Finite Group Databasebeta

← #30664993

Subgroups — #30664993

Order 16768.

208 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 49 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#22553907C211yesmaximal, characteristic
2#22553648C211yesmaximal, characteristic, complemented
3#22553628C211yesmaximal, characteristic
4#22553252C211yesmaximal, characteristic, complemented
5#22553048C211yesmaximal, characteristic, complemented
6#22553037C211yesmaximal, characteristic, complemented
7#327×C131C211yesmaximal, characteristic
8#16862239—21no—
9#16862236—21no—
10#16862227—21no—
11#16862227—21no—
12#16862225C2211yescharacteristic
13#16862223—21no—
14D4×Dic131—21no—
15C2×C4⋊Dic131C2211yescharacteristic
16C2×Dic131⋊C4C2211yescharacteristic
17#16862180—21no—
18#16862177—21no—
19#16862171—21no—
20#16862167—21no—
21C22⋊(C131⋊C8)C2211yescharacteristic
22C131⋊#95—21no—
23C131⋊#95—21no—
24#121×C131C2211yescharacteristic
25C22⋊C8×C131C2211yescharacteristic
26#95×C131C2211yescharacteristic
27C2×Dic262D411yescharacteristic
28C22×Dic131—21no—
29C22×Dic131—21no—
30C22×Dic131D411yescharacteristic
31C22×Dic131D411yescharacteristic
32C2×C131⋊C8—21no—
33C4⋊Dic131—21no—
34C4⋊Dic131—21no—
35C4⋊Dic131—21no—
36C4⋊Dic131D411yescharacteristic
37Dic131⋊C4—21no—
38Dic131⋊C4—21no—
39Dic131⋊C4—21no—
40Dic131⋊C4—21no—
41Dic131⋊C4—21no—
42C22⋊Dic131—21no—
43C22⋊Dic131—21no—
44C22⋊Dic131—21no—
45C22⋊Dic131—21no—
46C4×Dic131—21no—
47C2×D4×C131—21no—
48C2×D4×C131D411yescharacteristic
49C22×C4×C131—21no—
50C22×C4×C131C2311yescharacteristic
51C2×C8×C131—21no—
52C4⋊C4×C131D411yescharacteristic
53C22⋊C4×C131—21no—
54Dic262—41no—
55C2×Dic131—21no—
56C2×Dic131—21no—
57C2×Dic131—21no—
58C2×Dic131—21no—
59C2×Dic131—21no—
60C2×Dic131—21no—
61C2×Dic131—41no—
62C2×Dic131—42no—
63C2×Dic131—41no—
64C2×Dic131—81no—
65C131⋊C8—41no—
66C23×C131—21no—
67C23×C131C2×D411yescharacteristic
68D4×C131—41no—
69D4×C131—41no—
70D4×C131—41no—
71C2×C4×C131—21no—
72C2×C4×C131—41no—
73C2×C4×C131—41no—
74C2×C4×C131—41no—
75C2×C4×C131—41no—
76C2×C4×C131C2×D411yescharacteristic
77C2×C4×C131C2×D411yescharacteristic
78C8×C131—41no—
79C22×C131—21no—
80C22×C131—21no—
81C22×C131—41no—
82C22×C131—41no—
83C22×C131—41no—
84C22×C131#12011yescharacteristic
85C22×C131#13511yescharacteristic
86C22×C131#13711yescharacteristic
87C4×C131—21no—
88C4×C131—21no—
89C4×C131—41no—
90C4×C131—81no—
91Dic131—41no—
92Dic131—41no—
93Dic131—41no—
94Dic131—41no—
95Dic131—81no—
96C2×C131—21no—
97C2×C131—21no—
98C2×C131—81no—
99C2×C131#45211yescharacteristic
100C2×C131#45411yescharacteristic
101C2×C131#45811yescharacteristic
102C131#161511yescharacteristic, complemented, Sylow
103#1615—1311nomaximal, Sylow
104#539—1311no—
105#484—1311no—
106#481—1311no—
107#400—1311no—
108#328—1311no—
109#327D13111yescharacteristic
110#324—1311no—
111#126—2621no—
112#125—2621no—
113#123—2621no—
114#123—2621no—
115C22⋊Q8—2621no—
116C22⋊Q8—1311no—
117#121C2×D13111yescharacteristic
118C4×D4—2621no—
119C2×C4⋊C4—1311no—
120C2×C4⋊C4—1311no—
121#107—2621no—
122#106—2621no—
123#103—2621no—
124#102—2621no—
125C22⋊C8—1311no—
126C22⋊C8C2×D13111yescharacteristic
127#95—2621no—
128#95—2621no—
129#95C2×D13111yescharacteristic
130C2×Q8—1311no—
131C2×D4—21no—
132C2×D4D52411yescharacteristic
133C22×C4—21no—
134C22×C4—2621no—
135C22×C4—2621no—
136C22×C4—1311no—
137C22×C4—1311no—
138C22×C4C22×D13111yescharacteristic
139C2×C8—21no—
140C2×C8—2621no—
141C4⋊C4—2621no—
142C4⋊C4—2621no—
143C4⋊C4—2621no—
144C4⋊C4—2621no—
145C4⋊C4—2621no—
146C4⋊C4—2621no—
147C4⋊C4—2621no—
148C4⋊C4—2621no—
149C4⋊C4—1311no—
150C4⋊C4D52411yescharacteristic
151C22⋊C4—21no—
152C22⋊C4—2621no—
153C22⋊C4—2621no—
154C22⋊C4—2621no—
155C22⋊C4—2621no—
156C42—2621no—
157C23—21no—
158C23D4×D13111yescharacteristic
159Q8—5241no—
160D4—41no—
161D4—41no—
162D4—41no—
163C2×C4—21no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—2621no—
169C2×C4—2621no—
170C2×C4—2621no—
171C2×C4—2621no—
172C2×C4—2621no—
173C2×C4—2621no—
174C2×C4—5241no—
175C2×C4—5242no—
176C2×C4—5241no—
177C2×C4—10481no—
178C2×C4D4×D13111yescharacteristic
179C2×C4C2×D52411yescharacteristic
180C8—41no—
181C8—5241no—
182C22—21no—
183C22—21no—
184C22—41no—
185C22—41no—
186C22—41no—
187C22#1686221611yescharacteristic
188C22#1686226511yescharacteristic
189C22#1686227511yescharacteristic
190C4—21no—
191C4—21no—
192C4—41no—
193C4—81no—
194C4—5241no—
195C4—5241no—
196C4—5241no—
197C4—5241no—
198C4—10481no—
199C2—21no—
200C2—21no—
201C2—81no—
202C2#2255347511yescharacteristic
203C2#2255348611yescharacteristic
204C2#2255350311yescharacteristic
subgroupG/H
centerC22#16862216
commutator, cosocleC22×C4×C131C23
FrattiniC22×C4C22×D131
Fitting#327×C131C2
socleC22×C131#120
prime radical#121×C131C22
hypercenter#327D131
hypocenterC131#1615
LCSG ▷ C22×C4×C131 ▷ C22×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #22553648C2semidirect
4. #22553252C2semidirect
5. #22553048C2semidirect
6. #22553037C2semidirect
102. C131#1615semidirect

Maximal subgroups

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

subgroupindex
#225539072
#225536482
#225536282
#225532522
#225530482
#225530372
#327×C1312
#1615131