Finite Group Databasebeta

← #37582998

Subgroups — #37582998

Order 24448.

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

normal subgroups only

This group has 564 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×#20259353C211yesmaximal, characteristic, complemented
2C2×#20259347C211yesmaximal, characteristic, complemented
3C2×#20259342C211yesmaximal, characteristic, complemented
4C22×#15690274C211yesmaximal, characteristic, complemented
5#27133232C211yesmaximal, characteristic, complemented
6#27133055C212yesmaximal, complemented
7#27133053C212yesmaximal, complemented
8#27133045C212yesmaximal, complemented
9#27133037C212yesmaximal, complemented
10#27132619C211yesmaximal, characteristic, complemented
11#536×C191C211yesmaximal, characteristic, complemented
12C2×#15690287C2211yescharacteristic
13C22×D764C2211yescharacteristic
14#20259353—24no—
15#20259347—24no—
16#20259342—24no—
17C2×#15690277C2211yescharacteristic, complemented
18C2×#15690277C2212yescomplemented
19C2×#15690275C2211yescharacteristic, complemented
20C2×#15690274—24no—
21C2×#15690274C2212yes—
22C2×#15690274C2212yes—
23C2×#15690274C2212yes—
24C2×#15690274C2211yescharacteristic, complemented
25C2×#15690274C2212yescomplemented
26C2×#15690272C2212yes—
27C2×#15690272C2211yescharacteristic, complemented
28C2×#15690271C2211yescharacteristic
29C22×C191⋊C8C2211yescharacteristic
30#20259319C2212yes—
31#20259310C2212yes—
32#20259297—22no—
33#20259292—22no—
34#20259233C2212yescomplemented
35C4×C191⋊C8C2212yescomplemented
36C2×#41×C191C2211yescharacteristic
37C22×Q8×C191C2211yescharacteristic
38C4⋊Q8×C191C2212yescomplemented
39#124×C191C2212yescomplemented
40#123×C191—22no—
41C22⋊Q8×C191—22no—
42#117×C191C2211yescharacteristic
43C23×D191—21no—
44#15690287—22no—
45#15690287—22no—
46C2×Dic382C2311yescharacteristic
47C2×D764—22no—
48C2×D764—22no—
49C2×D764—21no—
50C2×D764—24no—
51C2×D764C2312yes—
52C2×D764C2311yescharacteristic
53C2×C22⋊D191—21no—
54C2×C4×D191—21no—
55#15690277—22no—
56#15690277—24no—
57#15690275—22no—
58#15690274—24no—
59#15690274—24no—
60#15690274—24no—
61#15690274—22no—
62#15690274—24no—
63#15690272—24no—
64#15690272—22no—
65#15690271D414yescomplemented
66C2×C191⋊C8D414yescomplemented
67C2×C191⋊C8C2312yes—
68C2×C191⋊C8C2311yescharacteristic
69C2×C191⋊C8C2311yescharacteristic
70(C2×D191)⋊C4—24no—
71#41×C191—22no—
72#41×C191—22no—
73C2×Q8×C191—21no—
74C2×Q8×C191—22no—
75C2×Q8×C191—22no—
76C2×Q8×C191C2312yes—
77C2×Q8×C191C2311yescharacteristic
78C2×D4×C191—21no—
79C2×D4×C191C2311yescharacteristic
80C22×C4×C191—21no—
81C22×C4×C191—21no—
82C22×C4×C191C2311yescharacteristic
83C4⋊C4×C191—24no—
84C4⋊C4×C191D412yes—
85C22⋊C4×C191—24no—
86C22⋊C4×C191D412yes—
87C42×C191C2312yes—
88C22×D191—22no—
89C22×D191—21no—
90C22×D191—42no—
91C22×D191—82no—
92Dic382—21no—
93Dic382—21no—
94D764—22no—
95D764—21no—
96D764—22no—
97D764—21no—
98D764—42no—
99D764—84no—
100C22⋊D191—42no—
101C2×Dic191—21no—
102C4×D191—42no—
103C191⋊C8C2×D412yes—
104C191⋊C8C2×D412yes—
105C191⋊C8C2×D414yes—
106C23×C191—21no—
107C23×C191C2×D411yescharacteristic
108Q8×C191—22no—
109Q8×C191—22no—
110Q8×C191—21no—
111Q8×C191—21no—
112Q8×C191—42no—
113D4×C191—21no—
114D4×C191—21no—
115D4×C191—42no—
116C2×C4×C191—22no—
117C2×C4×C191—21no—
118C2×C4×C191—42no—
119C2×C4×C191—42no—
120C2×C4×C191C2×D412yes—
121C2×C4×C191C2×D412yes—
122C2×C4×C191C2×D414yes—
123C2×C4×C191C2×D411yescharacteristic
124C2×C4×C191C2411yescharacteristic
125C22×C191—22no—
126C22×C191—41no—
127C22×C191—42no—
128C22×C191#12712yes—
129C22×C191C22×D411yescharacteristic
130C2×D191—41no—
131C2×D191—42no—
132C2×D191—42no—
133C2×D191—44no—
134C2×D191—84no—
135C4×C191—24no—
136C4×C191—41no—
137C4×C191—42no—
138C4×C191#12712yes—
139C4×C191C22×D411yescharacteristic
140C4×C191C22×D411yescharacteristic
141Dic191—41no—
142C2×C191—22no—
143C2×C191—81no—
144C2×C191C2×#12711yescharacteristic
145C2×C191#57712yes—
146D191—81no—
147D191—82no—
148C191#315411yescharacteristic, complemented, Sylow
149#3154—1911nomaximal, Sylow
150C2×#137—1911no—
151C2×AGL(1,ℤ/8)—1911no—
152C2×#135—1911no—
153C22×SD16—1911no—
154#536D19111yescharacteristic, complemented
155#535—1911no—
156#497—1912no—
157#496—1912no—
158#495—1912no—
159#492—1912no—
160#405—1911no—
161C2×#41—1911no—
162C2×#41C2×D19111yescharacteristic
163C22×Q8C2×D19111yescharacteristic
164C22×D4—1911no—
165#137—3824no—
166AGL(1,ℤ/8)—3824no—
167#135—3824no—
168C2×Q16—1911no—
169C2×Q16—1912no—
170C2×SD16—3824no—
171C2×SD16—1912no—
172C2×SD16—1912no—
173C2×SD16—1912no—
174C2×SD16—1911no—
175C2×SD16—1912no—
176C2×SD16—1911no—
177C2×D8—1912no—
178C2×D8—1911no—
179C2×M16—1911no—
180C22×C8—1911no—
181C4⋊Q8C2×D19112yescomplemented
182#127—1912no—
183#124—1912no—
184#124C2×D19112yescomplemented
185#123—22no—
186C22⋊Q8—22no—
187#121—3822no—
188#120—3822no—
189#117C2×D19111yescharacteristic
190#97—1912no—
191C4×C8—1912no—
192C24—3821no—
193#41—22no—
194#41—22no—
195#41—3822no—
196#41—3822no—
197C2×Q8—21no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—1911no—
201C2×Q8C22×D19112yes—
202C2×Q8C22×D19111yescharacteristic
203C2×D4—21no—
204C2×D4—3821no—
205C2×D4—3822no—
206C2×D4—3822no—
207C2×D4—3824no—
208C2×D4—3821no—
209C2×D4—1912no—
210C2×D4—1911no—
211C2×D4C22×D19111yescharacteristic
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—3821no—
215C22×C4C22×D19111yescharacteristic
216Q16—3822no—
217Q16—3824no—
218SD16—3822no—
219SD16—3824no—
220SD16—3824no—
221SD16—3822no—
222SD16—3824no—
223SD16—3824no—
224D8—3822no—
225D8—3824no—
226M16—1914no—
227C2×C8—1914no—
228C2×C8—1911no—
229C2×C8—1912no—
230C2×C8—1911no—
231C4⋊C4—24no—
232C4⋊C4C22⋊D19112yes—
233C22⋊C4—24no—
234C22⋊C4—3824no—
235C22⋊C4C22⋊D19112yes—
236C42C22×D19112yes—
237C23—21no—
238C23—3822no—
239C23—3821no—
240C23—15282no—
241C23—7642no—
242C23C2×C22⋊D19111yescharacteristic
243Q8—22no—
244Q8—21no—
245Q8—22no—
246Q8—21no—
247Q8—42no—
248Q8—3821no—
249Q8—3821no—
250D4—21no—
251D4—21no—
252D4—42no—
253D4—3822no—
254D4—3821no—
255D4—3822no—
256D4—3821no—
257D4—15284no—
258D4—7642no—
259D4—7642no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—3821no—
265C2×C4—7642no—
266C2×C4D4×D19112yes—
267C2×C4D4×D19112yes—
268C2×C4C2×C22⋊D19114yes—
269C2×C4C2×C22⋊D19111yescharacteristic
270C2×C4C23×D19111yescharacteristic
271C8—1912no—
272C8—1912no—
273C8—1914no—
274C22—22no—
275C22—41no—
276C22—42no—
277C22—15284no—
278C22—7641no—
279C22—7642no—
280C22—7642no—
281C22—7644no—
282C22#2025931812yes—
283C22C22×C22⋊D19111yescharacteristic
284C4—24no—
285C4—41no—
286C4—42no—
287C4—7641no—
288C4#2025931812yes—
289C4C2×D4×D19111yescharacteristic
290C4C2×D4×D19111yescharacteristic
291C2—22no—
292C2—81no—
293C2—15282no—
294C2—15281no—
295C2C2×#2025931811yescharacteristic
296C2#2713347112yes—
subgroupG/H
centerC22C22×C22⋊D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#536×C191C2
socleC22×C191C22×D4
hypercenter#536D191
hypocenterC191#3154
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ #117 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C2×#20259353C2semidirect
2. C2×#20259347C2semidirect
3. C2×#20259342C2semidirect
4. C22×#15690274C2semidirect
5. #27133232C2semidirect
6. #27133055C2semidirect
7. #27133053C2semidirect
8. #27133045C2semidirect
9. #27133037C2semidirect
10. #27132619C2semidirect
11. #536×C191C2semidirect
17. C2×#15690277C22semidirect
18. C2×#15690277C22semidirect
19. C2×#15690275C22semidirect
24. C2×#15690274C22semidirect
25. C2×#15690274C22semidirect
27. C2×#15690272C22semidirect
34. #20259233C22semidirect
35. C4×C191⋊C8C22semidirect
38. C4⋊Q8×C191C22semidirect
39. #124×C191C22semidirect
65. #15690271D4semidirect
66. C2×C191⋊C8D4semidirect
148. C191#3154semidirect
154. #536D191semidirect
181. C4⋊Q8C2×D191semidirect
184. #124C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593532
C2×#202593472
C2×#202593422
C22×#156902742
#271332322
#271330552
#271330532
#271330452
#271330372
#271326192
#536×C1912
#3154191