Finite Group Databasebeta

← #26201583

Subgroups — #26201583

Order 12032.

602 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254333C211yesmaximal, characteristic
2C2×#15689408C211yesmaximal, characteristic
3#20252469C211yesmaximal, characteristic
4#20242848C214yesmaximal, complemented
5#20242756C214yesmaximal, complemented
6#20242164C211yesmaximal, characteristic
7#20242074C211yesmaximal, characteristic
8#20242045C211yesmaximal, characteristic
9C47×#3078C211yesmaximal, characteristic
10C2×#13094024C2211yescharacteristic
11#15689502C2212yes—
12#15689475C2212yes—
13#15689408—22no—
14#15689408C2214yes—
15#15689333—22no—
16#15689278—22no—
17Q8×C47⋊C8—22no—
18Q8×C47⋊C8C2214yes—
19#15689176—22no—
20#15689176C2211yescharacteristic
21#15689176C2211yescharacteristic
22C2×#13093955—21no—
23C2×C4⋊(C47⋊C8)C2211yescharacteristic
24C2×#13093948—21no—
25#15688875C2214yes—
26#15688872C2212yes—
27#15688869C2212yes—
28#15688868C2214yes—
29C47×#533C2211yescharacteristic
30C2×C4×Q8×C47C2211yescharacteristic
31C47×#479—22no—
32C47×#477C2212yes—
33C47×#475C2212yes—
34C47×#428—21no—
35C47×#423C2211yescharacteristic
36C47×#419—21no—
37C22×Dic94—21no—
38C2×#10625337—24no—
39C22×C47⋊C8—21no—
40#13094024—21no—
41#13094024—21no—
42#13094024—21no—
43#13094024C2312yes—
44C2×C4⋊Dic47—21no—
45C2×C4⋊Dic47—21no—
46#13093955—24no—
47C4⋊(C47⋊C8)—21no—
48C4⋊(C47⋊C8)—24no—
49C4⋊(C47⋊C8)D412yes—
50C4⋊(C47⋊C8)C2312yes—
51C4⋊(C47⋊C8)C2312yes—
52#13093948—24no—
53#13093947—24no—
54C22⋊(C47⋊C8)—21no—
55C22⋊(C47⋊C8)D412yes—
56C4×C47⋊C8—21no—
57C4×C47⋊C8—24no—
58C4×C47⋊C8C2312yes—
59C22×Q8×C47D411yescharacteristic
60C2×M16×C47—21no—
61C4⋊Q8×C47C2311yescharacteristic
62C4⋊Q8×C47C2311yescharacteristic
63#125×C47—21no—
64C22⋊Q8×C47—21no—
65C22⋊Q8×C47—21no—
66C4×Q8×C47—21no—
67C4×Q8×C47—21no—
68C4×Q8×C47—21no—
69C4×Q8×C47—21no—
70C4×Q8×C47D412yes—
71C4×Q8×C47C2312yes—
72#117×C47—21no—
73C2×C4⋊C4×C47—21no—
74C2×C4⋊C4×C47D411yescharacteristic
75C2×C42×C47—21no—
76C2×C42×C47C2311yescharacteristic
77#107×C47—22no—
78#106×C47—22no—
79C4⋊C8×C47—21no—
80C4⋊C8×C47C2312yes—
81#103×C47—22no—
82#103×C47—22no—
83C2×Q8×C47—21no—
84C2×Q8×C47—22no—
85C2×Q8×C47—21no—
86C2×Q8×C47—21no—
87C2×Q8×C47Q1614yescomplemented
88C2×Q8×C47C2×D412yes—
89C22×C4×C47—21no—
90C22×C4×C47—21no—
91C22×C4×C47C2×D411yescharacteristic
92C22×C4×C47C2×D411yescharacteristic
93C22×C4×C47#4111yescharacteristic
94C22×Dic47—41no—
95C2×Dic94—22no—
96C2×Dic94—41no—
97C2×Dic94—82no—
98M16×C47—42no—
99C2×C8×C47—22no—
100C4⋊C4×C47—22no—
101C4⋊C4×C47—21no—
102C4⋊C4×C47—22no—
103C4⋊C4×C47—21no—
104C4⋊C4×C47—21no—
105C4⋊C4×C47—21no—
106C4⋊C4×C47—21no—
107C4⋊C4×C47—41no—
108C4⋊C4×C47—41no—
109C4⋊C4×C47C2×D412yes—
110C22⋊C4×C47—41no—
111C42×C47—22no—
112C42×C47—22no—
113C42×C47—41no—
114C42×C47C2×D411yescharacteristic
115C42×C47C2×D411yescharacteristic
116C42×C47#4111yescharacteristic
117C42×C47C2411yescharacteristic
118#10625337—84no—
119C4⋊Dic47—22no—
120C4⋊Dic47—22no—
121C4⋊Dic47—41no—
122C4⋊Dic47—41no—
123C2×C47⋊C8—22no—
124C2×C47⋊C8—22no—
125C2×C47⋊C8—42no—
126C2×C47⋊C8C2×D414yes—
127C23×C47#12111yescharacteristic
128Q8×C47—22no—
129Q8×C47—24no—
130Q8×C47—42no—
131Q8×C47—81no—
132Q8×C47C2×Q1614yes—
133C2×C4×C47—22no—
134C2×C4×C47—24no—
135C2×C4×C47—22no—
136C2×C4×C47—22no—
137C2×C4×C47—41no—
138C2×C4×C47—41no—
139C2×C4×C47—41no—
140C2×C4×C47—81no—
141C2×C4×C47#12111yescharacteristic
142C2×C4×C47#12111yescharacteristic
143C2×C4×C47#12111yescharacteristic
144C2×C4×C47C2×Q1612yes—
145C2×C4×C47#13512yes—
146C2×C4×C47C22×D411yescharacteristic
147C2×C4×C47C22×D411yescharacteristic
148C2×C4×C47C2×#4111yescharacteristic
149C2×Dic47—42no—
150C2×Dic47—82no—
151Dic94—44no—
152Dic94—82no—
153C8×C47—42no—
154C47⋊C8—24no—
155C47⋊C8—42no—
156C47⋊C8—42no—
157#19399—471nomaximal, Sylow
158C22×C47—21no—
159C22×C47—21no—
160C22×C47#46712yes—
161C22×C47C2×#12111yescharacteristic
162C4×C47—21no—
163C4×C47—24no—
164C4×C47—21no—
165C4×C47—21no—
166C4×C47—42no—
167C4×C47—81no—
168C4×C47—81no—
169C4×C47#46712yes—
170C4×C47C22×Q1611yescharacteristic
171C4×C47C2×#13511yescharacteristic
172C4×C47#57712yes—
173Dic47—82no—
174#3209—471no—
175#3078D4711yescharacteristic
176C2×#475—471no—
177#2982—471no—
178#1684—474no—
179#1670—474no—
180#1571—471no—
181#1549—471no—
182#1542—471no—
183C2×C47—22no—
184C2×C47#304311yescharacteristic
185C2×C47C2×#46711yescharacteristic
186C2×C47#306911yescharacteristic
187#533C2×D4711yescharacteristic
188C2×C4⋊Q8—471no—
189C2×C4×Q8C2×D4711yescharacteristic
190#494—472no—
191#487—472no—
192#479—22no—
193#477C2×D4712yes—
194#475—474no—
195#475—942no—
196#475C2×D4712yes—
197#462—942no—
198#451—942no—
199C8×Q8—474no—
200C8×Q8—942no—
201#431—471no—
202#431—471no—
203#431—942no—
204#428—21no—
205C2×#106—941no—
206#423C2×D4711yescharacteristic
207C2×C4⋊C8—471no—
208#419—21no—
209C2×#103—941no—
210#335—474no—
211#333—472no—
212#332—472no—
213#332—474no—
214C47#1939911yescharacteristic, complemented, Sylow
215C22×Q8—941no—
216C22×Q8C22⋊D4711yescharacteristic
217C2×Q16—944no—
218C2×M16—21no—
219C22×C8—941no—
220C4⋊Q8—472no—
221C4⋊Q8—941no—
222C4⋊Q8—941no—
223C4⋊Q8—941no—
224C4⋊Q8C22×D4711yescharacteristic
225C4⋊Q8C22×D4711yescharacteristic
226#125—21no—
227C22⋊Q8—21no—
228C22⋊Q8—21no—
229C4×Q8—21no—
230C4×Q8—21no—
231C4×Q8—21no—
232C4×Q8—21no—
233C4×Q8C22⋊D4712yes—
234C4×Q8C22×D4712yes—
235#117—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—941no—
238C2×C4⋊C4—941no—
239C2×C4⋊C4C22⋊D4711yescharacteristic
240C2×C42—21no—
241C2×C42C22×D4711yescharacteristic
242#107—22no—
243#106—22no—
244#106—944no—
245C4⋊C8—21no—
246C4⋊C8—472no—
247C4⋊C8—472no—
248C4⋊C8—472no—
249C4⋊C8—944no—
250C4⋊C8—941no—
251C4⋊C8C22×D4712yes—
252#103—22no—
253#103—22no—
254#103—944no—
255#103—944no—
256C22⋊C8—472no—
257C22⋊C8—941no—
258C4×C8—472no—
259C4×C8—944no—
260C4×C8—941no—
261C2×Q8—21no—
262C2×Q8—22no—
263C2×Q8—21no—
264C2×Q8—21no—
265C2×Q8—942no—
266C2×Q8—3762no—
267C2×Q8—1881no—
268C2×Q8#1062533714yescomplemented
269C2×Q8C2×C22⋊D4712yes—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—1881no—
273C22×C4D4×D4711yescharacteristic
274C22×C4#1062535211yescharacteristic
275C22×C4C2×C22⋊D4711yescharacteristic
276Q16—3764no—
277M16—42no—
278C2×C8—22no—
279C2×C8—474no—
280C2×C8—942no—
281C2×C8—942no—
282C2×C8—1882no—
283C4⋊C4—22no—
284C4⋊C4—21no—
285C4⋊C4—22no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—942no—
293C4⋊C4—942no—
294C4⋊C4—1881no—
295C4⋊C4—1881no—
296C4⋊C4C2×C22⋊D4712yes—
297C22⋊C4—41no—
298C42—22no—
299C42—22no—
300C42—41no—
301C42D4×D4711yescharacteristic
302C42#1062535211yescharacteristic
303C42C2×C22⋊D4711yescharacteristic
304C42C23×D4711yescharacteristic
305C23#1309399611yescharacteristic
306Q8—24no—
307Q8—22no—
308Q8—42no—
309Q8—81no—
310Q8—3762no—
311Q8—1884no—
312Q8C2×#1062533714yes—
313C2×C4—22no—
314C2×C4—24no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—81no—
321C2×C4—3762no—
322C2×C4—1882no—
323C2×C4#1309399611yescharacteristic
324C2×C4#1309399611yescharacteristic
325C2×C4#1309399611yescharacteristic
326C2×C4C2×#1062533712yes—
327C2×C4#1309404412yes—
328C2×C4C2×D4×D4711yescharacteristic
329C2×C4C22×C22⋊D4711yescharacteristic
330C2×C4C2×#1062535211yescharacteristic
331C8—42no—
332C8—944no—
333C8—1882no—
334C8—1882no—
335C22—21no—
336C22—21no—
337C22#1568936212yes—
338C22C2×#1309399611yescharacteristic
339C4—24no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—42no—
344C4—81no—
345C4—81no—
346C4—3762no—
347C4#1568936212yes—
348C4C22×#1062533711yescharacteristic
349C4C2×#1309404411yescharacteristic
350C4#1568993212yes—
351C2—22no—
352C2#2025291111yescharacteristic
353C2C2×#1568936211yescharacteristic
354C2#2025312711yescharacteristic
subgroupG/H
centerC22C2×#13093996
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#3078C2
socleC22×C47C2×#121
prime radicalC23×C47#121
hypercenter#3078D47
hypocenterC47#19399
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3078

Complements

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

NH ≅ G/Ntype
4. #20242848C2semidirect
5. #20242756C2semidirect
87. C2×Q8×C47Q16semidirect
214. C47#19399semidirect
268. C2×Q8#10625337semidirect

Maximal subgroups

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

subgroupindex
#202543332
C2×#156894082
#202524692
#202428482
#202427562
#202421642
#202420742
#202420452
C47×#30782
#1939947