Finite Group Databasebeta

← #30637164

Subgroups — #30637164

Order 16256.

246 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 89 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#22549933C211yesmaximal, characteristic, complemented
2#22549929C211yesmaximal, characteristic, complemented
3Q8⋊(C127⋊C8)C212yesmaximal, complemented
4D4⋊(C127⋊C8)C212yesmaximal, complemented
5C4×#41×C127C211yesmaximal, characteristic
6C4⋊(C127⋊C8)—21no—
7C4⋊(C127⋊C8)C2212yes—
8C22⋊(C127⋊C8)—21no—
9C22⋊(C127⋊C8)—21no—
10#16861120—21no—
11C4×C127⋊C8C2212yes—
12C2×#41×C127C411yescharacteristic
13C4×Q8×C127C411yescharacteristic
14C4×Q8×C127C2211yescharacteristic
15C4×D4×C127—21no—
16C4×D4×C127—21no—
17C4×D4×C127C411yescharacteristic
18C4×D4×C127C2211yescharacteristic
19#117×C127—21no—
20#117×C127C411yescharacteristic
21C2×C42×C127—21no—
22C2×C42×C127C2211yescharacteristic
23C2×C127⋊C8—22no—
24C2×C127⋊C8—22no—
25#41×C127—22no—
26#41×C127—22no—
27C2×Q8×C127C2×C411yescharacteristic
28C2×D4×C127—21no—
29C2×D4×C127C2×C411yescharacteristic
30C22×C4×C127—21no—
31C22×C4×C127—21no—
32C22×C4×C127—21no—
33C22×C4×C127C2×C411yescharacteristic
34C22×C4×C127D411yescharacteristic
35C22×C4×C127D411yescharacteristic
36C4⋊C4×C127—21no—
37C4⋊C4×C127—21no—
38C4⋊C4×C127C2×C411yescharacteristic
39C4⋊C4×C127C2×C411yescharacteristic
40C22⋊C4×C127—21no—
41C22⋊C4×C127—21no—
42C22⋊C4×C127—21no—
43C42×C127—21no—
44C42×C127—21no—
45C42×C127—21no—
46C42×C127C2×C411yescharacteristic
47C42×C127D411yescharacteristic
48C42×C127D411yescharacteristic
49C42×C127C2311yescharacteristic
50C127⋊C8—42no—
51C127⋊C8—42no—
52C23×C127—21no—
53C23×C127C22⋊C411yescharacteristic
54Q8×C127—21no—
55Q8×C127M1612yes—
56D4×C127—24no—
57D4×C127—21no—
58D4×C127M1612yes—
59C2×C4×C127—24no—
60C2×C4×C127—21no—
61C2×C4×C127—21no—
62C2×C4×C127—21no—
63C2×C4×C127—22no—
64C2×C4×C127—21no—
65C2×C4×C127—21no—
66C2×C4×C127—41no—
67C2×C4×C127—42no—
68C2×C4×C127—42no—
69C2×C4×C127C22⋊C411yescharacteristic
70C2×C4×C127C22⋊C411yescharacteristic
71C2×C4×C127C22⋊C411yescharacteristic
72C2×C4×C127C22×C411yescharacteristic
73C2×C4×C127C2×D411yescharacteristic
74C2×C4×C127C2×D411yescharacteristic
75C22×C127—21no—
76C22×C127—22no—
77C22×C127—21no—
78C22×C127—21no—
79C22×C127—41no—
80C22×C127—41no—
81C22×C127C2×C22⋊C411yescharacteristic
82C4×C127—21no—
83C4×C127—22no—
84C4×C127—21no—
85C4×C127—21no—
86C4×C127—21no—
87C4×C127—41no—
88C4×C127—41no—
89C4×C127C2×M1611yescharacteristic
90C4×C127C2×M1611yescharacteristic
91C4×C127#13512yes—
92C2×C127—41no—
93C2×C127—42no—
94C2×C127#40711yescharacteristic
95C2×C127#41611yescharacteristic
96C2×C127#42111yescharacteristic
97#1488—1271nomaximal, Sylow
98C127#148811yescharacteristic, complemented, Sylow
99C4×#41D12711yescharacteristic
100#432—1271no—
101#431—1271no—
102Q8⋊C8—1272no—
103D4⋊C8—1272no—
104C2×#41Dic12711yescharacteristic
105C4×Q8Dic12711yescharacteristic
106C4×Q8C2×D12711yescharacteristic
107C4×D4—21no—
108C4×D4—21no—
109C4×D4Dic12711yescharacteristic
110C4×D4C2×D12711yescharacteristic
111#117—21no—
112#117Dic12711yescharacteristic
113C2×C42—21no—
114C2×C42C2×D12711yescharacteristic
115C4⋊C8—1272no—
116C4⋊C8—2541no—
117C22⋊C8—2541no—
118C22⋊C8—2541no—
119#97—2541no—
120C4×C8—1272no—
121#41—22no—
122#41—22no—
123C2×Q8C2×Dic12711yescharacteristic
124C2×D4—21no—
125C2×D4C2×Dic12711yescharacteristic
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4C2×Dic12711yescharacteristic
130C22×C4C22⋊D12711yescharacteristic
131C22×C4C22⋊D12711yescharacteristic
132C2×C8—2542no—
133C2×C8—2542no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4C2×Dic12711yescharacteristic
137C4⋊C4C2×Dic12711yescharacteristic
138C22⋊C4—21no—
139C22⋊C4—21no—
140C22⋊C4—21no—
141C42—21no—
142C42—21no—
143C42—21no—
144C42C2×Dic12711yescharacteristic
145C42C22⋊D12711yescharacteristic
146C42C22⋊D12711yescharacteristic
147C42C22×D12711yescharacteristic
148C23—21no—
149C23C22⋊Dic12711yescharacteristic
150Q8—21no—
151Q8#1452323212yes—
152D4—24no—
153D4—21no—
154D4#1452323212yes—
155C2×C4—24no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—22no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—41no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4C22⋊Dic12711yescharacteristic
166C2×C4C22⋊Dic12711yescharacteristic
167C2×C4C22⋊Dic12711yescharacteristic
168C2×C4C22×Dic12711yescharacteristic
169C2×C4C2×C22⋊D12711yescharacteristic
170C2×C4C2×C22⋊D12711yescharacteristic
171C8—5082no—
172C8—5082no—
173C22—21no—
174C22—22no—
175C22—21no—
176C22—21no—
177C22—41no—
178C22—41no—
179C22C2×C22⋊Dic12711yescharacteristic
180C4—21no—
181C4—22no—
182C4—21no—
183C4—21no—
184C4—21no—
185C4—41no—
186C4—41no—
187C4C2×#1452323211yescharacteristic
188C4C2×#1452323211yescharacteristic
189C4#1686122912yes—
190C2—41no—
191C2—42no—
192C2#2254984011yescharacteristic
193C2#2254987211yescharacteristic
194C2#2254989311yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D127
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
FittingC4×#41×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC2×#41×C127C4
cosocleC42×C127C23
hypercenterC4×#41D127
hypocenterC127#1488
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#41

Complements

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

NH ≅ G/Ntype
1. #22549933C2semidirect
2. #22549929C2semidirect
3. Q8⋊(C127⋊C8)C2semidirect
4. D4⋊(C127⋊C8)C2semidirect
98. C127#1488semidirect

Maximal subgroups

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

subgroupindex
#225499332
#225499292
Q8⋊(C127⋊C8)2
D4⋊(C127⋊C8)2
C4×#41×C1272
#1488127