Finite Group Databasebeta

← #30639997

Subgroups — #30639997

Order 16256.

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

normal subgroups only

This group has 256 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×#14523238C211yesmaximal, characteristic
2#22549903C211yesmaximal, characteristic
3#22549884C211yesmaximal, characteristic
4C2×#16861128C211yesmaximal, characteristic
5C2×#16861127C211yesmaximal, characteristic
6#22549640C211yesmaximal, characteristic
7#419×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9C2×#14523238—22no—
10C2×#14523238—22no—
11C2×#14523238—22no—
12C2×#14523232C2211yescharacteristic
13C2×#14523232C2211yescharacteristic
14C22×C127⋊C8C2211yescharacteristic
15C4⋊(C127⋊C8)—22no—
16#16861132—22no—
17#16861128—22no—
18#16861127—22no—
19C22×Q8×C127C2211yescharacteristic
20C2×M16×C127C2211yescharacteristic
21#117×C127C2211yescharacteristic
22#103×C127—22no—
23C2×Dic254—21no—
24C2×Dic254—21no—
25C2×Dic254—21no—
26C2×Dic254—21no—
27C2×Dic254—21no—
28C2×Dic254D412yes—
29C22×Dic127—21no—
30#14523238—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—
40C2×Q8×C127—21no—
41C2×Q8×C127—21no—
42C2×Q8×C127—21no—
43C2×Q8×C127—21no—
44C2×Q8×C127—21no—
45C2×Q8×C127D412yes—
46C22×C4×C127—21no—
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#4111yescharacteristic
65Q8×C127—22no—
66Q8×C127—22no—
67Q8×C127—41no—
68Q8×C127—41no—
69C2×C4×C127—22no—
70C2×C4×C127—22no—
71C2×C4×C127—41no—
72C2×C4×C127—41no—
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—41no—
80C22×C127—22no—
81C22×C127#12111yescharacteristic
82C22×C127#12311yescharacteristic
83C22×C127#12311yescharacteristic
84C4×C127—42no—
85C4×C127—42no—
86C4×C127#12011yescharacteristic
87C4×C127#12111yescharacteristic
88C4×C127#12111yescharacteristic
89C4×C127#12411yescharacteristic
90Dic127—42no—
91C2×C127—21no—
92C2×C127—21no—
93C2×C127#39411yescharacteristic
94C2×C127#47312yes—
95#2047—1271nomaximal, Sylow
96C127#204711yescharacteristic, complemented, Sylow
97C22×Q16—1271no—
98#424—1271no—
99#419—1271no—
100#419D12711yescharacteristic
101C2×#101—1271no—
102C2×#101—1271no—
103#341—1271no—
104C22×Q8—1271no—
105C22×Q8C2×D12711yescharacteristic
106C2×Q16—2542no—
107C2×Q16—2542no—
108C2×Q16—2542no—
109C2×M16—1271no—
110C2×M16—1271no—
111C2×M16C2×D12711yescharacteristic
112C22×C8—1271no—
113#117C2×D12711yescharacteristic
114C4⋊C8—2542no—
115#103—22no—
116#103—2542no—
117#101—2542no—
118#101—2542no—
119C2×Q8—21no—
120C2×Q8—21no—
121C2×Q8—21no—
122C2×Q8—21no—
123C2×Q8—21no—
124C2×Q8—1272no—
125C2×Q8—2541no—
126C2×Q8—2541no—
127C2×Q8—2541no—
128C2×Q8—2541no—
129C2×Q8—2541no—
130C2×Q8D50812yes—
131C22×C4—21no—
132C22×C4—2541no—
133C22×C4C22×D12711yescharacteristic
134Q16—2548no—
135M16—21no—
136M16—21no—
137M16—2541no—
138M16—2541no—
139M16—2541no—
140M16—2541no—
141C2×C8—21no—
142C2×C8—1272no—
143C2×C8—2541no—
144C2×C8—2541no—
145C2×C8—2541no—
146C2×C8—2541no—
147C4⋊C4C22⋊D12712yes—
148C22⋊C4—21no—
149C42—21no—
150C23#1452324711yescharacteristic
151Q8—22no—
152Q8—22no—
153Q8—41no—
154Q8—41no—
155Q8—5081no—
156Q8—5081no—
157Q8—2542no—
158Q8—2542no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—5081no—
164C2×C4—5081no—
165C2×C4—2542no—
166C2×C4D4×D12711yescharacteristic
167C2×C4D4×D12711yescharacteristic
168C2×C4C2×C22⋊D12711yescharacteristic
169C2×C4C2×D50811yescharacteristic
170C2×C4#1452324711yescharacteristic
171C2×C4#1452324811yescharacteristic
172C8—41no—
173C8—5081no—
174C8—5081no—
175C8—2542no—
176C22—22no—
177C22#1686118111yescharacteristic
178C22#1686119111yescharacteristic
179C22#1686119311yescharacteristic
180C4—42no—
181C4—42no—
182C4—5082no—
183C4#1686117911yescharacteristic
184C4#1686118011yescharacteristic
185C4#1686118311yescharacteristic
186C4#1686119511yescharacteristic
187C2—21no—
188C2—21no—
189C2#2254979211yescharacteristic
190C2#2255014912yes—
subgroupG/H
centerC22#16861181
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#419×C127C2
socleC22×C127#121
prime radicalC23×C127#41
hypercenter#419D127
hypocenterC127#2047
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
96. C127#2047semidirect

Maximal subgroups

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

subgroupindex
C22×#145232382
#225499032
#225498842
C2×#168611282
C2×#168611272
#225496402
#419×C1272
#2047127