Finite Group Databasebeta

← #27151251

Subgroups — #27151251

Order 12416.

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

normal subgroups only

This group has 494 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×#15690835C211yesmaximal, characteristic
2C22×#13095500C211yesmaximal, characteristic, complemented
3#20261306C211yesmaximal, characteristic, complemented
4#20261209C211yesmaximal, characteristic, complemented
5#20260949C214yesmaximal, complemented
6#20260939C214yesmaximal, complemented
7#20260761C211yesmaximal, characteristic, complemented
8#20260743C211yesmaximal, characteristic, complemented
9#416×C97C211yesmaximal, characteristic, complemented
10C22×Dic194C2211yescharacteristic
11C22×D388C2211yescharacteristic
12#15690835—24no—
13C2×#13095500—24no—
14C2×#13095500C2214yes—
15C2×#13095498C2212yes—
16C2×#13095496C2212yes—
17C2×#13095495C2211yescharacteristic
18#15690808C2212yescomplemented
19#15690797C2212yescomplemented
20#15690791—22no—
21#15690789—22no—
22C4×D388—22no—
23#15690772C2214yes—
24#15690769C2211yescharacteristic
25C22⋊C4×D97—22no—
26#15690739C2214yescomplemented
27#15690733C2214yescomplemented
28C2×#41×C97C2211yescharacteristic
29C22×C8×C97C2211yescharacteristic
30#117×C97C2211yescharacteristic
31#103×C97C2212yescomplemented
32#102×C97C2212yescomplemented
33C23×D97—21no—
34C2×Dic194—22no—
35C2×Dic194—22no—
36C2×Dic194—21no—
37C2×Dic194C2312yes—
38C2×D388—21no—
39C2×D388—22no—
40C2×D388D414yes—
41C2×D388C2312yes—
42C22×Dic97—21no—
43C2×C4×D97—24no—
44#13095500—28no—
45#13095498—24no—
46#13095496—24no—
47#13095495—22no—
48C2×#11715895C2312yes—
49C4⋊Dic97D412yes—
50#13095490—24no—
51C22⋊Dic97—24no—
52C22⋊Dic97D412yes—
53#13095488—24no—
54C4×Dic97C2312yes—
55#41×C97—22no—
56#41×C97—22no—
57C2×Q8×C97C2311yescharacteristic
58C2×D4×C97—21no—
59C2×D4×C97C2311yescharacteristic
60C22×C4×C97—21no—
61C22×C4×C97C2311yescharacteristic
62C2×C8×C97—22no—
63C2×C8×C97C2312yes—
64C4⋊C4×C97C2312yes—
65C22⋊C4×C97—21no—
66C42×C97—21no—
67C23×C97—21no—
68C23×C97C2×D411yescharacteristic
69C22×D97—22no—
70C22×D97—24no—
71C22×D97—82no—
72Q8×C97—21no—
73Q8×C97—21no—
74D4×C97—21no—
75D4×C97—21no—
76D4×C97—42no—
77C2×C4×C97—22no—
78C2×C4×C97—21no—
79C2×C4×C97—42no—
80C2×C4×C97C2×D411yescharacteristic
81C2×C4×C97C2×D412yes—
82C2×C4×C97#4112yes—
83C2×C4×C97C2411yescharacteristic
84C2×Dic97—22no—
85C2×Dic97—42no—
86C2×Dic97C2×D414yes—
87D388—24no—
88D388—22no—
89D388C2×D414yes—
90C4×D97—84no—
91Dic194—22no—
92Dic194—22no—
93Dic194—42no—
94C8×C97—22no—
95#11715895—22no—
96C22×C97—22no—
97C22×C97—41no—
98C22×C97—42no—
99C22×C97#12112yes—
100C22×C97C22×D411yescharacteristic
101C2×D97—24no—
102C2×D97—44no—
103C2×D97—84no—
104C4×C97—41no—
105C4×C97—42no—
106C4×C97#12112yes—
107C4×C97C22×D411yescharacteristic
108C4×C97C2×#4111yescharacteristic
109Dic97—24no—
110Dic97—42no—
111C2×C97—22no—
112C2×C97—81no—
113C2×C97C2×#12111yescharacteristic
114C2×C97#57712yes—
115D97—44no—
116#3042—971nomaximal, Sylow
117C97#304211yescharacteristic, complemented, Sylow
118C2×AGL(1,ℤ/8)—971no—
119C22×SD16—971no—
120#536—971no—
121#518—971no—
122#463—974no—
123#461—974no—
124#424—971no—
125#419—971no—
126#416D9711yescharacteristic, complemented
127C2×#41C2×D9711yescharacteristic
128C22×Q8—971no—
129C22×D4—971no—
130AGL(1,ℤ/8)—1944no—
131C2×SD16—972no—
132C2×SD16—974no—
133C2×SD16—1944no—
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#117C2×D9711yescharacteristic
145C2×C22⋊C4—1942no—
146C4⋊C8—974no—
147#103—974no—
148#103C2×D9712yescomplemented
149#102C2×D9712yescomplemented
150C24—1941no—
151#41—22no—
152#41—22no—
153C2×Q8—972no—
154C2×Q8—1942no—
155C2×Q8—1942no—
156C2×Q8—1941no—
157C2×Q8C22×D9711yescharacteristic
158C2×D4—21no—
159C2×D4—974no—
160C2×D4—972no—
161C2×D4—1941no—
162C2×D4—1942no—
163C2×D4C22×D9711yescharacteristic
164C22×C4—21no—
165C22×C4—1944no—
166C22×C4—1941no—
167C22×C4C22×D9711yescharacteristic
168SD16—1948no—
169SD16—1944no—
170D8—1944no—
171M16—1942no—
172C2×C8—22no—
173C2×C8—972no—
174C2×C8C22×D9712yes—
175C4⋊C4—972no—
176C4⋊C4—1944no—
177C4⋊C4C22×D9712yes—
178C22⋊C4—21no—
179C22⋊C4—972no—
180C22⋊C4—1944no—
181C22⋊C4—1944no—
182C42—21no—
183C42—972no—
184C23—21no—
185C23—7762no—
186C23—1944no—
187C23—1942no—
188C23D4×D9711yescharacteristic
189Q8—21no—
190Q8—21no—
191Q8—3882no—
192Q8—1942no—
193Q8—1942no—
194D4—21no—
195D4—21no—
196D4—42no—
197D4—974no—
198D4—1944no—
199D4—1942no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—42no—
203C2×C4—7764no—
204C2×C4—974no—
205C2×C4—3882no—
206C2×C4—1942no—
207C2×C4D4×D9711yescharacteristic
208C2×C4D4×D9712yes—
209C2×C4#1309551112yes—
210C2×C4C23×D9711yescharacteristic
211C8—22no—
212C8—1942no—
213C22—22no—
214C22—41no—
215C22—42no—
216C22—7764no—
217C22—3884no—
218C22—1944no—
219C22#1569078312yes—
220C22C2×D4×D9711yescharacteristic
221C4—41no—
222C4—42no—
223C4—3882no—
224C4—1944no—
225C4#1569078312yes—
226C4C2×D4×D9711yescharacteristic
227C4C2×#1309551111yescharacteristic
228C2—22no—
229C2—81no—
230C2—3884no—
231C2C2×#1569078311yescharacteristic
232C2#2026154212yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#416×C97C2
socleC22×C97C22×D4
prime radicalC2×#15690835C2
hypercenter#416D97
hypocenterC97#3042
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. C22×#13095500C2semidirect
3. #20261306C2semidirect
4. #20261209C2semidirect
5. #20260949C2semidirect
6. #20260939C2semidirect
7. #20260761C2semidirect
8. #20260743C2semidirect
9. #416×C97C2semidirect
18. #15690808C22semidirect
19. #15690797C22semidirect
26. #15690739C22semidirect
27. #15690733C22semidirect
31. #103×C97C22semidirect
32. #102×C97C22semidirect
117. C97#3042semidirect
126. #416D97semidirect
148. #103C2×D97semidirect
149. #102C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908352
C22×#130955002
#202613062
#202612092
#202609492
#202609392
#202607612
#202607432
#416×C972
#304297