Finite Group Databasebeta

← #37582201

Subgroups — #37582201

Order 24448.

504 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 141 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×#20259330C211yesmaximal, characteristic, complemented
3#27133244C211yesmaximal, characteristic, complemented
4#27133232C211yesmaximal, characteristic, complemented
5#27132917C212yesmaximal, complemented
6#27132910C212yesmaximal, complemented
7#27132900C212yesmaximal, complemented
8#27132893C212yesmaximal, complemented
9#27132657C211yesmaximal, characteristic, complemented
10C2×#20259242C211yesmaximal, characteristic, complemented
11#428×C191C211yesmaximal, characteristic, complemented
12C2×#15690287C2211yescharacteristic
13C2×#15690287C2211yescharacteristic
14C22×D764C2211yescharacteristic
15#20259350—24no—
16C2×#15690276C2212yes—
17C2×D1528C2212yes—
18#20259330—24no—
19C2×#15690270C2212yescomplemented
20C22×C191⋊C8C2211yescharacteristic
21C2×C8×D191C2212yescomplemented
22#20259319—21no—
23#20259310—21no—
24#20259302—21no—
25#20259302C2212yescomplemented
26#20259297—21no—
27#20259297C2212yes—
28#20259297C2212yes—
29#20259297C2212yescomplemented
30#20259296—22no—
31#20259292—22no—
32#20259245C2211yescharacteristic, complemented
33#20259245C2211yescharacteristic, complemented
34#20259242C2212yes—
35#20259242C2212yes—
36#20259242C2211yescharacteristic, complemented
37#20259242C2211yescharacteristic, complemented
38C2×M16×C191C2211yescharacteristic
39#117×C191C2211yescharacteristic
40C2×C4⋊C4×C191C2211yescharacteristic
41#107×C191C2212yescomplemented
42#106×C191C2212yescomplemented
43C23×D191—21no—
44#15690287—22no—
45#15690287—22no—
46#15690287—22no—
47#15690287—22no—
48C2×Dic382D411yescharacteristic
49C2×Dic382C2311yescharacteristic
50C2×D764—22no—
51C2×D764—22no—
52C2×D764—22no—
53C2×D764—22no—
54C2×D764—21no—
55C2×D764—22no—
56C2×D764D411yescharacteristic
57C2×D764C2311yescharacteristic
58C2×D764C2312yes—
59C2×C22⋊D191—21no—
60C2×C22⋊D191D412yes—
61C2×C4×D191—21no—
62C2×C4×D191C2312yes—
63#15690276—24no—
64D1528—24no—
65#15690270—24no—
66C2×C191⋊C8—22no—
67C2×C191⋊C8C2311yescharacteristic
68C2×C191⋊C8C2311yescharacteristic
69C8×D191—24no—
70C4⋊Dic191—22no—
71(C2×D191)⋊C4—22no—
72(C2×D191)⋊C4—22no—
73(C2×D191)⋊C4—22no—
74(C2×D191)⋊C4—22no—
75C22×C4×C191—21no—
76C22×C4×C191C2311yescharacteristic
77M16×C191D414yescomplemented
78C2×C8×C191C2312yes—
79C4⋊C4×C191—21no—
80C4⋊C4×C191C2311yescharacteristic
81C4⋊C4×C191C2312yes—
82C4⋊C4×C191C2311yescharacteristic
83C22⋊C4×C191—21no—
84C42×C191—21no—
85C22×D191—21no—
86C22×D191—22no—
87C22×D191—42no—
88C22×D191—81no—
89C22×D191—81no—
90C22×D191C2×D412yes—
91Dic382—22no—
92Dic382—21no—
93Dic382—21no—
94D764—22no—
95D764—21no—
96D764—22no—
97D764—22no—
98D764—21no—
99D764—42no—
100D764—82no—
101D764—82no—
102D764—82no—
103C22⋊D191—24no—
104C22⋊D191—42no—
105C2×Dic191—21no—
106C2×Dic191C2×D412yes—
107C4×D191—22no—
108C4×D191—22no—
109C4×D191—42no—
110C191⋊C8—21no—
111C191⋊C8—21no—
112C23×C191C2×D411yescharacteristic
113C2×C4×C191—21no—
114C2×C4×C191—21no—
115C2×C4×C191—22no—
116C2×C4×C191—81no—
117C2×C4×C191C2×D411yescharacteristic
118C2×C4×C191C2×D412yes—
119C2×C4×C191#4112yes—
120C2×C4×C191C2411yescharacteristic
121C8×C191C2×D414yes—
122C22×C191—21no—
123C22×C191—21no—
124C22×C191#12112yes—
125C22×C191C22×D411yescharacteristic
126C2×D191—22no—
127C2×D191—22no—
128C2×D191—22no—
129C2×D191—41no—
130C2×D191—42no—
131C2×D191—41no—
132C2×D191—42no—
133C2×D191—42no—
134C2×D191—41no—
135C2×D191—84no—
136C4×C191—42no—
137C4×C191—41no—
138C4×C191—41no—
139C4×C191#12112yes—
140C4×C191C22×D411yescharacteristic
141C4×C191C2×#4111yescharacteristic
142Dic191—22no—
143Dic191—41no—
144C2×C191—22no—
145C2×C191C2×#12111yescharacteristic
146C2×C191#57611yescharacteristic
147C2×C191#57711yescharacteristic
148D191—42no—
149D191—81no—
150D191—82no—
151C191#306211yescharacteristic, complemented, Sylow
152#3062—1911nomaximal, Sylow
153C2×AGL(1,ℤ/8)—1911no—
154C2×#131—1911no—
155#537—1911no—
156#535—1911no—
157#469—1912no—
158#468—1912no—
159#466—1912no—
160#465—1912no—
161#428D19111yescharacteristic, complemented
162#416—1911no—
163C2×#102—1911no—
164C2×#41—1911no—
165C2×#41—1911no—
166C22×D4—1911no—
167AGL(1,ℤ/8)—3824no—
168C2×SD16—1912no—
169C2×D8—1912no—
170#131—3824no—
171C2×M16—1912no—
172C2×M16C2×D19111yescharacteristic
173C22×C8—1911no—
174C22×C8—1912no—
175#127—3821no—
176#124—3821no—
177C22⋊Q8—3821no—
178C22⋊Q8—1912no—
179#121—3821no—
180#121—3822no—
181#121—1912no—
182#121—1912no—
183#121—1912no—
184#120—3822no—
185#117C2×D19111yescharacteristic
186C2×C4⋊C4C2×D19111yescharacteristic
187#107C2×D19112yescomplemented
188#106C2×D19112yescomplemented
189#103—1911no—
190#103—1911no—
191#102—1912no—
192#102—1912no—
193#102—1911no—
194#102—1911no—
195C24—3821no—
196#41—3822no—
197#41—3822no—
198#41—3822no—
199#41—3822no—
200C2×Q8—1911no—
201C2×Q8—1911no—
202C2×D4—3822no—
203C2×D4—3822no—
204C2×D4—3822no—
205C2×D4—3821no—
206C2×D4—3821no—
207C2×D4—3822no—
208C2×D4—3822no—
209C2×D4—1912no—
210C2×D4—1911no—
211C2×D4—1911no—
212C2×D4—1912no—
213C22×C4—21no—
214C22×C4—3821no—
215C22×C4—1912no—
216C22×C4C22×D19111yescharacteristic
217SD16—3824no—
218D8—3824no—
219M16—3824no—
220M16D76414yescomplemented
221C2×C8—3822no—
222C2×C8—3824no—
223C2×C8—1911no—
224C2×C8—1911no—
225C2×C8C22×D19112yes—
226C4⋊C4—21no—
227C4⋊C4—3822no—
228C4⋊C4C22×D19112yes—
229C4⋊C4C22×D19111yescharacteristic
230C4⋊C4C22×D19111yescharacteristic
231C22⋊C4—21no—
232C22⋊C4—3822no—
233C22⋊C4—3822no—
234C22⋊C4—3822no—
235C22⋊C4—3822no—
236C42—21no—
237C23—3821no—
238C23—3822no—
239C23—1912no—
240C23—15281no—
241C23—15281no—
242C23—7642no—
243C23D4×D19111yescharacteristic
244Q8—3822no—
245Q8—3821no—
246Q8—3821no—
247D4—3824no—
248D4—3821no—
249D4—3822no—
250D4—3822no—
251D4—3821no—
252D4—3822no—
253D4—15282no—
254D4—15282no—
255D4—15282no—
256D4—7642no—
257D4—7642no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—81no—
262C2×C4—3822no—
263C2×C4—3822no—
264C2×C4—3821no—
265C2×C4—1912no—
266C2×C4—7642no—
267C2×C4D4×D19111yescharacteristic
268C2×C4C2×D76412yes—
269C2×C4#1569028812yes—
270C2×C4C23×D19111yescharacteristic
271C8—3821no—
272C8—3821no—
273C8C2×D76414yes—
274C22—21no—
275C22—21no—
276C22—3822no—
277C22—3822no—
278C22—3822no—
279C22—15284no—
280C22—7641no—
281C22—7642no—
282C22—7642no—
283C22—7641no—
284C22—7641no—
285C22—7642no—
286C22#2025929712yes—
287C22C2×D4×D19111yescharacteristic
288C4—42no—
289C4—41no—
290C4—41no—
291C4—3822no—
292C4—7641no—
293C4#2025929712yes—
294C4C22×D76411yescharacteristic
295C4C2×#1569028811yescharacteristic
296C2—22no—
297C2—15282no—
298C2—15281no—
299C2—7642no—
300C2C2×#2025929711yescharacteristic
301C2#2713346711yescharacteristic
302C2#2713347211yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#428×C191C2
socleC22×C191C22×D4
hypercenter#428D191
hypocenterC191#3062
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#20259350C2semidirect
2. C2×#20259330C2semidirect
3. #27133244C2semidirect
4. #27133232C2semidirect
5. #27132917C2semidirect
6. #27132910C2semidirect
7. #27132900C2semidirect
8. #27132893C2semidirect
9. #27132657C2semidirect
10. C2×#20259242C2semidirect
11. #428×C191C2semidirect
19. C2×#15690270C22semidirect
21. C2×C8×D191C22semidirect
25. #20259302C22semidirect
29. #20259297C22semidirect
32. #20259245C22semidirect
33. #20259245C22semidirect
36. #20259242C22semidirect
37. #20259242C22semidirect
41. #107×C191C22semidirect
42. #106×C191C22semidirect
77. M16×C191D4semidirect
151. C191#3062semidirect
161. #428D191semidirect
187. #107C2×D191semidirect
188. #106C2×D191semidirect
220. M16D764semidirect

Maximal subgroups

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

subgroupindex
C2×#202593502
C2×#202593302
#271332442
#271332322
#271329172
#271329102
#271329002
#271328932
#271326572
C2×#202592422
#428×C1912
#3062191