Finite Group Databasebeta

← #30639992

Subgroups — #30639992

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×#14523236C211yesmaximal, characteristic
2#22549903C211yesmaximal, characteristic, complemented
3#22549884C211yesmaximal, characteristic, complemented
4C2×#16861127C211yesmaximal, characteristic, complemented
5C2×#16861126C211yesmaximal, characteristic
6#22549640C211yesmaximal, characteristic, complemented
7#418×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9C2×#14523236—22no—
10C2×#14523236—22no—
11C2×#14523236—22no—
12C2×#14523232C2211yescharacteristic
13C2×#14523232C2211yescharacteristic
14C22×C127⋊C8C2211yescharacteristic
15C4⋊(C127⋊C8)—22no—
16#16861132—22no—
17#16861127—22no—
18#16861126—22no—
19C22×D4×C127C2211yescharacteristic
20C2×M16×C127C2211yescharacteristic
21#117×C127C2211yescharacteristic
22#102×C127—22no—
23C2×Dic254—21no—
24C2×Dic254—21no—
25C2×Dic254—21no—
26C2×Dic254—21no—
27C2×Dic254—21no—
28C2×Dic254D412yes—
29C22×Dic127—21no—
30#14523236—28no—
31#14523232—21no—
32#14523232—21no—
33#14523232—21no—
34#14523232—21no—
35C2×C127⋊C8—21no—
36C2×C127⋊C8—21no—
37C2×C127⋊C8—21no—
38C2×C127⋊C8—21no—
39C2×C127⋊C8D412yes—
40C24×C127—21no—
41C2×D4×C127—21no—
42C2×D4×C127—21no—
43C2×D4×C127—21no—
44C2×D4×C127—21no—
45C2×D4×C127—21no—
46C2×D4×C127D412yes—
47C22×C4×C127C2311yescharacteristic
48M16×C127—21no—
49M16×C127—21no—
50C2×C8×C127—21no—
51C4⋊C4×C127D412yes—
52C22⋊C4×C127—21no—
53C42×C127—21no—
54Dic254—22no—
55Dic254—22no—
56Dic254—41no—
57Dic254—41no—
58C2×Dic127—22no—
59C2×Dic127—41no—
60C2×Dic127—41no—
61C127⋊C8—22no—
62C127⋊C8—41no—
63C127⋊C8—41no—
64C23×C127—22no—
65C23×C127—41no—
66C23×C127—41no—
67C23×C127—82no—
68C23×C127#4111yescharacteristic
69D4×C127—22no—
70D4×C127—22no—
71D4×C127—41no—
72D4×C127—41no—
73C2×C4×C127—22no—
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—41no—
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×SD16—1271no—
103#424—1271no—
104#419—1271no—
105#418D12711yescharacteristic
106C2×#101—1271no—
107C2×#100—1271no—
108#341—1271no—
109C22×Q8—1271no—
110C22×D4C2×D12711yescharacteristic
111C2×SD16—2542no—
112C2×SD16—2542no—
113C2×SD16—2542no—
114C2×M16—1271no—
115C2×M16—1271no—
116C2×M16C2×D12711yescharacteristic
117C22×C8—1271no—
118#117C2×D12711yescharacteristic
119C4⋊C8—2542no—
120#103—2542no—
121#102—22no—
122#101—2542no—
123#100—2542no—
124C24—21no—
125C2×Q8—1272no—
126C2×Q8—2541no—
127C2×Q8—2541no—
128C2×Q8—2541no—
129C2×Q8—2541no—
130C2×Q8—2541no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4D50812yes—
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—1272no—
148C2×C8—2541no—
149C2×C8—2541no—
150C2×C8—2541no—
151C2×C8—2541no—
152C4⋊C4C22⋊D12712yes—
153C22⋊C4—21no—
154C42—21no—
155C23—22no—
156C23—41no—
157C23—41no—
158C23—82no—
159C23#1452324711yescharacteristic
160Q8—5081no—
161Q8—5081no—
162Q8—2542no—
163Q8—2542no—
164D4—22no—
165D4—22no—
166D4—41no—
167D4—41no—
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#1452324711yescharacteristic
177C2×C4#1452324811yescharacteristic
178C8—41no—
179C8—5081no—
180C8—5081no—
181C8—2542no—
182C22—22no—
183C22—42no—
184C22—82no—
185C22—82no—
186C22—82no—
187C22#1686118111yescharacteristic
188C22#1686119111yescharacteristic
189C22#1686119311yescharacteristic
190C4—42no—
191C4—5082no—
192C4#1686117911yescharacteristic
193C4#1686118011yescharacteristic
194C4#1686118311yescharacteristic
195C4#1686119511yescharacteristic
196C2—21no—
197C2—21no—
198C2—82no—
199C2#2254979211yescharacteristic
200C2#2255013712yes—
subgroupG/H
centerC22#16861181
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#418×C127C2
socleC22×C127#121
prime radicalC22×D4×C127C22
hypercenter#418D127
hypocenterC127#2046
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. #22549903C2semidirect
3. #22549884C2semidirect
4. C2×#16861127C2semidirect
6. #22549640C2semidirect
101. C127#2046semidirect

Maximal subgroups

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

subgroupindex
C22×#145232362
#225499032
#225498842
C2×#168611272
C2×#168611262
#225496402
#418×C1272
#2046127