Finite Group Databasebeta

← #37582036

Subgroups — #37582036

Order 24448.

490 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259351C211yesmaximal, characteristic, complemented
2C2×SD16×D191C211yesmaximal, characteristic, complemented
3#27133226C211yesmaximal, characteristic
4#27133139C211yesmaximal, characteristic, complemented
5#27132886C211yesmaximal, characteristic, complemented
6#27132885C211yesmaximal, characteristic, complemented
7#27132884C211yesmaximal, characteristic, complemented
8#27132883C211yesmaximal, characteristic, complemented
9#27132862C211yesmaximal, characteristic, complemented
10#27132861C211yesmaximal, characteristic, complemented
11#27132860C211yesmaximal, characteristic, complemented
12#27132688C211yesmaximal, characteristic, complemented
13#27132666C211yesmaximal, characteristic, complemented
14#27132656C211yesmaximal, characteristic, complemented
15#460×C191C211yesmaximal, characteristic, complemented
16C2×#15690286C2211yescharacteristic
17C2×Q8×D191C2211yescharacteristic
18C2×D4×D191C2211yescharacteristic
19#20259351—24no—
20C2×Dic764C2211yescharacteristic, complemented
21C2×#15690277C2211yescharacteristic, complemented
22C2×#15690276C2211yescharacteristic
23C2×#15690275C2211yescharacteristic
24C2×#15690275C2211yescharacteristic, complemented
25C2×#15690274C2211yescharacteristic
26SD16×D191—24no—
27C2×#15690270C2211yescharacteristic
28C2×C8×D191C2211yescharacteristic
29#20259318C2211yescharacteristic
30#20259310C2211yescharacteristic
31#20259309C2211yescharacteristic
32#20259294—21no—
33#20259294—21no—
34#20259290—21no—
35#20259290—21no—
36Q8×Dic191C2211yescharacteristic, complemented
37C4×Dic382—21no—
38C4×Dic382C2211yescharacteristic, complemented
39#20259287—21no—
40#20259287C2211yescharacteristic, complemented
41#20259280—21no—
42#20259280C2211yescharacteristic
43#20259279—21no—
44#20259279C2211yescharacteristic
45C4⋊(C191⋊C8)C2211yescharacteristic, complemented
46Dic191⋊C8C2211yescharacteristic, complemented
47Dic191⋊C8C2211yescharacteristic, complemented
48#20259247C2211yescharacteristic, complemented
49#20259245C2211yescharacteristic, complemented
50#20259244C2211yescharacteristic
51#20259243C2211yescharacteristic
52#20259243C2211yescharacteristic, complemented
53#20259242C2211yescharacteristic
54C2×SD16×C191C2211yescharacteristic
55C2×SD16×C191C2211yescharacteristic, complemented
56#127×C191C2211yescharacteristic
57C4×Q8×C191C2211yescharacteristic, complemented
58C4⋊C8×C191C2211yescharacteristic, complemented
59#102×C191C2211yescharacteristic
60#102×C191C2211yescharacteristic, complemented
61C23×D191—21no—
62#15690286—22no—
63#15690286—22no—
64C2×Dic382—21no—
65C2×Dic382C2311yescharacteristic
66C2×Dic382C2311yescharacteristic
67Q8×D191—22no—
68Q8×D191D414yescomplemented
69C2×D764C2311yescharacteristic
70C2×C22⋊D191—21no—
71C2×C22⋊D191—21no—
72C2×C22⋊D191—21no—
73C2×C22⋊D191—21no—
74D4×D191—22no—
75D4×D191—22no—
76C22×Dic191—21no—
77C2×C4×D191—21no—
78C2×C4×D191C2311yescharacteristic
79Dic764—22no—
80#15690277—22no—
81#15690276—22no—
82#15690275—22no—
83#15690275—22no—
84#15690274—22no—
85#15690270—22no—
86C2×C191⋊C8C2311yescharacteristic
87C2×C191⋊C8C2311yescharacteristic
88C8×D191—22no—
89C4⋊Dic191—21no—
90C4⋊Dic191C2311yescharacteristic
91Dic191⋊C4—21no—
92Dic191⋊C4—21no—
93Dic191⋊C4D411yescharacteristic
94Dic191⋊C4D411yescharacteristic
95C22⋊Dic191—21no—
96C22⋊Dic191—21no—
97(C2×D191)⋊C4—21no—
98(C2×D191)⋊C4—21no—
99(C2×D191)⋊C4—21no—
100(C2×D191)⋊C4D411yescharacteristic
101(C2×D191)⋊C4D411yescharacteristic
102C4×Dic191—21no—
103C4×Dic191—21no—
104C4×Dic191—21no—
105C4×Dic191—21no—
106C4×Dic191C2311yescharacteristic
107C2×Q8×C191C2311yescharacteristic
108C2×D4×C191—21no—
109C2×D4×C191—21no—
110C2×D4×C191C2311yescharacteristic
111C2×D4×C191C2311yescharacteristic
112SD16×C191—22no—
113SD16×C191—22no—
114C2×C8×C191C2311yescharacteristic
115C2×C8×C191C2311yescharacteristic
116C4⋊C4×C191—21no—
117C4⋊C4×C191C2311yescharacteristic
118C42×C191—21no—
119C42×C191C2311yescharacteristic
120C22×D191—21no—
121C22×D191—42no—
122C22×D191—81no—
123C22×D191—81no—
124C22×D191C2×D411yescharacteristic
125Dic382—21no—
126Dic382—24no—
127Dic382—21no—
128Dic382—21no—
129Dic382C2×D412yes—
130D764—21no—
131D764—21no—
132C22⋊D191—42no—
133C22⋊D191—42no—
134C22⋊D191—81no—
135C22⋊D191—81no—
136C2×Dic191—21no—
137C2×Dic191—21no—
138C2×Dic191—21no—
139C2×Dic191—42no—
140C2×Dic191C2×D411yescharacteristic
141C2×Dic191C2×D411yescharacteristic
142C2×Dic191C2×D411yescharacteristic
143C4×D191—24no—
144C4×D191C2×D412yes—
145C4×D191#4112yes—
146C191⋊C8—21no—
147C191⋊C8—21no—
148C23×C191—21no—
149C23×C191—21no—
150Q8×C191—21no—
151Q8×C191C2×D412yes—
152D4×C191—21no—
153D4×C191—21no—
154D4×C191—21no—
155D4×C191—21no—
156D4×C191—81no—
157D4×C191—81no—
158C2×C4×C191—21no—
159C2×C4×C191—21no—
160C2×C4×C191C2×D411yescharacteristic
161C2×C4×C191C2×D411yescharacteristic
162C2×C4×C191C2411yescharacteristic
163C8×C191—21no—
164C8×C191—21no—
165C22×C191—41no—
166C22×C191—41no—
167C22×C191—41no—
168C22×C191—41no—
169C22×C191—41no—
170C22×C191—41no—
171C22×C191C22×D411yescharacteristic
172C2×D191—21no—
173C2×D191—21no—
174C2×D191—41no—
175C2×D191—41no—
176C2×D191—41no—
177C2×D191—84no—
178C2×D191#12112yes—
179C4×C191—22no—
180C4×C191—21no—
181C4×C191—21no—
182C4×C191—41no—
183C4×C191C22×D411yescharacteristic
184C4×C191C2×#4111yescharacteristic
185Dic191—22no—
186Dic191—21no—
187Dic191—21no—
188Dic191—41no—
189Dic191—41no—
190Dic191#12112yes—
191C2×C191—81no—
192C2×C191—81no—
193C2×C191C2×#12111yescharacteristic
194C2×C191#57711yescharacteristic
195C2×C191#57711yescharacteristic
196D191—22no—
197D191—81no—
198C191#304311yescharacteristic, complemented, Sylow
199#3043—1911nomaximal, Sylow
200C2×#137—1911no—
201C22×SD16—1911no—
202#535—1911no—
203#519—1911no—
204#464—1911no—
205#464—1911no—
206#464—1911no—
207#464—1911no—
208#460—1911no—
209#460—1911no—
210#460—1911no—
211#460D19111yescharacteristic, complemented
212#424—1911no—
213#418—1911no—
214#416—1911no—
215C2×#41—1911no—
216C22×Q8—1911no—
217C22×D4—1911no—
218#137—3824no—
219C2×Q16—1911no—
220C2×Q16—1911no—
221C2×SD16—3824no—
222C2×SD16—1911no—
223C2×SD16—1911no—
224C2×SD16—1911no—
225C2×SD16—1911no—
226C2×SD16C2×D19111yescharacteristic
227C2×SD16C2×D19111yescharacteristic, complemented
228C2×M16—1911no—
229C22×C8—1911no—
230#127—1911no—
231#127C2×D19111yescharacteristic
232#124—1911no—
233#124—1911no—
234#121—3821no—
235#121—3821no—
236#120—3821no—
237#120—3821no—
238C4×Q8—3821no—
239C4×Q8—3821no—
240C4×Q8—1911no—
241C4×Q8—1911no—
242C4×Q8—1911no—
243C4×Q8C2×D19111yescharacteristic, complemented
244#117—3821no—
245#117—3821no—
246#117—1911no—
247#117—1911no—
248C4⋊C8—1911no—
249C4⋊C8—1911no—
250C4⋊C8—1911no—
251C4⋊C8C2×D19111yescharacteristic, complemented
252#103—1911no—
253#103—1911no—
254#102—1911no—
255#102—1911no—
256#102—1911no—
257#102—1911no—
258#102C2×D19111yescharacteristic
259#102C2×D19111yescharacteristic, complemented
260C24—3821no—
261#41—3822no—
262#41—3822no—
263C2×Q8—3821no—
264C2×Q8—3822no—
265C2×Q8—1914no—
266C2×Q8—1911no—
267C2×Q8—1911no—
268C2×Q8C22×D19111yescharacteristic
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—3821no—
272C2×D4—3822no—
273C2×D4—3822no—
274C2×D4—3821no—
275C2×D4—3821no—
276C2×D4—3821no—
277C2×D4—1911no—
278C2×D4C22×D19111yescharacteristic
279C2×D4C22×D19111yescharacteristic
280C22×C4—3821no—
281C22×C4—3821no—
282C22×C4—1911no—
283Q16—3822no—
284Q16—3822no—
285SD16—22no—
286SD16—22no—
287SD16—3822no—
288SD16—3822no—
289SD16—3822no—
290SD16—3822no—
291M16—3822no—
292C2×C8—3822no—
293C2×C8—1911no—
294C2×C8—1911no—
295C2×C8C22×D19111yescharacteristic
296C2×C8C22×D19111yescharacteristic
297C4⋊C4—21no—
298C4⋊C4—3821no—
299C4⋊C4—3821no—
300C4⋊C4—3821no—
301C4⋊C4—1911no—
302C4⋊C4—1911no—
303C4⋊C4—1911no—
304C4⋊C4C22×D19111yescharacteristic
305C22⋊C4—3821no—
306C22⋊C4—3821no—
307C22⋊C4—3821no—
308C22⋊C4—3821no—
309C22⋊C4—3821no—
310C22⋊C4—1911no—
311C22⋊C4—1911no—
312C42—21no—
313C42—3821no—
314C42—3821no—
315C42—3821no—
316C42—3821no—
317C42—1911no—
318C42C22×D19111yescharacteristic
319C23—21no—
320C23—21no—
321C23—3821no—
322C23—1911no—
323C23—15281no—
324C23—15281no—
325C23—7642no—
326Q8—21no—
327Q8—3824no—
328Q8—3821no—
329Q8—3821no—
330Q8—3821no—
331Q8—1912no—
332Q8D4×D19112yes—
333D4—21no—
334D4—21no—
335D4—21no—
336D4—21no—
337D4—81no—
338D4—81no—
339D4—3821no—
340D4—3821no—
341D4—15281no—
342D4—15281no—
343D4—7642no—
344D4—7642no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—3824no—
348C2×C4—3821no—
349C2×C4—3821no—
350C2×C4—3821no—
351C2×C4—1911no—
352C2×C4—1911no—
353C2×C4—1911no—
354C2×C4—1912no—
355C2×C4—1912no—
356C2×C4—7642no—
357C2×C4D4×D19111yescharacteristic
358C2×C4D4×D19111yescharacteristic
359C2×C4C23×D19111yescharacteristic
360C8—21no—
361C8—21no—
362C8—3821no—
363C8—3821no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—3821no—
371C22—3821no—
372C22—1912no—
373C22—15284no—
374C22—7641no—
375C22—7641no—
376C22—7641no—
377C22C2×D4×D19111yescharacteristic
378C4—21no—
379C4—21no—
380C4—22no—
381C4—41no—
382C4—3821no—
383C4—3821no—
384C4—3822no—
385C4—1912no—
386C4—7641no—
387C4—7641no—
388C4C2×D4×D19111yescharacteristic
389C4#41×D19111yescharacteristic
390C2—81no—
391C2—81no—
392C2—3822no—
393C2—15281no—
394C2#121×D19111yescharacteristic
395C2#2713346911yescharacteristic
396C2#2713347311yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#460×C191C2
socleC22×C191C22×D4
prime radical#27133226C2
hypercenter#460D191
hypocenterC191#3043
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#20259351C2semidirect
2. C2×SD16×D191C2semidirect
4. #27133139C2semidirect
5. #27132886C2semidirect
6. #27132885C2semidirect
7. #27132884C2semidirect
8. #27132883C2semidirect
9. #27132862C2semidirect
10. #27132861C2semidirect
11. #27132860C2semidirect
12. #27132688C2semidirect
13. #27132666C2semidirect
14. #27132656C2semidirect
15. #460×C191C2semidirect
20. C2×Dic764C22semidirect
21. C2×#15690277C22semidirect
24. C2×#15690275C22semidirect
36. Q8×Dic191C22semidirect
38. C4×Dic382C22semidirect
40. #20259287C22semidirect
45. C4⋊(C191⋊C8)C22semidirect
46. Dic191⋊C8C22semidirect
47. Dic191⋊C8C22semidirect
48. #20259247C22semidirect
49. #20259245C22semidirect
52. #20259243C22semidirect
55. C2×SD16×C191C22semidirect
57. C4×Q8×C191C22semidirect
58. C4⋊C8×C191C22semidirect
60. #102×C191C22semidirect
68. Q8×D191D4semidirect
198. C191#3043semidirect
211. #460D191semidirect
227. C2×SD16C2×D191semidirect
243. C4×Q8C2×D191semidirect
251. C4⋊C8C2×D191semidirect
259. #102C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593512
C2×SD16×D1912
#271332262
#271331392
#271328862
#271328852
#271328842
#271328832
#271328622
#271328612
#271328602
#271326882
#271326662
#271326562
#460×C1912
#3043191