Finite Group Databasebeta

← #26112034

Subgroups — #26112034

Order 12032.

946 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15688983C211yesmaximal, characteristic
2#20243298C211yesmaximal, characteristic
3C2×#15688860C212yesmaximal, complemented
4#20241266C212yesmaximal, complemented
5C47×#2848C211yesmaximal, characteristic
6C22×C4⋊Dic47—21no—
7C22×C22⋊Dic47—21no—
8C2×C22⋊(C47⋊C8)—22no—
9C2×C22⋊(C47⋊C8)C2212yes—
10#15688983—24no—
11C2×C47⋊#95C2212yes—
12#15688860—24no—
13C24×C4×C47C2211yescharacteristic
14C2×C22⋊Q8×C47C412yescomplemented
15C2×C22⋊Q8×C47C2211yescharacteristic
16C47×#385—22no—
17C47×#385—21no—
18C47×#380C412yescomplemented
19C47×#380C2211yescharacteristic
20C47×#379—21no—
21C23×Dic47—22no—
22C22×C47⋊C8—22no—
23C2×C4⋊Dic47—24no—
24C2×C4⋊Dic47—44no—
25C2×C22⋊Dic47—24no—
26C2×C22⋊Dic47—42no—
27C2×C22⋊Dic47—42no—
28C22⋊(C47⋊C8)—22no—
29C22⋊(C47⋊C8)—22no—
30C22⋊(C47⋊C8)—44no—
31C47⋊#95—22no—
32C47⋊#95D414yes—
33C25×C47D411yescharacteristic
34C22×Q8×C47C2×C411yescharacteristic
35C23×C4×C47—22no—
36C23×C4×C47—22no—
37C23×C4×C47—44no—
38C23×C4×C47C2×C412yes—
39C23×C4×C47D411yescharacteristic
40C23×C4×C47D412yes—
41C23×C4×C47C2311yescharacteristic
42C22⋊Q8×C47—22no—
43C22⋊Q8×C47—22no—
44C22⋊Q8×C47—44no—
45C2×C4⋊C4×C47—22no—
46C2×C4⋊C4×C47—21no—
47C2×C4⋊C4×C47C2×C411yescharacteristic
48C2×C4⋊C4×C47C2×C412yes—
49C2×C22⋊C4×C47—22no—
50C2×C22⋊C4×C47—22no—
51C2×C22⋊C4×C47—21no—
52C2×C22⋊C4×C47—21no—
53#95×C47—22no—
54#95×C47—21no—
55C24×C47—21no—
56C24×C47—21no—
57C24×C47—21no—
58C24×C47—21no—
59C24×C47—42no—
60C24×C47—42no—
61C24×C47C22⋊C412yes—
62C24×C47SD1612yes—
63C24×C47Q1612yes—
64C24×C47C2×D411yescharacteristic
65C2×Q8×C47—22no—
66C2×Q8×C47—42no—
67C22×C4×C47—21no—
68C22×C4×C47—21no—
69C22×C4×C47—24no—
70C22×C4×C47—24no—
71C22×C4×C47—24no—
72C22×C4×C47—24no—
73C22×C4×C47—24no—
74C22×C4×C47—21no—
75C22×C4×C47—21no—
76C22×C4×C47—22no—
77C22×C4×C47—42no—
78C22×C4×C47—42no—
79C22×C4×C47—42no—
80C22×C4×C47—48no—
81C22×C4×C47—48no—
82C22×C4×C47C22⋊C412yes—
83C22×C4×C47C22×C411yescharacteristic
84C22×C4×C47C2×D411yescharacteristic
85C22×C4×C47C2×D412yes—
86C22×C4×C47#4112yes—
87C22×Dic47—24no—
88C22×Dic47—44no—
89C22×Dic47—44no—
90C22×Dic47—42no—
91C22×Dic47—42no—
92C4⋊C4×C47—22no—
93C4⋊C4×C47—42no—
94C4⋊C4×C47—44no—
95C4⋊C4×C47—42no—
96C22⋊C4×C47—44no—
97C22⋊C4×C47—44no—
98C22⋊C4×C47—41no—
99C22⋊C4×C47—41no—
100C22⋊C4×C47—41no—
101C22⋊C4×C47—41no—
102C22⋊Dic47—48no—
103C4⋊Dic47—48no—
104C2×C47⋊C8—44no—
105C2×C47⋊C8—42no—
106C23×C47—22no—
107C23×C47—22no—
108C23×C47—22no—
109C23×C47—22no—
110C23×C47—22no—
111C23×C47—22no—
112C23×C47—22no—
113C23×C47—22no—
114C23×C47—22no—
115C23×C47—22no—
116C23×C47—22no—
117C23×C47—22no—
118C23×C47—44no—
119C23×C47—44no—
120C23×C47—42no—
121C23×C47—42no—
122C23×C47—42no—
123C23×C47—42no—
124C23×C47—44no—
125C23×C47—44no—
126C23×C47#10314yes—
127C23×C47C2×C22⋊C411yescharacteristic
128C23×C47#12011yescharacteristic
129C23×C47#12311yescharacteristic
130C23×C47C2×SD1611yescharacteristic
131C23×C47C2×Q1611yescharacteristic
132C23×C47AGL(1,ℤ/8)11yescharacteristic
133C23×C47#13711yescharacteristic
134Q8×C47—82no—
135C2×C4×C47—24no—
136C2×C4×C47—24no—
137C2×C4×C47—22no—
138C2×C4×C47—22no—
139C2×C4×C47—22no—
140C2×C4×C47—22no—
141C2×C4×C47—42no—
142C2×C4×C47—42no—
143C2×C4×C47—48no—
144C2×C4×C47—48no—
145C2×C4×C47—41no—
146C2×C4×C47—41no—
147C2×C4×C47—42no—
148C2×C4×C47—42no—
149C2×C4×C47—42no—
150C2×C4×C47—42no—
151C2×C4×C47—42no—
152C2×C4×C47#9914yes—
153C2×C4×C47C4×D412yes—
154C2×C4×C47#12112yes—
155C2×Dic47—44no—
156C2×Dic47—44no—
157C2×Dic47—44no—
158C2×Dic47—84no—
159C2×Dic47—84no—
160C47⋊C8—82no—
161#9151—471nomaximal, Sylow
162C22×C47—21no—
163C22×C47—21no—
164C22×C47—21no—
165C22×C47—21no—
166C22×C47—22no—
167C22×C47—22no—
168C22×C47—22no—
169C22×C47—22no—
170C22×C47—24no—
171C22×C47—24no—
172C22×C47—24no—
173C22×C47—44no—
174C22×C47—44no—
175C22×C47—44no—
176C22×C47—44no—
177C22×C47—42no—
178C22×C47—42no—
179C22×C47—42no—
180C22×C47—42no—
181C22×C47#38611yescharacteristic
182C22×C47C2×#9912yes—
183C22×C47C2×#10311yescharacteristic
184C22×C47#41711yescharacteristic
185C22×C47#42112yes—
186C22×C47#45011yescharacteristic
187C22×C47#45111yescharacteristic
188C22×C47#48311yescharacteristic
189C22×C47#48411yescharacteristic
190C4×C47—22no—
191C4×C47—24no—
192C4×C47—22no—
193C4×C47—81no—
194C4×C47—81no—
195C4×C47—82no—
196Dic47—84no—
197#2848D4711yescharacteristic
198C2×#380—471no—
199#1779—471no—
200C2×#328—472no—
201#1294—472no—
202C2×C47—21no—
203C2×C47—21no—
204C2×C47—21no—
205C2×C47—21no—
206C2×C47—44no—
207C2×C47#151312yes—
208C2×C47#152012yes—
209C2×C47#187211yescharacteristic
210C2×C47#189711yescharacteristic
211C2×C47#189811yescharacteristic
212C24×C4C2×D4711yescharacteristic
213C2×C22⋊Q8Dic4712yescomplemented
214C2×C22⋊Q8C2×D4711yescharacteristic
215C22×C4⋊C4—941no—
216C22×C22⋊C4—941no—
217C2×C22⋊C8—472no—
218C2×C22⋊C8—942no—
219#385—21no—
220#385—22no—
221#380—944no—
222#380Dic4712yescomplemented
223#380C2×D4711yescharacteristic
224#379—21no—
225C2×#95—472no—
226#328—944no—
227C47#915111yescharacteristic, complemented, Sylow
228C25C22⋊D4711yescharacteristic
229C22×Q8C2×Dic4711yescharacteristic
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—44no—
233C23×C4—942no—
234C23×C4C2×Dic4712yes—
235C23×C4C22⋊D4711yescharacteristic
236C23×C4C22⋊D4712yes—
237C23×C4C22×D4711yescharacteristic
238C22×C8—942no—
239C22⋊Q8—22no—
240C22⋊Q8—22no—
241C22⋊Q8—44no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—944no—
245C2×C4⋊C4—1884no—
246C2×C4⋊C4C2×Dic4711yescharacteristic
247C2×C4⋊C4C2×Dic4712yes—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—944no—
253C2×C22⋊C4—1882no—
254C2×C22⋊C4—1882no—
255C22⋊C8—942no—
256C22⋊C8—942no—
257C22⋊C8—1884no—
258#95—22no—
259#95—21no—
260#95—474no—
261#95—942no—
262C24—21no—
263C24—21no—
264C24—21no—
265C24—21no—
266C24—42no—
267C24—42no—
268C24#1062533612yes—
269C24#1062533712yes—
270C24C22⋊Dic4712yes—
271C24C2×C22⋊D4711yescharacteristic
272C2×Q8—22no—
273C2×Q8—42no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—48no—
288C22×C4—48no—
289C22×C4—944no—
290C22×C4—1884no—
291C22×C4—1884no—
292C22×C4—1882no—
293C22×C4—1882no—
294C22×C4C22⋊Dic4712yes—
295C22×C4D4×D4712yes—
296C22×C4#1062535212yes—
297C22×C4C22×Dic4711yescharacteristic
298C22×C4C2×C22⋊D4711yescharacteristic
299C2×C8—1884no—
300C2×C8—1882no—
301C4⋊C4—22no—
302C4⋊C4—42no—
303C4⋊C4—44no—
304C4⋊C4—42no—
305C4⋊C4—1888no—
306C22⋊C4—44no—
307C22⋊C4—44no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—1888no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—44no—
326C23—44no—
327C23—42no—
328C23—42no—
329C23—44no—
330C23—44no—
331C23—42no—
332C23—42no—
333C23#1309394814yes—
334C23C2×C22⋊Dic4711yescharacteristic
335C23#1309399211yescharacteristic
336C23#1309400511yescharacteristic
337C23C2×#1062533611yescharacteristic
338C23C2×#1062533711yescharacteristic
339C23#1309404911yescharacteristic
340C23#1309405411yescharacteristic
341Q8—82no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—48no—
351C2×C4—48no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—3764no—
360C2×C4—3764no—
361C2×C4—1884no—
362C2×C4—1884no—
363C2×C4—1884no—
364C2×C4#1309393914yes—
365C2×C4D4×Dic4712yes—
366C2×C4#1309399612yes—
367C8—3762no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—24no—
377C22—24no—
378C22—24no—
379C22—44no—
380C22—44no—
381C22—44no—
382C22—44no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—42no—
387C22#1568900311yescharacteristic
388C22C2×#1309393912yes—
389C22C2×#1309394811yescharacteristic
390C22#1568912411yescharacteristic
391C22#1568914012yes—
392C22#1568927311yescharacteristic
393C22#1568927811yescharacteristic
394C22#1568945111yescharacteristic
395C22#1568945611yescharacteristic
396C4—22no—
397C4—24no—
398C4—22no—
399C4—82no—
400C4—81no—
401C4—81no—
402C4—3764no—
403C2—21no—
404C2—21no—
405C2—21no—
406C2—21no—
407C2—44no—
408C2#2024193112yes—
409C2#2024196012yes—
410C2#2024369611yescharacteristic
411C2#2024383711yescharacteristic
412C2#2024384211yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2848C2
socleC23×C47C2×C22⋊C4
prime radicalC25×C47D4
cosocleC23×C4×C47C23
hypercenter#2848D47
hypocenterC47#9151
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2848

Complements

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

NH ≅ G/Ntype
3. C2×#15688860C2semidirect
4. #20241266C2semidirect
14. C2×C22⋊Q8×C47C4semidirect
18. C47×#380C4semidirect
213. C2×C22⋊Q8Dic47semidirect
222. #380Dic47semidirect
227. C47#9151semidirect

Maximal subgroups

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

subgroupindex
C2×#156889832
#202432982
C2×#156888602
#202412662
C47×#28482
#915147