Finite Group Databasebeta

← #27151215

Subgroups — #27151215

Order 12416.

584 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 137 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×#15690838C211yesmaximal, characteristic, complemented
2C2×#15690836C211yesmaximal, characteristic, complemented
3C2×#15690806C211yesmaximal, characteristic, complemented
4#20261203C211yesmaximal, characteristic, complemented
5#20260934C214yesmaximal, complemented
6#20260929C214yesmaximal, complemented
7#20260758C211yesmaximal, characteristic, complemented
8#20260739C211yesmaximal, characteristic, complemented
9#418×C97C211yesmaximal, characteristic, complemented
10C2×#13095511C2211yescharacteristic
11C22×D388C2211yescharacteristic
12C22×C22⋊D97—22no—
13#15690838—24no—
14#15690836—24no—
15C2×#13095500C2212yescomplemented
16C2×#13095499C2212yescomplemented
17C2×D776C2212yescomplemented
18C2×#13095496C2212yescomplemented
19C2×#13095495C2211yescharacteristic
20#15690806—21no—
21#15690806—21no—
22#15690806C2214yes—
23#15690775C2212yescomplemented
24#15690772C2212yescomplemented
25#15690770—22no—
26#15690767—21no—
27C2×C4×Dic97C2211yescharacteristic
28C42×D97—21no—
29#15690739C2214yescomplemented
30#15690730C2214yescomplemented
31C22×D4×C97C2211yescharacteristic
32C2×M16×C97C2211yescharacteristic
33#117×C97C2211yescharacteristic
34#102×C97C2214yescomplemented
35C23×D97—21no—
36#13095511—22no—
37#13095511D414yescomplemented
38C2×Dic194C2311yescharacteristic
39C2×D388—22no—
40C2×D388—22no—
41C2×D388—21no—
42C2×D388C2312yes—
43C2×D388C2311yescharacteristic
44C2×C22⋊D97—21no—
45C2×C22⋊D97—24no—
46C2×C22⋊D97—24no—
47C2×C22⋊D97—48no—
48C22×Dic97D412yes—
49C2×C4×D97—21no—
50C2×C4×D97—22no—
51#13095500—24no—
52#13095499—24no—
53D776—24no—
54#13095496—24no—
55#13095495—22no—
56C2×#11715895C2312yes—
57#13095490—22no—
58#13095488—22no—
59C4×Dic97—22no—
60C4×Dic97D411yescharacteristic
61C4×Dic97D411yescharacteristic
62C4×Dic97C2312yes—
63C24×C97—21no—
64C2×D4×C97—22no—
65C2×D4×C97—21no—
66C2×D4×C97—22no—
67C2×D4×C97C2312yes—
68C22×C4×C97C2311yescharacteristic
69M16×C97—22no—
70C2×C8×C97C2312yes—
71C4⋊C4×C97C2312yes—
72C22⋊C4×C97—21no—
73C42×C97—21no—
74C23×C97—22no—
75C23×C97—44no—
76C23×C97—42no—
77C23×C97C2×D411yescharacteristic
78C22×D97—21no—
79C22×D97—22no—
80C22×D97—44no—
81C22×D97—42no—
82D4×C97—22no—
83D4×C97—22no—
84D4×C97—42no—
85C2×C4×C97—22no—
86C2×C4×C97C2×D411yescharacteristic
87C2×C4×C97C2×D412yes—
88C2×C4×C97#4112yes—
89C2×C4×C97C2411yescharacteristic
90C22⋊D97—24no—
91C22⋊D97—48no—
92C22⋊D97—48no—
93C2×Dic97—21no—
94C2×Dic97—24no—
95C2×Dic97C2×D412yes—
96C2×Dic97C2×D412yes—
97D388—22no—
98D388—21no—
99D388—22no—
100D388—42no—
101D388C2×D412yescomplemented
102C4×D97—24no—
103C4×D97—44no—
104Dic194—21no—
105Dic194C2×D412yescomplemented
106C8×C97—22no—
107#11715895—22no—
108C22×C97—22no—
109C22×C97—42no—
110C22×C97—44no—
111C22×C97—84no—
112C22×C97#12112yes—
113C22×C97C22×D411yescharacteristic
114C2×D97—22no—
115C2×D97—42no—
116C2×D97—42no—
117C2×D97—44no—
118C2×D97—84no—
119C4×C97—42no—
120C4×C97#12112yes—
121C4×C97C22×D411yescharacteristic
122C4×C97C2×#4111yescharacteristic
123Dic97—22no—
124Dic97—22no—
125Dic97AGL(1,ℤ/8)14yescomplemented
126C2×C97—22no—
127C2×C97—82no—
128C2×C97C2×#12111yescharacteristic
129C2×C97C2×AGL(1,ℤ/8)12yes—
130D97—42no—
131D97—82no—
132#3038—971nomaximal, Sylow
133C97#303811yescharacteristic, complemented, Sylow
134C2×AGL(1,ℤ/8)—971no—
135C2×AGL(1,ℤ/8)—971no—
136C2×#127—971no—
137C4×#41—971no—
138#460—974no—
139#459—974no—
140#423—971no—
141#418—971no—
142#418D9711yescharacteristic, complemented
143C2×#41—971no—
144C22×D4—971no—
145C22×D4—1942no—
146C22×D4C2×D9711yescharacteristic
147AGL(1,ℤ/8)—1944no—
148AGL(1,ℤ/8)—1944no—
149C2×SD16—972no—
150C2×SD16—972no—
151C2×D8—972no—
152C2×D8—972no—
153C2×M16—971no—
154C2×M16C2×D9711yescharacteristic
155#127—974no—
156#127—1941no—
157#127—1941no—
158C4×Q8—972no—
159C4×D4—972no—
160C4×D4—1942no—
161#117—1941no—
162#117C2×D9711yescharacteristic
163C2×C42—971no—
164C2×C42—1941no—
165C4⋊C8—974no—
166#102—974no—
167#102C2×D9714yescomplemented
168C24—21no—
169C24—1941no—
170#41—974no—
171#41—1942no—
172C2×Q8—971no—
173C2×D4—22no—
174C2×D4—21no—
175C2×D4—22no—
176C2×D4—971no—
177C2×D4—972no—
178C2×D4—3888no—
179C2×D4—1944no—
180C2×D4—1944no—
181C2×D4—1941no—
182C2×D4—1941no—
183C2×D4—1942no—
184C2×D4—1942no—
185C2×D4C22×D9712yes—
186C22×C4—972no—
187C22×C4—1942no—
188C22×C4—1941no—
189C22×C4C22×D9711yescharacteristic
190SD16—1944no—
191SD16—1944no—
192D8—1944no—
193D8—1944no—
194M16—22no—
195M16—1942no—
196C2×C8—972no—
197C2×C8C22×D9712yes—
198C4⋊C4—1942no—
199C4⋊C4C22×D9712yes—
200C22⋊C4—21no—
201C22⋊C4—1942no—
202C42—21no—
203C42—971no—
204C42—971no—
205C42—972no—
206C42—1942no—
207C23—22no—
208C23—44no—
209C23—42no—
210C23—3884no—
211C23—3882no—
212C23—1942no—
213C23—1941no—
214C23D4×D9711yescharacteristic
215Q8—972no—
216Q8—1941no—
217D4—22no—
218D4—22no—
219D4—42no—
220D4—972no—
221D4—3888no—
222D4—3888no—
223D4—3882no—
224D4—1944no—
225D4—1942no—
226D4—1942no—
227D4—1941no—
228C2×C4—22no—
229C2×C4—972no—
230C2×C4—972no—
231C2×C4—3884no—
232C2×C4—1944no—
233C2×C4—1941no—
234C2×C4—1944no—
235C2×C4D4×D9711yescharacteristic
236C2×C4D4×D9712yes—
237C2×C4#1309551112yes—
238C2×C4C23×D9711yescharacteristic
239C8—22no—
240C8—1942no—
241C22—22no—
242C22—42no—
243C22—44no—
244C22—84no—
245C22—7764no—
246C22—3882no—
247C22—3884no—
248C22—3882no—
249C22—1942no—
250C22#1569078312yes—
251C22C2×D4×D9711yescharacteristic
252C4—42no—
253C4—974no—
254C4—1942no—
255C4—1942no—
256C4#1569078312yes—
257C4C2×D4×D9711yescharacteristic
258C4C2×#1309551111yescharacteristic
259C2—22no—
260C2—82no—
261C2—7762no—
262C2—3882no—
263C2C2×#1569078311yescharacteristic
264C2AGL(1,ℤ/8)×D9712yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#418×C97C2
socleC22×C97C22×D4
hypercenter#418D97
hypocenterC97#3038
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×#15690838C2semidirect
2. C2×#15690836C2semidirect
3. C2×#15690806C2semidirect
4. #20261203C2semidirect
5. #20260934C2semidirect
6. #20260929C2semidirect
7. #20260758C2semidirect
8. #20260739C2semidirect
9. #418×C97C2semidirect
15. C2×#13095500C22semidirect
16. C2×#13095499C22semidirect
17. C2×D776C22semidirect
18. C2×#13095496C22semidirect
23. #15690775C22semidirect
24. #15690772C22semidirect
29. #15690739C22semidirect
30. #15690730C22semidirect
34. #102×C97C22semidirect
37. #13095511D4semidirect
101. D388C2×D4semidirect
105. Dic194C2×D4semidirect
125. Dic97AGL(1,ℤ/8)semidirect
133. C97#3038semidirect
142. #418D97semidirect
167. #102C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908382
C2×#156908362
C2×#156908062
#202612032
#202609342
#202609292
#202607582
#202607392
#418×C972
#303897