Finite Group Databasebeta

← #26115452

Subgroups — #26115452

Order 12032.

460 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253947C211yesmaximal, characteristic
2#20244331C211yesmaximal, characteristic, complemented
3#20244306C211yesmaximal, characteristic
4#20241886C211yesmaximal, characteristic, complemented
5#20241882C211yesmaximal, characteristic
6#20241677C211yesmaximal, characteristic, complemented
7C4×#107×C47C211yesmaximal, characteristic, complemented
8#15689680C2211yescharacteristic
9#15689673C2211yescharacteristic
10#15689521C411yescharacteristic, complemented
11#15689496C411yescharacteristic, complemented
12#15689474C411yescharacteristic, complemented
13#15689470C411yescharacteristic, complemented
14#15689352—21no—
15#15689352—21no—
16C4×#10625325—21no—
17C4×#10625325—21no—
18#15689176C2211yescharacteristic
19C2×#13093952—22no—
20C4×#10625329C2211yescharacteristic
21#15688883—22no—
22#15688859—22no—
23#15688856—22no—
24#15688845—21no—
25#15688844—21no—
26C2×#107×C47—21no—
27C2×C4×C8×C47C2211yescharacteristic
28C4×C4⋊C4×C47C2211yescharacteristic
29C4×C4⋊C4×C47C2211yescharacteristic
30C47×#340—21no—
31C2×#10625350—21no—
32C2×#10625325—21no—
33C2×#10625325—21no—
34C2×#10625329D412yes—
35#13094024C2×C411yescharacteristic
36#13094024C2×C411yescharacteristic
37#13094024C2×C411yescharacteristic
38#13094021C2×C411yescharacteristic
39#13094016—21no—
40#13094012—21no—
41#13094003—21no—
42#13094003—21no—
43#13093998—21no—
44#13093998—21no—
45C4×Dic94—21no—
46C4×Dic94—21no—
47C4×D188—21no—
48C4×D188—21no—
49#13093983—21no—
50#13093956—21no—
51#13093956—21no—
52C4⋊(C47⋊C8)—21no—
53#13093952—44no—
54#13093949—21no—
55#13093949—21no—
56#13093946—21no—
57#13093946—21no—
58C22⋊(C47⋊C8)D412yes—
59#13093935—21no—
60C4×C47⋊C8—21no—
61C4×C47⋊C8—21no—
62C22×C8×C47D411yescharacteristic
63C22×C8×C47D411yescharacteristic
64C2×C4⋊C4×C47—21no—
65C2×C4⋊C4×C47—21no—
66C2×C42×C47—21no—
67C2×C42×C47—21no—
68C2×C42×C47C2311yescharacteristic
69#107×C47—24no—
70C4×C8×C47C2×C411yescharacteristic
71C4×C8×C47C2×C411yescharacteristic
72C4×C8×C47D411yescharacteristic
73C4×C8×C47D411yescharacteristic
74#95×C47—21no—
75#95×C47—21no—
76C22×C4×C47—41no—
77C22×C4×C47—41no—
78C22×C4×C47C2×D411yescharacteristic
79C22×C4×C47C2×D411yescharacteristic
80C22×C4×C47C2×D411yescharacteristic
81C2×C22⋊D47—41no—
82#10625350—42no—
83#10625350—42no—
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—22no—
92C2×C8×C47—41no—
93C2×C8×C47C22⋊C411yescharacteristic
94C2×C8×C47C22⋊C411yescharacteristic
95C2×C8×C47C22⋊C411yescharacteristic
96C2×C8×C47C22⋊C411yescharacteristic
97C4⋊C4×C47—22no—
98C4⋊C4×C47—22no—
99C4⋊C4×C47—41no—
100C4⋊C4×C47—41no—
101C42×C47—42no—
102C42×C47—42no—
103C42×C47C22×C411yescharacteristic
104C42×C47C2×D411yescharacteristic
105C42×C47#4111yescharacteristic
106C42×C47#4111yescharacteristic
107C22⋊Dic47—41no—
108(C2×D47)⋊C4—41no—
109(C2×D47)⋊C4—41no—
110C4⋊Dic47—21no—
111C4⋊Dic47—21no—
112C4⋊Dic47—21no—
113C4⋊Dic47—21no—
114C4⋊Dic47—21no—
115C4⋊Dic47—21no—
116Dic47⋊C4—41no—
117Dic47⋊C4—41no—
118C4×Dic47—41no—
119#10625329—22no—
120#10625329—22no—
121C2×C47⋊C8—22no—
122C2×C47⋊C8—22no—
123#10625325—42no—
124#10625325—42no—
125C23×C47#12711yescharacteristic
126C22×D47—41no—
127C2×C4×C47—21no—
128C2×C4×C47—21no—
129C2×C4×C47—41no—
130C2×C4×C47—42no—
131C2×C4×C47—42no—
132C2×C4×C47—81no—
133C2×C4×C47—81no—
134C2×C4×C47—81no—
135C2×C4×C47—81no—
136C2×C4×C47C2×C22⋊C411yescharacteristic
137C2×C4×C47C4×D411yescharacteristic
138C2×C4×C47C4×D411yescharacteristic
139C2×C4×C47#12111yescharacteristic
140C2×C4×C47#12111yescharacteristic
141C2×C4×C47#12411yescharacteristic
142C2×C4×C47AGL(1,ℤ/8)12yes—
143C2×C4×C47#13712yes—
144C22⋊D47—81no—
145C22⋊D47—81no—
146C2×Dic47—41no—
147C2×Dic47—41no—
148C2×Dic47—41no—
149D188—41no—
150D188—41no—
151D188—42no—
152D188—81no—
153C4×D47—82no—
154Dic94—41no—
155Dic94—41no—
156Dic94—42no—
157Dic94—81no—
158C8×C47—21no—
159C8×C47—21no—
160C8×C47—21no—
161C8×C47#10412yescomplemented
162C47⋊C8—42no—
163C47⋊C8—42no—
164#9722—471nomaximal, Sylow
165C22×C47—21no—
166C22×C47—21no—
167C22×C47#39011yescharacteristic
168C22×C47#49611yescharacteristic
169C22×C47#49711yescharacteristic
170C2×D47—81no—
171C2×D47—81no—
172C2×D47—81no—
173C4×C47—21no—
174C4×C47—21no—
175C4×C47—21no—
176C4×C47—82no—
177C4×C47—82no—
178C4×C47C2×#10411yescharacteristic
179C4×C47#42111yescharacteristic
180C4×C47#44012yes—
181C4×C47#46812yes—
182Dic47—81no—
183Dic47—81no—
184Dic47—81no—
185#3167—471no—
186#1986—471no—
187#1980—471no—
188C4×#107D4711yescharacteristic, complemented
189#1503—471no—
190#1502—471no—
191#1451—471no—
192C2×C47—21no—
193C2×C47—21no—
194C2×C47#197311yescharacteristic
195C2×C47#197611yescharacteristic
196C2×C47#198011yescharacteristic
197D47—161no—
198#533—471no—
199#532—471no—
200#498—471no—
201#492—471no—
202#487—471no—
203#486—471no—
204#465—941no—
205#465—941no—
206C4×SD16—941no—
207C4×SD16—941no—
208#431—471no—
209C2×#107—21no—
210C2×#104—942no—
211C4×M16—471no—
212C2×C4×C8C2×D4711yescharacteristic
213C4×C4⋊C4C2×D4711yescharacteristic
214C4×C4⋊C4C2×D4711yescharacteristic
215#340—21no—
216#339—942no—
217#328—942no—
218#327—942no—
219#322—941no—
220#322—941no—
221C47#972211yescharacteristic, complemented, Sylow
222C2×#41—941no—
223C2×SD16—941no—
224C2×SD16—941no—
225C2×M16—472no—
226C22×C8C22⋊D4711yescharacteristic
227C22×C8C22⋊D4711yescharacteristic
228C4⋊Q8—471no—
229C4⋊Q8—471no—
230C4⋊Q8—471no—
231#127—471no—
232#125—941no—
233#124—941no—
234C22⋊Q8—941no—
235C22⋊Q8—941no—
236#121—941no—
237#121—941no—
238C4×Q8—941no—
239C4×Q8—941no—
240C4×D4—941no—
241C4×D4—941no—
242#117—941no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42C22×D4711yescharacteristic
248#107—24no—
249#106—941no—
250#106—941no—
251C4⋊C8—941no—
252#104—1884no—
253#103—941no—
254#103—941no—
255#102—941no—
256#102—941no—
257C22⋊C8—472no—
258#97—941no—
259C4×C8—941no—
260C4×C8—941no—
261C4×C8C4×D4711yescharacteristic
262C4×C8C4×D4711yescharacteristic
263C4×C8D18811yescharacteristic
264C4×C8D18811yescharacteristic
265#95—21no—
266#95—21no—
267#41—1882no—
268#41—1882no—
269C2×Q8—941no—
270C2×Q8—941no—
271C2×Q8—941no—
272C2×D4—941no—
273C2×D4—941no—
274C2×D4—941no—
275C2×D4—1881no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—1881no—
279C22×C4D4×D4711yescharacteristic
280C22×C4D4×D4711yescharacteristic
281C22×C4C2×C22⋊D4711yescharacteristic
282SD16—1882no—
283SD16—1882no—
284M16—942no—
285M16—942no—
286C2×C8—22no—
287C2×C8—41no—
288C2×C8—942no—
289C2×C8—942no—
290C2×C8(C2×D47)⋊C411yescharacteristic
291C2×C8(C2×D47)⋊C411yescharacteristic
292C2×C8(C2×D47)⋊C411yescharacteristic
293C2×C8(C2×D47)⋊C411yescharacteristic
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—941no—
299C4⋊C4—941no—
300C4⋊C4—941no—
301C4⋊C4—941no—
302C4⋊C4—941no—
303C4⋊C4—941no—
304C4⋊C4—1881no—
305C4⋊C4—1881no—
306C22⋊C4—1881no—
307C22⋊C4—1881no—
308C22⋊C4—1881no—
309C42—42no—
310C42—42no—
311C42—1881no—
312C42C2×C4×D4711yescharacteristic
313C42C2×D18811yescharacteristic
314C42#1062535411yescharacteristic
315C42#1062535411yescharacteristic
316C23—1881no—
317C23#1309402011yescharacteristic
318Q8—3761no—
319Q8—1882no—
320Q8—1881no—
321Q8—1881no—
322D4—3761no—
323D4—3761no—
324D4—3761no—
325D4—1882no—
326D4—1881no—
327D4—1881no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—3762no—
338C2×C4—1881no—
339C2×C4—1881no—
340C2×C4—1881no—
341C2×C4C2×(C2×D47)⋊C411yescharacteristic
342C2×C4#1309398611yescharacteristic
343C2×C4#1309398611yescharacteristic
344C2×C4#1309399911yescharacteristic
345C2×C4#1309399911yescharacteristic
346C2×C4#1309401311yescharacteristic
347C2×C4#1309404812yes—
348C2×C4#1309405612yes—
349C8—21no—
350C8—21no—
351C8—21no—
352C8—1882no—
353C8—1882no—
354C8#1309395212yescomplemented
355C22—21no—
356C22—21no—
357C22—3761no—
358C22—3761no—
359C22—3761no—
360C22#1568902411yescharacteristic
361C22#1568951111yescharacteristic
362C22#1568951611yescharacteristic
363C4—21no—
364C4—21no—
365C4—21no—
366C4—82no—
367C4—82no—
368C4—3761no—
369C4—3761no—
370C4—3761no—
371C4C2×#1309395211yescharacteristic
372C4#1568914111yescharacteristic
373C4#1568922212yes—
374C4#1568937012yes—
375C2—21no—
376C2—21no—
377C2—7521no—
378C2#2024426711yescharacteristic
379C2#2024428411yescharacteristic
380C2#2024430611yescharacteristic
subgroupG/H
centerC2×C4#13093999
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC4×#107×C47C2
socleC22×C47#390
prime radical#15689673C22
cosocleC2×C42×C47C23
hypercenterC4×#107D47
hypocenterC47#9722
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#107

Complements

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

NH ≅ G/Ntype
2. #20244331C2semidirect
4. #20241886C2semidirect
6. #20241677C2semidirect
7. C4×#107×C47C2semidirect
10. #15689521C4semidirect
11. #15689496C4semidirect
12. #15689474C4semidirect
13. #15689470C4semidirect
161. C8×C47#104semidirect
188. C4×#107D47semidirect
221. C47#9722semidirect
354. C8#13093952semidirect

Maximal subgroups

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

subgroupindex
#202539472
#202443312
#202443062
#202418862
#202418822
#202416772
C4×#107×C472
#972247