Finite Group Databasebeta

← #30665151

Subgroups — #30665151

Order 16768.

250 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 55 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#22553982C211yesmaximal, characteristic
2#22553589C211yesmaximal, characteristic, complemented
3#22553519C211yesmaximal, characteristic, complemented
4#22553056C211yesmaximal, characteristic, complemented
5#22553044C211yesmaximal, characteristic, complemented
6#22553041C211yesmaximal, characteristic, complemented
7#486×C131C211yesmaximal, characteristic, complemented
8C2×#14523512—21no—
9C2×#14523503—21no—
10#16862242—21no—
11#16862218—22no—
12#16862217—21no—
13C4×Dic262C2211yescharacteristic
14C4×D524C2211yescharacteristic
15D4×Dic131—21no—
16#16862179—21no—
17C4⋊(C131⋊C8)C2211yescharacteristic
18C4⋊(C131⋊C8)C2211yescharacteristic
19#16862170—21no—
20#16862166—21no—
21C4⋊Q8×C131C2211yescharacteristic
22#127×C131C2211yescharacteristic
23#102×C131—22no—
24C4×C8×C131C2211yescharacteristic
25#14523512—42no—
26#14523512—42no—
27C2×Dic262D411yescharacteristic
28C2×D524D411yescharacteristic
29C2×C22⋊D131—22no—
30C2×C22⋊D131—22no—
31C22×Dic131—22no—
32C2×C4×D131—21no—
33#14523503—42no—
34C2×C131⋊C8—21no—
35C2×C131⋊C8—21no—
36C4⋊Dic131D411yescharacteristic
37C4⋊Dic131D411yescharacteristic
38Dic131⋊C4—21no—
39C22⋊Dic131—22no—
40(C2×D131)⋊C4—21no—
41C4×Dic131—21no—
42C2×Q8×C131—21no—
43C2×D4×C131—21no—
44C2×D4×C131—21no—
45C2×D4×C131D412yes—
46C2×C8×C131—21no—
47C4⋊C4×C131—21no—
48C4⋊C4×C131—21no—
49C42×C131C2311yescharacteristic
50C22×D131—21no—
51Dic262—21no—
52Dic262D812yes—
53D524—21no—
54D524—21no—
55C22⋊D131—44no—
56C22⋊D131—82no—
57C2×Dic131—21no—
58C2×Dic131—21no—
59C2×Dic131—21no—
60C2×Dic131—44no—
61C4×D131—41no—
62C4×D131—41no—
63C131⋊C8—41no—
64C131⋊C8—41no—
65C23×C131—22no—
66Q8×C131—41no—
67Q8×C131—41no—
68D4×C131—42no—
69D4×C131—41no—
70D4×C131—41no—
71D4×C131—41no—
72D4×C131—41no—
73C2×C4×C131—41no—
74C2×C4×C131C2×D411yescharacteristic
75C2×C4×C131C2×D411yescharacteristic
76C2×C4×C131C2×D411yescharacteristic
77C8×C131—41no—
78C22×C131—42no—
79C22×C131—42no—
80C22×C131—42no—
81C22×C131#12011yescharacteristic
82C2×D131—41no—
83C2×D131—41no—
84C2×D131—41no—
85C4×C131—21no—
86C4×C131—81no—
87C4×C131C2×D811yescharacteristic
88C4×C131#13511yescharacteristic
89C4×C131AGL(1,ℤ/8)11yescharacteristic
90C4×C131#13711yescharacteristic
91Dic131—22no—
92Dic131—41no—
93Dic131—41no—
94C2×C131—82no—
95C2×C131#44711yescharacteristic
96C2×C131#45211yescharacteristic
97C2×C131#45511yescharacteristic
98D131—81no—
99C131#164111yescharacteristic, complemented, Sylow
100#1641—1311nomaximal, Sylow
101#550—1311no—
102#486D13111yescharacteristic, complemented
103#474—1311no—
104#462—1311no—
105#331—1311no—
106Q8⋊C8—1311no—
107D4⋊C8—1311no—
108C2×#41—2621no—
109C2×Q16—2621no—
110C4⋊Q8C2×D13111yescharacteristic
111#127—2621no—
112#127C2×D13111yescharacteristic
113#121—2622no—
114#121—2621no—
115C4×Q8—1311no—
116C4×D4—2621no—
117C4×D4—1311no—
118#107—2621no—
119C4⋊C8—1311no—
120C4⋊C8—1311no—
121#103—2621no—
122#102—22no—
123#102—2621no—
124C4×C8C2×D13111yescharacteristic
125#41—5242no—
126#41—5242no—
127C2×Q8—21no—
128C2×Q8—1311no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—2622no—
132C2×D4—2622no—
133C2×D4—1311no—
134C2×D4C22⋊D13112yes—
135C22×C4—2622no—
136C22×C4—2621no—
137Q16—5242no—
138C2×C8—21no—
139C2×C8—2621no—
140C2×C8—2621no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—2621no—
144C4⋊C4—1311no—
145C4⋊C4—1311no—
146C22⋊C4—2622no—
147C22⋊C4—2621no—
148C42—2621no—
149C42C22×D13111yescharacteristic
150C23—22no—
151C23—2621no—
152Q8—41no—
153Q8—41no—
154Q8—2621no—
155Q8—1312no—
156D4—42no—
157D4—41no—
158D4—41no—
159D4—41no—
160D4—41no—
161D4—2621no—
162D4—2621no—
163D4—5244no—
164D4—10482no—
165C2×C4—41no—
166C2×C4—2621no—
167C2×C4—2621no—
168C2×C4—2621no—
169C2×C4—5244no—
170C2×C4—5241no—
171C2×C4—5241no—
172C2×C4D4×D13111yescharacteristic
173C2×C4D4×D13111yescharacteristic
174C2×C4C2×C22⋊D13111yescharacteristic
175C8—41no—
176C8—5241no—
177C8—5241no—
178C22—42no—
179C22—42no—
180C22—42no—
181C22—5241no—
182C22—5241no—
183C22—5241no—
184C22#1686221411yescharacteristic
185C4—21no—
186C4—2622no—
187C4—81no—
188C4—5241no—
189C4—5241no—
190C4D8×D13111yescharacteristic
191C4#1686226711yescharacteristic
192C4#1686227011yescharacteristic
193C4#1686227511yescharacteristic
194C2—82no—
195C2—10481no—
196C2#2255344411yescharacteristic
197C2#2255347111yescharacteristic
198C2#2255348911yescharacteristic
subgroupG/H
centerC22#16862214
commutator, cosocleC42×C131C23
FrattiniC42C22×D131
Fitting#486×C131C2
socleC22×C131#120
prime radical#22553982C2
hypercenter#486D131
hypocenterC131#1641
LCSG ▷ C42×C131 ▷ C22×C131 ▷ C131
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
2. #22553589C2semidirect
3. #22553519C2semidirect
4. #22553056C2semidirect
5. #22553044C2semidirect
6. #22553041C2semidirect
7. #486×C131C2semidirect
99. C131#1641semidirect
102. #486D131semidirect

Maximal subgroups

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

subgroupindex
#225539822
#225535892
#225535192
#225530562
#225530442
#225530412
#486×C1312
#1641131