Finite Group Databasebeta

← #26133228

Subgroups — #26133228

Order 12032.

530 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689361C211yesmaximal, characteristic
2C2×#15689160C211yesmaximal, characteristic
3C2×#15688927C211yesmaximal, characteristic, complemented
4C2×(C2×D47)⋊C16C211yesmaximal, characteristic
5#20241480C211yesmaximal, characteristic, complemented
6#20241479C211yesmaximal, characteristic, complemented
7C2×C47×#358C211yesmaximal, characteristic, complemented
8C22×C8×D47C2211yescharacteristic
9C2×#13094000—21no—
10C2×#13093982—21no—
11C22×C47⋊C16C2211yescharacteristic
12#15689361—22no—
13#15689361C414yescomplemented
14#15689160—22no—
15#15689160—22no—
16C2×#13093958C2211yescharacteristic
17C2×#13093958C2211yescharacteristic
18C2×#13093955—21no—
19C2×#13093948—21no—
20#15688927—22no—
21(C2×D47)⋊C16—22no—
22C22×Q16×C47C2211yescharacteristic
23C22×C16×C47C2211yescharacteristic
24C2×#107×C47C2211yescharacteristic
25C47×#358—22no—
26C22×C4×D47D411yescharacteristic
27C22×C47⋊C8D411yescharacteristic
28C2×C8×D47—21no—
29C2×C8×D47—21no—
30C2×C8×D47—21no—
31C2×C8×D47—21no—
32C2×C8×D47C2×C412yes—
33C2×C8×D47D412yes—
34#13094000—24no—
35#13094000—42no—
36#13093982—24no—
37#13093982—42no—
38C2×C4⋊Dic47—21no—
39C2×C4⋊Dic47—21no—
40C2×Dic47⋊C4—41no—
41C2×(C2×D47)⋊C4—41no—
42C2×(C2×D47)⋊C4—41no—
43C2×C4×Dic47—41no—
44C2×C47⋊C16—21no—
45C2×C47⋊C16—21no—
46C2×C47⋊C16—21no—
47#13093958—21no—
48#13093958—21no—
49#13093958C2×C412yes—
50#13093958D412yes—
51#13093955—24no—
52#13093948—24no—
53C22×Q8×C47—21no—
54C2×Q16×C47—21no—
55C2×Q16×C47—21no—
56C2×Q16×C47—21no—
57C2×Q16×C47—21no—
58C2×Q16×C47—21no—
59C2×Q16×C47C2×C412yes—
60C22×C8×C47C2311yescharacteristic
61C2×C4⋊C4×C47—21no—
62C2×C16×C47—21no—
63C2×C16×C47—21no—
64C2×C16×C47—21no—
65#107×C47—21no—
66#107×C47D412yes—
67C23×D47D811yescharacteristic
68C2×Q8×C47—22no—
69C2×Q8×C47—41no—
70C2×Q8×C47—41no—
71C2×Q8×C47—41no—
72C2×Q8×C47—41no—
73C2×Q8×C47—41no—
74C22×C4×C47—41no—
75C22×C4×C47—41no—
76C22×C4×C47C2×D411yescharacteristic
77C22×Dic47—41no—
78C22×Dic47—41no—
79C22×Dic47D811yescharacteristic
80C2×C4×D47—21no—
81C2×C4×D47—21no—
82C2×C4×D47—21no—
83C2×C4×D47—21no—
84C2×C4×D47C22⋊C412yes—
85C2×C4×D47SD1612yes—
86Q16×C47—22no—
87Q16×C47—22no—
88Q16×C47—41no—
89Q16×C47—41no—
90C2×C8×C47C22×C411yescharacteristic
91C2×C8×C47C2×D411yescharacteristic
92C2×C8×C47C2×D411yescharacteristic
93C2×C8×C47C2×D411yescharacteristic
94C2×C8×C47#4111yescharacteristic
95C2×C8×C47#4111yescharacteristic
96C4⋊C4×C47—22no—
97C4⋊C4×C47—41no—
98(C2×D47)⋊C4—44no—
99(C2×D47)⋊C4—44no—
100C4⋊Dic47—22no—
101C4⋊Dic47—22no—
102C4⋊Dic47—41no—
103C4⋊Dic47—41no—
104Dic47⋊C4—44no—
105C4×Dic47—44no—
106C2×C47⋊C8—21no—
107C2×C47⋊C8—21no—
108C2×C47⋊C8C22⋊C412yes—
109C8×D47—22no—
110C8×D47—22no—
111C8×D47—22no—
112C8×D47—22no—
113C16×C47—41no—
114C47⋊C16—41no—
115C23×C47C2×D811yescharacteristic
116C22×D47—21no—
117C22×D47—21no—
118C22×D47—21no—
119C22×D47—21no—
120C22×D47—21no—
121C22×D47—21no—
122C22×D47#10212yes—
123Q8×C47—42no—
124Q8×C47—42no—
125Q8×C47—81no—
126Q8×C47—81no—
127C2×C4×C47—42no—
128C2×C4×C47—42no—
129C2×C4×C47—81no—
130C2×C4×C47—81no—
131C2×C4×C47—81no—
132C2×C4×C47—81no—
133C2×C4×C47C2×C22⋊C411yescharacteristic
134C2×C4×C47#12011yescharacteristic
135C2×C4×C47#12311yescharacteristic
136C2×C4×C47C2×SD1611yescharacteristic
137C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
138C2×C4×C47#13711yescharacteristic
139C2×Dic47—21no—
140C2×Dic47—21no—
141C2×Dic47—42no—
142C2×Dic47—42no—
143C2×Dic47—81no—
144C2×Dic47—81no—
145C2×Dic47—81no—
146C2×Dic47—81no—
147C2×Dic47#10212yes—
148C4×D47—22no—
149C4×D47—22no—
150C4×D47—22no—
151C4×D47—22no—
152C8×C47C4×D411yescharacteristic
153C8×C47C4×D411yescharacteristic
154C8×C47#12111yescharacteristic
155C8×C47#12111yescharacteristic
156C47⋊C8—22no—
157#12994—471nomaximal, Sylow
158C22×C47C2×#10211yescharacteristic
159C22×C47#44711yescharacteristic
160C22×C47#48011yescharacteristic
161C22×C47#50811yescharacteristic
162C22×C47#50811yescharacteristic
163C22×C47#51011yescharacteristic
164C22×C47#51011yescharacteristic
165C2×D47—22no—
166C2×D47—22no—
167C2×D47—22no—
168C2×D47—41no—
169C2×D47—41no—
170C2×D47—41no—
171C2×D47—41no—
172C4×C47—82no—
173C4×C47—82no—
174C4×C47#38611yescharacteristic
175C4×C47#41711yescharacteristic
176C4×C47#44811yescharacteristic
177C4×C47#48111yescharacteristic
178Dic47—22no—
179Dic47—82no—
180Dic47—82no—
181C2×#467—471no—
182C2×#427—471no—
183C2×#358—471no—
184C2×#358D4711yescharacteristic, complemented
185C2×C22⋊C16—471no—
186#1380—471no—
187#1380—471no—
188C2×C47#187411yescharacteristic
189C2×C47#213811yescharacteristic
190C2×C47#214111yescharacteristic
191C2×C47#218911yescharacteristic
192C2×C47#218911yescharacteristic
193C2×C47#223711yescharacteristic
194C2×C47#223711yescharacteristic
195D47—42no—
196C22×Q16C2×D4711yescharacteristic
197C23×C8—471no—
198C2×C22⋊Q8—941no—
199C2×#117—941no—
200C22×C16—471no—
201C22×C16C2×D4711yescharacteristic
202#467—474no—
203#467—942no—
204#427—942no—
205#427—942no—
206C2×#107—471no—
207C2×#107—471no—
208C2×#107C2×D4711yescharacteristic
209C2×#106—941no—
210C2×#103—941no—
211#358—22no—
212#358—942no—
213C22⋊C16—942no—
214C47#1299411yescharacteristic, complemented, Sylow
215C22×Q8—21no—
216C23×C4—471no—
217C2×Q16—21no—
218C2×Q16—21no—
219C2×Q16—21no—
220C2×Q16—21no—
221C2×Q16—21no—
222C2×Q16C4×D4712yes—
223C22×C8—471no—
224C22×C8—472no—
225C22×C8—472no—
226C22×C8—941no—
227C22×C8—941no—
228C22×C8—941no—
229C22×C8—941no—
230C22×C8C22×D4711yescharacteristic
231C22⋊Q8—944no—
232C22⋊Q8—1882no—
233#117—944no—
234#117—1882no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—941no—
237C2×C4⋊C4—941no—
238C2×C4⋊C4—1881no—
239C2×C22⋊C4—1881no—
240C2×C22⋊C4—1881no—
241C2×C42—1881no—
242C2×C16—21no—
243C2×C16—21no—
244C2×C16—21no—
245C2×C16—941no—
246C2×C16—941no—
247C2×C16—941no—
248#107—21no—
249#107—472no—
250#107—472no—
251#107—941no—
252#107—941no—
253#107C22⋊D4712yes—
254#106—944no—
255#103—944no—
256C24—471no—
257C2×Q8—22no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—471no—
266C22×C4—472no—
267C22×C4—472no—
268C22×C4—941no—
269C22×C4—941no—
270C22×C4—941no—
271C22×C4—941no—
272C22×C4—1881no—
273C22×C4—1881no—
274C22×C4D4×D4711yescharacteristic
275Q16—22no—
276Q16—22no—
277Q16—41no—
278Q16—41no—
279C2×C8—472no—
280C2×C8—941no—
281C2×C8—941no—
282C2×C8—942no—
283C2×C8—942no—
284C2×C8—942no—
285C2×C8—942no—
286C2×C8C2×C4×D4711yescharacteristic
287C2×C8#1062535011yescharacteristic
288C2×C8D4×D4711yescharacteristic
289C2×C8D4×D4711yescharacteristic
290C2×C8#1062535211yescharacteristic
291C2×C8C2×C22⋊D4711yescharacteristic
292C4⋊C4—22no—
293C4⋊C4—41no—
294C4⋊C4—942no—
295C4⋊C4—942no—
296C4⋊C4—1884no—
297C4⋊C4—1881no—
298C4⋊C4—1881no—
299C22⋊C4—1884no—
300C22⋊C4—1884no—
301C42—1884no—
302C16—41no—
303C16—1881no—
304C23—472no—
305C23—941no—
306C23—941no—
307C23—941no—
308C23—941no—
309C23—941no—
310C23—941no—
311C23D8×D4711yescharacteristic
312Q8—42no—
313Q8—42no—
314Q8—81no—
315Q8—81no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—472no—
323C2×C4—941no—
324C2×C4—941no—
325C2×C4—942no—
326C2×C4—942no—
327C2×C4—942no—
328C2×C4—942no—
329C2×C4—3761no—
330C2×C4—3761no—
331C2×C4—3761no—
332C2×C4—3761no—
333C2×C4—1882no—
334C2×C4—1882no—
335C2×C4C22⋊C4×D4711yescharacteristic
336C2×C4#1309399211yescharacteristic
337C2×C4#1309400611yescharacteristic
338C2×C4SD16×D4711yescharacteristic
339C2×C4#1309404811yescharacteristic
340C2×C4#1309405311yescharacteristic
341C8—942no—
342C8(C22⋊D47)⋊C411yescharacteristic
343C8C4×C22⋊D4711yescharacteristic
344C8#1309399511yescharacteristic
345C8#1309399711yescharacteristic
346C22—942no—
347C22—942no—
348C22—942no—
349C22—1881no—
350C22—1881no—
351C22—1881no—
352C22—1881no—
353C22#102×D4711yescharacteristic
354C22#1568925511yescharacteristic
355C22#1568943511yescharacteristic
356C22#1568955411yescharacteristic
357C22#1568955611yescharacteristic
358C22#1568956211yescharacteristic
359C22#1568956511yescharacteristic
360C4—82no—
361C4—82no—
362C4—942no—
363C4—3762no—
364C4—3762no—
365C4#1568899911yescharacteristic
366C4#1568912111yescharacteristic
367C4#1568926011yescharacteristic
368C4#1568944011yescharacteristic
369C2—1882no—
370C2#2024370211yescharacteristic
371C2#2024509411yescharacteristic
372C2#2024510611yescharacteristic
373C2#2024532511yescharacteristic
374C2#2024532711yescharacteristic
375C2#2024559411yescharacteristic
376C2#2024559511yescharacteristic
subgroupG/H
centerC23D8×D47
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
FittingC2×C47×#358C2
socleC23×C47C2×D8
prime radicalC23×D47D8
cosocleC22×C8×C47C23
hypercenterC2×#358D47
hypocenterC47#12994
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#358

Complements

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

NH ≅ G/Ntype
3. C2×#15688927C2semidirect
5. #20241480C2semidirect
6. #20241479C2semidirect
7. C2×C47×#358C2semidirect
13. #15689361C4semidirect
184. C2×#358D47semidirect
214. C47#12994semidirect

Maximal subgroups

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

subgroupindex
C2×#156893612
C2×#156891602
C2×#156889272
C2×(C2×D47)⋊C162
#202414802
#202414792
C2×C47×#3582
#1299447