Finite Group Databasebeta

← #30639993

Subgroups — #30639993

Order 16256.

276 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 69 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
1C22×#14523237C211yesmaximal, characteristic
2#22549884C211yesmaximal, characteristic, complemented
3#22549880C211yesmaximal, characteristic
4C2×#16861127C211yesmaximal, characteristic, complemented
5C2×#16861125C211yesmaximal, characteristic
6#22549641C211yesmaximal, characteristic, complemented
7#424×C127C211yesmaximal, characteristic, complemented
8C22×Dic254C2211yescharacteristic
9C22×D508C2211yescharacteristic
10C2×#14523237—22no—
11C2×#14523237—22no—
12C2×#14523237—22no—
13C2×#14523232C2211yescharacteristic
14C2×#14523232C2211yescharacteristic
15#16861132—22no—
16#16861129—22no—
17#16861127—22no—
18#16861125—22no—
19C2×M16×C127C2211yescharacteristic
20C22×C8×C127C2211yescharacteristic
21#117×C127C2211yescharacteristic
22C4⋊C8×C127—22no—
23C23×D127—21no—
24C2×Dic254—21no—
25C2×Dic254—21no—
26C2×Dic254—21no—
27C2×Dic254—21no—
28C2×Dic254—21no—
29C2×Dic254D412yes—
30C2×D508—21no—
31C2×D508—21no—
32C2×D508—21no—
33C2×D508—21no—
34C2×D508—21no—
35C2×D508D412yes—
36C22×Dic127—21no—
37#14523237—28no—
38#14523232—21no—
39#14523232—21no—
40#14523232—21no—
41#14523232—21no—
42C2×C127⋊C8—21no—
43C2×C127⋊C8—21no—
44C22×C4×C127C2311yescharacteristic
45M16×C127—21no—
46M16×C127—21no—
47C2×C8×C127—21no—
48C2×C8×C127—21no—
49C2×C8×C127—21no—
50C2×C8×C127D412yes—
51C4⋊C4×C127D412yes—
52C22⋊C4×C127—21no—
53C42×C127—21no—
54C22×D127—22no—
55C22×D127—41no—
56C22×D127—41no—
57C22×D127—82no—
58Dic254—22no—
59Dic254—22no—
60Dic254—41no—
61Dic254—41no—
62D508—22no—
63D508—22no—
64D508—41no—
65D508—41no—
66C2×Dic127—22no—
67C2×Dic127—41no—
68C2×Dic127—41no—
69C127⋊C8—41no—
70C127⋊C8—41no—
71C23×C127#4111yescharacteristic
72C2×C4×C127—22no—
73C2×C4×C127C2×D411yescharacteristic
74C2×C4×C127C2×D411yescharacteristic
75C2×C4×C127C2×D411yescharacteristic
76C2×C4×C127C2×D411yescharacteristic
77C2×C4×C127#4111yescharacteristic
78C2×C4×C127#4111yescharacteristic
79C8×C127—22no—
80C8×C127—41no—
81C22×C127—22no—
82C22×C127#12111yescharacteristic
83C22×C127#12311yescharacteristic
84C22×C127#12311yescharacteristic
85C2×D127—42no—
86C2×D127—82no—
87C2×D127—82no—
88C2×D127—82no—
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#39411yescharacteristic
98C2×C127#47112yes—
99D127—82no—
100#2046—1271nomaximal, Sylow
101C127#204611yescharacteristic, complemented, Sylow
102C22×SD16—1271no—
103#424D12711yescharacteristic, complemented
104#419—1271no—
105#418—1271no—
106C2×#101—1271no—
107C2×#100—1271no—
108#341—1271no—
109C22×Q8—1271no—
110C22×D4—1271no—
111C2×SD16—2542no—
112C2×SD16—2542no—
113C2×SD16—2542no—
114C2×M16—1271no—
115C2×M16—1271no—
116C2×M16C2×D12711yescharacteristic
117C22×C8C2×D12711yescharacteristic
118#117C2×D12711yescharacteristic
119C4⋊C8—22no—
120#103—2542no—
121#102—2542no—
122#101—2542no—
123#100—2542no—
124C24—2541no—
125C2×Q8—1272no—
126C2×Q8—2541no—
127C2×Q8—2541no—
128C2×Q8—2541no—
129C2×Q8—2541no—
130C2×Q8—2541no—
131C2×D4—1272no—
132C2×D4—2541no—
133C2×D4—2541no—
134C2×D4—2541no—
135C2×D4—2541no—
136C2×D4—2541no—
137C22×C4—2541no—
138C22×C4C22×D12711yescharacteristic
139SD16—2548no—
140M16—21no—
141M16—21no—
142M16—2541no—
143M16—2541no—
144M16—2541no—
145M16—2541no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—2541no—
150C2×C8—2541no—
151C2×C8C22⋊D12712yes—
152C4⋊C4D50812yes—
153C22⋊C4—21no—
154C42—21no—
155C23—10162no—
156C23—5081no—
157C23—5081no—
158C23—2542no—
159C23#1452324811yescharacteristic
160Q8—5081no—
161Q8—5081no—
162Q8—2542no—
163Q8—2542no—
164D4—5081no—
165D4—5081no—
166D4—2542no—
167D4—2542no—
168C2×C4—22no—
169C2×C4—5081no—
170C2×C4—5081no—
171C2×C4—2542no—
172C2×C4D4×D12711yescharacteristic
173C2×C4D4×D12711yescharacteristic
174C2×C4C2×C22⋊D12711yescharacteristic
175C2×C4C2×D50811yescharacteristic
176C2×C4#1452324911yescharacteristic
177C2×C4#1452324911yescharacteristic
178C8—22no—
179C8—41no—
180C8—5081no—
181C8—5081no—
182C22—22no—
183C22—10162no—
184C22—10162no—
185C22—10162no—
186C22—5082no—
187C22#1686118311yescharacteristic
188C22#1686119411yescharacteristic
189C22#1686119411yescharacteristic
190C4—42no—
191C4—5082no—
192C4#1686117711yescharacteristic
193C4#1686118411yescharacteristic
194C4#1686118411yescharacteristic
195C4#1686119811yescharacteristic
196C2—21no—
197C2—21no—
198C2—10162no—
199C2#2254979311yescharacteristic
200C2#2255013912yes—
subgroupG/H
centerC22#16861183
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#424×C127C2
socleC22×C127#121
prime radicalC22×D508C22
hypercenter#424D127
hypocenterC127#2046
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #424

Complements

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

NH ≅ G/Ntype
2. #22549884C2semidirect
4. C2×#16861127C2semidirect
6. #22549641C2semidirect
7. #424×C127C2semidirect
101. C127#2046semidirect
103. #424D127semidirect

Maximal subgroups

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

subgroupindex
C22×#145232372
#225498842
#225498802
C2×#168611272
C2×#168611252
#225496412
#424×C1272
#2046127