Finite Group Databasebeta

← #30639988

Subgroups — #30639988

Order 16256.

276 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 71 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#22549902C211yesmaximal, characteristic, complemented
2#22549885C211yesmaximal, characteristic, complemented
3#22549881C211yesmaximal, characteristic
4C2×#16861128C211yesmaximal, characteristic, complemented
5C2×#16861126C211yesmaximal, characteristic
6#22549641C211yesmaximal, characteristic, complemented
7C22×SD16×C127C211yesmaximal, characteristic
8C2×#14523232C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10C2×#14523232C2211yescharacteristic
11#16861168C2211yescharacteristic
12Dic127⋊C8—22no—
13#16861133—22no—
14#16861130—22no—
15#16861128—22no—
16#16861126—22no—
17C22×Q8×C127C2211yescharacteristic
18C22×D4×C127C2211yescharacteristic
19C2×SD16×C127—22no—
20C2×SD16×C127—22no—
21C2×SD16×C127—22no—
22C22×C8×C127C2211yescharacteristic
23#14523232—21no—
24#14523232—21no—
25#14523232—21no—
26#14523232—21no—
27#14523232—21no—
28#14523232—21no—
29C2×C127⋊C8—21no—
30C2×C127⋊C8—21no—
31C2×C127⋊C8—21no—
32C4⋊Dic127D412yes—
33C22⋊Dic127—21no—
34C4×Dic127—21no—
35C24×C127—21no—
36C2×Q8×C127—21no—
37C2×Q8×C127—21no—
38C2×Q8×C127—21no—
39C2×Q8×C127—21no—
40C2×Q8×C127—21no—
41C2×Q8×C127D412yes—
42C2×D4×C127—21no—
43C2×D4×C127—21no—
44C2×D4×C127—21no—
45C2×D4×C127—21no—
46C2×D4×C127—21no—
47C2×D4×C127D412yes—
48C22×C4×C127—21no—
49C22×C4×C127C2311yescharacteristic
50SD16×C127—28no—
51C2×C8×C127—21no—
52C2×C8×C127—21no—
53C2×C8×C127D412yes—
54C2×Dic127—22no—
55C127⋊C8—41no—
56C127⋊C8—41no—
57C127⋊C8—41no—
58C23×C127—22no—
59C23×C127—41no—
60C23×C127—41no—
61C23×C127—82no—
62C23×C127#4111yescharacteristic
63Q8×C127—22no—
64Q8×C127—22no—
65Q8×C127—41no—
66Q8×C127—41no—
67D4×C127—22no—
68D4×C127—22no—
69D4×C127—41no—
70D4×C127—41no—
71C2×C4×C127—22no—
72C2×C4×C127—41no—
73C2×C4×C127—41no—
74C2×C4×C127C2×D411yescharacteristic
75C2×C4×C127C2×D411yescharacteristic
76C2×C4×C127C2×D411yescharacteristic
77C2×C4×C127C2×D411yescharacteristic
78C2×C4×C127#4111yescharacteristic
79C2×C4×C127#4111yescharacteristic
80C8×C127—22no—
81C22×C127—22no—
82C22×C127—42no—
83C22×C127—82no—
84C22×C127—82no—
85C22×C127—82no—
86C22×C127#12111yescharacteristic
87C22×C127#12311yescharacteristic
88C22×C127#12311yescharacteristic
89C4×C127—42no—
90C4×C127#12011yescharacteristic
91C4×C127#12111yescharacteristic
92C4×C127#12111yescharacteristic
93C4×C127#12411yescharacteristic
94Dic127—42no—
95C2×C127—21no—
96C2×C127—21no—
97C2×C127—82no—
98C2×C127#39411yescharacteristic
99C2×C127#47112yes—
100#2046—1271nomaximal, Sylow
101C127#204611yescharacteristic, complemented, Sylow
102C22×SD16D12711yescharacteristic
103#424—1271no—
104#419—1271no—
105#418—1271no—
106C2×#101—1271no—
107C2×#100—1271no—
108#341—1271no—
109C22×Q8C2×D12711yescharacteristic
110C22×D4C2×D12711yescharacteristic
111C2×SD16—22no—
112C2×SD16—22no—
113C2×SD16—22no—
114C2×M16—1271no—
115C2×M16—1271no—
116C2×M16—1271no—
117C22×C8C2×D12711yescharacteristic
118#117—1271no—
119C4⋊C8—2542no—
120#103—2542no—
121#102—2542no—
122#101—2542no—
123#100—2542no—
124C24—21no—
125C2×Q8—21no—
126C2×Q8—21no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8C22⋊D12712yes—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4C22⋊D12712yes—
137C22×C4—21no—
138C22×C4C22×D12711yescharacteristic
139SD16—28no—
140M16—2541no—
141M16—2541no—
142M16—2541no—
143M16—2541no—
144M16—2541no—
145M16—2541no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—2541no—
149C2×C8—2541no—
150C2×C8—2541no—
151C2×C8C22⋊D12712yes—
152C4⋊C4—1272no—
153C22⋊C4—2541no—
154C42—2541no—
155C23—22no—
156C23—41no—
157C23—41no—
158C23—82no—
159C23#1452324711yescharacteristic
160Q8—22no—
161Q8—22no—
162Q8—41no—
163Q8—41no—
164D4—22no—
165D4—22no—
166D4—41no—
167D4—41no—
168C2×C4—22no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—2542no—
172C2×C4D4×D12711yescharacteristic
173C2×C4C2×C22⋊D12711yescharacteristic
174C2×C4C2×C22⋊D12711yescharacteristic
175C2×C4C2×C22⋊D12711yescharacteristic
176C2×C4#1452324711yescharacteristic
177C2×C4#1452324711yescharacteristic
178C8—22no—
179C8—5081no—
180C8—5081no—
181C8—5081no—
182C22—22no—
183C22—42no—
184C22—82no—
185C22—82no—
186C22—82no—
187C22#1686118011yescharacteristic
188C22#1686119011yescharacteristic
189C22#1686119011yescharacteristic
190C4—42no—
191C4—5082no—
192C4#1686117811yescharacteristic
193C4#1686118111yescharacteristic
194C4#1686118111yescharacteristic
195C4#1686119611yescharacteristic
196C2—21no—
197C2—21no—
198C2—82no—
199C2#2254979011yescharacteristic
200C2#2255013612yes—
subgroupG/H
centerC22#16861180
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC22×SD16×C127C2
socleC22×C127#121
prime radicalC22×D4×C127C22
hypercenterC22×SD16D127
hypocenterC127#2046
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. #22549902C2semidirect
2. #22549885C2semidirect
4. C2×#16861128C2semidirect
6. #22549641C2semidirect
101. C127#2046semidirect

Maximal subgroups

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

subgroupindex
#225499022
#225498852
#225498812
C2×#168611282
C2×#168611262
#225496412
C22×SD16×C1272
#2046127