Finite Group Databasebeta

← #26199414

Subgroups — #26199414

Order 12032.

714 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254516C211yesmaximal, characteristic, complemented
2#20254306C211yesmaximal, characteristic
3#20252101C211yesmaximal, characteristic, complemented
4#20242918C214yesmaximal, complemented
5#20242794C214yesmaximal, complemented
6#20242136C211yesmaximal, characteristic, complemented
7#20242069C211yesmaximal, characteristic, complemented
8#20242056C211yesmaximal, characteristic, complemented
9C2×C47×#486C211yesmaximal, characteristic, complemented
10C2×#13094030—21no—
11#15689673C2211yescharacteristic
12#15689519—21no—
13#15689499—21no—
14#15689471—21no—
15#15689469—21no—
16#15689441—22no—
17#15689410C2214yescomplemented
18#15689315C2214yes—
19#15689257—22no—
20#15689202—21no—
21#15689202C2212yescomplemented
22C8×D188—21no—
23C8×D188C2212yescomplemented
24#15689176C2211yescharacteristic
25#15689176C2211yescharacteristic
26#15689147C2211yescharacteristic
27#15689089—21no—
28#15688876C2212yescomplemented
29#15688874C2212yescomplemented
30#15688872C2214yescomplemented
31#15688867C2214yes—
32C2×C4⋊Q8×C47C2211yescharacteristic
33C2×#127×C47C2211yescharacteristic
34C47×#486—21no—
35C47×#486—21no—
36C47×#486C2212yes—
37C47×#486C2212yes—
38C2×#102×C47—21no—
39C2×#102×C47—21no—
40C2×C4×C8×C47C2211yescharacteristic
41C2×#10625350—21no—
42C2×#10625334—24no—
43#13094030—44no—
44C2×#10625329—21no—
45C2×#10625324—22no—
46C2×C8×D47—22no—
47#13094024D411yescharacteristic
48#13094021D411yescharacteristic
49#13094012—21no—
50#13093998—21no—
51#13093998D412yes—
52C4×D188—21no—
53C4×D188C2312yes—
54#13093955—24no—
55C4⋊(C47⋊C8)—21no—
56C4⋊(C47⋊C8)C2312yes—
57C4⋊(C47⋊C8)C2312yes—
58C4⋊(C47⋊C8)C2312yes—
59#13093945—24no—
60#13093944—24no—
61C22⋊(C47⋊C8)—21no—
62C22⋊(C47⋊C8)—21no—
63(C2×D47)⋊C8—22no—
64(C2×D47)⋊C8—22no—
65C4×C47⋊C8—21no—
66C4×C47⋊C8—21no—
67C22×Q8×C47—21no—
68C22×D4×C47—21no—
69C22×D4×C47—21no—
70C22×D4×C47—21no—
71C22×C8×C47D411yescharacteristic
72C22×C8×C47D411yescharacteristic
73C4⋊Q8×C47—21no—
74C4⋊Q8×C47—21no—
75C4⋊Q8×C47—21no—
76C4⋊Q8×C47C2312yes—
77#127×C47—21no—
78#127×C47—21no—
79#127×C47—21no—
80#127×C47C2312yes—
81C2×C4⋊C4×C47—21no—
82C2×C4⋊C4×C47—21no—
83C2×C42×C47C2311yescharacteristic
84#102×C47—22no—
85#102×C47—22no—
86#102×C47—22no—
87#102×C47—22no—
88C4×C8×C47D411yescharacteristic
89C4×C8×C47D411yescharacteristic
90C4×C8×C47C2311yescharacteristic
91C4×C8×C47C2311yescharacteristic
92C24×C47—41no—
93C2×Q8×C47—22no—
94C2×Q8×C47—41no—
95C2×Q8×C47—82no—
96C2×D4×C47—22no—
97C2×D4×C47—22no—
98C2×D4×C47—22no—
99C2×D4×C47—41no—
100C2×D4×C47—41no—
101C2×D4×C47—41no—
102C2×D4×C47—42no—
103C2×D4×C47—42no—
104C2×D4×C47—82no—
105C2×D4×C47—82no—
106C22×C4×C47—41no—
107C22×C4×C47C2×D411yescharacteristic
108C22×C4×C47C2×D411yescharacteristic
109C22×C4×C47#4111yescharacteristic
110C2×C22⋊D47—22no—
111#10625350—42no—
112#10625350—42no—
113C2×D188—21no—
114C2×D188—21no—
115C2×D188C2×D412yes—
116C2×C4×D47—22no—
117C2×Dic94—21no—
118C2×C8×C47—22no—
119C2×C8×C47—22no—
120C2×C8×C47C2×D411yescharacteristic
121C2×C8×C47C2×D411yescharacteristic
122C2×C8×C47C2×D411yescharacteristic
123C2×C8×C47C2×D411yescharacteristic
124C4⋊C4×C47—22no—
125C4⋊C4×C47—22no—
126C4⋊C4×C47—41no—
127C4⋊C4×C47—41no—
128C42×C47C2×D411yescharacteristic
129C42×C47C2×D411yescharacteristic
130C42×C47#4111yescharacteristic
131C42×C47C2411yescharacteristic
132#10625334—84no—
133(C2×D47)⋊C4—22no—
134(C2×D47)⋊C4—22no—
135C4⋊Dic47—21no—
136C4⋊Dic47C2×D412yes—
137#10625329—42no—
138C2×C47⋊C8—22no—
139C2×C47⋊C8—22no—
140C2×C47⋊C8—22no—
141#10625324—44no—
142C8×D47—44no—
143C23×C47—42no—
144C23×C47—82no—
145C23×C47—82no—
146C23×C47—82no—
147C23×C47#12111yescharacteristic
148C22×D47—22no—
149Q8×C47—82no—
150Q8×C47—82no—
151D4×C47—42no—
152D4×C47—42no—
153D4×C47—44no—
154D4×C47—82no—
155D4×C47—82no—
156D4×C47—82no—
157D4×C47—82no—
158C2×C4×C47—22no—
159C2×C4×C47—22no—
160C2×C4×C47—42no—
161C2×C4×C47—82no—
162C2×C4×C47#12111yescharacteristic
163C2×C4×C47#12111yescharacteristic
164C2×C4×C47#12111yescharacteristic
165C2×C4×C47#13512yes—
166C2×C4×C47AGL(1,ℤ/8)12yes—
167C2×C4×C47C22×D411yescharacteristic
168C2×C4×C47C22×D411yescharacteristic
169C2×C4×C47C2×#4111yescharacteristic
170C22⋊D47—44no—
171C2×Dic47—22no—
172D188—24no—
173D188—42no—
174D188—42no—
175C4×D47—42no—
176C4×D47—42no—
177Dic94—42no—
178C8×C47—21no—
179C8×C47—21no—
180C8×C47—21no—
181C8×C47—21no—
182C47⋊C8—42no—
183C47⋊C8—42no—
184C47⋊C8—42no—
185#19196—471nomaximal, Sylow
186C22×C47—21no—
187C22×C47—21no—
188C22×C47—82no—
189C22×C47—82no—
190C22×C47—82no—
191C22×C47—164no—
192C22×C47#46312yes—
193C22×C47C2×#12111yescharacteristic
194C2×D47—42no—
195C2×D47—42no—
196C2×D47—42no—
197C4×C47—21no—
198C4×C47—21no—
199C4×C47—82no—
200C4×C47#46312yes—
201C4×C47#47212yes—
202C4×C47C2×#13511yescharacteristic
203C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
204C4×C47#57712yes—
205Dic47—42no—
206#3227—471no—
207#3205—471no—
208C2×#486D4711yescharacteristic, complemented
209#2931—471no—
210#1697—474no—
211#1677—474no—
212#1564—471no—
213#1548—471no—
214#1545—471no—
215C2×C47—22no—
216C2×C47—162no—
217C2×C47C2×#46311yescharacteristic
218C2×C47#305711yescharacteristic
219C2×C47C2×#47211yescharacteristic
220D47—82no—
221C2×#131—941no—
222#532—471no—
223C2×C4⋊Q8C2×D4711yescharacteristic
224C2×#127C2×D4711yescharacteristic
225#498—941no—
226#493—941no—
227#486—21no—
228#486—21no—
229#486—941no—
230#486—941no—
231#486C2×D4712yes—
232#486C2×D4712yes—
233#481—942no—
234#476—474no—
235#459—474no—
236#447—942no—
237#436—472no—
238#436—941no—
239C8×D4—472no—
240C8×D4—941no—
241#431—471no—
242#431—471no—
243#423—471no—
244C2×#102—21no—
245C2×#102—21no—
246#408—941no—
247C2×C4×C8C2×D4711yescharacteristic
248#335—472no—
249#334—472no—
250#333—474no—
251#331—474no—
252C47#1919611yescharacteristic, complemented, Sylow
253C2×#41—941no—
254C22×Q8—21no—
255C22×D4—21no—
256C22×D4—21no—
257C22×D4—21no—
258C2×D8—944no—
259#131—1884no—
260C2×M16—941no—
261C2×M16—942no—
262C22×C8—942no—
263C22×C8C22⋊D4711yescharacteristic
264C22×C8C22⋊D4711yescharacteristic
265C4⋊Q8—21no—
266C4⋊Q8—21no—
267C4⋊Q8—21no—
268C4⋊Q8—471no—
269C4⋊Q8C22×D4712yes—
270#127—21no—
271#127—21no—
272#127—21no—
273#127—471no—
274#127C22×D4712yes—
275#124—941no—
276#121—472no—
277#121—941no—
278C4×D4—472no—
279C4×D4—941no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C42C22×D4711yescharacteristic
283#106—944no—
284C4⋊C8—472no—
285C4⋊C8—472no—
286C4⋊C8—472no—
287C4⋊C8—941no—
288#102—22no—
289#102—22no—
290#102—22no—
291#102—22no—
292#102—944no—
293#102—944no—
294C22⋊C8—941no—
295C22⋊C8—941no—
296C22⋊C8—942no—
297C22⋊C8—942no—
298C4×C8—941no—
299C4×C8—941no—
300C4×C8C22⋊D4711yescharacteristic
301C4×C8C22⋊D4711yescharacteristic
302C4×C8C22×D4711yescharacteristic
303C4×C8C22×D4711yescharacteristic
304C24—41no—
305#41—1882no—
306#41—1882no—
307C2×Q8—22no—
308C2×Q8—41no—
309C2×Q8—82no—
310C2×Q8—941no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—82no—
320C2×D4—82no—
321C2×D4—472no—
322C2×D4—942no—
323C2×D4—941no—
324C2×D4—941no—
325C22×C4—41no—
326C22×C4—942no—
327C22×C4D4×D4711yescharacteristic
328C22×C4#1062535211yescharacteristic
329C22×C4C2×C22⋊D4711yescharacteristic
330D8—3764no—
331M16—1884no—
332M16—1882no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—942no—
336C2×C8—942no—
337C2×C8—942no—
338C2×C8—1884no—
339C2×C8C2×C22⋊D4711yescharacteristic
340C2×C8C2×C22⋊D4711yescharacteristic
341C2×C8C2×C22⋊D4711yescharacteristic
342C2×C8C2×C22⋊D4711yescharacteristic
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—472no—
348C4⋊C4—941no—
349C22⋊C4—942no—
350C22⋊C4—942no—
351C42D4×D4711yescharacteristic
352C42#1062535211yescharacteristic
353C42C2×C22⋊D4711yescharacteristic
354C42C23×D4711yescharacteristic
355C23—42no—
356C23—82no—
357C23—82no—
358C23—82no—
359C23—942no—
360C23#1309399511yescharacteristic
361Q8—82no—
362Q8—82no—
363Q8—1882no—
364D4—42no—
365D4—42no—
366D4—44no—
367D4—82no—
368D4—82no—
369D4—82no—
370D4—82no—
371D4—944no—
372D4—1884no—
373D4—1882no—
374D4—1882no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—42no—
378C2×C4—82no—
379C2×C4—942no—
380C2×C4—1882no—
381C2×C4—1882no—
382C2×C4#1309399511yescharacteristic
383C2×C4#1309399511yescharacteristic
384C2×C4#1309399511yescharacteristic
385C2×C4#1309404412yes—
386C2×C4#1309404912yes—
387C2×C4C2×D4×D4711yescharacteristic
388C2×C4C22×C22⋊D4711yescharacteristic
389C2×C4C2×#1062535211yescharacteristic
390C8—21no—
391C8—21no—
392C8—21no—
393C8—21no—
394C8—1882no—
395C8—1882no—
396C8—1882no—
397C22—21no—
398C22—21no—
399C22—82no—
400C22—82no—
401C22—82no—
402C22—164no—
403C22—1882no—
404C22—1882no—
405C22—1882no—
406C22#1568933512yes—
407C22C2×#1309399511yescharacteristic
408C4—21no—
409C4—21no—
410C4—82no—
411C4—1882no—
412C4#1568933512yes—
413C4#1568939012yes—
414C4C2×#1309404411yescharacteristic
415C4C2×#1309404911yescharacteristic
416C4#1568992912yes—
417C2—22no—
418C2—162no—
419C2—3762no—
420C2C2×#1568933511yescharacteristic
421C2#2025302311yescharacteristic
422C2C2×#1568939011yescharacteristic
subgroupG/H
centerC22C2×#13093995
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC2×C47×#486C2
socleC22×C47C2×#121
prime radical#20254306C2
hypercenterC2×#486D47
hypocenterC47#19196
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#486

Complements

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

NH ≅ G/Ntype
1. #20254516C2semidirect
3. #20252101C2semidirect
4. #20242918C2semidirect
5. #20242794C2semidirect
6. #20242136C2semidirect
7. #20242069C2semidirect
8. #20242056C2semidirect
9. C2×C47×#486C2semidirect
17. #15689410C22semidirect
21. #15689202C22semidirect
23. C8×D188C22semidirect
28. #15688876C22semidirect
29. #15688874C22semidirect
30. #15688872C22semidirect
208. C2×#486D47semidirect
252. C47#19196semidirect

Maximal subgroups

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

subgroupindex
#202545162
#202543062
#202521012
#202429182
#202427942
#202421362
#202420692
#202420562
C2×C47×#4862
#1919647