Finite Group Databasebeta

← #30648261

Subgroups — #30648261

Order 16256.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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×#16861242C211yesmaximal, characteristic
2C22×Dic508C211yesmaximal, characteristic, complemented
3#22550452C211yesmaximal, characteristic
4#22550353C211yesmaximal, characteristic, complemented
5#22550099C214yesmaximal, complemented
6#22550087C214yesmaximal, complemented
7#22549886C211yesmaximal, characteristic, complemented
8#22549873C211yesmaximal, characteristic
9#424×C127C211yesmaximal, characteristic, complemented
10C2×#14523248C2211yescharacteristic
11C22×Dic254C2211yescharacteristic
12C22×Dic254C2211yescharacteristic
13#16861242—24no—
14C2×Dic508—24no—
15C2×Dic508C2212yes—
16C2×Dic508C2214yes—
17C2×#14523237C2212yes—
18#16861209C2212yes—
19#16861197C2212yes—
20#16861193—22no—
21#16861189—22no—
22C4×Dic254C2214yes—
23#16861174—22no—
24#16861168C2211yescharacteristic
25#16861165—22no—
26#16861134C2212yes—
27#16861134C2214yes—
28#16861131C2212yes—
29C2×M16×C127C2211yescharacteristic
30C22×C8×C127C2211yescharacteristic
31#117×C127C2211yescharacteristic
32C4⋊C8×C127C2214yes—
33#14523248—22no—
34#14523248—22no—
35C2×Dic254—21no—
36C2×Dic254—21no—
37C2×Dic254—22no—
38C2×Dic254—22no—
39C2×Dic254—22no—
40C2×Dic254D414yes—
41C2×Dic254C2312yes—
42C2×Dic254C2311yescharacteristic
43C2×Dic254C2312yes—
44C2×D508C2311yescharacteristic
45C2×C22⋊D127—21no—
46C22×Dic127—21no—
47C22×Dic127—21no—
48C2×C4×D127—21no—
49Dic508—28no—
50Dic508—24no—
51#14523237—24no—
52C4⋊Dic127—24no—
53C4⋊Dic127C2312yes—
54Dic127⋊C4—24no—
55C22⋊Dic127—21no—
56(C2×D127)⋊C4—24no—
57C4×Dic127—24no—
58C4×Dic127—21no—
59C22×C4×C127C2311yescharacteristic
60M16×C127—22no—
61C2×C8×C127—22no—
62C2×C8×C127C2312yes—
63C2×C8×C127C2312yes—
64C4⋊C4×C127D412yes—
65C22⋊C4×C127D412yes—
66C42×C127C2312yes—
67C22×D127—21no—
68Dic254—24no—
69Dic254—22no—
70Dic254—22no—
71Dic254—21no—
72Dic254—22no—
73Dic254—21no—
74Dic254—42no—
75Dic254C2×D414yes—
76D508—21no—
77D508—21no—
78C22⋊D127—42no—
79C2×Dic127—24no—
80C2×Dic127—22no—
81C2×Dic127—22no—
82C2×Dic127—22no—
83C2×Dic127—21no—
84C2×Dic127—42no—
85C4×D127—42no—
86C23×C127C2×D411yescharacteristic
87C2×C4×C127C2×D412yes—
88C2×C4×C127C2×D414yes—
89C2×C4×C127C2×D411yescharacteristic
90C2×C4×C127#4112yes—
91C2×C4×C127C2411yescharacteristic
92C8×C127—22no—
93C8×C127—22no—
94C22×C127—22no—
95C22×C127#12112yes—
96C22×C127C22×D411yescharacteristic
97C2×D127—41no—
98C2×D127—42no—
99C4×C127—24no—
100C4×C127#12112yes—
101C4×C127C22×D411yescharacteristic
102C4×C127C2×#4111yescharacteristic
103Dic127—24no—
104Dic127—41no—
105Dic127—42no—
106Dic127—42no—
107C2×C127—22no—
108C2×C127C2×#12111yescharacteristic
109C2×C127#57812yes—
110D127—81no—
111#3044—1271nomaximal, Sylow
112C127#304411yescharacteristic, complemented, Sylow
113C2×#137—1271no—
114C22×Q16—1271no—
115#536—1271no—
116#519—1271no—
117#464—1274no—
118#462—1274no—
119#424D12711yescharacteristic, complemented
120#419—1271no—
121#416—1271no—
122C2×#41—1271no—
123C22×Q8—1271no—
124C22×Q8—1271no—
125#137—2544no—
126C2×Q16—1274no—
127C2×Q16—1272no—
128C2×Q16—2544no—
129C2×SD16—1272no—
130C2×M16C2×D12711yescharacteristic
131C22×C8C2×D12711yescharacteristic
132C4⋊Q8—1272no—
133#124—1272no—
134#123—2542no—
135C22⋊Q8—2542no—
136C4×Q8—1274no—
137C4×Q8—2542no—
138#117—1271no—
139#117—2542no—
140#117C2×D12711yescharacteristic
141C4⋊C8C2×D12714yes—
142#103—1272no—
143#103—1274no—
144#102—1272no—
145#41—2542no—
146#41—2542no—
147C2×Q8—1274no—
148C2×Q8—1272no—
149C2×Q8—1272no—
150C2×Q8—1271no—
151C2×Q8—2542no—
152C2×Q8—2542no—
153C2×Q8—2541no—
154C2×Q8—2541no—
155C2×Q8—2542no—
156C2×D4—1271no—
157C2×D4—2541no—
158C22×C4—2541no—
159C22×C4—2541no—
160C22×C4—2541no—
161C22×C4C22×D12711yescharacteristic
162Q16—2548no—
163Q16—2544no—
164SD16—2544no—
165M16—22no—
166C2×C8—22no—
167C2×C8C22×D12712yes—
168C2×C8C22×D12712yes—
169C4⋊C4—1272no—
170C4⋊C4—2544no—
171C4⋊C4—2544no—
172C4⋊C4D50812yes—
173C22⋊C4—2541no—
174C22⋊C4—2544no—
175C22⋊C4D50812yes—
176C42—2541no—
177C42—2544no—
178C42C22×D12712yes—
179C23—2541no—
180C23C2×D50811yescharacteristic
181Q8—1274no—
182Q8—5082no—
183Q8—2542no—
184Q8—2541no—
185Q8—2542no—
186Q8—2541no—
187Q8—2544no—
188Q8—2542no—
189D4—5082no—
190D4—2541no—
191D4—2541no—
192C2×C4—5082no—
193C2×C4—5082no—
194C2×C4—2542no—
195C2×C4—2542no—
196C2×C4—2541no—
197C2×C4—2544no—
198C2×C4—2542no—
199C2×C4D4×D12712yes—
200C2×C4C2×D50814yes—
201C2×C4C2×D50811yescharacteristic
202C2×C4#1452324912yes—
203C2×C4C23×D12711yescharacteristic
204C8—22no—
205C8—22no—
206C22—22no—
207C22—5081no—
208C22—5082no—
209C22#1686118412yes—
210C22C22×D50811yescharacteristic
211C4—24no—
212C4—5081no—
213C4—5082no—
214C4—5082no—
215C4—2544no—
216C4#1686118412yes—
217C4C2×D4×D12711yescharacteristic
218C4C2×#1452324911yescharacteristic
219C2—22no—
220C2—10161no—
221C2C2×#1686118411yescharacteristic
222C2#2255069312yes—
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#424×C127C2
socleC22×C127C22×D4
prime radicalC2×#14523248C22
hypercenter#424D127
hypocenterC127#3044
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
2. C22×Dic508C2semidirect
4. #22550353C2semidirect
5. #22550099C2semidirect
6. #22550087C2semidirect
7. #22549886C2semidirect
9. #424×C127C2semidirect
112. C127#3044semidirect
119. #424D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612422
C22×Dic5082
#225504522
#225503532
#225500992
#225500872
#225498862
#225498732
#424×C1272
#3044127