Finite Group Databasebeta

← #26133215

Subgroups — #26133215

Order 12032.

626 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689360C211yesmaximal, characteristic
2C2×#107×D47C211yesmaximal, characteristic, complemented
3C2×#15688925C211yesmaximal, characteristic, complemented
4C2×#15688923C211yesmaximal, characteristic, complemented
5C2×(C2×D47)⋊C16C211yesmaximal, characteristic, complemented
6#20241479C211yesmaximal, characteristic, complemented
7C47×#1380C211yesmaximal, characteristic, complemented
8C22×D376C2211yescharacteristic
9C22×C8×D47C2211yescharacteristic
10C2×#13093999—21no—
11C2×C4⋊C4×D47—21no—
12C22×C47⋊C16C2211yescharacteristic
13#15689360—22no—
14#15689360C414yescomplemented
15C2×#13093958C2211yescharacteristic
16C2×#13093957—21no—
17#107×D47—22no—
18#107×D47—22no—
19C2×#13093944—21no—
20#15688925—22no—
21#15688923—22no—
22(C2×D47)⋊C16—22no—
23C22×C16×C47C2211yescharacteristic
24C2×#107×C47C2211yescharacteristic
25C2×#107×C47C2211yescharacteristic
26C22×D188—21no—
27C22×D188—41no—
28C22×C4×D47—41no—
29C22×C4×D47D411yescharacteristic
30C2×D376—21no—
31C2×D376—21no—
32C2×D376—21no—
33C2×D376—21no—
34C2×D376—21no—
35C2×D376C2×C412yescomplemented
36C22×C47⋊C8D411yescharacteristic
37C2×C8×D47—21no—
38C2×C8×D47—21no—
39C2×C8×D47—21no—
40C2×C8×D47—21no—
41C2×C8×D47C2×C412yes—
42C2×C8×D47D412yes—
43#13093999—24no—
44#13093999—42no—
45C2×C4⋊Dic47—21no—
46C2×Dic47⋊C4—41no—
47C4⋊C4×D47—24no—
48C4⋊C4×D47—42no—
49C2×(C2×D47)⋊C4—41no—
50C2×C47⋊C16—21no—
51C2×C47⋊C16—21no—
52C2×C47⋊C16—21no—
53#13093958—21no—
54#13093958D412yescomplemented
55#13093957—24no—
56#13093944—24no—
57C22×C8×C47C2311yescharacteristic
58C2×C4⋊C4×C47—21no—
59C2×C4⋊C4×C47—21no—
60C2×C16×C47—21no—
61C2×C16×C47—21no—
62C2×C16×C47—21no—
63#107×C47—21no—
64#107×C47—21no—
65#107×C47C2×C412yescomplemented
66#107×C47D412yescomplemented
67C23×D47—41no—
68C23×D47D811yescharacteristic
69C22×C4×C47—41no—
70C22×C4×C47—41no—
71C22×C4×C47C2×D411yescharacteristic
72C22×Dic47—41no—
73C22×Dic47D811yescharacteristic
74C2×D188—22no—
75C2×D188—41no—
76C2×D188—41no—
77C2×D188—41no—
78C2×D188—41no—
79C2×D188—44no—
80C2×D188—41no—
81C2×D188—82no—
82C2×D188—82no—
83C2×C4×D47—21no—
84C2×C4×D47—21no—
85C2×C4×D47—21no—
86C2×C4×D47—21no—
87C2×C4×D47—44no—
88C2×C4×D47—82no—
89C2×C4×D47—82no—
90C2×C4×D47C22⋊C412yes—
91C2×C4×D47SD1612yes—
92C2×C8×C47C22×C411yescharacteristic
93C2×C8×C47C2×D411yescharacteristic
94C2×C8×C47C2×D411yescharacteristic
95C2×C8×C47C2×D411yescharacteristic
96C2×C8×C47#4111yescharacteristic
97C2×C8×C47#4111yescharacteristic
98C4⋊C4×C47—22no—
99C4⋊C4×C47—22no—
100C4⋊C4×C47—41no—
101C4⋊C4×C47—41no—
102(C2×D47)⋊C4—44no—
103C4⋊Dic47—22no—
104C4⋊Dic47—41no—
105Dic47⋊C4—44no—
106C2×C47⋊C8—21no—
107C2×C47⋊C8—21no—
108C2×C47⋊C8C22⋊C412yes—
109D376—22no—
110D376—22no—
111D376—41no—
112D376—41no—
113C8×D47—22no—
114C8×D47—22no—
115C8×D47—22no—
116C8×D47—22no—
117C16×C47—41no—
118C47⋊C16—41no—
119C23×C47C2×D811yescharacteristic
120C22×D47—21no—
121C22×D47—21no—
122C22×D47—21no—
123C22×D47—21no—
124C22×D47—42no—
125C22×D47—81no—
126C22×D47—81no—
127C22×D47—81no—
128C22×D47—81no—
129C22×D47—81no—
130C22×D47—81no—
131C22×D47#10212yes—
132C22×D47D1612yes—
133C22×D47SD3212yes—
134C2×C4×C47—42no—
135C2×C4×C47—42no—
136C2×C4×C47—81no—
137C2×C4×C47—81no—
138C2×C4×C47—81no—
139C2×C4×C47—81no—
140C2×C4×C47C2×C22⋊C411yescharacteristic
141C2×C4×C47#12011yescharacteristic
142C2×C4×C47#12311yescharacteristic
143C2×C4×C47C2×SD1611yescharacteristic
144C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C47#13711yescharacteristic
146C2×Dic47—21no—
147C2×Dic47—21no—
148C2×Dic47—42no—
149C2×Dic47—81no—
150C2×Dic47—81no—
151C2×Dic47#10212yes—
152D188—42no—
153D188—42no—
154D188—88no—
155D188—81no—
156D188—81no—
157C4×D47—22no—
158C4×D47—22no—
159C4×D47—22no—
160C4×D47—22no—
161C4×D47—88no—
162C8×C47C4×D411yescharacteristic
163C8×C47C4×D411yescharacteristic
164C8×C47#12111yescharacteristic
165C8×C47#12111yescharacteristic
166C47⋊C8—22no—
167#12991—471nomaximal, Sylow
168C22×C47C2×#10211yescharacteristic
169C22×C47#44711yescharacteristic
170C22×C47#48011yescharacteristic
171C22×C47C2×D1611yescharacteristic
172C22×C47C2×SD3211yescharacteristic
173C22×C47#50911yescharacteristic
174C22×C47#51011yescharacteristic
175C2×D47—22no—
176C2×D47—22no—
177C2×D47—22no—
178C2×D47—22no—
179C2×D47—22no—
180C2×D47—22no—
181C2×D47—82no—
182C2×D47—82no—
183C2×D47—82no—
184C2×D47—161no—
185C2×D47—161no—
186C2×D47—161no—
187C2×D47—161no—
188C2×D47#35714yescomplemented
189C4×C47—82no—
190C4×C47—82no—
191C4×C47#38611yescharacteristic
192C4×C47#41711yescharacteristic
193C4×C47#44811yescharacteristic
194C4×C47#48111yescharacteristic
195Dic47—22no—
196Dic47—82no—
197C2×#466—471no—
198C22×#107—471no—
199C2×#357—471no—
200C2×#357—471no—
201C2×C22⋊C16—471no—
202#1380—471no—
203#1380D4711yescharacteristic, complemented
204C2×C47#187411yescharacteristic
205C2×C47C2×#35711yescharacteristic
206C2×C47#213911yescharacteristic
207C2×C47#218411yescharacteristic
208C2×C47#218511yescharacteristic
209C2×C47#223211yescharacteristic
210C2×C47#223311yescharacteristic
211D47—24no—
212D47—162no—
213C22×D8—471no—
214C23×C8—471no—
215C2×#121—941no—
216C22×C4⋊C4—941no—
217C22×C16—471no—
218C22×C16C2×D4711yescharacteristic
219#466—474no—
220#466—942no—
221C2×#107—471no—
222C2×#107—942no—
223C2×#107—941no—
224C2×#107—942no—
225C2×#107C2×D4711yescharacteristic
226C2×#107C2×D4711yescharacteristic
227C2×#102—941no—
228#357—942no—
229#357—942no—
230C22⋊C16—942no—
231C47#1299111yescharacteristic, complemented, Sylow
232C22×D4—941no—
233C22×D4—1881no—
234C23×C4—471no—
235C23×C4—1881no—
236C2×D8—472no—
237C2×D8—941no—
238C2×D8—941no—
239C2×D8—941no—
240C2×D8—941no—
241C2×D8—941no—
242C22×C8—471no—
243C22×C8—472no—
244C22×C8—472no—
245C22×C8—941no—
246C22×C8—941no—
247C22×C8—941no—
248C22×C8—941no—
249C22×C8C22×D4711yescharacteristic
250#121—944no—
251#121—1882no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—944no—
255C2×C4⋊C4—941no—
256C2×C4⋊C4—1882no—
257C2×C4⋊C4—1881no—
258C2×C22⋊C4—1881no—
259C2×C16—21no—
260C2×C16—21no—
261C2×C16—21no—
262C2×C16—941no—
263C2×C16—941no—
264C2×C16—941no—
265#107—21no—
266#107—21no—
267#107—472no—
268#107—944no—
269#107—941no—
270#107C4×D4712yescomplemented
271#107D18812yescomplemented
272#102—944no—
273C24—471no—
274C24—1881no—
275C2×D4—942no—
276C2×D4—3762no—
277C2×D4—3762no—
278C2×D4—1884no—
279C2×D4—1881no—
280C2×D4—1881no—
281C2×D4—1881no—
282C2×D4—1881no—
283C2×D4—1881no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—471no—
287C22×C4—472no—
288C22×C4—472no—
289C22×C4—941no—
290C22×C4—941no—
291C22×C4—941no—
292C22×C4—941no—
293C22×C4—3762no—
294C22×C4—3762no—
295C22×C4—1884no—
296C22×C4—1881no—
297C22×C4D4×D4711yescharacteristic
298D8—942no—
299D8—942no—
300D8—1881no—
301D8—1881no—
302C2×C8—472no—
303C2×C8—941no—
304C2×C8—941no—
305C2×C8—942no—
306C2×C8—942no—
307C2×C8—942no—
308C2×C8—942no—
309C2×C8C2×C4×D4711yescharacteristic
310C2×C8C2×D18811yescharacteristic
311C2×C8#1062535011yescharacteristic
312C2×C8D4×D4711yescharacteristic
313C2×C8D4×D4711yescharacteristic
314C2×C8#1062535411yescharacteristic
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—942no—
320C4⋊C4—1884no—
321C4⋊C4—1881no—
322C22⋊C4—1884no—
323C16—41no—
324C16—1881no—
325C23—472no—
326C23—472no—
327C23—472no—
328C23—941no—
329C23—941no—
330C23—941no—
331C23—941no—
332C23—3761no—
333C23—3761no—
334C23—3761no—
335C23—3761no—
336C23—3761no—
337C23—3761no—
338C23—1882no—
339C23D8×D4711yescharacteristic
340D4—3768no—
341D4—3761no—
342D4—3761no—
343D4—1882no—
344D4—1882no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—472no—
352C2×C4—941no—
353C2×C4—941no—
354C2×C4—942no—
355C2×C4—942no—
356C2×C4—942no—
357C2×C4—942no—
358C2×C4—3768no—
359C2×C4—3761no—
360C2×C4—3761no—
361C2×C4—1882no—
362C2×C4C22⋊C4×D4711yescharacteristic
363C2×C4#1309399411yescharacteristic
364C2×C4#1309400911yescharacteristic
365C2×C4SD16×D4711yescharacteristic
366C2×C4#1309405111yescharacteristic
367C2×C4#1309405611yescharacteristic
368C8—942no—
369C8#1309398611yescharacteristic
370C8C4×D18811yescharacteristic
371C8#1309399711yescharacteristic
372C8#1309399911yescharacteristic
373C22—474no—
374C22—942no—
375C22—942no—
376C22—942no—
377C22—942no—
378C22—942no—
379C22—942no—
380C22—3762no—
381C22—3762no—
382C22—3762no—
383C22—7521no—
384C22—7521no—
385C22—7521no—
386C22—7521no—
387C22#102×D4711yescharacteristic
388C22#1568925811yescharacteristic
389C22#1568943811yescharacteristic
390C22D16×D4711yescharacteristic
391C22SD32×D4711yescharacteristic
392C22#1568956011yescharacteristic
393C22#1568956511yescharacteristic
394C4—82no—
395C4—82no—
396C4—942no—
397C4—3762no—
398C4#1568900111yescharacteristic
399C4#1568912311yescharacteristic
400C4#1568926311yescharacteristic
401C4#1568944311yescharacteristic
402C2—944no—
403C2—7522no—
404C2#2024370411yescharacteristic
405C2#357×D4711yescharacteristic
406C2#2024509911yescharacteristic
407C2#2024530111yescharacteristic
408C2#2024530611yescharacteristic
409C2#2024557011yescharacteristic
410C2#2024557511yescharacteristic
subgroupG/H
centerC23D8×D47
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
FittingC47×#1380C2
socleC23×C47C2×D8
prime radicalC2×#15689360C2
cosocleC22×C8×C47C23
hypercenter#1380D47
hypocenterC47#12991
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ #1380

Complements

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

NH ≅ G/Ntype
2. C2×#107×D47C2semidirect
3. C2×#15688925C2semidirect
4. C2×#15688923C2semidirect
5. C2×(C2×D47)⋊C16C2semidirect
6. #20241479C2semidirect
7. C47×#1380C2semidirect
14. #15689360C4semidirect
35. C2×D376C2×C4semidirect
54. #13093958D4semidirect
65. #107×C47C2×C4semidirect
66. #107×C47D4semidirect
188. C2×D47#357semidirect
203. #1380D47semidirect
231. C47#12991semidirect
270. #107C4×D47semidirect
271. #107D188semidirect

Maximal subgroups

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

subgroupindex
C2×#156893602
C2×#107×D472
C2×#156889252
C2×#156889232
C2×(C2×D47)⋊C162
#202414792
C47×#13802
#1299147