Finite Group Databasebeta

← #26201563

Subgroups — #26201563

Order 12032.

656 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
1#20254315C211yesmaximal, characteristic
2C2×#15689408C211yesmaximal, characteristic, complemented
3#20252860C211yesmaximal, characteristic
4#20242818C214yesmaximal, complemented
5#20242736C214yesmaximal, complemented
6#20242159C211yesmaximal, characteristic, complemented
7#20242072C211yesmaximal, characteristic, complemented
8#20242035C211yesmaximal, characteristic
9C47×#2982C211yesmaximal, characteristic, complemented
10C2×#13094055—21no—
11#15689673C2211yescharacteristic
12C2×#13094024C2211yescharacteristic
13#15689496C2212yes—
14#15689474C2212yes—
15#15689408—22no—
16#15689408C2214yes—
17#15689345—22no—
18#15689333C2212yes—
19#15689329—22no—
20#15689323C2212yes—
21#15689264—22no—
22C2×#13093955—21no—
23#15689147C2211yescharacteristic
24C2×C4⋊(C47⋊C8)C2211yescharacteristic
25#15689122—21no—
26C2×#13093948—21no—
27#15688874C2214yes—
28#15688870C2214yes—
29#15688868C2212yes—
30#15688866C2212yes—
31C2×C4×Q8×C47C2211yescharacteristic
32C8×Q8×C47—22no—
33C8×Q8×C47C2214yes—
34C47×#431—22no—
35C47×#431C2211yescharacteristic
36C47×#431C2211yescharacteristic
37C2×#10625350—21no—
38C22×Dic94—21no—
39#13094055—44no—
40C2×#10625337—22no—
41C2×#10625325—24no—
42C2×#10625336—22no—
43C2×#10625329—21no—
44C22×C47⋊C8—21no—
45#13094024—21no—
46#13094024—21no—
47#13094024—21no—
48#13094024C2311yescharacteristic
49#13094024C2312yes—
50#13094021C2311yescharacteristic
51#13094012—21no—
52#13093998—21no—
53#13093998—21no—
54C4×D188—21no—
55C4×D188—21no—
56C2×C4⋊Dic47—21no—
57C2×C4⋊Dic47—21no—
58#13093955—24no—
59C4⋊(C47⋊C8)—21no—
60C4⋊(C47⋊C8)—21no—
61C4⋊(C47⋊C8)C2312yes—
62C4⋊(C47⋊C8)C2312yes—
63#13093949—24no—
64#13093948—24no—
65#13093947—22no—
66#13093944—22no—
67C22×Q8×C47D411yescharacteristic
68C4×Q8×C47—21no—
69C4×Q8×C47—21no—
70C4×Q8×C47D412yes—
71C4×Q8×C47C2312yes—
72C2×C4⋊C4×C47—21no—
73C2×C4⋊C4×C47D411yescharacteristic
74C2×C42×C47—21no—
75C2×C42×C47C2311yescharacteristic
76C4⋊C8×C47—24no—
77C4⋊C8×C47D412yes—
78C4⋊C8×C47C2312yes—
79C22⋊C8×C47—21no—
80C22⋊C8×C47D412yes—
81C4×C8×C47—24no—
82C4×C8×C47—21no—
83C4×C8×C47C2312yes—
84C2×Q8×C47—21no—
85C2×Q8×C47—22no—
86C2×Q8×C47SD1614yescomplemented
87C2×Q8×C47C2×D412yes—
88C22×C4×C47—21no—
89C22×C4×C47—21no—
90C22×C4×C47C2×D411yescharacteristic
91C22×C4×C47C2×D411yescharacteristic
92C22×C4×C47#4111yescharacteristic
93C2×C22⋊D47—41no—
94C22×Dic47—41no—
95#10625350—42no—
96#10625350—42no—
97C2×D188—21no—
98C2×D188—21no—
99C2×D188—21no—
100C2×C4×D47—41no—
101C2×Dic94—21no—
102C2×Dic94—22no—
103C2×Dic94—41no—
104C2×Dic94—82no—
105C2×C8×C47—22no—
106C2×C8×C47C2×D414yes—
107C4⋊C4×C47—22no—
108C4⋊C4×C47—21no—
109C4⋊C4×C47—22no—
110C4⋊C4×C47C2×D412yes—
111C42×C47—22no—
112C42×C47—22no—
113C42×C47C2×D411yescharacteristic
114C42×C47C2×D411yescharacteristic
115C42×C47#4111yescharacteristic
116C42×C47C2411yescharacteristic
117#10625337—44no—
118#10625336—44no—
119(C2×D47)⋊C4—41no—
120(C2×D47)⋊C4—41no—
121C4⋊Dic47—22no—
122C4⋊Dic47—21no—
123C4⋊Dic47—22no—
124C4⋊Dic47—21no—
125C4⋊Dic47—41no—
126C4⋊Dic47—41no—
127#10625329—42no—
128C2×C47⋊C8—22no—
129C2×C47⋊C8—22no—
130C2×C47⋊C8—42no—
131#10625325—84no—
132C23×C47#12111yescharacteristic
133C22×D47—41no—
134Q8×C47—22no—
135Q8×C47—24no—
136Q8×C47C2×SD1614yes—
137C2×C4×C47—22no—
138C2×C4×C47—24no—
139C2×C4×C47—22no—
140C2×C4×C47—22no—
141C2×C4×C47—41no—
142C2×C4×C47—81no—
143C2×C4×C47#12111yescharacteristic
144C2×C4×C47#12111yescharacteristic
145C2×C4×C47#12111yescharacteristic
146C2×C4×C47C2×SD1612yes—
147C2×C4×C47#13512yes—
148C2×C4×C47C22×D411yescharacteristic
149C2×C4×C47C22×D411yescharacteristic
150C2×C4×C47C2×#4111yescharacteristic
151C22⋊D47—82no—
152C2×Dic47—42no—
153C2×Dic47—41no—
154C2×Dic47—82no—
155D188—41no—
156D188—41no—
157D188—42no—
158D188—81no—
159C4×D47—82no—
160Dic94—41no—
161Dic94—41no—
162Dic94—44no—
163Dic94—82no—
164C8×C47—24no—
165C8×C47—42no—
166C47⋊C8—42no—
167C47⋊C8—42no—
168#19397—471nomaximal, Sylow
169C22×C47—21no—
170C22×C47—21no—
171C22×C47#46512yes—
172C22×C47C2×#12111yescharacteristic
173C2×D47—81no—
174C2×D47—81no—
175C2×D47—81no—
176C4×C47—21no—
177C4×C47—24no—
178C4×C47—21no—
179C4×C47—21no—
180C4×C47—42no—
181C4×C47#46512yes—
182C4×C47C22×SD1611yescharacteristic
183C4×C47C2×#13511yescharacteristic
184C4×C47#57712yes—
185Dic47—82no—
186Dic47—81no—
187#3206—471no—
188C2×#475—471no—
189#3037—471no—
190#2982D4711yescharacteristic, complemented
191#1680—474no—
192#1666—474no—
193#1570—471no—
194#1549—471no—
195#1540—471no—
196C2×C47—22no—
197C2×C47#304311yescharacteristic
198C2×C47C2×#46511yescharacteristic
199C2×C47#306911yescharacteristic
200D47—161no—
201C2×#137—941no—
202#532—471no—
203C2×C4⋊Q8—471no—
204C2×C4×Q8C2×D4711yescharacteristic
205#492—472no—
206#487—472no—
207#475—474no—
208#475—942no—
209#464—942no—
210#462—472no—
211#461—942no—
212#460—472no—
213#448—942no—
214C8×Q8—22no—
215C8×Q8C2×D4714yes—
216#431—22no—
217#431C2×D4711yescharacteristic
218#431C2×D4711yescharacteristic
219C2×#106—941no—
220#423—471no—
221C2×C4⋊C8—471no—
222#417—941no—
223C2×#103—941no—
224#334—474no—
225#332—472no—
226#332—474no—
227#331—472no—
228C47#1939711yescharacteristic, complemented, Sylow
229C2×#41—941no—
230C22×Q8—941no—
231C22×Q8D18811yescharacteristic
232#137—1884no—
233C2×Q16—942no—
234C2×SD16—942no—
235C2×SD16—944no—
236C2×M16—941no—
237C22×C8—941no—
238C4⋊Q8—472no—
239C4⋊Q8—471no—
240C4⋊Q8—941no—
241C4⋊Q8—941no—
242C4⋊Q8—941no—
243#127—471no—
244#124—941no—
245#121—941no—
246#121—941no—
247C4×Q8—21no—
248C4×Q8—21no—
249C4×Q8D18812yes—
250C4×Q8C22×D4712yes—
251C4×D4—941no—
252C4×D4—941no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—941no—
255C2×C4⋊C4—941no—
256C2×C4⋊C4D18811yescharacteristic
257C2×C42—21no—
258C2×C42C22×D4711yescharacteristic
259#106—944no—
260C4⋊C8—24no—
261C4⋊C8—472no—
262C4⋊C8—472no—
263C4⋊C8—941no—
264C4⋊C8—941no—
265C4⋊C8C22⋊D4712yes—
266C4⋊C8C22×D4712yes—
267#103—944no—
268#103—944no—
269#103—942no—
270#102—942no—
271C22⋊C8—21no—
272C22⋊C8C22⋊D4712yes—
273C4×C8—24no—
274C4×C8—21no—
275C4×C8C22×D4712yes—
276#41—1882no—
277#41—1882no—
278C2×Q8—21no—
279C2×Q8—22no—
280C2×Q8—942no—
281C2×Q8—941no—
282C2×Q8—3762no—
283C2×Q8—1881no—
284C2×Q8#1062532514yescomplemented
285C2×Q8C2×D18812yes—
286C2×D4—941no—
287C2×D4—941no—
288C2×D4—941no—
289C2×D4—1881no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—1881no—
293C22×C4—1881no—
294C22×C4C2×D18811yescharacteristic
295C22×C4#1062535011yescharacteristic
296C22×C4C2×C22⋊D4711yescharacteristic
297Q16—1884no—
298SD16—3764no—
299SD16—1884no—
300M16—1882no—
301C2×C8—22no—
302C2×C8—942no—
303C2×C8—942no—
304C2×C8—1882no—
305C2×C8C2×C22⋊D4714yes—
306C4⋊C4—22no—
307C4⋊C4—21no—
308C4⋊C4—22no—
309C4⋊C4—941no—
310C4⋊C4—942no—
311C4⋊C4—942no—
312C4⋊C4—941no—
313C4⋊C4—1881no—
314C4⋊C4—1881no—
315C4⋊C4C2×D18812yes—
316C22⋊C4—1881no—
317C22⋊C4—1881no—
318C42—22no—
319C42—22no—
320C42C2×D18811yescharacteristic
321C42#1062535011yescharacteristic
322C42C2×C22⋊D4711yescharacteristic
323C42C23×D4711yescharacteristic
324C23—1881no—
325C23#1309399811yescharacteristic
326Q8—24no—
327Q8—22no—
328Q8—3762no—
329Q8—1881no—
330Q8—1881no—
331Q8—1884no—
332Q8C2×#1062532514yes—
333D4—3761no—
334D4—3762no—
335D4—1881no—
336D4—1881no—
337D4—1882no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—81no—
344C2×C4—3762no—
345C2×C4—3762no—
346C2×C4—1881no—
347C2×C4—1882no—
348C2×C4#1309399811yescharacteristic
349C2×C4#1309399811yescharacteristic
350C2×C4#1309399811yescharacteristic
351C2×C4C2×#1062532512yes—
352C2×C4#1309404612yes—
353C2×C4C22×C22⋊D4711yescharacteristic
354C2×C4C22×D18811yescharacteristic
355C2×C4C2×#1062535011yescharacteristic
356C8—24no—
357C8—42no—
358C8—1882no—
359C8—1882no—
360C22—21no—
361C22—21no—
362C22—3761no—
363C22—3761no—
364C22—3761no—
365C22#1568935212yes—
366C22C2×#1309399811yescharacteristic
367C4—24no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—42no—
372C4—3761no—
373C4—3762no—
374C4#1568935212yes—
375C4C22×#1062532511yescharacteristic
376C4C2×#1309404611yescharacteristic
377C4#1568993112yes—
378C2—22no—
379C2—7521no—
380C2#2025290911yescharacteristic
381C2C2×#1568935211yescharacteristic
382C2#2025312511yescharacteristic
subgroupG/H
centerC22C2×#13093998
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#2982C2
socleC22×C47C2×#121
prime radical#15689673C22
hypercenter#2982D47
hypocenterC47#19397
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
2. C2×#15689408C2semidirect
4. #20242818C2semidirect
5. #20242736C2semidirect
6. #20242159C2semidirect
7. #20242072C2semidirect
9. C47×#2982C2semidirect
86. C2×Q8×C47SD16semidirect
190. #2982D47semidirect
228. C47#19397semidirect
284. C2×Q8#10625325semidirect

Maximal subgroups

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

subgroupindex
#202543152
C2×#156894082
#202528602
#202428182
#202427362
#202421592
#202420722
#202420352
C47×#29822
#1939747