Finite Group Databasebeta

← #30639417

Subgroups — #30639417

Order 16256.

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

normal subgroups only

This group has 248 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×#16861133C211yesmaximal, characteristic
2C2×#16861119C211yesmaximal, characteristic
3#22549783C211yesmaximal, characteristic
4#22549772C211yesmaximal, characteristic
5#22549639C211yesmaximal, characteristic
6#22549638C211yesmaximal, characteristic
7C2×#103×C127C211yesmaximal, characteristic
8C22×C127⋊C8C2211yescharacteristic
9C2×C4⋊Dic127C2211yescharacteristic
10C2×C4⋊Dic127C2211yescharacteristic
11C2×Dic127⋊C4—21no—
12C2×C4×Dic127C2211yescharacteristic
13#16861133C414yescomplemented
14#16861119—22no—
15C127⋊#95—21no—
16C127⋊#95—21no—
17C127⋊#95—21no—
18C22×Q8×C127C2211yescharacteristic
19C22×C8×C127C2211yescharacteristic
20C2×C4⋊C4×C127C2211yescharacteristic
21#103×C127—22no—
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⋊Dic127C2×C412yes—
33Dic127⋊C4—42no—
34C4×Dic127—21no—
35C4×Dic127—21no—
36C2×Q8×C127—21no—
37C2×Q8×C127—21no—
38C2×Q8×C127—21no—
39C2×Q8×C127—21no—
40C2×Q8×C127—21no—
41C2×Q8×C127C2×C412yes—
42C22×C4×C127—21no—
43C22×C4×C127—21no—
44C22×C4×C127C2311yescharacteristic
45C2×C8×C127—21no—
46C2×C8×C127—21no—
47C2×C8×C127D412yes—
48C4⋊C4×C127—21no—
49C4⋊C4×C127—21no—
50C2×Dic127—22no—
51C2×Dic127—21no—
52C2×Dic127—21no—
53C2×Dic127—21no—
54C2×Dic127—21no—
55C2×Dic127—21no—
56C2×Dic127—21no—
57C2×Dic127—41no—
58C2×Dic127—41no—
59C2×Dic127—41no—
60C2×Dic127—41no—
61C2×Dic127—41no—
62C127⋊C8—22no—
63C23×C127C2×D411yescharacteristic
64Q8×C127—22no—
65Q8×C127—22no—
66Q8×C127—41no—
67Q8×C127—41no—
68C2×C4×C127—22no—
69C2×C4×C127—41no—
70C2×C4×C127—41no—
71C2×C4×C127—41no—
72C2×C4×C127—41no—
73C2×C4×C127—41no—
74C2×C4×C127C22×C411yescharacteristic
75C2×C4×C127C2×D411yescharacteristic
76C2×C4×C127#4111yescharacteristic
77C2×C4×C127#4111yescharacteristic
78C2×C4×C127#4111yescharacteristic
79C2×C4×C127#4111yescharacteristic
80C8×C127—22no—
81C22×C127C4×D411yescharacteristic
82C22×C127#12111yescharacteristic
83C22×C127#12411yescharacteristic
84C22×C127AGL(1,ℤ/8)11yescharacteristic
85C22×C127#13711yescharacteristic
86C22×C127#13711yescharacteristic
87C22×C127#13711yescharacteristic
88C4×C127—42no—
89C4×C127—81no—
90C4×C127#11711yescharacteristic
91C4×C127C4×D411yescharacteristic
92C4×C127#12311yescharacteristic
93C4×C127#12611yescharacteristic
94Dic127—42no—
95Dic127—41no—
96Dic127—41no—
97Dic127—81no—
98C2×C127#38811yescharacteristic
99C2×C127#44011yescharacteristic
100C2×C127#44111yescharacteristic
101C2×C127#46811yescharacteristic
102C2×C127#47011yescharacteristic
103C2×C127#48911yescharacteristic
104C2×C127#49111yescharacteristic
105#1935—1271nomaximal, Sylow
106C127#193511yescharacteristic, complemented, Sylow
107C2×#103—1271no—
108C2×#103D12711yescharacteristic
109C2×#97—1271no—
110#391—1271no—
111#389—1271no—
112#340—1271no—
113#340—1271no—
114C22×Q8C2×D12711yescharacteristic
115C22×C8—1271no—
116C22×C8C2×D12711yescharacteristic
117C2×C4⋊C4—1271no—
118C2×C4⋊C4—1271no—
119C2×C4⋊C4—2541no—
120C2×C4⋊C4C2×D12711yescharacteristic
121C2×C42—1271no—
122#103—22no—
123#103—1274no—
124#97—2542no—
125#95—2541no—
126#95—2541no—
127#95—2541no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8C4×D12712yes—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—1271no—
137C22×C4—1271no—
138C22×C4—2541no—
139C22×C4—2541no—
140C22×C4C22×D12711yescharacteristic
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—1272no—
144C2×C8—2541no—
145C2×C8—2541no—
146C2×C8C22⋊D12712yes—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—1272no—
150C4⋊C4—5082no—
151C4⋊C4—2541no—
152C4⋊C4—2541no—
153C4⋊C4—2541no—
154C42—2541no—
155C42—2541no—
156C23D4×D12711yescharacteristic
157Q8—22no—
158Q8—22no—
159Q8—41no—
160Q8—41no—
161C2×C4—22no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—5081no—
168C2×C4—5081no—
169C2×C4—5081no—
170C2×C4—5081no—
171C2×C4—5081no—
172C2×C4—2541no—
173C2×C4—2541no—
174C2×C4—2541no—
175C2×C4—2541no—
176C2×C4—2542no—
177C2×C4—2541no—
178C2×C4—2541no—
179C2×C4C2×C4×D12711yescharacteristic
180C2×C4C2×C22⋊D12711yescharacteristic
181C2×C4#1452324711yescharacteristic
182C2×C4#1452324711yescharacteristic
183C2×C4#1452324711yescharacteristic
184C2×C4#1452324811yescharacteristic
185C8—22no—
186C8—2542no—
187C22(C22⋊D127)⋊C411yescharacteristic
188C22#1686118011yescharacteristic
189C22#1686119511yescharacteristic
190C22#1686123611yescharacteristic
191C22#1686123811yescharacteristic
192C22#1686124111yescharacteristic
193C22#1686124111yescharacteristic
194C4—42no—
195C4—81no—
196C4—10161no—
197C4—5081no—
198C4—5081no—
199C4—5082no—
200C4#1686116511yescharacteristic
201C4C4×C22⋊D12711yescharacteristic
202C4#1686119011yescharacteristic
203C4#1686120211yescharacteristic
204C2#2254976411yescharacteristic
205C2#2254997911yescharacteristic
206C2#2254998011yescharacteristic
207C2#2255011911yescharacteristic
208C2#2255013111yescharacteristic
209C2#2255023711yescharacteristic
210C2#2255024311yescharacteristic
subgroupG/H
centerC23D4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#103×C127C2
socle, prime radicalC23×C127C2×D4
cosocleC22×C4×C127C23
hypercenterC2×#103D127
hypocenterC127#1935
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
13. #16861133C4semidirect
106. C127#1935semidirect

Maximal subgroups

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

subgroupindex
C2×#168611332
C2×#168611192
#225497832
#225497722
#225496392
#225496382
C2×#103×C1272
#1935127