Finite Group Databasebeta

← #30665127

Subgroups — #30665127

Order 16768.

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

normal subgroups only

This group has 228 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#22553977C211yesmaximal, characteristic
2#22553664C211yesmaximal, characteristic, complemented
3#22553608C211yesmaximal, characteristic, complemented
4#22553603C211yesmaximal, characteristic
5#22553042C211yesmaximal, characteristic, complemented
6#22553039C211yesmaximal, characteristic
7#332×C131C211yesmaximal, characteristic, complemented
8C2×Q8×D131—21no—
9#16862246C2211yescharacteristic
10#16862245—21no—
11#16862224—22no—
12#16862222—21no—
13C4×Dic262C2211yescharacteristic
14C4×D524C2211yescharacteristic
15#16862208—21no—
16#16862180—21no—
17#16862178—21no—
18#16862171—21no—
19#16862169—22no—
20#16862168—21no—
21C4×C131⋊C8C2211yescharacteristic
22C4⋊Q8×C131C2211yescharacteristic
23C4⋊C8×C131C2211yescharacteristic
24C4⋊C8×C131C2211yescharacteristic
25C2×Dic262—22no—
26C2×Dic262—21no—
27C2×Dic262D411yescharacteristic
28Q8×D131—42no—
29Q8×D131—42no—
30C2×D524D411yescharacteristic
31C2×C4×D131—22no—
32C2×C4×D131—21no—
33C2×C131⋊C8—21no—
34C4⋊Dic131—21no—
35C4⋊Dic131—21no—
36C4⋊Dic131D411yescharacteristic
37C4⋊Dic131D411yescharacteristic
38Dic131⋊C4—22no—
39Dic131⋊C4—21no—
40(C2×D131)⋊C4—22no—
41(C2×D131)⋊C4—21no—
42C4×Dic131—21no—
43C2×Q8×C131—21no—
44C2×C8×C131—21no—
45C2×C8×C131—21no—
46C4⋊C4×C131—21no—
47C4⋊C4×C131D412yes—
48C42×C131C2311yescharacteristic
49C22×D131—21no—
50Dic262—21no—
51Dic262—44no—
52Dic262—42no—
53Dic262SD1612yes—
54D524—21no—
55D524Q1612yes—
56C2×Dic131—21no—
57C2×Dic131—21no—
58C2×Dic131—21no—
59C2×Dic131—41no—
60C4×D131—44no—
61C4×D131—41no—
62C4×D131—41no—
63C131⋊C8—41no—
64Q8×C131—41no—
65Q8×C131—41no—
66C2×C4×C131—22no—
67C2×C4×C131C2×D411yescharacteristic
68C2×C4×C131C2×D411yescharacteristic
69C2×C4×C131C2×D411yescharacteristic
70C8×C131—41no—
71C8×C131—41no—
72C22×C131#12011yescharacteristic
73C2×D131—22no—
74C2×D131—41no—
75C2×D131—41no—
76C4×C131—21no—
77C4×C131—42no—
78C4×C131C2×SD1611yescharacteristic
79C4×C131C2×Q1611yescharacteristic
80C4×C131AGL(1,ℤ/8)11yescharacteristic
81C4×C131#13711yescharacteristic
82Dic131—22no—
83Dic131—41no—
84Dic131—41no—
85Dic131—81no—
86C2×C131#45011yescharacteristic
87C2×C131#45111yescharacteristic
88C2×C131#45411yescharacteristic
89D131—42no—
90C131#163711yescharacteristic, complemented, Sylow
91#1637—1311nomaximal, Sylow
92D4×Q8—1311no—
93#487—1311no—
94#477—1311no—
95#476—1311no—
96#332D13111yescharacteristic, complemented
97Q8⋊C8—1311no—
98D4⋊C8—1311no—
99C22×Q8—2621no—
100C4⋊Q8—2621no—
101C4⋊Q8—1311no—
102C4⋊Q8C2×D13111yescharacteristic
103C22⋊Q8—2621no—
104C22⋊Q8—2622no—
105C4×Q8—1311no—
106C4×D4—2621no—
107C4×D4—1311no—
108#107—2621no—
109#106—2621no—
110C4⋊C8C2×D13111yescharacteristic
111C4⋊C8C2×D13111yescharacteristic
112#103—2621no—
113#103—2622no—
114#102—2621no—
115C4×C8—1311no—
116C2×Q8—21no—
117C2×Q8—2622no—
118C2×Q8—2621no—
119C2×Q8—5242no—
120C2×Q8—5242no—
121C2×Q8—1311no—
122C2×D4—1311no—
123C22×C4—2622no—
124C22×C4—2621no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—2621no—
128C4⋊C4—21no—
129C4⋊C4—2622no—
130C4⋊C4—2621no—
131C4⋊C4—2621no—
132C4⋊C4—2621no—
133C4⋊C4—1311no—
134C4⋊C4—1311no—
135C4⋊C4D52412yes—
136C22⋊C4—2622no—
137C22⋊C4—2621no—
138C42—2621no—
139C42C22×D13111yescharacteristic
140C23—2621no—
141Q8—41no—
142Q8—41no—
143Q8—2621no—
144Q8—5242no—
145Q8—5244no—
146Q8—1312no—
147D4—2621no—
148D4—1312no—
149C2×C4—22no—
150C2×C4—2621no—
151C2×C4—2621no—
152C2×C4—2621no—
153C2×C4—5244no—
154C2×C4—5241no—
155C2×C4—5241no—
156C2×C4—5241no—
157C2×C4D4×D13111yescharacteristic
158C2×C4D4×D13111yescharacteristic
159C2×C4C2×D52411yescharacteristic
160C8—41no—
161C8—41no—
162C8—5241no—
163C22—2622no—
164C22—5241no—
165C22—5241no—
166C22#1686221611yescharacteristic
167C4—21no—
168C4—42no—
169C4—2622no—
170C4—5241no—
171C4—5241no—
172C4—10481no—
173C4SD16×D13111yescharacteristic
174C4Q16×D13111yescharacteristic
175C4#1686227011yescharacteristic
176C4#1686227511yescharacteristic
177C2—5242no—
178C2#2255346411yescharacteristic
179C2#2255346911yescharacteristic
180C2#2255348711yescharacteristic
subgroupG/H
centerC22#16862216
commutator, cosocleC42×C131C23
FrattiniC42C22×D131
Fitting#332×C131C2
socleC22×C131#120
prime radicalC2×D524D4
hypercenter#332D131
hypocenterC131#1637
LCSG ▷ C42×C131 ▷ C22×C131 ▷ C131
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #22553664C2semidirect
3. #22553608C2semidirect
5. #22553042C2semidirect
7. #332×C131C2semidirect
90. C131#1637semidirect
96. #332D131semidirect

Maximal subgroups

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

subgroupindex
#225539772
#225536642
#225536082
#225536032
#225530422
#225530392
#332×C1312
#1637131