Finite Group Databasebeta

← #37583389

Subgroups — #37583389

Order 24448.

410 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27133236C211yesmaximal, characteristic, complemented
2#27133230C211yesmaximal, characteristic
3#27133005C211yesmaximal, characteristic, complemented
4#27133001C211yesmaximal, characteristic, complemented
5#27132986C211yesmaximal, characteristic, complemented
6#27132985C211yesmaximal, characteristic, complemented
7#27132908C211yesmaximal, characteristic, complemented
8#27132906C211yesmaximal, characteristic, complemented
9#27132891C211yesmaximal, characteristic, complemented
10#27132889C211yesmaximal, characteristic, complemented
11#27132814C211yesmaximal, characteristic, complemented
12#27132800C211yesmaximal, characteristic, complemented
13#27132800C211yesmaximal, characteristic, complemented
14#27132628C211yesmaximal, characteristic, complemented
15#450×C191C211yesmaximal, characteristic, complemented
16C2×#15690287C2211yescharacteristic
17C2×Q8×D191C2211yescharacteristic, complemented
18C2×#15690275C2211yescharacteristic
19C2×#15690275C2211yescharacteristic
20C2×#15690270C2211yescharacteristic, complemented
21C2×C8×D191C2211yescharacteristic, complemented
22#20259321—21no—
23#20259318—21no—
24#20259309—21no—
25#20259308—21no—
26#20259307—21no—
27#20259304—21no—
28#20259302C2211yescharacteristic, complemented
29#20259301C2211yescharacteristic
30#20259298C2211yescharacteristic, complemented
31#20259293C2211yescharacteristic
32#20259293C2211yescharacteristic
33#20259293C2211yescharacteristic
34#20259293C2211yescharacteristic
35#20259291—21no—
36#20259291—21no—
37#20259281C2211yescharacteristic
38#20259280C2211yescharacteristic, complemented
39#20259256C2211yescharacteristic, complemented
40#20259255C2211yescharacteristic, complemented
41#20259254C2211yescharacteristic, complemented
42#20259253C2211yescharacteristic, complemented
43#20259247C2211yescharacteristic, complemented
44#20259247C2211yescharacteristic, complemented
45#20259246C2211yescharacteristic, complemented
46#20259246C2211yescharacteristic, complemented
47#20259243C2211yescharacteristic
48#20259243C2211yescharacteristic
49C22⋊(C191⋊C8)C2211yescharacteristic
50(C2×D191)⋊C8C2211yescharacteristic, complemented
51(C2×D191)⋊C8C2211yescharacteristic, complemented
52C22×D4×C191C2211yescharacteristic
53C2×SD16×C191C2211yescharacteristic
54C2×SD16×C191C2211yescharacteristic
55C22⋊Q8×C191C2211yescharacteristic
56#102×C191C2211yescharacteristic
57#102×C191C2211yescharacteristic
58C22⋊C8×C191C2211yescharacteristic
59#15690287—41no—
60#15690287—41no—
61C2×Dic382—21no—
62C2×Dic382D411yescharacteristic
63C2×Dic382C2311yescharacteristic
64Q8×D191—41no—
65Q8×D191—41no—
66C2×D764D411yescharacteristic
67C2×C22⋊D191—21no—
68C2×C22⋊D191—21no—
69C2×C22⋊D191—21no—
70C2×C22⋊D191—21no—
71C2×C22⋊D191D411yescharacteristic
72C2×C22⋊D191D411yescharacteristic
73C2×C4×D191—21no—
74C2×C4×D191C2311yescharacteristic
75C2×C4×D191C2311yescharacteristic
76#15690275—41no—
77#15690275—41no—
78#15690270—41no—
79C2×C191⋊C8C2311yescharacteristic
80C2×C191⋊C8C2311yescharacteristic
81C8×D191—41no—
82C4⋊Dic191—21no—
83C4⋊Dic191C2311yescharacteristic
84C4⋊Dic191C2311yescharacteristic
85C4⋊Dic191C2311yescharacteristic
86Dic191⋊C4—21no—
87Dic191⋊C4—21no—
88C22⋊Dic191—21no—
89C22⋊Dic191—21no—
90C22⋊Dic191—21no—
91C22⋊Dic191—21no—
92C22⋊Dic191—21no—
93C22⋊Dic191—21no—
94(C2×D191)⋊C4—21no—
95(C2×D191)⋊C4—21no—
96(C2×D191)⋊C4—21no—
97C4×Dic191—21no—
98C4×Dic191—21no—
99C24×C191—21no—
100C2×Q8×C191C2311yescharacteristic
101C2×D4×C191—21no—
102C2×D4×C191—41no—
103C2×D4×C191—41no—
104C2×D4×C191C2311yescharacteristic
105C2×D4×C191C2311yescharacteristic
106C22×C4×C191C2311yescharacteristic
107SD16×C191—41no—
108SD16×C191—41no—
109C2×C8×C191C2311yescharacteristic
110C2×C8×C191C2311yescharacteristic
111C4⋊C4×C191—21no—
112C4⋊C4×C191C2311yescharacteristic
113C22⋊C4×C191—21no—
114C22×D191C2×D411yescharacteristic
115C22×D191C2×D411yescharacteristic
116Dic382—21no—
117Dic382—21no—
118Dic382—41no—
119Dic382—42no—
120D764—41no—
121C22⋊D191—41no—
122C22⋊D191—41no—
123C22⋊D191—81no—
124C22⋊D191—81no—
125C22⋊D191—81no—
126C22⋊D191—81no—
127C2×Dic191—21no—
128C2×Dic191—21no—
129C2×Dic191—21no—
130C2×Dic191—21no—
131C2×Dic191C2×D411yescharacteristic
132C2×Dic191C2×D411yescharacteristic
133C4×D191—21no—
134C4×D191—21no—
135C4×D191—21no—
136C4×D191—21no—
137C4×D191—42no—
138C191⋊C8—21no—
139C191⋊C8—21no—
140C23×C191—21no—
141C23×C191—21no—
142C23×C191—42no—
143C23×C191—81no—
144C23×C191—81no—
145C23×C191C2×D411yescharacteristic
146Q8×C191—21no—
147Q8×C191—21no—
148D4×C191—21no—
149D4×C191—21no—
150D4×C191—21no—
151D4×C191—21no—
152D4×C191—42no—
153C2×C4×C191—21no—
154C2×C4×C191—21no—
155C2×C4×C191—21no—
156C2×C4×C191—21no—
157C2×C4×C191C2×D411yescharacteristic
158C2×C4×C191C2411yescharacteristic
159C8×C191—21no—
160C8×C191—21no—
161C22×C191—21no—
162C22×C191—21no—
163C22×C191—21no—
164C22×C191—41no—
165C22×C191—41no—
166C22×C191—41no—
167C22×C191—41no—
168C22×C191—41no—
169C22×C191—41no—
170C22×C191—84no—
171C22×C191C22×D411yescharacteristic
172C2×D191—21no—
173C2×D191—21no—
174C2×D191—21no—
175C2×D191—21no—
176C2×D191—21no—
177C2×D191—21no—
178C4×C191—21no—
179C4×C191—41no—
180C4×C191—41no—
181C4×C191H(5,𝔽2)11yescharacteristic
182C4×C191H(5,𝔽2)11yescharacteristic
183Dic191—21no—
184Dic191—21no—
185Dic191—41no—
186Dic191—41no—
187Dic191—41no—
188Dic191—41no—
189C2×C191—41no—
190C2×C191—81no—
191C2×C191—81no—
192C2×C191#53411yescharacteristic
193C2×C191#57711yescharacteristic
194C2×C191#57711yescharacteristic
195D191—41no—
196D191—41no—
197C191#320011yescharacteristic, complemented, Sylow
198#3200—1911nomaximal, Sylow
199#536—1911no—
200#535—1911no—
201#485—1911no—
202#485—1911no—
203#482—1911no—
204#482—1911no—
205#468—1911no—
206#468—1911no—
207#465—1911no—
208#465—1911no—
209#450—1911no—
210#450D19111yescharacteristic, complemented
211#448—1911no—
212#448—1911no—
213#408—1911no—
214C2×#41—1911no—
215C22×Q8—1911no—
216C22×D4C2×D19111yescharacteristic
217C2×SD16—1911no—
218C2×SD16—1911no—
219C2×SD16C2×D19111yescharacteristic
220C2×SD16C2×D19111yescharacteristic
221C2×M16—1911no—
222C22×C8—1911no—
223C4⋊Q8—3821no—
224#127—3821no—
225#124—3821no—
226#124—3821no—
227#123—3821no—
228#123—3821no—
229C22⋊Q8—1911no—
230C22⋊Q8—1911no—
231C22⋊Q8—1911no—
232C22⋊Q8C2×D19111yescharacteristic
233#121—1911no—
234#121—1911no—
235#121—1911no—
236#121—1911no—
237#120—3821no—
238#120—3821no—
239#117—1911no—
240#117—1911no—
241#107—1911no—
242#107—1911no—
243#106—1911no—
244#106—1911no—
245#103—1911no—
246#103—1911no—
247#103—1911no—
248#103—1911no—
249#102—1911no—
250#102—1911no—
251#102C2×D19111yescharacteristic
252#102C2×D19111yescharacteristic
253C22⋊C8—1911no—
254C22⋊C8—1911no—
255C22⋊C8—1911no—
256C22⋊C8C2×D19111yescharacteristic
257C24—21no—
258#41—7641no—
259#41—7641no—
260C2×Q8—3821no—
261C2×Q8—1911no—
262C2×Q8—1911no—
263C2×Q8—7641no—
264C2×Q8—7641no—
265C2×Q8C22×D19111yescharacteristic
266C2×D4—21no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—3821no—
270C2×D4—3821no—
271C2×D4—3821no—
272C2×D4—3821no—
273C2×D4—1911no—
274C2×D4—1911no—
275C2×D4—1911no—
276C2×D4C22×D19111yescharacteristic
277C2×D4C22×D19111yescharacteristic
278C22×C4—3821no—
279C22×C4—1911no—
280C22×C4—1911no—
281C22×C4C22×D19111yescharacteristic
282SD16—41no—
283SD16—41no—
284SD16—7641no—
285SD16—7641no—
286M16—7641no—
287C2×C8—1911no—
288C2×C8—1911no—
289C2×C8—7641no—
290C2×C8C22×D19111yescharacteristic
291C2×C8C22×D19111yescharacteristic
292C4⋊C4—21no—
293C4⋊C4—3821no—
294C4⋊C4—3821no—
295C4⋊C4—3821no—
296C4⋊C4—1911no—
297C4⋊C4—1911no—
298C4⋊C4—1911no—
299C4⋊C4C22×D19111yescharacteristic
300C22⋊C4—21no—
301C22⋊C4—3821no—
302C22⋊C4—3821no—
303C22⋊C4—3821no—
304C22⋊C4—3821no—
305C22⋊C4—3821no—
306C22⋊C4—3821no—
307C22⋊C4—3821no—
308C22⋊C4—3821no—
309C22⋊C4—3821no—
310C42—3821no—
311C42—3821no—
312C23—21no—
313C23—21no—
314C23—42no—
315C23—81no—
316C23—81no—
317C23—1911no—
318C23—1911no—
319C23D4×D19111yescharacteristic
320Q8—21no—
321Q8—21no—
322Q8—3821no—
323Q8—3821no—
324Q8—7641no—
325Q8—7642no—
326D4—21no—
327D4—21no—
328D4—21no—
329D4—21no—
330D4—42no—
331D4—15281no—
332D4—15281no—
333D4—15281no—
334D4—15281no—
335D4—7641no—
336D4—7641no—
337D4—7641no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—3821no—
343C2×C4—3821no—
344C2×C4—3821no—
345C2×C4—3821no—
346C2×C4—3821no—
347C2×C4—3821no—
348C2×C4—3821no—
349C2×C4—3821no—
350C2×C4—1911no—
351C2×C4—1911no—
352C2×C4—7642no—
353C2×C4D4×D19111yescharacteristic
354C2×C4C23×D19111yescharacteristic
355C8—21no—
356C8—21no—
357C8—3821no—
358C8—3821no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—84no—
369C22—3821no—
370C22—3821no—
371C22—3821no—
372C22—3821no—
373C22—3821no—
374C22—3821no—
375C22C2×D4×D19111yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—3821no—
380C4—3821no—
381C4—7641no—
382C4—7641no—
383C4—7641no—
384C4—7641no—
385C4#2025936711yescharacteristic
386C4#2025936711yescharacteristic
387C2—41no—
388C2—81no—
389C2—81no—
390C2—7641no—
391C2—7641no—
392C2#2713322311yescharacteristic
393C2#2713346911yescharacteristic
394C2#2713347311yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#450×C191C2
socleC22×C191C22×D4
prime radical#27133230C2
hypercenter#450D191
hypocenterC191#3200
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #27133236C2semidirect
3. #27133005C2semidirect
4. #27133001C2semidirect
5. #27132986C2semidirect
6. #27132985C2semidirect
7. #27132908C2semidirect
8. #27132906C2semidirect
9. #27132891C2semidirect
10. #27132889C2semidirect
11. #27132814C2semidirect
12. #27132800C2semidirect
13. #27132800C2semidirect
14. #27132628C2semidirect
15. #450×C191C2semidirect
17. C2×Q8×D191C22semidirect
20. C2×#15690270C22semidirect
21. C2×C8×D191C22semidirect
28. #20259302C22semidirect
30. #20259298C22semidirect
38. #20259280C22semidirect
39. #20259256C22semidirect
40. #20259255C22semidirect
41. #20259254C22semidirect
42. #20259253C22semidirect
43. #20259247C22semidirect
44. #20259247C22semidirect
45. #20259246C22semidirect
46. #20259246C22semidirect
50. (C2×D191)⋊C8C22semidirect
51. (C2×D191)⋊C8C22semidirect
197. C191#3200semidirect
210. #450D191semidirect

Maximal subgroups

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

subgroupindex
#271332362
#271332302
#271330052
#271330012
#271329862
#271329852
#271329082
#271329062
#271328912
#271328892
#271328142
#271328002
#271326282
#450×C1912
#3200191