Finite Group Databasebeta

← #30639163

Subgroups — #30639163

Order 16256.

360 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 109 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
1C2×#16861188C211yesmaximal, characteristic
2C2×#16861134C212yesmaximal, complemented
3#22549736C211yesmaximal, characteristic
4#22549639C212yesmaximal, complemented
5C22×M16×C127C211yesmaximal, characteristic
6C22×Dic254C2211yescharacteristic
7#16861188—22no—
8#16861188C414yescomplemented
9C2×C4⋊Dic127C2212yes—
10C2×C4⋊Dic127C2211yescharacteristic
11C2×Dic127⋊C4—21no—
12C2×C22⋊Dic127—21no—
13C2×C22⋊Dic127—21no—
14#16861134—24no—
15C127⋊#95—21no—
16C23×C4×C127C2211yescharacteristic
17C2×M16×C127—22no—
18C2×M16×C127—24no—
19C22×C8×C127C2212yes—
20C2×Dic254—21no—
21C2×Dic254—22no—
22C2×Dic254—22no—
23C2×Dic254C2×C412yes—
24C22×Dic127—21no—
25C22×Dic127—21no—
26C22×Dic127—22no—
27C4⋊Dic127—21no—
28C4⋊Dic127—22no—
29C4⋊Dic127—22no—
30C4⋊Dic127C2×C412yes—
31Dic127⋊C4—24no—
32C22⋊Dic127—24no—
33C22⋊Dic127—42no—
34C24×C127D411yescharacteristic
35C22×C4×C127—22no—
36C22×C4×C127—22no—
37C22×C4×C127C2×C412yes—
38C22×C4×C127D412yes—
39C22×C4×C127D411yescharacteristic
40C22×C4×C127C2311yescharacteristic
41M16×C127—28no—
42C2×C8×C127—22no—
43C2×C8×C127—22no—
44C2×C8×C127D414yes—
45Dic254—24no—
46Dic254—41no—
47Dic254—41no—
48C2×Dic127—22no—
49C2×Dic127—22no—
50C2×Dic127—42no—
51C2×Dic127—41no—
52C2×Dic127—41no—
53C2×Dic127—42no—
54C2×Dic127—42no—
55C2×Dic127—42no—
56C23×C127—22no—
57C23×C127—21no—
58C23×C127—21no—
59C23×C127—22no—
60C23×C127C22⋊C412yes—
61C23×C127C2×D411yescharacteristic
62C2×C4×C127—22no—
63C2×C4×C127—24no—
64C2×C4×C127—22no—
65C2×C4×C127—22no—
66C2×C4×C127C22⋊C412yes—
67C2×C4×C127C22×C411yescharacteristic
68C2×C4×C127C2×D412yes—
69C2×C4×C127C2×D411yescharacteristic
70C2×C4×C127#4112yes—
71C8×C127—24no—
72C22×C127—22no—
73C22×C127—22no—
74C22×C127—22no—
75C22×C127—42no—
76C22×C127—42no—
77C22×C127C2×C22⋊C411yescharacteristic
78C22×C127#12112yes—
79C22×C127AGL(1,ℤ/8)11yescharacteristic
80C22×C127#13711yescharacteristic
81C22×C127#13712yes—
82C4×C127—22no—
83C4×C127C4×D412yes—
84C4×C127#12011yescharacteristic
85C4×C127#12311yescharacteristic
86Dic127—42no—
87Dic127—42no—
88Dic127—82no—
89C2×C127—42no—
90C2×C127#38611yescharacteristic
91C2×C127#41711yescharacteristic
92C2×C127#41911yescharacteristic
93C2×C127#46812yes—
94C2×C127#47012yes—
95#1894—1271nomaximal, Sylow
96C127#189411yescharacteristic, complemented, Sylow
97C22×M16D12711yescharacteristic
98C2×C22⋊Q8—1271no—
99C2×#103—1272no—
100#380—1271no—
101#340—1272no—
102C22×Q8—1271no—
103C23×C4C2×D12711yescharacteristic
104C2×M16—22no—
105C2×M16—24no—
106C22×C8C2×D12712yes—
107C22⋊Q8—1274no—
108C22⋊Q8—2542no—
109C2×C4⋊C4—1271no—
110C2×C4⋊C4—1272no—
111C2×C4⋊C4—2541no—
112C2×C22⋊C4—2541no—
113C2×C22⋊C4—2541no—
114#103—2544no—
115#95—2541no—
116C24D50811yescharacteristic
117C2×Q8—1272no—
118C2×Q8—2542no—
119C2×Q8—2542no—
120C2×Q8—2541no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—2542no—
124C22×C4—2541no—
125C22×C4—2541no—
126C22×C4C4×D12712yes—
127C22×C4C22⋊D12712yes—
128C22×C4D50811yescharacteristic
129C22×C4C22×D12711yescharacteristic
130M16—28no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8C22⋊D12714yes—
134C4⋊C4—1272no—
135C4⋊C4—2542no—
136C4⋊C4—2542no—
137C4⋊C4—2544no—
138C4⋊C4—2541no—
139C22⋊C4—5082no—
140C22⋊C4—2544no—
141C23—21no—
142C23—21no—
143C23—22no—
144C23—22no—
145C23(C2×D127)⋊C412yes—
146C23C2×D50811yescharacteristic
147Q8—5081no—
148Q8—5081no—
149Q8—2544no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—24no—
153C2×C4—22no—
154C2×C4—5082no—
155C2×C4—5082no—
156C2×C4—5082no—
157C2×C4—5081no—
158C2×C4—5081no—
159C2×C4—5082no—
160C2×C4—2542no—
161C2×C4—2542no—
162C2×C4(C2×D127)⋊C412yes—
163C2×C4C2×C4×D12711yescharacteristic
164C2×C4C2×C22⋊D12711yescharacteristic
165C2×C4C2×C22⋊D12712yes—
166C2×C4#1452324812yes—
167C8—24no—
168C22—22no—
169C22—22no—
170C22—22no—
171C22—42no—
172C22—42no—
173C22C2×(C2×D127)⋊C411yescharacteristic
174C22#1686118312yes—
175C22#1686123711yescharacteristic
176C22#1686124212yes—
177C22#1686124211yescharacteristic
178C4—22no—
179C4—10162no—
180C4—5082no—
181C4—5082no—
182C4C4×C22⋊D12712yes—
183C4#1686117811yescharacteristic
184C4#1686119211yescharacteristic
185C2—42no—
186C2#2254975511yescharacteristic
187C2#2254987811yescharacteristic
188C2#2254988611yescharacteristic
189C2#2255012212yes—
190C2#2255013412yes—
subgroupG/H
centerC23C2×D508
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC22×M16×C127C2
socleC23×C127C2×D4
prime radicalC24×C127D4
cosocleC22×C4×C127C23
hypercenterC22×M16D127
hypocenterC127#1894
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
2. C2×#16861134C2semidirect
4. #22549639C2semidirect
8. #16861188C4semidirect
96. C127#1894semidirect

Maximal subgroups

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

subgroupindex
C2×#168611882
C2×#168611342
#225497362
#225496392
C22×M16×C1272
#1894127