Finite Group Databasebeta

← #30665000

Subgroups — #30665000

Order 16768.

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

normal subgroups only

This group has 246 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#22553893C211yesmaximal, characteristic
2#22553631C212yesmaximal, complemented
3#22553251C211yesmaximal, characteristic, complemented
4#22553047C212yesmaximal, complemented
5#324×C131C211yesmaximal, characteristic, complemented
6C2×#14523513—21no—
7#16862226—22no—
8#16862221—22no—
9#16862220—21no—
10#16862220C2212yes—
11C2×C4⋊Dic131C2211yescharacteristic
12#16862178—22no—
13#16862168—22no—
14C131⋊#95—21no—
15C131⋊#95—21no—
16C131⋊#95C2211yescharacteristic
17C2×C4⋊C4×C131C2211yescharacteristic
18C22⋊C8×C131C2212yes—
19#14523513—42no—
20#14523513—42no—
21C2×Dic262—21no—
22C2×D524—21no—
23C2×D524—21no—
24C2×D524D412yes—
25C2×C22⋊D131—22no—
26C22×Dic131—21no—
27C22×Dic131—21no—
28C2×C4×D131—22no—
29C4⋊Dic131—21no—
30C4⋊Dic131—22no—
31C4⋊Dic131D412yes—
32(C2×D131)⋊C4—22no—
33(C2×D131)⋊C4—22no—
34C22×C4×C131D411yescharacteristic
35C22×C4×C131D411yescharacteristic
36C22×C4×C131C2311yescharacteristic
37C2×C8×C131—22no—
38C4⋊C4×C131—22no—
39C22×D131—22no—
40Dic262—42no—
41D524—42no—
42D524—42no—
43D524—81no—
44C22⋊D131—42no—
45C22⋊D131—42no—
46C2×Dic131—22no—
47C2×Dic131—41no—
48C2×Dic131—41no—
49C2×Dic131—42no—
50C2×Dic131—41no—
51C2×Dic131—41no—
52C4×D131—44no—
53C23×C131C2×D411yescharacteristic
54C2×C4×C131—21no—
55C2×C4×C131—21no—
56C2×C4×C131—21no—
57C2×C4×C131—41no—
58C2×C4×C131—41no—
59C2×C4×C131SD1612yes—
60C2×C4×C131C2×D412yes—
61C8×C131—42no—
62C22×C131—21no—
63C22×C131—21no—
64C22×C131#12011yescharacteristic
65C22×C131C2×SD1611yescharacteristic
66C22×C131AGL(1,ℤ/8)11yescharacteristic
67C2×D131—42no—
68C2×D131—42no—
69C2×D131—42no—
70C4×C131—22no—
71C4×C131—22no—
72C4×C131—41no—
73Dic131—42no—
74Dic131—81no—
75Dic131—81no—
76C2×C131—21no—
77C2×C131—21no—
78C2×C131#45011yescharacteristic
79C2×C131#45411yescharacteristic
80C2×C131#45811yescharacteristic
81D131—82no—
82C131#161611yescharacteristic, complemented, Sylow
83#1616—1311nomaximal, Sylow
84#537—1311no—
85#481—1312no—
86#400—1311no—
87#327—1312no—
88#324D13111yescharacteristic, complemented
89C2×#41—2621no—
90C22⋊Q8—2622no—
91#121—2622no—
92#121—2621no—
93#121—1312no—
94C2×C4⋊C4—1311no—
95C2×C4⋊C4C2×D13111yescharacteristic
96#106—2622no—
97#102—2622no—
98C22⋊C8C2×D13112yes—
99#95—2621no—
100#95—2621no—
101#95—1311no—
102#41—5242no—
103#41—5242no—
104C2×Q8—2621no—
105C2×D4—2621no—
106C2×D4—2621no—
107C2×D4—2622no—
108C2×D4—1312no—
109C22×C4—2622no—
110C22×C4—2621no—
111C22×C4—2621no—
112C22×C4D52411yescharacteristic
113C22×C4D52411yescharacteristic
114C22×C4C22×D13111yescharacteristic
115C2×C8—22no—
116C4⋊C4—22no—
117C4⋊C4—2621no—
118C4⋊C4—2622no—
119C4⋊C4—1312no—
120C22⋊C4—2622no—
121C22⋊C4—2622no—
122C23—2622no—
123C23C2×D52411yescharacteristic
124Q8—5242no—
125D4—5242no—
126D4—5242no—
127D4—5242no—
128D4—5242no—
129D4—10481no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—2622no—
136C2×C4—5244no—
137C2×C4—5241no—
138C2×C4—5241no—
139C2×C4—5242no—
140C2×C4—5241no—
141C2×C4—5241no—
142C2×C4#1452350212yes—
143C2×C4D4×D13112yes—
144C8—42no—
145C22—21no—
146C22—21no—
147C22—5242no—
148C22—5242no—
149C22—5242no—
150C22#1686221611yescharacteristic
151C22C2×#1452350211yescharacteristic
152C22#1686227411yescharacteristic
153C4—22no—
154C4—22no—
155C4—41no—
156C4—5242no—
157C4—10481no—
158C4—10481no—
159C2—21no—
160C2—21no—
161C2—10482no—
162C2#2255346511yescharacteristic
163C2#2255348711yescharacteristic
164C2#2255350311yescharacteristic
subgroupG/H
centerC22#16862216
commutator, cosocleC22×C4×C131C23
FrattiniC22×C4C22×D131
Fitting#324×C131C2
socleC22×C131#120
prime radical#22553893C2
hypercenter#324D131
hypocenterC131#1616
LCSG ▷ C22×C4×C131 ▷ C22×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #22553631C2semidirect
3. #22553251C2semidirect
4. #22553047C2semidirect
5. #324×C131C2semidirect
82. C131#1616semidirect
88. #324D131semidirect

Maximal subgroups

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

subgroupindex
#225538932
#225536312
#225532512
#225530472
#324×C1312
#1616131