Finite Group Databasebeta

← #37584116

Subgroups — #37584116

Order 24448.

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

normal subgroups only

This group has 546 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×#20259348C212yesmaximal, complemented
2#27133318C211yesmaximal, characteristic, complemented
3#27133035C211yesmaximal, characteristic, complemented
4#27132907C212yesmaximal, complemented
5#27132868C211yesmaximal, characteristic, complemented
6#27132855C211yesmaximal, characteristic, complemented
7#27132814C212yesmaximal, complemented
8#27132806C212yesmaximal, complemented
9C4×#15690272C211yesmaximal, characteristic, complemented
10#27132741C211yesmaximal, characteristic, complemented
11D42×C191C211yesmaximal, characteristic, complemented
12C2×#15690287C2212yescomplemented
13#20259348—28no—
14C2×#15690275C2212yes—
15C2×#15690275C2212yescomplemented
16C2×#15690275C2212yescomplemented
17C2×#15690272C2211yescharacteristic
18C2×#15690272C2212yescomplemented
19C2×#15690271C2212yescomplemented
20#20259322C2211yescharacteristic, complemented
21#20259304—22no—
22#20259301C2212yescomplemented
23#20259293—21no—
24C4×D764C2211yescharacteristic, complemented
25C4⋊C4×D191—21no—
26#20259255C2211yescharacteristic, complemented
27C4⋊(C191⋊C8)C2211yescharacteristic, complemented
28#20259245C2212yescomplemented
29#20259243C2211yescharacteristic
30#20259243C2212yescomplemented
31#20259242C2211yescharacteristic, complemented
32C22⋊(C191⋊C8)C2212yescomplemented
33C4×C191⋊C8C2211yescharacteristic, complemented
34C22×D4×C191—21no—
35C22×D4×C191C2212yes—
36#127×C191C2211yescharacteristic
37#121×C191—21no—
38#121×C191C2212yes—
39#120×C191—22no—
40C4×D4×C191C2211yescharacteristic
41C4×D4×C191C2211yescharacteristic
42#15690287—24no—
43#15690287—42no—
44C2×Dic382C2312yes—
45C2×D764C2311yescharacteristic
46C2×C22⋊D191—22no—
47C2×C4×D191—22no—
48C2×C4×D191—21no—
49#15690275—24no—
50#15690275—24no—
51#15690275—42no—
52#15690272—24no—
53#15690272D414yescomplemented
54#15690271—24no—
55C2×C191⋊C8C2312yes—
56C2×C191⋊C8C2311yescharacteristic
57C2×C191⋊C8C2311yescharacteristic
58C4⋊Dic191—21no—
59C4⋊Dic191—21no—
60C4⋊Dic191C2311yescharacteristic
61Dic191⋊C4—22no—
62C22⋊Dic191—22no—
63(C2×D191)⋊C4—21no—
64C24×C191—22no—
65C2×D4×C191—24no—
66C2×D4×C191—22no—
67C2×D4×C191—22no—
68C2×D4×C191—21no—
69C2×D4×C191—21no—
70C2×D4×C191—42no—
71C2×D4×C191—44no—
72C2×D4×C191D411yescharacteristic
73C2×D4×C191C2312yes—
74C2×D4×C191C2311yescharacteristic
75C22×C4×C191—21no—
76C22×C4×C191C2312yes—
77C4⋊C4×C191D411yescharacteristic
78C4⋊C4×C191C2311yescharacteristic
79C22⋊C4×C191—21no—
80C22⋊C4×C191—22no—
81C22⋊C4×C191D412yes—
82C42×C191C2311yescharacteristic
83C22×D191—21no—
84Dic382—22no—
85Dic382—22no—
86D764—21no—
87D764C2×D412yescomplemented
88C22⋊D191—44no—
89C2×Dic191—21no—
90C2×Dic191—22no—
91C4×D191—44no—
92C4×D191—42no—
93C191⋊C8—22no—
94C191⋊C8—21no—
95C191⋊C8C2×D412yescomplemented
96C23×C191—21no—
97C23×C191—22no—
98C23×C191—44no—
99C23×C191—44no—
100C23×C191—82no—
101C23×C191C2×D412yes—
102D4×C191—22no—
103D4×C191—22no—
104D4×C191—21no—
105D4×C191—22no—
106D4×C191—44no—
107D4×C191—44no—
108D4×C191—42no—
109D4×C191—81no—
110D4×C191C2×D412yes—
111C2×C4×C191—22no—
112C2×C4×C191—22no—
113C2×C4×C191—21no—
114C2×C4×C191—42no—
115C2×C4×C191C2×D411yescharacteristic
116C2×C4×C191C2×D412yes—
117C2×C4×C191C2×D411yescharacteristic
118C2×C4×C191C2411yescharacteristic
119C22×C191—22no—
120C22×C191—22no—
121C22×C191—22no—
122C22×C191—22no—
123C22×C191—41no—
124C22×C191—41no—
125C22×C191—42no—
126C22×C191—42no—
127C22×C191—42no—
128C22×C191—88no—
129C22×C191C22×D411yescharacteristic
130C2×D191—22no—
131C2×D191—41no—
132C2×D191—41no—
133C4×C191—22no—
134C4×C191—21no—
135C4×C191—41no—
136C4×C191AGL(1,ℤ/8)12yescomplemented
137C4×C191C22×D411yescharacteristic
138C4×C191H(5,𝔽2)11yescharacteristic
139Dic191—42no—
140Dic191—41no—
141C2×C191—42no—
142C2×C191—42no—
143C2×C191—82no—
144C2×C191D4211yescharacteristic
145C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C191#57711yescharacteristic
147D191—42no—
148C191#327211yescharacteristic, complemented, Sylow
149#3272—1911nomaximal, Sylow
150C2×AGL(1,ℤ/8)—1912no—
151#547—1911no—
152D42D19111yescharacteristic, complemented
153#492—1911no—
154#468—1912no—
155#461—1911no—
156#459—1911no—
157#450—1912no—
158#449—1912no—
159C4×D8—1911no—
160#436—1911no—
161C2×#41—1912no—
162C22×D4—21no—
163C22×D4C2×D19112yes—
164AGL(1,ℤ/8)—3828no—
165C2×SD16—1912no—
166C2×SD16—1912no—
167C2×SD16—1912no—
168C2×D8—1912no—
169C2×D8—1911no—
170C2×M16—1912no—
171C4⋊Q8—1911no—
172#127C2×D19111yescharacteristic
173#123—3822no—
174C22⋊Q8—1912no—
175#121—21no—
176#121—3821no—
177#121C2×D19112yes—
178#120—22no—
179C4×D4—1911no—
180C4×D4C2×D19111yescharacteristic
181C4×D4C2×D19111yescharacteristic
182C2×C4⋊C4—3821no—
183#107—1911no—
184C4⋊C8—1911no—
185#103—1912no—
186#102—1911no—
187#102—1912no—
188#102—1911no—
189C22⋊C8—1912no—
190C4×C8—1911no—
191C24—22no—
192#41—3824no—
193#41—7642no—
194C2×Q8—1912no—
195C2×D4—24no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—42no—
201C2×D4—44no—
202C2×D4—3822no—
203C2×D4—1911no—
204C2×D4C22⋊D19111yescharacteristic
205C2×D4C22×D19112yes—
206C2×D4C22×D19111yescharacteristic
207C22×C4—21no—
208C22×C4—3821no—
209C22×C4—3822no—
210C22×C4C22×D19112yes—
211SD16—3824no—
212SD16—3824no—
213SD16—7642no—
214D8—3824no—
215D8—1914no—
216M16—3824no—
217C2×C8—1911no—
218C2×C8—1912no—
219C2×C8—1911no—
220C4⋊C4—3822no—
221C4⋊C4—3821no—
222C4⋊C4—3821no—
223C4⋊C4—1911no—
224C4⋊C4C22⋊D19111yescharacteristic
225C4⋊C4C22×D19111yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—22no—
228C22⋊C4—3822no—
229C22⋊C4—3821no—
230C22⋊C4C22⋊D19112yes—
231C42C22×D19111yescharacteristic
232C23—21no—
233C23—22no—
234C23—44no—
235C23—44no—
236C23—82no—
237C23—3821no—
238C23C2×C22⋊D19112yes—
239Q8—3822no—
240Q8—3822no—
241D4—22no—
242D4—22no—
243D4—21no—
244D4—22no—
245D4—44no—
246D4—44no—
247D4—42no—
248D4—81no—
249D4—3821no—
250D4—1912no—
251D4—7644no—
252D4D4×D19112yes—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—42no—
257C2×C4—3821no—
258C2×C4—3822no—
259C2×C4—7644no—
260C2×C4—7642no—
261C2×C4C2×C22⋊D19111yescharacteristic
262C2×C4C2×C22⋊D19112yes—
263C2×C4C2×C22⋊D19111yescharacteristic
264C2×C4C23×D19111yescharacteristic
265C8—3822no—
266C8—3821no—
267C8—1912no—
268C22—22no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—41no—
273C22—41no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—88no—
278C22—3822no—
279C22—7641no—
280C22—7641no—
281C22C22×C22⋊D19111yescharacteristic
282C4—21no—
283C4—22no—
284C4—41no—
285C4—7641no—
286C4—7642no—
287C4#2025934812yescomplemented
288C4C2×D4×D19111yescharacteristic
289C4#2025936711yescharacteristic
290C2—42no—
291C2—42no—
292C2—82no—
293C2—7642no—
294C2D4×C22⋊D19111yescharacteristic
295C2C2×#2025934811yescharacteristic
296C2#2713347411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
FittingD42×C191C2
socleC22×C191C22×D4
hypercenterD42D191
hypocenterC191#3272
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C2×#20259348C2semidirect
2. #27133318C2semidirect
3. #27133035C2semidirect
4. #27132907C2semidirect
5. #27132868C2semidirect
6. #27132855C2semidirect
7. #27132814C2semidirect
8. #27132806C2semidirect
9. C4×#15690272C2semidirect
10. #27132741C2semidirect
11. D42×C191C2semidirect
12. C2×#15690287C22semidirect
15. C2×#15690275C22semidirect
16. C2×#15690275C22semidirect
18. C2×#15690272C22semidirect
19. C2×#15690271C22semidirect
20. #20259322C22semidirect
22. #20259301C22semidirect
24. C4×D764C22semidirect
26. #20259255C22semidirect
27. C4⋊(C191⋊C8)C22semidirect
28. #20259245C22semidirect
30. #20259243C22semidirect
31. #20259242C22semidirect
32. C22⋊(C191⋊C8)C22semidirect
33. C4×C191⋊C8C22semidirect
53. #15690272D4semidirect
87. D764C2×D4semidirect
95. C191⋊C8C2×D4semidirect
136. C4×C191AGL(1,ℤ/8)semidirect
148. C191#3272semidirect
152. D42D191semidirect
287. C4#20259348semidirect

Maximal subgroups

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

subgroupindex
C2×#202593482
#271333182
#271330352
#271329072
#271328682
#271328552
#271328142
#271328062
C4×#156902722
#271327412
D42×C1912
#3272191