Finite Group Databasebeta

← #30639487

Subgroups — #30639487

Order 16256.

256 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 107 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×#16861143C211yesmaximal, characteristic
2C2×#16861142C211yesmaximal, characteristic
3C2×Dic127⋊C8C211yesmaximal, characteristic
4#22549745C211yesmaximal, characteristic
5C4⋊C4×Dic127C211yesmaximal, characteristic
6#22549638C211yesmaximal, characteristic
7#340×C127C211yesmaximal, characteristic
8C22×C127⋊C8C2211yescharacteristic
9C2×C4⋊Dic127C2211yescharacteristic
10C2×C4⋊Dic127C2211yescharacteristic
11C2×Dic127⋊C4—21no—
12C2×Dic127⋊C4—21no—
13C2×C4×Dic127—21no—
14C2×C4×Dic127C2211yescharacteristic
15#16861143—22no—
16#16861142—22no—
17Dic127⋊C8C414yescomplemented
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⋊C8—21no—
27C2×C127⋊C8—21no—
28C2×C127⋊C8C2×C412yes—
29C4⋊Dic127—21no—
30C4⋊Dic127—21no—
31C4⋊Dic127—21no—
32C4⋊Dic127D412yes—
33Dic127⋊C4—42no—
34Dic127⋊C4—42no—
35C4×Dic127—42no—
36C4×Dic127C2×C412yes—
37C4×Dic127Q812yes—
38C22×C4×C127—21no—
39C22×C4×C127—21no—
40C22×C4×C127C2311yescharacteristic
41C2×C8×C127—21no—
42C2×C8×C127—21no—
43C2×C8×C127C2×C412yes—
44C4⋊C4×C127—21no—
45C4⋊C4×C127—21no—
46C4⋊C4×C127—21no—
47C4⋊C4×C127Q812yes—
48C2×Dic127—21no—
49C2×Dic127—21no—
50C2×Dic127—22no—
51C2×Dic127—41no—
52C2×Dic127—41no—
53C2×Dic127—41no—
54C2×Dic127—41no—
55C2×Dic127—41no—
56C2×Dic127C4⋊C412yes—
57C2×Dic127C4⋊C412yes—
58C2×Dic127D812yes—
59C2×Dic127Q1612yes—
60C127⋊C8—22no—
61C23×C127C2×D411yescharacteristic
62C2×C4×C127—22no—
63C2×C4×C127—41no—
64C2×C4×C127—41no—
65C2×C4×C127—41no—
66C2×C4×C127—41no—
67C2×C4×C127—41no—
68C2×C4×C127C22×C411yescharacteristic
69C2×C4×C127C2×D411yescharacteristic
70C2×C4×C127C2×Q811yescharacteristic
71C2×C4×C127C2×Q811yescharacteristic
72C2×C4×C127#4111yescharacteristic
73C2×C4×C127#4111yescharacteristic
74C8×C127—22no—
75C22×C127C2×C4⋊C411yescharacteristic
76C22×C127#12111yescharacteristic
77C22×C127C22⋊Q811yescharacteristic
78C22×C127C2×D811yescharacteristic
79C22×C127C2×Q1611yescharacteristic
80C22×C127AGL(1,ℤ/8)11yescharacteristic
81C22×C127#13711yescharacteristic
82C4×C127—42no—
83C4×C127—81no—
84C4×C127C4×D411yescharacteristic
85C4×C127C4×Q811yescharacteristic
86C4×C127#12511yescharacteristic
87C4×C127C4⋊Q811yescharacteristic
88Dic127—22no—
89Dic127—42no—
90Dic127—81no—
91Dic127#10714yescomplemented
92C2×C127#38911yescharacteristic
93C2×C127C2×#10711yescharacteristic
94C2×C127#42811yescharacteristic
95C2×C127#45911yescharacteristic
96C2×C127#46211yescharacteristic
97C2×C127#47411yescharacteristic
98C2×C127#47711yescharacteristic
99#1947—1271nomaximal, Sylow
100C127#194711yescharacteristic, complemented, Sylow
101C2×#107—1271no—
102C2×#107—1271no—
103C2×C4⋊C8—1271no—
104#384—1271no—
105C4×C4⋊C4—1271no—
106#340—1271no—
107#340D12711yescharacteristic
108C22×C8—1271no—
109C22×C8C2×D12711yescharacteristic
110C2×C4⋊C4—1271no—
111C2×C4⋊C4—1271no—
112C2×C4⋊C4—2541no—
113C2×C4⋊C4—2541no—
114C2×C4⋊C4C2×D12711yescharacteristic
115C2×C4⋊C4C2×D12711yescharacteristic
116C2×C42—1271no—
117C2×C42—2541no—
118#107—2542no—
119#107—2542no—
120C4⋊C8—1274no—
121#95—2541no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—1271no—
125C22×C4—1271no—
126C22×C4—2541no—
127C22×C4—2541no—
128C22×C4C22×D12711yescharacteristic
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—1272no—
132C2×C8—2541no—
133C2×C8—2541no—
134C2×C8C4×D12712yes—
135C4⋊C4—21no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—1272no—
139C4⋊C4—5082no—
140C4⋊C4—5082no—
141C4⋊C4—2541no—
142C4⋊C4—2541no—
143C4⋊C4—2541no—
144C4⋊C4Dic25412yes—
145C42—1272no—
146C42—1272no—
147C42—5082no—
148C23D4×D12711yescharacteristic
149C2×C4—22no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—1272no—
156C2×C4—1272no—
157C2×C4—1272no—
158C2×C4—1272no—
159C2×C4—5081no—
160C2×C4—5081no—
161C2×C4—5081no—
162C2×C4—5081no—
163C2×C4—5081no—
164C2×C4—2542no—
165C2×C4—2541no—
166C2×C4—2541no—
167C2×C4C2×C4×D12711yescharacteristic
168C2×C4D4×D12711yescharacteristic
169C2×C4Q8×D12711yescharacteristic
170C2×C4C2×Dic25411yescharacteristic
171C2×C4#1452324711yescharacteristic
172C2×C4#1452324811yescharacteristic
173C8—22no—
174C8—2542no—
175C22C4⋊C4×D12711yescharacteristic
176C22#1686118211yescharacteristic
177C22#1686118611yescharacteristic
178C22D8×D12711yescharacteristic
179C22Q16×D12711yescharacteristic
180C22#1686123611yescharacteristic
181C22#1686123811yescharacteristic
182C4—42no—
183C4—81no—
184C4—1274no—
185C4—10161no—
186C4—5082no—
187C4—2542no—
188C4(C22⋊D127)⋊C411yescharacteristic
189C4C4×Dic25411yescharacteristic
190C4#1686119911yescharacteristic
191C4#1686120811yescharacteristic
192C2#2254977311yescharacteristic
193C2#107×D12711yescharacteristic
194C2#2254991511yescharacteristic
195C2#2255007011yescharacteristic
196C2#2255008511yescharacteristic
197C2#2255015411yescharacteristic
198C2#2255016911yescharacteristic
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#1947
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
17. Dic127⋊C8C4semidirect
91. Dic127#107semidirect
100. C127#1947semidirect

Maximal subgroups

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

subgroupindex
C2×#168611432
C2×#168611422
C2×Dic127⋊C82
#225497452
C4⋊C4×Dic1272
#225496382
#340×C1272
#1947127