Finite Group Databasebeta

← #26182335

Subgroups — #26182335

Order 12032.

940 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689745C211yesmaximal, characteristic
2#20252469C211yesmaximal, characteristic, complemented
3#20241758C214yesmaximal, complemented
4#20241741C214yesmaximal, complemented
5#20241708C212yesmaximal, complemented
6#20241700C212yesmaximal, complemented
7C47×#2983C211yesmaximal, characteristic, complemented
8#15689745—24no—
9#15689745—24no—
10C2×#13094023C411yescharacteristic
11C2×#13094018C2212yes—
12C2×#13094013C411yescharacteristic
13C2×#13094012C2212yes—
14C2×#13094009C412yes—
15C2×#13094002C412yes—
16C2×#13094000C411yescharacteristic
17C2×#13093999C411yescharacteristic
18C2×C4×D188C2211yescharacteristic
19C2×#13093981C2211yescharacteristic
20Q8×C47⋊C8—24no—
21#15689176C2212yes—
22#15689176C2214yes—
23#15688848C2214yes—
24#15688846C2214yes—
25#15688846C2218yes—
26C2×C4×Q8×C47C2211yescharacteristic
27C47×#446—24no—
28C47×#432C2214yes—
29C47×#431C2212yes—
30C22×Dic94C2×C412yes—
31C22×D188C2×C411yescharacteristic
32C22×D188C2×C412yes—
33C22×C4×D47C2×C411yescharacteristic
34C22×C4×D47C2×C412yes—
35#13094023—24no—
36#13094018—24no—
37#13094018—24no—
38#13094013—24no—
39#13094012—24no—
40#13094012—24no—
41#13094009—28no—
42#13094002—28no—
43#13094000—24no—
44#13093999—24no—
45C4×D188—22no—
46C4×D188—22no—
47#13093981—22no—
48#13093981—22no—
49C2×C4⋊Dic47C2×C411yescharacteristic
50C2×Dic47⋊C4C2×C411yescharacteristic
51C2×Dic47⋊C4C2×C412yes—
52C2×Dic47⋊C4C2×C412yes—
53C2×Dic47⋊C4C2311yescharacteristic
54C2×(C2×D47)⋊C4C2×C411yescharacteristic
55C2×(C2×D47)⋊C4C2×C412yes—
56C2×(C2×D47)⋊C4C2×C412yes—
57C2×(C2×D47)⋊C4C2×C412yes—
58C2×(C2×D47)⋊C4C2×C412yes—
59C2×(C2×D47)⋊C4C2311yescharacteristic
60C2×(C2×D47)⋊C4C2312yes—
61C2×C4×Dic47C2×C411yescharacteristic
62C4⋊(C47⋊C8)—22no—
63C4⋊(C47⋊C8)—24no—
64C22⋊(C47⋊C8)C2314yes—
65C4×C47⋊C8—22no—
66C4×C47⋊C8—24no—
67C22×Q8×C47C2×C411yescharacteristic
68C4×Q8×C47D414yes—
69C4×Q8×C47D414yes—
70C2×C4⋊C4×C47C2×C411yescharacteristic
71C2×C4⋊C4×C47C2×C412yes—
72C2×C42×C47C2311yescharacteristic
73C2×C42×C47C2312yes—
74C4⋊C8×C47—22no—
75C4⋊C8×C47—24no—
76C22⋊C8×C47C2314yes—
77#97×C47—24no—
78C4×C8×C47—22no—
79C23×D47C22×C411yescharacteristic
80C23×D47C22×C412yes—
81C2×Q8×C47—24no—
82C2×Q8×C47C22⋊C414yes—
83C22×C4×C47C22×C412yes—
84C22×C4×C47C22×C411yescharacteristic
85C22×C4×C47C22×C412yes—
86C22×C4×C47C22×C411yescharacteristic
87C22×C4×C47C2411yescharacteristic
88C22×Dic47C22×C411yescharacteristic
89C22×Dic47C22×C411yescharacteristic
90C22×Dic47C22×C411yescharacteristic
91C22×Dic47C22×C412yes—
92C2×D188—22no—
93C2×D188—24no—
94C2×D188—24no—
95C2×D188—82no—
96C2×C4×D47—22no—
97C2×C4×D47—24no—
98C2×C4×D47—24no—
99C2×C4×D47—82no—
100C2×Dic94—24no—
101C2×Dic94—82no—
102C2×C8×C47—24no—
103C4⋊C4×C47C22⋊C414yes—
104C4⋊C4×C47C22⋊C418yes—
105C42×C47C2×D412yes—
106C42×C47C2×D414yes—
107C42×C47C2×D412yes—
108C42×C47C2×D414yes—
109(C2×D47)⋊C4—22no—
110(C2×D47)⋊C4—24no—
111(C2×D47)⋊C4—24no—
112(C2×D47)⋊C4—24no—
113(C2×D47)⋊C4—24no—
114(C2×D47)⋊C4—21no—
115(C2×D47)⋊C4—21no—
116(C2×D47)⋊C4—22no—
117(C2×D47)⋊C4—22no—
118C4⋊Dic47—22no—
119Dic47⋊C4—22no—
120Dic47⋊C4—24no—
121Dic47⋊C4—24no—
122Dic47⋊C4—21no—
123Dic47⋊C4—21no—
124C4×Dic47—22no—
125C2×C47⋊C8—24no—
126C23×C47C23×C411yescharacteristic
127C22×D47—21no—
128C22×D47—22no—
129C22×D47—22no—
130C22×D47—24no—
131C22×D47—42no—
132C22×D47—41no—
133C22×D47—41no—
134C22×D47—42no—
135Q8×C47—28no—
136C2×C4×C47—24no—
137C2×C4×C47—42no—
138C2×C4×C47—41no—
139C2×C4×C47—41no—
140C2×C4×C47C2×C22⋊C414yes—
141C2×C4×C47C2×C22⋊C414yes—
142C2×C4×C47C2×C22⋊C412yes—
143C2×C4×C47C2×C22⋊C412yes—
144C2×C4×C47#13114yes—
145C2×C4×C47C22×D411yescharacteristic
146C2×C4×C47C22×D411yescharacteristic
147C2×Dic47—21no—
148C2×Dic47—21no—
149C2×Dic47—21no—
150C2×Dic47—22no—
151C2×Dic47—22no—
152C2×Dic47—41no—
153C2×Dic47—42no—
154C2×Dic47—41no—
155D188—44no—
156D188—84no—
157C4×D47—44no—
158C4×D47—84no—
159Dic94—84no—
160C8×C47—44no—
161C47⋊C8—44no—
162#17390—471nomaximal, Sylow
163C22×C47—21no—
164C22×C47—21no—
165C22×C47C22×C22⋊C411yescharacteristic
166C22×C47C2×#13112yes—
167C2×D47—41no—
168C2×D47—42no—
169C2×D47—41no—
170C2×D47—42no—
171C2×D47—42no—
172C2×D47—41no—
173C2×D47—82no—
174C2×D47—84no—
175C4×C47—24no—
176C4×C47—21no—
177C4×C47—21no—
178C4×C47—24no—
179C4×C47—22no—
180C4×C47#40814yes—
181Dic47—41no—
182Dic47—41no—
183Dic47—41no—
184Dic47—42no—
185C2×#542—471no—
186#2983D4711yescharacteristic, complemented
187#2982—471no—
188#1473—474no—
189#1469—474no—
190#1460—472no—
191#1458—472no—
192C2×C47—22no—
193C2×C47C2×#40811yescharacteristic
194C2×C47#288411yescharacteristic
195C2×C47#288811yescharacteristic
196D47—81no—
197D47—82no—
198#542—944no—
199#542—944no—
200C2×C4⋊Q8—471no—
201C2×#126—472no—
202C2×#124—471no—
203C2×#124—472no—
204C2×#123—472no—
205C2×C22⋊Q8—471no—
206C2×C22⋊Q8—472no—
207C2×#121—471no—
208C2×C4×Q8C2×D4711yescharacteristic
209C2×C4×D4—471no—
210C2×#117—471no—
211#446—24no—
212C8×Q8—944no—
213#432C2×D4714yes—
214#431—472no—
215#431—474no—
216#431C2×D4712yes—
217#324—474no—
218#323—474no—
219#323—478no—
220C47#1739011yescharacteristic, complemented, Sylow
221C22×Q8—472no—
222C22×Q8C4×D4711yescharacteristic
223C22×D4—471no—
224C22×D4—472no—
225C23×C4—471no—
226C23×C4—472no—
227C4⋊Q8—944no—
228#126—944no—
229#126—944no—
230#124—944no—
231#124—944no—
232#124—944no—
233#123—948no—
234C22⋊Q8—944no—
235C22⋊Q8—948no—
236#121—944no—
237C4×Q8D18814yes—
238C4×Q8C22⋊D4714yes—
239C4×D4—942no—
240C4×D4—942no—
241#117—942no—
242#117—942no—
243C2×C4⋊C4—471no—
244C2×C4⋊C4—471no—
245C2×C4⋊C4—472no—
246C2×C4⋊C4—472no—
247C2×C4⋊C4—471no—
248C2×C4⋊C4C4×D4711yescharacteristic
249C2×C4⋊C4C4×D4712yes—
250C2×C22⋊C4—471no—
251C2×C22⋊C4—472no—
252C2×C22⋊C4—472no—
253C2×C22⋊C4—472no—
254C2×C22⋊C4—472no—
255C2×C22⋊C4—471no—
256C2×C22⋊C4—472no—
257C2×C42—471no—
258C2×C42C22×D4711yescharacteristic
259C2×C42C22×D4712yes—
260C4⋊C8—22no—
261C4⋊C8—24no—
262C4⋊C8—942no—
263C4⋊C8—944no—
264C22⋊C8—474no—
265C22⋊C8C22×D4714yes—
266#97—24no—
267C4×C8—22no—
268C4×C8—942no—
269C4×C8—944no—
270C24—471no—
271C24—472no—
272C2×Q8—24no—
273C2×Q8—944no—
274C2×Q8—3762no—
275C2×Q8(C2×D47)⋊C414yes—
276C2×D4—944no—
277C2×D4—942no—
278C2×D4—944no—
279C2×D4—3762no—
280C22×C4—471no—
281C22×C4—471no—
282C22×C4—471no—
283C22×C4—472no—
284C22×C4—944no—
285C22×C4—942no—
286C22×C4—944no—
287C22×C4—3762no—
288C22×C4C2×C4×D4712yes—
289C22×C4C2×C4×D4711yescharacteristic
290C22×C4C2×C4×D4712yes—
291C22×C4C2×C4×D4711yescharacteristic
292C22×C4C23×D4711yescharacteristic
293C2×C8—24no—
294C2×C8—944no—
295C4⋊C4—942no—
296C4⋊C4—944no—
297C4⋊C4—944no—
298C4⋊C4—942no—
299C4⋊C4—941no—
300C4⋊C4—941no—
301C4⋊C4(C2×D47)⋊C414yes—
302C4⋊C4(C2×D47)⋊C418yes—
303C22⋊C4—942no—
304C22⋊C4—944no—
305C22⋊C4—944no—
306C22⋊C4—944no—
307C22⋊C4—944no—
308C22⋊C4—941no—
309C22⋊C4—942no—
310C22⋊C4—941no—
311C22⋊C4—942no—
312C42—942no—
313C42C2×D18812yes—
314C42C2×D18814yes—
315C42C2×C22⋊D4712yes—
316C42C2×C22⋊D4714yes—
317C23—942no—
318C23—944no—
319C23—941no—
320C23—942no—
321C23—1882no—
322C23—1881no—
323C23—1881no—
324C23—1882no—
325C23C22×C4×D4711yescharacteristic
326Q8—28no—
327Q8—3764no—
328D4—3764no—
329D4—1884no—
330C2×C4—24no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—941no—
335C2×C4—941no—
336C2×C4—941no—
337C2×C4—942no—
338C2×C4—942no—
339C2×C4—3764no—
340C2×C4—1884no—
341C2×C4—1881no—
342C2×C4—1882no—
343C2×C4—1881no—
344C2×C4C2×(C2×D47)⋊C414yes—
345C2×C4C2×(C2×D47)⋊C412yes—
346C2×C4C2×(C2×D47)⋊C414yes—
347C2×C4C2×(C2×D47)⋊C412yes—
348C2×C4#1309403214yes—
349C2×C4C22×C22⋊D4711yescharacteristic
350C2×C4C22×D18811yescharacteristic
351C8—44no—
352C8—1884no—
353C22—21no—
354C22—21no—
355C22—3762no—
356C22—3764no—
357C22—1881no—
358C22—1882no—
359C22—1881no—
360C22—1882no—
361C22—1882no—
362C22—1881no—
363C22C22×(C2×D47)⋊C411yescharacteristic
364C22C2×#1309403212yes—
365C4—24no—
366C4—21no—
367C4—21no—
368C4—22no—
369C4—24no—
370C4—1881no—
371C4—1881no—
372C4—1882no—
373C4—1881no—
374C4#1568909114yes—
375C2—22no—
376C2—3761no—
377C2—3762no—
378C2C2×#1568909111yescharacteristic
379C2#2025185111yescharacteristic
380C2#2025186611yescharacteristic
subgroupG/H
centerC22C22×(C2×D47)⋊C4
commutatorC23×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2983C2
socleC22×C47C22×C22⋊C4
prime radicalC2×#13093999C4
cosocleC22×C4×C47C24
hypercenter#2983D47
hypocenterC47#17390
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #2983

Complements

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

NH ≅ G/Ntype
2. #20252469C2semidirect
3. #20241758C2semidirect
4. #20241741C2semidirect
5. #20241708C2semidirect
6. #20241700C2semidirect
7. C47×#2983C2semidirect
186. #2983D47semidirect
220. C47#17390semidirect

Maximal subgroups

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

subgroupindex
C2×#156897452
#202524692
#202417582
#202417412
#202417082
#202417002
C47×#29832
#1739047