Finite Group Databasebeta

← #30664372

Subgroups — #30664372

Order 16768.

234 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 77 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#22553846C211yesmaximal, characteristic
2C4×#14523497C211yesmaximal, characteristic, complemented
3#22553042C212yesmaximal, complemented
4#22553039C212yesmaximal, complemented
5C2×C4⋊C8×C131C211yesmaximal, characteristic, complemented
6C2×#14523497—21no—
7#16862240—21no—
8#16862238C411yescharacteristic
9#16862234C411yescharacteristic
10#16862229—21no—
11#16862225C411yescharacteristic
12#16862220C411yescharacteristic
13C4×Dic262C2211yescharacteristic
14C4×D524C2211yescharacteristic
15C4×C22⋊D131—21no—
16#16862203—21no—
17#16862157—21no—
18C4×C131⋊C8C2212yes—
19C22×C8×C131—21no—
20C2×C42×C131C2211yescharacteristic
21C4⋊C8×C131—21no—
22C4⋊C8×C131C2212yes—
23C2×Dic262C2×C411yescharacteristic
24C2×D524C2×C411yescharacteristic
25C2×C22⋊D131—21no—
26C2×C4×D131—21no—
27#14523497—24no—
28C2×C131⋊C8—22no—
29C4⋊Dic131C2×C411yescharacteristic
30C4⋊Dic131C2×C411yescharacteristic
31Dic131⋊C4—21no—
32Dic131⋊C4—21no—
33Dic131⋊C4—21no—
34Dic131⋊C4—21no—
35C22⋊Dic131—21no—
36(C2×D131)⋊C4—21no—
37(C2×D131)⋊C4—21no—
38(C2×D131)⋊C4—21no—
39(C2×D131)⋊C4—21no—
40C4×Dic131—21no—
41C22×C4×C131C2×C411yescharacteristic
42C22×C4×C131D411yescharacteristic
43C22×C4×C131D411yescharacteristic
44C2×C8×C131—22no—
45C2×C8×C131—42no—
46C42×C131C2×C411yescharacteristic
47C42×C131D411yescharacteristic
48C42×C131D411yescharacteristic
49C42×C131C2311yescharacteristic
50C22×D131—21no—
51Dic262—22no—
52D524—22no—
53C22⋊D131—42no—
54C2×Dic131—21no—
55C2×Dic131—21no—
56C2×Dic131—21no—
57C4×D131—41no—
58C4×D131—41no—
59C131⋊C8—24no—
60C23×C131C22⋊C411yescharacteristic
61C2×C4×C131—22no—
62C2×C4×C131—22no—
63C2×C4×C131—21no—
64C2×C4×C131—21no—
65C2×C4×C131C22⋊C411yescharacteristic
66C2×C4×C131C22⋊C411yescharacteristic
67C2×C4×C131C22⋊C411yescharacteristic
68C2×C4×C131C22×C411yescharacteristic
69C2×C4×C131C2×D411yescharacteristic
70C2×C4×C131C2×D411yescharacteristic
71C8×C131—42no—
72C22×C131—21no—
73C22×C131—21no—
74C22×C131#10412yes—
75C22×C131C2×C22⋊C411yescharacteristic
76C2×D131—41no—
77C2×D131—41no—
78C2×D131—41no—
79C4×C131—21no—
80C4×C131—21no—
81C4×C131—21no—
82C4×C131—21no—
83C4×C131#13112yes—
84C4×C131AGL(1,ℤ/8)11yescharacteristic
85C4×C131#13711yescharacteristic
86Dic131—41no—
87Dic131—41no—
88Dic131—41no—
89C2×C131—22no—
90C2×C131#40811yescharacteristic
91C2×C131#41711yescharacteristic
92C2×C131C2×#10411yescharacteristic
93D131—81no—
94C131#148311yescharacteristic, complemented, Sylow
95#1483—1311nomaximal, Sylow
96#529—1311no—
97C2×C4⋊C8D13111yescharacteristic, complemented
98C4×M16—1311no—
99Q8⋊C8—1312no—
100D4⋊C8—1312no—
101C2×M16—2621no—
102C22×C8—21no—
103#126—2621no—
104#125—1311no—
105#124—1311no—
106#123—2621no—
107C22⋊Q8—1311no—
108#121—1311no—
109C4×Q8—1311no—
110C4×D4—2621no—
111C4×D4—1311no—
112#117—2621no—
113C2×C42C2×D13111yescharacteristic
114C4⋊C8—21no—
115C4⋊C8C2×D13112yes—
116#97—2621no—
117C4×C8—1312no—
118C2×Q8—1311no—
119C2×D4—2621no—
120C2×D4—1311no—
121C22×C4—2621no—
122C22×C4C4×D13111yescharacteristic
123C22×C4C22⋊D13111yescharacteristic
124C22×C4D52411yescharacteristic
125M16—2624no—
126C2×C8—22no—
127C2×C8—42no—
128C2×C8—2622no—
129C4⋊C4—2621no—
130C4⋊C4—2621no—
131C4⋊C4—2621no—
132C4⋊C4—2621no—
133C4⋊C4—1311no—
134C4⋊C4—1311no—
135C22⋊C4—2621no—
136C22⋊C4—2621no—
137C22⋊C4—2621no—
138C22⋊C4—2621no—
139C22⋊C4—2621no—
140C42—2621no—
141C42C4×D13111yescharacteristic
142C42C22⋊D13111yescharacteristic
143C42D52411yescharacteristic
144C42C22×D13111yescharacteristic
145C23—2621no—
146C23(C2×D131)⋊C411yescharacteristic
147Q8—2622no—
148D4—2622no—
149D4—5242no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—2621no—
155C2×C4—2621no—
156C2×C4—2621no—
157C2×C4—5241no—
158C2×C4—5241no—
159C2×C4(C2×D131)⋊C411yescharacteristic
160C2×C4(C2×D131)⋊C411yescharacteristic
161C2×C4(C2×D131)⋊C411yescharacteristic
162C2×C4C2×C4×D13111yescharacteristic
163C2×C4C2×C22⋊D13111yescharacteristic
164C2×C4C2×D52411yescharacteristic
165C8—42no—
166C8—2624no—
167C22—21no—
168C22—21no—
169C22—5241no—
170C22—5241no—
171C22—5241no—
172C22#1686217212yes—
173C22C2×(C2×D131)⋊C411yescharacteristic
174C4—21no—
175C4—21no—
176C4—21no—
177C4—21no—
178C4—5241no—
179C4—5241no—
180C4—5241no—
181C4#1686225212yes—
182C4#1686227211yescharacteristic
183C4#1686227711yescharacteristic
184C2—22no—
185C2—10481no—
186C2#2255327811yescharacteristic
187C2#2255331111yescharacteristic
188C2C2×#1686217211yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D131
commutatorC2×C4×C131C22×C4
FrattiniC42C22×D131
FittingC2×C4⋊C8×C131C2
socleC22×C131C2×C22⋊C4
prime radical#16862220C4
cosocleC42×C131C23
hypercenterC2×C4⋊C8D131
hypocenterC131#1483
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4⋊C8

Complements

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

NH ≅ G/Ntype
2. C4×#14523497C2semidirect
3. #22553042C2semidirect
4. #22553039C2semidirect
5. C2×C4⋊C8×C131C2semidirect
94. C131#1483semidirect
97. C2×C4⋊C8D131semidirect

Maximal subgroups

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

subgroupindex
#225538462
C4×#145234972
#225530422
#225530392
C2×C4⋊C8×C1312
#1483131