Finite Group Databasebeta

← #30637882

Subgroups — #30637882

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#22550228C211yesmaximal, characteristic, complemented
3#22550167C211yesmaximal, characteristic, complemented
4#22550167C211yesmaximal, characteristic, complemented
5#22549603C211yesmaximal, characteristic, complemented
6#22549603C211yesmaximal, characteristic, complemented
7#332×C127C211yesmaximal, characteristic, complemented
8C2×#14523249—21no—
9#16861209C2211yescharacteristic
10#16861194—22no—
11#16861185—21no—
12#16861184—21no—
13C4×D508C2211yescharacteristic
14C4×D508C2211yescharacteristic
15C4⋊C4×D127—21no—
16#16861141—21no—
17#16861141—21no—
18#16861132—22no—
19#16861131—21no—
20#16861131—21no—
21C4×C127⋊C8C2211yescharacteristic
22C4⋊Q8×C127C2211yescharacteristic
23C4⋊C8×C127C2211yescharacteristic
24C4⋊C8×C127C2211yescharacteristic
25#14523249—42no—
26#14523249—42no—
27C2×Dic254—21no—
28C2×D508—22no—
29C2×D508D411yescharacteristic
30C2×D508D411yescharacteristic
31C2×C4×D127—22no—
32C2×C4×D127—21no—
33C2×C4×D127—21no—
34C2×C127⋊C8—21no—
35C4⋊Dic127—21no—
36C4⋊Dic127—21no—
37C4⋊Dic127D411yescharacteristic
38C4⋊Dic127D411yescharacteristic
39Dic127⋊C4—22no—
40(C2×D127)⋊C4—22no—
41(C2×D127)⋊C4—21no—
42(C2×D127)⋊C4—21no—
43C2×Q8×C127—21no—
44C2×C8×C127—21no—
45C2×C8×C127—21no—
46C4⋊C4×C127—21no—
47C4⋊C4×C127D412yes—
48C42×C127C2311yescharacteristic
49C22×D127—21no—
50C22×D127—21no—
51Dic254—42no—
52D508—21no—
53D508—21no—
54D508—21no—
55D508—44no—
56D508SD1612yes—
57C2×Dic127—21no—
58C2×Dic127—21no—
59C2×Dic127—41no—
60C4×D127—44no—
61C4×D127—41no—
62C4×D127—41no—
63C4×D127—41no—
64C4×D127—41no—
65C127⋊C8—41no—
66Q8×C127—41no—
67Q8×C127—41no—
68C2×C4×C127—22no—
69C2×C4×C127C2×D411yescharacteristic
70C2×C4×C127C2×D411yescharacteristic
71C2×C4×C127C2×D411yescharacteristic
72C8×C127—41no—
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×C127C2×SD1611yescharacteristic
84C4×C127#13511yescharacteristic
85C4×C127AGL(1,ℤ/8)11yescharacteristic
86C4×C127#13711yescharacteristic
87Dic127—41no—
88Dic127—41no—
89Dic127—81no—
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#487—1271no—
99#476—1271no—
100#476—1271no—
101#332D12711yescharacteristic, complemented
102D4⋊C8—1271no—
103D4⋊C8—1271no—
104C2×#41—2541no—
105C4⋊Q8—1271no—
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⋊C8C2×D12711yescharacteristic
116C4⋊C8C2×D12711yescharacteristic
117#103—2542no—
118#102—2541no—
119#102—2541no—
120C4×C8—1271no—
121#41—5082no—
122#41—5082no—
123C2×Q8—21no—
124C2×Q8—2541no—
125C2×D4—1271no—
126C2×D4—1271no—
127C2×D4—2542no—
128C22×C4—2542no—
129C22×C4—2541no—
130C22×C4—2541no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—2541no—
134C4⋊C4—21no—
135C4⋊C4—1271no—
136C4⋊C4—1271no—
137C4⋊C4—2542no—
138C4⋊C4—2541no—
139C4⋊C4—2541no—
140C4⋊C4D50812yes—
141C22⋊C4—2542no—
142C22⋊C4—2541no—
143C22⋊C4—2541no—
144C42C22×D12711yescharacteristic
145C23—2541no—
146C23—2541no—
147Q8—41no—
148Q8—41no—
149Q8—5082no—
150D4—1272no—
151D4—5084no—
152D4—2541no—
153D4—2541no—
154D4—2541no—
155C2×C4—22no—
156C2×C4—5084no—
157C2×C4—5081no—
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×D50811yescharacteristic
167C8—41no—
168C8—41no—
169C8—5081no—
170C22—5081no—
171C22—5081no—
172C22—5081no—
173C22—5081no—
174C22—5081no—
175C22—2542no—
176C22#1686117911yescharacteristic
177C4—21no—
178C4—42no—
179C4—10161no—
180C4—5081no—
181C4—5081no—
182C4SD16×D12711yescharacteristic
183C4#1686123011yescharacteristic
184C4#1686123311yescharacteristic
185C4#1686124111yescharacteristic
186C2—10161no—
187C2—5082no—
188C2#2255002811yescharacteristic
189C2#2255004111yescharacteristic
190C2#2255005111yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#332×C127C2
socleC22×C127#120
prime radical#22550529C2
hypercenter#332D127
hypocenterC127#1635
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #22550228C2semidirect
3. #22550167C2semidirect
4. #22550167C2semidirect
5. #22549603C2semidirect
6. #22549603C2semidirect
7. #332×C127C2semidirect
96. C127#1635semidirect
101. #332D127semidirect

Maximal subgroups

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

subgroupindex
#225505292
#225502282
#225501672
#225496032
#332×C1272
#1635127