Finite Group Databasebeta

← #30637796

Subgroups — #30637796

Order 16256.

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

normal subgroups only

This group has 278 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#22550546C211yesmaximal, characteristic, complemented
2#22550252C211yesmaximal, characteristic, complemented
3#22550077C211yesmaximal, characteristic, complemented
4#22550072C211yesmaximal, characteristic, complemented
5#22549606C211yesmaximal, characteristic, complemented
6#22549603C211yesmaximal, characteristic, complemented
7#331×C127C211yesmaximal, characteristic, complemented
8C2×#14523247—21no—
9C2×#14523237—21no—
10C2×D1016—21no—
11C2×#14523233—22no—
12#16861206C2211yescharacteristic
13#16861205—21no—
14#16861181—22no—
15#16861180—21no—
16C4×Dic254C2211yescharacteristic
17C4×D508C2211yescharacteristic
18D4×Dic127—21no—
19#16861131—21no—
20#16861131—21no—
21C4×C127⋊C8C2211yescharacteristic
22#127×C127C2211yescharacteristic
23C4⋊C8×C127C2211yescharacteristic
24C4⋊C8×C127C2211yescharacteristic
25#14523247—42no—
26#14523247—42no—
27C2×Dic254D411yescharacteristic
28C2×D508—21no—
29C2×D508—21no—
30C2×D508—21no—
31C2×D508D411yescharacteristic
32C2×C22⋊D127—22no—
33C2×C22⋊D127—22no—
34C22×Dic127—22no—
35C2×C4×D127—21no—
36#14523237—42no—
37D1016—42no—
38#14523233—44no—
39C2×C127⋊C8—21no—
40C4⋊Dic127D411yescharacteristic
41C4⋊Dic127D411yescharacteristic
42Dic127⋊C4—21no—
43C22⋊Dic127—22no—
44(C2×D127)⋊C4—21no—
45C4×Dic127—21no—
46C2×D4×C127—21no—
47C2×D4×C127—21no—
48C2×D4×C127D412yes—
49C2×C8×C127—21no—
50C2×C8×C127—21no—
51C42×C127C2311yescharacteristic
52C22×D127—21no—
53C22×D127—41no—
54Dic254—21no—
55Dic254D812yescomplemented
56D508—21no—
57D508—21no—
58D508—42no—
59D508—41no—
60D508—41no—
61D508—41no—
62D508—41no—
63C22⋊D127—44no—
64C22⋊D127—82no—
65C2×Dic127—21no—
66C2×Dic127—21no—
67C2×Dic127—21no—
68C2×Dic127—44no—
69C4×D127—41no—
70C4×D127—41no—
71C127⋊C8—22no—
72C23×C127—22no—
73D4×C127—42no—
74D4×C127—41no—
75D4×C127—41no—
76D4×C127—41no—
77D4×C127—41no—
78C2×C4×C127C2×D411yescharacteristic
79C2×C4×C127C2×D411yescharacteristic
80C2×C4×C127C2×D411yescharacteristic
81C8×C127—41no—
82C8×C127—41no—
83C22×C127—42no—
84C22×C127—42no—
85C22×C127—42no—
86C22×C127#12011yescharacteristic
87C2×D127—41no—
88C2×D127—41no—
89C2×D127—41no—
90C2×D127—81no—
91C2×D127—81no—
92C2×D127—81no—
93C4×C127—21no—
94C4×C127C2×D811yescharacteristic
95C4×C127#13511yescharacteristic
96C4×C127AGL(1,ℤ/8)11yescharacteristic
97C4×C127AGL(1,ℤ/8)11yescharacteristic
98Dic127—22no—
99Dic127—41no—
100Dic127—41no—
101C2×C127—82no—
102C2×C127#44711yescharacteristic
103C2×C127#44911yescharacteristic
104C2×C127#45311yescharacteristic
105D127—81no—
106D127—161no—
107#1621—1271nomaximal, Sylow
108C127#162111yescharacteristic, complemented, Sylow
109#550—1271no—
110#493—1271no—
111#460—1271no—
112#459—1271no—
113#331D12711yescharacteristic, complemented
114Q8⋊C8—1271no—
115D4⋊C8—1271no—
116C2×#41—2541no—
117C2×SD16—2541no—
118C2×D8—2541no—
119C2×D8—2542no—
120#127—1271no—
121#127—2541no—
122#127C2×D12711yescharacteristic
123#121—2541no—
124#121—2542no—
125C4×Q8—1271no—
126C4×D4—1271no—
127C4×D4—2541no—
128C4⋊C8C2×D12711yescharacteristic
129C4⋊C8C2×D12711yescharacteristic
130#102—2541no—
131#102—2541no—
132C4×C8—1271no—
133#41—5082no—
134#41—5082no—
135C2×Q8—1271no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—1271no—
139C2×D4—2542no—
140C2×D4—2542no—
141C2×D4—2541no—
142C2×D4—2541no—
143C2×D4—2541no—
144C2×D4D50812yes—
145C22×C4—2542no—
146C22×C4—2541no—
147SD16—5082no—
148D8—5082no—
149D8—5084no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—2541no—
153C4⋊C4—1271no—
154C4⋊C4—1271no—
155C4⋊C4—2541no—
156C22⋊C4—2542no—
157C22⋊C4—2541no—
158C42—2541no—
159C42C22×D12711yescharacteristic
160C23—22no—
161C23—5081no—
162C23—2541no—
163Q8—1272no—
164Q8—2541no—
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—5084no—
180C2×C4—5081no—
181C2×C4—5081no—
182C2×C4—2541no—
183C2×C4—2541no—
184C2×C4—2541no—
185C2×C4D4×D12711yescharacteristic
186C2×C4D4×D12711yescharacteristic
187C2×C4C2×D50811yescharacteristic
188C8—41no—
189C8—41no—
190C8—2542no—
191C22—42no—
192C22—42no—
193C22—42no—
194C22—10161no—
195C22—10161no—
196C22—10161no—
197C22—5081no—
198C22—5081no—
199C22—5081no—
200C22#1686117911yescharacteristic
201C4—21no—
202C4—5081no—
203C4—5081no—
204C4—2542no—
205C4D8×D12711yescharacteristic
206C4#1686123011yescharacteristic
207C4#1686123311yescharacteristic
208C4#1686123611yescharacteristic
209C2—82no—
210C2—20321no—
211C2—10161no—
212C2#2255001111yescharacteristic
213C2#2255002211yescharacteristic
214C2#2255004611yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#331×C127C2
socleC22×C127#120
hypercenter#331D127
hypocenterC127#1621
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
1. #22550546C2semidirect
2. #22550252C2semidirect
3. #22550077C2semidirect
4. #22550072C2semidirect
5. #22549606C2semidirect
6. #22549603C2semidirect
7. #331×C127C2semidirect
55. Dic254D8semidirect
108. C127#1621semidirect
113. #331D127semidirect

Maximal subgroups

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

subgroupindex
#225505462
#225502522
#225500772
#225500722
#225496062
#225496032
#331×C1272
#1621127