Finite Group Databasebeta

← #37583005

Subgroups — #37583005

Order 24448.

596 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 149 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×#20259342C211yesmaximal, characteristic, complemented
3C22×#15690272C211yesmaximal, characteristic, complemented
4#27133230C211yesmaximal, characteristic, complemented
5#27133227C211yesmaximal, characteristic, complemented
6#27133051C212yesmaximal, complemented
7#27133049C212yesmaximal, complemented
8#27133044C212yesmaximal, complemented
9#27133038C212yesmaximal, complemented
10#27132621C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C191C211yesmaximal, characteristic, complemented
12C2×#15690287C2211yescharacteristic
13C22×D764C2211yescharacteristic
14#20259350—24no—
15#20259342—24no—
16C2×#15690277C2211yescharacteristic, complemented
17C2×#15690276C2212yescomplemented
18C2×#15690275C2211yescharacteristic, complemented
19C2×#15690274C2211yescharacteristic, complemented
20C2×D1528C2212yescomplemented
21C2×#15690272—24no—
22C2×#15690272C2212yes—
23C2×#15690272C2212yes—
24C2×#15690272C2211yescharacteristic, complemented
25C22×C191⋊C8C2211yescharacteristic
26#20259318C2212yescomplemented
27#20259318C2212yescomplemented
28#20259311C2212yescomplemented
29#20259309C2212yescomplemented
30#20259296—22no—
31#20259293—22no—
32#20259291—22no—
33#20259290—22no—
34#20259281C2211yescharacteristic
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×D764C2312yes—
51C2×D764C2311yescharacteristic
52C2×C22⋊D191—24no—
53C2×C22⋊D191—24no—
54C2×C22⋊D191—21no—
55C2×C4×D191—21no—
56#15690277—22no—
57#15690276—24no—
58#15690275—22no—
59#15690274—22no—
60D1528—24no—
61#15690272—24no—
62#15690272—24no—
63#15690272—22no—
64C2×C191⋊C8D414yescomplemented
65C2×C191⋊C8C2311yescharacteristic
66C2×C191⋊C8C2311yescharacteristic
67C4⋊Dic191D412yes—
68C22⋊Dic191—24no—
69C22⋊Dic191D412yes—
70(C2×D191)⋊C4—24no—
71C4×Dic191C2312yes—
72C24×C191—21no—
73#41×C191—22no—
74#41×C191—22no—
75C2×Q8×C191C2311yescharacteristic
76C2×D4×C191—21no—
77C2×D4×C191—22no—
78C2×D4×C191—22no—
79C2×D4×C191—21no—
80C2×D4×C191C2312yes—
81C2×D4×C191C2311yescharacteristic
82C22×C4×C191—21no—
83C22×C4×C191C2311yescharacteristic
84SD16×C191—24no—
85D8×C191—24no—
86M16×C191D414yescomplemented
87C2×C8×C191C2312yes—
88C22×D191—22no—
89C22×D191—21no—
90C22×D191—42no—
91C22×D191—82no—
92Dic382—21no—
93Dic382—21no—
94D764—22no—
95D764—21no—
96D764—21no—
97D764—22no—
98D764—42no—
99C22⋊D191—42no—
100C22⋊D191—84no—
101C22⋊D191—84no—
102C2×Dic191—21no—
103C2×Dic191C2×D414yes—
104C4×D191—42no—
105C191⋊C8C2×D412yes—
106C191⋊C8C2×D412yes—
107C23×C191—21no—
108C23×C191—22no—
109C23×C191—42no—
110C23×C191—82no—
111C23×C191C2×D411yescharacteristic
112Q8×C191—21no—
113Q8×C191—21no—
114D4×C191—21no—
115D4×C191—22no—
116D4×C191—22no—
117D4×C191—21no—
118D4×C191—42no—
119D4×C191—42no—
120C2×C4×C191—21no—
121C2×C4×C191—42no—
122C2×C4×C191C2×D411yescharacteristic
123C2×C4×C191C2×D412yes—
124C2×C4×C191C2×D412yes—
125C2×C4×C191C2411yescharacteristic
126C8×C191C2×D414yes—
127C22×C191—22no—
128C22×C191—41no—
129C22×C191—42no—
130C22×C191—44no—
131C22×C191—42no—
132C22×C191—84no—
133C22×C191#12712yes—
134C22×C191C22×D411yescharacteristic
135C2×D191—41no—
136C2×D191—42no—
137C2×D191—42no—
138C2×D191—44no—
139C2×D191—84no—
140C4×C191—41no—
141C4×C191#12712yes—
142C4×C191C22×D411yescharacteristic
143C4×C191C22×D411yescharacteristic
144Dic191—24no—
145Dic191—41no—
146C2×C191—22no—
147C2×C191—81no—
148C2×C191—82no—
149C2×C191C2×#12711yescharacteristic
150C2×C191#57612yes—
151D191—81no—
152D191—82no—
153C191#315511yescharacteristic, complemented, Sylow
154#3155—1911nomaximal, Sylow
155C2×AGL(1,ℤ/8)—1911no—
156C2×AGL(1,ℤ/8)D19111yescharacteristic, complemented
157C2×#135—1911no—
158C22×D8—1911no—
159#535—1911no—
160#535—1911no—
161#496—1912no—
162#496—1912no—
163#495—1912no—
164#493—1912no—
165#405—1911no—
166C2×#41—1911no—
167C2×#41C2×D19111yescharacteristic
168C22×D4—1911no—
169C22×D4C2×D19111yescharacteristic
170AGL(1,ℤ/8)—24no—
171AGL(1,ℤ/8)—3824no—
172#135—3824no—
173C2×Q16—1911no—
174C2×SD16—1911no—
175C2×SD16—1911no—
176C2×SD16—1912no—
177C2×SD16C2×D19112yescomplemented
178C2×D8—3824no—
179C2×D8—1912no—
180C2×D8—1912no—
181C2×D8—1911no—
182C2×D8—1912no—
183C2×D8C2×D19112yescomplemented
184C2×M16C2×D19111yescharacteristic
185C22×C8—1911no—
186#127—1912no—
187#127—1912no—
188#124—1912no—
189#124—1912no—
190#121—3822no—
191#121—3822no—
192#120—3822no—
193#120—3822no—
194#117—1911no—
195#97—1912no—
196C4×C8—1912no—
197C24—21no—
198C24—3821no—
199#41—22no—
200#41—22no—
201#41—3822no—
202#41—3822no—
203C2×Q8—1911no—
204C2×Q8C22×D19111yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—3821no—
210C2×D4—3824no—
211C2×D4—3824no—
212C2×D4—3822no—
213C2×D4—3821no—
214C2×D4—3822no—
215C2×D4—1912no—
216C2×D4—1911no—
217C2×D4C22×D19112yes—
218C2×D4C22×D19111yescharacteristic
219C22×C4—21no—
220C22×C4—3821no—
221C22×C4C22×D19111yescharacteristic
222Q16—3822no—
223SD16—24no—
224SD16—3822no—
225SD16—3822no—
226SD16—3824no—
227D8—24no—
228D8—3824no—
229D8—3824no—
230D8—3822no—
231D8—3824no—
232M16C22⋊D19114yescomplemented
233C2×C8—1914no—
234C2×C8—1911no—
235C2×C8—1911no—
236C2×C8C22×D19112yes—
237C4⋊C4—1912no—
238C22⋊C4—3824no—
239C22⋊C4—3824no—
240C22⋊C4—1912no—
241C42—1912no—
242C23—22no—
243C23—21no—
244C23—42no—
245C23—82no—
246C23—3822no—
247C23—3821no—
248C23—15282no—
249C23—7642no—
250C23D4×D19111yescharacteristic
251Q8—21no—
252Q8—21no—
253Q8—3821no—
254Q8—3821no—
255D4—22no—
256D4—21no—
257D4—21no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—3822no—
262D4—3821no—
263D4—3821no—
264D4—3822no—
265D4—15284no—
266D4—15284no—
267D4—7642no—
268D4—7642no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—3821no—
272C2×C4—1914no—
273C2×C4—7642no—
274C2×C4D4×D19112yes—
275C2×C4D4×D19111yescharacteristic
276C2×C4C2×C22⋊D19112yes—
277C2×C4C23×D19111yescharacteristic
278C8—1912no—
279C8—1912no—
280C8C2×C22⋊D19114yes—
281C22—22no—
282C22—41no—
283C22—42no—
284C22—42no—
285C22—44no—
286C22—84no—
287C22—15284no—
288C22—7641no—
289C22—7642no—
290C22—7642no—
291C22—7644no—
292C22#2025931812yes—
293C22C2×D4×D19111yescharacteristic
294C4—41no—
295C4—3824no—
296C4—7641no—
297C4#2025931812yes—
298C4C2×D4×D19111yescharacteristic
299C4C22×C22⋊D19111yescharacteristic
300C2—22no—
301C2—82no—
302C2—81no—
303C2—15282no—
304C2—15281no—
305C2C2×#2025931811yescharacteristic
306C2#2713346712yes—
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#3155
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×#20259342C2semidirect
3. C22×#15690272C2semidirect
4. #27133230C2semidirect
5. #27133227C2semidirect
6. #27133051C2semidirect
7. #27133049C2semidirect
8. #27133044C2semidirect
9. #27133038C2semidirect
10. #27132621C2semidirect
11. C2×AGL(1,ℤ/8)×C191C2semidirect
16. C2×#15690277C22semidirect
17. C2×#15690276C22semidirect
18. C2×#15690275C22semidirect
19. C2×#15690274C22semidirect
20. C2×D1528C22semidirect
24. C2×#15690272C22semidirect
26. #20259318C22semidirect
27. #20259318C22semidirect
28. #20259311C22semidirect
29. #20259309C22semidirect
35. #20259232C22semidirect
36. C8×Dic191C22semidirect
40. C2×SD16×C191C22semidirect
41. C2×D8×C191C22semidirect
64. C2×C191⋊C8D4semidirect
86. M16×C191D4semidirect
153. C191#3155semidirect
156. C2×AGL(1,ℤ/8)D191semidirect
177. C2×SD16C2×D191semidirect
183. C2×D8C2×D191semidirect
232. M16C22⋊D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593502
C2×#202593422
C22×#156902722
#271332302
#271332272
#271330512
#271330492
#271330442
#271330382
#271326212
C2×AGL(1,ℤ/8)×C1912
#3155191