Finite Group Databasebeta

← #30665129

Subgroups — #30665129

Order 16768.

228 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 59 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
1D4×Dic262C211yesmaximal, characteristic
2#22553663C211yesmaximal, characteristic, complemented
3#22553608C211yesmaximal, characteristic, complemented
4#22553602C211yesmaximal, characteristic
5#22553059C211yesmaximal, characteristic, complemented
6#22553044C211yesmaximal, characteristic, complemented
7D4⋊C8×C131C211yesmaximal, characteristic
8C22×Dic262—21no—
9#16862246C2211yescharacteristic
10#16862246C2211yescharacteristic
11#16862245—21no—
12#16862225—21no—
13#16862223—22no—
14C4×Dic262C2211yescharacteristic
15D4×Dic131—21no—
16#16862180—21no—
17#16862177—21no—
18C4⋊(C131⋊C8)C2211yescharacteristic
19#16862171—21no—
20#16862171—22no—
21#16862167—21no—
22C4×D4×C131C2211yescharacteristic
23C4⋊C8×C131C2211yescharacteristic
24C4×C8×C131C2211yescharacteristic
25C2×Dic262—22no—
26C2×Dic262—21no—
27C2×Dic262—21no—
28C2×Dic262—42no—
29C2×Dic262—42no—
30C2×Dic262D411yescharacteristic
31C22×Dic131—22no—
32C2×C131⋊C8—21no—
33C4⋊Dic131—21no—
34C4⋊Dic131—21no—
35C4⋊Dic131—21no—
36C4⋊Dic131D411yescharacteristic
37C4⋊Dic131D412yes—
38Dic131⋊C4—22no—
39Dic131⋊C4—21no—
40C22⋊Dic131—22no—
41C4×Dic131—21no—
42C2×D4×C131D411yescharacteristic
43C22×C4×C131—21no—
44C2×C8×C131—21no—
45C2×C8×C131—21no—
46C4⋊C4×C131D411yescharacteristic
47C22⋊C4×C131—21no—
48C42×C131C2311yescharacteristic
49Dic262—21no—
50Dic262—44no—
51Dic262—41no—
52Dic262—41no—
53Dic262—42no—
54Dic262SD1612yes—
55C2×Dic131—21no—
56C2×Dic131—21no—
57C2×Dic131—22no—
58C2×Dic131—44no—
59C2×Dic131—41no—
60C131⋊C8—41no—
61C23×C131—21no—
62D4×C131—21no—
63D4×C131Q1612yes—
64C2×C4×C131—21no—
65C2×C4×C131—41no—
66C2×C4×C131—41no—
67C2×C4×C131C2×D411yescharacteristic
68C2×C4×C131C2×D411yescharacteristic
69C2×C4×C131C2×D411yescharacteristic
70C8×C131—41no—
71C8×C131—41no—
72C22×C131—22no—
73C22×C131—41no—
74C22×C131—41no—
75C22×C131#12011yescharacteristic
76C4×C131—21no—
77C4×C131—41no—
78C4×C131C2×SD1611yescharacteristic
79C4×C131C2×Q1611yescharacteristic
80C4×C131AGL(1,ℤ/8)11yescharacteristic
81C4×C131#13711yescharacteristic
82Dic131—22no—
83Dic131—42no—
84Dic131—41no—
85Dic131—81no—
86C2×C131—42no—
87C2×C131#45011yescharacteristic
88C2×C131#45111yescharacteristic
89C2×C131#45411yescharacteristic
90C131#163711yescharacteristic, complemented, Sylow
91#1637—1311nomaximal, Sylow
92D4×Q8—1311no—
93#487—1311no—
94#477—1311no—
95#476—1311no—
96#332—1311no—
97Q8⋊C8—1311no—
98D4⋊C8D13111yescharacteristic
99C22×Q8—2621no—
100C4⋊Q8—2621no—
101C4⋊Q8—1311no—
102C4⋊Q8—1311no—
103C22⋊Q8—2622no—
104C22⋊Q8—2621no—
105C4×Q8—1311no—
106C4×D4—2621no—
107C4×D4C2×D13111yescharacteristic
108#107—2621no—
109#106—2621no—
110C4⋊C8—1311no—
111C4⋊C8C2×D13111yescharacteristic
112#103—2621no—
113#103—2622no—
114#102—2621no—
115C4×C8C2×D13111yescharacteristic
116C2×Q8—2622no—
117C2×Q8—2621no—
118C2×Q8—2621no—
119C2×Q8—5242no—
120C2×Q8—5242no—
121C2×Q8—1311no—
122C2×D4D52411yescharacteristic
123C22×C4—21no—
124C22×C4—2622no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—2621no—
128C4⋊C4—2622no—
129C4⋊C4—2621no—
130C4⋊C4—2621no—
131C4⋊C4—2621no—
132C4⋊C4—2621no—
133C4⋊C4—1312no—
134C4⋊C4—1311no—
135C4⋊C4D52411yescharacteristic
136C22⋊C4—21no—
137C22⋊C4—2622no—
138C42—2621no—
139C42C22×D13111yescharacteristic
140C23—21no—
141Q8—2621no—
142Q8—5242no—
143Q8—5244no—
144Q8—5241no—
145Q8—5241no—
146Q8—1312no—
147D4—21no—
148D4Dic52412yes—
149C2×C4—21no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—2621no—
153C2×C4—2621no—
154C2×C4—2622no—
155C2×C4—5244no—
156C2×C4—5241no—
157C2×C4D4×D13111yescharacteristic
158C2×C4D4×D13111yescharacteristic
159C2×C4C2×D52411yescharacteristic
160C8—41no—
161C8—41no—
162C8—5241no—
163C22—22no—
164C22—41no—
165C22—41no—
166C22#1686221611yescharacteristic
167C4—21no—
168C4—41no—
169C4—2622no—
170C4—5242no—
171C4—5241no—
172C4—10481no—
173C4SD16×D13111yescharacteristic
174C4C2×Dic52411yescharacteristic
175C4#1686227011yescharacteristic
176C4#1686227911yescharacteristic
177C2—42no—
178C2#2255346411yescharacteristic
179C2#2255347011yescharacteristic
180C2#2255348611yescharacteristic
subgroupG/H
centerC22#16862216
commutator, cosocleC42×C131C23
FrattiniC42C22×D131
FittingD4⋊C8×C131C2
socleC22×C131#120
prime radicalC2×D4×C131D4
hypercenterD4⋊C8D131
hypocenterC131#1637
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
3. #22553608C2semidirect
5. #22553059C2semidirect
6. #22553044C2semidirect
90. C131#1637semidirect

Maximal subgroups

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

subgroupindex
D4×Dic2622
#225536632
#225536082
#225536022
#225530592
#225530442
D4⋊C8×C1312
#1637131