Finite Group Databasebeta

← #26112187

Subgroups — #26112187

Order 12032.

566 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20246298C211yesmaximal, characteristic
2#20241952C211yesmaximal, characteristic, complemented
3#20241887C211yesmaximal, characteristic, complemented
4#20241779C211yesmaximal, characteristic
5#20241289C211yesmaximal, characteristic, complemented
6#20241248C211yesmaximal, characteristic, complemented
7C47×#3077C211yesmaximal, characteristic
8C2×D4×Dic47—21no—
9#15689176C2211yescharacteristic
10#15689176C2211yescharacteristic
11#15689022—21no—
12#15688998—22no—
13#15688989C2211yescharacteristic
14#15688983—21no—
15C4×C4⋊Dic47C2211yescharacteristic
16C2×C47⋊#95—21no—
17#15688860—22no—
18#15688860—21no—
19#15688860—21no—
20D4⋊(C47⋊C8)—22no—
21C47×#533C2211yescharacteristic
22C2×C4×D4×C47C2211yescharacteristic
23C47×#478—22no—
24C47×#476C412yescomplemented
25C47×#474C412yescomplemented
26C47×#428—21no—
27C47×#423C2211yescharacteristic
28C47×#418—21no—
29C23×Dic47—22no—
30D4×Dic47—42no—
31D4×Dic47—42no—
32C2×C4⋊Dic47—21no—
33C2×C4⋊Dic47—21no—
34C2×C4⋊Dic47—22no—
35C2×C22⋊Dic47—22no—
36C2×C4×Dic47—21no—
37C4⋊(C47⋊C8)—21no—
38C4⋊(C47⋊C8)—21no—
39C22⋊(C47⋊C8)—21no—
40C22⋊(C47⋊C8)—21no—
41C4×C47⋊C8—21no—
42C4×C47⋊C8—21no—
43C47⋊#95—22no—
44C47⋊#95—21no—
45C47⋊#95D411yescharacteristic
46C47⋊#95D411yescharacteristic
47C47⋊#95D412yes—
48C22×D4×C47D411yescharacteristic
49C23×C4×C47—21no—
50C2×M16×C47—21no—
51C4⋊Q8×C47C2×C411yescharacteristic, complemented
52C4⋊Q8×C47C2×C411yescharacteristic, complemented
53#125×C47—21no—
54C22⋊Q8×C47—21no—
55C22⋊Q8×C47—21no—
56C4×Q8×C47—21no—
57C4×Q8×C47—21no—
58C4×D4×C47—21no—
59C4×D4×C47—21no—
60C4×D4×C47C2×C412yes—
61C4×D4×C47D412yes—
62#117×C47—21no—
63C2×C4⋊C4×C47D411yescharacteristic
64C2×C22⋊C4×C47—21no—
65C2×C42×C47C2311yescharacteristic
66#107×C47—22no—
67#106×C47—22no—
68C4⋊C8×C47—21no—
69C4⋊C8×C47C2×C412yescomplemented
70#102×C47—22no—
71#102×C47—22no—
72C24×C47—21no—
73C2×Q8×C47—21no—
74C2×Q8×C47—21no—
75C2×D4×C47—21no—
76C2×D4×C47—21no—
77C2×D4×C47—21no—
78C2×D4×C47C22⋊C412yes—
79C2×D4×C47SD1612yes—
80C2×D4×C47Q1612yes—
81C22×C4×C47—21no—
82C22×C4×C47—22no—
83C22×C4×C47—22no—
84C22×C4×C47—82no—
85C22×C4×C47C2×D411yescharacteristic
86C22×C4×C47C2×D411yescharacteristic
87C22×C4×C47C2×D411yescharacteristic
88C22×Dic47—21no—
89C22×Dic47—22no—
90C22×Dic47—21no—
91C22×Dic47—44no—
92C22×Dic47—44no—
93C22×Dic47—82no—
94M16×C47—41no—
95M16×C47—41no—
96C2×C8×C47—22no—
97C4⋊C4×C47—21no—
98C4⋊C4×C47—21no—
99C4⋊C4×C47—21no—
100C4⋊C4×C47—21no—
101C4⋊C4×C47—21no—
102C4⋊C4×C47—41no—
103C4⋊C4×C47—41no—
104C4⋊C4×C47C22⋊C412yes—
105C22⋊C4×C47—22no—
106C22⋊C4×C47—22no—
107C22⋊C4×C47—41no—
108C42×C47—41no—
109C42×C47C22×C411yescharacteristic
110C42×C47C2×D411yescharacteristic
111C42×C47#4111yescharacteristic
112C42×C47#4111yescharacteristic
113C22⋊Dic47—44no—
114C4⋊Dic47—42no—
115C4⋊Dic47—41no—
116C4⋊Dic47—41no—
117C4⋊Dic47—82no—
118C4×Dic47—41no—
119C4×Dic47—41no—
120C2×C47⋊C8—41no—
121C2×C47⋊C8—41no—
122C23×C47—22no—
123C23×C47—22no—
124C23×C47—22no—
125C23×C47—42no—
126C23×C47—42no—
127C23×C47#12011yescharacteristic
128Q8×C47—81no—
129Q8×C47—81no—
130D4×C47—22no—
131D4×C47—22no—
132D4×C47—22no—
133D4×C47#10314yes—
134C2×C4×C47—22no—
135C2×C4×C47—41no—
136C2×C4×C47—41no—
137C2×C4×C47—42no—
138C2×C4×C47—41no—
139C2×C4×C47—41no—
140C2×C4×C47—82no—
141C2×C4×C47—82no—
142C2×C4×C47C2×C22⋊C411yescharacteristic
143C2×C4×C47C4×D411yescharacteristic
144C2×C4×C47C4×D411yescharacteristic
145C2×C4×C47#12111yescharacteristic
146C2×C4×C47#12111yescharacteristic
147C2×C4×C47#12311yescharacteristic
148C2×C4×C47C2×SD1611yescharacteristic
149C2×C4×C47C2×Q1611yescharacteristic
150C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
151C2×C4×C47#13711yescharacteristic
152C2×Dic47—22no—
153C2×Dic47—41no—
154C2×Dic47—41no—
155C2×Dic47—41no—
156C2×Dic47—41no—
157C2×Dic47—42no—
158C2×Dic47—42no—
159C2×Dic47—41no—
160C2×Dic47—42no—
161C2×Dic47—88no—
162C8×C47—42no—
163C47⋊C8—81no—
164C47⋊C8—81no—
165#9175—471nomaximal, Sylow
166C22×C47—21no—
167C22×C47—21no—
168C22×C47—24no—
169C22×C47—44no—
170C22×C47—44no—
171C22×C47—42no—
172C22×C47—42no—
173C22×C47#38611yescharacteristic
174C22×C47#45011yescharacteristic
175C22×C47#45111yescharacteristic
176C4×C47—21no—
177C4×C47—21no—
178C4×C47—21no—
179C4×C47—21no—
180C4×C47—42no—
181C4×C47—81no—
182C4×C47—81no—
183C4×C47C2×#10311yescharacteristic
184C4×C47#41711yescharacteristic
185C4×C47#48311yescharacteristic
186C4×C47#48411yescharacteristic
187Dic47—42no—
188Dic47—81no—
189Dic47—82no—
190#3077D4711yescharacteristic
191#2349—471no—
192#1518—471no—
193#1503—471no—
194#1478—471no—
195#1303—471no—
196#1286—471no—
197C2×C47—21no—
198C2×C47—21no—
199C2×C47—44no—
200C2×C47#187211yescharacteristic
201C2×C47#190411yescharacteristic
202C2×C47#190611yescharacteristic
203#533C2×D4711yescharacteristic
204C2×C4×D4—941no—
205C2×C4×D4C2×D4711yescharacteristic
206#478—22no—
207#476Dic4712yescomplemented
208#474Dic4712yescomplemented
209#431—471no—
210#431—471no—
211#428—21no—
212#423C2×D4711yescharacteristic
213#418—21no—
214#389—941no—
215#385—942no—
216#382—471no—
217#380—941no—
218C4×C4⋊C4—471no—
219C2×#95—941no—
220#328—941no—
221#328—941no—
222#328—942no—
223D4⋊C8—942no—
224C47#917511yescharacteristic, complemented, Sylow
225C22×D4C22⋊D4711yescharacteristic
226C23×C4—21no—
227C23×C4—942no—
228C2×M16—21no—
229C4⋊Q8C2×Dic4711yescharacteristic, complemented
230C4⋊Q8C2×Dic4711yescharacteristic, complemented
231#125—21no—
232C22⋊Q8—21no—
233C22⋊Q8—21no—
234C4×Q8—21no—
235C4×Q8—21no—
236C4×D4—21no—
237C4×D4—21no—
238C4×D4—1882no—
239C4×D4—1882no—
240C4×D4C2×Dic4712yes—
241C4×D4C22⋊D4712yes—
242#117—21no—
243C2×C4⋊C4—942no—
244C2×C4⋊C4—941no—
245C2×C4⋊C4—941no—
246C2×C4⋊C4C22⋊D4711yescharacteristic
247C2×C22⋊C4—21no—
248C2×C22⋊C4—942no—
249C2×C42—941no—
250C2×C42C22×D4711yescharacteristic
251#107—22no—
252#106—22no—
253C4⋊C8—21no—
254C4⋊C8—941no—
255C4⋊C8—941no—
256C4⋊C8C2×Dic4712yescomplemented
257#102—22no—
258#102—22no—
259C22⋊C8—941no—
260C22⋊C8—941no—
261C4×C8—941no—
262C4×C8—941no—
263#95—471no—
264#95—471no—
265#95—472no—
266#95—942no—
267#95—941no—
268C24—21no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4#1062533612yes—
275C2×D4#1062533712yes—
276C2×D4C22⋊Dic4712yes—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—82no—
281C22×C4—941no—
282C22×C4—942no—
283C22×C4—941no—
284C22×C4—3762no—
285C22×C4—1884no—
286C22×C4—1884no—
287C22×C4D4×D4711yescharacteristic
288C22×C4D4×D4711yescharacteristic
289C22×C4C2×C22⋊D4711yescharacteristic
290M16—41no—
291M16—41no—
292C2×C8—22no—
293C2×C8—1881no—
294C2×C8—1881no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—3762no—
303C4⋊C4—1882no—
304C4⋊C4—1881no—
305C4⋊C4—1881no—
306C4⋊C4C22⋊Dic4712yes—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—41no—
310C22⋊C4—1884no—
311C42—41no—
312C42—1881no—
313C42—1881no—
314C42#1062535211yescharacteristic
315C42#1062535211yescharacteristic
316C42C22×Dic4711yescharacteristic
317C42C2×C22⋊D4711yescharacteristic
318C23—22no—
319C23—22no—
320C23—22no—
321C23—42no—
322C23—42no—
323C23#1309399211yescharacteristic
324Q8—81no—
325Q8—81no—
326D4—22no—
327D4—22no—
328D4—22no—
329D4#1309394814yes—
330C2×C4—22no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—942no—
339C2×C4—3768no—
340C2×C4—1881no—
341C2×C4—1882no—
342C2×C4—1881no—
343C2×C4—1882no—
344C2×C4—1881no—
345C2×C4—1882no—
346C2×C4—1881no—
347C2×C4—1881no—
348C2×C4C2×C22⋊Dic4711yescharacteristic
349C2×C4D4×Dic4711yescharacteristic
350C2×C4D4×Dic4711yescharacteristic
351C2×C4#1309399611yescharacteristic
352C2×C4#1309399611yescharacteristic
353C2×C4#1309400511yescharacteristic
354C2×C4C2×#1062533611yescharacteristic
355C2×C4C2×#1062533711yescharacteristic
356C2×C4#1309404911yescharacteristic
357C2×C4#1309405411yescharacteristic
358C8—42no—
359C8—3761no—
360C8—3761no—
361C22—21no—
362C22—21no—
363C22—24no—
364C22—44no—
365C22—44no—
366C22—42no—
367C22—42no—
368C22#1568900311yescharacteristic
369C22#1568927311yescharacteristic
370C22#1568927811yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—42no—
376C4—81no—
377C4—81no—
378C4—3762no—
379C4—3761no—
380C4—1882no—
381C4C2×#1309394811yescharacteristic
382C4#1568912411yescharacteristic
383C4#1568945111yescharacteristic
384C4#1568945611yescharacteristic
385C2—21no—
386C2—21no—
387C2—44no—
388C2#2024369611yescharacteristic
389C2#2024387711yescharacteristic
390C2#2024389111yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#3077C2
socleC22×C47#386
prime radicalC22×D4×C47D4
cosocleC2×C42×C47C23
hypercenter#3077D47
hypocenterC47#9175
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3077

Complements

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

NH ≅ G/Ntype
2. #20241952C2semidirect
3. #20241887C2semidirect
5. #20241289C2semidirect
6. #20241248C2semidirect
24. C47×#476C4semidirect
25. C47×#474C4semidirect
51. C4⋊Q8×C47C2×C4semidirect
52. C4⋊Q8×C47C2×C4semidirect
69. C4⋊C8×C47C2×C4semidirect
207. #476Dic47semidirect
208. #474Dic47semidirect
224. C47#9175semidirect
229. C4⋊Q8C2×Dic47semidirect
230. C4⋊Q8C2×Dic47semidirect
256. C4⋊C8C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
#202462982
#202419522
#202418872
#202417792
#202412892
#202412482
C47×#30772
#917547