Finite Group Databasebeta

← #30637957

Subgroups — #30637957

Order 16256.

212 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 53 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#22550591C211yesmaximal, characteristic
2#22550228C211yesmaximal, characteristic, complemented
3#22550078C211yesmaximal, characteristic, complemented
4#22549624C211yesmaximal, characteristic, complemented
5D4⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
6#22549603C211yesmaximal, characteristic, complemented
7#474×C127C211yesmaximal, characteristic, complemented
8C2×#14523235—21no—
9#16861209C2211yescharacteristic
10#16861195—22no—
11#16861183—21no—
12#16861180—21no—
13C4×D508C2211yescharacteristic
14#16861168—21no—
15#16861167—21no—
16C4⋊(C127⋊C8)C2211yescharacteristic
17#16861133—21no—
18#16861132—22no—
19C4×C127⋊C8C2211yescharacteristic
20C4⋊Q8×C127C2211yescharacteristic
21C4×D4×C127C2211yescharacteristic
22#107×C127—21no—
23C4⋊C8×C127C2211yescharacteristic
24#102×C127—21no—
25C2×Dic254D412yes—
26C2×D508D411yescharacteristic
27C2×C22⋊D127—22no—
28C2×C4×D127—21no—
29#14523235—42no—
30C2×C127⋊C8—21no—
31C2×C127⋊C8—21no—
32C4⋊Dic127—21no—
33C4⋊Dic127—21no—
34C4⋊Dic127D411yescharacteristic
35C22⋊Dic127—22no—
36(C2×D127)⋊C4—22no—
37(C2×D127)⋊C4—21no—
38C4×Dic127—22no—
39C2×Q8×C127—21no—
40C2×D4×C127D411yescharacteristic
41C22×C4×C127—21no—
42C2×C8×C127—21no—
43C4⋊C4×C127—21no—
44C4⋊C4×C127—21no—
45C4⋊C4×C127D411yescharacteristic
46C22⋊C4×C127—21no—
47C42×C127C2311yescharacteristic
48C22×D127—21no—
49Dic254—42no—
50D508—21no—
51D508—21no—
52C22⋊D127—82no—
53C2×Dic127—21no—
54C2×Dic127—22no—
55C4×D127—41no—
56C4×D127—41no—
57C127⋊C8—41no—
58C127⋊C8—41no—
59C23×C127—21no—
60Q8×C127—41no—
61Q8×C127—41no—
62D4×C127—21no—
63D4×C127—21no—
64C2×C4×C127—21no—
65C2×C4×C127—41no—
66C2×C4×C127—41no—
67C2×C4×C127—41no—
68C2×C4×C127C2×D411yescharacteristic
69C2×C4×C127C2×D411yescharacteristic
70C2×C4×C127C2×D411yescharacteristic
71C8×C127—41no—
72C22×C127—41no—
73C22×C127—41no—
74C22×C127—41no—
75C22×C127#12011yescharacteristic
76C2×D127—41no—
77C2×D127—41no—
78C2×D127—41no—
79C4×C127—21no—
80C4×C127—41no—
81C4×C127—81no—
82C4×C127#13511yescharacteristic
83C4×C127#13511yescharacteristic
84C4×C127AGL(1,ℤ/8)11yescharacteristic
85C4×C127#13711yescharacteristic
86Dic127—41no—
87Dic127—42no—
88C2×C127—81no—
89C2×C127#44911yescharacteristic
90C2×C127#45211yescharacteristic
91C2×C127#45511yescharacteristic
92D127—81no—
93#1646—1271nomaximal, Sylow
94C127#164611yescharacteristic, complemented, Sylow
95#555—1271no—
96#487—1271no—
97#474D12711yescharacteristic, complemented
98#461—1271no—
99#333—1271no—
100D4⋊C8—1271no—
101D4⋊C8—1271no—
102C2×SD16—2541no—
103C4⋊Q8—1271no—
104C4⋊Q8C2×D12711yescharacteristic
105#124—2542no—
106#121—2541no—
107#121—2541no—
108C4×D4—1271no—
109C4×D4C2×D12711yescharacteristic
110#117—2541no—
111#117—2541no—
112#107—21no—
113C4⋊C8—1271no—
114C4⋊C8C2×D12711yescharacteristic
115#103—2542no—
116#103—2541no—
117#102—21no—
118C4×C8—1271no—
119C2×Q8—21no—
120C2×Q8—1272no—
121C2×D4—1271no—
122C2×D4—2542no—
123C2×D4C22⋊D12711yescharacteristic
124C22×C4—21no—
125C22×C4—2541no—
126SD16—5082no—
127C2×C8—21no—
128C2×C8—2541no—
129C2×C8—2541no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—1271no—
133C4⋊C4—2541no—
134C4⋊C4—2541no—
135C4⋊C4C22⋊D12711yescharacteristic
136C22⋊C4—21no—
137C22⋊C4—2542no—
138C22⋊C4—2542no—
139C22⋊C4—2541no—
140C42—2542no—
141C42C22×D12711yescharacteristic
142C23—21no—
143C23—2541no—
144Q8—41no—
145Q8—41no—
146Q8—5082no—
147D4—21no—
148D4—21no—
149D4—10162no—
150D4—2541no—
151D4—2541no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—5081no—
157C2×C4—5081no—
158C2×C4—2541no—
159C2×C4—2542no—
160C2×C4D4×D12711yescharacteristic
161C2×C4D4×D12711yescharacteristic
162C2×C4C2×C22⋊D12711yescharacteristic
163C8—41no—
164C8—5081no—
165C8—5081no—
166C22—41no—
167C22—41no—
168C22—41no—
169C22—5081no—
170C22—5081no—
171C22—5081no—
172C22#1686117711yescharacteristic
173C4—21no—
174C4—41no—
175C4—81no—
176C4—5081no—
177C4—5082no—
178C4#1686122811yescharacteristic
179C4#1686122911yescharacteristic
180C4#1686123311yescharacteristic
181C4#1686124011yescharacteristic
182C2—81no—
183C2—10161no—
184C2#2255001811yescharacteristic
185C2#2255003611yescharacteristic
186C2#2255005211yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#474×C127C2
socleC22×C127#120
prime radical#22550591C2
hypercenter#474D127
hypocenterC127#1646
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
2. #22550228C2semidirect
3. #22550078C2semidirect
4. #22549624C2semidirect
5. D4⋊(C127⋊C8)C2semidirect
6. #22549603C2semidirect
7. #474×C127C2semidirect
94. C127#1646semidirect
97. #474D127semidirect

Maximal subgroups

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

subgroupindex
#225505912
#225502282
#225500782
#225496242
D4⋊(C127⋊C8)2
#225496032
#474×C1272
#1646127