Finite Group Databasebeta

← #30664484

Subgroups — #30664484

Order 16768.

242 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 73 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
1#22553846C211yesmaximal, characteristic
2#22553369C211yesmaximal, characteristic, complemented
3#22553048C212yesmaximal, complemented
4#22553045C212yesmaximal
5C4×C4⋊C4×C131C211yesmaximal, characteristic, complemented
6#16862240—21no—
7#16862238C411yescharacteristic, complemented
8#16862234C411yescharacteristic, complemented
9#16862229—21no—
10#16862225C2211yescharacteristic
11#16862220C2211yescharacteristic
12C4×Dic262C411yescharacteristic, complemented
13C4×D524C411yescharacteristic, complemented
14C4×C22⋊D131—21no—
15#16862203—21no—
16C4⋊(C131⋊C8)—21no—
17C22⋊(C131⋊C8)C2212yes—
18#16862157—21no—
19C2×C4⋊C4×C131—21no—
20C2×C42×C131—21no—
21C2×C42×C131C2211yescharacteristic
22#95×C131C2212yes—
23C2×Dic262C2×C411yescharacteristic
24C2×D524C2×C411yescharacteristic
25C2×C22⋊D131—21no—
26C2×C4×D131—21no—
27C2×C131⋊C8—22no—
28C4⋊Dic131C2×C411yescharacteristic
29C4⋊Dic131C2×C411yescharacteristic
30Dic131⋊C4—21no—
31Dic131⋊C4—21no—
32Dic131⋊C4—21no—
33Dic131⋊C4—21no—
34C22⋊Dic131—21no—
35(C2×D131)⋊C4—21no—
36(C2×D131)⋊C4—21no—
37(C2×D131)⋊C4—21no—
38(C2×D131)⋊C4—21no—
39C4×Dic131—21no—
40C22×C4×C131—22no—
41C22×C4×C131D411yescharacteristic
42C22×C4×C131D411yescharacteristic
43C22×C4×C131C2311yescharacteristic
44C4⋊C4×C131—24no—
45C42×C131—42no—
46C42×C131C2×C411yescharacteristic
47C42×C131C2×C411yescharacteristic
48C42×C131D411yescharacteristic
49C42×C131D411yescharacteristic
50C22×D131—21no—
51Dic262—41no—
52D524—41no—
53C22⋊D131—41no—
54C22⋊D131—41no—
55C2×Dic131—21no—
56C2×Dic131—21no—
57C2×Dic131—21no—
58C4×D131—42no—
59C131⋊C8—42no—
60C23×C131C2×D411yescharacteristic
61C2×C4×C131—21no—
62C2×C4×C131—21no—
63C2×C4×C131—22no—
64C2×C4×C131—24no—
65C2×C4×C131—41no—
66C2×C4×C131—42no—
67C2×C4×C131—42no—
68C2×C4×C131C22⋊C411yescharacteristic
69C2×C4×C131C22⋊C411yescharacteristic
70C2×C4×C131C22⋊C411yescharacteristic
71C2×C4×C131C22⋊C411yescharacteristic
72C2×C4×C131C22×C411yescharacteristic
73C2×C4×C131C2×D411yescharacteristic
74C22×C131—21no—
75C22×C131—21no—
76C22×C131C2×C22⋊C411yescharacteristic
77C22×C131AGL(1,ℤ/8)11yescharacteristic
78C22×C131#13711yescharacteristic
79C2×D131—41no—
80C2×D131—41no—
81C2×D131—41no—
82C4×C131—21no—
83C4×C131—21no—
84C4×C131—21no—
85C4×C131—21no—
86C4×C131—21no—
87C4×C131—44no—
88C4×C131#10412yes—
89Dic131—41no—
90Dic131—41no—
91Dic131—41no—
92C2×C131—21no—
93C2×C131—21no—
94C2×C131#41011yescharacteristic
95C2×C131#41711yescharacteristic
96C2×C131C2×#10411yescharacteristic
97D131—81no—
98C131#150911yescharacteristic, complemented, Sylow
99#1509—1311nomaximal, Sylow
100#529—1311no—
101#432—1311no—
102C4×C4⋊C4D13111yescharacteristic, complemented
103#328—1312no—
104#327—1312no—
105#126—2621no—
106#125—1311no—
107#124—1311no—
108#123—2621no—
109C22⋊Q8—1311no—
110#121—1311no—
111C4×Q8—1311no—
112C4×D4—2621no—
113C4×D4—1311no—
114#117—2621no—
115C2×C4⋊C4—21no—
116C2×C42—21no—
117C2×C42C2×D13111yescharacteristic
118C4⋊C8—2621no—
119C22⋊C8—1312no—
120#97—2621no—
121#95C2×D13112yes—
122C2×Q8—1311no—
123C2×D4—2621no—
124C2×D4—1311no—
125C22×C4—22no—
126C22×C4—2621no—
127C22×C4C22⋊D13111yescharacteristic
128C22×C4C22⋊D13111yescharacteristic
129C22×C4C22×D13111yescharacteristic
130C2×C8—2622no—
131C4⋊C4—24no—
132C4⋊C4—2621no—
133C4⋊C4—2621no—
134C4⋊C4—2621no—
135C4⋊C4—2621no—
136C4⋊C4—1311no—
137C4⋊C4—1311no—
138C22⋊C4—2621no—
139C22⋊C4—2621no—
140C22⋊C4—2621no—
141C22⋊C4—2621no—
142C22⋊C4—2621no—
143C42—42no—
144C42—2621no—
145C42C4×D13111yescharacteristic
146C42C4×D13111yescharacteristic
147C42D52411yescharacteristic
148C42D52411yescharacteristic
149C23—2621no—
150C23C2×C22⋊D13111yescharacteristic
151Q8—5241no—
152D4—5241no—
153D4—5241no—
154D4—5241no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—24no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—2621no—
163C2×C4—2621no—
164C2×C4—2621no—
165C2×C4—5242no—
166C2×C4(C2×D131)⋊C411yescharacteristic
167C2×C4(C2×D131)⋊C411yescharacteristic
168C2×C4(C2×D131)⋊C411yescharacteristic
169C2×C4(C2×D131)⋊C411yescharacteristic
170C2×C4C2×C4×D13111yescharacteristic
171C2×C4C2×D52411yescharacteristic
172C8—5242no—
173C22—21no—
174C22—21no—
175C22—5241no—
176C22—5241no—
177C22—5241no—
178C22C2×(C2×D131)⋊C411yescharacteristic
179C22#1686227211yescharacteristic
180C22#1686227711yescharacteristic
181C4—21no—
182C4—21no—
183C4—21no—
184C4—21no—
185C4—21no—
186C4—44no—
187C4—5241no—
188C4—5241no—
189C4—5241no—
190C4#1686217412yes—
191C2—21no—
192C2—21no—
193C2—10481no—
194C2#2255328511yescharacteristic
195C2#2255331111yescharacteristic
196C2C2×#1686217411yescharacteristic
subgroupG/H
centerC22C2×(C2×D131)⋊C4
commutatorC2×C4×C131C22×C4
FrattiniC22×C4C22×D131
FittingC4×C4⋊C4×C131C2
socleC22×C131C2×C22⋊C4
prime radical#16862220C22
cosocleC22×C4×C131C23
hypercenterC4×C4⋊C4D131
hypocenterC131#1509
LCSG ▷ C2×C4×C131 ▷ C22×C131 ▷ C131
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #22553369C2semidirect
3. #22553048C2semidirect
5. C4×C4⋊C4×C131C2semidirect
7. #16862238C4semidirect
8. #16862234C4semidirect
12. C4×Dic262C4semidirect
13. C4×D524C4semidirect
98. C131#1509semidirect
102. C4×C4⋊C4D131semidirect

Maximal subgroups

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

subgroupindex
#225538462
#225533692
#225530482
#225530452
C4×C4⋊C4×C1312
#1509131