Finite Group Databasebeta

← #26115629

Subgroups — #26115629

Order 12032.

428 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
1#20253711C211yesmaximal, characteristic
2#20242136C211yesmaximal, characteristic, complemented
3#20241886C211yesmaximal, characteristic, complemented
4#20241882C211yesmaximal, characteristic
5#20241790C211yesmaximal, characteristic, complemented
6#20241786C211yesmaximal, characteristic
7C8×C4⋊C4×C47C211yesmaximal, characteristic, complemented
8#15689680C2211yescharacteristic
9#15689673C2211yescharacteristic
10#15689478—21no—
11#15689478—21no—
12#15689474C411yescharacteristic, complemented
13#15689474C411yescharacteristic, complemented
14#15689470C411yescharacteristic, complemented
15#15689470C411yescharacteristic, complemented
16#15689176C2211yescharacteristic
17#15689176C2211yescharacteristic
18#15689120—21no—
19#15689120—21no—
20#15688876—21no—
21#15688874—21no—
22#15688859—22no—
23#15688856—22no—
24#15688853—22no—
25#15688850—22no—
26C2×C4⋊C8×C47C2211yescharacteristic
27C2×C4×C8×C47—21no—
28C2×C4×C8×C47C2211yescharacteristic
29C4×C4⋊C4×C47C2211yescharacteristic
30C47×#336—21no—
31C2×#10625350—21no—
32#13094024C2×C411yescharacteristic
33#13094024C2×C411yescharacteristic
34#13094024C2×C411yescharacteristic
35#13094021C2×C411yescharacteristic
36#13094016—21no—
37#13094012—21no—
38#13094003—21no—
39#13094003—21no—
40#13093998—21no—
41#13093998—21no—
42C4×Dic94—21no—
43C4×Dic94—21no—
44C4×D188—21no—
45C4×D188—21no—
46#13093983—21no—
47C4⋊(C47⋊C8)—21no—
48C4⋊(C47⋊C8)—21no—
49#13093949—21no—
50#13093949—21no—
51#13093949—21no—
52#13093949—21no—
53#13093946—21no—
54#13093946—21no—
55#13093946—21no—
56#13093946—21no—
57C22⋊(C47⋊C8)—21no—
58C22⋊(C47⋊C8)—21no—
59C4×C47⋊C8—21no—
60C4×C47⋊C8—21no—
61C22×C8×C47—21no—
62C22×C8×C47D411yescharacteristic
63C22×C8×C47D411yescharacteristic
64C2×C4⋊C4×C47—21no—
65C2×C42×C47—21no—
66C2×C42×C47C2311yescharacteristic
67C4⋊C8×C47—21no—
68C4⋊C8×C47D412yes—
69C4×C8×C47—42no—
70C4×C8×C47C2×C411yescharacteristic
71C4×C8×C47C2×C411yescharacteristic
72C4×C8×C47D411yescharacteristic
73C4×C8×C47D411yescharacteristic
74#95×C47D412yes—
75C22×C4×C47—22no—
76C22×C4×C47C2×D411yescharacteristic
77C22×C4×C47C2×D411yescharacteristic
78C22×C4×C47#4111yescharacteristic
79C2×C22⋊D47—41no—
80#10625350—41no—
81#10625350—41no—
82#10625350—41no—
83#10625350—41no—
84C2×D188—21no—
85C2×D188—21no—
86C2×D188—21no—
87C2×C4×D47—41no—
88C2×Dic94—21no—
89C2×Dic94—21no—
90C2×Dic94—21no—
91C2×C8×C47—21no—
92C2×C8×C47—21no—
93C2×C8×C47—21no—
94C2×C8×C47—21no—
95C2×C8×C47—22no—
96C2×C8×C47—41no—
97C2×C8×C47—41no—
98C2×C8×C47C22⋊C411yescharacteristic
99C2×C8×C47C22⋊C411yescharacteristic
100C2×C8×C47C22⋊C411yescharacteristic
101C2×C8×C47C22⋊C411yescharacteristic
102C4⋊C4×C47—24no—
103C42×C47—41no—
104C42×C47—41no—
105C42×C47C22×C411yescharacteristic
106C42×C47C2×D411yescharacteristic
107C42×C47C2×D411yescharacteristic
108C42×C47#4111yescharacteristic
109C22⋊Dic47—41no—
110(C2×D47)⋊C4—41no—
111(C2×D47)⋊C4—41no—
112C4⋊Dic47—21no—
113C4⋊Dic47—21no—
114C4⋊Dic47—21no—
115C4⋊Dic47—21no—
116C4⋊Dic47—21no—
117C4⋊Dic47—21no—
118Dic47⋊C4—41no—
119Dic47⋊C4—41no—
120C4×Dic47—41no—
121C2×C47⋊C8—41no—
122C2×C47⋊C8—41no—
123C23×C47#12111yescharacteristic
124C22×D47—41no—
125C2×C4×C47—22no—
126C2×C4×C47—24no—
127C2×C4×C47—41no—
128C2×C4×C47—42no—
129C2×C4×C47—42no—
130C2×C4×C47C2×C22⋊C411yescharacteristic
131C2×C4×C47C4×D411yescharacteristic
132C2×C4×C47C4×D411yescharacteristic
133C2×C4×C47#12111yescharacteristic
134C2×C4×C47#12411yescharacteristic
135C2×C4×C47#12711yescharacteristic
136C2×C4×C47#13511yescharacteristic
137C2×C4×C47#13511yescharacteristic
138C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
139C2×C4×C47#13711yescharacteristic
140C22⋊D47—81no—
141C22⋊D47—81no—
142C2×Dic47—41no—
143C2×Dic47—41no—
144C2×Dic47—41no—
145D188—41no—
146D188—41no—
147D188—42no—
148D188—81no—
149C4×D47—81no—
150C4×D47—81no—
151Dic94—41no—
152Dic94—41no—
153Dic94—42no—
154Dic94—81no—
155C8×C47—21no—
156C8×C47—21no—
157C8×C47—21no—
158C8×C47—21no—
159C8×C47—42no—
160C47⋊C8—81no—
161C47⋊C8—81no—
162#9751—471nomaximal, Sylow
163C22×C47—21no—
164C22×C47—21no—
165C22×C47#39011yescharacteristic
166C22×C47#46311yescharacteristic
167C22×C47#46411yescharacteristic
168C2×D47—81no—
169C2×D47—81no—
170C2×D47—81no—
171C4×C47—21no—
172C4×C47—21no—
173C4×C47—21no—
174C4×C47—44no—
175C4×C47#41611yescharacteristic
176C4×C47#41711yescharacteristic
177C4×C47#44312yes—
178C4×C47#46311yescharacteristic
179C4×C47#46411yescharacteristic
180Dic47—81no—
181Dic47—81no—
182Dic47—81no—
183#3131—471no—
184C8×C4⋊C4D4711yescharacteristic, complemented
185#1564—471no—
186#1503—471no—
187#1502—471no—
188#1481—471no—
189#1480—471no—
190C2×C47—21no—
191C2×C47—21no—
192C2×C47#197411yescharacteristic
193C2×C47#197611yescharacteristic
194C2×C47#198311yescharacteristic
195D47—161no—
196#533—471no—
197#532—471no—
198#488—941no—
199#488—941no—
200#487—471no—
201#487—471no—
202#486—471no—
203#486—471no—
204#431—471no—
205#431—471no—
206C2×C4⋊C8C2×D4711yescharacteristic
207#416—941no—
208#416—941no—
209C2×C4×C8—21no—
210C2×C4×C8C2×D4711yescharacteristic
211C4×C4⋊C4C2×D4711yescharacteristic
212#336—21no—
213#335—941no—
214#334—941no—
215#328—942no—
216#327—942no—
217Q8⋊C8—942no—
218D4⋊C8—942no—
219C47#975111yescharacteristic, complemented, Sylow
220C2×#41—941no—
221C22×C8—21no—
222C22×C8C22⋊D4711yescharacteristic
223C22×C8C22⋊D4711yescharacteristic
224C4⋊Q8—471no—
225C4⋊Q8—471no—
226C4⋊Q8—471no—
227#127—471no—
228#125—941no—
229#124—941no—
230C22⋊Q8—941no—
231C22⋊Q8—941no—
232#121—941no—
233#121—941no—
234C4×Q8—941no—
235C4×Q8—941no—
236C4×D4—941no—
237C4×D4—941no—
238#117—941no—
239C2×C4⋊C4—21no—
240C2×C42—21no—
241C2×C42C22×D4711yescharacteristic
242C4⋊C8—21no—
243C4⋊C8—941no—
244C4⋊C8—941no—
245C4⋊C8D18812yes—
246#103—941no—
247#103—941no—
248#103—941no—
249#103—941no—
250#102—941no—
251#102—941no—
252#102—941no—
253#102—941no—
254C22⋊C8—941no—
255C22⋊C8—941no—
256C4×C8—42no—
257C4×C8—941no—
258C4×C8—941no—
259C4×C8C4×D4711yescharacteristic
260C4×C8C4×D4711yescharacteristic
261C4×C8D18811yescharacteristic
262C4×C8D18811yescharacteristic
263#95D18812yes—
264#41—1881no—
265#41—1881no—
266#41—1881no—
267#41—1881no—
268C2×Q8—941no—
269C2×Q8—941no—
270C2×Q8—941no—
271C2×D4—941no—
272C2×D4—941no—
273C2×D4—941no—
274C2×D4—1881no—
275C22×C4—22no—
276C22×C4—1881no—
277C22×C4C2×D18811yescharacteristic
278C22×C4#1062535011yescharacteristic
279C22×C4C2×C22⋊D4711yescharacteristic
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—22no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—1881no—
288C2×C8—1881no—
289C2×C8(C2×D47)⋊C411yescharacteristic
290C2×C8(C2×D47)⋊C411yescharacteristic
291C2×C8(C2×D47)⋊C411yescharacteristic
292C2×C8(C2×D47)⋊C411yescharacteristic
293C4⋊C4—24no—
294C4⋊C4—941no—
295C4⋊C4—941no—
296C4⋊C4—941no—
297C4⋊C4—941no—
298C4⋊C4—941no—
299C4⋊C4—941no—
300C4⋊C4—1881no—
301C4⋊C4—1881no—
302C22⋊C4—1881no—
303C22⋊C4—1881no—
304C22⋊C4—1881no—
305C42—41no—
306C42—41no—
307C42—1881no—
308C42C2×C4×D4711yescharacteristic
309C42C2×D18811yescharacteristic
310C42C2×D18811yescharacteristic
311C42#1062535011yescharacteristic
312C23—1881no—
313C23#1309399811yescharacteristic
314Q8—3761no—
315Q8—1882no—
316Q8—1881no—
317Q8—1881no—
318D4—3761no—
319D4—3761no—
320D4—3761no—
321D4—1882no—
322D4—1881no—
323D4—1881no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—3761no—
330C2×C4—3761no—
331C2×C4—1881no—
332C2×C4—1881no—
333C2×C4—1881no—
334C2×C4C2×(C2×D47)⋊C411yescharacteristic
335C2×C4C4×D18811yescharacteristic
336C2×C4C4×D18811yescharacteristic
337C2×C4#1309399811yescharacteristic
338C2×C4#1309401211yescharacteristic
339C2×C4#1309402111yescharacteristic
340C2×C4#1309404411yescharacteristic
341C2×C4#1309404411yescharacteristic
342C2×C4#1309405011yescharacteristic
343C2×C4#1309405511yescharacteristic
344C8—21no—
345C8—21no—
346C8—21no—
347C8—21no—
348C8—42no—
349C8—3761no—
350C8—3761no—
351C22—21no—
352C22—21no—
353C22—3761no—
354C22—3761no—
355C22—3761no—
356C22#1568902511yescharacteristic
357C22#1568933811yescharacteristic
358C22#1568934511yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—44no—
363C4—3761no—
364C4—3761no—
365C4—3761no—
366C4#1568911711yescharacteristic
367C4#1568912211yescharacteristic
368C4#1568924112yes—
369C4#1568933811yescharacteristic
370C4#1568934511yescharacteristic
371C2—21no—
372C2—21no—
373C2—7521no—
374C2#2024427611yescharacteristic
375C2#2024428611yescharacteristic
376C2#2024432011yescharacteristic
subgroupG/H
centerC2×C4#13093998
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC8×C4⋊C4×C47C2
socleC22×C47#390
prime radical#15689673C22
cosocleC2×C42×C47C23
hypercenterC8×C4⋊C4D47
hypocenterC47#9751
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C8×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #20242136C2semidirect
3. #20241886C2semidirect
5. #20241790C2semidirect
7. C8×C4⋊C4×C47C2semidirect
12. #15689474C4semidirect
13. #15689474C4semidirect
14. #15689470C4semidirect
15. #15689470C4semidirect
184. C8×C4⋊C4D47semidirect
219. C47#9751semidirect

Maximal subgroups

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

subgroupindex
#202537112
#202421362
#202418862
#202418822
#202417902
#202417862
C8×C4⋊C4×C472
#975147