Finite Group Databasebeta

← #30637921

Subgroups — #30637921

Order 16256.

250 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 57 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#22550548C211yesmaximal, characteristic
2#22550224C211yesmaximal, characteristic, complemented
3#22550153C211yesmaximal, characteristic, complemented
4#22549619C211yesmaximal, characteristic, complemented
5Q8⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
6#22549603C211yesmaximal, characteristic, complemented
7#462×C127C211yesmaximal, characteristic, complemented
8C2×#14523249—21no—
9#16861206—21no—
10#16861206C2211yescharacteristic
11#16861184—22no—
12#16861184—21no—
13C4×D508—21no—
14C4×D508C2211yescharacteristic
15#16861142—21no—
16C4⋊(C127⋊C8)C2211yescharacteristic
17#16861129—21no—
18#16861129—22no—
19C4×C127⋊C8C2211yescharacteristic
20C2×Q16×C127—21no—
21C4⋊Q8×C127C2211yescharacteristic
22C4×Q8×C127C2211yescharacteristic
23C4⋊C8×C127C2211yescharacteristic
24#103×C127—21no—
25#14523249—42no—
26#14523249—42no—
27C2×D508—22no—
28C2×D508—22no—
29C2×D508—21no—
30C2×D508—21no—
31C2×D508D411yescharacteristic
32C2×D508D412yes—
33C2×C4×D127—22no—
34C2×C4×D127—21no—
35C2×C127⋊C8—21no—
36C2×C127⋊C8—21no—
37C4⋊Dic127D411yescharacteristic
38(C2×D127)⋊C4—22no—
39(C2×D127)⋊C4—21no—
40C2×Q8×C127—21no—
41C2×Q8×C127D411yescharacteristic
42Q16×C127—42no—
43C2×C8×C127—21no—
44C4⋊C4×C127—21no—
45C4⋊C4×C127—21no—
46C4⋊C4×C127—21no—
47C4⋊C4×C127D411yescharacteristic
48C42×C127—21no—
49C42×C127C2311yescharacteristic
50C22×D127—21no—
51C22×D127—22no—
52D508—21no—
53D508—21no—
54D508—42no—
55D508—44no—
56D508—41no—
57D508—41no—
58D508—41no—
59D508—41no—
60D508—82no—
61C2×Dic127—21no—
62C4×D127—44no—
63C4×D127—41no—
64C4×D127—41no—
65C127⋊C8—41no—
66C127⋊C8—41no—
67Q8×C127—21no—
68Q8×C127—41no—
69Q8×C127—41no—
70Q8×C127D812yes—
71C2×C4×C127—21no—
72C2×C4×C127—21no—
73C2×C4×C127—41no—
74C2×C4×C127C2×D411yescharacteristic
75C2×C4×C127C2×D411yescharacteristic
76C2×C4×C127C2×D411yescharacteristic
77C8×C127—41no—
78C22×C127#12011yescharacteristic
79C2×D127—42no—
80C2×D127—41no—
81C2×D127—41no—
82C2×D127—42no—
83C2×D127—42no—
84C2×D127—41no—
85C4×C127—22no—
86C4×C127—21no—
87C4×C127—41no—
88C4×C127—81no—
89C4×C127C2×D811yescharacteristic
90C4×C127#13511yescharacteristic
91C4×C127AGL(1,ℤ/8)11yescharacteristic
92C4×C127#13711yescharacteristic
93Dic127—41no—
94C2×C127#44711yescharacteristic
95C2×C127#45211yescharacteristic
96C2×C127#45511yescharacteristic
97D127—81no—
98D127—82no—
99#1641—1271nomaximal, Sylow
100C127#164111yescharacteristic, complemented, Sylow
101#550—1271no—
102#486—1271no—
103#474—1271no—
104#462D12711yescharacteristic, complemented
105#331—1271no—
106Q8⋊C8—1271no—
107D4⋊C8—1271no—
108C2×#41—2541no—
109C2×Q16—21no—
110C4⋊Q8C2×D12711yescharacteristic
111#127—1271no—
112#127—2541no—
113#121—2542no—
114#121—2541no—
115C4×Q8C2×D12711yescharacteristic
116C4×D4—1271no—
117C4×D4—2541no—
118#107—2541no—
119C4⋊C8—1271no—
120C4⋊C8C2×D12711yescharacteristic
121#103—21no—
122#102—2541no—
123#102—2542no—
124C4×C8—1271no—
125#41—5082no—
126#41—5082no—
127C2×Q8—21no—
128C2×Q8C22⋊D12711yescharacteristic
129C2×D4—1272no—
130C2×D4—1271no—
131C2×D4—2542no—
132C2×D4—2542no—
133C2×D4—2541no—
134C2×D4—2541no—
135C22×C4—2542no—
136C22×C4—2541no—
137Q16—42no—
138C2×C8—21no—
139C2×C8—2541no—
140C2×C8—2541no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—1271no—
145C4⋊C4C22⋊D12711yescharacteristic
146C22⋊C4—2542no—
147C22⋊C4—2541no—
148C42—21no—
149C42C22×D12711yescharacteristic
150C23—2542no—
151C23—2541no—
152Q8—21no—
153Q8—41no—
154Q8—41no—
155Q8#1452323312yes—
156D4—10162no—
157D4—5082no—
158D4—5084no—
159D4—5081no—
160D4—5081no—
161D4—5081no—
162D4—5081no—
163D4—2541no—
164D4—2541no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—41no—
168C2×C4—5084no—
169C2×C4—5081no—
170C2×C4—5081no—
171C2×C4—2541no—
172C2×C4D4×D12711yescharacteristic
173C2×C4D4×D12711yescharacteristic
174C2×C4C2×C22⋊D12711yescharacteristic
175C8—41no—
176C8—5081no—
177C8—5081no—
178C22—5082no—
179C22—5081no—
180C22—5081no—
181C22—5082no—
182C22—5082no—
183C22—5081no—
184C22#1686117711yescharacteristic
185C4—21no—
186C4—22no—
187C4—41no—
188C4—81no—
189C4—5081no—
190C4C2×#1452323311yescharacteristic
191C4#1686123211yescharacteristic
192C4#1686123411yescharacteristic
193C4#1686124111yescharacteristic
194C2—10162no—
195C2—10161no—
196C2#2255000911yescharacteristic
197C2#2255003711yescharacteristic
198C2#2255005211yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#462×C127C2
socleC22×C127#120
prime radical#22550548C2
hypercenter#462D127
hypocenterC127#1641
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
2. #22550224C2semidirect
3. #22550153C2semidirect
4. #22549619C2semidirect
5. Q8⋊(C127⋊C8)C2semidirect
6. #22549603C2semidirect
7. #462×C127C2semidirect
100. C127#1641semidirect
104. #462D127semidirect

Maximal subgroups

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

subgroupindex
#225505482
#225502242
#225501532
#225496192
Q8⋊(C127⋊C8)2
#225496032
#462×C1272
#1641127