Finite Group Databasebeta

← #26133697

Subgroups — #26133697

Order 12032.

456 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20256491C211yesmaximal, characteristic, complemented
2#20253108C211yesmaximal, characteristic
3C2×#15688932C211yesmaximal, characteristic, complemented
4#20245115C211yesmaximal, characteristic, complemented
5#20241501C211yesmaximal, characteristic, complemented
6#20241486C211yesmaximal, characteristic, complemented
7C47×#1815C211yesmaximal, characteristic, complemented
8#15689921—21no—
9C2×#13094046C2211yescharacteristic
10#15689540—22no—
11#15689540—22no—
12C2×#13093964—22no—
13C2×#13093964C2211yescharacteristic
14C2×#13093964C2211yescharacteristic
15#15689398C412yescomplemented
16#15689393C412yescomplemented
17#15689386—22no—
18C2×#13093960C2211yescharacteristic
19#15689105—21no—
20#15688932—22no—
21#15688929—22no—
22C2×#131×C47C2211yescharacteristic
23C2×#108×C47C2211yescharacteristic
24C2×#106×C47C2211yescharacteristic
25C47×#417—21no—
26C47×#343—21no—
27C2×#10625350—21no—
28#13094055—41no—
29#13094054—22no—
30#13094050—41no—
31#13094049—22no—
32#13094046—21no—
33#13094046—21no—
34#13094046—21no—
35#13094046—21no—
36#13094044—42no—
37C2×Dic188C2×C411yescharacteristic
38C2×#10625325—21no—
39C2×D376C2×C411yescharacteristic
40C2×#10625329—21no—
41#13093964—24no—
42#13093964—21no—
43#13093964—21no—
44#13093964—21no—
45#13093964D412yes—
46C2×C47⋊C16—24no—
47C2×C47⋊C16—21no—
48C2×C47⋊C16D412yes—
49#13093960—21no—
50#13093960C2×C412yes—
51#13093942—22no—
52#13093940—22no—
53C2×#41×C47D411yescharacteristic
54#131×C47—21no—
55#131×C47—21no—
56#131×C47C2×C412yes—
57#131×C47D412yes—
58C2×M16×C47—21no—
59C2×M16×C47—21no—
60C2×M16×C47D411yescharacteristic
61C22×C8×C47—21no—
62C22×C8×C47C2311yescharacteristic
63C2×C4⋊C4×C47—21no—
64#108×C47—21no—
65#108×C47—21no—
66#106×C47—21no—
67#106×C47—21no—
68#103×C47—41no—
69#102×C47—41no—
70#41×C47—21no—
71#41×C47—21no—
72#41×C47C22⋊C412yes—
73#41×C47SD1612yes—
74C2×Q8×C47D811yescharacteristic
75C2×D4×C47—21no—
76C2×D4×C47D811yescharacteristic
77C22×C4×C47—21no—
78C22×C4×C47—41no—
79C22×C4×C47C2×D411yescharacteristic
80C2×C22⋊D47—41no—
81#10625350—41no—
82#10625350—41no—
83#10625350—41no—
84#10625350—41no—
85C2×D188—21no—
86C2×C4×D47—41no—
87C2×Dic94—21no—
88M16×C47—21no—
89M16×C47—22no—
90M16×C47—22no—
91M16×C47—41no—
92M16×C47—41no—
93M16×C47C22⋊C412yes—
94C2×C8×C47—21no—
95C2×C8×C47—22no—
96C2×C8×C47—22no—
97C2×C8×C47—41no—
98C2×C8×C47C22×C411yescharacteristic
99C2×C8×C47C2×D411yescharacteristic
100C2×C8×C47C2×D411yescharacteristic
101C2×C8×C47C2×D411yescharacteristic
102C2×C8×C47#4111yescharacteristic
103C2×C8×C47#4111yescharacteristic
104C4⋊C4×C47—41no—
105C4⋊C4×C47—41no—
106#10625337—42no—
107#10625336—42no—
108#10625335—42no—
109#10625334—42no—
110#10625329—22no—
111#10625329—41no—
112C2×C47⋊C8—41no—
113Dic188—21no—
114Dic188—21no—
115D376—21no—
116D376—21no—
117#10625325—41no—
118#10625325—41no—
119C47⋊C16—22no—
120C47⋊C16—22no—
121C23×C47—21no—
122C23×C47#13511yescharacteristic
123C22×D47—41no—
124Q8×C47—21no—
125Q8×C47#10212yes—
126D4×C47—22no—
127D4×C47—22no—
128D4×C47—21no—
129D4×C47#10212yes—
130C2×C4×C47—21no—
131C2×C4×C47—22no—
132C2×C4×C47—22no—
133C2×C4×C47—81no—
134C2×C4×C47—161no—
135C2×C4×C47C2×C22⋊C411yescharacteristic
136C2×C4×C47#12111yescharacteristic
137C2×C4×C47#12111yescharacteristic
138C2×C4×C47C2×D811yescharacteristic
139C2×C4×C47C2×SD1611yescharacteristic
140C2×C4×C47#13511yescharacteristic
141C22⋊D47—81no—
142C22⋊D47—81no—
143C2×Dic47—41no—
144D188—41no—
145D188—41no—
146C4×D47—81no—
147C4×D47—81no—
148Dic94—41no—
149Dic94—41no—
150C8×C47—22no—
151C8×C47—81no—
152C8×C47C4×D411yescharacteristic
153C8×C47C4×D411yescharacteristic
154C8×C47#12011yescharacteristic
155C8×C47#12311yescharacteristic
156C47⋊C8—42no—
157#13070—471nomaximal, Sylow
158C22×C47—22no—
159C22×C47—41no—
160C22×C47—81no—
161C22×C47#41611yescharacteristic
162C22×C47#46511yescharacteristic
163C22×C47#46611yescharacteristic
164C2×D47—81no—
165C2×D47—161no—
166C4×C47—22no—
167C4×C47—161no—
168C4×C47#38611yescharacteristic
169C4×C47C2×#10211yescharacteristic
170C4×C47#46511yescharacteristic
171C4×C47#46611yescharacteristic
172Dic47—81no—
173#3407—471no—
174#3067—471no—
175C2×#360—471no—
176#2143—471no—
177#1815D4711yescharacteristic, complemented
178#1388—471no—
179#1383—471no—
180C2×C47—21no—
181C2×C47—21no—
182C2×C47—21no—
183C2×C47—42no—
184C2×C47#189311yescharacteristic
185D47—161no—
186#575—941no—
187C2×#135—471no—
188C2×#131C2×D4711yescharacteristic
189#504—942no—
190#504—942no—
191C2×M32—471no—
192C2×M32—471no—
193C2×M32—942no—
194#473—472no—
195#472—472no—
196#471—942no—
197C2×#108—471no—
198C2×#108C2×D4711yescharacteristic
199C2×#106C2×D4711yescharacteristic
200#417—21no—
201#413—941no—
202#360—942no—
203#359—942no—
204#343—21no—
205C47#1307011yescharacteristic, complemented, Sylow
206C2×#41—941no—
207C2×#41C22⋊D4711yescharacteristic
208#137—942no—
209#137—1881no—
210AGL(1,ℤ/8)—942no—
211AGL(1,ℤ/8)—1881no—
212#135—941no—
213#135—941no—
214#135—941no—
215#135—941no—
216#135—1882no—
217C2×Q16—471no—
218C2×SD16—941no—
219C2×D8—471no—
220#131—21no—
221#131—21no—
222#131C4×D4712yes—
223#131D18812yes—
224C2×M16—21no—
225C2×M16—21no—
226C2×M16—941no—
227C2×M16C22⋊D4711yescharacteristic
228C22×C8—21no—
229C22×C8C22×D4711yescharacteristic
230C2×C4⋊C4—21no—
231M32—472no—
232M32—944no—
233M32—941no—
234M32—941no—
235M32—941no—
236C2×C16—472no—
237C2×C16—944no—
238C2×C16—941no—
239#108—21no—
240#108—21no—
241#108—472no—
242#108—941no—
243#106—21no—
244#106—21no—
245#103—41no—
246#102—41no—
247#101—942no—
248#100—942no—
249#41—21no—
250#41—21no—
251#41—1881no—
252#41—1881no—
253#41—1881no—
254#41—1881no—
255#41(C2×D47)⋊C412yes—
256#41#1062533612yes—
257C2×Q8—941no—
258C2×Q8#1062533411yescharacteristic
259C2×D4—21no—
260C2×D4—941no—
261C2×D4—1881no—
262C2×D4#1062533411yescharacteristic
263C22×C4—21no—
264C22×C4—41no—
265C22×C4—1881no—
266C22×C4C2×C22⋊D4711yescharacteristic
267Q16—941no—
268Q16—941no—
269Q16—1882no—
270SD16—1882no—
271SD16—1882no—
272SD16—1881no—
273SD16—1881no—
274D8—941no—
275D8—941no—
276D8—1882no—
277M16—21no—
278M16—22no—
279M16—22no—
280M16—41no—
281M16—41no—
282M16—942no—
283M16—1881no—
284M16(C2×D47)⋊C412yes—
285C2×C8—21no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—41no—
289C2×C8—1881no—
290C2×C8C2×C4×D4711yescharacteristic
291C2×C8C2×D18811yescharacteristic
292C2×C8D4×D4711yescharacteristic
293C2×C8D4×D4711yescharacteristic
294C2×C8#1062535211yescharacteristic
295C2×C8#1062535211yescharacteristic
296C4⋊C4—41no—
297C4⋊C4—41no—
298C16—942no—
299C16—942no—
300C23—21no—
301C23—1881no—
302C23#1309404411yescharacteristic
303Q8—21no—
304Q8—1881no—
305Q8—1881no—
306Q8#1309394412yes—
307D4—22no—
308D4—22no—
309D4—21no—
310D4—3761no—
311D4—3761no—
312D4—1881no—
313D4—1881no—
314D4#1309394412yes—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—81no—
319C2×C4—161no—
320C2×C4—3761no—
321C2×C4—3761no—
322C2×C4—1881no—
323C2×C4C2×(C2×D47)⋊C411yescharacteristic
324C2×C4#1309399611yescharacteristic
325C2×C4#1309399611yescharacteristic
326C2×C4C2×#1062533411yescharacteristic
327C2×C4C2×#1062533611yescharacteristic
328C2×C4#1309404411yescharacteristic
329C8—22no—
330C8—81no—
331C8—1882no—
332C8(C22⋊D47)⋊C411yescharacteristic
333C8(C22⋊D47)⋊C411yescharacteristic
334C8#1309399411yescharacteristic
335C8#1309400811yescharacteristic
336C22—22no—
337C22—41no—
338C22—81no—
339C22—3761no—
340C22—7521no—
341C22#1568911711yescharacteristic
342C22#1568935411yescharacteristic
343C22#1568935711yescharacteristic
344C4—22no—
345C4—161no—
346C4—3761no—
347C4#1568900011yescharacteristic
348C4C2×#1309394411yescharacteristic
349C4#1568935411yescharacteristic
350C4#1568935711yescharacteristic
351C2—21no—
352C2—21no—
353C2—21no—
354C2—42no—
355C2—7521no—
356C2#2024381411yescharacteristic
subgroupG/H
centerC2#20243814
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
FittingC47×#1815C2
socleC2×C47#1893
prime radical#20253108C2
cosocleC22×C8×C47C23
hypercenter#1815D47
hypocenterC47#13070
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ #1815

Complements

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

NH ≅ G/Ntype
1. #20256491C2semidirect
3. C2×#15688932C2semidirect
4. #20245115C2semidirect
5. #20241501C2semidirect
6. #20241486C2semidirect
7. C47×#1815C2semidirect
15. #15689398C4semidirect
16. #15689393C4semidirect
177. #1815D47semidirect
205. C47#13070semidirect

Maximal subgroups

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

subgroupindex
#202564912
#202531082
C2×#156889322
#202451152
#202415012
#202414862
C47×#18152
#1307047