Finite Group Databasebeta

← #26199436

Subgroups — #26199436

Order 12032.

726 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254310C211yesmaximal, characteristic, complemented
2#20254303C211yesmaximal, characteristic
3C2×#15689470C211yesmaximal, characteristic, complemented
4#20242949C214yesmaximal, complemented
5#20242825C214yesmaximal, complemented
6#20242136C211yesmaximal, characteristic, complemented
7#20242074C211yesmaximal, characteristic, complemented
8#20242057C211yesmaximal, characteristic, complemented
9C47×#2931C211yesmaximal, characteristic, complemented
10C2×#13094024C2211yescharacteristic
11C2×#13094021C2211yescharacteristic
12#15689499—21no—
13#15689495—21no—
14#15689475—21no—
15#15689471—21no—
16#15689470—21no—
17#15689470—21no—
18#15689470C2212yes—
19#15689470C2212yes—
20#15689328C2214yescomplemented
21#15689318C2214yes—
22#15689261—22no—
23#15689256—22no—
24#15689176C2211yescharacteristic
25#15689176C2211yescharacteristic
26C2×#13093946—21no—
27C2×#13093946—21no—
28#15688876C2212yescomplemented
29#15688874C2212yescomplemented
30#15688868C2214yescomplemented
31#15688866C2214yes—
32C2×#131×C47—21no—
33C47×#532C2211yescharacteristic
34C47×#436—21no—
35C47×#436C2212yescomplemented
36C8×D4×C47—21no—
37C8×D4×C47C2212yescomplemented
38C47×#423C2211yescharacteristic
39C47×#408—21no—
40C2×C4×C8×C47C2211yescharacteristic
41C22×Dic94—21no—
42C22×D188—21no—
43C22×D188—21no—
44C22×D188—21no—
45C2×#10625335—24no—
46C2×#10625334—24no—
47#13094024—21no—
48#13094024—21no—
49#13094024—21no—
50#13094024C2312yes—
51#13094021—21no—
52#13094021—21no—
53#13094021—21no—
54#13094021C2312yes—
55C2×C4⋊Dic47—21no—
56C2×C4⋊Dic47—21no—
57C4⋊(C47⋊C8)C2312yes—
58C4⋊(C47⋊C8)C2312yes—
59#13093947—24no—
60#13093946—22no—
61#13093946—22no—
62#13093946—22no—
63#13093946—22no—
64#13093944—24no—
65C22⋊(C47⋊C8)—21no—
66C22⋊(C47⋊C8)—21no—
67C4×C47⋊C8—21no—
68C4×C47⋊C8—21no—
69C2×#41×C47—21no—
70#131×C47—44no—
71C2×M16×C47—21no—
72C2×M16×C47—22no—
73C22×C8×C47—22no—
74C22×C8×C47D411yescharacteristic
75C22×C8×C47D411yescharacteristic
76C4⋊Q8×C47D411yescharacteristic
77#127×C47D411yescharacteristic
78#124×C47—21no—
79#121×C47—21no—
80#121×C47D412yes—
81C4×D4×C47—21no—
82C4×D4×C47C2312yes—
83C2×C42×C47C2311yescharacteristic
84C4⋊C8×C47—21no—
85C4⋊C8×C47C2312yes—
86C22⋊C8×C47—22no—
87C22⋊C8×C47—22no—
88C4×C8×C47D411yescharacteristic
89C4×C8×C47D411yescharacteristic
90C4×C8×C47C2311yescharacteristic
91C4×C8×C47C2311yescharacteristic
92C23×D47—41no—
93#41×C47—42no—
94#41×C47—42no—
95C2×Q8×C47—21no—
96C2×D4×C47—22no—
97C2×D4×C47—21no—
98C2×D4×C47—21no—
99C2×D4×C47C2×D412yes—
100C22×C4×C47—22no—
101C22×C4×C47C2×D411yescharacteristic
102C22×C4×C47C2×D411yescharacteristic
103C22×C4×C47#4111yescharacteristic
104C22×Dic47—41no—
105C2×D188—22no—
106C2×D188—22no—
107C2×D188—22no—
108C2×D188—42no—
109C2×D188—42no—
110C2×D188—41no—
111C2×D188—41no—
112C2×D188—41no—
113C2×D188—82no—
114C2×D188—82no—
115C2×Dic94—22no—
116C2×Dic94—41no—
117C2×Dic94—82no—
118M16×C47—44no—
119M16×C47—42no—
120C2×C8×C47—22no—
121C2×C8×C47—22no—
122C2×C8×C47—22no—
123C2×C8×C47—44no—
124C2×C8×C47C2×D411yescharacteristic
125C2×C8×C47C2×D411yescharacteristic
126C2×C8×C47C2×D411yescharacteristic
127C2×C8×C47C2×D411yescharacteristic
128C4⋊C4×C47—21no—
129C4⋊C4×C47C2×D412yes—
130C22⋊C4×C47—22no—
131C22⋊C4×C47—22no—
132C42×C47C2×D411yescharacteristic
133C42×C47C2×D411yescharacteristic
134C42×C47#4111yescharacteristic
135C42×C47C2411yescharacteristic
136#10625335—84no—
137#10625334—84no—
138C4⋊Dic47—22no—
139C4⋊Dic47—22no—
140C4⋊Dic47—41no—
141C4⋊Dic47—41no—
142C2×C47⋊C8—22no—
143C2×C47⋊C8—22no—
144C23×C47—22no—
145C23×C47#12111yescharacteristic
146C22×D47—42no—
147C22×D47—82no—
148C22×D47—82no—
149C22×D47—82no—
150Q8×C47—42no—
151D4×C47—24no—
152D4×C47—44no—
153D4×C47—42no—
154D4×C47—42no—
155C2×C4×C47—22no—
156C2×C4×C47—22no—
157C2×C4×C47—22no—
158C2×C4×C47—42no—
159C2×C4×C47—42no—
160C2×C4×C47#12111yescharacteristic
161C2×C4×C47#12111yescharacteristic
162C2×C4×C47#12111yescharacteristic
163C2×C4×C47#13512yes—
164C2×C4×C47AGL(1,ℤ/8)12yes—
165C2×C4×C47C22×D411yescharacteristic
166C2×C4×C47C22×D411yescharacteristic
167C2×C4×C47C2×#4111yescharacteristic
168C2×Dic47—42no—
169C2×Dic47—82no—
170D188—42no—
171D188—42no—
172D188—44no—
173D188—82no—
174D188—82no—
175D188—82no—
176D188—82no—
177Dic94—44no—
178Dic94—82no—
179C8×C47—21no—
180C8×C47—21no—
181C8×C47—21no—
182C8×C47—21no—
183C8×C47—42no—
184C47⋊C8—42no—
185C47⋊C8—42no—
186#19198—471nomaximal, Sylow
187C22×C47—21no—
188C22×C47—21no—
189C22×C47—42no—
190C22×C47—42no—
191C22×C47—42no—
192C22×C47#46312yes—
193C22×C47C2×#12111yescharacteristic
194C2×D47—82no—
195C2×D47—82no—
196C2×D47—82no—
197C2×D47—164no—
198C4×C47—21no—
199C4×C47—21no—
200C4×C47—42no—
201C4×C47#46312yes—
202C4×C47#47112yes—
203C4×C47C2×#13511yescharacteristic
204C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
205C4×C47#57612yes—
206Dic47—82no—
207#3206—471no—
208#3205—471no—
209C2×#486—471no—
210#2931D4711yescharacteristic, complemented
211#1701—474no—
212#1681—474no—
213#1564—471no—
214#1549—471no—
215#1545—471no—
216C2×C47—22no—
217C2×C47—82no—
218C2×C47C2×#46311yescharacteristic
219C2×C47#305811yescharacteristic
220C2×C47C2×#47111yescharacteristic
221D47—162no—
222C2×#131—21no—
223#532C2×D4711yescharacteristic
224C2×C4⋊Q8—471no—
225C2×#127—471no—
226#493—941no—
227#492—941no—
228#487—941no—
229#486—472no—
230#486—472no—
231#486—941no—
232#486—941no—
233#486—941no—
234#461—474no—
235#459—474no—
236#448—942no—
237#447—942no—
238#436—21no—
239#436C2×D4712yescomplemented
240C8×D4—21no—
241C8×D4C2×D4712yescomplemented
242#431—471no—
243#431—471no—
244#423C2×D4711yescharacteristic
245C2×#102—941no—
246C2×#102—941no—
247#408—21no—
248C2×C4×C8C2×D4711yescharacteristic
249#335—472no—
250#334—472no—
251#332—474no—
252#331—474no—
253C47#1919811yescharacteristic, complemented, Sylow
254C2×#41—21no—
255C22×Q8—941no—
256C22×D4—941no—
257C22×D4—941no—
258C22×D4—941no—
259C2×SD16—944no—
260C2×D8—944no—
261#131—44no—
262C2×M16—21no—
263C2×M16—22no—
264C22×C8—22no—
265C22×C8C22⋊D4711yescharacteristic
266C22×C8C22⋊D4711yescharacteristic
267C4⋊Q8—472no—
268C4⋊Q8—941no—
269C4⋊Q8—941no—
270C4⋊Q8—941no—
271C4⋊Q8D18811yescharacteristic
272#127—472no—
273#127—941no—
274#127—941no—
275#127—941no—
276#127D18811yescharacteristic
277#124—21no—
278#121—21no—
279#121D18812yes—
280C4×D4—21no—
281C4×D4C22×D4712yes—
282C2×C4⋊C4—941no—
283C2×C4⋊C4—941no—
284C2×C42C22×D4711yescharacteristic
285C4⋊C8—21no—
286C4⋊C8—472no—
287C4⋊C8—472no—
288C4⋊C8C22×D4712yes—
289#103—944no—
290#102—942no—
291#102—942no—
292#102—942no—
293#102—942no—
294#102—944no—
295C22⋊C8—22no—
296C22⋊C8—22no—
297C22⋊C8—941no—
298C22⋊C8—941no—
299C4×C8—941no—
300C4×C8—941no—
301C4×C8C22⋊D4711yescharacteristic
302C4×C8C22⋊D4711yescharacteristic
303C4×C8C22×D4711yescharacteristic
304C4×C8C22×D4711yescharacteristic
305C24—1881no—
306#41—42no—
307#41—42no—
308C2×Q8—21no—
309C2×Q8—942no—
310C2×Q8—3762no—
311C2×Q8—1881no—
312C2×D4—22no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—942no—
316C2×D4—942no—
317C2×D4—942no—
318C2×D4—3762no—
319C2×D4—3762no—
320C2×D4—1881no—
321C2×D4—1881no—
322C2×D4—1881no—
323C2×D4—1882no—
324C2×D4—1882no—
325C2×D4C2×D18812yes—
326C22×C4—22no—
327C22×C4—1881no—
328C22×C4C2×D18811yescharacteristic
329C22×C4#1062535011yescharacteristic
330C22×C4C2×C22⋊D4711yescharacteristic
331SD16—3764no—
332D8—3764no—
333M16—44no—
334M16—42no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—44no—
339C2×C8—942no—
340C2×C8—942no—
341C2×C8C2×C22⋊D4711yescharacteristic
342C2×C8C2×C22⋊D4711yescharacteristic
343C2×C8C2×C22⋊D4711yescharacteristic
344C2×C8C2×C22⋊D4711yescharacteristic
345C4⋊C4—21no—
346C4⋊C4—942no—
347C4⋊C4—942no—
348C4⋊C4—1881no—
349C4⋊C4—1881no—
350C4⋊C4C2×D18812yes—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C42C2×D18811yescharacteristic
354C42#1062535011yescharacteristic
355C42C2×C22⋊D4711yescharacteristic
356C42C23×D4711yescharacteristic
357C23—22no—
358C23—3762no—
359C23—3762no—
360C23—3762no—
361C23—1882no—
362C23#1309399811yescharacteristic
363Q8—42no—
364Q8—3762no—
365Q8—1884no—
366D4—24no—
367D4—44no—
368D4—42no—
369D4—42no—
370D4—3762no—
371D4—3762no—
372D4—3762no—
373D4—3762no—
374D4—1882no—
375D4—1882no—
376D4—1884no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—3762no—
383C2×C4—1882no—
384C2×C4#1309399811yescharacteristic
385C2×C4#1309399811yescharacteristic
386C2×C4#1309399811yescharacteristic
387C2×C4#1309404412yes—
388C2×C4#1309405012yes—
389C2×C4C22×C22⋊D4711yescharacteristic
390C2×C4C22×D18811yescharacteristic
391C2×C4C2×#1062535011yescharacteristic
392C8—21no—
393C8—21no—
394C8—21no—
395C8—21no—
396C8—42no—
397C8—1882no—
398C8—1882no—
399C22—21no—
400C22—21no—
401C22—42no—
402C22—42no—
403C22—42no—
404C22—3762no—
405C22—3762no—
406C22—3762no—
407C22—7524no—
408C22#1568933812yes—
409C22C2×#1309399811yescharacteristic
410C4—21no—
411C4—21no—
412C4—42no—
413C4—3762no—
414C4#1568933812yes—
415C4#1568938612yes—
416C4C2×#1309404411yescharacteristic
417C4C2×#1309405011yescharacteristic
418C4#1568992812yes—
419C2—22no—
420C2—82no—
421C2—7522no—
422C2C2×#1568933811yescharacteristic
423C2#2025303611yescharacteristic
424C2C2×#1568938611yescharacteristic
subgroupG/H
centerC22C2×#13093998
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#2931C2
socleC22×C47C2×#121
prime radical#20254303C2
hypercenter#2931D47
hypocenterC47#19198
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2931

Complements

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

NH ≅ G/Ntype
1. #20254310C2semidirect
3. C2×#15689470C2semidirect
4. #20242949C2semidirect
5. #20242825C2semidirect
6. #20242136C2semidirect
7. #20242074C2semidirect
8. #20242057C2semidirect
9. C47×#2931C2semidirect
20. #15689328C22semidirect
28. #15688876C22semidirect
29. #15688874C22semidirect
30. #15688868C22semidirect
35. C47×#436C22semidirect
37. C8×D4×C47C22semidirect
210. #2931D47semidirect
239. #436C2×D47semidirect
241. C8×D4C2×D47semidirect
253. C47#19198semidirect

Maximal subgroups

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

subgroupindex
#202543102
#202543032
C2×#156894702
#202429492
#202428252
#202421362
#202420742
#202420572
C47×#29312
#1919847