Finite Group Databasebeta

← #30640088

Subgroups — #30640088

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×#16861145C211yesmaximal, characteristic, complemented
2#22549916C211yesmaximal, characteristic, complemented
3#22549902C211yesmaximal, characteristic, complemented
4#22549881C211yesmaximal, characteristic
5C2×#16861126C211yesmaximal, characteristic
6#22549645C211yesmaximal, characteristic, complemented
7C2×#102×C127C211yesmaximal, characteristic
8C2×#14523232C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10C2×#14523232C2211yescharacteristic
11#16861168C2211yescharacteristic
12#16861145—22no—
13#16861142—21no—
14#16861140—21no—
15Dic127⋊C8—22no—
16#16861130—21no—
17#16861130—21no—
18#16861126—22no—
19C22×D4×C127C2211yescharacteristic
20C22×C8×C127C2211yescharacteristic
21C2×C4⋊C4×C127C2211yescharacteristic
22#102×C127—22no—
23#14523232—21no—
24#14523232—21no—
25#14523232—21no—
26#14523232—21no—
27#14523232—21no—
28#14523232D412yes—
29C2×C127⋊C8—21no—
30C2×C127⋊C8—21no—
31C2×C127⋊C8—21no—
32C4⋊Dic127D411yescharacteristic
33C4⋊Dic127D411yescharacteristic
34C22⋊Dic127—21no—
35C4×Dic127—21no—
36C24×C127—21no—
37C2×D4×C127—21no—
38C2×D4×C127—21no—
39C2×D4×C127—21no—
40C2×D4×C127—21no—
41C2×D4×C127—21no—
42C2×D4×C127Q812yes—
43C22×C4×C127—21no—
44C22×C4×C127C2311yescharacteristic
45C2×C8×C127—21no—
46C2×C8×C127—21no—
47C2×C8×C127D412yes—
48C4⋊C4×C127—21no—
49C4⋊C4×C127—21no—
50C2×Dic127—21no—
51C2×Dic127—21no—
52C127⋊C8—22no—
53C127⋊C8—41no—
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—41no—
66C2×C4×C127—81no—
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—22no—
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—81no—
85C4×C127#12111yescharacteristic
86C4×C127#12111yescharacteristic
87C4×C127C22⋊Q811yescharacteristic
88C4×C127C22⋊Q811yescharacteristic
89Dic127—41no—
90Dic127—41no—
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#418—1271no—
103C2×#102D12711yescharacteristic
104C2×#100—1271no—
105#343—1271no—
106C22×D4C2×D12711yescharacteristic
107C2×M16—1271no—
108C2×M16—1271no—
109C2×M16—1271no—
110C22×C8C2×D12711yescharacteristic
111#117—1271no—
112C2×C4⋊C4C2×D12711yescharacteristic
113#108—2542no—
114#107—2541no—
115#106—2541no—
116C4⋊C8—2542no—
117#102—22no—
118#102—2541no—
119#102—2541no—
120#100—2542no—
121C24—21no—
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4Dic25412yes—
128C22×C4—21no—
129C22×C4C22×D12711yescharacteristic
130M16—1272no—
131M16—2541no—
132M16—2541no—
133M16—2541no—
134M16—2541no—
135M16—2541no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—2541no—
139C2×C8—2541no—
140C2×C8—2541no—
141C2×C8C22⋊D12712yes—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—1271no—
145C4⋊C4—1271no—
146C22⋊C4—2541no—
147C42—2541no—
148C23—22no—
149C23—41no—
150C23—41no—
151C23—81no—
152C23—81no—
153C23#1452324711yescharacteristic
154D4—22no—
155D4—22no—
156D4—41no—
157D4—41no—
158C2×C4—41no—
159C2×C4—81no—
160C2×C4—2541no—
161C2×C4—2541no—
162C2×C4D4×D12711yescharacteristic
163C2×C4D4×D12711yescharacteristic
164C2×C4C2×C22⋊D12711yescharacteristic
165C2×C4C2×Dic25411yescharacteristic
166C2×C4#1452324711yescharacteristic
167C2×C4#1452324811yescharacteristic
168C8—22no—
169C8—5081no—
170C8—5081no—
171C8—2542no—
172C22—21no—
173C22—21no—
174C22—42no—
175C22—82no—
176C22—82no—
177C22—81no—
178C22—81no—
179C22#1686118011yescharacteristic
180C22#1686118611yescharacteristic
181C22#1686120211yescharacteristic
182C4—81no—
183C4—5081no—
184C4—5081no—
185C4#1686118111yescharacteristic
186C4#1686118211yescharacteristic
187C4#1686118611yescharacteristic
188C4#1686118811yescharacteristic
189C2—21no—
190C2—21no—
191C2—82no—
192C2#2254979911yescharacteristic
193C2#2255013611yescharacteristic
194C2#2255014311yescharacteristic
subgroupG/H
centerC22#16861180
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#102×C127C2
socleC22×C127#121
prime radicalC22×D4×C127C22
hypercenterC2×#102D127
hypocenterC127#2063
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C2×#16861145C2semidirect
2. #22549916C2semidirect
3. #22549902C2semidirect
6. #22549645C2semidirect
98. C127#2063semidirect

Maximal subgroups

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

subgroupindex
C2×#168611452
#225499162
#225499022
#225498812
C2×#168611262
#225496452
C2×#102×C1272
#2063127