Finite Group Databasebeta

← #30665243

Subgroups — #30665243

Order 16768.

240 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 55 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#22553967C211yesmaximal, characteristic
2#22553708C211yesmaximal, characteristic, complemented
3#22553589C211yesmaximal, characteristic, complemented
4#22553057C211yesmaximal, characteristic, complemented
5D4⋊(C131⋊C8)C211yesmaximal, characteristic, complemented
6#22553039C211yesmaximal, characteristic, complemented
7#461×C131C211yesmaximal, characteristic, complemented
8C2×#14523513—21no—
9#16862246C2211yescharacteristic
10#16862228—22no—
11#16862225—21no—
12#16862217—21no—
13C4×D524C2211yescharacteristic
14C4⋊C4×D131—21no—
15#16862179—21no—
16#16862177—22no—
17C4⋊(C131⋊C8)C2211yescharacteristic
18#16862166—21no—
19C4×C131⋊C8C2211yescharacteristic
20C2×SD16×C131—21no—
21C4⋊Q8×C131C2211yescharacteristic
22C4×D4×C131C2211yescharacteristic
23C4⋊C8×C131C2211yescharacteristic
24#103×C131—21no—
25#14523513—42no—
26#14523513—42no—
27C2×Dic262—21no—
28C2×D524D411yescharacteristic
29C2×C22⋊D131—22no—
30C2×C4×D131—22no—
31C2×C4×D131—21no—
32C2×C131⋊C8—21no—
33C2×C131⋊C8—21no—
34C4⋊Dic131—21no—
35C4⋊Dic131D411yescharacteristic
36C4⋊Dic131D412yes—
37Dic131⋊C4—22no—
38C22⋊Dic131—22no—
39(C2×D131)⋊C4—21no—
40C2×Q8×C131—21no—
41C2×D4×C131D411yescharacteristic
42C22×C4×C131—21no—
43SD16×C131—42no—
44C2×C8×C131—21no—
45C4⋊C4×C131—21no—
46C4⋊C4×C131—21no—
47C4⋊C4×C131D411yescharacteristic
48C22⋊C4×C131—21no—
49C42×C131C2311yescharacteristic
50C22×D131—21no—
51Dic262—41no—
52Dic262—41no—
53D524—21no—
54D524SD1612yescomplemented
55C22⋊D131—44no—
56C2×Dic131—21no—
57C2×Dic131—22no—
58C4×D131—44no—
59C4×D131—41no—
60C4×D131—41no—
61C131⋊C8—22no—
62C131⋊C8—41no—
63C23×C131—21no—
64Q8×C131—41no—
65Q8×C131—41no—
66D4×C131—21no—
67D4×C131—21no—
68C2×C4×C131—21no—
69C2×C4×C131—41no—
70C2×C4×C131—41no—
71C2×C4×C131—41no—
72C2×C4×C131C2×D411yescharacteristic
73C2×C4×C131C2×D411yescharacteristic
74C2×C4×C131C2×D411yescharacteristic
75C8×C131—41no—
76C22×C131—41no—
77C22×C131—41no—
78C22×C131—41no—
79C22×C131#12011yescharacteristic
80C2×D131—22no—
81C2×D131—41no—
82C2×D131—41no—
83C4×C131—21no—
84C4×C131—41no—
85C4×C131—81no—
86C4×C131C2×SD1611yescharacteristic
87C4×C131#13511yescharacteristic
88C4×C131AGL(1,ℤ/8)11yescharacteristic
89C4×C131AGL(1,ℤ/8)11yescharacteristic
90Dic131—41no—
91Dic131—42no—
92C2×C131—81no—
93C2×C131#44911yescharacteristic
94C2×C131#45011yescharacteristic
95C2×C131#45611yescharacteristic
96D131—42no—
97C131#165411yescharacteristic, complemented, Sylow
98#1654—1311nomaximal, Sylow
99#547—1311no—
100#498—1311no—
101#474—1311no—
102#461D13111yescharacteristic, complemented
103#332—1311no—
104D4⋊C8—1311no—
105D4⋊C8—1311no—
106C2×#41—2621no—
107C2×SD16—21no—
108C4⋊Q8—1311no—
109C4⋊Q8C2×D13111yescharacteristic
110#123—2622no—
111C22⋊Q8—2621no—
112#121—2621no—
113C4×D4—1311no—
114C4×D4C2×D13111yescharacteristic
115C2×C4⋊C4—2621no—
116#107—2621no—
117#106—2622no—
118C4⋊C8—1311no—
119C4⋊C8C2×D13111yescharacteristic
120#103—21no—
121#102—2621no—
122C4×C8—1311no—
123#41—5242no—
124#41—5242no—
125C2×Q8—21no—
126C2×Q8—2621no—
127C2×D4—2622no—
128C2×D4—1311no—
129C2×D4C22⋊D13111yescharacteristic
130C22×C4—21no—
131C22×C4—2622no—
132C22×C4—2621no—
133SD16—42no—
134C2×C8—21no—
135C2×C8—2621no—
136C2×C8—2621no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—2622no—
140C4⋊C4—2621no—
141C4⋊C4—1312no—
142C4⋊C4—1311no—
143C4⋊C4C22⋊D13111yescharacteristic
144C22⋊C4—21no—
145C22⋊C4—2622no—
146C22⋊C4—2621no—
147C42C22×D13111yescharacteristic
148C23—21no—
149C23—2621no—
150Q8—41no—
151Q8—41no—
152Q8—5241no—
153Q8—5241no—
154D4—21no—
155D4—21no—
156D4—2621no—
157D4—5244no—
158D4—1312no—
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—2621no—
164C2×C4—2622no—
165C2×C4—5244no—
166C2×C4—5241no—
167C2×C4—5241no—
168C2×C4D4×D13111yescharacteristic
169C2×C4D4×D13111yescharacteristic
170C2×C4C2×C22⋊D13111yescharacteristic
171C8—41no—
172C8—2622no—
173C8—5241no—
174C22—41no—
175C22—41no—
176C22—41no—
177C22—2622no—
178C22—5241no—
179C22—5241no—
180C22#1686221411yescharacteristic
181C4—21no—
182C4—41no—
183C4—81no—
184C4—5241no—
185C4—5242no—
186C4SD16×D13111yescharacteristic
187C4#1686226611yescharacteristic
188C4#1686227211yescharacteristic
189C4#1686227311yescharacteristic
190C2—81no—
191C2—5242no—
192C2#2255345511yescharacteristic
193C2#2255346111yescharacteristic
194C2#2255349311yescharacteristic
subgroupG/H
centerC22#16862214
commutator, cosocleC42×C131C23
FrattiniC42C22×D131
Fitting#461×C131C2
socleC22×C131#120
prime radical#22553967C2
hypercenter#461D131
hypocenterC131#1654
LCSG ▷ C42×C131 ▷ C22×C131 ▷ C131
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. #22553708C2semidirect
3. #22553589C2semidirect
4. #22553057C2semidirect
5. D4⋊(C131⋊C8)C2semidirect
6. #22553039C2semidirect
7. #461×C131C2semidirect
54. D524SD16semidirect
97. C131#1654semidirect
102. #461D131semidirect

Maximal subgroups

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

subgroupindex
#225539672
#225537082
#225535892
#225530572
D4⋊(C131⋊C8)2
#225530392
#461×C1312
#1654131