Finite Group Databasebeta

← #37582968

Subgroups — #37582968

Order 24448.

656 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 153 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×#20259350C211yesmaximal, characteristic, complemented
2C2×#20259348C211yesmaximal, characteristic, complemented
3C2×#20259347C211yesmaximal, characteristic, complemented
4#27133209C211yesmaximal, characteristic, complemented
5C2×#20259318C211yesmaximal, characteristic, complemented
6#27133051C212yesmaximal, complemented
7#27133049C212yesmaximal, complemented
8#27133038C212yesmaximal, complemented
9#27133034C212yesmaximal, complemented
10M16×Dic191C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C191C211yesmaximal, characteristic, complemented
12C2×#15690287C2211yescharacteristic
13C22×D764C2211yescharacteristic
14C22×C22⋊D191—22no—
15#20259350—24no—
16#20259348—24no—
17#20259347—24no—
18C2×#15690276C2212yescomplemented
19C2×#15690275C2212yescomplemented
20C2×#15690274C2212yescomplemented
21C2×D1528C2212yescomplemented
22C2×#15690272C2212yescomplemented
23C2×#15690272C2212yescomplemented
24C2×#15690271C2211yescharacteristic
25#20259320C2211yescharacteristic, complemented
26#20259318—22no—
27#20259318C2212yes—
28#20259318C2212yes—
29#20259318C2211yescharacteristic, complemented
30#20259311C2211yescharacteristic, complemented
31#20259309C2211yescharacteristic, complemented
32#20259294—22no—
33C4×C22⋊D191—22no—
34C2×C4×Dic191C2211yescharacteristic
35#20259232C2212yescomplemented
36C8×Dic191C2212yescomplemented
37C2×#41×C191C2211yescharacteristic
38C22×D4×C191C2211yescharacteristic
39AGL(1,ℤ/8)×C191—24no—
40C2×SD16×C191C2212yescomplemented
41C2×D8×C191C2212yescomplemented
42C2×M16×C191C2211yescharacteristic
43C23×D191—21no—
44#15690287—22no—
45#15690287—22no—
46C2×Dic382C2311yescharacteristic
47C2×D764—22no—
48C2×D764—22no—
49C2×D764—21no—
50C2×D764C2311yescharacteristic
51C2×D764C2312yes—
52C2×C22⋊D191—24no—
53C2×C22⋊D191—24no—
54C2×C22⋊D191—22no—
55C2×C22⋊D191—21no—
56C2×C22⋊D191—48no—
57C22×Dic191D412yes—
58C2×C4×D191—21no—
59#15690276—24no—
60#15690275—24no—
61#15690274—24no—
62D1528—24no—
63#15690272—24no—
64#15690272—24no—
65#15690271D414yescomplemented
66C2×C191⋊C8C2312yes—
67Dic191⋊C4—22no—
68C22⋊Dic191—22no—
69(C2×D191)⋊C4—22no—
70C4×Dic191D412yes—
71C4×Dic191C2311yescharacteristic
72C4×Dic191C2311yescharacteristic
73C24×C191—21no—
74#41×C191—22no—
75#41×C191—22no—
76C2×Q8×C191C2311yescharacteristic
77C2×D4×C191—21no—
78C2×D4×C191—22no—
79C2×D4×C191—22no—
80C2×D4×C191—21no—
81C2×D4×C191C2311yescharacteristic
82C2×D4×C191C2312yes—
83C22×C4×C191—21no—
84C22×C4×C191C2311yescharacteristic
85SD16×C191—24no—
86D8×C191—24no—
87M16×C191D414yescomplemented
88C2×C8×C191C2312yes—
89C22×D191—21no—
90C22×D191—22no—
91C22×D191—44no—
92C22×D191—42no—
93Dic382—21no—
94Dic382—21no—
95D764—21no—
96D764—22no—
97D764—21no—
98D764—22no—
99D764—42no—
100C22⋊D191—48no—
101C22⋊D191—42no—
102C22⋊D191—44no—
103C22⋊D191—48no—
104C2×Dic191—24no—
105C2×Dic191—21no—
106C2×Dic191C2×D412yes—
107C2×Dic191C2×D412yes—
108C4×D191—42no—
109C191⋊C8C2×D414yescomplemented
110C23×C191—21no—
111C23×C191—22no—
112C23×C191—44no—
113C23×C191—42no—
114C23×C191C2×D411yescharacteristic
115Q8×C191—21no—
116Q8×C191—21no—
117D4×C191—21no—
118D4×C191—22no—
119D4×C191—22no—
120D4×C191—21no—
121D4×C191—42no—
122D4×C191—42no—
123C2×C4×C191—21no—
124C2×C4×C191—42no—
125C2×C4×C191C2×D411yescharacteristic
126C2×C4×C191C2×D412yes—
127C2×C4×C191C2×D412yes—
128C2×C4×C191C2411yescharacteristic
129C8×C191C2×D414yescomplemented
130C22×C191—22no—
131C22×C191—41no—
132C22×C191—42no—
133C22×C191—44no—
134C22×C191—42no—
135C22×C191—84no—
136C22×C191#12712yes—
137C22×C191C22×D411yescharacteristic
138C2×D191—41no—
139C2×D191—42no—
140C2×D191—44no—
141C2×D191—42no—
142C2×D191—84no—
143C4×C191—41no—
144C4×C191#12712yes—
145C4×C191C22×D411yescharacteristic
146C4×C191C22×D411yescharacteristic
147Dic191—22no—
148Dic191—41no—
149Dic191AGL(1,ℤ/8)14yescomplemented
150C2×C191—22no—
151C2×C191—81no—
152C2×C191—82no—
153C2×C191C2×#12711yescharacteristic
154C2×C191C2×AGL(1,ℤ/8)12yes—
155D191—81no—
156D191—82no—
157C191#315111yescharacteristic, complemented, Sylow
158#3151—1911nomaximal, Sylow
159C2×AGL(1,ℤ/8)—1911no—
160C2×AGL(1,ℤ/8)—1911no—
161C2×AGL(1,ℤ/8)—1911no—
162C2×AGL(1,ℤ/8)D19111yescharacteristic, complemented
163#532—1911no—
164C2×#127—1911no—
165#496—1912no—
166#496—1912no—
167#493—1912no—
168#492—1912no—
169C4×M16—1911no—
170C2×#41—1911no—
171C2×#41C2×D19111yescharacteristic
172C22×D4—3822no—
173C22×D4—1911no—
174C22×D4C2×D19111yescharacteristic
175AGL(1,ℤ/8)—24no—
176AGL(1,ℤ/8)—3824no—
177AGL(1,ℤ/8)—3824no—
178AGL(1,ℤ/8)—3824no—
179C2×SD16—1912no—
180C2×SD16—1912no—
181C2×SD16—1912no—
182C2×SD16C2×D19112yescomplemented
183C2×D8—1912no—
184C2×D8—1912no—
185C2×D8—1912no—
186C2×D8C2×D19112yescomplemented
187C2×M16—1911no—
188C2×M16C2×D19111yescharacteristic
189C4⋊Q8—1911no—
190#127—3822no—
191#127—1912no—
192#127—1912no—
193#127—1911no—
194#124—1911no—
195#124—1911no—
196#121—3822no—
197C4×D4—3822no—
198C2×C42—1911no—
199#97—1912no—
200C4×C8—1912no—
201C24—21no—
202C24—3821no—
203#41—22no—
204#41—22no—
205#41—3822no—
206#41—3822no—
207C2×Q8—1911no—
208C2×Q8C22×D19111yescharacteristic
209C2×D4—21no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—22no—
213C2×D4—3821no—
214C2×D4—3822no—
215C2×D4—3824no—
216C2×D4—3824no—
217C2×D4—3822no—
218C2×D4—3821no—
219C2×D4—3822no—
220C2×D4—1911no—
221C2×D4—1912no—
222C2×D4—7648no—
223C2×D4C22×D19111yescharacteristic
224C2×D4C22×D19112yes—
225C22×C4—21no—
226C22×C4—3821no—
227C22×C4—1912no—
228C22×C4C22×D19111yescharacteristic
229SD16—24no—
230SD16—3824no—
231SD16—3824no—
232SD16—3824no—
233D8—24no—
234D8—3824no—
235D8—3824no—
236D8—3824no—
237M16—1914no—
238M16C22⋊D19114yescomplemented
239C2×C8—1912no—
240C2×C8C22×D19112yes—
241C4⋊C4—3822no—
242C22⋊C4—3822no—
243C22⋊C4—3822no—
244C42—1912no—
245C42—1911no—
246C42—1911no—
247C23—21no—
248C23—22no—
249C23—44no—
250C23—42no—
251C23—3821no—
252C23—3822no—
253C23—7644no—
254C23—7642no—
255C23D4×D19111yescharacteristic
256Q8—21no—
257Q8—21no—
258Q8—3821no—
259Q8—3821no—
260D4—21no—
261D4—22no—
262D4—21no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—3821no—
267D4—3822no—
268D4—3821no—
269D4—3822no—
270D4—7648no—
271D4—7648no—
272D4—7644no—
273D4—7642no—
274D4—7642no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—3824no—
278C2×C4—3821no—
279C2×C4—1912no—
280C2×C4—1912no—
281C2×C4—7642no—
282C2×C4D4×D19112yes—
283C2×C4D4×D19111yescharacteristic
284C2×C4C2×C22⋊D19112yes—
285C2×C4C23×D19111yescharacteristic
286C8—1914no—
287C8C2×C22⋊D19114yescomplemented
288C22—22no—
289C22—44no—
290C22—42no—
291C22—41no—
292C22—42no—
293C22—84no—
294C22—15284no—
295C22—7644no—
296C22—7642no—
297C22—7641no—
298C22—7642no—
299C22#2025931812yes—
300C22C2×D4×D19111yescharacteristic
301C4—41no—
302C4—3822no—
303C4—1914no—
304C4—7641no—
305C4#2025931812yes—
306C4C2×D4×D19111yescharacteristic
307C4C22×C22⋊D19111yescharacteristic
308C2—22no—
309C2—82no—
310C2—81no—
311C2—15282no—
312C2—15281no—
313C2C2×#2025931811yescharacteristic
314C2AGL(1,ℤ/8)×D19112yes—
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
FittingC2×AGL(1,ℤ/8)×C191C2
socleC22×C191C22×D4
hypercenterC2×AGL(1,ℤ/8)D191
hypocenterC191#3151
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#20259350C2semidirect
2. C2×#20259348C2semidirect
3. C2×#20259347C2semidirect
4. #27133209C2semidirect
5. C2×#20259318C2semidirect
6. #27133051C2semidirect
7. #27133049C2semidirect
8. #27133038C2semidirect
9. #27133034C2semidirect
10. M16×Dic191C2semidirect
11. C2×AGL(1,ℤ/8)×C191C2semidirect
18. C2×#15690276C22semidirect
19. C2×#15690275C22semidirect
20. C2×#15690274C22semidirect
21. C2×D1528C22semidirect
22. C2×#15690272C22semidirect
23. C2×#15690272C22semidirect
25. #20259320C22semidirect
29. #20259318C22semidirect
30. #20259311C22semidirect
31. #20259309C22semidirect
35. #20259232C22semidirect
36. C8×Dic191C22semidirect
40. C2×SD16×C191C22semidirect
41. C2×D8×C191C22semidirect
65. #15690271D4semidirect
87. M16×C191D4semidirect
109. C191⋊C8C2×D4semidirect
129. C8×C191C2×D4semidirect
149. Dic191AGL(1,ℤ/8)semidirect
157. C191#3151semidirect
162. C2×AGL(1,ℤ/8)D191semidirect
182. C2×SD16C2×D191semidirect
186. C2×D8C2×D191semidirect
238. M16C22⋊D191semidirect
287. C8C2×C22⋊D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593502
C2×#202593482
C2×#202593472
#271332092
C2×#202593182
#271330512
#271330492
#271330382
#271330342
M16×Dic1912
C2×AGL(1,ℤ/8)×C1912
#3151191