Finite Group Databasebeta

← C22×#15688856 #26183355

Subgroups — C22×#15688856 #26183355

Order 12032.

2190 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093998C211yesmaximal, characteristic
2C22×C22⋊(C47⋊C8)C211yesmaximal, characteristic, complemented
3C2×#15688856C2112yesmaximal, direct product factor
4C22×#95×C47C211yesmaximal, characteristic, complemented
5C23×D188C411yescharacteristic, complemented
6C23×C22⋊D47—21no—
7C23×C47⋊C8—21no—
8C2×#13093998—23no—
9C2×#13093998—23no—
10C2×#13093998C416yescomplemented
11C2×#13093998C2216yes—
12C22×C4⋊Dic47C411yescharacteristic, complemented
13C22×(C2×D47)⋊C4—21no—
14C2×C22⋊(C47⋊C8)—23no—
15C2×C22⋊(C47⋊C8)C2216yescomplemented
16#15688856C22116yesdirect product factor
17C24×C4×C47—21no—
18C24×C4×C47C2211yescharacteristic
19C2×#95×C47—23no—
20C2×#95×C47C2216yescomplemented
21C24×D47—21no—
22C22×D188—23no—
23C22×D188—44no—
24C22×D188C2×C416yescomplemented
25C22×C22⋊D47—26no—
26C22×C22⋊D47—26no—
27C22×C22⋊D47—44no—
28C22×C22⋊D47—44no—
29C23×Dic47—21no—
30C22×C47⋊C8—26no—
31C22×C47⋊C8—44no—
32#13093998—212no—
33#13093998—26no—
34#13093998—26no—
35#13093998C2×C4112yescomplemented
36#13093998C2314yes—
37C2×C4⋊Dic47—23no—
38C2×C4⋊Dic47C2×C416yescomplemented
39C2×(C2×D47)⋊C4—26no—
40C2×(C2×D47)⋊C4—26no—
41C22⋊(C47⋊C8)—26no—
42C22⋊(C47⋊C8)C2314yescomplemented
43C25×C47D411yescharacteristic
44C23×C4×C47—28no—
45C23×C4×C47—26no—
46C23×C4×C47—44no—
47C23×C4×C47—44no—
48C23×C4×C47—44no—
49C23×C4×C47C2×C411yescharacteristic
50C23×C4×C47C2×C413yes—
51C23×C4×C47D411yescharacteristic
52C23×C4×C47D413yes—
53C23×C4×C47D413yes—
54C23×C4×C47C2313yes—
55#95×C47—26no—
56#95×C47C2314yescomplemented
57C24×C47—24no—
58C24×C47—24no—
59C24×C47C22⋊C413yes—
60C24×C47C22⋊C411yescharacteristic
61C24×C47D814yes—
62C24×C47SD1614yes—
63C24×C47C2×D413yes—
64C23×D47—26no—
65C23×D47—44no—
66C23×D47—44no—
67C23×D47—44no—
68C22×C4×C47—24no—
69C22×C4×C47—24no—
70C22×C4×C47—224no—
71C22×C4×C47—24no—
72C22×C4×C47—46no—
73C22×C4×C47—46no—
74C22×C4×C47—46no—
75C22×C4×C47—46no—
76C22×C4×C47—432no—
77C22×C4×C47—412no—
78C22×C4×C47—412no—
79C22×C4×C47C22⋊C413yes—
80C22×C4×C47C22⋊C413yes—
81C22×C4×C47C22⋊C416yes—
82C22×C4×C47C22×C413yes—
83C22×C4×C47C22×C413yes—
84C22×C4×C47C2×D413yes—
85C22×C4×C47C2×D413yes—
86C22×C4×C47C2×D413yes—
87C22×C4×C47C2411yescharacteristic
88C2×C22⋊D47—212no—
89C2×C22⋊D47—24no—
90C2×C22⋊D47—424no—
91C2×C22⋊D47—412no—
92C2×C22⋊D47—412no—
93C22×Dic47—26no—
94C22×Dic47—44no—
95C2×D188—26no—
96C2×D188—412no—
97C2×D188—412no—
98C2×D188C22×C414yescomplemented
99(C2×D47)⋊C4—212no—
100(C2×D47)⋊C4—24no—
101C4⋊Dic47—26no—
102C4⋊Dic47C22×C414yescomplemented
103C2×C47⋊C8—24no—
104C2×C47⋊C8—412no—
105C23×C47—216no—
106C23×C47—216no—
107C23×C47—26no—
108C23×C47—26no—
109C23×C47—26no—
110C23×C47—26no—
111C23×C47#9914yes—
112C23×C47#102112yes—
113C23×C47#10214yes—
114C23×C47#10414yes—
115C23×C47C2×C22⋊C413yes—
116C23×C47C2×C22⋊C413yes—
117C23×C47C2×D816yes—
118C23×C47C2×SD1616yes—
119C23×C47C22×D411yescharacteristic
120C22×D47—24no—
121C22×D47—412no—
122C22×D47—412no—
123C22×D47—412no—
124C22×D47—816no—
125C2×C4×C47—26no—
126C2×C4×C47—26no—
127C2×C4×C47—26no—
128C2×C4×C47—26no—
129C2×C4×C47—28no—
130C2×C4×C47—46no—
131C2×C4×C47—43no—
132C2×C4×C47—43no—
133C2×C4×C47—43no—
134C2×C4×C47—41no—
135C2×C4×C47—44no—
136C2×C4×C47—448no—
137C2×C4×C47—44no—
138C2×C4×C47C2×C22⋊C413yes—
139C2×C4×C47C2×C22⋊C413yes—
140C2×C4×C47C23×C411yescharacteristic
141C2×C4×C47C22×D411yescharacteristic
142C22⋊D47—44no—
143C22⋊D47—44no—
144C22⋊D47—424no—
145C2×Dic47—24no—
146C2×Dic47—412no—
147D188—44no—
148D188—412no—
149C47⋊C8—44no—
150C22×#327—471nomaximal, Sylow
151C22×C47—21no—
152C22×C47—23no—
153C22×C47—21no—
154C22×C47—23no—
155C22×C47—212no—
156C22×C47—224no—
157C22×C47—212no—
158C22×C47#327116yescomplemented
159C22×C47C2×#9916yes—
160C22×C47C2×#10216yes—
161C22×C47C2×#10216yes—
162C22×C47C2×#10416yes—
163C22×C47C22×C22⋊C411yescharacteristic
164C22×C47C22×D811yescharacteristic
165C22×C47C22×SD1611yescharacteristic
166C2×D47—44no—
167C2×D47—44no—
168C2×D47—44no—
169C2×D47—824no—
170C4×C47—21no—
171C4×C47—23no—
172C4×C47—21no—
173C4×C47—23no—
174C4×C47—48no—
175Dic47—44no—
176C22×#121—471no—
177C22×C22⋊C8—471no—
178C22×#95D4711yescharacteristic, complemented
179C2×#327—4712no—
180C2×C47—21no—
181C2×C47—21no—
182C2×C47—26no—
183C2×C47C2×#327112yescomplemented
184C2×C47C22×#9911yescharacteristic
185C2×C47C22×#10211yescharacteristic
186C2×C47C22×#10411yescharacteristic
187D47—84no—
188C23×D4—471no—
189C23×D4—941no—
190C24×C4—21no—
191C24×C4C2×D4711yescharacteristic
192C23×C8—941no—
193C2×#121—476no—
194C2×#121—476no—
195C2×#121—943no—
196C2×#121—943no—
197C22×C4⋊C4—471no—
198C22×C22⋊C4—941no—
199C2×C22⋊C8—476no—
200C2×C22⋊C8—943no—
201C2×#95—23no—
202C2×#95C2×D4716yescomplemented
203#327—4716no—
204C47C22×#32711yescharacteristic, complemented, Sylow
205C25—941no—
206C25C22⋊D4711yescharacteristic
207C22×D4—476no—
208C22×D4—946no—
209C22×D4—946no—
210C22×D4—943no—
211C22×D4—1884no—
212C22×D4—1884no—
213C22×D4—1884no—
214C23×C4—28no—
215C23×C4—26no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4—941no—
220C23×C4C4×D4711yescharacteristic
221C23×C4C4×D4713yes—
222C23×C4D18811yescharacteristic
223C23×C4D18813yes—
224C23×C4C22⋊D4713yes—
225C23×C4C22×D4713yes—
226C22×C8—946no—
227C22×C8—1884no—
228#121—4712no—
229#121—474no—
230#121—9412no—
231#121—946no—
232#121—946no—
233C2×C4⋊C4—476no—
234C2×C4⋊C4—943no—
235C2×C22⋊C4—946no—
236C2×C22⋊C4—946no—
237C22⋊C8—474no—
238C22⋊C8—946no—
239#95—26no—
240#95C22×D4714yescomplemented
241C24—24no—
242C24—24no—
243C24—946no—
244C24—1884no—
245C24—1884no—
246C24—1884no—
247C24(C2×D47)⋊C413yes—
248C24(C2×D47)⋊C411yescharacteristic
249C24#1062533414yes—
250C24#1062533614yes—
251C24C2×C22⋊D4713yes—
252C2×D4—474no—
253C2×D4—9412no—
254C2×D4—944no—
255C2×D4—946no—
256C2×D4—18812no—
257C2×D4—18812no—
258C2×D4—18824no—
259C2×D4—18812no—
260C2×D4—18812no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—224no—
264C22×C4—24no—
265C22×C4—46no—
266C22×C4—46no—
267C22×C4—46no—
268C22×C4—46no—
269C22×C4—412no—
270C22×C4—432no—
271C22×C4—412no—
272C22×C4—946no—
273C22×C4—1884no—
274C22×C4(C2×D47)⋊C413yes—
275C22×C4(C2×D47)⋊C413yes—
276C22×C4(C2×D47)⋊C416yes—
277C22×C4C2×C4×D4713yes—
278C22×C4C2×C4×D4713yes—
279C22×C4C2×D18813yes—
280C22×C4C2×D18813yes—
281C22×C4C2×C22⋊D4713yes—
282C22×C4C23×D4711yescharacteristic
283C2×C8—944no—
284C2×C8—18812no—
285C4⋊C4—474no—
286C4⋊C4—946no—
287C22⋊C4—9412no—
288C22⋊C4—944no—
289C23—26no—
290C23—26no—
291C23—216no—
292C23—216no—
293C23—26no—
294C23—26no—
295C23—944no—
296C23—37616no—
297C23—18812no—
298C23—18812no—
299C23—18812no—
300C23#1309393814yes—
301C23#13093944112yes—
302C23#1309394414yes—
303C23#1309395214yes—
304C23C2×(C2×D47)⋊C413yes—
305C23C2×(C2×D47)⋊C413yes—
306C23C2×#1062533416yes—
307C23C2×#1062533616yes—
308C23C22×C22⋊D4711yescharacteristic
309D4—1884no—
310D4—18812no—
311D4—18824no—
312D4—1884no—
313D4—1884no—
314C2×C4—26no—
315C2×C4—26no—
316C2×C4—26no—
317C2×C4—26no—
318C2×C4—28no—
319C2×C4—46no—
320C2×C4—43no—
321C2×C4—43no—
322C2×C4—43no—
323C2×C4—41no—
324C2×C4—44no—
325C2×C4—448no—
326C2×C4—44no—
327C2×C4—944no—
328C2×C4—18812no—
329C2×C4C2×(C2×D47)⋊C413yes—
330C2×C4C2×(C2×D47)⋊C413yes—
331C2×C4C22×C4×D4711yescharacteristic
332C2×C4C22×D18811yescharacteristic
333C8—1884no—
334C22—21no—
335C22—23no—
336C22—21no—
337C22—23no—
338C22—212no—
339C22—224no—
340C22—212no—
341C22—37624no—
342C22—1884no—
343C22—1884no—
344C22—1884no—
345C22#15688856116yesdirect product factor
346C22C2×#1309393816yes—
347C22C2×#1309394416yes—
348C22C2×#1309394416yes—
349C22C2×#1309395216yes—
350C22C22×(C2×D47)⋊C411yescharacteristic
351C22C22×#1062533411yescharacteristic
352C22C22×#1062533611yescharacteristic
353C4—21no—
354C4—23no—
355C4—23no—
356C4—21no—
357C4—48no—
358C4—1884no—
359C2—21no—
360C2—26no—
361C2—21no—
362C2—3764no—
363C2C2×#15688856112yesdirect product factor
364C2C22×#1309393811yescharacteristic
365C2C22×#1309394411yescharacteristic
366C2C22×#1309395211yescharacteristic
subgroupG/H
centerC24(C2×D47)⋊C4
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC22×#95×C47C2
socleC24×C47C22⋊C4
prime radicalC22×#13093998C2
cosocleC22×C4×C47C24
hypercenterC22×#95D47
hypocenterC47C22×#327
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C24 ◁ C24×C4 ◁ C22×#95

Complements

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

NH ≅ G/Ntype
2. C22×C22⋊(C47⋊C8)C2semidirect
3. C2×#15688856C2direct (N × H)
4. C22×#95×C47C2semidirect
5. C23×D188C4semidirect
10. C2×#13093998C4semidirect
12. C22×C4⋊Dic47C4semidirect
15. C2×C22⋊(C47⋊C8)C22semidirect
16. #15688856C22direct (N × H)
20. C2×#95×C47C22semidirect
24. C22×D188C2×C4semidirect
35. #13093998C2×C4semidirect
38. C2×C4⋊Dic47C2×C4semidirect
42. C22⋊(C47⋊C8)C23semidirect
56. #95×C47C23semidirect
98. C2×D188C22×C4semidirect
102. C4⋊Dic47C22×C4semidirect
158. C22×C47#327semidirect
178. C22×#95D47semidirect
183. C2×C47C2×#327semidirect
202. C2×#95C2×D47semidirect
204. C47C22×#327semidirect
240. #95C22×D47semidirect
345. C22#15688856direct (N × H)
363. C2C2×#15688856direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130939982
C22×C22⋊(C47⋊C8)2
C2×#156888562
C22×#95×C472
C22×#32747