Finite Group Databasebeta

← #27153524

Subgroups — #27153524

Order 12416.

454 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#20261549C211yesmaximal, characteristic, complemented
2#20261547C212yesmaximal, complemented
3#20261537C211yesmaximal, characteristic, complemented
4#20261119C211yesmaximal, characteristic, complemented
5#20261011C211yesmaximal, characteristic, complemented
6#20261006C211yesmaximal, characteristic, complemented
7#20260998C212yesmaximal, complemented
8#20260914C212yesmaximal, complemented
9#20260909C212yesmaximal, complemented
10#20260853C211yesmaximal, characteristic, complemented
11#443×C97C211yesmaximal, characteristic, complemented
12#15690857C2212yescomplemented
13#15690856C2212yes—
14#15690843—22no—
15#15690843C2212yes—
16#15690840C2212yescomplemented
17#15690839—22no—
18#15690838—22no—
19#15690838C2212yescomplemented
20#15690837—22no—
21#15690835C2212yes—
22#15690833—21no—
23#15690832—22no—
24#15690832C2211yescharacteristic
25#15690829—21no—
26C2×Dic388C2211yescharacteristic, complemented
27Q16×D97—21no—
28C2×#13095500—22no—
29C2×#13095500C2212yescomplemented
30C2×D776C2211yescharacteristic
31D8×D97—21no—
32#15690816C2211yescharacteristic
33#15690816C2211yescharacteristic, complemented
34#15690798C2212yescomplemented
35#15690746C2211yescharacteristic, complemented
36#15690738C2211yescharacteristic
37#15690738C2211yescharacteristic, complemented
38#15690728C2212yescomplemented
39#15690726C2212yescomplemented
40#135×C97C2211yescharacteristic
41#131×C97C2212yescomplemented
42#108×C97C2211yescharacteristic, complemented
43#104×C97C2212yescomplemented
44C4×C8×C97C2211yescharacteristic, complemented
45#13095512—22no—
46#13095511—22no—
47#13095511—22no—
48#13095511C2311yescharacteristic
49#13095511C2311yescharacteristic
50#13095510—22no—
51#13095510—42no—
52C2×Dic194—22no—
53C2×Dic194C2312yes—
54Q8×D97—22no—
55C2×D388—22no—
56C2×D388C2312yes—
57D4×D97—22no—
58D4×D97—42no—
59Dic388—22no—
60Dic388—21no—
61Dic388—21no—
62Dic388D411yescharacteristic
63#13095501—22no—
64#13095500—22no—
65#13095500—22no—
66#13095500—22no—
67#13095500—22no—
68#13095500D411yescharacteristic
69#13095500D411yescharacteristic
70#13095499—22no—
71#13095498—22no—
72D776—22no—
73D776—21no—
74D776—21no—
75D776D411yescharacteristic
76#13095496—22no—
77#13095495C2311yescharacteristic
78#13095495C2311yescharacteristic
79#13095494—21no—
80#13095494—21no—
81C8×D97—21no—
82C8×D97—21no—
83#13095488—42no—
84#41×C97C2312yes—
85Q16×C97D411yescharacteristic
86SD16×C97D412yes—
87D8×C97D411yescharacteristic
88M16×C97—22no—
89M16×C97C2312yes—
90C2×C8×C97—22no—
91C2×C8×C97C2311yescharacteristic
92C2×C8×C97C2311yescharacteristic
93C42×C97C2311yescharacteristic
94C22×D97—22no—
95C22×D97—42no—
96Q8×C97C2×D412yes—
97D4×C97—22no—
98D4×C97C2×D412yes—
99C2×C4×C97—22no—
100C2×C4×C97—21no—
101C2×C4×C97C2411yescharacteristic
102C22⋊D97—21no—
103C22⋊D97—21no—
104C22⋊D97—42no—
105C22⋊D97—42no—
106C2×Dic97—22no—
107C2×Dic97—42no—
108D388—22no—
109D388—22no—
110D388—42no—
111D388C2×D411yescharacteristic
112D388C2×D411yescharacteristic
113C4×D97—21no—
114C4×D97—21no—
115C4×D97—42no—
116C4×D97—42no—
117Dic194—22no—
118Dic194—22no—
119Dic194—42no—
120Dic194C2×D411yescharacteristic
121Dic194C2×D411yescharacteristic
122C8×C97—22no—
123C8×C97C2×D411yescharacteristic
124C8×C97C2×D411yescharacteristic
125C8×C97C2×D411yescharacteristic
126C8×C97C2×D411yescharacteristic
127#11715895—21no—
128#11715895—21no—
129C22×C97—22no—
130C22×C97H(5,𝔽2)11yescharacteristic
131C2×D97—21no—
132C2×D97—21no—
133C2×D97—42no—
134C2×D97—82no—
135C2×D97—162no—
136C4×C97—22no—
137C4×C97—41no—
138C4×C97C22×D411yescharacteristic
139C4×C97C22×D411yescharacteristic
140Dic97—21no—
141Dic97—21no—
142Dic97—42no—
143C2×C97—21no—
144C2×C97—42no—
145C2×C97D4211yescharacteristic
146D97—41no—
147D97—41no—
148D97—82no—
149#3290—971nomaximal, Sylow
150C97#329011yescharacteristic, complemented, Sylow
151#578—971no—
152#577—972no—
153#576—971no—
154#495—971no—
155#473—971no—
156#472—971no—
157#471—972no—
158#455—972no—
159#454—972no—
160#443—971no—
161#443D9711yescharacteristic, complemented
1622−1+4—972no—
163H(5,𝔽2)—972no—
164#137—972no—
165#137—972no—
166#137—1942no—
167#137—1942no—
168AGL(1,ℤ/8)—972no—
169AGL(1,ℤ/8)—972no—
170AGL(1,ℤ/8)—1942no—
171AGL(1,ℤ/8)—1942no—
172#135—971no—
173#135—1942no—
174#135—1941no—
175#135—1941no—
176#135C2×D9711yescharacteristic
177C2×Q16—971no—
178C2×Q16—1941no—
179C2×SD16—972no—
180C2×SD16—1942no—
181C2×D8—971no—
182C2×D8—1941no—
183#131—971no—
184#131—971no—
185#131C2×D9712yescomplemented
186#124—972no—
187#108—971no—
188#108C2×D9711yescharacteristic, complemented
189#104—971no—
190#104—971no—
191#104C2×D9712yescomplemented
192#101—972no—
193#100—972no—
194C4×C8C2×D9711yescharacteristic, complemented
195#41—971no—
196#41—971no—
197#41—3882no—
198#41—1942no—
199#41—1942no—
200#41—1942no—
201#41—1942no—
202#41C22×D9712yes—
203C2×Q8—972no—
204C2×Q8—1942no—
205C2×Q8—1942no—
206C2×D4—972no—
207C2×D4—3882no—
208C2×D4—1942no—
209C2×D4—1942no—
210Q16—971no—
211Q16—1942no—
212Q16—1942no—
213Q16—1941no—
214Q16—1941no—
215Q16D38811yescharacteristic
216SD16—971no—
217SD16—971no—
218SD16—1942no—
219SD16—1942no—
220SD16—1942no—
221SD16—1942no—
222SD16—1942no—
223SD16—1942no—
224SD16D38812yes—
225D8—971no—
226D8—1942no—
227D8—1942no—
228D8—1941no—
229D8—1941no—
230D8D38811yescharacteristic
231M16—22no—
232M16—971no—
233M16—971no—
234M16—1941no—
235M16—1941no—
236M16C22×D9712yes—
237C2×C8—22no—
238C2×C8—1941no—
239C2×C8—1941no—
240C2×C8C22×D9711yescharacteristic
241C2×C8C22×D9711yescharacteristic
242C22⋊C4—3882no—
243C42C22×D9711yescharacteristic
244C23—3882no—
245C23—1942no—
246Q8—971no—
247Q8—971no—
248Q8—3882no—
249Q8—1942no—
250Q8—1942no—
251Q8C2×D38812yes—
252D4—22no—
253D4—971no—
254D4—971no—
255D4—3882no—
256D4—3882no—
257D4—3882no—
258D4—1941no—
259D4—1941no—
260D4—1942no—
261D4—1942no—
262D4C2×D38812yes—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—3882no—
266C2×C4—3882no—
267C2×C4—3882no—
268C2×C4—1942no—
269C2×C4—1941no—
270C2×C4—1941no—
271C2×C4C23×D9711yescharacteristic
272C8—22no—
273C8—1941no—
274C8—1941no—
275C8D4×D9711yescharacteristic
276C8D4×D9711yescharacteristic
277C8C2×D38811yescharacteristic
278C8C2×D38811yescharacteristic
279C22—22no—
280C22—7762no—
281C22—15522no—
282C22—3882no—
283C22—1941no—
284C22—1941no—
285C22#1569085611yescharacteristic
286C4—22no—
287C4—41no—
288C4—3882no—
289C4—1941no—
290C4—1941no—
291C4C2×D4×D9711yescharacteristic
292C4C22×D38811yescharacteristic
293C2—21no—
294C2—42no—
295C2—7762no—
296C2—3881no—
297C2—3881no—
298C2D4×D38811yescharacteristic
subgroupG/H
centerC2D4×D388
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#443×C97C2
socleC2×C97D42
hypercenter#443D97
hypocenterC97#3290
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443

Complements

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

NH ≅ G/Ntype
1. #20261549C2semidirect
2. #20261547C2semidirect
3. #20261537C2semidirect
4. #20261119C2semidirect
5. #20261011C2semidirect
6. #20261006C2semidirect
7. #20260998C2semidirect
8. #20260914C2semidirect
9. #20260909C2semidirect
10. #20260853C2semidirect
11. #443×C97C2semidirect
12. #15690857C22semidirect
16. #15690840C22semidirect
19. #15690838C22semidirect
26. C2×Dic388C22semidirect
29. C2×#13095500C22semidirect
33. #15690816C22semidirect
34. #15690798C22semidirect
35. #15690746C22semidirect
37. #15690738C22semidirect
38. #15690728C22semidirect
39. #15690726C22semidirect
41. #131×C97C22semidirect
42. #108×C97C22semidirect
43. #104×C97C22semidirect
44. C4×C8×C97C22semidirect
150. C97#3290semidirect
161. #443D97semidirect
185. #131C2×D97semidirect
188. #108C2×D97semidirect
191. #104C2×D97semidirect
194. C4×C8C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202615492
#202615472
#202615372
#202611192
#202610112
#202610062
#202609982
#202609142
#202609092
#202608532
#443×C972
#329097