Finite Group Databasebeta

← #26201641

Subgroups — #26201641

Order 12032.

590 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254536C211yesmaximal, characteristic
2C2×#15689418C211yesmaximal, characteristic
3#20252469C211yesmaximal, characteristic
4#20242844C214yesmaximal, complemented
5#20242790C214yesmaximal, complemented
6#20242164C211yesmaximal, characteristic
7#20242070C211yesmaximal, characteristic
8#20242045C211yesmaximal, characteristic
9C47×#3078C211yesmaximal, characteristic
10C2×#13094024C2211yescharacteristic
11#15689526C2212yes—
12#15689473C2212yes—
13#15689456—22no—
14#15689418—22no—
15#15689418—22no—
16#15689418C2214yes—
17Q8×C47⋊C8—22no—
18Q8×C47⋊C8C2214yes—
19#15689176—22no—
20#15689176C2211yescharacteristic
21#15689176C2211yescharacteristic
22C2×#13093957—21no—
23C2×C4⋊(C47⋊C8)C2211yescharacteristic
24C2×#13093948—21no—
25#15688875C2214yes—
26#15688872C2212yes—
27#15688871C2214yes—
28#15688869C2212yes—
29C47×#533C2211yescharacteristic
30C2×C4×Q8×C47C2211yescharacteristic
31C47×#479—22no—
32C47×#477C2212yes—
33C47×#475C2212yes—
34C47×#428—21no—
35C47×#423C2211yescharacteristic
36C47×#419—21no—
37C22×Dic94—21no—
38C22×C47⋊C8—21no—
39#13094024—21no—
40#13094024—21no—
41#13094024—21no—
42#13094024C2312yes—
43C2×C4⋊Dic47—21no—
44C2×C4⋊Dic47—21no—
45#13093957—24no—
46#13093957—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#13093948—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×C47Q1614yes—
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
118C4⋊Dic47—22no—
119C4⋊Dic47—22no—
120C4⋊Dic47—41no—
121C4⋊Dic47—41no—
122C2×C47⋊C8—22no—
123C2×C47⋊C8—22no—
124C2×C47⋊C8—42no—
125C2×C47⋊C8C2×D414yes—
126C23×C47#12111yescharacteristic
127Q8×C47—22no—
128Q8×C47—24no—
129Q8×C47—42no—
130Q8×C47—81no—
131Q8×C47C2×Q1614yes—
132C2×C4×C47—22no—
133C2×C4×C47—24no—
134C2×C4×C47—22no—
135C2×C4×C47—22no—
136C2×C4×C47—41no—
137C2×C4×C47—41no—
138C2×C4×C47—41no—
139C2×C4×C47—81no—
140C2×C4×C47#12111yescharacteristic
141C2×C4×C47#12111yescharacteristic
142C2×C4×C47#12111yescharacteristic
143C2×C4×C47C2×Q1612yes—
144C2×C4×C47#13512yes—
145C2×C4×C47C22×D411yescharacteristic
146C2×C4×C47C22×D411yescharacteristic
147C2×C4×C47C2×#4111yescharacteristic
148C2×Dic47—42no—
149C2×Dic47—82no—
150Dic94—82no—
151Dic94—82no—
152C8×C47—42no—
153C47⋊C8—24no—
154C47⋊C8—42no—
155C47⋊C8—42no—
156#19405—471nomaximal, Sylow
157C22×C47—21no—
158C22×C47—21no—
159C22×C47#46712yes—
160C22×C47C2×#12111yescharacteristic
161C4×C47—21no—
162C4×C47—24no—
163C4×C47—21no—
164C4×C47—21no—
165C4×C47—42no—
166C4×C47—81no—
167C4×C47—81no—
168C4×C47#46712yes—
169C4×C47C22×Q1611yescharacteristic
170C4×C47C2×#13511yescharacteristic
171C4×C47#57812yes—
172Dic47—82no—
173#3230—471no—
174#3078D4711yescharacteristic
175C2×#477—471no—
176#2982—471no—
177#1684—474no—
178#1676—474no—
179#1571—471no—
180#1548—471no—
181#1542—471no—
182C2×C47—22no—
183C2×C47#304411yescharacteristic
184C2×C47C2×#46711yescharacteristic
185C2×C47#306811yescharacteristic
186#533C2×D4711yescharacteristic
187C2×C4⋊Q8—471no—
188C2×C4×Q8C2×D4711yescharacteristic
189#500—472no—
190#487—472no—
191#484—942no—
192#479—22no—
193#477—474no—
194#477—942no—
195#477—942no—
196#477C2×D4712yes—
197#475C2×D4712yes—
198C8×Q8—474no—
199C8×Q8—942no—
200#431—471no—
201#431—471no—
202#431—942no—
203#428—21no—
204C2×#107—941no—
205#423C2×D4711yescharacteristic
206C2×C4⋊C8—471no—
207#419—21no—
208C2×#103—941no—
209#335—474no—
210#333—472no—
211#333—474no—
212#332—472no—
213C47#1940511yescharacteristic, complemented, Sylow
214C22×Q8—941no—
215C22×Q8C22⋊D4711yescharacteristic
216C2×M16—21no—
217C22×C8—941no—
218C4⋊Q8—472no—
219C4⋊Q8—941no—
220C4⋊Q8—941no—
221C4⋊Q8—941no—
222C4⋊Q8C22×D4711yescharacteristic
223C4⋊Q8C22×D4711yescharacteristic
224#125—21no—
225C22⋊Q8—21no—
226C22⋊Q8—21no—
227C4×Q8—21no—
228C4×Q8—21no—
229C4×Q8—21no—
230C4×Q8—21no—
231C4×Q8C22⋊D4712yes—
232C4×Q8C22×D4712yes—
233#117—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—941no—
236C2×C4⋊C4—941no—
237C2×C4⋊C4C22⋊D4711yescharacteristic
238C2×C42—21no—
239C2×C42C22×D4711yescharacteristic
240#107—22no—
241#107—944no—
242#107—944no—
243#106—22no—
244C4⋊C8—21no—
245C4⋊C8—472no—
246C4⋊C8—472no—
247C4⋊C8—472no—
248C4⋊C8—944no—
249C4⋊C8—941no—
250C4⋊C8C22×D4712yes—
251#103—22no—
252#103—22no—
253#103—944no—
254#103—944no—
255C22⋊C8—472no—
256C22⋊C8—941no—
257C4×C8—472no—
258C4×C8—944no—
259C4×C8—941no—
260C2×Q8—21no—
261C2×Q8—22no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—942no—
265C2×Q8—3762no—
266C2×Q8—1881no—
267C2×Q8#1062533714yes—
268C2×Q8C2×C22⋊D4712yes—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—1881no—
272C22×C4D4×D4711yescharacteristic
273C22×C4#1062535211yescharacteristic
274C22×C4C2×C22⋊D4711yescharacteristic
275M16—42no—
276C2×C8—22no—
277C2×C8—474no—
278C2×C8—942no—
279C2×C8—942no—
280C2×C8—1882no—
281C4⋊C4—22no—
282C4⋊C4—21no—
283C4⋊C4—22no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—942no—
291C4⋊C4—942no—
292C4⋊C4—1881no—
293C4⋊C4—1881no—
294C4⋊C4C2×C22⋊D4712yes—
295C22⋊C4—41no—
296C42—22no—
297C42—22no—
298C42—41no—
299C42D4×D4711yescharacteristic
300C42#1062535211yescharacteristic
301C42C2×C22⋊D4711yescharacteristic
302C42C23×D4711yescharacteristic
303C23#1309399611yescharacteristic
304Q8—24no—
305Q8—22no—
306Q8—42no—
307Q8—81no—
308Q8—3762no—
309Q8—3762no—
310Q8C2×#1062533714yes—
311C2×C4—22no—
312C2×C4—24no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—81no—
319C2×C4—3762no—
320C2×C4—1882no—
321C2×C4#1309399611yescharacteristic
322C2×C4#1309399611yescharacteristic
323C2×C4#1309399611yescharacteristic
324C2×C4C2×#1062533712yes—
325C2×C4#1309404412yes—
326C2×C4C2×D4×D4711yescharacteristic
327C2×C4C22×C22⋊D4711yescharacteristic
328C2×C4C2×#1062535211yescharacteristic
329C8—42no—
330C8—944no—
331C8—1882no—
332C8—1882no—
333C22—21no—
334C22—21no—
335C22#1568936212yes—
336C22C2×#1309399611yescharacteristic
337C4—24no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—42no—
342C4—81no—
343C4—81no—
344C4—3762no—
345C4#1568936212yes—
346C4C22×#1062533711yescharacteristic
347C4C2×#1309404411yescharacteristic
348C4#1568993912yes—
349C2—22no—
350C2#2025292011yescharacteristic
351C2C2×#1568936211yescharacteristic
352C2#2025311811yescharacteristic
subgroupG/H
centerC22C2×#13093996
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#3078C2
socleC22×C47C2×#121
prime radicalC23×C47#121
hypercenter#3078D47
hypocenterC47#19405
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. #20242844C2semidirect
5. #20242790C2semidirect
213. C47#19405semidirect

Maximal subgroups

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

subgroupindex
#202545362
C2×#156894182
#202524692
#202428442
#202427902
#202421642
#202420702
#202420452
C47×#30782
#1940547