Finite Group Databasebeta

← #30676425

Subgroups — #30676425

Order 16768.

572 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16862277C212yesmaximal, complemented
2C2×#16862272C212yesmaximal, complemented
3#22553862C211yesmaximal, characteristic, complemented
4#22553707C212yesmaximal, complemented
5#22553702C212yesmaximal, complemented
6#22553694C214yesmaximal, complemented
7C4×#14523497C211yesmaximal, characteristic, complemented
8C2×#124×C131C211yesmaximal, characteristic, complemented
9C2×#14523513C2212yes—
10#16862277—28no—
11#16862272—28no—
12C2×#14523503C2214yescomplemented
13C2×#14523501C2214yes—
14C2×#14523500C2214yescomplemented
15C2×#14523498C2214yes—
16C2×#14523497C2212yes—
17#16862246C2211yescharacteristic, complemented
18#16862243C2211yescharacteristic, complemented
19#16862234C2212yescomplemented
20#16862220—22no—
21C4×D524—22no—
22#16862157C2212yescomplemented
23C4×C131⋊C8C2212yescomplemented
24C22×Q8×C131C2211yescharacteristic
25C22×D4×C131C2211yescharacteristic
26#124×C131—22no—
27#124×C131C2212yes—
28#124×C131C2212yes—
29C2×C22⋊C4×C131—22no—
30C2×C42×C131C2211yescharacteristic
31#14523513—24no—
32#14523513—24no—
33C2×Dic262C2312yes—
34C2×D524—22no—
35C2×D524C2312yes—
36C2×C22⋊D131—22no—
37C2×C4×D131—22no—
38#14523503—28no—
39#14523501—28no—
40#14523500—28no—
41#14523498—28no—
42#14523497D418yescomplemented
43C2×C131⋊C8C2314yes—
44C4⋊Dic131—22no—
45(C2×D131)⋊C4—24no—
46C24×C131—21no—
47C2×Q8×C131—21no—
48C2×Q8×C131—24no—
49C2×Q8×C131C2312yes—
50C2×D4×C131—24no—
51C2×D4×C131—21no—
52C2×D4×C131C2312yes—
53C22×C4×C131—21no—
54C22×C4×C131D412yes—
55C22×C4×C131C2311yescharacteristic
56C22⋊C4×C131—24no—
57C22⋊C4×C131—24no—
58C42×C131D412yes—
59C42×C131C2311yescharacteristic
60C42×C131C2311yescharacteristic
61C22×D131—22no—
62Dic262—22no—
63Dic262—22no—
64D524—22no—
65D524—22no—
66D524—44no—
67C22⋊D131—44no—
68C2×Dic131—22no—
69C4×D131—44no—
70C131⋊C8C2×D418yes—
71C23×C131—22no—
72C23×C131—44no—
73C23×C131—42no—
74C23×C131C2×D411yescharacteristic
75Q8×C131—24no—
76Q8×C131—42no—
77D4×C131—24no—
78D4×C131—42no—
79C2×C4×C131—22no—
80C2×C4×C131—42no—
81C2×C4×C131—42no—
82C2×C4×C131C2×D414yes—
83C2×C4×C131C2×D412yes—
84C2×C4×C131C2×D412yes—
85C2×C4×C131C2×D411yescharacteristic
86C2×C4×C131C2411yescharacteristic
87C22×C131—22no—
88C22×C131—42no—
89C22×C131—44no—
90C22×C131—84no—
91C22×C131#12712yes—
92C22×C131C22×D411yescharacteristic
93C2×D131—42no—
94C2×D131—44no—
95C4×C131—22no—
96C4×C131—22no—
97C4×C131—42no—
98C4×C131#12712yes—
99C4×C131C22×D412yes—
100Dic131—42no—
101C2×C131—22no—
102C2×C131—82no—
103C2×C131C2×#12711yescharacteristic
104C2×C131#57512yes—
105D131—82no—
106C131#315311yescharacteristic, complemented, Sylow
107#3153—1311nomaximal, Sylow
108C2×#137—1312no—
109C2×AGL(1,ℤ/8)—1312no—
110#532—1311no—
111C2×#124D13111yescharacteristic, complemented
112#497—1312no—
113#496—1312no—
114#495—1314no—
115C4×M16—1311no—
116C2×#41—1312no—
117C22×Q8C2×D13111yescharacteristic
118C22×D4C2×D13111yescharacteristic
119#137—2628no—
120AGL(1,ℤ/8)—2628no—
121C2×Q16—1314no—
122C2×SD16—1314no—
123C2×SD16—1314no—
124C2×D8—1314no—
125C2×M16—1312no—
126C4⋊Q8—1311no—
127#127—1311no—
128#124—22no—
129#124—1312no—
130#124C2×D13112yes—
131#124C2×D13112yes—
132#121—2622no—
133C4×D4—2622no—
134C2×C22⋊C4—22no—
135C2×C42C2×D13111yescharacteristic
136#97—1312no—
137C4×C8—1312no—
138C24—21no—
139#41—2624no—
140#41—2624no—
141C2×Q8—21no—
142C2×Q8—24no—
143C2×Q8—1312no—
144C2×Q8C22×D13112yes—
145C2×D4—24no—
146C2×D4—21no—
147C2×D4—2622no—
148C2×D4—2622no—
149C2×D4—1312no—
150C2×D4C22×D13112yes—
151C22×C4—21no—
152C22×C4—2622no—
153C22×C4C22⋊D13112yes—
154C22×C4C22×D13111yescharacteristic
155Q16—2628no—
156SD16—2628no—
157SD16—2628no—
158D8—2628no—
159M16—1318no—
160C2×C8—1314no—
161C4⋊C4—2622no—
162C22⋊C4—24no—
163C22⋊C4—24no—
164C22⋊C4—2624no—
165C42C22⋊D13112yes—
166C42C22×D13111yescharacteristic
167C42C22×D13111yescharacteristic
168C23—22no—
169C23—44no—
170C23—42no—
171C23—2622no—
172C23C2×C22⋊D13111yescharacteristic
173Q8—24no—
174Q8—42no—
175Q8—2622no—
176Q8—2622no—
177D4—24no—
178D4—42no—
179D4—2622no—
180D4—2622no—
181D4—5244no—
182D4—5244no—
183C2×C4—22no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—2622no—
187C2×C4—5244no—
188C2×C4D4×D13114yes—
189C2×C4C2×C22⋊D13112yes—
190C2×C4C2×C22⋊D13112yes—
191C2×C4C2×C22⋊D13111yescharacteristic
192C2×C4C23×D13111yescharacteristic
193C8—1318no—
194C22—22no—
195C22—42no—
196C22—44no—
197C22—84no—
198C22—5242no—
199C22—5244no—
200C22#1686224212yes—
201C22C22×C22⋊D13111yescharacteristic
202C4—22no—
203C4—22no—
204C4—42no—
205C4—5242no—
206C4#1686224212yes—
207C4C2×D4×D13112yes—
208C2—22no—
209C2—82no—
210C2—10482no—
211C2C2×#1686224211yescharacteristic
212C2#2255411012yes—
subgroupG/H
centerC22C22×C22⋊D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
FittingC2×#124×C131C2
socleC22×C131C22×D4
hypercenterC2×#124D131
hypocenterC131#3153
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#16862277C2semidirect
2. C2×#16862272C2semidirect
3. #22553862C2semidirect
4. #22553707C2semidirect
5. #22553702C2semidirect
6. #22553694C2semidirect
7. C4×#14523497C2semidirect
8. C2×#124×C131C2semidirect
12. C2×#14523503C22semidirect
14. C2×#14523500C22semidirect
17. #16862246C22semidirect
18. #16862243C22semidirect
19. #16862234C22semidirect
22. #16862157C22semidirect
23. C4×C131⋊C8C22semidirect
42. #14523497D4semidirect
106. C131#3153semidirect
111. C2×#124D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622772
C2×#168622722
#225538622
#225537072
#225537022
#225536942
C4×#145234972
C2×#124×C1312
#3153131