Finite Group Databasebeta

← #26201447

Subgroups — #26201447

Order 12032.

708 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254315C211yesmaximal, characteristic, complemented
2C2×#15689403C211yesmaximal, characteristic, complemented
3#20252852C211yesmaximal, characteristic
4#20242811C214yesmaximal, complemented
5#20242729C214yesmaximal, complemented
6#20242159C211yesmaximal, characteristic, complemented
7#20242072C211yesmaximal, characteristic, complemented
8#20242035C211yesmaximal, characteristic, complemented
9C47×#2972C211yesmaximal, characteristic, complemented
10C2×#13094050—21no—
11#15689673C2211yescharacteristic
12C2×#13094024C2211yescharacteristic
13#15689496C2212yescomplemented
14#15689474C2212yescomplemented
15#15689403—22no—
16#15689403C2214yes—
17#15689338—22no—
18#15689329—22no—
19#15689328C2212yes—
20#15689318C2212yes—
21#15689264—22no—
22C2×#13093957—21no—
23#15689147C2211yescharacteristic
24C2×C4⋊(C47⋊C8)C2211yescharacteristic
25#15689122—21no—
26C2×#13093945—21no—
27#15688874C2214yescomplemented
28#15688870C2214yescomplemented
29#15688868C2212yescomplemented
30#15688866C2212yescomplemented
31C2×C4×D4×C47C2211yescharacteristic
32C8×D4×C47—22no—
33C8×D4×C47C2214yes—
34C47×#431C2211yescharacteristic
35C47×#431C2211yescharacteristic
36C2×C22⋊C8×C47—22no—
37C2×#10625350—21no—
38C22×Dic94—21no—
39#13094050—44no—
40C2×#10625325—24no—
41C2×#10625335—22no—
42C2×#10625334—22no—
43C2×#10625329—21no—
44C22×C47⋊C8—21no—
45#13094024—21no—
46#13094024—21no—
47#13094024—21no—
48#13094024C2311yescharacteristic
49#13094024C2312yes—
50#13094021C2311yescharacteristic
51#13094012—21no—
52#13093998—21no—
53#13093998—21no—
54C4×D188—21no—
55C4×D188—21no—
56C2×C4⋊Dic47—21no—
57C2×C4⋊Dic47—21no—
58#13093957—24no—
59C4⋊(C47⋊C8)—21no—
60C4⋊(C47⋊C8)—21no—
61C4⋊(C47⋊C8)C2312yes—
62C4⋊(C47⋊C8)C2312yes—
63#13093949—24no—
64#13093947—22no—
65#13093945—24no—
66#13093944—22no—
67C22×D4×C47D411yescharacteristic
68C23×C4×C47—21no—
69C22×C8×C47—24no—
70C4×D4×C47—21no—
71C4×D4×C47—21no—
72C4×D4×C47D412yes—
73C4×D4×C47C2312yes—
74C2×C4⋊C4×C47D411yescharacteristic
75C2×C22⋊C4×C47—21no—
76C2×C42×C47C2311yescharacteristic
77C4⋊C8×C47D412yes—
78C4⋊C8×C47C2312yes—
79C22⋊C8×C47—21no—
80C22⋊C8×C47—24no—
81C22⋊C8×C47D412yes—
82C4×C8×C47—21no—
83C4×C8×C47C2312yes—
84C24×C47—21no—
85C2×D4×C47—21no—
86C2×D4×C47—22no—
87C2×D4×C47SD1614yescomplemented
88C2×D4×C47C2×D412yes—
89C22×C4×C47—21no—
90C22×C4×C47—22no—
91C22×C4×C47—22no—
92C22×C4×C47—82no—
93C22×C4×C47C2×D411yescharacteristic
94C22×C4×C47C2×D411yescharacteristic
95C22×C4×C47#4111yescharacteristic
96C2×C22⋊D47—41no—
97C22×Dic47—41no—
98#10625350—42no—
99#10625350—42no—
100C2×D188—21no—
101C2×D188—21no—
102C2×D188—21no—
103C2×C4×D47—41no—
104C2×Dic94—21no—
105C2×Dic94—22no—
106C2×Dic94—41no—
107C2×Dic94—82no—
108C2×C8×C47—22no—
109C2×C8×C47—84no—
110C2×C8×C47C2×D414yes—
111C4⋊C4×C47—21no—
112C4⋊C4×C47C2×D412yes—
113C22⋊C4×C47—22no—
114C22⋊C4×C47—22no—
115C42×C47C2×D411yescharacteristic
116C42×C47C2×D411yescharacteristic
117C42×C47#4111yescharacteristic
118C42×C47C2411yescharacteristic
119#10625335—44no—
120#10625334—44no—
121(C2×D47)⋊C4—41no—
122(C2×D47)⋊C4—41no—
123C4⋊Dic47—22no—
124C4⋊Dic47—21no—
125C4⋊Dic47—22no—
126C4⋊Dic47—21no—
127C4⋊Dic47—41no—
128C4⋊Dic47—41no—
129#10625329—42no—
130C2×C47⋊C8—22no—
131C2×C47⋊C8—22no—
132C2×C47⋊C8—42no—
133#10625325—84no—
134C23×C47—22no—
135C23×C47—24no—
136C23×C47—81no—
137C23×C47—81no—
138C23×C47#12111yescharacteristic
139C22×D47—41no—
140D4×C47—22no—
141D4×C47—24no—
142D4×C47C2×SD1614yes—
143C2×C4×C47—22no—
144C2×C4×C47—22no—
145C2×C4×C47—41no—
146C2×C4×C47—44no—
147C2×C4×C47—82no—
148C2×C4×C47—81no—
149C2×C4×C47#12111yescharacteristic
150C2×C4×C47#12111yescharacteristic
151C2×C4×C47#12111yescharacteristic
152C2×C4×C47C2×SD1612yes—
153C2×C4×C47#13512yes—
154C2×C4×C47C22×D411yescharacteristic
155C2×C4×C47C22×D411yescharacteristic
156C2×C4×C47C2×#4111yescharacteristic
157C22⋊D47—82no—
158C2×Dic47—42no—
159C2×Dic47—41no—
160C2×Dic47—82no—
161D188—41no—
162D188—41no—
163D188—42no—
164D188—81no—
165C4×D47—82no—
166Dic94—41no—
167Dic94—41no—
168Dic94—44no—
169Dic94—82no—
170C8×C47—24no—
171C8×C47—42no—
172C47⋊C8—42no—
173C47⋊C8—42no—
174#19385—471nomaximal, Sylow
175C22×C47—21no—
176C22×C47—21no—
177C22×C47—24no—
178C22×C47—42no—
179C22×C47—42no—
180C22×C47—84no—
181C22×C47#46512yes—
182C22×C47C2×#12111yescharacteristic
183C2×D47—81no—
184C2×D47—81no—
185C2×D47—81no—
186C4×C47—21no—
187C4×C47—21no—
188C4×C47—21no—
189C4×C47—42no—
190C4×C47#46512yes—
191C4×C47C22×SD1611yescharacteristic
192C4×C47C2×#13511yescharacteristic
193C4×C47#57612yes—
194Dic47—82no—
195Dic47—81no—
196#3206—471no—
197C2×#474—471no—
198#3036—471no—
199#2972D4711yescharacteristic, complemented
200#1679—474no—
201#1665—474no—
202#1570—471no—
203#1549—471no—
204#1540—471no—
205C2×C47—22no—
206C2×C47—44no—
207C2×C47#304111yescharacteristic
208C2×C47C2×#46511yescharacteristic
209C2×C47#306911yescharacteristic
210D47—161no—
211C2×AGL(1,ℤ/8)—941no—
212#532—471no—
213C2×C4⋊Q8—471no—
214C2×C4×D4C2×D4711yescharacteristic
215#492—472no—
216#487—472no—
217#474—474no—
218#474—942no—
219#463—942no—
220#461—472no—
221#461—942no—
222#459—472no—
223#448—942no—
224C8×D4—22no—
225C8×D4C2×D4714yes—
226#431C2×D4711yescharacteristic
227#431C2×D4711yescharacteristic
228C2×#107—941no—
229#423—471no—
230C2×C4⋊C8—471no—
231#417—941no—
232C2×#102—941no—
233C2×C22⋊C8—22no—
234#334—474no—
235#332—472no—
236#332—474no—
237#331—472no—
238C47#1938511yescharacteristic, complemented, Sylow
239C2×#41—941no—
240C22×Q8—941no—
241C22×D4D18811yescharacteristic
242C23×C4—21no—
243AGL(1,ℤ/8)—1884no—
244C2×SD16—942no—
245C2×SD16—944no—
246C2×D8—942no—
247C2×M16—941no—
248C22×C8—24no—
249C22×C8—941no—
250C4⋊Q8—472no—
251C4⋊Q8—471no—
252C4⋊Q8—941no—
253C4⋊Q8—941no—
254C4⋊Q8—941no—
255#127—471no—
256#124—941no—
257#121—941no—
258#121—941no—
259C4×D4—21no—
260C4×D4—21no—
261C4×D4—941no—
262C4×D4—941no—
263C4×D4D18812yes—
264C4×D4C22×D4712yes—
265C2×C4⋊C4—941no—
266C2×C4⋊C4—941no—
267C2×C4⋊C4D18811yescharacteristic
268C2×C22⋊C4—21no—
269C2×C42C22×D4711yescharacteristic
270#107—944no—
271C4⋊C8—472no—
272C4⋊C8—472no—
273C4⋊C8—941no—
274C4⋊C8—941no—
275C4⋊C8C22⋊D4712yes—
276C4⋊C8C22×D4712yes—
277#103—944no—
278#103—942no—
279#102—944no—
280#102—942no—
281C22⋊C8—21no—
282C22⋊C8—24no—
283C22⋊C8C22⋊D4712yes—
284C4×C8—21no—
285C4×C8C22×D4712yes—
286C24—21no—
287#41—1882no—
288#41—1882no—
289C2×Q8—942no—
290C2×Q8—941no—
291C2×Q8—3762no—
292C2×Q8—1881no—
293C2×D4—21no—
294C2×D4—22no—
295C2×D4—941no—
296C2×D4—941no—
297C2×D4—941no—
298C2×D4—1881no—
299C2×D4#1062532514yescomplemented
300C2×D4C2×D18812yes—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—82no—
305C22×C4—1881no—
306C22×C4—1881no—
307C22×C4C2×D18811yescharacteristic
308C22×C4#1062535011yescharacteristic
309C22×C4C2×C22⋊D4711yescharacteristic
310SD16—3764no—
311SD16—1884no—
312D8—1884no—
313M16—1882no—
314C2×C8—22no—
315C2×C8—84no—
316C2×C8—942no—
317C2×C8—942no—
318C2×C8—1882no—
319C2×C8C2×C22⋊D4714yes—
320C4⋊C4—21no—
321C4⋊C4—941no—
322C4⋊C4—942no—
323C4⋊C4—942no—
324C4⋊C4—941no—
325C4⋊C4—1881no—
326C4⋊C4—1881no—
327C4⋊C4C2×D18812yes—
328C22⋊C4—22no—
329C22⋊C4—22no—
330C22⋊C4—1881no—
331C22⋊C4—1881no—
332C42C2×D18811yescharacteristic
333C42#1062535011yescharacteristic
334C42C2×C22⋊D4711yescharacteristic
335C42C23×D4711yescharacteristic
336C23—22no—
337C23—24no—
338C23—81no—
339C23—81no—
340C23—1881no—
341C23#1309399811yescharacteristic
342Q8—3762no—
343Q8—1881no—
344Q8—1881no—
345Q8—1884no—
346D4—24no—
347D4—22no—
348D4—3761no—
349D4—3762no—
350D4—1881no—
351D4—1881no—
352D4—1882no—
353D4C2×#1062532514yes—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—44no—
358C2×C4—82no—
359C2×C4—81no—
360C2×C4—3762no—
361C2×C4—3762no—
362C2×C4—1881no—
363C2×C4—1882no—
364C2×C4#1309399811yescharacteristic
365C2×C4#1309399811yescharacteristic
366C2×C4#1309399811yescharacteristic
367C2×C4C2×#1062532512yes—
368C2×C4#1309404612yes—
369C2×C4C22×C22⋊D4711yescharacteristic
370C2×C4C22×D18811yescharacteristic
371C2×C4C2×#1062535011yescharacteristic
372C8—24no—
373C8—42no—
374C8—1882no—
375C8—1882no—
376C22—21no—
377C22—21no—
378C22—24no—
379C22—42no—
380C22—42no—
381C22—84no—
382C22—3761no—
383C22—3761no—
384C22—3761no—
385C22#1568935212yes—
386C22C2×#1309399811yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—42no—
391C4—3761no—
392C4—3762no—
393C4#1568935212yes—
394C4C22×#1062532511yescharacteristic
395C4C2×#1309404611yescharacteristic
396C4#1568992612yes—
397C2—22no—
398C2—44no—
399C2—7521no—
400C2#2025289111yescharacteristic
401C2C2×#1568935211yescharacteristic
402C2#2025312511yescharacteristic
subgroupG/H
centerC22C2×#13093998
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#2972C2
socleC22×C47C2×#121
prime radical#20252852C2
hypercenter#2972D47
hypocenterC47#19385
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2972

Complements

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

NH ≅ G/Ntype
1. #20254315C2semidirect
2. C2×#15689403C2semidirect
4. #20242811C2semidirect
5. #20242729C2semidirect
6. #20242159C2semidirect
7. #20242072C2semidirect
8. #20242035C2semidirect
9. C47×#2972C2semidirect
13. #15689496C22semidirect
14. #15689474C22semidirect
27. #15688874C22semidirect
28. #15688870C22semidirect
29. #15688868C22semidirect
30. #15688866C22semidirect
87. C2×D4×C47SD16semidirect
199. #2972D47semidirect
238. C47#19385semidirect
299. C2×D4#10625325semidirect

Maximal subgroups

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

subgroupindex
#202543152
C2×#156894032
#202528522
#202428112
#202427292
#202421592
#202420722
#202420352
C47×#29722
#1938547