Finite Group Databasebeta

← #27151245

Subgroups — #27151245

Order 12416.

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

normal subgroups only

This group has 494 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×#15690836C211yesmaximal, characteristic
2#20261309C211yesmaximal, characteristic, complemented
3#20261210C211yesmaximal, characteristic, complemented
4#20260952C214yesmaximal, complemented
5#20260938C214yesmaximal, complemented
6#20260761C211yesmaximal, characteristic, complemented
7#20260744C211yesmaximal, characteristic, complemented
8#20260732C211yesmaximal, characteristic, complemented
9C22×SD16×C97C211yesmaximal, characteristic, complemented
10C2×#13095511C2211yescharacteristic
11#15690836—24no—
12C2×#13095499C2212yes—
13C2×#13095496C2212yes—
14C2×#13095495C2211yescharacteristic
15#15690808C2212yescomplemented
16#15690799C2212yescomplemented
17#15690793—22no—
18#15690786—22no—
19D4×Dic97—22no—
20D4×Dic97C2214yes—
21#15690769C2211yescharacteristic
22#15690769C2211yescharacteristic
23C2×C22⋊Dic97—22no—
24#15690739C2214yescomplemented
25#15690735C2212yescomplemented
26#15690734C2214yescomplemented
27#15690732C2212yescomplemented
28C22×Q8×C97C2211yescharacteristic
29C22×D4×C97C2211yescharacteristic
30C2×SD16×C97—24no—
31C2×SD16×C97C2214yes—
32C22×C8×C97C2211yescharacteristic
33#13095511—22no—
34#13095511—22no—
35C2×Dic194C2311yescharacteristic
36C2×D388C2311yescharacteristic
37C2×C22⋊D97—21no—
38C22×Dic97—24no—
39C2×C4×D97—21no—
40#13095499—24no—
41#13095496—24no—
42#13095495—22no—
43C2×#11715895C2312yes—
44C4⋊Dic97D412yes—
45C4⋊Dic97C2312yes—
46#13095490—24no—
47C22⋊Dic97—21no—
48C22⋊Dic97—24no—
49C22⋊Dic97D412yes—
50#13095488—24no—
51C4×Dic97—21no—
52C4×Dic97C2312yes—
53C24×C97—21no—
54C2×Q8×C97—21no—
55C2×Q8×C97—22no—
56C2×Q8×C97—22no—
57C2×Q8×C97C2312yes—
58C2×D4×C97—21no—
59C2×D4×C97—22no—
60C2×D4×C97D414yes—
61C2×D4×C97C2312yes—
62C22×C4×C97—21no—
63C22×C4×C97C2311yescharacteristic
64SD16×C97—28no—
65C2×C8×C97—22no—
66C2×C8×C97C2312yes—
67C23×C97—22no—
68C23×C97—24no—
69C23×C97—82no—
70C23×C97C2×D411yescharacteristic
71C22×D97—21no—
72Q8×C97—22no—
73Q8×C97—22no—
74Q8×C97—42no—
75D4×C97—24no—
76D4×C97—22no—
77D4×C97C2×D414yes—
78C2×C4×C97—22no—
79C2×C4×C97—42no—
80C2×C4×C97C2×D411yescharacteristic
81C2×C4×C97C2×D412yes—
82C2×C4×C97#4112yes—
83C2×C4×C97C2411yescharacteristic
84C22⋊D97—42no—
85C2×Dic97—22no—
86C2×Dic97—21no—
87C2×Dic97—84no—
88C2×Dic97C2×D414yes—
89D388—21no—
90D388—21no—
91C4×D97—42no—
92Dic194—21no—
93Dic194—21no—
94C8×C97—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—42no—
105C4×C97#12112yes—
106C4×C97C22×D411yescharacteristic
107C4×C97C2×#4111yescharacteristic
108Dic97—24no—
109Dic97—41no—
110Dic97—42no—
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×SD16D9711yescharacteristic, complemented
120#536—971no—
121#518—971no—
122#463—974no—
123#461—974no—
124#424—971no—
125#419—971no—
126#416—971no—
127C2×#41—971no—
128C22×Q8C2×D9711yescharacteristic
129C22×D4C2×D9711yescharacteristic
130AGL(1,ℤ/8)—1944no—
131C2×SD16—24no—
132C2×SD16—972no—
133C2×SD16C2×D9714yes—
134C2×D8—972no—
135C2×M16—971no—
136C22×C8C2×D9711yescharacteristic
137C4⋊Q8—972no—
138#124—972no—
139#123—1942no—
140C22⋊Q8—1942no—
141C4×D4—974no—
142C4×D4—1942no—
143#117—971no—
144#117—971no—
145C2×C22⋊C4—1942no—
146C4⋊C8—974no—
147#103—974no—
148#103—972no—
149#102—972no—
150C24—21no—
151#41—1942no—
152#41—1942no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—971no—
157C2×Q8C22×D9712yes—
158C2×D4—21no—
159C2×D4—22no—
160C2×D4—971no—
161C2×D4—1941no—
162C2×D4C22⋊D9714yes—
163C2×D4C22×D9712yes—
164C22×C4—21no—
165C22×C4—1944no—
166C22×C4—1941no—
167C22×C4C22×D9711yescharacteristic
168SD16—28no—
169SD16—1944no—
170D8—1944no—
171M16—1942no—
172C2×C8—22no—
173C2×C8—972no—
174C2×C8C22×D9712yes—
175C4⋊C4—972no—
176C4⋊C4—972no—
177C4⋊C4—1944no—
178C22⋊C4—972no—
179C22⋊C4—1944no—
180C22⋊C4—1944no—
181C22⋊C4—1941no—
182C42—972no—
183C42—1941no—
184C23—22no—
185C23—24no—
186C23—82no—
187C23—1941no—
188C23D4×D9711yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—42no—
192Q8—1941no—
193Q8—1941no—
194D4—24no—
195D4—22no—
196D4—3882no—
197D4—1941no—
198D4—1941no—
199D4C2×C22⋊D9714yes—
200C2×C4—22no—
201C2×C4—42no—
202C2×C4—7764no—
203C2×C4—974no—
204C2×C4—3882no—
205C2×C4—1942no—
206C2×C4—1941no—
207C2×C4D4×D9711yescharacteristic
208C2×C4C2×C22⋊D9712yes—
209C2×C4#1309551012yes—
210C2×C4C23×D9711yescharacteristic
211C8—22no—
212C8—1942no—
213C22—22no—
214C22—24no—
215C22—44no—
216C22—84no—
217C22—3881no—
218C22—3882no—
219C22#1569078212yes—
220C22C2×D4×D9711yescharacteristic
221C4—42no—
222C4—3881no—
223C4—3882no—
224C4—1944no—
225C4#1569078212yes—
226C4C22×C22⋊D9711yescharacteristic
227C4C2×#1309551011yescharacteristic
228C2—22no—
229C2—44no—
230C2—7761no—
231C2C2×#1569078211yescharacteristic
232C2#2026154112yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC22×SD16×C97C2
socleC22×C97C22×D4
prime radicalC2×#15690836C2
hypercenterC22×SD16D97
hypocenterC97#3042
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
2. #20261309C2semidirect
3. #20261210C2semidirect
4. #20260952C2semidirect
5. #20260938C2semidirect
6. #20260761C2semidirect
7. #20260744C2semidirect
8. #20260732C2semidirect
9. C22×SD16×C97C2semidirect
15. #15690808C22semidirect
16. #15690799C22semidirect
24. #15690739C22semidirect
25. #15690735C22semidirect
26. #15690734C22semidirect
27. #15690732C22semidirect
117. C97#3042semidirect
119. C22×SD16D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908362
#202613092
#202612102
#202609522
#202609382
#202607612
#202607442
#202607322
C22×SD16×C972
#304297