Finite Group Databasebeta

← #481×D191 #37582416

Subgroups — #481×D191 #37582416

Order 24448.

512 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 135 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
1#121×D191C211yesmaximal, characteristic
2C2×C4⋊C4×D191C211yesmaximal, characteristic, complemented
3#27132983C212yesmaximal, complemented
4#27132982C211yesmaximal, characteristic, complemented
5#27132981C211yesmaximal, characteristic, complemented
6#27132980C212yesmaximal, complemented
7#27132979C211yesmaximal, characteristic, complemented
8#106×D191C212yesmaximal, complemented
9#102×D191C212yesmaximal, complemented
10C22⋊C8×D191C211yesmaximal, characteristic, complemented
11#481×C191C211yesmaximal, characteristic, complemented
12C2×D4×D191—21no—
13C2×D4×D191C2211yescharacteristic
14C22×C4×D191—21no—
15C22×C4×D191C2211yescharacteristic
16C2×C8×D191C2212yes—
17#20259297C2211yescharacteristic
18#20259296C2211yescharacteristic
19#20259295—21no—
20#20259294—21no—
21#20259293C2211yescharacteristic
22C2×C4⋊Dic191C2211yescharacteristic, complemented
23C2×Dic191⋊C4—21no—
24C4⋊C4×D191—21no—
25C4⋊C4×D191—21no—
26C4⋊C4×D191C2211yescharacteristic
27C4⋊C4×D191C2212yes—
28C4⋊C4×D191C2212yescomplemented
29C22⋊C4×D191—21no—
30#20259254C2212yescomplemented
31#20259253C2212yescomplemented
32#20259253C2212yescomplemented
33#20259244C2212yescomplemented
34#20259243C2212yescomplemented
35#20259242C2212yescomplemented
36C22⋊(C191⋊C8)C2211yescharacteristic, complemented
37(C2×D191)⋊C8C2212yescomplemented
38#121×C191C2211yescharacteristic
39C2×C4⋊C4×C191C2211yescharacteristic, complemented
40#106×C191C2212yescomplemented
41#102×C191C2212yescomplemented
42C22⋊C8×C191C2211yescharacteristic, complemented
43C23×D191—21no—
44C23×D191D411yescharacteristic
45C2×D764—21no—
46C2×D764C2311yescharacteristic
47C2×C22⋊D191—21no—
48C2×C22⋊D191—21no—
49C2×C22⋊D191—21no—
50D4×D191—24no—
51D4×D191—44no—
52C22×Dic191—21no—
53C22×Dic191D411yescharacteristic
54C2×C4×D191—24no—
55C2×C4×D191—22no—
56C2×C4×D191—22no—
57C2×C4×D191—21no—
58C2×C4×D191—44no—
59C2×C4×D191D411yescharacteristic
60C2×C4×D191D411yescharacteristic
61C2×C4×D191C2311yescharacteristic
62C2×C4×D191C2311yescharacteristic
63C2×C191⋊C8C2312yes—
64C8×D191—24no—
65C4⋊Dic191—21no—
66C4⋊Dic191C2312yes—
67C4⋊Dic191C2311yescharacteristic
68Dic191⋊C4—22no—
69Dic191⋊C4—22no—
70Dic191⋊C4—21no—
71C22⋊Dic191—21no—
72(C2×D191)⋊C4—21no—
73(C2×D191)⋊C4—21no—
74C2×D4×C191—21no—
75C2×D4×C191C2311yescharacteristic
76C22×C4×C191—21no—
77C22×C4×C191C2311yescharacteristic
78C2×C8×C191C2312yes—
79C4⋊C4×C191—21no—
80C4⋊C4×C191C2312yes—
81C4⋊C4×C191C2311yescharacteristic
82C22⋊C4×C191—21no—
83C22×D191—22no—
84C22×D191—22no—
85C22×D191—21no—
86C22×D191—42no—
87C22×D191—42no—
88C22×D191—42no—
89C22×D191SD1614yes—
90C22×D191C2×D411yescharacteristic
91C22×D191C2×D411yescharacteristic
92D764—22no—
93D764—42no—
94C22⋊D191—42no—
95C22⋊D191—42no—
96C22⋊D191—42no—
97C2×Dic191—22no—
98C2×Dic191—21no—
99C2×Dic191—22no—
100C2×Dic191—42no—
101C2×Dic191C2×D411yescharacteristic
102C2×Dic191C2×D411yescharacteristic
103C4×D191—22no—
104C4×D191—22no—
105C4×D191—22no—
106C4×D191—44no—
107C4×D191—42no—
108C4×D191—44no—
109C4×D191#4114yes—
110C191⋊C8—22no—
111C23×C191—21no—
112C23×C191C2×D411yescharacteristic
113D4×C191—22no—
114D4×C191—42no—
115C2×C4×C191—22no—
116C2×C4×C191—21no—
117C2×C4×C191—22no—
118C2×C4×C191—42no—
119C2×C4×C191C2×D411yescharacteristic
120C2×C4×C191C2411yescharacteristic
121C8×C191—22no—
122C22×C191—21no—
123C22×C191—21no—
124C22×C191—41no—
125C22×C191—41no—
126C22×C191—41no—
127C22×C191C2×SD1612yes—
128C22×C191C22×D411yescharacteristic
129C2×D191—21no—
130C2×D191—21no—
131C2×D191—28no—
132C2×D191—41no—
133C2×D191—41no—
134C2×D191—41no—
135C2×D191—84no—
136C2×D191#12312yes—
137C2×D191C2×SD1612yes—
138C2×D191C2×SD1612yes—
139C2×D191AGL(1,ℤ/8)12yes—
140C4×C191—21no—
141C4×C191—41no—
142C4×C191—42no—
143C4×C191C2×#4112yes—
144Dic191—21no—
145Dic191—41no—
146Dic191—42no—
147Dic191#12312yes—
148C2×C191—22no—
149C2×C191—81no—
150C2×C191C2×#12311yescharacteristic
151C2×C191C22×SD1611yescharacteristic
152C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
153D191—22no—
154D191—81no—
155D191#48114yesdirect product factor
156C191C2×#48111yescharacteristic, complemented, Sylow
157C2×#481—1911nomaximal, Sylow
158C2×#121—1911no—
159C22×C4⋊C4—1911no—
160#481—1911no—
161#481—1911no—
162#481—1912no—
163#481—1912no—
164#481—1911no—
165#481D19111yescharacteristic, direct product factor
166C2×#106—1912no—
167C2×#102—1912no—
168C2×C22⋊C8—1911no—
169C22×D4—3821no—
170C22×D4—1911no—
171C23×C4—3821no—
172C23×C4—1911no—
173C22×C8—1912no—
174#121—3821no—
175#121—3821no—
176#121—1911no—
177#121—1911no—
178#121—1911no—
179#121C2×D19111yescharacteristic
180C2×C4⋊C4—3821no—
181C2×C4⋊C4—3821no—
182C2×C4⋊C4—3821no—
183C2×C4⋊C4—1911no—
184C2×C4⋊C4—1912no—
185C2×C4⋊C4—1912no—
186C2×C4⋊C4—1911no—
187C2×C4⋊C4C2×D19111yescharacteristic, complemented
188C2×C22⋊C4—3821no—
189#106—1912no—
190#106—1912no—
191#106—1912no—
192#106C2×D19112yescomplemented
193#102—1912no—
194#102—1912no—
195#102—1912no—
196#102C2×D19112yescomplemented
197C22⋊C8—1911no—
198C22⋊C8—1912no—
199C22⋊C8C2×D19111yescharacteristic, complemented
200C24—3821no—
201C24—1911no—
202C2×D4—21no—
203C2×D4—3821no—
204C2×D4—3821no—
205C2×D4—3821no—
206C2×D4—3824no—
207C2×D4—3821no—
208C2×D4—1911no—
209C2×D4—7644no—
210C2×D4C22×D19111yescharacteristic
211C22×C4—21no—
212C22×C4—3824no—
213C22×C4—3821no—
214C22×C4—3822no—
215C22×C4—3822no—
216C22×C4—3821no—
217C22×C4—1911no—
218C22×C4—1911no—
219C22×C4—1911no—
220C22×C4—1911no—
221C22×C4—1911no—
222C22×C4—7644no—
223C22×C4C22×D19111yescharacteristic
224C2×C8—3824no—
225C2×C8—1912no—
226C2×C8C22×D19112yes—
227C4⋊C4—21no—
228C4⋊C4—3822no—
229C4⋊C4—3822no—
230C4⋊C4—3821no—
231C4⋊C4—3821no—
232C4⋊C4—1912no—
233C4⋊C4—1911no—
234C4⋊C4C22×D19112yes—
235C4⋊C4C22×D19111yescharacteristic
236C22⋊C4—21no—
237C22⋊C4—3821no—
238C22⋊C4—3821no—
239C22⋊C4—3821no—
240C23—21no—
241C23—3822no—
242C23—3822no—
243C23—3821no—
244C23—1914no—
245C23—1911no—
246C23—1911no—
247C23—7642no—
248C23—7642no—
249C23—7642no—
250C23D4×D19111yescharacteristic
251D4—22no—
252D4—42no—
253D4—3822no—
254D4—7642no—
255D4—7642no—
256D4—7642no—
257D4—7642no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—42no—
262C2×C4—3822no—
263C2×C4—3822no—
264C2×C4—3822no—
265C2×C4—3822no—
266C2×C4—3821no—
267C2×C4—3822no—
268C2×C4—1911no—
269C2×C4—1911no—
270C2×C4—1914no—
271C2×C4—7644no—
272C2×C4—7644no—
273C2×C4—7642no—
274C2×C4—7642no—
275C2×C4D4×D19111yescharacteristic
276C2×C4C23×D19111yescharacteristic
277C8—22no—
278C8—3822no—
279C22—21no—
280C22—21no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—3828no—
285C22—3821no—
286C22—3821no—
287C22—1912no—
288C22—1912no—
289C22—1912no—
290C22—1912no—
291C22—15284no—
292C22—7641no—
293C22—7641no—
294C22—7641no—
295C22SD16×D19112yes—
296C22C2×D4×D19111yescharacteristic
297C4—21no—
298C4—42no—
299C4—41no—
300C4—3821no—
301C4—1912no—
302C4—7642no—
303C4—7641no—
304C4#41×D19112yes—
305C2—22no—
306C2—81no—
307C2—3822no—
308C2—1914no—
309C2—15281no—
310C2#123×D19111yescharacteristic
311C2C2×SD16×D19111yescharacteristic
312C2AGL(1,ℤ/8)×D19111yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#481×C191C2
socleC22×C191C22×D4
prime radical#121×D191C2
hypercenter#481D191
hypocenterC191C2×#481
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D191C2semidirect
3. #27132983C2semidirect
4. #27132982C2semidirect
5. #27132981C2semidirect
6. #27132980C2semidirect
7. #27132979C2semidirect
8. #106×D191C2semidirect
9. #102×D191C2semidirect
10. C22⋊C8×D191C2semidirect
11. #481×C191C2semidirect
22. C2×C4⋊Dic191C22semidirect
28. C4⋊C4×D191C22semidirect
30. #20259254C22semidirect
31. #20259253C22semidirect
32. #20259253C22semidirect
33. #20259244C22semidirect
34. #20259243C22semidirect
35. #20259242C22semidirect
36. C22⋊(C191⋊C8)C22semidirect
37. (C2×D191)⋊C8C22semidirect
39. C2×C4⋊C4×C191C22semidirect
40. #106×C191C22semidirect
41. #102×C191C22semidirect
42. C22⋊C8×C191C22semidirect
155. D191#481direct (N × H)
156. C191C2×#481semidirect
165. #481D191direct (N × H)
187. C2×C4⋊C4C2×D191semidirect
192. #106C2×D191semidirect
196. #102C2×D191semidirect
199. C22⋊C8C2×D191semidirect

Maximal subgroups

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

subgroupindex
#121×D1912
C2×C4⋊C4×D1912
#271329832
#271329822
#271329812
#271329802
#271329792
#106×D1912
#102×D1912
C22⋊C8×D1912
#481×C1912
C2×#481191