Finite Group Databasebeta

← #30637845

Subgroups — #30637845

Order 16256.

314 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 59 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
1D4×D508C211yesmaximal, characteristic
2#22550250C211yesmaximal, characteristic, complemented
3#22550163C211yesmaximal, characteristic, complemented
4#22549619C211yesmaximal, characteristic, complemented
5D4⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
6#22549603C211yesmaximal, characteristic, complemented
7#476×C127C211yesmaximal, characteristic, complemented
8C22×D508—21no—
9C2×D4×D127—21no—
10C2×#14523235—22no—
11#16861206C2211yescharacteristic
12#16861184—21no—
13#16861183—21no—
14#16861179—22no—
15C4×D508C2211yescharacteristic
16#16861140—21no—
17C4⋊(C127⋊C8)C2211yescharacteristic, complemented
18#16861129—21no—
19C4×C127⋊C8C2211yescharacteristic, complemented
20C4⋊Q8×C127C2211yescharacteristic, complemented
21C4×D4×C127C2211yescharacteristic
22#106×C127—21no—
23C4⋊C8×C127C2211yescharacteristic, complemented
24#102×C127—21no—
25C23×D127—22no—
26C2×D508—22no—
27C2×D508—21no—
28C2×D508—21no—
29C2×D508—42no—
30C2×D508—42no—
31C2×D508D411yescharacteristic
32C2×D508D412yes—
33C2×C22⋊D127—22no—
34D4×D127—42no—
35D4×D127—42no—
36C2×C4×D127—21no—
37#14523235—44no—
38C2×C127⋊C8—21no—
39C2×C127⋊C8—21no—
40C4⋊Dic127D411yescharacteristic
41(C2×D127)⋊C4—22no—
42(C2×D127)⋊C4—21no—
43C2×Q8×C127—21no—
44C2×D4×C127D411yescharacteristic
45C22×C4×C127—21no—
46C2×C8×C127—21no—
47C4⋊C4×C127—21no—
48C4⋊C4×C127—21no—
49C4⋊C4×C127D411yescharacteristic
50C22⋊C4×C127—21no—
51C42×C127C2311yescharacteristic
52C22×D127—21no—
53C22×D127—22no—
54C22×D127—44no—
55C22×D127—44no—
56C22×D127—82no—
57D508—21no—
58D508—42no—
59D508—44no—
60D508—41no—
61D508—41no—
62D508—41no—
63D508—41no—
64D508SD1612yescomplemented
65C22⋊D127—44no—
66C2×Dic127—21no—
67C4×D127—41no—
68C4×D127—41no—
69C127⋊C8—22no—
70C127⋊C8—41no—
71C23×C127—21no—
72Q8×C127—42no—
73D4×C127—21no—
74D4×C127SD1612yescomplemented
75C2×C4×C127—21no—
76C2×C4×C127—41no—
77C2×C4×C127—41no—
78C2×C4×C127—41no—
79C2×C4×C127C2×D411yescharacteristic
80C2×C4×C127C2×D411yescharacteristic
81C2×C4×C127C2×D411yescharacteristic
82C8×C127—41no—
83C22×C127—22no—
84C22×C127—41no—
85C22×C127—41no—
86C22×C127#12011yescharacteristic
87C2×D127—22no—
88C2×D127—41no—
89C2×D127—41no—
90C2×D127—42no—
91C2×D127—42no—
92C2×D127—42no—
93C2×D127—88no—
94C4×C127—21no—
95C4×C127—41no—
96C4×C127—81no—
97C4×C127C2×SD1611yescharacteristic
98C4×C127C2×SD1611yescharacteristic
99C4×C127AGL(1,ℤ/8)11yescharacteristic
100C4×C127AGL(1,ℤ/8)11yescharacteristic
101Dic127—41no—
102C2×C127—42no—
103C2×C127#45011yescharacteristic
104C2×C127#45011yescharacteristic
105C2×C127#45311yescharacteristic
106D127—42no—
107D127—82no—
108#1629—1271nomaximal, Sylow
109C127#162911yescharacteristic, complemented, Sylow
110D42—1271no—
111#492—1271no—
112#476—1271no—
113#476D12711yescharacteristic, complemented
114#331—1271no—
115D4⋊C8—1271no—
116D4⋊C8—1271no—
117C22×D4—2541no—
118C22×D4—2541no—
119C2×SD16—2542no—
120C4⋊Q8C2×D12711yescharacteristic, complemented
121#127—1271no—
122#121—2541no—
123#121—2541no—
124#120—2542no—
125C4×D4—1271no—
126C4×D4C2×D12711yescharacteristic
127#106—21no—
128#106—2541no—
129C4⋊C8—1271no—
130C4⋊C8C2×D12711yescharacteristic, complemented
131#102—21no—
132#102—2541no—
133C4×C8—1271no—
134C24—2542no—
135C2×Q8—21no—
136C2×D4—1271no—
137C2×D4—1272no—
138C2×D4—5082no—
139C2×D4—5082no—
140C2×D4—5082no—
141C2×D4—5082no—
142C2×D4—2542no—
143C2×D4—2542no—
144C2×D4—2541no—
145C2×D4—2541no—
146C2×D4C22⋊D12711yescharacteristic
147C22×C4—21no—
148C22×C4—2541no—
149SD16—5084no—
150C2×C8—21no—
151C2×C8—2541no—
152C2×C8—2541no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—1271no—
156C4⋊C4C22⋊D12711yescharacteristic
157C22⋊C4—21no—
158C22⋊C4—2542no—
159C22⋊C4—2541no—
160C42C22×D12711yescharacteristic
161C23—21no—
162C23—10162no—
163C23—5084no—
164C23—5084no—
165C23—2542no—
166C23—2541no—
167Q8—42no—
168D4—21no—
169D4—1272no—
170D4—5082no—
171D4—5084no—
172D4—5084no—
173D4—5081no—
174D4—5081no—
175D4—5081no—
176D4—5081no—
177D4—2541no—
178D4#1452323512yescomplemented
179C2×C4—21no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—5081no—
184C2×C4—5081no—
185C2×C4—2541no—
186C2×C4D4×D12711yescharacteristic
187C2×C4D4×D12711yescharacteristic
188C2×C4C2×C22⋊D12711yescharacteristic
189C8—41no—
190C8—5081no—
191C8—2542no—
192C22—22no—
193C22—41no—
194C22—41no—
195C22—10168no—
196C22—5082no—
197C22—5082no—
198C22—5082no—
199C22—5081no—
200C22—5081no—
201C22—2542no—
202C22#1686117711yescharacteristic
203C4—21no—
204C4—41no—
205C4—81no—
206C4—5081no—
207C4SD16×D12711yescharacteristic
208C4C2×#1452323511yescharacteristic
209C4#1686123411yescharacteristic
210C4#1686123611yescharacteristic
211C2—42no—
212C2—10162no—
213C2—5082no—
214C2#2255002511yescharacteristic
215C2#2255002611yescharacteristic
216C2#2255004211yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#476×C127C2
socleC22×C127#120
prime radicalD4×D508C2
hypercenter#476D127
hypocenterC127#1629
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
2. #22550250C2semidirect
3. #22550163C2semidirect
4. #22549619C2semidirect
5. D4⋊(C127⋊C8)C2semidirect
6. #22549603C2semidirect
7. #476×C127C2semidirect
17. C4⋊(C127⋊C8)C22semidirect
19. C4×C127⋊C8C22semidirect
20. C4⋊Q8×C127C22semidirect
23. C4⋊C8×C127C22semidirect
64. D508SD16semidirect
74. D4×C127SD16semidirect
109. C127#1629semidirect
113. #476D127semidirect
120. C4⋊Q8C2×D127semidirect
130. C4⋊C8C2×D127semidirect
178. D4#14523235semidirect

Maximal subgroups

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

subgroupindex
D4×D5082
#225502502
#225501632
#225496192
D4⋊(C127⋊C8)2
#225496032
#476×C1272
#1629127