Finite Group Databasebeta

← #30639740

Subgroups — #30639740

Order 16256.

392 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C22×Dic508C211yesmaximal, characteristic, complemented
2C2×#16861188C212yesmaximal
3C2×#16861134C212yesmaximal, complemented
4#22549639C211yesmaximal, characteristic, complemented
5C2×C22⋊C8×C127C211yesmaximal, characteristic
6C22×Dic254C2212yes—
7C2×Dic508—24no—
8C2×Dic508—22no—
9#16861188—24no—
10#16861188—24no—
11C2×C4⋊Dic127C2212yes—
12C2×Dic127⋊C4—22no—
13C2×C22⋊Dic127—22no—
14#16861134—24no—
15C23×C4×C127C2211yescharacteristic
16C22×C8×C127C2211yescharacteristic
17C22×C8×C127C2211yescharacteristic
18C22⋊C8×C127—22no—
19C2×Dic254—22no—
20C2×Dic254—22no—
21C2×Dic254—22no—
22C2×Dic254—22no—
23C2×Dic254—22no—
24C2×Dic254D414yes—
25C22×Dic127—22no—
26C22×Dic127—22no—
27Dic508—28no—
28C4⋊Dic127—22no—
29C4⋊Dic127D414yes—
30Dic127⋊C4—28no—
31C22⋊Dic127—28no—
32C24×C127D411yescharacteristic
33C22×C4×C127—22no—
34C22×C4×C127—22no—
35C22×C4×C127—21no—
36C22×C4×C127—21no—
37C22×C4×C127D411yescharacteristic
38C22×C4×C127C2311yescharacteristic
39C2×C8×C127—22no—
40C2×C8×C127—21no—
41C2×C8×C127—22no—
42C2×C8×C127D412yes—
43Dic254—24no—
44Dic254—24no—
45Dic254—42no—
46Dic254—42no—
47C2×Dic127—24no—
48C2×Dic127—24no—
49C2×Dic127—42no—
50C2×Dic127—42no—
51C2×Dic127—42no—
52C2×Dic127—42no—
53C23×C127—22no—
54C23×C127—21no—
55C23×C127—21no—
56C23×C127—22no—
57C23×C127Q1612yescomplemented
58C23×C127C2×D411yescharacteristic
59C2×C4×C127—22no—
60C2×C4×C127—21no—
61C2×C4×C127—41no—
62C2×C4×C127—42no—
63C2×C4×C127—41no—
64C2×C4×C127C2×D412yes—
65C2×C4×C127C2×D412yes—
66C2×C4×C127C2×D411yescharacteristic
67C2×C4×C127#4111yescharacteristic
68C8×C127—22no—
69C8×C127—41no—
70C22×C127—24no—
71C22×C127—22no—
72C22×C127—41no—
73C22×C127—42no—
74C22×C127—41no—
75C22×C127#12012yes—
76C22×C127#12111yescharacteristic
77C22×C127C2×Q1611yescharacteristic
78C22×C127#13511yescharacteristic
79C22×C127#13712yes—
80C4×C127—41no—
81C4×C127#12011yescharacteristic
82C4×C127#12112yes—
83C4×C127#12711yescharacteristic
84Dic127—44no—
85Dic127—44no—
86C2×C127—42no—
87C2×C127#39211yescharacteristic
88C2×C127#45112yes—
89C2×C127#45212yes—
90C2×C127#46711yescharacteristic
91C2×C127#47011yescharacteristic
92#2001—1271nomaximal, Sylow
93C127#200111yescharacteristic, complemented, Sylow
94C22×Q16—1271no—
95C2×C22⋊Q8—1272no—
96C2×#103—1272no—
97C2×C22⋊C8D12711yescharacteristic
98#340—1271no—
99C22×Q8—1272no—
100C23×C4C2×D12711yescharacteristic
101C2×Q16—2544no—
102C2×Q16—2542no—
103C22×C8C2×D12711yescharacteristic
104C22×C8C2×D12711yescharacteristic
105C22⋊Q8—2544no—
106C22⋊Q8—2544no—
107C2×C4⋊C4—1272no—
108C2×C4⋊C4—2542no—
109C2×C22⋊C4—2542no—
110#103—2544no—
111C22⋊C8—22no—
112C24D50811yescharacteristic
113C2×Q8—1274no—
114C2×Q8—2542no—
115C2×Q8—2542no—
116C2×Q8—2542no—
117C2×Q8—2542no—
118C2×Q8—2542no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—2542no—
124C22×C4—2542no—
125C22×C4D50811yescharacteristic
126C22×C4C22×D12711yescharacteristic
127Q16—2548no—
128C2×C8—22no—
129C2×C8—22no—
130C2×C8—21no—
131C2×C8C22⋊D12712yes—
132C4⋊C4—1274no—
133C4⋊C4—2548no—
134C4⋊C4—2542no—
135C22⋊C4—2548no—
136C23—21no—
137C23—22no—
138C23—22no—
139C23—21no—
140C23Dic50812yescomplemented
141C23C2×D50811yescharacteristic
142Q8—5082no—
143Q8—5082no—
144Q8—2544no—
145Q8—2544no—
146C2×C4—22no—
147C2×C4—21no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—42no—
151C2×C4—5082no—
152C2×C4—5082no—
153C2×C4—5082no—
154C2×C4—5082no—
155C2×C4—2544no—
156C2×C4—2544no—
157C2×C4D4×D12712yes—
158C2×C4D4×D12712yes—
159C2×C4C2×C22⋊D12711yescharacteristic
160C2×C4#1452324811yescharacteristic
161C8—22no—
162C8—41no—
163C22—24no—
164C22—22no—
165C22—42no—
166C22—41no—
167C22—41no—
168C22#1686117912yes—
169C22#1686118311yescharacteristic
170C22C2×Dic50811yescharacteristic
171C22#1686123111yescharacteristic
172C22#1686124212yes—
173C4—41no—
174C4—5084no—
175C4—5084no—
176C4#1686117711yescharacteristic
177C4#1686118212yes—
178C4#1686120511yescharacteristic
179C2—42no—
180C2#2254978511yescharacteristic
181C2#2255003412yes—
182C2#2255004012yes—
183C2#2255011711yescharacteristic
184C2#2255013411yescharacteristic
subgroupG/H
centerC23C2×D508
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×C22⋊C8×C127C2
socleC23×C127C2×D4
prime radicalC24×C127D4
hypercenterC2×C22⋊C8D127
hypocenterC127#2001
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
1. C22×Dic508C2semidirect
3. C2×#16861134C2semidirect
4. #22549639C2semidirect
57. C23×C127Q16semidirect
93. C127#2001semidirect
140. C23Dic508semidirect

Maximal subgroups

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

subgroupindex
C22×Dic5082
C2×#168611882
C2×#168611342
#225496392
C2×C22⋊C8×C1272
#2001127