Finite Group Databasebeta

← #37582009

Subgroups — #37582009

Order 24448.

500 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×#20259346C211yesmaximal, characteristic
3#124×D191C211yesmaximal, characteristic, complemented
4C4×#15690286C211yesmaximal, characteristic, complemented
5#27132888C211yesmaximal, characteristic, complemented
6#27132886C211yesmaximal, characteristic, complemented
7#27132884C211yesmaximal, characteristic, complemented
8#27132882C211yesmaximal, characteristic, complemented
9#27132880C211yesmaximal, characteristic, complemented
10#27132878C211yesmaximal, characteristic, complemented
11#27132876C211yesmaximal, characteristic, complemented
12#27132685C211yesmaximal, characteristic, complemented
13#27132670C211yesmaximal, characteristic, complemented
14#27132666C211yesmaximal, characteristic, complemented
15#463×C191C211yesmaximal, characteristic, complemented
16C2×#15690286C2211yescharacteristic
17C2×Q8×D191C2211yescharacteristic
18C2×D4×D191C2211yescharacteristic
19#20259351—24no—
20#20259346—24no—
21C2×Dic764C2211yescharacteristic, complemented
22C2×#15690277C2211yescharacteristic, complemented
23C2×#15690276C2211yescharacteristic
24C2×#15690275C2211yescharacteristic
25C2×#15690275C2211yescharacteristic, complemented
26C2×#15690272C2211yescharacteristic
27C2×#15690270C2211yescharacteristic
28C2×#15690270C2211yescharacteristic
29#20259311C2211yescharacteristic
30#20259310C2211yescharacteristic
31#20259309C2211yescharacteristic
32#20259308—21no—
33#20259308—21no—
34C4×Dic382C2211yescharacteristic, complemented
35#20259287C2211yescharacteristic, complemented
36D4×Dic191C2211yescharacteristic, complemented
37C4×C22⋊D191—21no—
38(C22⋊D191)⋊C4—21no—
39#20259280C2211yescharacteristic
40#20259278—21no—
41C22⋊C4×D191—21no—
42C22⋊C4×D191—21no—
43C2×C4×Dic191—21no—
44C42×D191C2211yescharacteristic
45C4⋊(C191⋊C8)C2211yescharacteristic, complemented
46Dic191⋊C8C2211yescharacteristic, complemented
47Dic191⋊C8C2211yescharacteristic, complemented
48#20259247C2211yescharacteristic, complemented
49#20259246C2211yescharacteristic, complemented
50#20259245C2211yescharacteristic, complemented
51#20259244C2211yescharacteristic
52#20259243C2211yescharacteristic
53#20259242C2211yescharacteristic
54C2×SD16×C191C2211yescharacteristic, complemented
55C2×D8×C191C2211yescharacteristic
56#124×C191C2211yescharacteristic
57C4×D4×C191C2211yescharacteristic, complemented
58C4⋊C8×C191C2211yescharacteristic, complemented
59#103×C191C2211yescharacteristic, complemented
60#102×C191C2211yescharacteristic
61C23×D191—21no—
62#15690286—22no—
63#15690286D414yes—
64C2×Dic382—21no—
65C2×Dic382C2311yescharacteristic
66C2×Dic382C2311yescharacteristic
67Q8×D191—22no—
68Q8×D191—22no—
69C2×D764C2311yescharacteristic
70C2×C22⋊D191—21no—
71C2×C22⋊D191—21no—
72D4×D191—22no—
73D4×D191—22no—
74C22×Dic191—21no—
75C22×Dic191—21no—
76C2×C4×D191—21no—
77C2×C4×D191D411yescharacteristic
78C2×C4×D191D411yescharacteristic
79C2×C4×D191C2311yescharacteristic
80Dic764—22no—
81#15690277—22no—
82#15690276—22no—
83#15690275—22no—
84#15690275—22no—
85#15690272—22no—
86#15690270—22no—
87#15690270—22no—
88C2×C191⋊C8C2311yescharacteristic
89C2×C191⋊C8C2311yescharacteristic
90C4⋊Dic191C2311yescharacteristic
91Dic191⋊C4—21no—
92Dic191⋊C4—21no—
93C22⋊Dic191—21no—
94C22⋊Dic191—21no—
95C22⋊Dic191—21no—
96(C2×D191)⋊C4—21no—
97(C2×D191)⋊C4—21no—
98(C2×D191)⋊C4—21no—
99(C2×D191)⋊C4—21no—
100(C2×D191)⋊C4—21no—
101C4×Dic191—21no—
102C4×Dic191—21no—
103C4×Dic191—21no—
104C4×Dic191D411yescharacteristic
105C4×Dic191D411yescharacteristic
106C4×Dic191C2311yescharacteristic
107C2×Q8×C191C2311yescharacteristic
108C2×D4×C191C2311yescharacteristic
109C2×D4×C191C2311yescharacteristic
110C22×C4×C191—21no—
111SD16×C191—22no—
112D8×C191—22no—
113C2×C8×C191C2311yescharacteristic
114C2×C8×C191C2311yescharacteristic
115C4⋊C4×C191C2311yescharacteristic
116C22⋊C4×C191—21no—
117C22⋊C4×C191—21no—
118C22⋊C4×C191—21no—
119C42×C191C2311yescharacteristic
120C22×D191—21no—
121C22×D191—42no—
122C22×D191—42no—
123C22×D191—42no—
124C22×D191C2×D411yescharacteristic
125Dic382—21no—
126Dic382—21no—
127Dic382—21no—
128Dic382—42no—
129Dic382C2×D412yes—
130D764—21no—
131D764—21no—
132C22⋊D191—24no—
133C22⋊D191—42no—
134C2×Dic191—21no—
135C2×Dic191—24no—
136C2×Dic191—21no—
137C2×Dic191—21no—
138C2×Dic191—42no—
139C2×Dic191C2×D411yescharacteristic
140C2×Dic191C2×D411yescharacteristic
141C2×Dic191C2×D411yescharacteristic
142C4×D191—41no—
143C4×D191—41no—
144C4×D191—42no—
145C4×D191C2×D412yes—
146C4×D191#4112yes—
147C191⋊C8—21no—
148C191⋊C8—21no—
149C23×C191—21no—
150C23×C191—21no—
151Q8×C191—21no—
152Q8×C191—21no—
153D4×C191—21no—
154D4×C191—21no—
155D4×C191—21no—
156D4×C191C2×D412yes—
157C2×C4×C191—21no—
158C2×C4×C191—21no—
159C2×C4×C191—42no—
160C2×C4×C191C2×D411yescharacteristic
161C2×C4×C191C2×D411yescharacteristic
162C2×C4×C191C2411yescharacteristic
163C8×C191—21no—
164C8×C191—21no—
165C22×C191—22no—
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—21no—
180C4×C191—21no—
181C4×C191—41no—
182C4×C191—41no—
183C4×C191C22×D411yescharacteristic
184C4×C191C2×#4111yescharacteristic
185Dic191—22no—
186Dic191—21no—
187Dic191—21no—
188Dic191—41no—
189Dic191—41no—
190Dic191#12112yes—
191C2×C191—42no—
192C2×C191—81no—
193C2×C191C2×#12111yescharacteristic
194C2×C191#57511yescharacteristic
195C2×C191#57511yescharacteristic
196D191—22no—
197D191—81no—
198C191#304011yescharacteristic, complemented, Sylow
199#3040—1911nomaximal, Sylow
200C2×#137—1911no—
201C2×AGL(1,ℤ/8)—1911no—
202C2×#124—1911no—
203C4×#41—1911no—
204#464—1911no—
205#464—1911no—
206#464—1911no—
207#464—1911no—
208#463—1911no—
209#463—1911no—
210#463—1911no—
211#463D19111yescharacteristic, complemented
212#423—1911no—
213#419—1911no—
214#418—1911no—
215C2×#41—1911no—
216C22×Q8—1911no—
217C22×D4—1911no—
218#137—3824no—
219AGL(1,ℤ/8)—3824no—
220C2×Q16—1911no—
221C2×Q16—1911no—
222C2×SD16—1911no—
223C2×SD16—1911no—
224C2×SD16—1911no—
225C2×SD16C2×D19111yescharacteristic, complemented
226C2×D8—1911no—
227C2×D8C2×D19111yescharacteristic
228C2×M16—1911no—
229C2×M16—1911no—
230#124—3821no—
231#124—3821no—
232#124—1911no—
233#124—1911no—
234#124—1911no—
235#124C2×D19111yescharacteristic
236C4×Q8—1911no—
237C4×Q8—1911no—
238C4×D4—3821no—
239C4×D4—3821no—
240C4×D4—1911no—
241C4×D4C2×D19111yescharacteristic, complemented
242#117—3821no—
243#117—1911no—
244C2×C22⋊C4—3821no—
245C2×C22⋊C4—3821no—
246C2×C42—3821no—
247C2×C42—1911no—
248C4⋊C8—1911no—
249C4⋊C8—1911no—
250C4⋊C8—1911no—
251C4⋊C8C2×D19111yescharacteristic, complemented
252#103—1911no—
253#103—1911no—
254#103—1911no—
255#103C2×D19111yescharacteristic, complemented
256#102—1911no—
257#102—1911no—
258#102—1911no—
259#102C2×D19111yescharacteristic
260C24—3821no—
261#41—3822no—
262#41—1914no—
263C2×Q8—3821no—
264C2×Q8—3822no—
265C2×Q8—3822no—
266C2×Q8—1911no—
267C2×Q8—1911no—
268C2×Q8C22×D19111yescharacteristic
269C2×D4—3821no—
270C2×D4—3822no—
271C2×D4—3822no—
272C2×D4—3821no—
273C2×D4—1911no—
274C2×D4C22×D19111yescharacteristic
275C2×D4C22×D19111yescharacteristic
276C22×C4—21no—
277C22×C4—3821no—
278C22×C4—3821no—
279C22×C4—3821no—
280C22×C4—1911no—
281C22×C4—1911no—
282C22×C4—1911no—
283Q16—3822no—
284Q16—3822no—
285SD16—22no—
286SD16—3822no—
287SD16—3822no—
288SD16—3822no—
289D8—22no—
290D8—3822no—
291M16—3822no—
292M16—3822no—
293C2×C8—1911no—
294C2×C8—1911no—
295C2×C8C22×D19111yescharacteristic
296C2×C8C22×D19111yescharacteristic
297C4⋊C4—3821no—
298C4⋊C4—3821no—
299C4⋊C4—1911no—
300C4⋊C4C22×D19111yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—21no—
304C22⋊C4—3821no—
305C22⋊C4—3821no—
306C22⋊C4—3821no—
307C22⋊C4—3821no—
308C22⋊C4—3821no—
309C22⋊C4—3821no—
310C22⋊C4—3821no—
311C22⋊C4—3821no—
312C42—3821no—
313C42—3821no—
314C42—3821no—
315C42—1911no—
316C42—1911no—
317C42—1911no—
318C42C22×D19111yescharacteristic
319C23—21no—
320C23—21no—
321C23—3821no—
322C23—1911no—
323C23—7642no—
324C23—7642no—
325C23—7642no—
326Q8—21no—
327Q8—21no—
328Q8—3821no—
329Q8—3821no—
330Q8—3821no—
331Q8—1912no—
332Q8—7642no—
333D4—21no—
334D4—21no—
335D4—21no—
336D4—3824no—
337D4—3821no—
338D4—3821no—
339D4—7642no—
340D4D4×D19112yes—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—42no—
344C2×C4—3821no—
345C2×C4—3824no—
346C2×C4—3821no—
347C2×C4—3821no—
348C2×C4—1911no—
349C2×C4—1911no—
350C2×C4—1911no—
351C2×C4—1912no—
352C2×C4—1912no—
353C2×C4—7641no—
354C2×C4—7641no—
355C2×C4—7642no—
356C2×C4—7642no—
357C2×C4D4×D19111yescharacteristic
358C2×C4D4×D19111yescharacteristic
359C2×C4C23×D19111yescharacteristic
360C8—21no—
361C8—21no—
362C8—3821no—
363C8—3821no—
364C22—22no—
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—41no—
381C4—41no—
382C4—3821no—
383C4—3821no—
384C4—3822no—
385C4—1912no—
386C4—7641no—
387C4—7641no—
388C4C2×D4×D19111yescharacteristic
389C4#41×D19111yescharacteristic
390C2—42no—
391C2—81no—
392C2—3822no—
393C2—15281no—
394C2#121×D19111yescharacteristic
395C2#2713345911yescharacteristic
396C2#2713346011yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#463×C191C2
socleC22×C191C22×D4
prime radicalC2×#20259346C2
hypercenter#463D191
hypocenterC191#3040
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#20259351C2semidirect
3. #124×D191C2semidirect
4. C4×#15690286C2semidirect
5. #27132888C2semidirect
6. #27132886C2semidirect
7. #27132884C2semidirect
8. #27132882C2semidirect
9. #27132880C2semidirect
10. #27132878C2semidirect
11. #27132876C2semidirect
12. #27132685C2semidirect
13. #27132670C2semidirect
14. #27132666C2semidirect
15. #463×C191C2semidirect
21. C2×Dic764C22semidirect
22. C2×#15690277C22semidirect
25. C2×#15690275C22semidirect
34. C4×Dic382C22semidirect
35. #20259287C22semidirect
36. D4×Dic191C22semidirect
45. C4⋊(C191⋊C8)C22semidirect
46. Dic191⋊C8C22semidirect
47. Dic191⋊C8C22semidirect
48. #20259247C22semidirect
49. #20259246C22semidirect
50. #20259245C22semidirect
54. C2×SD16×C191C22semidirect
57. C4×D4×C191C22semidirect
58. C4⋊C8×C191C22semidirect
59. #103×C191C22semidirect
198. C191#3040semidirect
211. #463D191semidirect
225. C2×SD16C2×D191semidirect
241. C4×D4C2×D191semidirect
251. C4⋊C8C2×D191semidirect
255. #103C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593512
C2×#202593462
#124×D1912
C4×#156902862
#271328882
#271328862
#271328842
#271328822
#271328802
#271328782
#271328762
#271326852
#271326702
#271326662
#463×C1912
#3040191