Finite Group Databasebeta

← C2×#20260933 #27151166

Subgroups — C2×#20260933 #27151166

Order 12416.

584 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 165 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
1C22×#13095498C211yesmaximal, characteristic, complemented
2C2×#15690806C211yesmaximal, characteristic
3C2×Q8×Dic97C211yesmaximal, characteristic, complemented
4#20260933C218yesmaximal, direct product factor
5C2×#15690739C211yesmaximal, characteristic, complemented
6C2×#15690732C211yesmaximal, characteristic, complemented
7C2×#15690731C211yesmaximal, characteristic, complemented
8C22×SD16×C97C211yesmaximal, characteristic, complemented
9C22×D388C2211yescharacteristic
10C22×C22⋊D97—21no—
11C22×C22⋊D97—21no—
12C2×#13095498—24no—
13C2×#13095498C2214yescomplemented
14C22×#11715895C2211yescharacteristic
15#15690806—22no—
16#15690806C2214yes—
17Q8×Dic97—22no—
18Q8×Dic97C2214yescomplemented
19C2×C4⋊Dic97—21no—
20C2×C4⋊Dic97C2211yescharacteristic
21C2×C4×Dic97—21no—
22C2×C4×Dic97C2211yescharacteristic
23#15690739C2214yescomplemented
24#15690732C2214yescomplemented
25#15690731C2214yescomplemented
26C22×Q8×C97C2211yescharacteristic
27C22×D4×C97C2211yescharacteristic
28C2×SD16×C97—24no—
29C2×SD16×C97C2214yescomplemented
30C22×C8×C97C2211yescharacteristic
31C23×D97—21no—
32C2×D388—22no—
33C2×D388—22no—
34C2×D388—21no—
35C2×D388C2312yes—
36C2×C22⋊D97—24no—
37C2×C22⋊D97—24no—
38C2×C22⋊D97—44no—
39C2×C22⋊D97—44no—
40C22×Dic97—21no—
41C22×Dic97D411yescharacteristic
42C22×Dic97D411yescharacteristic
43#13095498—28no—
44C2×#11715895—22no—
45C2×#11715895C2312yes—
46C4⋊Dic97—21no—
47C4⋊Dic97—24no—
48C4⋊Dic97C2312yes—
49C4×Dic97—24no—
50C4×Dic97D412yes—
51C4×Dic97C2312yes—
52C24×C97—21no—
53C2×Q8×C97—21no—
54C2×Q8×C97—22no—
55C2×Q8×C97D414yescomplemented
56C2×Q8×C97C2312yes—
57C2×D4×C97—22no—
58C2×D4×C97—21no—
59C2×D4×C97—22no—
60C2×D4×C97C2312yes—
61C22×C4×C97—21no—
62C22×C4×C97C2311yescharacteristic
63SD16×C97—28no—
64C2×C8×C97—22no—
65C2×C8×C97C2312yes—
66C23×C97—22no—
67C23×C97—42no—
68C23×C97—42no—
69C23×C97—42no—
70C23×C97C2×D411yescharacteristic
71C22×D97—22no—
72C22×D97—42no—
73C22×D97—42no—
74C22×D97—42no—
75Q8×C97—24no—
76Q8×C97—22no—
77Q8×C97C2×D414yescomplemented
78D4×C97—22no—
79D4×C97—22no—
80D4×C97—42no—
81C2×C4×C97—24no—
82C2×C4×C97—22no—
83C2×C4×C97C2×D411yescharacteristic
84C2×C4×C97C2×D412yes—
85C2×C4×C97#4112yes—
86C2×C4×C97C2411yescharacteristic
87C22⋊D97—48no—
88C22⋊D97—48no—
89C2×Dic97—22no—
90C2×Dic97—22no—
91C2×Dic97—42no—
92C2×Dic97SD1614yescomplemented
93C2×Dic97C2×D412yes—
94C2×Dic97C2×D412yes—
95D388—22no—
96D388—22no—
97D388—42no—
98C8×C97—22no—
99#11715895—22no—
100C22×C97—42no—
101C22×C97—42no—
102C22×C97—42no—
103C22×C97—84no—
104C22×C97#12112yes—
105C22×C97C2×SD1612yes—
106C22×C97AGL(1,ℤ/8)12yes—
107C22×C97C22×D411yescharacteristic
108C2×D97—42no—
109C2×D97—42no—
110C2×D97—42no—
111C2×D97—84no—
112C4×C97—24no—
113C4×C97#12112yes—
114C4×C97C22×D411yescharacteristic
115C4×C97C2×#4111yescharacteristic
116Dic97—22no—
117Dic97—42no—
118Dic97C2×SD1614yescomplemented
119C2×C97—82no—
120C2×C97#46014yescomplemented
121C2×C97C2×#12111yescharacteristic
122C2×C97C22×SD1611yescharacteristic
123C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
124D97—82no—
125C2×#460—971nomaximal, Sylow
126C97C2×#46011yescharacteristic, complemented, Sylow
127C22×SD16—971no—
128C22×SD16D9711yescharacteristic, complemented
129C2×#127—971no—
130C2×C4×Q8—971no—
131#460—978no—
132C2×C4⋊C8—971no—
133C2×#102—971no—
134C2×#102—971no—
135C22×Q8C2×D9711yescharacteristic
136C22×D4—971no—
137C22×D4—1941no—
138C22×D4—1941no—
139C22×D4C2×D9711yescharacteristic
140C2×SD16—24no—
141C2×SD16—974no—
142C2×SD16—1944no—
143C2×SD16C2×D9714yescomplemented
144C22×C8—971no—
145C22×C8C2×D9711yescharacteristic
146#127—974no—
147#127—1942no—
148C4×Q8—974no—
149C4×Q8—1942no—
150C2×C4⋊C4—971no—
151C2×C4⋊C4—1941no—
152C2×C42—971no—
153C2×C42—1941no—
154C4⋊C8—974no—
155#102—974no—
156#102—974no—
157C24—21no—
158C24—1941no—
159C2×Q8—21no—
160C2×Q8—22no—
161C2×Q8C22⋊D9714yescomplemented
162C2×Q8C22×D9712yes—
163C2×D4—21no—
164C2×D4—22no—
165C2×D4—22no—
166C2×D4—972no—
167C2×D4—3884no—
168C2×D4—3884no—
169C2×D4—1944no—
170C2×D4—1944no—
171C2×D4—1941no—
172C2×D4—1942no—
173C2×D4—1942no—
174C2×D4C22×D9712yes—
175C22×C4—21no—
176C22×C4—971no—
177C22×C4—971no—
178C22×C4—1941no—
179C22×C4C22×D9711yescharacteristic
180SD16—28no—
181SD16—1948no—
182C2×C8—22no—
183C2×C8—972no—
184C2×C8—1942no—
185C2×C8C22×D9712yes—
186C4⋊C4—972no—
187C4⋊C4—1944no—
188C4⋊C4—1941no—
189C42—972no—
190C42—972no—
191C42—1944no—
192C23—22no—
193C23—42no—
194C23—42no—
195C23—42no—
196C23—3882no—
197C23—3882no—
198C23—3882no—
199C23—1942no—
200C23D4×D9711yescharacteristic
201Q8—24no—
202Q8—22no—
203Q8C2×C22⋊D9714yescomplemented
204D4—22no—
205D4—22no—
206D4—42no—
207D4—3888no—
208D4—3888no—
209D4—3882no—
210D4—1942no—
211D4—1942no—
212C2×C4—24no—
213C2×C4—22no—
214C2×C4—974no—
215C2×C4—972no—
216C2×C4—972no—
217C2×C4—3882no—
218C2×C4—1942no—
219C2×C4—1942no—
220C2×C4D4×D9711yescharacteristic
221C2×C4C2×C22⋊D9712yes—
222C2×C4#1309551012yes—
223C2×C4C23×D9711yescharacteristic
224C8—22no—
225C8—1942no—
226C22—42no—
227C22—42no—
228C22—42no—
229C22—84no—
230C22—7764no—
231C22—3882no—
232C22—3882no—
233C22—3882no—
234C22#1569078212yes—
235C22SD16×D9712yes—
236C22#1569083712yes—
237C22C2×D4×D9711yescharacteristic
238C4—24no—
239C4—974no—
240C4—3882no—
241C4—1942no—
242C4#1569078212yes—
243C4C22×C22⋊D9711yescharacteristic
244C4C2×#1309551011yescharacteristic
245C2—82no—
246C2—7762no—
247C2#2026093314yesdirect product factor
248C2C2×#1569078211yescharacteristic
249C2C2×SD16×D9711yescharacteristic
250C2C2×#1569083711yescharacteristic
subgroupG/H
centerC23D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC22×SD16×C97C2
socleC23×C97C2×D4
prime radicalC2×#15690806C2
hypercenterC22×SD16D97
hypocenterC97C2×#460
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C22×#13095498C2semidirect
3. C2×Q8×Dic97C2semidirect
4. #20260933C2direct (N × H)
5. C2×#15690739C2semidirect
6. C2×#15690732C2semidirect
7. C2×#15690731C2semidirect
8. C22×SD16×C97C2semidirect
13. C2×#13095498C22semidirect
18. Q8×Dic97C22semidirect
23. #15690739C22semidirect
24. #15690732C22semidirect
25. #15690731C22semidirect
29. C2×SD16×C97C22semidirect
55. C2×Q8×C97D4semidirect
77. Q8×C97C2×D4semidirect
92. C2×Dic97SD16semidirect
118. Dic97C2×SD16semidirect
120. C2×C97#460semidirect
126. C97C2×#460semidirect
128. C22×SD16D97semidirect
143. C2×SD16C2×D97semidirect
161. C2×Q8C22⋊D97semidirect
203. Q8C2×C22⋊D97semidirect
247. C2#20260933direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130954982
C2×#156908062
C2×Q8×Dic972
#202609332
C2×#156907392
C2×#156907322
C2×#156907312
C22×SD16×C972
C2×#46097