Finite Group Databasebeta

← #30665293

Subgroups — #30665293

Order 16768.

184 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 45 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#22554066C211yesmaximal, characteristic
2#22553680C211yesmaximal, characteristic, complemented
3#22553674C211yesmaximal, characteristic
4#22553054C211yesmaximal, characteristic, complemented
5#22553053C211yesmaximal, characteristic, complemented
6#22553051C211yesmaximal, characteristic
7#501×C131C211yesmaximal, characteristic, complemented
8#16862245—21no—
9#16862241—21no—
10#16862241—21no—
11#16862238C2211yescharacteristic
12#16862234C2211yescharacteristic
13#16862231—21no—
14#16862226—21no—
15#16862224—21no—
16#16862170—21no—
17#16862170—21no—
18#16862169—21no—
19#16862166—21no—
20#16862157C2211yescharacteristic
21#16862157C2211yescharacteristic
22C4⋊Q8×C131C2211yescharacteristic
23#125×C131C2211yescharacteristic
24#107×C131—21no—
25#106×C131—21no—
26#97×C131C2211yescharacteristic
27C2×Dic262—21no—
28C2×Dic262D411yescharacteristic
29C2×D524D411yescharacteristic
30C2×C4×D131—21no—
31C2×C131⋊C8—21no—
32C2×C131⋊C8—21no—
33C4⋊Dic131—21no—
34C4⋊Dic131D411yescharacteristic
35C4⋊Dic131D411yescharacteristic
36Dic131⋊C4—21no—
37Dic131⋊C4—21no—
38Dic131⋊C4—21no—
39(C2×D131)⋊C4—21no—
40(C2×D131)⋊C4—21no—
41(C2×D131)⋊C4—21no—
42(C2×D131)⋊C4—21no—
43(C2×D131)⋊C4—21no—
44C4×Dic131—21no—
45C2×Q8×C131—21no—
46C2×C8×C131—21no—
47C4⋊C4×C131—21no—
48C4⋊C4×C131—21no—
49C4⋊C4×C131—21no—
50C4⋊C4×C131—21no—
51C4⋊C4×C131D411yescharacteristic
52C4⋊C4×C131D411yescharacteristic
53C42×C131C2311yescharacteristic
54C22×D131—21no—
55Dic262—41no—
56Dic262—81no—
57D524—41no—
58C2×Dic131—21no—
59C2×Dic131—21no—
60C2×Dic131—21no—
61C4×D131—81no—
62C131⋊C8—41no—
63C131⋊C8—41no—
64Q8×C131—42no—
65C2×C4×C131—21no—
66C2×C4×C131—21no—
67C2×C4×C131—41no—
68C2×C4×C131C2×D411yescharacteristic
69C2×C4×C131C2×D411yescharacteristic
70C2×C4×C131C2×D411yescharacteristic
71C8×C131—22no—
72C22×C131#12011yescharacteristic
73C2×D131—41no—
74C2×D131—41no—
75C2×D131—41no—
76C4×C131—21no—
77C4×C131—21no—
78C4×C131—21no—
79C4×C131—41no—
80C4×C131—41no—
81C4×C131—81no—
82Dic131—41no—
83Dic131—41no—
84Dic131—41no—
85C2×C131#45411yescharacteristic
86C2×C131#45511yescharacteristic
87C2×C131#45611yescharacteristic
88D131—81no—
89C131#166411yescharacteristic, complemented, Sylow
90#1664—1311nomaximal, Sylow
91#563—1311no—
92#501D13111yescharacteristic, complemented
93#491—1311no—
94#489—1311no—
95#330—1311no—
96#330—1311no—
97#329—1311no—
98C4⋊Q8—2621no—
99C4⋊Q8C2×D13111yescharacteristic
100#126—2621no—
101#126—2621no—
102#125—1311no—
103#125C2×D13111yescharacteristic
104#124—1311no—
105#123—2621no—
106C22⋊Q8—2621no—
107C22⋊Q8—2621no—
108#107—21no—
109#106—21no—
110#103—2621no—
111#103—2621no—
112#103—2621no—
113#102—2621no—
114#97—1311no—
115#97—1311no—
116#97C2×D13111yescharacteristic
117C2×Q8—21no—
118C2×Q8—2621no—
119C2×Q8—1311no—
120C2×D4—1311no—
121C22×C4—2621no—
122C2×C8—21no—
123C2×C8—2621no—
124C2×C8—2621no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—2621no—
130C4⋊C4—2621no—
131C4⋊C4—2621no—
132C4⋊C4—2621no—
133C4⋊C4—1311no—
134C4⋊C4—1311no—
135C4⋊C4C22⋊D13111yescharacteristic
136C4⋊C4C22⋊D13111yescharacteristic
137C22⋊C4—2621no—
138C22⋊C4—2621no—
139C22⋊C4—2621no—
140C22⋊C4—2621no—
141C22⋊C4—2621no—
142C42—2621no—
143C42C22×D13111yescharacteristic
144C23—2621no—
145Q8—42no—
146Q8—5241no—
147Q8—10481no—
148D4—5241no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—2621no—
153C2×C4—2621no—
154C2×C4—2621no—
155C2×C4—10481no—
156C2×C4D4×D13111yescharacteristic
157C2×C4D4×D13111yescharacteristic
158C2×C4C2×C22⋊D13111yescharacteristic
159C8—22no—
160C8—5241no—
161C8—5241no—
162C22—5241no—
163C22—5241no—
164C22—5241no—
165C22#1686221411yescharacteristic
166C4—21no—
167C4—21no—
168C4—21no—
169C4—41no—
170C4—41no—
171C4—81no—
172C4—5241no—
173C4—5241no—
174C4—5241no—
175C2—10481no—
176C2#2255348311yescharacteristic
177C2#2255348811yescharacteristic
178C2#2255349411yescharacteristic
subgroupG/H
centerC22#16862214
commutator, cosocleC42×C131C23
FrattiniC42C22×D131
Fitting#501×C131C2
socleC22×C131#120
prime radicalC2×D524D4
hypercenter#501D131
hypocenterC131#1664
LCSG ▷ C42×C131 ▷ C22×C131 ▷ C131
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
2. #22553680C2semidirect
4. #22553054C2semidirect
5. #22553053C2semidirect
7. #501×C131C2semidirect
89. C131#1664semidirect
92. #501D131semidirect

Maximal subgroups

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

subgroupindex
#225540662
#225536802
#225536742
#225530542
#225530532
#225530512
#501×C1312
#1664131