Finite Group Databasebeta

← #30665187

Subgroups — #30665187

Order 16768.

212 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 53 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#22554027C211yesmaximal, characteristic
2#22553664C211yesmaximal, characteristic, complemented
3#22553589C211yesmaximal, characteristic, complemented
4#22553060C211yesmaximal, characteristic, complemented
5D4⋊(C131⋊C8)C211yesmaximal, characteristic, complemented
6#22553039C211yesmaximal, characteristic, complemented
7#461×C131C211yesmaximal, characteristic, complemented
8#16862246C2211yescharacteristic
9#16862232—22no—
10#16862220—21no—
11#16862217—21no—
12C4×D524C2211yescharacteristic
13#16862205—21no—
14#16862204—21no—
15#16862179—21no—
16C4⋊(C131⋊C8)C2211yescharacteristic
17#16862169—22no—
18#16862166—21no—
19C4×C131⋊C8C2211yescharacteristic
20C2×SD16×C131—21no—
21C4⋊Q8×C131C2211yescharacteristic
22C4×D4×C131C2211yescharacteristic
23C4⋊C8×C131C2211yescharacteristic
24#103×C131—21no—
25C2×Dic262D412yes—
26C2×D524D411yescharacteristic
27C2×C22⋊D131—22no—
28C2×C4×D131—21no—
29C2×C131⋊C8—21no—
30C2×C131⋊C8—21no—
31C4⋊Dic131—21no—
32C4⋊Dic131—21no—
33C4⋊Dic131D411yescharacteristic
34C22⋊Dic131—22no—
35(C2×D131)⋊C4—22no—
36(C2×D131)⋊C4—21no—
37C4×Dic131—22no—
38C2×Q8×C131—21no—
39C2×D4×C131D411yescharacteristic
40C22×C4×C131—21no—
41SD16×C131—42no—
42C2×C8×C131—21no—
43C4⋊C4×C131—21no—
44C4⋊C4×C131—21no—
45C4⋊C4×C131D411yescharacteristic
46C22⋊C4×C131—21no—
47C42×C131C2311yescharacteristic
48C22×D131—21no—
49Dic262—42no—
50D524—21no—
51D524—21no—
52C22⋊D131—82no—
53C2×Dic131—21no—
54C2×Dic131—22no—
55C4×D131—41no—
56C4×D131—41no—
57C131⋊C8—41no—
58C131⋊C8—41no—
59C23×C131—21no—
60Q8×C131—41no—
61Q8×C131—41no—
62D4×C131—21no—
63D4×C131—21no—
64C2×C4×C131—21no—
65C2×C4×C131—41no—
66C2×C4×C131—41no—
67C2×C4×C131—41no—
68C2×C4×C131C2×D411yescharacteristic
69C2×C4×C131C2×D411yescharacteristic
70C2×C4×C131C2×D411yescharacteristic
71C8×C131—41no—
72C22×C131—41no—
73C22×C131—41no—
74C22×C131—41no—
75C22×C131#12011yescharacteristic
76C2×D131—41no—
77C2×D131—41no—
78C2×D131—41no—
79C4×C131—21no—
80C4×C131—41no—
81C4×C131—81no—
82C4×C131#13511yescharacteristic
83C4×C131#13511yescharacteristic
84C4×C131AGL(1,ℤ/8)11yescharacteristic
85C4×C131#13711yescharacteristic
86Dic131—41no—
87Dic131—42no—
88C2×C131—81no—
89C2×C131#44911yescharacteristic
90C2×C131#45211yescharacteristic
91C2×C131#45511yescharacteristic
92D131—81no—
93C131#164611yescharacteristic, complemented, Sylow
94#1646—1311nomaximal, Sylow
95#555—1311no—
96#487—1311no—
97#474—1311no—
98#461D13111yescharacteristic, complemented
99#333—1311no—
100D4⋊C8—1311no—
101D4⋊C8—1311no—
102C2×SD16—21no—
103C4⋊Q8—1311no—
104C4⋊Q8C2×D13111yescharacteristic
105#124—2622no—
106#121—2621no—
107#121—2621no—
108C4×D4—1311no—
109C4×D4C2×D13111yescharacteristic
110#117—2621no—
111#117—2621no—
112#107—2621no—
113C4⋊C8—1311no—
114C4⋊C8C2×D13111yescharacteristic
115#103—21no—
116#103—2622no—
117#102—2621no—
118C4×C8—1311no—
119C2×Q8—21no—
120C2×Q8—1312no—
121C2×D4—2622no—
122C2×D4—1311no—
123C2×D4C22⋊D13111yescharacteristic
124C22×C4—21no—
125C22×C4—2621no—
126SD16—42no—
127C2×C8—21no—
128C2×C8—2621no—
129C2×C8—2621no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—2621no—
133C4⋊C4—2621no—
134C4⋊C4—1311no—
135C4⋊C4C22⋊D13111yescharacteristic
136C22⋊C4—21no—
137C22⋊C4—2622no—
138C22⋊C4—2622no—
139C22⋊C4—2621no—
140C42—2622no—
141C42C22×D13111yescharacteristic
142C23—21no—
143C23—2621no—
144Q8—41no—
145Q8—41no—
146Q8—5242no—
147D4—21no—
148D4—21no—
149D4—2621no—
150D4—2621no—
151D4—10482no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—2621no—
157C2×C4—2622no—
158C2×C4—5241no—
159C2×C4—5241no—
160C2×C4D4×D13111yescharacteristic
161C2×C4D4×D13111yescharacteristic
162C2×C4C2×C22⋊D13111yescharacteristic
163C8—41no—
164C8—5241no—
165C8—5241no—
166C22—41no—
167C22—41no—
168C22—41no—
169C22—5241no—
170C22—5241no—
171C22—5241no—
172C22#1686221411yescharacteristic
173C4—21no—
174C4—41no—
175C4—81no—
176C4—5241no—
177C4—5242no—
178C4#1686226611yescharacteristic
179C4#1686226711yescharacteristic
180C4#1686227211yescharacteristic
181C4#1686227511yescharacteristic
182C2—81no—
183C2—10481no—
184C2#2255345511yescharacteristic
185C2#2255347111yescharacteristic
186C2#2255348811yescharacteristic
subgroupG/H
centerC22#16862214
commutator, cosocleC42×C131C23
FrattiniC42C22×D131
Fitting#461×C131C2
socleC22×C131#120
prime radical#22554027C2
hypercenter#461D131
hypocenterC131#1646
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. #22553664C2semidirect
3. #22553589C2semidirect
4. #22553060C2semidirect
5. D4⋊(C131⋊C8)C2semidirect
6. #22553039C2semidirect
7. #461×C131C2semidirect
93. C131#1646semidirect
98. #461D131semidirect

Maximal subgroups

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

subgroupindex
#225540272
#225536642
#225535892
#225530602
D4⋊(C131⋊C8)2
#225530392
#461×C1312
#1646131