Finite Group Databasebeta

← #37582203

Subgroups — #37582203

Order 24448.

504 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259347C211yesmaximal, characteristic, complemented
2C2×#20259328C211yesmaximal, characteristic, complemented
3#27133241C211yesmaximal, characteristic, complemented
4#27133232C211yesmaximal, characteristic, complemented
5#27132915C212yesmaximal, complemented
6#27132912C212yesmaximal, complemented
7#27132898C212yesmaximal, complemented
8#27132895C212yesmaximal, complemented
9#27132706C211yesmaximal, characteristic, complemented
10C2×#20259244C211yesmaximal, characteristic, complemented
11#416×C191C211yesmaximal, characteristic, complemented
12C2×#15690287C2211yescharacteristic
13C22×D764C2211yescharacteristic
14#20259347—24no—
15C2×#15690274C2212yes—
16C2×#15690272C2212yes—
17#20259328—24no—
18C2×#15690271C2211yescharacteristic
19C2×#15690270C2212yescomplemented
20C2×C8×D191C2212yescomplemented
21#20259319—21no—
22#20259310—21no—
23#20259301—21no—
24#20259298C2212yescomplemented
25#20259297C2212yes—
26#20259297C2212yes—
27#20259296—21no—
28#20259294—22no—
29#20259293C2212yescomplemented
30#20259292—22no—
31C2×C4⋊Dic191C2211yescharacteristic
32#20259255C2212yescomplemented
33#20259253C2212yescomplemented
34#20259244C2212yes—
35#20259244C2212yes—
36C2×#41×C191C2211yescharacteristic
37C22×C8×C191C2211yescharacteristic
38#117×C191C2211yescharacteristic
39#103×C191C2211yescharacteristic, complemented
40#103×C191C2211yescharacteristic, complemented
41#102×C191C2211yescharacteristic, complemented
42#102×C191C2211yescharacteristic, complemented
43C23×D191—21no—
44#15690287—22no—
45#15690287—22no—
46C2×Dic382D411yescharacteristic
47C2×D764—22no—
48C2×D764—22no—
49C2×D764—22no—
50C2×D764—21no—
51C2×D764—22no—
52C2×D764D411yescharacteristic
53C2×D764C2312yes—
54C2×C22⋊D191—22no—
55C2×C22⋊D191D412yes—
56C22×Dic191—21no—
57C2×C4×D191C2312yes—
58#15690274—24no—
59#15690272—24no—
60#15690271D414yescomplemented
61#15690270—24no—
62C2×C191⋊C8C2312yes—
63C8×D191—24no—
64C4⋊Dic191—21no—
65C4⋊Dic191C2312yes—
66Dic191⋊C4—22no—
67C22⋊Dic191—22no—
68(C2×D191)⋊C4—22no—
69(C2×D191)⋊C4—22no—
70(C2×D191)⋊C4—22no—
71#41×C191—22no—
72#41×C191—22no—
73C2×Q8×C191C2311yescharacteristic
74C2×D4×C191—21no—
75C2×D4×C191C2311yescharacteristic
76C22×C4×C191—21no—
77C22×C4×C191C2311yescharacteristic
78C2×C8×C191—22no—
79C2×C8×C191C2311yescharacteristic
80C2×C8×C191C2311yescharacteristic
81C4⋊C4×C191C2311yescharacteristic
82C4⋊C4×C191C2311yescharacteristic
83C22⋊C4×C191—21no—
84C42×C191—21no—
85C22×D191—22no—
86C22×D191—42no—
87C22×D191—81no—
88C22×D191—81no—
89C22×D191C2×D412yes—
90Dic382—22no—
91D764—22no—
92D764—22no—
93D764—22no—
94D764—42no—
95D764—82no—
96D764—82no—
97C22⋊D191—24no—
98C22⋊D191—82no—
99C2×Dic191—22no—
100C2×Dic191—81no—
101C2×Dic191C2×D412yes—
102C4×D191—22no—
103C4×D191—22no—
104C191⋊C8C2×D414yes—
105C23×C191—21no—
106C23×C191C2×D411yescharacteristic
107Q8×C191—21no—
108Q8×C191—21no—
109D4×C191—21no—
110D4×C191—21no—
111D4×C191—42no—
112C2×C4×C191—21no—
113C2×C4×C191—21no—
114C2×C4×C191—21no—
115C2×C4×C191—42no—
116C2×C4×C191C2×D411yescharacteristic
117C2×C4×C191C2×D412yes—
118C2×C4×C191#4112yes—
119C2×C4×C191C2411yescharacteristic
120C8×C191—21no—
121C8×C191—21no—
122C22×C191—21no—
123C22×C191—21no—
124C22×C191—41no—
125C22×C191—41no—
126C22×C191—41no—
127C22×C191#12112yes—
128C22×C191C22×D411yescharacteristic
129C2×D191—22no—
130C2×D191—22no—
131C2×D191—22no—
132C2×D191—42no—
133C2×D191—42no—
134C2×D191—42no—
135C2×D191—84no—
136C4×C191—41no—
137C4×C191—41no—
138C4×C191—41no—
139C4×C191#12112yes—
140C4×C191C22×D411yescharacteristic
141C4×C191C2×#4111yescharacteristic
142Dic191—22no—
143Dic191—42no—
144C2×C191—22no—
145C2×C191—81no—
146C2×C191C2×#12111yescharacteristic
147C2×C191#57611yescharacteristic
148C2×C191#57711yescharacteristic
149D191—42no—
150D191—82no—
151C191#306211yescharacteristic, complemented, Sylow
152#3062—1911nomaximal, Sylow
153C2×AGL(1,ℤ/8)—1911no—
154C2×#131—1911no—
155#537—1911no—
156#535—1911no—
157#469—1912no—
158#468—1912no—
159#466—1912no—
160#465—1912no—
161#428—1911no—
162#416D19111yescharacteristic, complemented
163C2×#102—1911no—
164C2×#41—1911no—
165C2×#41C2×D19111yescharacteristic
166C22×D4—1911no—
167AGL(1,ℤ/8)—3824no—
168C2×SD16—1912no—
169C2×D8—1912no—
170#131—3824no—
171C2×M16—1911no—
172C2×M16—1912no—
173C22×C8—1912no—
174C22×C8C2×D19111yescharacteristic
175#127—3821no—
176#124—3821no—
177C22⋊Q8—3821no—
178C22⋊Q8—1912no—
179#121—3821no—
180#121—3822no—
181#121—1912no—
182#121—1912no—
183#121—1912no—
184#120—3822no—
185#117C2×D19111yescharacteristic
186C2×C4⋊C4—1911no—
187#107—1912no—
188#106—1912no—
189#103C2×D19111yescharacteristic, complemented
190#103C2×D19111yescharacteristic, complemented
191#102—1912no—
192#102—1912no—
193#102C2×D19111yescharacteristic, complemented
194#102C2×D19111yescharacteristic, complemented
195C24—3821no—
196#41—22no—
197#41—22no—
198#41—3822no—
199#41—3822no—
200C2×Q8—1911no—
201C2×Q8C22×D19111yescharacteristic
202C2×D4—21no—
203C2×D4—3822no—
204C2×D4—3822no—
205C2×D4—3822no—
206C2×D4—3822no—
207C2×D4—3821no—
208C2×D4—3822no—
209C2×D4—1912no—
210C2×D4—1911no—
211C2×D4—1912no—
212C2×D4C22×D19111yescharacteristic
213C22×C4—21no—
214C22×C4—3821no—
215C22×C4—1912no—
216C22×C4C22×D19111yescharacteristic
217SD16—3824no—
218D8—3824no—
219M16—3824no—
220M16—1914no—
221C2×C8—22no—
222C2×C8—3824no—
223C2×C8—1912no—
224C2×C8C22×D19111yescharacteristic
225C2×C8C22×D19111yescharacteristic
226C4⋊C4—3822no—
227C4⋊C4—3821no—
228C4⋊C4—1912no—
229C4⋊C4C22×D19111yescharacteristic
230C4⋊C4C22×D19111yescharacteristic
231C22⋊C4—21no—
232C22⋊C4—3822no—
233C22⋊C4—3822no—
234C22⋊C4—3822no—
235C22⋊C4—3822no—
236C42—21no—
237C23—21no—
238C23—3822no—
239C23—1912no—
240C23—15281no—
241C23—15281no—
242C23—7642no—
243C23D4×D19111yescharacteristic
244Q8—21no—
245Q8—21no—
246Q8—3822no—
247D4—21no—
248D4—21no—
249D4—42no—
250D4—3824no—
251D4—3822no—
252D4—3822no—
253D4—3822no—
254D4—15282no—
255D4—15282no—
256D4—15282no—
257D4—7642no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—3822no—
263C2×C4—3822no—
264C2×C4—3822no—
265C2×C4—1912no—
266C2×C4—15281no—
267C2×C4D4×D19112yes—
268C2×C4D4×D19111yescharacteristic
269C2×C4#1569028712yes—
270C2×C4C23×D19111yescharacteristic
271C8—21no—
272C8—21no—
273C8—1914no—
274C22—21no—
275C22—21no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—3822no—
280C22—3822no—
281C22—3822no—
282C22—15284no—
283C22—7642no—
284C22—7642no—
285C22—7642no—
286C22#2025929512yes—
287C22C2×D4×D19111yescharacteristic
288C4—41no—
289C4—41no—
290C4—41no—
291C4—3822no—
292C4—7642no—
293C4#2025929512yes—
294C4C2×D4×D19111yescharacteristic
295C4C2×#1569028711yescharacteristic
296C2—22no—
297C2—81no—
298C2—15282no—
299C2—7642no—
300C2C2×#2025929511yescharacteristic
301C2#2713346511yescharacteristic
302C2#2713347011yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#416×C191C2
socleC22×C191C22×D4
hypercenter#416D191
hypocenterC191#3062
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#20259347C2semidirect
2. C2×#20259328C2semidirect
3. #27133241C2semidirect
4. #27133232C2semidirect
5. #27132915C2semidirect
6. #27132912C2semidirect
7. #27132898C2semidirect
8. #27132895C2semidirect
9. #27132706C2semidirect
10. C2×#20259244C2semidirect
11. #416×C191C2semidirect
19. C2×#15690270C22semidirect
20. C2×C8×D191C22semidirect
24. #20259298C22semidirect
29. #20259293C22semidirect
32. #20259255C22semidirect
33. #20259253C22semidirect
39. #103×C191C22semidirect
40. #103×C191C22semidirect
41. #102×C191C22semidirect
42. #102×C191C22semidirect
60. #15690271D4semidirect
151. C191#3062semidirect
162. #416D191semidirect
189. #103C2×D191semidirect
190. #103C2×D191semidirect
193. #102C2×D191semidirect
194. #102C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593472
C2×#202593282
#271332412
#271332322
#271329152
#271329122
#271328982
#271328952
#271327062
C2×#202592442
#416×C1912
#3062191