Finite Group Databasebeta

← #27151256

Subgroups — #27151256

Order 12416.

490 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 133 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×#15690843C211yesmaximal, characteristic, complemented
2C22×#13095499C211yesmaximal, characteristic, complemented
3#20261302C211yesmaximal, characteristic
4#20261212C211yesmaximal, characteristic, complemented
5#20260956C214yesmaximal, complemented
6#20260934C214yesmaximal, complemented
7#20260763C211yesmaximal, characteristic, complemented
8#20260729C211yesmaximal, characteristic, complemented
9#418×C97C211yesmaximal, characteristic, complemented
10C2×#13095511C2211yescharacteristic
11C22×Dic194C2211yescharacteristic
12#15690843—24no—
13C2×Dic388C2212yescomplemented
14C2×#13095500C2212yescomplemented
15C2×#13095499—24no—
16C2×#13095499C2214yes—
17C22×#11715895C2211yescharacteristic
18#15690806C2212yes—
19#15690797C2212yes—
20#15690781—22no—
21#15690779—22no—
22C4×Dic194—22no—
23#15690775C2214yescomplemented
24#15690769C2211yescharacteristic
25#15690766—22no—
26#15690739C2214yescomplemented
27#15690733C2212yescomplemented
28#15690730C2212yescomplemented
29C22×D4×C97C2211yescharacteristic
30C2×M16×C97C2211yescharacteristic
31#117×C97C2211yescharacteristic
32#102×C97C2214yes—
33#13095511—22no—
34#13095511—22no—
35C2×Dic194—21no—
36C2×Dic194—22no—
37C2×Dic194D414yescomplemented
38C2×Dic194C2311yescharacteristic
39C2×Dic194C2312yes—
40C2×D388C2311yescharacteristic
41C2×C22⋊D97—21no—
42C2×C22⋊D97—24no—
43C22×Dic97—21no—
44C2×C4×D97—21no—
45Dic388—24no—
46#13095500—24no—
47#13095499—28no—
48C2×#11715895—22no—
49C2×#11715895C2312yes—
50C4⋊Dic97D412yes—
51#13095490—24no—
52C22⋊Dic97—24no—
53C22⋊Dic97D412yes—
54C4×Dic97—24no—
55C4×Dic97C2312yes—
56C24×C97—21no—
57C2×D4×C97—22no—
58C2×D4×C97—21no—
59C2×D4×C97—22no—
60C2×D4×C97C2312yes—
61C22×C4×C97C2311yescharacteristic
62M16×C97—22no—
63C2×C8×C97C2312yes—
64C4⋊C4×C97C2312yes—
65C22⋊C4×C97—21no—
66C42×C97—21no—
67C23×C97—22no—
68C23×C97—42no—
69C23×C97—82no—
70C23×C97C2×D411yescharacteristic
71C22×D97—21no—
72D4×C97—22no—
73D4×C97—22no—
74D4×C97—42no—
75C2×C4×C97—22no—
76C2×C4×C97C2×D411yescharacteristic
77C2×C4×C97C2×D412yes—
78C2×C4×C97#4112yes—
79C2×C4×C97C2411yescharacteristic
80C22⋊D97—42no—
81C22⋊D97—84no—
82C2×Dic97—24no—
83C2×Dic97—22no—
84C2×Dic97—21no—
85C2×Dic97C2×D414yes—
86D388—21no—
87D388—21no—
88C4×D97—42no—
89Dic194—24no—
90Dic194—22no—
91Dic194—21no—
92Dic194—21no—
93Dic194C2×D414yes—
94C8×C97—22no—
95#11715895—22no—
96C22×C97—22no—
97C22×C97—42no—
98C22×C97—44no—
99C22×C97—84no—
100C22×C97#12112yes—
101C22×C97C22×D411yescharacteristic
102C2×D97—41no—
103C2×D97—42no—
104C4×C97—42no—
105C4×C97#12112yes—
106C4×C97C22×D411yescharacteristic
107C4×C97C2×#4111yescharacteristic
108Dic97—24no—
109Dic97—24no—
110Dic97—41no—
111C2×C97—22no—
112C2×C97—82no—
113C2×C97C2×#12111yescharacteristic
114C2×C97#57712yes—
115D97—81no—
116#3043—971nomaximal, Sylow
117C97#304311yescharacteristic, complemented, Sylow
118C2×#137—971no—
119C22×SD16—971no—
120#535—971no—
121#519—971no—
122#464—974no—
123#460—974no—
124#424—971no—
125#418D9711yescharacteristic, complemented
126#416—971no—
127C2×#41—971no—
128C22×Q8—971no—
129C22×D4C2×D9711yescharacteristic
130#137—1944no—
131C2×Q16—972no—
132C2×SD16—974no—
133C2×SD16—972no—
134C2×SD16—1944no—
135C2×M16C2×D9711yescharacteristic
136C22×C8—971no—
137#127—972no—
138#124—972no—
139#121—1942no—
140#120—1942no—
141C4×Q8—974no—
142C4×Q8—1942no—
143#117—971no—
144#117—1942no—
145#117C2×D9711yescharacteristic
146C4⋊C8—974no—
147#103—972no—
148#102—972no—
149#102C2×D9714yes—
150C24—21no—
151#41—1942no—
152#41—1942no—
153C2×Q8—974no—
154C2×Q8—972no—
155C2×Q8—971no—
156C2×Q8—1941no—
157C2×Q8—1942no—
158C2×D4—22no—
159C2×D4—21no—
160C2×D4—22no—
161C2×D4—971no—
162C2×D4—1944no—
163C2×D4—1941no—
164C2×D4C22×D9712yes—
165C22×C4—1941no—
166C22×C4—1941no—
167C22×C4C22×D9711yescharacteristic
168Q16—1944no—
169SD16—1944no—
170SD16—1948no—
171M16—22no—
172C2×C8—972no—
173C2×C8—1942no—
174C2×C8C22×D9712yes—
175C4⋊C4—972no—
176C4⋊C4—1944no—
177C4⋊C4C22×D9712yes—
178C22⋊C4—21no—
179C22⋊C4—972no—
180C22⋊C4—1944no—
181C42—21no—
182C42—972no—
183C42—1944no—
184C23—22no—
185C23—42no—
186C23—82no—
187C23—1941no—
188C23D4×D9711yescharacteristic
189Q8—974no—
190Q8—1941no—
191Q8—1941no—
192Q8—1944no—
193Q8—1942no—
194D4—22no—
195D4—22no—
196D4—42no—
197D4—7764no—
198D4—3882no—
199D4—1941no—
200D4—1941no—
201C2×C4—22no—
202C2×C4—974no—
203C2×C4—3882no—
204C2×C4—1941no—
205C2×C4—1944no—
206C2×C4—1942no—
207C2×C4D4×D9711yescharacteristic
208C2×C4D4×D9712yes—
209C2×C4#1309551112yes—
210C2×C4C23×D9711yescharacteristic
211C8—22no—
212C8—1942no—
213C22—22no—
214C22—42no—
215C22—44no—
216C22—84no—
217C22—3881no—
218C22—3882no—
219C22#1569078312yes—
220C22C2×D4×D9711yescharacteristic
221C4—42no—
222C4—3881no—
223C4—1944no—
224C4—1944no—
225C4#1569078312yes—
226C4C2×D4×D9711yescharacteristic
227C4C2×#1309551111yescharacteristic
228C2—22no—
229C2—82no—
230C2—7761no—
231C2C2×#1569078311yescharacteristic
232C2#2026154512yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#418×C97C2
socleC22×C97C22×D4
prime radical#20261302C2
hypercenter#418D97
hypocenterC97#3043
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C2×#15690843C2semidirect
2. C22×#13095499C2semidirect
4. #20261212C2semidirect
5. #20260956C2semidirect
6. #20260934C2semidirect
7. #20260763C2semidirect
8. #20260729C2semidirect
9. #418×C97C2semidirect
13. C2×Dic388C22semidirect
14. C2×#13095500C22semidirect
23. #15690775C22semidirect
26. #15690739C22semidirect
27. #15690733C22semidirect
28. #15690730C22semidirect
37. C2×Dic194D4semidirect
117. C97#3043semidirect
125. #418D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908432
C22×#130954992
#202613022
#202612122
#202609562
#202609342
#202607632
#202607292
#418×C972
#304397