Finite Group Databasebeta

← #27151248

Subgroups — #27151248

Order 12416.

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

normal subgroups only

This group has 494 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×#15690836C211yesmaximal, characteristic
2C22×#13095499C211yesmaximal, characteristic, complemented
3#20261311C211yesmaximal, characteristic, complemented
4#20260950C214yesmaximal, complemented
5#20260940C214yesmaximal, complemented
6#20260762C211yesmaximal, characteristic, complemented
7#20260743C211yesmaximal, characteristic, complemented
8#20260729C211yesmaximal, characteristic, complemented
9#518×C97C211yesmaximal, characteristic, complemented
10C2×#13095511C2211yescharacteristic
11C22×Dic194C2211yescharacteristic
12#15690836—24no—
13C2×#13095499—24no—
14C2×#13095499C2212yes—
15C2×#13095499C2214yes—
16C2×#13095496C2212yes—
17C2×#13095495C2211yescharacteristic
18C22×#11715895C2211yescharacteristic
19#15690810C2212yescomplemented
20#15690798C2212yescomplemented
21#15690794—22no—
22#15690790—22no—
23C4⋊#11715895C2214yescomplemented
24#15690733C2212yescomplemented
25#15690733C2214yescomplemented
26#15690730C2212yescomplemented
27C22×D4×C97C2211yescharacteristic
28C4×D4×C97—22no—
29C4×D4×C97C2214yes—
30#117×C97C2211yescharacteristic
31#117×C97C2211yescharacteristic
32C2×C22⋊C4×C97—22no—
33#13095511—22no—
34#13095511—22no—
35C2×Dic194—22no—
36C2×Dic194—22no—
37C2×Dic194—21no—
38C2×Dic194C2312yes—
39C2×Dic194C2311yescharacteristic
40C2×D388C2311yescharacteristic
41C2×C22⋊D97—21no—
42C22×Dic97—21no—
43C2×C4×D97—21no—
44#13095499—28no—
45#13095499—24no—
46#13095496—24no—
47#13095495—22no—
48C2×#11715895—22no—
49C2×#11715895C2312yes—
50C2×#11715895C2312yes—
51C4⋊Dic97—24no—
52#13095488—24no—
53C24×C97—21no—
54C2×D4×C97—21no—
55C2×D4×C97—22no—
56C2×D4×C97D414yes—
57C2×D4×C97C2312yes—
58C22×C4×C97—24no—
59C22×C4×C97C2311yescharacteristic
60C4⋊C4×C97D412yes—
61C4⋊C4×C97C2312yes—
62C22⋊C4×C97—21no—
63C22⋊C4×C97—24no—
64C22⋊C4×C97D412yes—
65C42×C97—21no—
66C42×C97C2312yes—
67C23×C97—22no—
68C23×C97—24no—
69C23×C97—82no—
70C23×C97C2×D411yescharacteristic
71C22×D97—21no—
72D4×C97—24no—
73D4×C97—22no—
74D4×C97C2×D414yes—
75C2×C4×C97—22no—
76C2×C4×C97—84no—
77C2×C4×C97C2×D414yes—
78C2×C4×C97C2×D411yescharacteristic
79C2×C4×C97C2×D412yes—
80C2×C4×C97#4112yes—
81C2×C4×C97C2411yescharacteristic
82C22⋊D97—42no—
83C2×Dic97—22no—
84C2×Dic97—21no—
85C2×Dic97—42no—
86D388—21no—
87D388—21no—
88C4×D97—42no—
89Dic194—22no—
90Dic194—21no—
91Dic194—22no—
92Dic194—21no—
93Dic194—42no—
94#11715895—22no—
95#11715895—22no—
96C22×C97—22no—
97C22×C97—24no—
98C22×C97—44no—
99C22×C97—84no—
100C22×C97#12112yes—
101C22×C97C22×D411yescharacteristic
102C2×D97—41no—
103C2×D97—42no—
104C4×C97—24no—
105C4×C97—42no—
106C4×C97#12112yes—
107C4×C97C22×D411yescharacteristic
108C4×C97C2×#4111yescharacteristic
109Dic97—42no—
110Dic97—41no—
111C2×C97—22no—
112C2×C97—44no—
113C2×C97C2×#12111yescharacteristic
114C2×C97#57712yes—
115D97—81no—
116#3042—971nomaximal, Sylow
117C97#304211yescharacteristic, complemented, Sylow
118C2×AGL(1,ℤ/8)—971no—
119C22×SD16—971no—
120#536—971no—
121#518D9711yescharacteristic, complemented
122#463—974no—
123#461—974no—
124#424—971no—
125#419—971no—
126#416—971no—
127C2×#41—971no—
128C22×Q8—971no—
129C22×D4C2×D9711yescharacteristic
130AGL(1,ℤ/8)—1944no—
131C2×SD16—974no—
132C2×SD16—972no—
133C2×SD16—1944no—
134C2×D8—972no—
135C2×M16—971no—
136C22×C8—971no—
137C4⋊Q8—972no—
138#124—972no—
139#123—1942no—
140C22⋊Q8—1942no—
141C4×D4—22no—
142C4×D4C2×D9714yes—
143#117C2×D9711yescharacteristic
144#117C2×D9711yescharacteristic
145C2×C22⋊C4—22no—
146C4⋊C8—974no—
147#103—972no—
148#103—974no—
149#102—972no—
150C24—21no—
151#41—1942no—
152#41—1942no—
153C2×Q8—972no—
154C2×Q8—971no—
155C2×Q8—1942no—
156C2×Q8—1942no—
157C2×Q8—1941no—
158C2×D4—21no—
159C2×D4—22no—
160C2×D4—971no—
161C2×D4—1941no—
162C2×D4D38814yes—
163C2×D4C22×D9712yes—
164C22×C4—24no—
165C22×C4—1941no—
166C22×C4—1941no—
167C22×C4C22×D9711yescharacteristic
168SD16—1948no—
169SD16—1944no—
170D8—1944no—
171M16—1942no—
172C2×C8—972no—
173C2×C8—972no—
174C2×C8—1942no—
175C4⋊C4—1944no—
176C4⋊C4C22⋊D9712yes—
177C4⋊C4C22×D9712yes—
178C22⋊C4—21no—
179C22⋊C4—24no—
180C22⋊C4—1944no—
181C22⋊C4C22⋊D9712yes—
182C42—21no—
183C42C22×D9712yes—
184C23—22no—
185C23—24no—
186C23—82no—
187C23—1941no—
188C23C2×C22⋊D9711yescharacteristic
189Q8—3882no—
190Q8—1942no—
191Q8—1941no—
192Q8—1942no—
193Q8—1941no—
194D4—24no—
195D4—22no—
196D4—3882no—
197D4—1941no—
198D4—1941no—
199D4C2×D38814yes—
200C2×C4—22no—
201C2×C4—84no—
202C2×C4—3882no—
203C2×C4—3882no—
204C2×C4—1942no—
205C2×C4—1941no—
206C2×C4C2×C22⋊D9714yes—
207C2×C4C2×C22⋊D9711yescharacteristic
208C2×C4C2×D38812yes—
209C2×C4#1309551112yes—
210C2×C4C23×D9711yescharacteristic
211C8—1942no—
212C8—1942no—
213C22—22no—
214C22—24no—
215C22—44no—
216C22—84no—
217C22—3881no—
218C22—3882no—
219C22#1569078412yes—
220C22C22×C22⋊D9711yescharacteristic
221C4—24no—
222C4—42no—
223C4—3881no—
224C4—3882no—
225C4#1569078412yes—
226C4C22×D38811yescharacteristic
227C4C2×#1309551111yescharacteristic
228C2—22no—
229C2—44no—
230C2—7761no—
231C2C2×#1569078411yescharacteristic
232C2#2026154612yes—
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#518×C97C2
socleC22×C97C22×D4
prime radicalC2×#15690836C2
hypercenter#518D97
hypocenterC97#3042
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
2. C22×#13095499C2semidirect
3. #20261311C2semidirect
4. #20260950C2semidirect
5. #20260940C2semidirect
6. #20260762C2semidirect
7. #20260743C2semidirect
8. #20260729C2semidirect
9. #518×C97C2semidirect
19. #15690810C22semidirect
20. #15690798C22semidirect
23. C4⋊#11715895C22semidirect
24. #15690733C22semidirect
25. #15690733C22semidirect
26. #15690730C22semidirect
117. C97#3042semidirect
121. #518D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908362
C22×#130954992
#202613112
#202609502
#202609402
#202607622
#202607432
#202607292
#518×C972
#304297