Finite Group Databasebeta

← #37584120

Subgroups — #37584120

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×#20259347C212yesmaximal, complemented
2D4×D764C211yesmaximal, characteristic, complemented
3#27133037C211yesmaximal, characteristic, complemented
4#27132911C212yesmaximal, complemented
5#27132865C211yesmaximal, characteristic, complemented
6#27132858C211yesmaximal, characteristic, complemented
7#27132813C212yesmaximal, complemented
8#27132808C212yesmaximal, complemented
9C4×#15690272C211yesmaximal, characteristic, complemented
10#27132741C211yesmaximal, characteristic, complemented
11#547×C191C211yesmaximal, characteristic, complemented
12C22×D764C2212yes—
13C2×D4×D191—21no—
14#20259347—28no—
15C2×#15690274C2212yes—
16C2×#15690274C2212yescomplemented
17C2×#15690274C2212yescomplemented
18C2×#15690272C2211yescharacteristic
19C2×#15690272C2212yescomplemented
20C2×#15690271C2212yescomplemented
21#20259319C2211yescharacteristic
22#20259297—21no—
23#20259296C2212yes—
24#20259292—22no—
25C4×D764C2211yescharacteristic
26#20259255C2211yescharacteristic, complemented
27C4⋊(C191⋊C8)C2211yescharacteristic, complemented
28#20259246C2212yescomplemented
29#20259243C2211yescharacteristic, complemented
30#20259242C2211yescharacteristic
31#20259242C2212yescomplemented
32C22⋊(C191⋊C8)C2212yescomplemented
33C4×C191⋊C8C2211yescharacteristic, complemented
34C2×#41×C191C2212yescomplemented
35C4⋊Q8×C191C2211yescharacteristic, complemented
36#123×C191—22no—
37C22⋊Q8×C191C2212yescomplemented
38#121×C191—21no—
39C4×D4×C191C2211yescharacteristic
40C4×D4×C191C2211yescharacteristic, complemented
41C2×C4⋊C4×C191—21no—
42C23×D191—22no—
43C2×D764—24no—
44C2×D764—21no—
45C2×D764—22no—
46C2×D764—21no—
47C2×D764—42no—
48C2×D764C2312yes—
49C2×D764C2311yescharacteristic
50C2×C22⋊D191—22no—
51D4×D191—44no—
52C2×C4×D191—21no—
53#15690274—24no—
54#15690274—24no—
55#15690274—42no—
56#15690272—24no—
57#15690272D414yescomplemented
58#15690271—24no—
59C2×C191⋊C8C2312yes—
60C2×C191⋊C8C2311yescharacteristic
61C2×C191⋊C8C2311yescharacteristic
62C4⋊Dic191C2311yescharacteristic
63(C2×D191)⋊C4—21no—
64(C2×D191)⋊C4—22no—
65#41×C191—24no—
66#41×C191—42no—
67C2×Q8×C191C2312yes—
68C2×D4×C191—22no—
69C2×D4×C191D411yescharacteristic
70C2×D4×C191C2311yescharacteristic
71C22×C4×C191—21no—
72C22×C4×C191—22no—
73C22×C4×C191C2312yes—
74C4⋊C4×C191—21no—
75C4⋊C4×C191—22no—
76C4⋊C4×C191—21no—
77C4⋊C4×C191D411yescharacteristic
78C4⋊C4×C191C2311yescharacteristic
79C22⋊C4×C191—22no—
80C22⋊C4×C191—21no—
81C22⋊C4×C191D412yes—
82C42×C191C2311yescharacteristic
83C22×D191—21no—
84C22×D191—22no—
85C22×D191—44no—
86C22×D191—44no—
87C22×D191—82no—
88D764—22no—
89D764—21no—
90D764—22no—
91D764—44no—
92D764—42no—
93D764—81no—
94D764C2×D412yes—
95C22⋊D191—44no—
96C2×Dic191—21no—
97C4×D191—42no—
98C191⋊C8—22no—
99C191⋊C8—21no—
100C191⋊C8C2×D412yescomplemented
101C23×C191—21no—
102C23×C191C2×D412yes—
103Q8×C191—22no—
104Q8×C191—22no—
105D4×C191—22no—
106D4×C191—21no—
107D4×C191—44no—
108D4×C191C2×D412yescomplemented
109C2×C4×C191—22no—
110C2×C4×C191—22no—
111C2×C4×C191—21no—
112C2×C4×C191—22no—
113C2×C4×C191—44no—
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×C191C22×D411yescharacteristic
126C2×D191—22no—
127C2×D191—41no—
128C2×D191—41no—
129C2×D191—42no—
130C2×D191—42no—
131C2×D191—42no—
132C2×D191—88no—
133C4×C191—22no—
134C4×C191—21no—
135C4×C191—42no—
136C4×C191—41no—
137C4×C191AGL(1,ℤ/8)12yescomplemented
138C4×C191C22×D411yescharacteristic
139C4×C191H(5,𝔽2)11yescharacteristic
140Dic191—41no—
141C2×C191—42no—
142C2×C191—42no—
143C2×C191D4211yescharacteristic
144C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
145C2×C191#57711yescharacteristic
146D191—42no—
147D191—82no—
148C191#327211yescharacteristic, complemented, Sylow
149#3272—1911nomaximal, Sylow
150C2×AGL(1,ℤ/8)—1912no—
151#547D19111yescharacteristic, complemented
152D42—1911no—
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×#41C2×D19112yescomplemented
162C22×D4—3821no—
163C22×D4—1912no—
164AGL(1,ℤ/8)—3828no—
165C2×SD16—1912no—
166C2×SD16—1912no—
167C2×SD16—1912no—
168C2×D8—1912no—
169C2×D8—1911no—
170C2×M16—1912no—
171C4⋊Q8C2×D19111yescharacteristic, complemented
172#127—1911no—
173#123—22no—
174C22⋊Q8C2×D19112yescomplemented
175#121—21no—
176#121—3821no—
177#121—1912no—
178#120—3822no—
179C4×D4—1911no—
180C4×D4C2×D19111yescharacteristic
181C4×D4C2×D19111yescharacteristic, complemented
182C2×C4⋊C4—21no—
183#107—1911no—
184C4⋊C8—1911no—
185#103—1912no—
186#102—1911no—
187#102—1911no—
188#102—1912no—
189C22⋊C8—1912no—
190C4×C8—1911no—
191C24—3822no—
192#41—24no—
193#41—42no—
194C2×Q8C22×D19112yes—
195C2×D4—22no—
196C2×D4—3824no—
197C2×D4—3822no—
198C2×D4—3821no—
199C2×D4—3822no—
200C2×D4—3821no—
201C2×D4—1912no—
202C2×D4—1911no—
203C2×D4—7642no—
204C2×D4—7644no—
205C2×D4C22⋊D19111yescharacteristic
206C2×D4C22×D19111yescharacteristic
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—3821no—
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—21no—
221C4⋊C4—22no—
222C4⋊C4—21no—
223C4⋊C4—1911no—
224C4⋊C4C22⋊D19111yescharacteristic
225C4⋊C4C22×D19111yescharacteristic
226C22⋊C4—22no—
227C22⋊C4—21no—
228C22⋊C4—3821no—
229C22⋊C4—3822no—
230C22⋊C4C22⋊D19112yes—
231C42C22×D19111yescharacteristic
232C23—21no—
233C23—3821no—
234C23—3822no—
235C23—15282no—
236C23—7644no—
237C23—7644no—
238C23C2×C22⋊D19112yes—
239Q8—22no—
240Q8—22no—
241D4—22no—
242D4—21no—
243D4—44no—
244D4—3822no—
245D4—3821no—
246D4—3822no—
247D4—1912no—
248D4—15281no—
249D4—7644no—
250D4—7644no—
251D4—7642no—
252D4D4×D19112yescomplemented
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—44no—
258C2×C4—42no—
259C2×C4—3821no—
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—3822no—
275C22—15288no—
276C22—7642no—
277C22—7642no—
278C22—7642no—
279C22—7641no—
280C22—7641no—
281C22C22×C22⋊D19111yescharacteristic
282C4—21no—
283C4—22no—
284C4—41no—
285C4—42no—
286C4—7641no—
287C4#2025934712yescomplemented
288C4C2×D4×D19111yescharacteristic
289C4#2025936711yescharacteristic
290C2—42no—
291C2—42no—
292C2—15282no—
293C2—7642no—
294C2D4×C22⋊D19111yescharacteristic
295C2C2×#2025934711yescharacteristic
296C2#2713347111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#547×C191C2
socleC22×C191C22×D4
hypercenter#547D191
hypocenterC191#3272
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C2×#20259347C2semidirect
2. D4×D764C2semidirect
3. #27133037C2semidirect
4. #27132911C2semidirect
5. #27132865C2semidirect
6. #27132858C2semidirect
7. #27132813C2semidirect
8. #27132808C2semidirect
9. C4×#15690272C2semidirect
10. #27132741C2semidirect
11. #547×C191C2semidirect
16. C2×#15690274C22semidirect
17. C2×#15690274C22semidirect
19. C2×#15690272C22semidirect
20. C2×#15690271C22semidirect
26. #20259255C22semidirect
27. C4⋊(C191⋊C8)C22semidirect
28. #20259246C22semidirect
29. #20259243C22semidirect
31. #20259242C22semidirect
32. C22⋊(C191⋊C8)C22semidirect
33. C4×C191⋊C8C22semidirect
34. C2×#41×C191C22semidirect
35. C4⋊Q8×C191C22semidirect
37. C22⋊Q8×C191C22semidirect
40. C4×D4×C191C22semidirect
57. #15690272D4semidirect
100. C191⋊C8C2×D4semidirect
108. D4×C191C2×D4semidirect
137. C4×C191AGL(1,ℤ/8)semidirect
148. C191#3272semidirect
151. #547D191semidirect
161. C2×#41C2×D191semidirect
171. C4⋊Q8C2×D191semidirect
174. C22⋊Q8C2×D191semidirect
181. C4×D4C2×D191semidirect
252. D4D4×D191semidirect
287. C4#20259347semidirect

Maximal subgroups

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

subgroupindex
C2×#202593472
D4×D7642
#271330372
#271329112
#271328652
#271328582
#271328132
#271328082
C4×#156902722
#271327412
#547×C1912
#3272191