Finite Group Databasebeta

← #30639457

Subgroups — #30639457

Order 16256.

248 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 103 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×#16861140C211yesmaximal, characteristic
2C2×#16861140C211yesmaximal, characteristic
3C2×C8×Dic127C211yesmaximal, characteristic
4#22549775C211yesmaximal, characteristic
5#22549772C211yesmaximal, characteristic
6#22549639C211yesmaximal, characteristic
7#340×C127C211yesmaximal, characteristic
8C22×C127⋊C8C2211yescharacteristic
9C2×C4⋊Dic127—21no—
10C2×C4⋊Dic127C2211yescharacteristic
11C2×C4⋊Dic127C2211yescharacteristic
12C2×Dic127⋊C4—21no—
13C2×C4×Dic127C2211yescharacteristic
14#16861140—22no—
15#16861140C414yescomplemented
16C8×Dic127—22no—
17C127⋊#95—21no—
18C127⋊#95—21no—
19C22×C8×C127C2211yescharacteristic
20C2×C4⋊C4×C127C2211yescharacteristic
21C2×C4⋊C4×C127C2211yescharacteristic
22C22×Dic127—21no—
23C22×Dic127—21no—
24C22×Dic127D411yescharacteristic
25C22×Dic127D411yescharacteristic
26C2×C127⋊C8C2×C412yes—
27C2×C127⋊C8D412yes—
28C2×C127⋊C8Q812yes—
29C4⋊Dic127—21no—
30C4⋊Dic127—21no—
31C4⋊Dic127—21no—
32C4⋊Dic127—42no—
33C4⋊Dic127C2×C412yes—
34Dic127⋊C4—42no—
35C4×Dic127—21no—
36C4×Dic127—21no—
37C22×C4×C127—21no—
38C22×C4×C127—21no—
39C22×C4×C127C2311yescharacteristic
40C2×C8×C127—21no—
41C2×C8×C127—21no—
42C2×C8×C127Q812yes—
43C4⋊C4×C127—21no—
44C4⋊C4×C127—21no—
45C4⋊C4×C127—21no—
46C4⋊C4×C127C2×C412yes—
47C2×Dic127—21no—
48C2×Dic127—21no—
49C2×Dic127—21no—
50C2×Dic127—21no—
51C2×Dic127—22no—
52C2×Dic127—41no—
53C2×Dic127—41no—
54C2×Dic127—41no—
55C2×Dic127—41no—
56C2×Dic127—41no—
57C2×Dic127SD1612yes—
58C2×Dic127SD1612yes—
59C127⋊C8C4⋊C414yescomplemented
60C23×C127C2×D411yescharacteristic
61C2×C4×C127—22no—
62C2×C4×C127—41no—
63C2×C4×C127—41no—
64C2×C4×C127—41no—
65C2×C4×C127—41no—
66C2×C4×C127—41no—
67C2×C4×C127C22×C411yescharacteristic
68C2×C4×C127C2×D411yescharacteristic
69C2×C4×C127C2×Q811yescharacteristic
70C2×C4×C127C2×Q811yescharacteristic
71C2×C4×C127#4111yescharacteristic
72C2×C4×C127#4111yescharacteristic
73C8×C127—22no—
74C22×C127C4×D411yescharacteristic
75C22×C127#12111yescharacteristic
76C22×C127C4⋊Q811yescharacteristic
77C22×C127C2×SD1611yescharacteristic
78C22×C127C2×SD1611yescharacteristic
79C22×C127#13511yescharacteristic
80C22×C127#13511yescharacteristic
81C4×C127—42no—
82C4×C127—81no—
83C4×C127C2×C4⋊C411yescharacteristic
84C4×C127C4×Q811yescharacteristic
85C4×C127C22⋊Q811yescharacteristic
86C4×C127#12511yescharacteristic
87Dic127—22no—
88Dic127—22no—
89Dic127—42no—
90Dic127—81no—
91C2×C127#38911yescharacteristic
92C2×C127C4×SD1611yescharacteristic
93C2×C127C4×SD1611yescharacteristic
94C2×C127#46511yescharacteristic
95C2×C127#46511yescharacteristic
96C2×C127#49811yescharacteristic
97C2×C127#49911yescharacteristic
98#1941—1271nomaximal, Sylow
99C127#194111yescharacteristic, complemented, Sylow
100C2×#106—1271no—
101C2×#106—1271no—
102C2×C4×C8—1271no—
103#389—1271no—
104#389—1271no—
105#340—1271no—
106#340D12711yescharacteristic
107C22×C8—1271no—
108C22×C8C2×D12711yescharacteristic
109C2×C4⋊C4—1271no—
110C2×C4⋊C4—1271no—
111C2×C4⋊C4—2541no—
112C2×C4⋊C4—2541no—
113C2×C4⋊C4C2×D12711yescharacteristic
114C2×C4⋊C4C2×D12711yescharacteristic
115C2×C42—1271no—
116#106—1274no—
117#106—2542no—
118C4×C8—2542no—
119#95—2541no—
120#95—2541no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—1271no—
124C22×C4—1271no—
125C22×C4—2541no—
126C22×C4—2541no—
127C22×C4C22×D12711yescharacteristic
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—1272no—
131C2×C8—1272no—
132C2×C8—1272no—
133C2×C8Dic25412yes—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—21no—
137C4⋊C4—1272no—
138C4⋊C4—5082no—
139C4⋊C4—5082no—
140C4⋊C4—2541no—
141C4⋊C4—2541no—
142C4⋊C4—2541no—
143C4⋊C4C4×D12712yes—
144C42—2541no—
145C42—2541no—
146C23D4×D12711yescharacteristic
147C2×C4—22no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—1272no—
154C2×C4—1272no—
155C2×C4—5081no—
156C2×C4—5081no—
157C2×C4—5081no—
158C2×C4—5081no—
159C2×C4—5081no—
160C2×C4—2542no—
161C2×C4—2541no—
162C2×C4—2541no—
163C2×C4—2541no—
164C2×C4—2541no—
165C2×C4C2×C4×D12711yescharacteristic
166C2×C4D4×D12711yescharacteristic
167C2×C4Q8×D12711yescharacteristic
168C2×C4C2×Dic25411yescharacteristic
169C2×C4#1452324711yescharacteristic
170C2×C4#1452324811yescharacteristic
171C8—22no—
172C8—1274no—
173C22(C22⋊D127)⋊C411yescharacteristic
174C22#1686118211yescharacteristic
175C22#1686120811yescharacteristic
176C22SD16×D12711yescharacteristic
177C22SD16×D12711yescharacteristic
178C22#1686122811yescharacteristic
179C22#1686123211yescharacteristic
180C4—42no—
181C4—81no—
182C4—10161no—
183C4—5082no—
184C4—2542no—
185C4—2542no—
186C4C4⋊C4×D12711yescharacteristic
187C4C4×Dic25411yescharacteristic
188C4#1686118611yescharacteristic
189C4#1686119911yescharacteristic
190C2#2254977311yescharacteristic
191C2#2254996111yescharacteristic
192C2#2254996711yescharacteristic
193C2#2255010411yescharacteristic
194C2#2255010811yescharacteristic
195C2#2255027311yescharacteristic
196C2#2255027711yescharacteristic
subgroupG/H
centerC23D4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#340×C127C2
socle, prime radicalC23×C127C2×D4
cosocleC22×C4×C127C23
hypercenter#340D127
hypocenterC127#1941
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
15. #16861140C4semidirect
59. C127⋊C8C4⋊C4semidirect
99. C127#1941semidirect

Maximal subgroups

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

subgroupindex
C2×#168611402
C2×C8×Dic1272
#225497752
#225497722
#225496392
#340×C1272
#1941127