Finite Group Databasebeta

← #30637878

Subgroups — #30637878

Order 16256.

234 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 55 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#22550529C211yesmaximal, characteristic
2#22550163C211yesmaximal, characteristic, complemented
3#22550163C211yesmaximal, characteristic, complemented
4#22549622C211yesmaximal, characteristic, complemented
5#22549605C211yesmaximal, characteristic, complemented
6#22549605C211yesmaximal, characteristic, complemented
7#487×C127C211yesmaximal, characteristic, complemented
8C2×#14523249—21no—
9#16861194—22no—
10#16861185—21no—
11#16861184—21no—
12C4×D508C2211yescharacteristic
13C4×D508C2211yescharacteristic
14C4⋊C4×D127—21no—
15#16861140—21no—
16#16861140—21no—
17C4⋊(C127⋊C8)C2211yescharacteristic
18C4⋊(C127⋊C8)C2211yescharacteristic
19#16861129—21no—
20#16861129—21no—
21C4⋊Q8×C127C2211yescharacteristic
22C4⋊Q8×C127C2211yescharacteristic
23#103×C127—22no—
24C4×C8×C127C2211yescharacteristic
25#14523249—42no—
26#14523249—42no—
27C2×D508—22no—
28C2×D508D411yescharacteristic
29C2×D508D411yescharacteristic
30C2×C4×D127—22no—
31C2×C4×D127—21no—
32C2×C4×D127—21no—
33C2×C127⋊C8—21no—
34C2×C127⋊C8—21no—
35C4⋊Dic127D411yescharacteristic
36C4⋊Dic127D411yescharacteristic
37Dic127⋊C4—22no—
38(C2×D127)⋊C4—22no—
39(C2×D127)⋊C4—21no—
40(C2×D127)⋊C4—21no—
41C2×Q8×C127—21no—
42C2×Q8×C127—21no—
43C2×C8×C127—21no—
44C4⋊C4×C127—21no—
45C4⋊C4×C127—21no—
46C4⋊C4×C127—21no—
47C4⋊C4×C127D412yes—
48C42×C127C2311yescharacteristic
49C22×D127—21no—
50C22×D127—21no—
51D508—21no—
52D508—21no—
53D508—21no—
54D508—44no—
55D508SD1612yes—
56C2×Dic127—21no—
57C2×Dic127—21no—
58C4×D127—44no—
59C4×D127—41no—
60C4×D127—41no—
61C4×D127—41no—
62C4×D127—41no—
63C127⋊C8—41no—
64C127⋊C8—41no—
65Q8×C127—42no—
66Q8×C127—41no—
67Q8×C127—41no—
68C2×C4×C127—22no—
69C2×C4×C127—41no—
70C2×C4×C127C2×D411yescharacteristic
71C2×C4×C127C2×D411yescharacteristic
72C2×C4×C127C2×D411yescharacteristic
73C8×C127—41no—
74C22×C127#12011yescharacteristic
75C2×D127—22no—
76C2×D127—41no—
77C2×D127—41no—
78C2×D127—41no—
79C2×D127—41no—
80C2×D127—41no—
81C4×C127—21no—
82C4×C127—42no—
83C4×C127—81no—
84C4×C127C2×SD1611yescharacteristic
85C4×C127#13511yescharacteristic
86C4×C127AGL(1,ℤ/8)11yescharacteristic
87C4×C127#13711yescharacteristic
88Dic127—41no—
89Dic127—41no—
90C2×C127#45011yescharacteristic
91C2×C127#45211yescharacteristic
92C2×C127#45411yescharacteristic
93D127—42no—
94D127—81no—
95#1635—1271nomaximal, Sylow
96C127#163511yescharacteristic, complemented, Sylow
97#547—1271no—
98#487D12711yescharacteristic, complemented
99#476—1271no—
100#476—1271no—
101#332—1271no—
102D4⋊C8—1271no—
103D4⋊C8—1271no—
104C2×#41—2541no—
105C4⋊Q8C2×D12711yescharacteristic
106C4⋊Q8C2×D12711yescharacteristic
107#123—2542no—
108C22⋊Q8—2541no—
109#121—2541no—
110C4×D4—1271no—
111C4×D4—1271no—
112C2×C4⋊C4—2541no—
113#106—2541no—
114#106—2541no—
115C4⋊C8—1271no—
116C4⋊C8—1271no—
117#103—22no—
118#102—2541no—
119#102—2541no—
120C4×C8C2×D12711yescharacteristic
121#41—5082no—
122#41—5082no—
123C2×Q8—21no—
124C2×Q8—21no—
125C2×D4—1271no—
126C2×D4—1271no—
127C2×D4—2542no—
128C22×C4—2542no—
129C22×C4—2541no—
130C22×C4—2541no—
131C2×C8—21no—
132C2×C8—2541no—
133C2×C8—2541no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—21no—
137C4⋊C4—1271no—
138C4⋊C4—1271no—
139C4⋊C4—2542no—
140C4⋊C4C22⋊D12712yes—
141C22⋊C4—2542no—
142C22⋊C4—2541no—
143C22⋊C4—2541no—
144C42C22×D12711yescharacteristic
145C23—2541no—
146C23—2541no—
147Q8—42no—
148Q8—41no—
149Q8—41no—
150D4—1272no—
151D4—5084no—
152D4—2541no—
153D4—2541no—
154D4—2541no—
155C2×C4—22no—
156C2×C4—41no—
157C2×C4—5084no—
158C2×C4—5081no—
159C2×C4—5081no—
160C2×C4—5081no—
161C2×C4—5081no—
162C2×C4—2541no—
163C2×C4—2541no—
164C2×C4D4×D12711yescharacteristic
165C2×C4D4×D12711yescharacteristic
166C2×C4C2×C22⋊D12711yescharacteristic
167C8—41no—
168C8—5081no—
169C8—5081no—
170C22—5081no—
171C22—5081no—
172C22—5081no—
173C22—5081no—
174C22—5081no—
175C22—2542no—
176C22#1686117711yescharacteristic
177C4—21no—
178C4—42no—
179C4—81no—
180C4—5081no—
181C4—5081no—
182C4SD16×D12711yescharacteristic
183C4#1686123211yescharacteristic
184C4#1686123611yescharacteristic
185C4#1686124111yescharacteristic
186C2—10161no—
187C2—5082no—
188C2#2255002511yescharacteristic
189C2#2255003711yescharacteristic
190C2#2255004811yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#487×C127C2
socleC22×C127#120
prime radical#22550529C2
hypercenter#487D127
hypocenterC127#1635
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
2. #22550163C2semidirect
3. #22550163C2semidirect
4. #22549622C2semidirect
5. #22549605C2semidirect
6. #22549605C2semidirect
7. #487×C127C2semidirect
96. C127#1635semidirect
98. #487D127semidirect

Maximal subgroups

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

subgroupindex
#225505292
#225501632
#225496222
#225496052
#487×C1272
#1635127