Finite Group Databasebeta

← #30639985

Subgroups — #30639985

Order 16256.

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

normal subgroups only

This group has 296 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×#14523233C211yesmaximal, characteristic
2#22549903C211yesmaximal, characteristic, complemented
3#22549880C211yesmaximal, characteristic, complemented
4C2×#16861126C211yesmaximal, characteristic, complemented
5C2×#16861125C211yesmaximal, characteristic, complemented
6#22549640C211yesmaximal, characteristic, complemented
7#418×C127C211yesmaximal, characteristic, complemented
8C22×D508C2211yescharacteristic
9C2×#14523233—22no—
10C2×#14523233—22no—
11C2×#14523233—22no—
12C2×#14523232C2211yescharacteristic
13C2×#14523232C2211yescharacteristic
14C22×C127⋊C8C2211yescharacteristic
15C4⋊(C127⋊C8)—22no—
16#16861129—22no—
17#16861126—22no—
18#16861125—22no—
19C22×D4×C127C2211yescharacteristic
20C2×M16×C127C2211yescharacteristic
21#117×C127C2211yescharacteristic
22#102×C127—22no—
23C23×D127—21no—
24C2×D508—21no—
25C2×D508—21no—
26C2×D508—21no—
27C2×D508—21no—
28C2×D508—21no—
29C2×D508D412yes—
30#14523233—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—
54C22×D127—22no—
55C22×D127—41no—
56C22×D127—41no—
57C22×D127—82no—
58D508—22no—
59D508—22no—
60D508—41no—
61D508—41no—
62C127⋊C8—22no—
63C127⋊C8—41no—
64C127⋊C8—41no—
65C23×C127—22no—
66C23×C127—41no—
67C23×C127—41no—
68C23×C127—82no—
69C23×C127#4111yescharacteristic
70D4×C127—22no—
71D4×C127—22no—
72D4×C127—41no—
73D4×C127—41no—
74C2×C4×C127—22no—
75C2×C4×C127C2×D411yescharacteristic
76C2×C4×C127C2×D411yescharacteristic
77C2×C4×C127C2×D411yescharacteristic
78C2×C4×C127C2×D411yescharacteristic
79C2×C4×C127#4111yescharacteristic
80C2×C4×C127#4111yescharacteristic
81C8×C127—41no—
82C22×C127—22no—
83C22×C127—42no—
84C22×C127—82no—
85C22×C127—82no—
86C22×C127—82no—
87C22×C127#12111yescharacteristic
88C22×C127#12311yescharacteristic
89C22×C127#12311yescharacteristic
90C2×D127—42no—
91C2×D127—82no—
92C2×D127—82no—
93C2×D127—82no—
94C4×C127—42no—
95C4×C127#12011yescharacteristic
96C4×C127#12111yescharacteristic
97C4×C127#12111yescharacteristic
98C4×C127#12411yescharacteristic
99C2×C127—21no—
100C2×C127—21no—
101C2×C127—82no—
102C2×C127#39411yescharacteristic
103C2×C127#47212yes—
104D127—82no—
105#2045—1271nomaximal, Sylow
106C127#204511yescharacteristic, complemented, Sylow
107C22×D8—1271no—
108#424—1271no—
109#418—1271no—
110#418D12711yescharacteristic, complemented
111C2×#100—1271no—
112C2×#100—1271no—
113#341—1271no—
114C22×D4—1271no—
115C22×D4C2×D12711yescharacteristic
116C2×D8—2542no—
117C2×D8—2542no—
118C2×D8—2542no—
119C2×M16—1271no—
120C2×M16—1271no—
121C2×M16C2×D12711yescharacteristic
122C22×C8—1271no—
123#117C2×D12711yescharacteristic
124C4⋊C8—2542no—
125#102—22no—
126#102—2542no—
127#100—2542no—
128#100—2542no—
129C24—21no—
130C24—2541no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—1272no—
137C2×D4—2541no—
138C2×D4—2541no—
139C2×D4—2541no—
140C2×D4—2541no—
141C2×D4—2541no—
142C2×D4D50812yes—
143C22×C4C22×D12711yescharacteristic
144D8—2548no—
145M16—21no—
146M16—21no—
147M16—2541no—
148M16—2541no—
149M16—2541no—
150M16—2541no—
151C2×C8—21no—
152C2×C8—1272no—
153C2×C8—2541no—
154C2×C8—2541no—
155C2×C8—2541no—
156C2×C8—2541no—
157C4⋊C4C22⋊D12712yes—
158C22⋊C4—21no—
159C42—21no—
160C23—22no—
161C23—41no—
162C23—41no—
163C23—82no—
164C23—10162no—
165C23—5081no—
166C23—5081no—
167C23—2542no—
168C23#1452324711yescharacteristic
169D4—22no—
170D4—22no—
171D4—41no—
172D4—41no—
173D4—5081no—
174D4—5081no—
175D4—2542no—
176D4—2542no—
177C2×C4—22no—
178C2×C4D4×D12711yescharacteristic
179C2×C4D4×D12711yescharacteristic
180C2×C4C2×C22⋊D12711yescharacteristic
181C2×C4C2×D50811yescharacteristic
182C2×C4#1452324711yescharacteristic
183C2×C4#1452324811yescharacteristic
184C8—41no—
185C8—5081no—
186C8—5081no—
187C8—2542no—
188C22—22no—
189C22—42no—
190C22—82no—
191C22—82no—
192C22—82no—
193C22—10162no—
194C22—10162no—
195C22—10162no—
196C22—5082no—
197C22#1686118111yescharacteristic
198C22#1686119111yescharacteristic
199C22#1686119311yescharacteristic
200C4—42no—
201C4#1686117911yescharacteristic
202C4#1686118011yescharacteristic
203C4#1686118311yescharacteristic
204C4#1686119511yescharacteristic
205C2—21no—
206C2—21no—
207C2—82no—
208C2—10162no—
209C2#2254979211yescharacteristic
210C2#2255014412yes—
subgroupG/H
centerC22#16861181
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#418×C127C2
socleC22×C127#121
prime radicalC22×#14523233C2
hypercenter#418D127
hypocenterC127#2045
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. #22549880C2semidirect
4. C2×#16861126C2semidirect
5. C2×#16861125C2semidirect
6. #22549640C2semidirect
7. #418×C127C2semidirect
106. C127#2045semidirect
110. #418D127semidirect

Maximal subgroups

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

subgroupindex
C22×#145232332
#225499032
#225498802
C2×#168611262
C2×#168611252
#225496402
#418×C1272
#2045127