Finite Group Databasebeta

← #30640065

Subgroups — #30640065

Order 16256.

250 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 91 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×#16861142C211yesmaximal, characteristic
2C2×#16861140C211yesmaximal, characteristic
3#22549774C211yesmaximal, characteristic
4#22549638C211yesmaximal, characteristic
5#22549638C211yesmaximal, characteristic
6#22549631C211yesmaximal, characteristic
7C2×#125×C127C211yesmaximal, characteristic
8C22×C127⋊C8C2211yescharacteristic
9C22×C127⋊C8C2211yescharacteristic
10C2×C4⋊Dic127C2211yescharacteristic
11C2×C4⋊Dic127C2211yescharacteristic
12C2×Dic127⋊C4—21no—
13#16861142—22no—
14#16861140—22no—
15C127⋊#95—21no—
16#125×C127—22no—
17#125×C127—22no—
18C2×C4⋊C4×C127—21no—
19C2×C4⋊C4×C127—21no—
20C2×C4⋊C4×C127C2211yescharacteristic
21C2×C4⋊C4×C127C2211yescharacteristic
22C2×C42×C127C2211yescharacteristic
23C22×Dic127—21no—
24C22×Dic127—21no—
25C2×C127⋊C8—21no—
26C2×C127⋊C8—21no—
27C2×C127⋊C8—21no—
28C2×C127⋊C8—21no—
29C2×C127⋊C8Q812yes—
30C2×C127⋊C8Q812yes—
31C4⋊Dic127—21no—
32C4⋊Dic127—21no—
33C4⋊Dic127—21no—
34C4⋊Dic127Q812yes—
35Dic127⋊C4—42no—
36C22×C4×C127—21no—
37C22×C4×C127—21no—
38C22×C4×C127D411yescharacteristic
39C22×C4×C127D411yescharacteristic
40C22×C4×C127C2311yescharacteristic
41C4⋊C4×C127—24no—
42C4⋊C4×C127—24no—
43C4⋊C4×C127—21no—
44C4⋊C4×C127—21no—
45C4⋊C4×C127D412yes—
46C4⋊C4×C127D412yes—
47C42×C127—21no—
48C42×C127—21no—
49C2×Dic127—22no—
50C2×Dic127—41no—
51C2×Dic127—41no—
52C2×Dic127—41no—
53C2×Dic127—41no—
54C2×Dic127—41no—
55C127⋊C8—22no—
56C127⋊C8—22no—
57C23×C127C2×D411yescharacteristic
58C2×C4×C127—21no—
59C2×C4×C127—21no—
60C2×C4×C127—22no—
61C2×C4×C127—22no—
62C2×C4×C127—21no—
63C2×C4×C127—21no—
64C2×C4×C127—21no—
65C2×C4×C127—21no—
66C2×C4×C127—41no—
67C2×C4×C127—41no—
68C2×C4×C127—41no—
69C2×C4×C127—41no—
70C2×C4×C127C2×D411yescharacteristic
71C2×C4×C127C2×D411yescharacteristic
72C2×C4×C127C2×Q811yescharacteristic
73C2×C4×C127C2×Q811yescharacteristic
74C2×C4×C127C2×Q811yescharacteristic
75C2×C4×C127#4111yescharacteristic
76C22×C127#12011yescharacteristic
77C22×C127C22⋊Q811yescharacteristic
78C22×C127C4⋊Q811yescharacteristic
79C22×C127#13511yescharacteristic
80C22×C127#13511yescharacteristic
81C22×C127AGL(1,ℤ/8)11yescharacteristic
82C22×C127#13711yescharacteristic
83C4×C127—41no—
84C4×C127—42no—
85C4×C127—41no—
86C4×C127—42no—
87C4×C127C22⋊Q811yescharacteristic
88C4×C127C22⋊Q811yescharacteristic
89C4×C127C4⋊Q811yescharacteristic
90C4×C127C4⋊Q811yescharacteristic
91Dic127—42no—
92Dic127—81no—
93C2×C127#39511yescharacteristic
94C2×C127#44911yescharacteristic
95C2×C127#45211yescharacteristic
96C2×C127#47811yescharacteristic
97C2×C127#47911yescharacteristic
98C2×C127#49911yescharacteristic
99C2×C127#50111yescharacteristic
100#2059—1271nomaximal, Sylow
101C127#205911yescharacteristic, complemented, Sylow
102C2×#125D12711yescharacteristic
103C2×#107—1271no—
104C2×#106—1271no—
105#389—1271no—
106#340—1271no—
107#340—1271no—
108#336—1271no—
109C22×C8—1271no—
110C22×C8—1271no—
111#125—22no—
112#125—22no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—1271no—
116C2×C4⋊C4—1271no—
117C2×C4⋊C4—2541no—
118C2×C4⋊C4C2×D12711yescharacteristic
119C2×C4⋊C4C2×D12711yescharacteristic
120C2×C42C2×D12711yescharacteristic
121#107—2542no—
122#106—2542no—
123#95—2541no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—2541no—
127C22×C4—2541no—
128C22×C4C22⋊D12711yescharacteristic
129C22×C4C22⋊D12711yescharacteristic
130C22×C4C22×D12711yescharacteristic
131C2×C8—1272no—
132C2×C8—1272no—
133C2×C8—2541no—
134C2×C8—2541no—
135C2×C8—2541no—
136C2×C8—2541no—
137C4⋊C4—24no—
138C4⋊C4—24no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—1272no—
142C4⋊C4—5082no—
143C4⋊C4—2541no—
144C4⋊C4—2541no—
145C4⋊C4—2541no—
146C4⋊C4C22⋊D12712yes—
147C4⋊C4C22⋊D12712yes—
148C42—21no—
149C42—21no—
150C23C2×C22⋊D12711yescharacteristic
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—5081no—
164C2×C4—5081no—
165C2×C4—5081no—
166C2×C4—5081no—
167C2×C4—5081no—
168C2×C4—2542no—
169C2×C4C2×C22⋊D12711yescharacteristic
170C2×C4C2×C22⋊D12711yescharacteristic
171C2×C4Q8×D12711yescharacteristic
172C2×C4Q8×D12711yescharacteristic
173C2×C4Q8×D12711yescharacteristic
174C2×C4#1452324911yescharacteristic
175C8—2542no—
176C8—2542no—
177C22#1686117811yescharacteristic
178C22#1686118511yescharacteristic
179C22#1686120711yescharacteristic
180C22#1686122911yescharacteristic
181C22#1686122911yescharacteristic
182C22#1686123411yescharacteristic
183C22#1686123911yescharacteristic
184C4—41no—
185C4—42no—
186C4—41no—
187C4—42no—
188C4—10161no—
189C4—5082no—
190C4#1686118511yescharacteristic
191C4#1686118511yescharacteristic
192C4#1686120711yescharacteristic
193C4#1686120711yescharacteristic
194C2#2254979511yescharacteristic
195C2#2255002111yescharacteristic
196C2#2255003811yescharacteristic
197C2#2255017311yescharacteristic
198C2#2255018011yescharacteristic
199C2#2255027511yescharacteristic
200C2#2255028611yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#125×C127C2
socle, prime radicalC23×C127C2×D4
hypercenterC2×#125D127
hypocenterC127#2059
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
101. C127#2059semidirect

Maximal subgroups

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

subgroupindex
C2×#168611422
C2×#168611402
#225497742
#225496382
#225496312
C2×#125×C1272
#2059127