Finite Group Databasebeta

← #30639040

Subgroups — #30639040

Order 16256.

360 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 105 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×#16861168C211yesmaximal, characteristic
3C2×#16861134C212yesmaximal, complemented
4#22549639C212yesmaximal, complemented
5C2×C22⋊C8×C127C211yesmaximal, characteristic
6C22×Dic254C2211yescharacteristic
7#16861188—22no—
8#16861188C414yescomplemented
9#16861168—24no—
10C2×C4⋊Dic127C2211yescharacteristic
11C2×C4⋊Dic127C2212yes—
12C2×Dic127⋊C4—21no—
13C2×C22⋊Dic127—21no—
14C2×C22⋊Dic127—21no—
15C2×C4×Dic127—21no—
16#16861134—24no—
17C23×C4×C127C2211yescharacteristic
18C22×C8×C127C2212yes—
19C22⋊C8×C127—22no—
20C2×Dic254—21no—
21C2×Dic254—22no—
22C2×Dic254—22no—
23C2×Dic254C2×C412yes—
24C22×Dic127—21no—
25C22×Dic127—22no—
26C22×Dic127—21no—
27C4⋊Dic127—21no—
28C4⋊Dic127—22no—
29C4⋊Dic127C2×C412yes—
30C4⋊Dic127D414yes—
31Dic127⋊C4—24no—
32C22⋊Dic127—24no—
33C22⋊Dic127—24no—
34C4×Dic127—24no—
35C24×C127D411yescharacteristic
36C22×C4×C127—22no—
37C22×C4×C127—22no—
38C22×C4×C127C2×C412yes—
39C22×C4×C127D412yes—
40C22×C4×C127D411yescharacteristic
41C22×C4×C127C2311yescharacteristic
42C2×C8×C127—22no—
43C2×C8×C127—22no—
44C2×C8×C127—22no—
45Dic254—24no—
46Dic254—42no—
47C2×Dic127—22no—
48C2×Dic127—22no—
49C2×Dic127—24no—
50C2×Dic127—41no—
51C2×Dic127—41no—
52C2×Dic127—42no—
53C2×Dic127—42no—
54C2×Dic127—42no—
55C23×C127—21no—
56C23×C127—21no—
57C23×C127—22no—
58C23×C127—22no—
59C23×C127C22⋊C412yes—
60C23×C127C2×D411yescharacteristic
61C2×C4×C127—21no—
62C2×C4×C127—21no—
63C2×C4×C127—24no—
64C2×C4×C127—24no—
65C2×C4×C127C22⋊C412yes—
66C2×C4×C127C22×C411yescharacteristic
67C2×C4×C127C2×D412yes—
68C2×C4×C127C2×D411yescharacteristic
69C2×C4×C127#4112yes—
70C8×C127—42no—
71C22×C127—22no—
72C22×C127—22no—
73C22×C127—22no—
74C22×C127—42no—
75C22×C127—42no—
76C22×C127C2×C22⋊C411yescharacteristic
77C22×C127#12011yescharacteristic
78C22×C127#12311yescharacteristic
79C22×C127#13512yes—
80C22×C127#13712yes—
81C4×C127—22no—
82C4×C127C4×D412yes—
83C4×C127#12112yes—
84Dic127—42no—
85Dic127—42no—
86Dic127—44no—
87C2×C127—42no—
88C2×C127#38611yescharacteristic
89C2×C127#41611yescharacteristic
90C2×C127#41911yescharacteristic
91C2×C127#45212yes—
92C2×C127#48512yes—
93#1873—1271nomaximal, Sylow
94C127#187311yescharacteristic, complemented, Sylow
95C2×C22⋊Q8—1271no—
96C2×#117—1271no—
97C2×#103—1272no—
98C2×C22⋊C8D12711yescharacteristic
99#340—1272no—
100C22×Q8—1271no—
101C23×C4C2×D12711yescharacteristic
102C22×C8C2×D12712yes—
103C22⋊Q8—1274no—
104C22⋊Q8—2542no—
105#117—2544no—
106C2×C4⋊C4—1272no—
107C2×C4⋊C4—1271no—
108C2×C4⋊C4—2541no—
109C2×C22⋊C4—2541no—
110C2×C22⋊C4—2541no—
111C2×C42—2541no—
112#103—2544no—
113C22⋊C8—22no—
114C24D50811yescharacteristic
115C2×Q8—1272no—
116C2×Q8—2542no—
117C2×Q8—2542no—
118C2×Q8—2541no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—2542no—
122C22×C4—2541no—
123C22×C4—2541no—
124C22×C4C4×D12712yes—
125C22×C4C22⋊D12712yes—
126C22×C4D50811yescharacteristic
127C22×C4C22×D12711yescharacteristic
128C2×C8—22no—
129C2×C8—22no—
130C2×C8—22no—
131C4⋊C4—1272no—
132C4⋊C4—1274no—
133C4⋊C4—2544no—
134C4⋊C4—2542no—
135C4⋊C4—2541no—
136C22⋊C4—2544no—
137C22⋊C4—2544no—
138C42—2544no—
139C23—21no—
140C23—22no—
141C23—22no—
142C23—21no—
143C23(C2×D127)⋊C412yes—
144C23C2×D50811yescharacteristic
145Q8—5082no—
146Q8—2544no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—24no—
150C2×C4—24no—
151C2×C4—5081no—
152C2×C4—5082no—
153C2×C4—5082no—
154C2×C4—5082no—
155C2×C4—5081no—
156C2×C4—2542no—
157C2×C4—2542no—
158C2×C4—2544no—
159C2×C4(C2×D127)⋊C412yes—
160C2×C4C2×C4×D12711yescharacteristic
161C2×C4D4×D12712yes—
162C2×C4C2×C22⋊D12711yescharacteristic
163C2×C4#1452324712yes—
164C8—42no—
165C22—22no—
166C22—22no—
167C22—22no—
168C22—42no—
169C22—42no—
170C22C2×(C2×D127)⋊C411yescharacteristic
171C22#1686117911yescharacteristic
172C22#1686119311yescharacteristic
173C22#1686123112yes—
174C22#1686124212yes—
175C4—22no—
176C4—5084no—
177C4—5082no—
178C4—5082no—
179C4(C22⋊D127)⋊C412yes—
180C4#1686118112yes—
181C2—42no—
182C2#2254975311yescharacteristic
183C2#2254987311yescharacteristic
184C2#2254988611yescharacteristic
185C2#2255004012yes—
186C2#2255021812yes—
subgroupG/H
centerC23C2×D508
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×C22⋊C8×C127C2
socleC23×C127C2×D4
prime radicalC24×C127D4
cosocleC22×C4×C127C23
hypercenterC2×C22⋊C8D127
hypocenterC127#1873
LCSG ▷ C2×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
3. C2×#16861134C2semidirect
4. #22549639C2semidirect
8. #16861188C4semidirect
94. C127#1873semidirect

Maximal subgroups

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

subgroupindex
C2×#168611882
C2×#168611682
C2×#168611342
#225496392
C2×C22⋊C8×C1272
#1873127