Finite Group Databasebeta

← #30637793

Subgroups — #30637793

Order 16256.

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

normal subgroups only

This group has 278 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, complemented
2#22550252C211yesmaximal, characteristic, complemented
3#22550068C211yesmaximal, characteristic, complemented
4#22549619C211yesmaximal, characteristic, complemented
5Q8⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
6#22549603C211yesmaximal, characteristic, complemented
7#460×C127C211yesmaximal, characteristic, complemented
8C2×#14523249—21no—
9C2×#14523233—21no—
10C2×#14523233—22no—
11#16861206—21no—
12#16861206C2211yescharacteristic
13#16861184—22no—
14#16861184—21no—
15C4×D508—21no—
16C4×D508C2211yescharacteristic
17C4⋊(C127⋊C8)C2211yescharacteristic
18#16861130—21no—
19C4×C127⋊C8C2211yescharacteristic
20C2×SD16×C127—21no—
21#127×C127C2211yescharacteristic
22C4×Q8×C127C2211yescharacteristic
23C4⋊C8×C127C2211yescharacteristic
24#102×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—
35#14523233—44no—
36#14523233—42no—
37C2×C127⋊C8—21no—
38C2×C127⋊C8—21no—
39C4⋊Dic127D411yescharacteristic
40(C2×D127)⋊C4—22no—
41(C2×D127)⋊C4—21no—
42C2×Q8×C127D411yescharacteristic
43C2×D4×C127—21no—
44C2×D4×C127—21no—
45C2×D4×C127—21no—
46SD16×C127—42no—
47C2×C8×C127—21no—
48C4⋊C4×C127—21no—
49C4⋊C4×C127D411yescharacteristic
50C42×C127—21no—
51C42×C127C2311yescharacteristic
52C22×D127—21no—
53C22×D127—22no—
54D508—21no—
55D508—21no—
56D508—42no—
57D508—44no—
58D508—41no—
59D508—41no—
60D508—41no—
61D508—41no—
62D508—82no—
63C2×Dic127—21no—
64C4×D127—44no—
65C4×D127—41no—
66C4×D127—41no—
67C127⋊C8—22no—
68C127⋊C8—41no—
69C23×C127—41no—
70Q8×C127—21no—
71Q8×C127D812yescomplemented
72D4×C127—42no—
73D4×C127—41no—
74D4×C127—41no—
75D4×C127—41no—
76D4×C127—41no—
77C2×C4×C127—21no—
78C2×C4×C127—21no—
79C2×C4×C127C2×D411yescharacteristic
80C2×C4×C127C2×D411yescharacteristic
81C2×C4×C127C2×D411yescharacteristic
82C8×C127—41no—
83C22×C127—81no—
84C22×C127—81no—
85C22×C127—81no—
86C22×C127#12011yescharacteristic
87C2×D127—42no—
88C2×D127—41no—
89C2×D127—41no—
90C2×D127—42no—
91C2×D127—42no—
92C2×D127—41no—
93C4×C127—22no—
94C4×C127—21no—
95C4×C127—41no—
96C4×C127C2×D811yescharacteristic
97C4×C127#13511yescharacteristic
98C4×C127AGL(1,ℤ/8)11yescharacteristic
99C4×C127AGL(1,ℤ/8)11yescharacteristic
100Dic127—41no—
101C2×C127—161no—
102C2×C127#44711yescharacteristic
103C2×C127#44911yescharacteristic
104C2×C127#45311yescharacteristic
105D127—81no—
106D127—82no—
107#1621—1271nomaximal, Sylow
108C127#162111yescharacteristic, complemented, Sylow
109#550—1271no—
110#493—1271no—
111#460D12711yescharacteristic, complemented
112#459—1271no—
113#331—1271no—
114Q8⋊C8—1271no—
115D4⋊C8—1271no—
116C2×#41—2541no—
117C2×SD16—21no—
118C2×D8—2541no—
119C2×D8—2542no—
120#127—1271no—
121#127—2541no—
122#127C2×D12711yescharacteristic
123#121—2542no—
124#121—2541no—
125C4×Q8C2×D12711yescharacteristic
126C4×D4—1271no—
127C4×D4—2541no—
128C4⋊C8—1271no—
129C4⋊C8C2×D12711yescharacteristic
130#102—21no—
131#102—2541no—
132C4×C8—1271no—
133#41—5082no—
134#41—5082no—
135C2×Q8C22⋊D12711yescharacteristic
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—1272no—
140C2×D4—1271no—
141C2×D4—2542no—
142C2×D4—2542no—
143C2×D4—2541no—
144C2×D4—2541no—
145C22×C4—2542no—
146C22×C4—2541no—
147SD16—42no—
148D8—5082no—
149D8—5084no—
150C2×C8—21no—
151C2×C8—2541no—
152C2×C8—2541no—
153C4⋊C4—21no—
154C4⋊C4—1271no—
155C4⋊C4C22⋊D12711yescharacteristic
156C22⋊C4—2542no—
157C22⋊C4—2541no—
158C42—21no—
159C42C22×D12711yescharacteristic
160C23—41no—
161C23—2542no—
162C23—2541no—
163Q8—21no—
164Q8#1452323312yescomplemented
165D4—42no—
166D4—41no—
167D4—41no—
168D4—41no—
169D4—41no—
170D4—10162no—
171D4—5082no—
172D4—5084no—
173D4—5081no—
174D4—5081no—
175D4—5081no—
176D4—5081no—
177D4—2541no—
178D4—2541no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—5084no—
182C2×C4—5081no—
183C2×C4—5081no—
184C2×C4—2541no—
185C2×C4D4×D12711yescharacteristic
186C2×C4D4×D12711yescharacteristic
187C2×C4C2×C22⋊D12711yescharacteristic
188C8—41no—
189C8—5081no—
190C8—2542no—
191C22—81no—
192C22—81no—
193C22—81no—
194C22—5082no—
195C22—5081no—
196C22—5081no—
197C22—5082no—
198C22—5082no—
199C22—5081no—
200C22#1686117711yescharacteristic
201C4—21no—
202C4—22no—
203C4—41no—
204C4—5081no—
205C4C2×#1452323311yescharacteristic
206C4#1686123011yescharacteristic
207C4#1686123411yescharacteristic
208C4#1686123611yescharacteristic
209C2—161no—
210C2—10162no—
211C2—10161no—
212C2#2255000911yescharacteristic
213C2#2255002011yescharacteristic
214C2#2255004211yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#460×C127C2
socleC22×C127#120
hypercenter#460D127
hypocenterC127#1621
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. #22550548C2semidirect
2. #22550252C2semidirect
3. #22550068C2semidirect
4. #22549619C2semidirect
5. Q8⋊(C127⋊C8)C2semidirect
6. #22549603C2semidirect
7. #460×C127C2semidirect
71. Q8×C127D8semidirect
108. C127#1621semidirect
111. #460D127semidirect
164. Q8#14523233semidirect

Maximal subgroups

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

subgroupindex
#225505482
#225502522
#225500682
#225496192
Q8⋊(C127⋊C8)2
#225496032
#460×C1272
#1621127