Finite Group Databasebeta

← #30639310

Subgroups — #30639310

Order 16256.

230 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 99 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#22549775C211yesmaximal, characteristic
2C4⋊C4×Dic127C211yesmaximal, characteristic
3#22549639C211yesmaximal, characteristic
4#22549638C211yesmaximal, characteristic
5#22549638C211yesmaximal, characteristic
6#22549630C211yesmaximal, characteristic
7C2×#106×C127C211yesmaximal, characteristic
8C22×C127⋊C8C2211yescharacteristic
9C2×C4⋊Dic127—21no—
10C2×C4⋊Dic127C2211yescharacteristic
11C2×C4⋊Dic127C2211yescharacteristic
12C2×C4×Dic127—21no—
13C2×C4×Dic127C2211yescharacteristic
14C127⋊#95—21no—
15C127⋊#95—21no—
16C22×C8×C127C2211yescharacteristic
17C2×C4⋊C4×C127C2211yescharacteristic
18C2×C4⋊C4×C127C2211yescharacteristic
19#106×C127C414yescomplemented
20C22×Dic127—21no—
21C22×Dic127—21no—
22C22×Dic127D411yescharacteristic
23C22×Dic127D411yescharacteristic
24C2×C127⋊C8—21no—
25C2×C127⋊C8—21no—
26C2×C127⋊C8—21no—
27C4⋊Dic127—21no—
28C4⋊Dic127—21no—
29C4⋊Dic127—21no—
30C4⋊Dic127—42no—
31C4⋊Dic127Q812yes—
32C4×Dic127—21no—
33C4×Dic127—21no—
34C4×Dic127—42no—
35C22×C4×C127—21no—
36C22×C4×C127—21no—
37C22×C4×C127C2311yescharacteristic
38C2×C8×C127—21no—
39C2×C8×C127—21no—
40C2×C8×C127C2×C412yes—
41C4⋊C4×C127—21no—
42C4⋊C4×C127—21no—
43C4⋊C4×C127C2×C412yes—
44C4⋊C4×C127C2×C412yes—
45C2×Dic127—21no—
46C2×Dic127—21no—
47C2×Dic127—22no—
48C2×Dic127—21no—
49C2×Dic127—21no—
50C2×Dic127—21no—
51C2×Dic127—41no—
52C2×Dic127—41no—
53C2×Dic127—41no—
54C2×Dic127—41no—
55C2×Dic127—41no—
56C2×Dic127SD1612yes—
57C127⋊C8—41no—
58C23×C127C2×D411yescharacteristic
59C2×C4×C127—22no—
60C2×C4×C127—22no—
61C2×C4×C127—41no—
62C2×C4×C127—41no—
63C2×C4×C127—41no—
64C2×C4×C127—41no—
65C2×C4×C127C22×C411yescharacteristic
66C2×C4×C127C2×Q811yescharacteristic
67C2×C4×C127#4111yescharacteristic
68C2×C4×C127#4111yescharacteristic
69C2×C4×C127#4111yescharacteristic
70C2×C4×C127#4111yescharacteristic
71C8×C127—22no—
72C22×C127C4×D411yescharacteristic
73C22×C127C22⋊Q811yescharacteristic
74C22×C127#12311yescharacteristic
75C22×C127C2×SD1611yescharacteristic
76C22×C127#13511yescharacteristic
77C22×C127AGL(1,ℤ/8)11yescharacteristic
78C22×C127#13711yescharacteristic
79C4×C127—42no—
80C4×C127—42no—
81C4×C127#11711yescharacteristic
82C4×C127C4×Q811yescharacteristic
83C4×C127#12511yescharacteristic
84C4×C127#12611yescharacteristic
85Dic127—22no—
86Dic127—42no—
87Dic127—41no—
88Dic127—81no—
89C2×C127#38711yescharacteristic
90C2×C127C4×SD1611yescharacteristic
91C2×C127#44011yescharacteristic
92C2×C127#47511yescharacteristic
93C2×C127#47811yescharacteristic
94C2×C127#48111yescharacteristic
95C2×C127#48511yescharacteristic
96#1919—1271nomaximal, Sylow
97C127#191911yescharacteristic, complemented, Sylow
98C2×#106D12711yescharacteristic
99#389—1271no—
100C4×C4⋊C4—1271no—
101#340—1271no—
102#340—1271no—
103#340—1271no—
104#336—1271no—
105C22×C8—1271no—
106C22×C8C2×D12711yescharacteristic
107C2×C4⋊C4—1271no—
108C2×C4⋊C4—1271no—
109C2×C4⋊C4—2541no—
110C2×C4⋊C4C2×D12711yescharacteristic
111C2×C4⋊C4C2×D12711yescharacteristic
112C2×C42—1271no—
113C2×C42—2541no—
114#106Dic12714yescomplemented
115#95—2541no—
116#95—2541no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—1271no—
120C22×C4—1271no—
121C22×C4—2541no—
122C22×C4—2541no—
123C22×C4C22×D12711yescharacteristic
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—2541no—
127C2×C8—2541no—
128C2×C8—2541no—
129C2×C8C2×Dic12712yes—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—1272no—
133C4⋊C4—5082no—
134C4⋊C4—2541no—
135C4⋊C4—2541no—
136C4⋊C4—2541no—
137C4⋊C4C2×Dic12712yes—
138C4⋊C4C2×Dic12712yes—
139C42—5082no—
140C42—2541no—
141C42—2541no—
142C23D4×D12711yescharacteristic
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—1272no—
150C2×C4—5081no—
151C2×C4—5081no—
152C2×C4—5081no—
153C2×C4—5081no—
154C2×C4—5081no—
155C2×C4—2542no—
156C2×C4—2541no—
157C2×C4—2541no—
158C2×C4—2541no—
159C2×C4—2541no—
160C2×C4—2541no—
161C2×C4C22×Dic12711yescharacteristic
162C2×C4Q8×D12711yescharacteristic
163C2×C4#1452324711yescharacteristic
164C2×C4#1452324811yescharacteristic
165C2×C4#1452324811yescharacteristic
166C2×C4#1452324911yescharacteristic
167C8—22no—
168C8—5081no—
169C22D4×Dic12711yescharacteristic
170C22#1686118711yescharacteristic
171C22#1686119411yescharacteristic
172C22SD16×D12711yescharacteristic
173C22#1686123011yescharacteristic
174C22#1686123611yescharacteristic
175C22#1686123811yescharacteristic
176C4—42no—
177C4—42no—
178C4—10161no—
179C4—5082no—
180C4—5081no—
181C4—2542no—
182C4#1686116811yescharacteristic
183C4Q8×Dic12711yescharacteristic
184C4#1686119911yescharacteristic
185C4#1686120411yescharacteristic
186C2#2254976211yescharacteristic
187C2SD16×Dic12711yescharacteristic
188C2#2254997711yescharacteristic
189C2#2255016011yescharacteristic
190C2#2255017511yescharacteristic
191C2#2255019611yescharacteristic
192C2#2255022011yescharacteristic
subgroupG/H
centerC23D4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#106×C127C2
socle, prime radicalC23×C127C2×D4
cosocleC22×C4×C127C23
hypercenterC2×#106D127
hypocenterC127#1919
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#106

Complements

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

NH ≅ G/Ntype
19. #106×C127C4semidirect
97. C127#1919semidirect
114. #106Dic127semidirect

Maximal subgroups

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

subgroupindex
#225497752
C4⋊C4×Dic1272
#225496392
#225496382
#225496302
C2×#106×C1272
#1919127