
46C4×D188C2×C413yes—
47C4×D188C2313yes—
48D4×Dic47—23no—
49#13093986—23no—
50#13093986—23no—
51C4×C22⋊D47—23no—
52C4×C22⋊D47—23no—
53(C22⋊D47)⋊C4—26no—
54#13093983C2×C413yes—
55#13093982—23no—
56#13093981—23no—
57C2×C4⋊Dic47C2×C413yes—
58C4⋊C4×D47—23no—
59C2×(C2×D47)⋊C4—23no—
60C22⋊C4×D47—26no—
61C42×D47—23no—
62C4⋊(C47⋊C8)—23no—
63C4⋊(C47⋊C8)C2313yes—
64C4⋊(C47⋊C8)C2311yescharacteristic
65C22⋊(C47⋊C8)—23no—
66C4×C47⋊C8—23no—
67C2×#41×C47C2×C411yescharacteristic
68C22×C8×C47—23no—
69C4×Q8×C47D411yescharacteristic
70C4×Q8×C47D411yescharacteristic
71C4×D4×C47D413yes—
72C4×D4×C47D413yes—
73#117×C47C2×C413yes—
74C2×C42×C47C2313yes—
75C4⋊C8×C47—23no—
76C22⋊C8×C47—23no—
77C4×C8×C47—23no—
78C4×C8×C47C2311yescharacteristic
79C4×C8×C47C2313yes—
80C23×D47—23no—
81#41×C47C2×C814yes—
82#41×C47M1614yes—
83C2×Q8×C47C22⋊C411yescharacteristic
84C2×D4×C47C22⋊C413yes—
85C22×C4×C47C22×C413yes—
86C22×C4×C47C2×D413yes—
87C22×C4×C47C2×D413yes—
88C2×C22⋊D47—23no—
89C22×Dic47—23no—
90#10625354—28no—
91D4×D47—224no—
92#10625350—26no—
93#10625350C2×C8112yescomplemented
94C2×D188—23no—
95C2×D188—26no—
96C2×D188C2×C8112yescomplemented
97C2×D188C22×C413yes—
98C2×C4×D47—23no—
99C2×C4×D47—23no—
100C2×C4×D47—23no—
101C2×C4×D47—26no—
102C2×C4×D47—83no—
103C2×C4×D47—83no—
104C2×Dic94C22×C411yescharacteristic
105C2×C8×C47—21no—
106C2×C8×C47—23no—
107C2×C8×C47—46no—
108C4⋊C4×C47C22⋊C413yes—
109C4⋊C4×C47C22⋊C413yes—
110C22⋊C4×C47C22⋊C416yes—
111C42×C47C22×C413yes—
112C42×C47C2×D413yes—
113C42×C47C2×D413yes—
114C42×C47C2411yescharacteristic
115C22⋊Dic47—23no—
116(C2×D47)⋊C4—23no—
117(C2×D47)⋊C4—26no—
118(C2×D47)⋊C4—23no—
119C4⋊Dic47—23no—
120C4⋊Dic47C22×C413yes—
121C4⋊Dic47C22×C411yescharacteristic
122Dic47⋊C4—23no—
123Dic47⋊C4—21no—
124C4×Dic47—23no—
125C4×Dic47—23no—
126C4×Dic47—21no—
127C2×C47⋊C8—21no—
128C2×C47⋊C8—23no—
129C2×C47⋊C8—83no—
130C23×C47C2×C22⋊C413yes—
131C22×D47—23no—
132C22×D47—212no—
133C22×D47—83no—
134C22×D47—83no—
135Q8×C47C22⋊C814yes—
136D4×C47C22⋊C8112yes—
137C2×C4×C47—23no—
138C2×C4×C47—23no—
139C2×C4×C47—43no—
140C2×C4×C47C2×C22⋊C413yes—
141C2×C4×C47C2×C22⋊C413yes—
142C2×C4×C47C2×C22⋊C413yes—
143C2×C4×C47C22×C816yes—
144C2×C4×C47C2×M1616yes—
145C2×C4×C47C23×C411yescharacteristic
146C2×C4×C47C22×D411yescharacteristic
147C2×C4×C47C22×D411yescharacteristic
148C22⋊D47—212no—
149C2×Dic47—21no—
150C2×Dic47—21no—
151C2×Dic47—23no—
152C2×Dic47—83no—
153D188—23no—
154D188—212no—
155D188C22×C816yes—
156C4×D47—212no—
157C4×D47—43no—
158C4×D47—43no—
159C4×D47—83no—
160C4×D47—86no—
161Dic94—21no—
162Dic94C22×C812yes—
163C8×C47—22no—
164C8×C47—43no—
165C47⋊C8—43no—
166C47⋊C8—41no—
167#17438—471nomaximal, Sylow
168C22×C47—23no—
169C22×C47—23no—
170C22×C47C2×C22⋊C816yes—
171C22×C47C22×C22⋊C411yescharacteristic
172C2×D47—26no—
173C2×D47—43no—
174C2×D47—43no—
175C2×D47—812no—
176C4×C47—23no—
177C4×C47—21no—
178C4×C47—23no—
179C4×C47C2×C22⋊C816yes—
180C4×C47C23×C811yescharacteristic
181C4×C47C22×M1611yescharacteristic
182C4×C47#57611yescharacteristic
183C4×C47#57711yescharacteristic
184Dic47—22no—
185Dic47—43no—
186Dic47—41no—
187C4×H(5,𝔽2)—471no—
188#2988—471no—
189C8×#41D4711yescharacteristic, complemented
190#1478—473no—
191#1477—473no—
192#1476—473no—
193C2×D4⋊C8—473no—
194C2×C47—26no—
195C2×C47C22×C22⋊C811yescharacteristic
196C2×C47#289711yescharacteristic
197C2×C47#290111yescharacteristic
198D47—46no—
199C2×H(5,𝔽2)—471no—
200#520—471no—
201#520—473no—
202#520—941no—
203#518—473no—
204#518—943no—
205C4×#41—473no—
206C4×#41—941no—
207C4×#41C2×D4711yescharacteristic
208C2×C4×D4—473no—
209C2×C4×D4—943no—
210C8×Q8—21no—
211C8×Q8—941no—
212C8×D4—23no—
213C8×D4—943no—
214#431—473no—
215#431C2×D4713yescomplemented
216C2×C4⋊C8—473no—
217C2×C4×C8C2×D4713yescomplemented
218Q8⋊C8—471no—
219Q8⋊C8—473no—
220D4⋊C8—473no—
221D4⋊C8—476no—
222D4⋊C8—473no—
223C47#1743811yescharacteristic, complemented, Sylow
224H(5,𝔽2)—478no—
225H(5,𝔽2)—944no—
226C2×#41—473no—
227C2×#41—941no—
228C2×#41C4×D4711yescharacteristic
229C22×D4—473no—
230C22×D4—943no—
231C23×C4—943no—
232C22×C8—23no—
233C22×C8—943no—
234C4×Q8—473no—
235C4×Q8—471no—
236C4×Q8—941no—
237C4×Q8D18811yescharacteristic
238C4×Q8C22⋊D4711yescharacteristic
239C4×D4—476no—
240C4×D4—473no—
241C4×D4—473no—
242C4×D4—943no—
243C4×D4—943no—
244C4×D4—943no—
245C4×D4—943no—
246C4×D4—943no—
247C4×D4—946no—
248C4×D4—943no—
249C4×D4—943no—
250C4×D4D18813yes—
251C4×D4C22⋊D4713yes—
252#117—473no—
253#117—943no—
254#117—943no—
255#117C4×D4713yes—
256C2×C4⋊C4—473no—
257C2×C4⋊C4—943no—
258C2×C22⋊C4—943no—
259C2×C22⋊C4—946no—
260C2×C42—943no—
261C2×C42C22×D4713yes—
262C4⋊C8—23no—
263C4⋊C8—473no—
264C4⋊C8—471no—
265C4⋊C8—943no—
266C22⋊C8—23no—
267C22⋊C8—943no—
268C4×C8—23no—
269C4×C8—943no—
270C4×C8C22×D4711yescharacteristic
271C4×C8C22×D4713yes—
272C24—943no—
273#41—4712no—
274#41—948no—
275#41—946no—
276#41C8×D4714yes—
277#41#1062532414yes—
278C2×Q8—471no—
279C2×Q8(C2×D47)⋊C411yescharacteristic
280C2×D4—4712no—
281C2×D4—473no—
282C2×D4—943no—
283C2×D4—9424no—
284C2×D4—943no—
285C2×D4—946no—
286C2×D4(C2×D47)⋊C413yes—
287C22×C4—943no—
288C22×C4—943no—
289C22×C4—943no—
290C22×C4—946no—
291C22×C4—943no—
292C22×C4—3763no—
293C22×C4—3763no—
294C22×C4C2×C4×D4713yes—
295C22×C4C2×D18813yes—
296C22×C4C2×C22⋊D4713yes—
297C2×C8—21no—
298C2×C8—23no—
299C2×C8—46no—
300C2×C8—941no—
301C2×C8—943no—
302C2×C8—3763no—
303C4⋊C4—473no—
304C4⋊C4—471no—
305C4⋊C4—943no—
306C4⋊C4—943no—
307C4⋊C4—941no—
308C4⋊C4(C2×D47)⋊C413yes—
309C4⋊C4(C2×D47)⋊C413yes—
310C22⋊C4—943no—
311C22⋊C4—943no—
312C22⋊C4—946no—
313C22⋊C4—943no—
314C22⋊C4(C2×D47)⋊C416yes—
315C42—943no—
316C42—943no—
317C42—941no—
318C42C2×C4×D4713yes—
319C42C2×D18813yes—
320C42C2×C22⋊D4713yes—
321C42C23×D4711yescharacteristic
322C23—9412no—
323C23—943no—
324C23—3763no—
325C23—3763no—
326C23C2×(C2×D47)⋊C413yes—
327Q8—472no—
328Q8—941no—
329Q8(C2×D47)⋊C814yes—
330D4—476no—
331D4—9412no—
332D4—9412no—
333D4—943no—
334D4(C2×D47)⋊C8112yes—
335C2×C4—23no—
336C2×C4—23no—
337C2×C4—43no—
338C2×C4—941no—
339C2×C4—9412no—
340C2×C4—941no—
341C2×C4—943no—
342C2×C4—3763no—
343C2×C4—3766no—
344C2×C4—3763no—
345C2×C4—1883no—
346C2×C4—1883no—
347C2×C4C2×(C2×D47)⋊C413yes—
348C2×C4C2×(C2×D47)⋊C413yes—
349C2×C4C2×(C2×D47)⋊C413yes—
350C2×C4C2×C8×D4716yes—
351C2×C4C2×#1062532416yes—
352C2×C4C22×C4×D4711yescharacteristic
353C2×C4C22×C22⋊D4711yescharacteristic
354C2×C4C22×D18811yescharacteristic
355C8—22no—
356C8—43no—
357C8—1883no—
358C8—1881no—
359C22—23no—
360C22—23no—
361C22—946no—
362C22—37612no—
363C22—1883no—
364C22—1883no—
365C22C2×(C2×D47)⋊C816yes—
366C22C22×(C2×D47)⋊C411yescharacteristic
367C4—23no—
368C4—21no—
369C4—23no—
370C4—942no—
371C4—1883no—
372C4—1881no—
373C4C2×(C2×D47)⋊C816yes—
374C4C22×C8×D4711yescharacteristic
375C4C22×#1062532411yescharacteristic
376C4#1568992811yescharacteristic
377C4#1568993511yescharacteristic
378C2—26no—
379C2—1886no—
380C2C22×(C2×D47)⋊C811yescharacteristic
381C2#2025191511yescharacteristic
382C2#2025193611yescharacteristic
subgroupG/H
centerC2×C4C22×D188
commutatorC4×C47C23×C8
FrattiniC42C23×D47
FittingC8×#41×C47C2
socleC22×C47C22×C22⋊C4
prime radicalC2×#13094070C4
cosocleC42×C47C24
hypercenterC8×#41D47
hypocenterC47#17438
LCSG ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ C8×#41

Complements

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

NH ≅ G/Ntype
2. #20252499C2semidirect
3. #20241780C2semidirect
4. #20241776C2semidirect
5. #20241771C2semidirect
6. C2×#15688852C2semidirect
7. C8×#41×C47C2semidirect
20. #15689176C22semidirect
21. C2×C4⋊(C47⋊C8)C22semidirect
22. #15688855C22semidirect
23. #15688855C22semidirect
24. #15688852C22semidirect
25. #15688852C22semidirect
26. #15688852C22semidirect
30. C47×#431C22semidirect
31. C2×C4×C8×C47C22semidirect
33. #13094070C8semidirect
93. #10625350C2×C8semidirect
96. C2×D188C2×C8semidirect
189. C8×#41D47semidirect
215. #431C2×D47semidirect
217. C2×C4×C8C2×D47semidirect
223. C47#17438semidirect

Maximal subgroups

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

subgroupindex
C4×#130940702
#202524992
#202417802
#202417762
#202417712
C2×#156888522
C8×#41×C472
#1743847