Finite Group Databasebeta

← #30665178

Subgroups — #30665178

Order 16768.

208 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 57 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#22554019C211yesmaximal, characteristic
2#22553663C211yesmaximal, characteristic, complemented
3#22553592C211yesmaximal, characteristic
4#22553523C211yesmaximal, characteristic, complemented
5#22553060C211yesmaximal, characteristic, complemented
6#22553044C211yesmaximal, characteristic, complemented
7D4⋊C8×C131C211yesmaximal, characteristic
8C2×Dic524—21no—
9#16862246C2211yescharacteristic
10#16862246C2211yescharacteristic
11#16862237—21no—
12#16862225—21no—
13#16862223—22no—
14C4×Dic262C2211yescharacteristic
15D4×Dic131—21no—
16C2×C4⋊Dic131—21no—
17#16862179—21no—
18C4⋊(C131⋊C8)C2211yescharacteristic
19#16862171—22no—
20#16862171—21no—
21#16862167—21no—
22C4×D4×C131C2211yescharacteristic
23C4⋊C8×C131C2211yescharacteristic
24C4×C8×C131C2211yescharacteristic
25C2×Dic262—21no—
26C2×Dic262D411yescharacteristic
27C2×Dic262D412yes—
28C22×Dic131—22no—
29Dic524—42no—
30C2×C131⋊C8—21no—
31C4⋊Dic131—22no—
32C4⋊Dic131—21no—
33C4⋊Dic131—21no—
34C4⋊Dic131—21no—
35C4⋊Dic131—21no—
36C4⋊Dic131D411yescharacteristic
37Dic131⋊C4—22no—
38Dic131⋊C4—21no—
39C22⋊Dic131—22no—
40C4×Dic131—21no—
41C2×D4×C131D411yescharacteristic
42C22×C4×C131—21no—
43C2×C8×C131—21no—
44C2×C8×C131—21no—
45C4⋊C4×C131D411yescharacteristic
46C22⋊C4×C131—21no—
47C42×C131C2311yescharacteristic
48Dic262—21no—
49Dic262—21no—
50Dic262—42no—
51Dic262—41no—
52Dic262—41no—
53C2×Dic131—21no—
54C2×Dic131—21no—
55C2×Dic131—22no—
56C2×Dic131—41no—
57C2×Dic131—82no—
58C131⋊C8—41no—
59C23×C131—21no—
60D4×C131—21no—
61D4×C131Q1612yescomplemented
62C2×C4×C131—21no—
63C2×C4×C131—41no—
64C2×C4×C131—41no—
65C2×C4×C131C2×D411yescharacteristic
66C2×C4×C131C2×D411yescharacteristic
67C2×C4×C131C2×D411yescharacteristic
68C8×C131—41no—
69C8×C131—41no—
70C22×C131—22no—
71C22×C131—41no—
72C22×C131—41no—
73C22×C131#12011yescharacteristic
74C4×C131—21no—
75C4×C131—41no—
76C4×C131C2×Q1611yescharacteristic
77C4×C131#13511yescharacteristic
78C4×C131AGL(1,ℤ/8)11yescharacteristic
79C4×C131#13711yescharacteristic
80Dic131—41no—
81Dic131—41no—
82Dic131—42no—
83Dic131—81no—
84C2×C131—42no—
85C2×C131#44911yescharacteristic
86C2×C131#45111yescharacteristic
87C2×C131#45511yescharacteristic
88C131#164411yescharacteristic, complemented, Sylow
89#1644—1311nomaximal, Sylow
90#554—1311no—
91#487—1311no—
92#474—1311no—
93#462—1311no—
94#333—1311no—
95Q8⋊C8—1311no—
96D4⋊C8D13111yescharacteristic
97C2×Q16—2621no—
98C4⋊Q8—1311no—
99C4⋊Q8—1311no—
100#125—2621no—
101C22⋊Q8—2622no—
102C22⋊Q8—2621no—
103C4×Q8—1311no—
104C4×D4—2621no—
105C4×D4C2×D13111yescharacteristic
106C2×C4⋊C4—2621no—
107#107—2621no—
108C4⋊C8—1311no—
109C4⋊C8C2×D13111yescharacteristic
110#103—2622no—
111#103—2621no—
112#102—2621no—
113C4×C8C2×D13111yescharacteristic
114C2×Q8—2621no—
115C2×Q8—1311no—
116C2×Q8—1312no—
117C2×D4D52411yescharacteristic
118C22×C4—21no—
119C22×C4—2622no—
120Q16—5242no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—2621no—
124C4⋊C4—2622no—
125C4⋊C4—2622no—
126C4⋊C4—2621no—
127C4⋊C4—2621no—
128C4⋊C4—2621no—
129C4⋊C4—2621no—
130C4⋊C4—2621no—
131C4⋊C4—1311no—
132C4⋊C4D52411yescharacteristic
133C22⋊C4—21no—
134C22⋊C4—2622no—
135C42—2621no—
136C42C22×D13111yescharacteristic
137C23—21no—
138Q8—2621no—
139Q8—2621no—
140Q8—5242no—
141Q8—5241no—
142Q8—5241no—
143D4—21no—
144D4Dic52412yescomplemented
145C2×C4—21no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—2621no—
149C2×C4—2621no—
150C2×C4—2622no—
151C2×C4—5241no—
152C2×C4—10482no—
153C2×C4D4×D13111yescharacteristic
154C2×C4D4×D13111yescharacteristic
155C2×C4C2×D52411yescharacteristic
156C8—41no—
157C8—41no—
158C8—5241no—
159C22—22no—
160C22—41no—
161C22—41no—
162C22#1686221611yescharacteristic
163C4—21no—
164C4—41no—
165C4—5241no—
166C4—5241no—
167C4—5242no—
168C4—10481no—
169C4C2×Dic52411yescharacteristic
170C4#1686226711yescharacteristic
171C4#1686227011yescharacteristic
172C4#1686227911yescharacteristic
173C2—42no—
174C2#2255345811yescharacteristic
175C2#2255347011yescharacteristic
176C2#2255349111yescharacteristic
subgroupG/H
centerC22#16862216
commutator, cosocleC42×C131C23
FrattiniC42C22×D131
FittingD4⋊C8×C131C2
socleC22×C131#120
prime radicalC2×D4×C131D4
hypercenterD4⋊C8D131
hypocenterC131#1644
LCSG ▷ C42×C131 ▷ C22×C131 ▷ C131
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #22553663C2semidirect
4. #22553523C2semidirect
5. #22553060C2semidirect
6. #22553044C2semidirect
61. D4×C131Q16semidirect
88. C131#1644semidirect
144. D4Dic524semidirect

Maximal subgroups

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

subgroupindex
#225540192
#225536632
#225535922
#225535232
#225530602
#225530442
D4⋊C8×C1312
#1644131