Finite Group Databasebeta

← #30637917

Subgroups — #30637917

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#22550612C211yesmaximal, characteristic
2#22550222C211yesmaximal, characteristic
3#22550162C211yesmaximal, characteristic, complemented
4#22550087C211yesmaximal, characteristic, complemented
5#22549606C211yesmaximal, characteristic, complemented
6#22549606C211yesmaximal, characteristic, complemented
7#331×C127C211yesmaximal, characteristic
8C2×Dic508—21no—
9#16861209C2211yescharacteristic
10#16861196—21no—
11#16861196—21no—
12#16861190—22no—
13C4×Dic254C2211yescharacteristic
14C4×Dic254C2211yescharacteristic
15D4×Dic127—21no—
16D4×Dic127—21no—
17#16861141—21no—
18#16861134—21no—
19#16861134—21no—
20#16861130—22no—
21C4×C127⋊C8C2211yescharacteristic
22#127×C127C2211yescharacteristic
23C4⋊C8×C127C2211yescharacteristic
24C4⋊C8×C127C2211yescharacteristic
25C2×Dic254—21no—
26C2×Dic254D411yescharacteristic
27C2×Dic254D411yescharacteristic
28C22×Dic127—22no—
29Dic508—42no—
30C2×C127⋊C8—21no—
31C4⋊Dic127—21no—
32C4⋊Dic127—21no—
33C4⋊Dic127D411yescharacteristic
34C4⋊Dic127D411yescharacteristic
35Dic127⋊C4—21no—
36Dic127⋊C4—21no—
37C22⋊Dic127—22no—
38C22⋊Dic127—22no—
39C22⋊Dic127—22no—
40C4×Dic127—21no—
41C4×Dic127—21no—
42C2×D4×C127—21no—
43C2×D4×C127—21no—
44C2×D4×C127D412yes—
45C2×C8×C127—21no—
46C2×C8×C127—21no—
47C42×C127C2311yescharacteristic
48Dic254—21no—
49Dic254—21no—
50Dic254—21no—
51Dic254—21no—
52Dic254—41no—
53Dic254—41no—
54C2×Dic127—21no—
55C2×Dic127—21no—
56C2×Dic127—21no—
57C2×Dic127—21no—
58C2×Dic127—41no—
59C2×Dic127—82no—
60C127⋊C8—41no—
61C23×C127—22no—
62D4×C127—42no—
63D4×C127—41no—
64D4×C127—41no—
65D4×C127—41no—
66D4×C127—41no—
67C2×C4×C127C2×D411yescharacteristic
68C2×C4×C127C2×D411yescharacteristic
69C2×C4×C127C2×D411yescharacteristic
70C8×C127—41no—
71C8×C127—41no—
72C22×C127—42no—
73C22×C127—42no—
74C22×C127—42no—
75C22×C127#12011yescharacteristic
76C4×C127—21no—
77C4×C127#13511yescharacteristic
78C4×C127#13511yescharacteristic
79C4×C127AGL(1,ℤ/8)11yescharacteristic
80C4×C127#13711yescharacteristic
81Dic127—41no—
82Dic127—41no—
83Dic127—41no—
84Dic127—41no—
85Dic127—81no—
86C2×C127—82no—
87C2×C127#44911yescharacteristic
88C2×C127#45211yescharacteristic
89C2×C127#45511yescharacteristic
90#1640—1271nomaximal, Sylow
91C127#164011yescharacteristic, complemented, Sylow
92#559—1271no—
93#486—1271no—
94#475—1271no—
95#462—1271no—
96#331D12711yescharacteristic
97Q8⋊C8—1271no—
98Q8⋊C8—1271no—
99C2×Q16—2541no—
100C4⋊Q8—1271no—
101#127C2×D12711yescharacteristic
102#124—2541no—
103#124—2541no—
104#123—2542no—
105C4×Q8—1271no—
106C4×Q8—1271no—
107C4×D4—2541no—
108C4×D4—2541no—
109#106—2541no—
110C4⋊C8C2×D12711yescharacteristic
111C4⋊C8C2×D12711yescharacteristic
112#103—2541no—
113#103—2541no—
114#102—2542no—
115C4×C8—1271no—
116C2×Q8—1271no—
117C2×Q8—1271no—
118C2×Q8—2541no—
119C2×D4—21no—
120C2×D4—21no—
121C2×D4D50812yes—
122C22×C4—2542no—
123Q16—5082no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—2541no—
127C4⋊C4—1271no—
128C4⋊C4—1271no—
129C4⋊C4—2541no—
130C4⋊C4—2541no—
131C4⋊C4—2541no—
132C4⋊C4—2541no—
133C22⋊C4—2542no—
134C22⋊C4—2542no—
135C22⋊C4—2542no—
136C42—2541no—
137C42—2541no—
138C42C22×D12711yescharacteristic
139C23—22no—
140Q8—5081no—
141Q8—5081no—
142Q8—2541no—
143Q8—2541no—
144Q8—2541no—
145Q8—2541no—
146D4—42no—
147D4—41no—
148D4—41no—
149D4—41no—
150D4—41no—
151C2×C4—10162no—
152C2×C4—5081no—
153C2×C4—2541no—
154C2×C4—2541no—
155C2×C4—2541no—
156C2×C4—2541no—
157C2×C4D4×D12711yescharacteristic
158C2×C4D4×D12711yescharacteristic
159C2×C4C2×D50811yescharacteristic
160C8—41no—
161C8—41no—
162C8—5081no—
163C22—42no—
164C22—42no—
165C22—42no—
166C22#1686117911yescharacteristic
167C4—21no—
168C4—10161no—
169C4—5081no—
170C4—5081no—
171C4—5081no—
172C4—5081no—
173C4#1686122811yescharacteristic
174C4#1686123011yescharacteristic
175C4#1686123311yescharacteristic
176C4#1686123811yescharacteristic
177C2—82no—
178C2#2255002211yescharacteristic
179C2#2255003911yescharacteristic
180C2#2255005611yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#331×C127C2
socleC22×C127#120
prime radical#127×C127C22
hypercenter#331D127
hypocenterC127#1640
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
3. #22550162C2semidirect
4. #22550087C2semidirect
5. #22549606C2semidirect
6. #22549606C2semidirect
91. C127#1640semidirect

Maximal subgroups

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

subgroupindex
#225506122
#225502222
#225501622
#225500872
#225496062
#331×C1272
#1640127