Finite Group Databasebeta

← #27151214

Subgroups — #27151214

Order 12416.

584 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 145 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, complemented
2C2×#15690806C211yesmaximal, characteristic, complemented
3#41×Dic97C211yesmaximal, characteristic, complemented
4#20260933C214yesmaximal, complemented
5#20260928C214yesmaximal, complemented
6#20260758C211yesmaximal, characteristic, complemented
7#20260741C211yesmaximal, characteristic, complemented
8#20260740C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C97C211yesmaximal, characteristic, complemented
10C22×D388C2211yescharacteristic
11C22×C22⋊D97—22no—
12#15690835—24no—
13C2×#13095498C2212yescomplemented
14C2×#13095496C2212yescomplemented
15C2×#13095495C2211yescharacteristic
16#15690806—21no—
17#15690806—21no—
18#15690806C2214yes—
19Q8×Dic97C2212yescomplemented
20D4×Dic97—22no—
21D4×Dic97C2212yescomplemented
22#15690769—21no—
23#15690769C2211yescharacteristic
24C2×C4×Dic97—21no—
25C2×C4×Dic97C2211yescharacteristic
26#15690739C2214yescomplemented
27#15690732C2214yescomplemented
28#15690731C2214yescomplemented
29C2×#41×C97C2211yescharacteristic
30C22×D4×C97C2211yescharacteristic
31AGL(1,ℤ/8)×C97—24no—
32C2×SD16×C97C2212yescomplemented
33C2×D8×C97C2212yescomplemented
34C2×M16×C97C2211yescharacteristic
35C23×D97—21no—
36C2×D388—22no—
37C2×D388—22no—
38C2×D388—21no—
39C2×D388C2312yes—
40C2×C22⋊D97—24no—
41C2×C22⋊D97—24no—
42C2×C22⋊D97—48no—
43C22×Dic97—22no—
44C22×Dic97D412yes—
45#13095498—24no—
46#13095496—24no—
47#13095495—22no—
48C2×#11715895C2312yes—
49C4⋊Dic97—22no—
50C4⋊Dic97C2312yes—
51C22⋊Dic97—21no—
52C22⋊Dic97—22no—
53C4×Dic97—21no—
54C4×Dic97—22no—
55C4×Dic97D411yescharacteristic
56C4×Dic97D411yescharacteristic
57C4×Dic97C2312yes—
58C24×C97—21no—
59#41×C97—22no—
60#41×C97D414yescomplemented
61C2×Q8×C97C2311yescharacteristic
62C2×D4×C97—21no—
63C2×D4×C97—22no—
64C2×D4×C97—21no—
65C2×D4×C97—22no—
66C2×D4×C97C2312yes—
67C2×D4×C97C2311yescharacteristic
68C22×C4×C97—21no—
69C22×C4×C97C2311yescharacteristic
70SD16×C97—24no—
71D8×C97—24no—
72M16×C97—22no—
73C2×C8×C97C2312yes—
74C23×C97—21no—
75C23×C97—22no—
76C23×C97—44no—
77C23×C97—42no—
78C23×C97C2×D411yescharacteristic
79C22×D97—22no—
80C22×D97—44no—
81C22×D97—42no—
82Q8×C97—21no—
83Q8×C97C2×D412yescomplemented
84D4×C97—24no—
85D4×C97—22no—
86D4×C97—21no—
87D4×C97—22no—
88D4×C97—42no—
89D4×C97C2×D412yescomplemented
90C2×C4×C97—21no—
91C2×C4×C97—24no—
92C2×C4×C97C2×D411yescharacteristic
93C2×C4×C97C2×D412yes—
94C2×C4×C97#4112yes—
95C2×C4×C97C2411yescharacteristic
96C22⋊D97—48no—
97C22⋊D97—48no—
98C2×Dic97—24no—
99C2×Dic97—22no—
100C2×Dic97—44no—
101C2×Dic97C2×D412yes—
102C2×Dic97C2×D412yes—
103D388—22no—
104D388—22no—
105D388—42no—
106C8×C97—22no—
107#11715895—22no—
108C22×C97—22no—
109C22×C97—22no—
110C22×C97—42no—
111C22×C97—42no—
112C22×C97—44no—
113C22×C97—84no—
114C22×C97#12112yes—
115C22×C97C22×D411yescharacteristic
116C2×D97—42no—
117C2×D97—44no—
118C2×D97—84no—
119C4×C97—22no—
120C4×C97#12112yes—
121C4×C97C22×D411yescharacteristic
122C4×C97C2×#4111yescharacteristic
123Dic97—22no—
124Dic97—42no—
125Dic97AGL(1,ℤ/8)14yescomplemented
126C2×C97—22no—
127C2×C97—42no—
128C2×C97—82no—
129C2×C97C2×#12111yescharacteristic
130C2×C97C2×AGL(1,ℤ/8)12yes—
131D97—82no—
132#3038—971nomaximal, Sylow
133C97#303811yescharacteristic, complemented, Sylow
134C2×AGL(1,ℤ/8)—971no—
135C2×AGL(1,ℤ/8)D9711yescharacteristic, complemented
136C2×#127—971no—
137C4×#41—971no—
138#460—974no—
139#459—974no—
140#423—971no—
141#418—971no—
142#418—971no—
143C2×#41C2×D9711yescharacteristic
144C22×D4—971no—
145C22×D4—1942no—
146C22×D4C2×D9711yescharacteristic
147AGL(1,ℤ/8)—24no—
148AGL(1,ℤ/8)—1944no—
149C2×SD16—972no—
150C2×SD16C2×D9712yescomplemented
151C2×D8—972no—
152C2×D8C2×D9712yescomplemented
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—971no—
162#117—1941no—
163C2×C42—971no—
164C2×C42—1941no—
165C4⋊C8—974no—
166#102—974no—
167#102—974no—
168C24—21no—
169C24—1941no—
170#41—22no—
171#41C22⋊D9714yescomplemented
172C2×Q8C22×D9711yescharacteristic
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—972no—
178C2×D4—3888no—
179C2×D4—1944no—
180C2×D4—1944no—
181C2×D4—1941no—
182C2×D4—1942no—
183C2×D4—1942no—
184C2×D4C22×D9712yes—
185C2×D4C22×D9711yescharacteristic
186C22×C4—21no—
187C22×C4—972no—
188C22×C4—1942no—
189C22×C4C22×D9711yescharacteristic
190SD16—24no—
191SD16—1944no—
192D8—24no—
193D8—1944no—
194M16—22no—
195M16—1942no—
196C2×C8—972no—
197C2×C8C22×D9712yes—
198C4⋊C4—972no—
199C4⋊C4—1942no—
200C22⋊C4—1942no—
201C22⋊C4—1941no—
202C42—971no—
203C42—971no—
204C42—972no—
205C42—1942no—
206C42—1941no—
207C23—21no—
208C23—22no—
209C23—44no—
210C23—42no—
211C23—3884no—
212C23—3882no—
213C23—1942no—
214C23D4×D9711yescharacteristic
215Q8—21no—
216Q8C2×C22⋊D9712yescomplemented
217D4—24no—
218D4—22no—
219D4—21no—
220D4—22no—
221D4—42no—
222D4—3888no—
223D4—3888no—
224D4—3882no—
225D4—1942no—
226D4—1942no—
227D4C2×C22⋊D9712yescomplemented
228C2×C4—21no—
229C2×C4—24no—
230C2×C4—972no—
231C2×C4—972no—
232C2×C4—3884no—
233C2×C4—1944no—
234C2×C4—1942no—
235C2×C4D4×D9711yescharacteristic
236C2×C4C2×C22⋊D9712yes—
237C2×C4#1309551012yes—
238C2×C4C23×D9711yescharacteristic
239C8—22no—
240C8—1942no—
241C22—22no—
242C22—22no—
243C22—42no—
244C22—42no—
245C22—44no—
246C22—84no—
247C22—7764no—
248C22—3882no—
249C22—3884no—
250C22#1569078212yes—
251C22C2×D4×D9711yescharacteristic
252C4—22no—
253C4—974no—
254C4—3882no—
255C4—1942no—
256C4#1569078212yes—
257C4C22×C22⋊D9711yescharacteristic
258C4C2×#1309551011yescharacteristic
259C2—22no—
260C2—42no—
261C2—82no—
262C2—7762no—
263C2C2×#1569078211yescharacteristic
264C2AGL(1,ℤ/8)×D9712yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×AGL(1,ℤ/8)×C97C2
socleC22×C97C22×D4
hypercenterC2×AGL(1,ℤ/8)D97
hypocenterC97#3038
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#15690835C2semidirect
2. C2×#15690806C2semidirect
3. #41×Dic97C2semidirect
4. #20260933C2semidirect
5. #20260928C2semidirect
6. #20260758C2semidirect
7. #20260741C2semidirect
8. #20260740C2semidirect
9. C2×AGL(1,ℤ/8)×C97C2semidirect
13. C2×#13095498C22semidirect
14. C2×#13095496C22semidirect
19. Q8×Dic97C22semidirect
21. D4×Dic97C22semidirect
26. #15690739C22semidirect
27. #15690732C22semidirect
28. #15690731C22semidirect
32. C2×SD16×C97C22semidirect
33. C2×D8×C97C22semidirect
60. #41×C97D4semidirect
83. Q8×C97C2×D4semidirect
89. D4×C97C2×D4semidirect
125. Dic97AGL(1,ℤ/8)semidirect
133. C97#3038semidirect
135. C2×AGL(1,ℤ/8)D97semidirect
150. C2×SD16C2×D97semidirect
152. C2×D8C2×D97semidirect
171. #41C22⋊D97semidirect
216. Q8C2×C22⋊D97semidirect
227. D4C2×C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908352
C2×#156908062
#41×Dic972
#202609332
#202609282
#202607582
#202607412
#202607402
C2×AGL(1,ℤ/8)×C972
#303897