Finite Group Databasebeta

← #30676427

Subgroups — #30676427

Order 16768.

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

normal subgroups only

This group has 572 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
1C2×#16862279C212yesmaximal, complemented
2C2×#16862274C212yesmaximal, complemented
3#22553862C211yesmaximal, characteristic, complemented
4C2×#16862234C211yesmaximal, characteristic, complemented
5#22553706C212yesmaximal, complemented
6#22553701C212yesmaximal, complemented
7#22553696C214yesmaximal, complemented
8C4×M16×C131C211yesmaximal, characteristic, complemented
9C2×#14523513C2212yes—
10C22×Dic262C2211yescharacteristic
11C22×D524C2211yescharacteristic
12#16862279—28no—
13#16862274—28no—
14C2×Dic524C2214yescomplemented
15C2×#14523502C2214yes—
16C2×#14523502C2214yescomplemented
17C2×D1048C2214yes—
18#16862246C2211yescharacteristic, complemented
19#16862243C2211yescharacteristic, complemented
20#16862234—22no—
21#16862234C2212yes—
22#16862234C2212yes—
23#16862234C2212yescomplemented
24#16862220—22no—
25C4×D524—22no—
26C2×(C2×D131)⋊C4—22no—
27C2×M16×C131C2212yes—
28C2×C42×C131C2211yescharacteristic
29#97×C131C2212yescomplemented
30C4×C8×C131C2212yescomplemented
31C23×D131—21no—
32#14523513—24no—
33#14523513—24no—
34C2×Dic262—21no—
35C2×Dic262—24no—
36C2×Dic262C2312yes—
37C2×Dic262C2312yes—
38C2×D524—24no—
39C2×D524—22no—
40C2×D524—21no—
41C2×D524C2312yes—
42C2×D524C2312yes—
43C2×C22⋊D131—22no—
44C22×Dic131—21no—
45C2×C4×D131—22no—
46Dic524—28no—
47#14523502—28no—
48#14523502—28no—
49D1048—28no—
50C4⋊Dic131—22no—
51(C2×D131)⋊C4—24no—
52(C2×D131)⋊C4—24no—
53(C2×D131)⋊C4—24no—
54C22×C4×C131D412yes—
55C22×C4×C131C2311yescharacteristic
56M16×C131D418yescomplemented
57C2×C8×C131C2314yes—
58C42×C131D412yes—
59C42×C131C2311yescharacteristic
60C42×C131C2311yescharacteristic
61C22×D131—22no—
62C22×D131—22no—
63C22×D131—44no—
64C22×D131—42no—
65Dic262—22no—
66Dic262—24no—
67Dic262—22no—
68Dic262—42no—
69D524—22no—
70D524—24no—
71D524—22no—
72D524—44no—
73D524—42no—
74C22⋊D131—44no—
75C2×Dic131—22no—
76C2×Dic131—22no—
77C2×Dic131—42no—
78C4×D131—44no—
79C23×C131C2×D411yescharacteristic
80C2×C4×C131—42no—
81C2×C4×C131C2×D412yes—
82C2×C4×C131C2×D412yes—
83C2×C4×C131C2×D411yescharacteristic
84C2×C4×C131C2×D414yes—
85C2×C4×C131C2411yescharacteristic
86C8×C131C2×D418yes—
87C22×C131—22no—
88C22×C131#12712yes—
89C22×C131C22×D411yescharacteristic
90C2×D131—42no—
91C2×D131—42no—
92C2×D131—44no—
93C2×D131—44no—
94C2×D131—84no—
95C4×C131—22no—
96C4×C131—22no—
97C4×C131#12712yes—
98C4×C131C22×D412yes—
99Dic131—42no—
100Dic131—42no—
101C2×C131—22no—
102C2×C131C2×#12711yescharacteristic
103C2×C131#57512yes—
104D131—82no—
105D131—82no—
106C131#315311yescharacteristic, complemented, Sylow
107#3153—1311nomaximal, Sylow
108C2×#137—1312no—
109C2×AGL(1,ℤ/8)—1312no—
110#532—1311no—
111C2×#124—1311no—
112#497—1312no—
113#496—1312no—
114#495—1314no—
115C4×M16D13111yescharacteristic, complemented
116C2×#41—1312no—
117C22×Q8—1311no—
118C22×D4—1311no—
119#137—2628no—
120AGL(1,ℤ/8)—2628no—
121C2×Q16—1314no—
122C2×SD16—1314no—
123C2×SD16—1314no—
124C2×D8—1314no—
125C2×M16C2×D13112yes—
126C4⋊Q8—1311no—
127#127—1311no—
128#124—2622no—
129#124—1312no—
130#124—1312no—
131#124—1312no—
132#121—2622no—
133C4×D4—2622no—
134C2×C22⋊C4—2622no—
135C2×C42C2×D13111yescharacteristic
136#97C2×D13112yescomplemented
137C4×C8C2×D13112yescomplemented
138C24—2621no—
139#41—2624no—
140#41—2624no—
141C2×Q8—2621no—
142C2×Q8—2624no—
143C2×Q8—1312no—
144C2×Q8—1312no—
145C2×D4—2622no—
146C2×D4—2624no—
147C2×D4—2622no—
148C2×D4—2621no—
149C2×D4—1312no—
150C2×D4—1312no—
151C22×C4—2621no—
152C22×C4—2622no—
153C22×C4D52412yes—
154C22×C4C22×D13111yescharacteristic
155Q16—2628no—
156SD16—2628no—
157SD16—2628no—
158D8—2628no—
159M16D52418yescomplemented
160C2×C8C22×D13114yes—
161C4⋊C4—2622no—
162C22⋊C4—2624no—
163C22⋊C4—2624no—
164C22⋊C4—2624no—
165C42D52412yes—
166C42C22×D13111yescharacteristic
167C42C22×D13111yescharacteristic
168C23—2622no—
169C23—2622no—
170C23—5244no—
171C23—5242no—
172C23C2×D52411yescharacteristic
173Q8—2622no—
174Q8—2624no—
175Q8—2622no—
176Q8—5242no—
177D4—2622no—
178D4—2624no—
179D4—2622no—
180D4—5244no—
181D4—5244no—
182D4—5242no—
183C2×C4—42no—
184C2×C4—2622no—
185C2×C4—2622no—
186C2×C4—5242no—
187C2×C4—5244no—
188C2×C4C2×D52412yes—
189C2×C4C2×D52412yes—
190C2×C4C2×D52411yescharacteristic
191C2×C4C2×D52414yes—
192C2×C4C23×D13111yescharacteristic
193C8C2×D52418yes—
194C22—22no—
195C22—5244no—
196C22—5244no—
197C22—5242no—
198C22—5242no—
199C22—10484no—
200C22#1686224312yes—
201C22C22×D52411yescharacteristic
202C4—22no—
203C4—22no—
204C4—5242no—
205C4—5242no—
206C4#1686224312yes—
207C4C22×D52412yes—
208C2—22no—
209C2—10482no—
210C2—10482no—
211C2C2×#1686224311yescharacteristic
212C2#2255411212yes—
subgroupG/H
centerC22C22×D524
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
FittingC4×M16×C131C2
socleC22×C131C22×D4
hypercenterC4×M16D131
hypocenterC131#3153
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×M16

Complements

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

NH ≅ G/Ntype
1. C2×#16862279C2semidirect
2. C2×#16862274C2semidirect
3. #22553862C2semidirect
4. C2×#16862234C2semidirect
5. #22553706C2semidirect
6. #22553701C2semidirect
7. #22553696C2semidirect
8. C4×M16×C131C2semidirect
14. C2×Dic524C22semidirect
16. C2×#14523502C22semidirect
18. #16862246C22semidirect
19. #16862243C22semidirect
23. #16862234C22semidirect
29. #97×C131C22semidirect
30. C4×C8×C131C22semidirect
56. M16×C131D4semidirect
106. C131#3153semidirect
115. C4×M16D131semidirect
136. #97C2×D131semidirect
137. C4×C8C2×D131semidirect
159. M16D524semidirect

Maximal subgroups

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

subgroupindex
C2×#168622792
C2×#168622742
#225538622
C2×#168622342
#225537062
#225537012
#225536962
C4×M16×C1312
#3153131