Finite Group Databasebeta

← #30640090

Subgroups — #30640090

Order 16256.

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

normal subgroups only

This group has 228 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
1C2×#16861144C211yesmaximal, characteristic, complemented
2#22549919C211yesmaximal, characteristic, complemented
3#22549903C211yesmaximal, characteristic, complemented
4C2×#16861130C211yesmaximal, characteristic
5C2×#16861126C211yesmaximal, characteristic
6#22549644C211yesmaximal, characteristic, complemented
7#418×C127C211yesmaximal, characteristic
8C2×#14523232C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10C22×C127⋊C8C2211yescharacteristic
11C2×C4⋊Dic127C2211yescharacteristic
12#16861144—22no—
13#16861142—21no—
14#16861140—21no—
15C4⋊(C127⋊C8)—22no—
16#16861130—22no—
17#16861126—22no—
18C22×D4×C127C2211yescharacteristic
19C2×M16×C127C2211yescharacteristic
20#117×C127C2211yescharacteristic
21#102×C127—21no—
22#102×C127—21no—
23C22×Dic127—21no—
24#14523232—21no—
25#14523232—21no—
26#14523232—21no—
27#14523232D412yes—
28C2×C127⋊C8—21no—
29C2×C127⋊C8—21no—
30C2×C127⋊C8—21no—
31C2×C127⋊C8—21no—
32C2×C127⋊C8D412yes—
33C4⋊Dic127—21no—
34C4⋊Dic127—21no—
35C24×C127—21no—
36C2×D4×C127—21no—
37C2×D4×C127—21no—
38C2×D4×C127—21no—
39C2×D4×C127—21no—
40C2×D4×C127—21no—
41C2×D4×C127Q812yes—
42C22×C4×C127C2311yescharacteristic
43M16×C127—21no—
44M16×C127—21no—
45C2×C8×C127—21no—
46C4⋊C4×C127D411yescharacteristic
47C4⋊C4×C127D411yescharacteristic
48C22⋊C4×C127—21no—
49C42×C127—21no—
50C2×Dic127—41no—
51C2×Dic127—81no—
52C127⋊C8—22no—
53C127⋊C8—22no—
54C127⋊C8—41no—
55C23×C127—22no—
56C23×C127—41no—
57C23×C127—41no—
58C23×C127—81no—
59C23×C127—81no—
60C23×C127#4111yescharacteristic
61D4×C127—22no—
62D4×C127—22no—
63D4×C127—41no—
64D4×C127—41no—
65C2×C4×C127—21no—
66C2×C4×C127—21no—
67C2×C4×C127C2×D411yescharacteristic
68C2×C4×C127C2×D411yescharacteristic
69C2×C4×C127C2×D411yescharacteristic
70C2×C4×C127C2×Q811yescharacteristic
71C2×C4×C127#4111yescharacteristic
72C2×C4×C127#4111yescharacteristic
73C8×C127—41no—
74C22×C127—21no—
75C22×C127—21no—
76C22×C127—42no—
77C22×C127—82no—
78C22×C127—82no—
79C22×C127—81no—
80C22×C127—81no—
81C22×C127#12111yescharacteristic
82C22×C127C22⋊Q811yescharacteristic
83C22×C127#12611yescharacteristic
84C4×C127—41no—
85C4×C127—41no—
86C4×C127#12111yescharacteristic
87C4×C127#12111yescharacteristic
88C4×C127C22⋊Q811yescharacteristic
89C4×C127C22⋊Q811yescharacteristic
90Dic127—81no—
91C2×C127—21no—
92C2×C127—21no—
93C2×C127—82no—
94C2×C127#39611yescharacteristic
95C2×C127#47111yescharacteristic
96C2×C127#47211yescharacteristic
97#2063—1271nomaximal, Sylow
98C127#206311yescharacteristic, complemented, Sylow
99C2×#108—1271no—
100#428—1271no—
101#424—1271no—
102#418D12711yescharacteristic
103C2×#102—1271no—
104C2×#100—1271no—
105#343—1271no—
106C22×D4C2×D12711yescharacteristic
107C2×M16—1271no—
108C2×M16—1271no—
109C2×M16C2×D12711yescharacteristic
110C22×C8—1271no—
111#117C2×D12711yescharacteristic
112C2×C4⋊C4—1271no—
113#108—2542no—
114#107—2541no—
115#106—2541no—
116C4⋊C8—2542no—
117#102—21no—
118#102—21no—
119#102—2542no—
120#100—2542no—
121C24—21no—
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4Dic25412yes—
128C22×C4—2541no—
129C22×C4C22×D12711yescharacteristic
130M16—21no—
131M16—21no—
132M16—1272no—
133M16—2541no—
134M16—2541no—
135M16—2541no—
136C2×C8—21no—
137C2×C8—1272no—
138C2×C8—2541no—
139C2×C8—2541no—
140C2×C8—2541no—
141C2×C8—2541no—
142C4⋊C4—2541no—
143C4⋊C4—2541no—
144C4⋊C4C22⋊D12711yescharacteristic
145C4⋊C4C22⋊D12711yescharacteristic
146C22⋊C4—21no—
147C42—21no—
148C23—22no—
149C23—41no—
150C23—41no—
151C23—81no—
152C23—81no—
153C23#1452324711yescharacteristic
154D4—22no—
155D4—22no—
156D4—41no—
157D4—41no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—10161no—
161C2×C4—5081no—
162C2×C4D4×D12711yescharacteristic
163C2×C4D4×D12711yescharacteristic
164C2×C4C2×C22⋊D12711yescharacteristic
165C2×C4C2×Dic25411yescharacteristic
166C2×C4#1452324711yescharacteristic
167C2×C4#1452324811yescharacteristic
168C8—41no—
169C8—5081no—
170C8—2542no—
171C8—2542no—
172C22—21no—
173C22—21no—
174C22—42no—
175C22—82no—
176C22—82no—
177C22—81no—
178C22—81no—
179C22#1686118111yescharacteristic
180C22#1686118611yescharacteristic
181C22#1686120211yescharacteristic
182C4—41no—
183C4—41no—
184C4—10161no—
185C4#1686118011yescharacteristic
186C4#1686118211yescharacteristic
187C4#1686118611yescharacteristic
188C4#1686118811yescharacteristic
189C2—21no—
190C2—21no—
191C2—82no—
192C2#2254979911yescharacteristic
193C2#2255013711yescharacteristic
194C2#2255014411yescharacteristic
subgroupG/H
centerC22#16861181
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#418×C127C2
socleC22×C127#121
prime radicalC22×D4×C127C22
hypercenter#418D127
hypocenterC127#2063
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
1. C2×#16861144C2semidirect
2. #22549919C2semidirect
3. #22549903C2semidirect
6. #22549644C2semidirect
98. C127#2063semidirect

Maximal subgroups

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

subgroupindex
C2×#168611442
#225499192
#225499032
C2×#168611302
C2×#168611262
#225496442
#418×C1272
#2063127