Finite Group Databasebeta

← #30640417

Subgroups — #30640417

Order 16256.

230 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 65 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#22550450C211yesmaximal, characteristic
2C2×#16861128C211yesmaximal, characteristic, complemented
3#22549674C212yesmaximal, complemented
4#22549672C212yesmaximal, complemented
5C2×#101×C127C211yesmaximal, characteristic, complemented
6C2×#14523248C411yescharacteristic
7C2×#14523232—21no—
8#16861207C411yescharacteristic
9#16861207C2211yescharacteristic
10#16861198C411yescharacteristic
11#16861198C2211yescharacteristic
12#16861192—21no—
13#16861192—21no—
14#16861185—21no—
15#16861185—21no—
16#16861168C411yescharacteristic
17#16861128—21no—
18#16861128C2212yes—
19C22×Q8×C127C2211yescharacteristic
20C2×M16×C127—21no—
21#101×C127—21no—
22#101×C127C2212yes—
23#14523248—41no—
24#14523248—41no—
25C2×Dic254C2×C411yescharacteristic
26C2×D508C2×C411yescharacteristic
27C2×C22⋊D127—21no—
28C2×C4×D127—21no—
29#14523232—22no—
30#14523232—41no—
31C2×C127⋊C8—41no—
32C4⋊Dic127—21no—
33Dic127⋊C4—21no—
34Dic127⋊C4—21no—
35Dic127⋊C4—21no—
36Dic127⋊C4—21no—
37C22⋊Dic127—21no—
38(C2×D127)⋊C4—21no—
39(C2×D127)⋊C4—21no—
40(C2×D127)⋊C4—21no—
41(C2×D127)⋊C4—21no—
42C4×Dic127C2×C411yescharacteristic
43C4×Dic127C2×C411yescharacteristic
44C2×Q8×C127—41no—
45C2×Q8×C127—41no—
46C2×Q8×C127C2×C411yescharacteristic
47C2×Q8×C127D411yescharacteristic
48C2×Q8×C127D411yescharacteristic
49C2×Q8×C127C2311yescharacteristic
50C22×C4×C127C2×C411yescharacteristic
51C22×C4×C127D411yescharacteristic
52C22×C4×C127D411yescharacteristic
53M16×C127—22no—
54M16×C127—41no—
55C2×C8×C127—41no—
56C22×D127—21no—
57Dic254—41no—
58D508—41no—
59C22⋊D127—41no—
60C22⋊D127—41no—
61C2×Dic127—21no—
62C2×Dic127—21no—
63C2×Dic127—21no—
64C4×D127—42no—
65C127⋊C8—42no—
66C23×C127C22⋊C411yescharacteristic
67Q8×C127—41no—
68Q8×C127—41no—
69Q8×C127—41no—
70Q8×C127—41no—
71C2×C4×C127—21no—
72C2×C4×C127—21no—
73C2×C4×C127—21no—
74C2×C4×C127—21no—
75C2×C4×C127—21no—
76C2×C4×C127—21no—
77C2×C4×C127C22⋊C411yescharacteristic
78C2×C4×C127C22⋊C411yescharacteristic
79C2×C4×C127C22⋊C411yescharacteristic
80C2×C4×C127C22×C411yescharacteristic
81C2×C4×C127C2×D411yescharacteristic
82C2×C4×C127C2×D411yescharacteristic
83C8×C127—42no—
84C22×C127—41no—
85C22×C127#9911yescharacteristic
86C22×C127#9911yescharacteristic
87C22×C127C2×C22⋊C411yescharacteristic
88C2×D127—42no—
89C2×D127—41no—
90C4×C127—21no—
91C4×C127—21no—
92C4×C127—21no—
93C4×C127—21no—
94C4×C127—21no—
95C4×C127—21no—
96Dic127—41no—
97Dic127—41no—
98Dic127—41no—
99C2×C127—21no—
100C2×C127—21no—
101C2×C127—21no—
102C2×C127C2×#9911yescharacteristic
103D127—81no—
104#2133—1271nomaximal, Sylow
105C127#213311yescharacteristic, complemented, Sylow
106#536—1271no—
107C2×#101—1271no—
108C2×#101D12711yescharacteristic, complemented
109#356—1272no—
110#355—1272no—
111C2×#41—1271no—
112C22×Q8C2×D12711yescharacteristic
113C2×M16—21no—
114C2×M16—2541no—
115C4⋊Q8—1271no—
116C4⋊Q8—1271no—
117#124—1271no—
118#124—1271no—
119#123—2541no—
120#123—2541no—
121C22⋊Q8—2541no—
122C22⋊Q8—2541no—
123#117—1271no—
124#101—21no—
125#101—1272no—
126#101—2541no—
127#101C2×D12712yes—
128#41—5081no—
129#41—5081no—
130C2×Q8—41no—
131C2×Q8—41no—
132C2×Q8—1271no—
133C2×Q8C4×D12711yescharacteristic
134C2×Q8C22⋊D12711yescharacteristic
135C2×Q8D50811yescharacteristic
136C2×Q8C22×D12711yescharacteristic
137C2×D4—1271no—
138C2×D4—2541no—
139C22×C4—2541no—
140C22×C4C4×D12711yescharacteristic
141C22×C4C22⋊D12711yescharacteristic
142C22×C4D50811yescharacteristic
143M16—22no—
144M16—41no—
145M16—5081no—
146M16—2542no—
147C2×C8—41no—
148C2×C8—5081no—
149C4⋊C4—2541no—
150C4⋊C4—2541no—
151C4⋊C4—2541no—
152C4⋊C4—2541no—
153C4⋊C4—2541no—
154C22⋊C4—2541no—
155C22⋊C4—2541no—
156C22⋊C4—2541no—
157C22⋊C4—2541no—
158C22⋊C4—2541no—
159C42—1271no—
160C42—1271no—
161C23—2541no—
162C23(C2×D127)⋊C411yescharacteristic
163Q8—41no—
164Q8—41no—
165Q8—41no—
166Q8—41no—
167Q8—5081no—
168D4—5081no—
169D4—5081no—
170D4—5081no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—5082no—
178C2×C4—2541no—
179C2×C4—2541no—
180C2×C4—2541no—
181C2×C4(C2×D127)⋊C411yescharacteristic
182C2×C4(C2×D127)⋊C411yescharacteristic
183C2×C4(C2×D127)⋊C411yescharacteristic
184C2×C4C2×C4×D12711yescharacteristic
185C2×C4C2×C22⋊D12711yescharacteristic
186C2×C4C2×D50811yescharacteristic
187C8—42no—
188C8—5082no—
189C22—41no—
190C22—5082no—
191C22—5081no—
192C22#1686112311yescharacteristic
193C22#1686112311yescharacteristic
194C22C2×(C2×D127)⋊C411yescharacteristic
195C4—21no—
196C4—21no—
197C4—21no—
198C4—21no—
199C4—21no—
200C4—21no—
201C4—5081no—
202C4—5081no—
203C4—5081no—
204C2—21no—
205C2—21no—
206C2—21no—
207C2—10161no—
208C2C2×#1686112311yescharacteristic
subgroupG/H
centerC2C2×#16861123
commutatorC2×C4×C127C22×C4
FrattiniC2×Q8C22×D127
FittingC2×#101×C127C2
socleC2×C127C2×#99
prime radicalC2×#14523248C4
cosocleC2×Q8×C127C23
hypercenterC2×#101D127
hypocenterC127#2133
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C23 ◁ C22×Q8 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
2. C2×#16861128C2semidirect
3. #22549674C2semidirect
4. #22549672C2semidirect
5. C2×#101×C127C2semidirect
105. C127#2133semidirect
108. C2×#101D127semidirect

Maximal subgroups

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

subgroupindex
#225504502
C2×#168611282
#225496742
#225496722
C2×#101×C1272
#2133127