Finite Group Databasebeta

← #30637187

Subgroups — #30637187

Order 16256.

230 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 67 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#22550408C211yesmaximal, characteristic
2#22549932C211yesmaximal, characteristic, complemented
3C4×C8×D127C211yesmaximal, characteristic, complemented
4#22549608C211yesmaximal, characteristic, complemented
5#22549606C211yesmaximal, characteristic, complemented
6D4⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
7D4⋊C8×C127C211yesmaximal, characteristic, complemented
8C2×C8×D127—21no—
9#16861202—21no—
10#16861200C411yescharacteristic
11#16861196C411yescharacteristic
12#16861191—21no—
13#16861189C411yescharacteristic
14#16861180C411yescharacteristic
15C4×Dic254C2211yescharacteristic
16C4×C22⋊D127—21no—
17#16861168—21no—
18C42×D127C2211yescharacteristic
19C4⋊(C127⋊C8)C2211yescharacteristic
20(C2×D127)⋊C8—21no—
21#16861119—21no—
22C4×C127⋊C8C2211yescharacteristic
23C8×Dic127—21no—
24C4×D4×C127C2211yescharacteristic
25C4⋊C8×C127C2211yescharacteristic
26C4×C8×C127C2211yescharacteristic
27C2×Dic254C2×C411yescharacteristic
28C2×C22⋊D127—21no—
29C2×C4×D127C2×C411yescharacteristic
30C2×C4×D127D411yescharacteristic
31C2×C4×D127D411yescharacteristic
32C2×C127⋊C8—21no—
33C2×C127⋊C8—21no—
34C8×D127—24no—
35C4⋊Dic127—21no—
36C4⋊Dic127—21no—
37C4⋊Dic127C2×C411yescharacteristic
38Dic127⋊C4—21no—
39Dic127⋊C4—21no—
40C22⋊Dic127—21no—
41C22⋊Dic127—21no—
42C22⋊Dic127—21no—
43(C2×D127)⋊C4—21no—
44C4×Dic127—21no—
45C4×Dic127C2×C411yescharacteristic
46C4×Dic127D411yescharacteristic
47C4×Dic127D411yescharacteristic
48C2×D4×C127C2×C411yescharacteristic
49C22×C4×C127—21no—
50C2×C8×C127—21no—
51C2×C8×C127—21no—
52C4⋊C4×C127C2×C411yescharacteristic
53C22⋊C4×C127—21no—
54C42×C127C2311yescharacteristic
55C22×D127C22⋊C411yescharacteristic
56Dic254—21no—
57Dic254—21no—
58C22⋊D127—42no—
59C2×Dic127—21no—
60C2×Dic127—21no—
61C2×Dic127C22⋊C411yescharacteristic
62C2×Dic127C22⋊C411yescharacteristic
63C2×Dic127C22⋊C411yescharacteristic
64C4×D127—22no—
65C4×D127—21no—
66C4×D127—21no—
67C4×D127—21no—
68C4×D127—21no—
69C127⋊C8—22no—
70C127⋊C8—41no—
71C23×C127—21no—
72D4×C127—21no—
73D4×C127—21no—
74C2×C4×C127—21no—
75C2×C4×C127—41no—
76C2×C4×C127—41no—
77C2×C4×C127C22×C411yescharacteristic
78C2×C4×C127C2×D411yescharacteristic
79C2×C4×C127C2×D411yescharacteristic
80C8×C127—22no—
81C8×C127—41no—
82C22×C127—41no—
83C22×C127—41no—
84C22×C127—41no—
85C22×C127C2×C22⋊C411yescharacteristic
86C2×D127—21no—
87C2×D127—21no—
88C2×D127#10412yes—
89C4×C127—21no—
90C4×C127—41no—
91C4×C127#13111yescharacteristic
92C4×C127#13111yescharacteristic
93C4×C127#13511yescharacteristic
94C4×C127#13511yescharacteristic
95Dic127—21no—
96Dic127—21no—
97Dic127—21no—
98Dic127—41no—
99Dic127—41no—
100C2×C127—81no—
101C2×C127#40811yescharacteristic
102C2×C127#41611yescharacteristic
103C2×C127C2×#10411yescharacteristic
104D127—22no—
105#1494—1271nomaximal, Sylow
106C127#149411yescharacteristic, complemented, Sylow
107#529—1271no—
108#432—1271no—
109C2×C4×C8—1271no—
110Q8⋊C8—1271no—
111Q8⋊C8—1271no—
112D4⋊C8—1271no—
113D4⋊C8D12711yescharacteristic, complemented
114C22×C8—2541no—
115#126—2541no—
116#125—1271no—
117#124—1271no—
118#123—2541no—
119C22⋊Q8—1271no—
120#121—1271no—
121C4×Q8—1271no—
122C4×D4—2541no—
123C4×D4C2×D12711yescharacteristic
124#117—2541no—
125C2×C42—1271no—
126C4⋊C8—1271no—
127C4⋊C8C2×D12711yescharacteristic
128C22⋊C8—2541no—
129#97—2541no—
130C4×C8—1271no—
131C4×C8—2541no—
132C4×C8C2×D12711yescharacteristic
133C2×Q8—1271no—
134C2×D4—2541no—
135C2×D4C4×D12711yescharacteristic
136C22×C4—21no—
137C22×C4—1271no—
138C22×C4—1271no—
139C22×C4—1271no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—2544no—
143C2×C8—2541no—
144C2×C8—2541no—
145C4⋊C4—1271no—
146C4⋊C4—2541no—
147C4⋊C4—2541no—
148C4⋊C4—2541no—
149C4⋊C4—2541no—
150C4⋊C4C4×D12711yescharacteristic
151C22⋊C4—21no—
152C22⋊C4—2541no—
153C22⋊C4—2541no—
154C22⋊C4—2541no—
155C22⋊C4—2541no—
156C42—1271no—
157C42—1271no—
158C42—1271no—
159C42—2541no—
160C42C22×D12711yescharacteristic
161C23—21no—
162C23—1271no—
163Q8—2541no—
164Q8—2541no—
165D4—21no—
166D4—21no—
167D4—5082no—
168C2×C4—21no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—1271no—
172C2×C4—1271no—
173C2×C4—1271no—
174C2×C4—2542no—
175C2×C4—2541no—
176C2×C4—2541no—
177C2×C4—2541no—
178C2×C4—2541no—
179C2×C4—2541no—
180C2×C4—2541no—
181C2×C4C2×C4×D12711yescharacteristic
182C2×C4D4×D12711yescharacteristic
183C2×C4D4×D12711yescharacteristic
184C8—22no—
185C8—41no—
186C8—5081no—
187C8—2542no—
188C22—41no—
189C22—41no—
190C22—41no—
191C22—1272no—
192C22—2541no—
193C22—2541no—
194C22C22⋊C4×D12711yescharacteristic
195C4—21no—
196C4—41no—
197C4—5081no—
198C4—5081no—
199C4—2541no—
200C4—2541no—
201C4—2541no—
202C4#1686121511yescharacteristic
203C4#1686121711yescharacteristic
204C4#1686122811yescharacteristic
205C4#1686123011yescharacteristic
206C2—81no—
207C2—2542no—
208C2#2254984111yescharacteristic
209C2#2254986911yescharacteristic
210C2#104×D12711yescharacteristic
subgroupG/H
centerC2×C4D4×D127
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
FittingD4⋊C8×C127C2
socleC22×C127C2×C22⋊C4
prime radical#16861180C4
cosocleC42×C127C23
hypercenterD4⋊C8D127
hypocenterC127#1494
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #22549932C2semidirect
3. C4×C8×D127C2semidirect
4. #22549608C2semidirect
5. #22549606C2semidirect
6. D4⋊(C127⋊C8)C2semidirect
7. D4⋊C8×C127C2semidirect
106. C127#1494semidirect
113. D4⋊C8D127semidirect

Maximal subgroups

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

subgroupindex
#225504082
#225499322
C4×C8×D1272
#225496082
#225496062
D4⋊(C127⋊C8)2
D4⋊C8×C1272
#1494127