Finite Group Databasebeta

← #37582982

Subgroups — #37582982

Order 24448.

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

normal subgroups only

This group has 572 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×#20259355C211yesmaximal, characteristic, complemented
2C2×#20259352C211yesmaximal, characteristic, complemented
3C2×#20259348C211yesmaximal, characteristic, complemented
4#27133209C211yesmaximal, characteristic, complemented
5C2×#20259309C211yesmaximal, characteristic, complemented
6#27133054C212yesmaximal, complemented
7#27133051C212yesmaximal, complemented
8#27133046C212yesmaximal, complemented
9#27133044C212yesmaximal, complemented
10M16×Dic191C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C191C211yesmaximal, characteristic, complemented
12C2×#15690287C2211yescharacteristic
13C22×Dic382C2211yescharacteristic
14#20259355—24no—
15#20259352—24no—
16#20259348—24no—
17C2×Dic764C2212yescomplemented
18C2×#15690277C2212yescomplemented
19C2×#15690276C2212yescomplemented
20C2×#15690275C2212yes—
21C2×#15690275C2212yescomplemented
22C2×#15690272C2212yes—
23C2×#15690271C2211yescharacteristic
24#20259320C2211yescharacteristic, complemented
25#20259318C2211yescharacteristic, complemented
26#20259311C2211yescharacteristic, complemented
27#20259309—22no—
28#20259309C2212yes—
29#20259309C2212yes—
30#20259309C2211yescharacteristic, complemented
31#20259294—22no—
32C4×C22⋊D191—22no—
33C2×C22⋊Dic191—22no—
34C2×C4×Dic191C2211yescharacteristic
35#20259232C2212yescomplemented
36C8×Dic191C2212yescomplemented
37C2×#41×C191C2211yescharacteristic
38C22×D4×C191C2211yescharacteristic
39AGL(1,ℤ/8)×C191—24no—
40C2×SD16×C191C2212yes—
41C2×D8×C191C2212yes—
42C2×M16×C191C2211yescharacteristic
43#15690287—22no—
44#15690287—22no—
45C2×Dic382—21no—
46C2×Dic382—22no—
47C2×Dic382—22no—
48C2×Dic382C2311yescharacteristic
49C2×Dic382C2312yes—
50C2×D764C2311yescharacteristic
51C2×C22⋊D191—22no—
52C2×C22⋊D191—21no—
53C22×Dic191—21no—
54C22×Dic191D412yes—
55C2×C4×D191—21no—
56Dic764—24no—
57#15690277—24no—
58#15690276—24no—
59#15690275—24no—
60#15690275—24no—
61#15690272—24no—
62#15690271D414yescomplemented
63C2×C191⋊C8C2312yes—
64Dic191⋊C4—22no—
65C22⋊Dic191—24no—
66C22⋊Dic191—24no—
67C22⋊Dic191—22no—
68(C2×D191)⋊C4—22no—
69C4×Dic191D412yes—
70C4×Dic191C2311yescharacteristic
71C4×Dic191C2311yescharacteristic
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—21no—
89Dic382—21no—
90Dic382—22no—
91Dic382—22no—
92Dic382—21no—
93Dic382—42no—
94D764—21no—
95D764—21no—
96C22⋊D191—44no—
97C22⋊D191—42no—
98C2×Dic191—22no—
99C2×Dic191—21no—
100C2×Dic191—42no—
101C2×Dic191—42no—
102C2×Dic191C2×D412yes—
103C2×Dic191C2×D412yes—
104C4×D191—42no—
105C191⋊C8C2×D414yes—
106C23×C191—21no—
107C23×C191—22no—
108C23×C191—44no—
109C23×C191—42no—
110C23×C191C2×D411yescharacteristic
111Q8×C191—21no—
112Q8×C191—21no—
113D4×C191—21no—
114D4×C191—22no—
115D4×C191—22no—
116D4×C191—21no—
117D4×C191—42no—
118D4×C191—42no—
119C2×C4×C191—21no—
120C2×C4×C191—42no—
121C2×C4×C191C2×D411yescharacteristic
122C2×C4×C191C2×D412yes—
123C2×C4×C191C2×D412yes—
124C2×C4×C191C2411yescharacteristic
125C8×C191C2×D414yes—
126C22×C191—22no—
127C22×C191—41no—
128C22×C191—42no—
129C22×C191—44no—
130C22×C191—42no—
131C22×C191—84no—
132C22×C191#12712yes—
133C22×C191C22×D411yescharacteristic
134C2×D191—41no—
135C2×D191—42no—
136C4×C191—41no—
137C4×C191#12712yes—
138C4×C191C22×D411yescharacteristic
139C4×C191C22×D411yescharacteristic
140Dic191—22no—
141Dic191—22no—
142Dic191—41no—
143Dic191—42no—
144C2×C191—22no—
145C2×C191—81no—
146C2×C191—82no—
147C2×C191C2×#12711yescharacteristic
148C2×C191#57512yes—
149D191—81no—
150C191#315311yescharacteristic, complemented, Sylow
151#3153—1911nomaximal, Sylow
152C2×#137—1911no—
153C2×#137—1911no—
154C2×AGL(1,ℤ/8)—1911no—
155C2×AGL(1,ℤ/8)D19111yescharacteristic, complemented
156#532—1911no—
157C2×#124—1911no—
158#497—1912no—
159#496—1912no—
160#495—1912no—
161#495—1912no—
162C4×M16—1911no—
163C2×#41—1911no—
164C2×#41C2×D19111yescharacteristic
165C22×Q8—1911no—
166C22×D4C2×D19111yescharacteristic
167#137—3824no—
168#137—3824no—
169AGL(1,ℤ/8)—24no—
170AGL(1,ℤ/8)—3824no—
171C2×Q16—1912no—
172C2×Q16—1912no—
173C2×SD16—1912no—
174C2×SD16—1912no—
175C2×SD16—1912no—
176C2×SD16C2×D19112yes—
177C2×D8—1912no—
178C2×D8C2×D19112yes—
179C2×M16—1911no—
180C2×M16C2×D19111yescharacteristic
181C4⋊Q8—1911no—
182#127—1911no—
183#124—3822no—
184#124—1911no—
185#124—1912no—
186#124—1912no—
187#124—1911no—
188#121—3822no—
189C4×D4—3822no—
190C2×C22⋊C4—3822no—
191C2×C42—1911no—
192#97—1912no—
193C4×C8—1912no—
194C24—21no—
195#41—22no—
196#41—22no—
197#41—3822no—
198#41—3822no—
199C2×Q8—3821no—
200C2×Q8—3822no—
201C2×Q8—3822no—
202C2×Q8—1912no—
203C2×Q8—1911no—
204C2×Q8C22×D19111yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—3821no—
210C2×D4—3822no—
211C2×D4—1911no—
212C2×D4C22×D19112yes—
213C2×D4C22×D19111yescharacteristic
214C22×C4—21no—
215C22×C4—3821no—
216C22×C4—3821no—
217C22×C4—1912no—
218C22×C4C22×D19111yescharacteristic
219Q16—3824no—
220Q16—3824no—
221SD16—24no—
222SD16—3824no—
223SD16—3824no—
224SD16—3824no—
225D8—24no—
226D8—3824no—
227M16—1914no—
228M16C22⋊D19114yescomplemented
229C2×C8—1912no—
230C2×C8C22×D19112yes—
231C4⋊C4—3822no—
232C22⋊C4—3822no—
233C22⋊C4—3824no—
234C22⋊C4—3824no—
235C22⋊C4—3822no—
236C42—1912no—
237C42—1911no—
238C42—1911no—
239C23—22no—
240C23—21no—
241C23—44no—
242C23—42no—
243C23—3821no—
244C23D4×D19111yescharacteristic
245Q8—21no—
246Q8—21no—
247Q8—3821no—
248Q8—3822no—
249Q8—3822no—
250Q8—3821no—
251Q8—7642no—
252D4—22no—
253D4—21no—
254D4—21no—
255D4—22no—
256D4—42no—
257D4—42no—
258D4—3821no—
259D4—3821no—
260D4—7644no—
261D4—7642no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—3822no—
265C2×C4—3821no—
266C2×C4—1912no—
267C2×C4—1912no—
268C2×C4—7642no—
269C2×C4—7642no—
270C2×C4—7642no—
271C2×C4D4×D19112yes—
272C2×C4D4×D19111yescharacteristic
273C2×C4C2×C22⋊D19112yes—
274C2×C4C23×D19111yescharacteristic
275C8—1914no—
276C8C2×C22⋊D19114yes—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—44no—
282C22—84no—
283C22—7641no—
284C22—7642no—
285C22#2025931812yes—
286C22C2×D4×D19111yescharacteristic
287C4—41no—
288C4—3822no—
289C4—3822no—
290C4—7641no—
291C4—7642no—
292C4#2025931812yes—
293C4C2×D4×D19111yescharacteristic
294C4C22×C22⋊D19111yescharacteristic
295C2—22no—
296C2—82no—
297C2—81no—
298C2—15281no—
299C2C2×#2025931811yescharacteristic
300C2#2713345912yes—
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#3153
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×#20259355C2semidirect
2. C2×#20259352C2semidirect
3. C2×#20259348C2semidirect
4. #27133209C2semidirect
5. C2×#20259309C2semidirect
6. #27133054C2semidirect
7. #27133051C2semidirect
8. #27133046C2semidirect
9. #27133044C2semidirect
10. M16×Dic191C2semidirect
11. C2×AGL(1,ℤ/8)×C191C2semidirect
17. C2×Dic764C22semidirect
18. C2×#15690277C22semidirect
19. C2×#15690276C22semidirect
21. C2×#15690275C22semidirect
24. #20259320C22semidirect
25. #20259318C22semidirect
26. #20259311C22semidirect
30. #20259309C22semidirect
35. #20259232C22semidirect
36. C8×Dic191C22semidirect
62. #15690271D4semidirect
86. M16×C191D4semidirect
150. C191#3153semidirect
155. C2×AGL(1,ℤ/8)D191semidirect
228. M16C22⋊D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593552
C2×#202593522
C2×#202593482
#271332092
C2×#202593092
#271330542
#271330512
#271330462
#271330442
M16×Dic1912
C2×AGL(1,ℤ/8)×C1912
#3153191