Finite Group Databasebeta

← #27151200

Subgroups — #27151200

Order 12416.

548 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 153 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×#15690836C212yesmaximal, complemented
2#20261285C211yesmaximal, characteristic, complemented
3#20260950C214yesmaximal, complemented
4#20260940C212yesmaximal, complemented
5#20260930C212yesmaximal, complemented
6#20260759C211yesmaximal, characteristic, complemented
7#20260734C212yesmaximal, complemented
8C2×C4×D4×C97C211yesmaximal, characteristic, complemented
9C2×#13095511C2212yes—
10#15690836—28no—
11C2×#13095499C2214yes—
12C2×#13095496C2214yes—
13C2×#13095495C2212yes—
14#15690810C2211yescharacteristic, complemented
15#15690807C2211yescharacteristic, complemented
16#15690798C2212yescomplemented
17#15690784—21no—
18#15690784—21no—
19C4×D388—21no—
20C4×D388—21no—
21C4⋊#11715895C2212yescomplemented
22C4⋊#11715895C2212yescomplemented
23#15690733C2214yescomplemented
24#15690730C2214yescomplemented
25C22×D4×C97C2211yescharacteristic
26C23×C4×C97—21no—
27C4×D4×C97—21no—
28C4×D4×C97—21no—
29C4×D4×C97C2212yes—
30C4×D4×C97C2212yes—
31C2×C4⋊C4×C97C2211yescharacteristic
32C2×C22⋊C4×C97—21no—
33C2×C42×C97C2211yescharacteristic
34#13095511—24no—
35#13095511—24no—
36C2×Dic194C2312yes—
37C2×D388—21no—
38C2×D388—21no—
39C2×D388C2312yes—
40C2×C22⋊D97—22no—
41C2×C4×D97—22no—
42#13095499—28no—
43#13095496—28no—
44#13095495—24no—
45C2×#11715895C2314yes—
46C4⋊Dic97—21no—
47C4⋊Dic97—21no—
48#13095488—22no—
49#13095488—22no—
50C24×C97—21no—
51C2×D4×C97—21no—
52C2×D4×C97—22no—
53C2×D4×C97D414yescomplemented
54C2×D4×C97C2312yes—
55C22×C4×C97—21no—
56C22×C4×C97—22no—
57C22×C4×C97—22no—
58C22×C4×C97—44no—
59C22×C4×C97D411yescharacteristic
60C22×C4×C97D411yescharacteristic
61C22×C4×C97C2311yescharacteristic
62C4⋊C4×C97—21no—
63C4⋊C4×C97C2312yes—
64C22⋊C4×C97—22no—
65C22⋊C4×C97—22no—
66C42×C97D411yescharacteristic
67C42×C97D411yescharacteristic
68C42×C97C2311yescharacteristic
69C42×C97C2311yescharacteristic
70C23×C97—24no—
71C23×C97—22no—
72C23×C97—42no—
73C23×C97—42no—
74C23×C97C2×D411yescharacteristic
75C22×D97—22no—
76D4×C97—24no—
77D4×C97—22no—
78D4×C97C2×D414yes—
79C2×C4×C97—22no—
80C2×C4×C97—22no—
81C2×C4×C97—41no—
82C2×C4×C97—44no—
83C2×C4×C97—42no—
84C2×C4×C97—44no—
85C2×C4×C97C2×D411yescharacteristic
86C2×C4×C97C2×D411yescharacteristic
87C2×C4×C97C2×D411yescharacteristic
88C2×C4×C97C2×D411yescharacteristic
89C2×C4×C97C2×D411yescharacteristic
90C2×C4×C97C2×D412yes—
91C2×C4×C97#4112yes—
92C2×C4×C97C2411yescharacteristic
93C22⋊D97—44no—
94C2×Dic97—22no—
95D388—22no—
96D388—22no—
97D388—42no—
98D388—42no—
99C4×D97—44no—
100Dic194—22no—
101Dic194—22no—
102#11715895—24no—
103C22×C97—21no—
104C22×C97—21no—
105C22×C97—24no—
106C22×C97—48no—
107C22×C97—42no—
108C22×C97—42no—
109C22×C97#12112yes—
110C22×C97C22×D411yescharacteristic
111C2×D97—42no—
112C2×D97—42no—
113C2×D97—42no—
114C4×C97—21no—
115C4×C97—21no—
116C4×C97—21no—
117C4×C97—42no—
118C4×C97#12112yes—
119C4×C97AGL(1,ℤ/8)12yescomplemented
120C4×C97C22×D411yescharacteristic
121C4×C97C2×#4111yescharacteristic
122Dic97—42no—
123C2×C97—22no—
124C2×C97—44no—
125C2×C97C2×#12111yescharacteristic
126C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
127C2×C97#57511yescharacteristic
128D97—82no—
129#3036—971nomaximal, Sylow
130C97#303611yescharacteristic, complemented, Sylow
131C2×AGL(1,ℤ/8)—972no—
132#532—971no—
133C2×C4×D4D9711yescharacteristic, complemented
134#463—974no—
135#461—972no—
136#459—972no—
137#423—971no—
138#417—972no—
139C2×#41—972no—
140C22×D4C2×D9711yescharacteristic
141C23×C4—21no—
142AGL(1,ℤ/8)—1948no—
143C2×SD16—974no—
144C2×D8—974no—
145C2×M16—972no—
146C4⋊Q8—971no—
147#127—971no—
148#124—972no—
149#121—1941no—
150#121—1941no—
151C4×D4—21no—
152C4×D4—21no—
153C4×D4—1941no—
154C4×D4—1941no—
155C4×D4C2×D9712yes—
156C4×D4C2×D9712yes—
157C2×C4⋊C4C2×D9711yescharacteristic
158C2×C22⋊C4—21no—
159C2×C42C2×D9711yescharacteristic
160C4⋊C8—972no—
161C4⋊C8—972no—
162#103—974no—
163#102—974no—
164C24—21no—
165#41—1944no—
166#41—1944no—
167C2×Q8—972no—
168C2×D4—21no—
169C2×D4—22no—
170C2×D4—972no—
171C2×D4—1942no—
172C2×D4—1941no—
173C2×D4—1941no—
174C2×D4D38814yescomplemented
175C2×D4C22×D9712yes—
176C22×C4—21no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—44no—
180C22×C4—1942no—
181C22×C4C22⋊D9711yescharacteristic
182C22×C4C22⋊D9711yescharacteristic
183C22×C4C22×D9711yescharacteristic
184SD16—1948no—
185D8—1948no—
186M16—1944no—
187C2×C8—974no—
188C4⋊C4—21no—
189C4⋊C4—1941no—
190C4⋊C4—1941no—
191C4⋊C4C22×D9712yes—
192C22⋊C4—22no—
193C22⋊C4—22no—
194C22⋊C4—1942no—
195C22⋊C4—1942no—
196C42C22⋊D9711yescharacteristic
197C42C22⋊D9711yescharacteristic
198C42C22×D9711yescharacteristic
199C42C22×D9711yescharacteristic
200C23—24no—
201C23—22no—
202C23—42no—
203C23—42no—
204C23—1942no—
205C23C2×C22⋊D9711yescharacteristic
206Q8—1942no—
207Q8—1942no—
208D4—24no—
209D4—22no—
210D4—3882no—
211D4—3884no—
212D4—3882no—
213D4—1942no—
214D4—1942no—
215D4C2×D38814yes—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—41no—
219C2×C4—44no—
220C2×C4—42no—
221C2×C4—44no—
222C2×C4—3884no—
223C2×C4—1942no—
224C2×C4C2×C22⋊D9711yescharacteristic
225C2×C4C2×C22⋊D9711yescharacteristic
226C2×C4C2×C22⋊D9711yescharacteristic
227C2×C4C2×C22⋊D9711yescharacteristic
228C2×C4C2×C22⋊D9711yescharacteristic
229C2×C4C2×D38812yes—
230C2×C4#1309551112yes—
231C2×C4C23×D9711yescharacteristic
232C8—1944no—
233C22—21no—
234C22—21no—
235C22—24no—
236C22—48no—
237C22—42no—
238C22—42no—
239C22—3882no—
240C22—3882no—
241C22—3882no—
242C22#1569078412yes—
243C22C22×C22⋊D9711yescharacteristic
244C4—21no—
245C4—21no—
246C4—21no—
247C4—42no—
248C4—3882no—
249C4#1569078412yes—
250C4#1569083612yescomplemented
251C4C22×D38811yescharacteristic
252C4C2×#1309551111yescharacteristic
253C2—22no—
254C2—44no—
255C2—7762no—
256C2C2×#1569078411yescharacteristic
257C2C2×#1569083611yescharacteristic
258C2#2026153311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×C4×D4×C97C2
socleC22×C97C22×D4
hypercenterC2×C4×D4D97
hypocenterC97#3036
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#15690836C2semidirect
2. #20261285C2semidirect
3. #20260950C2semidirect
4. #20260940C2semidirect
5. #20260930C2semidirect
6. #20260759C2semidirect
7. #20260734C2semidirect
8. C2×C4×D4×C97C2semidirect
14. #15690810C22semidirect
15. #15690807C22semidirect
16. #15690798C22semidirect
21. C4⋊#11715895C22semidirect
22. C4⋊#11715895C22semidirect
23. #15690733C22semidirect
24. #15690730C22semidirect
53. C2×D4×C97D4semidirect
119. C4×C97AGL(1,ℤ/8)semidirect
130. C97#3036semidirect
133. C2×C4×D4D97semidirect
174. C2×D4D388semidirect
250. C4#15690836semidirect

Maximal subgroups

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

subgroupindex
C2×#156908362
#202612852
#202609502
#202609402
#202609302
#202607592
#202607342
C2×C4×D4×C972
#303697