Finite Group Databasebeta

← #30664487

Subgroups — #30664487

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
2C4×C4⋊Dic131C211yesmaximal, characteristic, complemented
3#22553050C212yesmaximal, complemented
4#22553047C212yesmaximal
5#432×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—
16C2×C4⋊Dic131—21no—
17C2×C4×Dic131—21no—
18C131⋊#95C2212yes—
19C2×C42×C131C2211yescharacteristic
20C4⋊C8×C131—21no—
21C22⋊C8×C131C2212yes—
22#97×C131—21no—
23C2×Dic262C2×C411yescharacteristic
24C2×D524C2×C411yescharacteristic
25C2×C22⋊D131—21no—
26C22×Dic131—22no—
27C2×C4×D131—21no—
28C4⋊Dic131—24no—
29C4⋊Dic131C2×C411yescharacteristic
30C4⋊Dic131C2×C411yescharacteristic
31Dic131⋊C4—21no—
32Dic131⋊C4—21no—
33Dic131⋊C4—21no—
34Dic131⋊C4—21no—
35C22⋊Dic131—21no—
36(C2×D131)⋊C4—21no—
37(C2×D131)⋊C4—21no—
38(C2×D131)⋊C4—21no—
39(C2×D131)⋊C4—21no—
40C4×Dic131—21no—
41C4×Dic131—42no—
42C22×C4×C131D411yescharacteristic
43C22×C4×C131D411yescharacteristic
44C22×C4×C131C2311yescharacteristic
45C2×C8×C131—22no—
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—24no—
56C2×Dic131—21no—
57C2×Dic131—21no—
58C2×Dic131—21no—
59C2×Dic131—42no—
60C2×Dic131—42no—
61C4×D131—42no—
62C23×C131C2×D411yescharacteristic
63C2×C4×C131—21no—
64C2×C4×C131—21no—
65C2×C4×C131—22no—
66C2×C4×C131—41no—
67C2×C4×C131C22⋊C411yescharacteristic
68C2×C4×C131C22⋊C411yescharacteristic
69C2×C4×C131C22⋊C411yescharacteristic
70C2×C4×C131C22⋊C411yescharacteristic
71C2×C4×C131C22×C411yescharacteristic
72C2×C4×C131C2×D411yescharacteristic
73C8×C131—42no—
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#10412yes—
88Dic131—44no—
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#432D13111yescharacteristic, complemented
102C4×C4⋊C4—1311no—
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—2621no—
116C2×C42—2621no—
117C2×C42C2×D13111yescharacteristic
118C4⋊C8—21no—
119C22⋊C8C2×D13112yes—
120#97—21no—
121#95—1312no—
122C2×Q8—1311no—
123C2×D4—2621no—
124C2×D4—1311no—
125C22×C4—2622no—
126C22×C4—2621no—
127C22×C4D52411yescharacteristic
128C22×C4D52411yescharacteristic
129C22×C4C22×D13111yescharacteristic
130C2×C8—22no—
131C4⋊C4—2624no—
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—2621no—
144C42—5242no—
145C42C4×D13111yescharacteristic
146C42C4×D13111yescharacteristic
147C42C22⋊D13111yescharacteristic
148C42C22⋊D13111yescharacteristic
149C23—2621no—
150C23C2×D52411yescharacteristic
151Q8—5241no—
152D4—5241no—
153D4—5241no—
154D4—5241no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—41no—
159C2×C4—2624no—
160C2×C4—2621no—
161C2×C4—2621no—
162C2×C4—2621no—
163C2×C4—5242no—
164C2×C4—5242no—
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×C22⋊D13111yescharacteristic
172C8—42no—
173C22—21no—
174C22—21no—
175C22—5241no—
176C22—5241no—
177C22—5241no—
178C22C2×(C2×D131)⋊C411yescharacteristic
179C22#1686227411yescharacteristic
180C22#1686227911yescharacteristic
181C4—21no—
182C4—21no—
183C4—21no—
184C4—21no—
185C4—21no—
186C4—5244no—
187C4—5241no—
188C4—5241no—
189C4—5241no—
190C4#1686217212yes—
191C2—21no—
192C2—21no—
193C2—10481no—
194C2#2255328511yescharacteristic
195C2#2255331411yescharacteristic
196C2C2×#1686217211yescharacteristic
subgroupG/H
centerC22C2×(C2×D131)⋊C4
commutatorC2×C4×C131C22×C4
FrattiniC22×C4C22×D131
Fitting#432×C131C2
socleC22×C131C2×C22⋊C4
prime radical#16862220C22
cosocleC22×C4×C131C23
hypercenter#432D131
hypocenterC131#1509
LCSG ▷ C2×C4×C131 ▷ C22×C131 ▷ C131
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
2. C4×C4⋊Dic131C2semidirect
3. #22553050C2semidirect
5. #432×C131C2semidirect
7. #16862238C4semidirect
8. #16862234C4semidirect
12. C4×Dic262C4semidirect
13. C4×D524C4semidirect
98. C131#1509semidirect
101. #432D131semidirect

Maximal subgroups

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

subgroupindex
#225538462
C4×C4⋊Dic1312
#225530502
#225530472
#432×C1312
#1509131