Finite Group Databasebeta

← #30648260

Subgroups — #30648260

Order 16256.

458 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16861240C211yesmaximal, characteristic
2C22×#14523238C211yesmaximal, characteristic, complemented
3#22550452C211yesmaximal, characteristic
4#22550098C214yesmaximal, complemented
5#22550086C214yesmaximal, complemented
6#22549903C211yesmaximal, characteristic, complemented
7#22549884C211yesmaximal, characteristic, complemented
8#22549870C211yesmaximal, characteristic
9#519×C127C211yesmaximal, characteristic, complemented
10C2×#14523248C2211yescharacteristic
11C22×Dic254C2211yescharacteristic
12#16861240—24no—
13C2×#14523238—24no—
14C2×#14523238C2212yes—
15C2×#14523238C2214yes—
16C2×#14523235C2212yes—
17C2×#14523232C2211yescharacteristic
18C22×C127⋊C8C2211yescharacteristic
19#16861209C2212yes—
20#16861197C2212yes—
21#16861193—22no—
22#16861189—22no—
23C4⋊(C127⋊C8)C2214yes—
24#16861132C2212yes—
25#16861132C2214yes—
26#16861129C2212yes—
27C22×Q8×C127C2211yescharacteristic
28C4×Q8×C127—22no—
29C4×Q8×C127C2214yes—
30#117×C127—22no—
31#117×C127C2211yescharacteristic
32#117×C127C2211yescharacteristic
33#14523248—22no—
34#14523248—22no—
35C2×Dic254—21no—
36C2×Dic254—22no—
37C2×Dic254—22no—
38C2×Dic254C2312yes—
39C2×Dic254C2311yescharacteristic
40C2×D508C2311yescharacteristic
41C2×C22⋊D127—21no—
42C22×Dic127—21no—
43C2×C4×D127—21no—
44#14523238—28no—
45#14523238—24no—
46#14523235—24no—
47#14523232—22no—
48C2×C127⋊C8—22no—
49C2×C127⋊C8C2312yes—
50C2×C127⋊C8C2312yes—
51C4⋊Dic127—24no—
52(C2×D127)⋊C4—24no—
53C2×Q8×C127—21no—
54C2×Q8×C127—22no—
55C2×Q8×C127D414yes—
56C2×Q8×C127C2312yes—
57C22×C4×C127—21no—
58C22×C4×C127C2311yescharacteristic
59C4⋊C4×C127—24no—
60C4⋊C4×C127D412yes—
61C4⋊C4×C127C2312yes—
62C22⋊C4×C127—21no—
63C22⋊C4×C127D412yes—
64C42×C127—24no—
65C42×C127—21no—
66C42×C127C2312yes—
67C22×D127—21no—
68Dic254—22no—
69Dic254—21no—
70Dic254—22no—
71Dic254—21no—
72Dic254—42no—
73D508—21no—
74D508—21no—
75C22⋊D127—42no—
76C2×Dic127—22no—
77C2×Dic127—21no—
78C2×Dic127—42no—
79C4×D127—42no—
80C127⋊C8—22no—
81C127⋊C8—22no—
82C23×C127C2×D411yescharacteristic
83Q8×C127—24no—
84Q8×C127—22no—
85Q8×C127C2×D414yes—
86C2×C4×C127—24no—
87C2×C4×C127—22no—
88C2×C4×C127—22no—
89C2×C4×C127C2×D414yes—
90C2×C4×C127C2×D411yescharacteristic
91C2×C4×C127C2×D412yes—
92C2×C4×C127#4112yes—
93C2×C4×C127C2411yescharacteristic
94C22×C127—22no—
95C22×C127#12112yes—
96C22×C127C22×D411yescharacteristic
97C2×D127—41no—
98C2×D127—42no—
99C4×C127—24no—
100C4×C127—24no—
101C4×C127—42no—
102C4×C127#12112yes—
103C4×C127C22×D411yescharacteristic
104C4×C127C2×#4111yescharacteristic
105Dic127—41no—
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#519D12711yescharacteristic, complemented
117#464—1274no—
118#462—1274no—
119#424—1271no—
120#419—1271no—
121#416—1271no—
122C2×#41—1271no—
123C22×Q8—1271no—
124C22×Q8C2×D12711yescharacteristic
125#137—2544no—
126C2×Q16—1274no—
127C2×Q16—1272no—
128C2×Q16—2544no—
129C2×SD16—1272no—
130C2×M16—1271no—
131C22×C8—1271no—
132C4⋊Q8—1272no—
133#124—1272no—
134#123—2542no—
135C22⋊Q8—2542no—
136C4×Q8—22no—
137C4×Q8C2×D12714yes—
138#117—22no—
139#117C2×D12711yescharacteristic
140#117C2×D12711yescharacteristic
141C4⋊C8—1274no—
142#103—1272no—
143#103—1274no—
144#102—1272no—
145#41—2542no—
146#41—2542no—
147C2×Q8—21no—
148C2×Q8—22no—
149C2×Q8—1272no—
150C2×Q8—1271no—
151C2×Q8—2542no—
152C2×Q8—2542no—
153C2×Q8—2541no—
154C2×Q8D50814yes—
155C2×Q8C22×D12712yes—
156C2×D4—1271no—
157C2×D4—2541no—
158C22×C4—21no—
159C22×C4—2541no—
160C22×C4—2541no—
161C22×C4C22×D12711yescharacteristic
162Q16—2548no—
163Q16—2544no—
164SD16—2544no—
165M16—2542no—
166C2×C8—1272no—
167C2×C8—1272no—
168C2×C8—2542no—
169C4⋊C4—24no—
170C4⋊C4—2544no—
171C4⋊C4C22⋊D12712yes—
172C4⋊C4C22×D12712yes—
173C22⋊C4—21no—
174C22⋊C4—2544no—
175C22⋊C4C22⋊D12712yes—
176C42—24no—
177C42—21no—
178C42C22×D12712yes—
179C23—2541no—
180C23C2×C22⋊D12711yescharacteristic
181Q8—22no—
182Q8—24no—
183Q8—5082no—
184Q8—2542no—
185Q8—2541no—
186Q8—2542no—
187Q8—2541no—
188Q8C2×D50814yes—
189D4—5082no—
190D4—2541no—
191D4—2541no—
192C2×C4—22no—
193C2×C4—24no—
194C2×C4—22no—
195C2×C4—5082no—
196C2×C4—5082no—
197C2×C4—2542no—
198C2×C4—2541no—
199C2×C4C2×C22⋊D12714yes—
200C2×C4C2×C22⋊D12711yescharacteristic
201C2×C4C2×D50812yes—
202C2×C4#1452324812yes—
203C2×C4C23×D12711yescharacteristic
204C8—2542no—
205C8—2542no—
206C22—22no—
207C22—5081no—
208C22—5082no—
209C22#1686118312yes—
210C22C22×C22⋊D12711yescharacteristic
211C4—24no—
212C4—24no—
213C4—42no—
214C4—5081no—
215C4—5082no—
216C4#1686118312yes—
217C4C22×D50811yescharacteristic
218C4C2×#1452324811yescharacteristic
219C2—22no—
220C2—10161no—
221C2C2×#1686118311yescharacteristic
222C2#2255069112yes—
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#519×C127C2
socleC22×C127C22×D4
prime radicalC2×#14523248C22
hypercenter#519D127
hypocenterC127#3044
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ #117 ◁ #519

Complements

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

NH ≅ G/Ntype
2. C22×#14523238C2semidirect
4. #22550098C2semidirect
5. #22550086C2semidirect
6. #22549903C2semidirect
7. #22549884C2semidirect
9. #519×C127C2semidirect
112. C127#3044semidirect
116. #519D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612402
C22×#145232382
#225504522
#225500982
#225500862
#225499032
#225498842
#225498702
#519×C1272
#3044127