Finite Group Databasebeta

← #30640087

Subgroups — #30640087

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#22549903C211yesmaximal, characteristic, complemented
3#22549880C211yesmaximal, characteristic
4C2×#16861129C211yesmaximal, characteristic
5C2×#16861125C211yesmaximal, characteristic
6#22549645C211yesmaximal, characteristic, complemented
7#428×C127C211yesmaximal, characteristic, complemented
8C22×D508C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10C2×#14523232C2211yescharacteristic
11C22×C127⋊C8C2211yescharacteristic
12#16861144—22no—
13C4⋊(C127⋊C8)—22no—
14#16861129—22no—
15#16861129—21no—
16#16861129—21no—
17#16861125—22no—
18C2×M16×C127C2211yescharacteristic
19#117×C127C2211yescharacteristic
20C2×C4⋊C4×C127C2211yescharacteristic
21#107×C127—21no—
22#106×C127—21no—
23C23×D127—21no—
24C2×D508—21no—
25C2×D508—21no—
26C2×D508—21no—
27C2×D508—21no—
28C2×D508—21no—
29C2×D508Q812yes—
30#14523232—21no—
31#14523232—21no—
32#14523232—21no—
33#14523232—21no—
34C2×C127⋊C8—21no—
35C2×C127⋊C8—21no—
36C2×C127⋊C8—21no—
37C2×C127⋊C8—21no—
38C2×C127⋊C8D412yes—
39C22×C4×C127—21no—
40C22×C4×C127C2311yescharacteristic
41M16×C127—21no—
42M16×C127D412yes—
43C2×C8×C127—21no—
44C4⋊C4×C127—21no—
45C4⋊C4×C127—21no—
46C4⋊C4×C127D411yescharacteristic
47C4⋊C4×C127D411yescharacteristic
48C22⋊C4×C127—21no—
49C42×C127—21no—
50C22×D127—22no—
51C22×D127—41no—
52C22×D127—41no—
53C22×D127—81no—
54C22×D127—81no—
55D508—22no—
56D508—22no—
57D508—41no—
58D508—41no—
59C127⋊C8—22no—
60C127⋊C8—41no—
61C127⋊C8—41no—
62C23×C127#4111yescharacteristic
63C2×C4×C127—21no—
64C2×C4×C127—21no—
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#12111yescharacteristic
77C22×C127C22⋊Q811yescharacteristic
78C22×C127#12611yescharacteristic
79C2×D127—42no—
80C2×D127—82no—
81C2×D127—81no—
82C2×D127—81no—
83C2×D127—82no—
84C4×C127—41no—
85C4×C127—41no—
86C4×C127—81no—
87C4×C127#12111yescharacteristic
88C4×C127#12111yescharacteristic
89C4×C127C22⋊Q811yescharacteristic
90C4×C127C22⋊Q811yescharacteristic
91C2×C127—21no—
92C2×C127—21no—
93C2×C127#39611yescharacteristic
94C2×C127#47111yescharacteristic
95C2×C127#47211yescharacteristic
96D127—82no—
97#2063—1271nomaximal, Sylow
98C127#206311yescharacteristic, complemented, Sylow
99C2×#108—1271no—
100#428D12711yescharacteristic, complemented
101#424—1271no—
102#418—1271no—
103C2×#102—1271no—
104C2×#100—1271no—
105#343—1271no—
106C22×D4—1271no—
107C2×M16—1271no—
108C2×M16—1271no—
109C2×M16C2×D12711yescharacteristic
110C22×C8—1271no—
111#117C2×D12711yescharacteristic
112C2×C4⋊C4C2×D12711yescharacteristic
113#108—2542no—
114#107—21no—
115#106—21no—
116C4⋊C8—2542no—
117#102—2542no—
118#102—2541no—
119#102—2541no—
120#100—2542no—
121C24—2541no—
122C2×D4—1272no—
123C2×D4—2541no—
124C2×D4—2541no—
125C2×D4—2541no—
126C2×D4—2541no—
127C2×D4—2541no—
128C22×C4—21no—
129C22×C4C22×D12711yescharacteristic
130M16—21no—
131M16—2541no—
132M16—2541no—
133M16—2541no—
134M16—2541no—
135M16D50812yes—
136C2×C8—21no—
137C2×C8—1272no—
138C2×C8—2541no—
139C2×C8—2541no—
140C2×C8—2541no—
141C2×C8—2541no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4C22⋊D12711yescharacteristic
145C4⋊C4C22⋊D12711yescharacteristic
146C22⋊C4—21no—
147C42—21no—
148C23—10161no—
149C23—10161no—
150C23—5081no—
151C23—5081no—
152C23—2542no—
153C23#1452324711yescharacteristic
154D4—5081no—
155D4—5081no—
156D4—2542no—
157D4—2542no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—81no—
162C2×C4D4×D12711yescharacteristic
163C2×C4C2×C22⋊D12711yescharacteristic
164C2×C4C2×D50811yescharacteristic
165C2×C4Q8×D12711yescharacteristic
166C2×C4#1452324811yescharacteristic
167C2×C4#1452324911yescharacteristic
168C8—22no—
169C8—5081no—
170C8—5081no—
171C8—2542no—
172C22—21no—
173C22—21no—
174C22—10162no—
175C22—10162no—
176C22—10161no—
177C22—10161no—
178C22—5082no—
179C22#1686118111yescharacteristic
180C22#1686118911yescharacteristic
181C22#1686120411yescharacteristic
182C4—41no—
183C4—41no—
184C4—81no—
185C4#1686118311yescharacteristic
186C4#1686118411yescharacteristic
187C4#1686118511yescharacteristic
188C4#1686118711yescharacteristic
189C2—21no—
190C2—21no—
191C2—10162no—
192C2#2254979811yescharacteristic
193C2#2255014111yescharacteristic
194C2#2255014411yescharacteristic
subgroupG/H
centerC22#16861181
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#428×C127C2
socleC22×C127#121
prime radicalC22×D508C22
hypercenter#428D127
hypocenterC127#2063
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#16861144C2semidirect
2. #22549903C2semidirect
6. #22549645C2semidirect
7. #428×C127C2semidirect
98. C127#2063semidirect
100. #428D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611442
#225499032
#225498802
C2×#168611292
C2×#168611252
#225496452
#428×C1272
#2063127