Finite Group Databasebeta

← #26201554

Subgroups — #26201554

Order 12032.

686 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254331C211yesmaximal, characteristic
2#20253193C211yesmaximal, characteristic, complemented
3C2×#15689323C211yesmaximal, characteristic
4#20242804C214yesmaximal, complemented
5#20242736C214yesmaximal, complemented
6#20242159C211yesmaximal, characteristic, complemented
7#20242055C211yesmaximal, characteristic
8#20242043C211yesmaximal, characteristic, complemented
9C47×#2982C211yesmaximal, characteristic, complemented
10C22×#10625336—21no—
11#15689680C2211yescharacteristic
12C2×#13094021C2211yescharacteristic
13#15689496C2212yes—
14#15689470C2212yes—
15#15689430—22no—
16#15689418C2212yescomplemented
17#15689408C2212yescomplemented
18#15689324—22no—
19#15689323—22no—
20#15689323C2214yes—
21#15689276—22no—
22#15689163—21no—
23#15689147C2211yescharacteristic
24C2×C4⋊(C47⋊C8)C2211yescharacteristic
25#15689132—21no—
26C2×#13093944—21no—
27#15688874C2214yescomplemented
28#15688871C2212yescomplemented
29#15688870C2212yescomplemented
30#15688866C2214yes—
31C2×C4×Q8×C47C2211yescharacteristic
32C8×Q8×C47—22no—
33C8×Q8×C47C2214yescomplemented
34C47×#431—22no—
35C47×#431C2211yescharacteristic
36C47×#431C2211yescharacteristic
37C22×D188—21no—
38C22×D188—21no—
39C22×D188—21no—
40C2×#10625325—24no—
41C2×#10625336—24no—
42C2×#10625336—44no—
43C2×#10625329—21no—
44C22×C47⋊C8—21no—
45#13094024C2311yescharacteristic
46#13094024C2311yescharacteristic
47#13094021—21no—
48#13094021—21no—
49#13094021—21no—
50#13094021C2312yes—
51#13094016—21no—
52#13094003—21no—
53#13094003—21no—
54C4×Dic94—21no—
55C4×Dic94—21no—
56#13093983—21no—
57#13093957—22no—
58#13093955—22no—
59C4⋊(C47⋊C8)—21no—
60C4⋊(C47⋊C8)—21no—
61C4⋊(C47⋊C8)C2312yes—
62C4⋊(C47⋊C8)C2312yes—
63#13093948—22no—
64#13093948—22no—
65#13093946—24no—
66#13093944—24no—
67C22×Q8×C47D411yescharacteristic
68C4×Q8×C47—21no—
69C4×Q8×C47—21no—
70C4×Q8×C47D412yes—
71C4×Q8×C47C2312yes—
72C2×C4⋊C4×C47—21no—
73C2×C4⋊C4×C47D411yescharacteristic
74C2×C42×C47—21no—
75C2×C42×C47C2311yescharacteristic
76C4⋊C8×C47—24no—
77C4⋊C8×C47D412yes—
78C4⋊C8×C47C2312yes—
79C22⋊C8×C47—21no—
80C22⋊C8×C47D412yes—
81C4×C8×C47—24no—
82C4×C8×C47—21no—
83C4×C8×C47C2312yes—
84C23×D47—41no—
85C2×Q8×C47—21no—
86C2×Q8×C47—22no—
87C2×Q8×C47SD1614yescomplemented
88C2×Q8×C47C2×D412yes—
89C22×C4×C47—21no—
90C22×C4×C47—21no—
91C22×C4×C47C2×D411yescharacteristic
92C22×C4×C47C2×D411yescharacteristic
93C22×C4×C47#4111yescharacteristic
94C2×D188—22no—
95C2×D188—22no—
96C2×D188—22no—
97C2×D188—42no—
98C2×D188—42no—
99C2×D188—41no—
100C2×D188—41no—
101C2×D188—41no—
102C2×D188—82no—
103C2×D188—82no—
104C2×Dic94—21no—
105C2×Dic94—21no—
106C2×C8×C47—22no—
107C2×C8×C47C2×D414yes—
108C4⋊C4×C47—22no—
109C4⋊C4×C47—21no—
110C4⋊C4×C47—22no—
111C4⋊C4×C47C2×D412yes—
112C42×C47—22no—
113C42×C47—22no—
114C42×C47C2×D411yescharacteristic
115C42×C47C2×D411yescharacteristic
116C42×C47#4111yescharacteristic
117C42×C47C2411yescharacteristic
118C22⋊Dic47—41no—
119#10625336—48no—
120C4⋊Dic47—21no—
121C4⋊Dic47—21no—
122C4⋊Dic47—21no—
123C4⋊Dic47—21no—
124Dic47⋊C4—41no—
125Dic47⋊C4—41no—
126C4×Dic47—41no—
127#10625329—42no—
128C2×C47⋊C8—22no—
129C2×C47⋊C8—22no—
130C2×C47⋊C8—42no—
131#10625325—84no—
132C23×C47#12111yescharacteristic
133C22×D47—42no—
134C22×D47—82no—
135C22×D47—82no—
136C22×D47—82no—
137Q8×C47—22no—
138Q8×C47—24no—
139Q8×C47C2×SD1614yes—
140C2×C4×C47—22no—
141C2×C4×C47—24no—
142C2×C4×C47—22no—
143C2×C4×C47—22no—
144C2×C4×C47—41no—
145C2×C4×C47—81no—
146C2×C4×C47#12111yescharacteristic
147C2×C4×C47#12111yescharacteristic
148C2×C4×C47#12111yescharacteristic
149C2×C4×C47C2×SD1612yes—
150C2×C4×C47#13512yes—
151C2×C4×C47C22×D411yescharacteristic
152C2×C4×C47C22×D411yescharacteristic
153C2×C4×C47C2×#4111yescharacteristic
154C2×Dic47—41no—
155C2×Dic47—41no—
156D188—42no—
157D188—42no—
158D188—44no—
159D188—82no—
160D188—82no—
161D188—82no—
162D188—82no—
163Dic94—42no—
164Dic94—81no—
165C8×C47—24no—
166C8×C47—42no—
167C47⋊C8—42no—
168C47⋊C8—42no—
169#19396—471nomaximal, Sylow
170C22×C47—21no—
171C22×C47—21no—
172C22×C47#46512yes—
173C22×C47C2×#12111yescharacteristic
174C2×D47—82no—
175C2×D47—82no—
176C2×D47—82no—
177C2×D47—164no—
178C4×C47—21no—
179C4×C47—24no—
180C4×C47—21no—
181C4×C47—21no—
182C4×C47—42no—
183C4×C47#46512yes—
184C4×C47C22×SD1611yescharacteristic
185C4×C47C2×#13511yescharacteristic
186C4×C47#57712yes—
187Dic47—81no—
188Dic47—81no—
189#3208—471no—
190#3078—471no—
191C2×#460—471no—
192#2982D4711yescharacteristic, complemented
193#1678—474no—
194#1666—474no—
195#1570—471no—
196#1545—471no—
197#1542—471no—
198C2×C47—22no—
199C2×C47#304311yescharacteristic
200C2×C47C2×#46511yescharacteristic
201C2×C47#306811yescharacteristic
202D47—162no—
203C22×SD16—941no—
204#533—471no—
205C2×#127—471no—
206C2×C4×Q8C2×D4711yescharacteristic
207#492—472no—
208#486—472no—
209#479—942no—
210#477—472no—
211#475—472no—
212#460—474no—
213#460—942no—
214#460—942no—
215#450—942no—
216C8×Q8—22no—
217C8×Q8C2×D4714yescomplemented
218#431—22no—
219#431C2×D4711yescharacteristic
220#431C2×D4711yescharacteristic
221#428—941no—
222#423—471no—
223C2×C4⋊C8—471no—
224#419—941no—
225C2×#102—941no—
226#334—474no—
227#333—472no—
228#332—472no—
229#331—474no—
230C47#1939611yescharacteristic, complemented, Sylow
231C22×Q8D18811yescharacteristic
232C22×D4—941no—
233C22×D4—941no—
234C22×D4—941no—
235C2×SD16—944no—
236C2×SD16—944no—
237C2×SD16—1884no—
238C2×M16—941no—
239C22×C8—941no—
240C4⋊Q8—471no—
241C4⋊Q8—471no—
242#127—472no—
243#127—941no—
244#127—941no—
245#127—941no—
246#125—941no—
247C22⋊Q8—941no—
248C22⋊Q8—941no—
249C4×Q8—21no—
250C4×Q8—21no—
251C4×Q8—941no—
252C4×Q8—941no—
253C4×Q8D18812yes—
254C4×Q8C22×D4712yes—
255#117—941no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4D18811yescharacteristic
258C2×C42—21no—
259C2×C42C22×D4711yescharacteristic
260#107—942no—
261#106—942no—
262C4⋊C8—24no—
263C4⋊C8—472no—
264C4⋊C8—472no—
265C4⋊C8—941no—
266C4⋊C8—941no—
267C4⋊C8C22⋊D4712yes—
268C4⋊C8C22×D4712yes—
269#103—942no—
270#103—942no—
271#102—944no—
272#102—944no—
273C22⋊C8—21no—
274C22⋊C8C22⋊D4712yes—
275C4×C8—24no—
276C4×C8—21no—
277C4×C8C22×D4712yes—
278C24—1881no—
279C2×Q8—21no—
280C2×Q8—22no—
281C2×Q8—941no—
282C2×Q8—941no—
283C2×Q8#1062532514yescomplemented
284C2×Q8C2×D18812yes—
285C2×D4—942no—
286C2×D4—942no—
287C2×D4—942no—
288C2×D4—3762no—
289C2×D4—3762no—
290C2×D4—1881no—
291C2×D4—1881no—
292C2×D4—1881no—
293C2×D4—1882no—
294C2×D4—1882no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4C2×D18811yescharacteristic
298C22×C4#1062535011yescharacteristic
299C22×C4C2×C22⋊D4711yescharacteristic
300SD16—3764no—
301SD16—1888no—
302M16—1882no—
303C2×C8—22no—
304C2×C8—942no—
305C2×C8—942no—
306C2×C8—1882no—
307C2×C8C2×C22⋊D4714yes—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—22no—
311C4⋊C4—941no—
312C4⋊C4—941no—
313C4⋊C4—941no—
314C4⋊C4—941no—
315C4⋊C4—1881no—
316C4⋊C4—1881no—
317C4⋊C4C2×D18812yes—
318C22⋊C4—1881no—
319C42—22no—
320C42—22no—
321C42—1881no—
322C42C2×D18811yescharacteristic
323C42#1062535011yescharacteristic
324C42C2×C22⋊D4711yescharacteristic
325C42C23×D4711yescharacteristic
326C23—3762no—
327C23—3762no—
328C23—3762no—
329C23—1882no—
330C23#1309399811yescharacteristic
331Q8—24no—
332Q8—22no—
333Q8—3761no—
334Q8—1882no—
335Q8C2×#1062532514yes—
336D4—3762no—
337D4—3762no—
338D4—3762no—
339D4—3762no—
340D4—1882no—
341D4—1882no—
342D4—1884no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—41no—
348C2×C4—81no—
349C2×C4—1881no—
350C2×C4—1881no—
351C2×C4#1309399811yescharacteristic
352C2×C4#1309399811yescharacteristic
353C2×C4#1309399811yescharacteristic
354C2×C4C2×#1062532512yes—
355C2×C4#1309404612yes—
356C2×C4C22×C22⋊D4711yescharacteristic
357C2×C4C22×D18811yescharacteristic
358C2×C4C2×#1062535011yescharacteristic
359C8—24no—
360C8—42no—
361C8—1882no—
362C8—1882no—
363C22—21no—
364C22—21no—
365C22—3762no—
366C22—3762no—
367C22—3762no—
368C22—7524no—
369C22#1568935212yes—
370C22C2×#1309399811yescharacteristic
371C4—24no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—42no—
376C4—3761no—
377C4—3761no—
378C4#1568935212yes—
379C4C22×#1062532511yescharacteristic
380C4C2×#1309404611yescharacteristic
381C4#1568993112yes—
382C2—22no—
383C2—7522no—
384C2#2025290911yescharacteristic
385C2C2×#1568935211yescharacteristic
386C2#2025311611yescharacteristic
subgroupG/H
centerC22C2×#13093998
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#2982C2
socleC22×C47C2×#121
prime radicalC2×#13094021C22
hypercenter#2982D47
hypocenterC47#19396
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
2. #20253193C2semidirect
4. #20242804C2semidirect
5. #20242736C2semidirect
6. #20242159C2semidirect
8. #20242043C2semidirect
9. C47×#2982C2semidirect
16. #15689418C22semidirect
17. #15689408C22semidirect
27. #15688874C22semidirect
28. #15688871C22semidirect
29. #15688870C22semidirect
33. C8×Q8×C47C22semidirect
87. C2×Q8×C47SD16semidirect
192. #2982D47semidirect
217. C8×Q8C2×D47semidirect
230. C47#19396semidirect
283. C2×Q8#10625325semidirect

Maximal subgroups

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

subgroupindex
#202543312
#202531932
C2×#156893232
#202428042
#202427362
#202421592
#202420552
#202420432
C47×#29822
#1939647