Finite Group Databasebeta

← #30665255

Subgroups — #30665255

Order 16768.

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

normal subgroups only

This group has 230 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#22554056C211yesmaximal, characteristic, complemented
2#22553702C211yesmaximal, characteristic, complemented
3#22553701C211yesmaximal, characteristic, complemented
4#22553677C211yesmaximal, characteristic, complemented
5#22553053C211yesmaximal, characteristic, complemented
6#22553051C211yesmaximal, characteristic, complemented
7#329×C131C211yesmaximal, characteristic, complemented
8C2×#14523502—21no—
9C2×#14523500—21no—
10C2×D1048—21no—
11C2×#14523498—21no—
12#16862243C2211yescharacteristic
13#16862238C2211yescharacteristic
14#16862234C2211yescharacteristic
15#16862230—21no—
16#16862228—21no—
17#16862223—21no—
18#16862222—21no—
19#16862219—21no—
20#16862218—21no—
21#16862168—21no—
22#16862168—21no—
23#16862157C2211yescharacteristic
24#124×C131C2211yescharacteristic
25#97×C131C2211yescharacteristic
26#97×C131C2211yescharacteristic
27C2×Dic262D411yescharacteristic
28C2×D524—21no—
29C2×D524—21no—
30C2×D524—21no—
31C2×D524D411yescharacteristic
32C2×C22⋊D131—21no—
33C2×C22⋊D131—21no—
34C22×Dic131—21no—
35C2×C4×D131—21no—
36#14523502—42no—
37#14523500—42no—
38D1048—42no—
39#14523498—42no—
40C2×C131⋊C8—21no—
41C4⋊Dic131—21no—
42C4⋊Dic131D411yescharacteristic
43C4⋊Dic131D411yescharacteristic
44Dic131⋊C4—21no—
45Dic131⋊C4—21no—
46Dic131⋊C4—21no—
47C22⋊Dic131—21no—
48(C2×D131)⋊C4—21no—
49(C2×D131)⋊C4—21no—
50(C2×D131)⋊C4—21no—
51C2×Q8×C131D411yescharacteristic
52C2×D4×C131D411yescharacteristic
53C2×C8×C131—21no—
54C2×C8×C131—21no—
55C22⋊C4×C131—21no—
56C22⋊C4×C131—21no—
57C42×C131C2311yescharacteristic
58C22×D131—21no—
59C22×D131—41no—
60Dic262—41no—
61D524—41no—
62D524—41no—
63D524—41no—
64D524—41no—
65D524—41no—
66D524—42no—
67C22⋊D131—81no—
68C22⋊D131—81no—
69C2×Dic131—21no—
70C2×Dic131—21no—
71C2×Dic131—21no—
72C2×Dic131—81no—
73C4×D131—81no—
74C131⋊C8—22no—
75C23×C131—21no—
76Q8×C131—41no—
77D4×C131—41no—
78C2×C4×C131—21no—
79C2×C4×C131C2×D411yescharacteristic
80C2×C4×C131C2×D411yescharacteristic
81C2×C4×C131C2×D411yescharacteristic
82C8×C131—22no—
83C8×C131—41no—
84C22×C131—41no—
85C22×C131—41no—
86C22×C131—41no—
87C22×C131#12011yescharacteristic
88C2×D131—41no—
89C2×D131—41no—
90C2×D131—41no—
91C2×D131—81no—
92C2×D131—81no—
93C2×D131—81no—
94C4×C131—21no—
95C4×C131—21no—
96C4×C131—21no—
97C4×C131—41no—
98Dic131—41no—
99Dic131—41no—
100Dic131—41no—
101C2×C131—81no—
102C2×C131#45311yescharacteristic
103C2×C131#45311yescharacteristic
104C2×C131#45411yescharacteristic
105D131—81no—
106D131—161no—
107C131#165611yescharacteristic, complemented, Sylow
108#1656—1311nomaximal, Sylow
109#562—1311no—
110#496—1311no—
111#496—1311no—
112#490—1311no—
113#330—1311no—
114#329—1311no—
115#329D13111yescharacteristic, complemented
116C2×SD16—2621no—
117C2×SD16—2621no—
118C2×D8—2621no—
119C2×D8—2621no—
120#127—1311no—
121#125—1311no—
122#124—1311no—
123#124C2×D13111yescharacteristic
124#123—2621no—
125#123—2621no—
126C22⋊Q8—2621no—
127C22⋊Q8—2621no—
128#121—2621no—
129#121—2621no—
130#102—2621no—
131#102—2621no—
132#97—1311no—
133#97C2×D13111yescharacteristic
134#97C2×D13111yescharacteristic
135C2×Q8—1311no—
136C2×Q8D52411yescharacteristic
137C2×D4—2621no—
138C2×D4—2621no—
139C2×D4—2621no—
140C2×D4—2621no—
141C2×D4—2621no—
142C2×D4—1311no—
143C2×D4D52411yescharacteristic
144C22×C4—2621no—
145C22×C4—2621no—
146SD16—5242no—
147SD16—5242no—
148D8—5242no—
149D8—5242no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—2621no—
153C4⋊C4—2621no—
154C4⋊C4—2621no—
155C4⋊C4—2621no—
156C4⋊C4—2621no—
157C4⋊C4—1311no—
158C4⋊C4—1311no—
159C22⋊C4—21no—
160C22⋊C4—21no—
161C22⋊C4—2621no—
162C22⋊C4—2621no—
163C22⋊C4—2621no—
164C22⋊C4—2621no—
165C42C22×D13111yescharacteristic
166C23—21no—
167C23—2621no—
168C23—5241no—
169Q8—41no—
170Q8—5241no—
171D4—41no—
172D4—5241no—
173D4—5241no—
174D4—5241no—
175D4—5241no—
176D4—5241no—
177D4—5242no—
178D4—10481no—
179D4—10481no—
180C2×C4—21no—
181C2×C4—2621no—
182C2×C4—2621no—
183C2×C4—2621no—
184C2×C4—10481no—
185C2×C4—10481no—
186C2×C4D4×D13111yescharacteristic
187C2×C4D4×D13111yescharacteristic
188C2×C4C2×D52411yescharacteristic
189C8—22no—
190C8—41no—
191C8—2622no—
192C22—41no—
193C22—41no—
194C22—41no—
195C22—5241no—
196C22—5241no—
197C22—5241no—
198C22—10481no—
199C22—10481no—
200C22—10481no—
201C22#1686221611yescharacteristic
202C4—21no—
203C4—21no—
204C4—21no—
205C4—41no—
206C4—5241no—
207C4—5241no—
208C4—5241no—
209C2—81no—
210C2—10481no—
211C2—20961no—
212C2#2255348111yescharacteristic
213C2#2255348211yescharacteristic
214C2#2255348611yescharacteristic
subgroupG/H
centerC22#16862216
commutator, cosocleC42×C131C23
FrattiniC42C22×D131
Fitting#329×C131C2
socleC22×C131#120
hypercenter#329D131
hypocenterC131#1656
LCSG ▷ C42×C131 ▷ C22×C131 ▷ C131
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
1. #22554056C2semidirect
2. #22553702C2semidirect
3. #22553701C2semidirect
4. #22553677C2semidirect
5. #22553053C2semidirect
6. #22553051C2semidirect
7. #329×C131C2semidirect
107. C131#1656semidirect
115. #329D131semidirect

Maximal subgroups

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

subgroupindex
#225540562
#225537022
#225537012
#225536772
#225530532
#225530512
#329×C1312
#1656131