Finite Group Databasebeta

← #30639825

Subgroups — #30639825

Order 16256.

390 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 89 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×#14523238C211yesmaximal, characteristic, complemented
2C22×#14523237C211yesmaximal, characteristic
3C2×#16861198C211yesmaximal, characteristic
4C2×#16861129C211yesmaximal, characteristic
5#22549781C211yesmaximal, characteristic, complemented
6#22549630C211yesmaximal, characteristic, complemented
7C2×#103×C127C211yesmaximal, characteristic, complemented
8C22×Dic254C2211yescharacteristic
9C22×D508C2211yescharacteristic
10C2×#14523238—22no—
11C2×#14523238—22no—
12C2×#14523238—22no—
13C2×#14523237—22no—
14C2×#14523237—22no—
15C2×#14523237—22no—
16C22×C127⋊C8C2211yescharacteristic
17#16861198—22no—
18#16861198—22no—
19C2×(C2×D127)⋊C4—22no—
20C2×C4×Dic127C2211yescharacteristic
21#16861129—22no—
22C127⋊#95—22no—
23C22×Q8×C127C2211yescharacteristic
24C22×C8×C127C2211yescharacteristic
25C2×C4⋊C4×C127C2211yescharacteristic
26#103×C127—22no—
27C23×D127—21no—
28C2×Dic254—21no—
29C2×Dic254—21no—
30C2×Dic254—21no—
31C2×Dic254—21no—
32C2×Dic254—21no—
33C2×Dic254D412yes—
34C2×D508—21no—
35C2×D508—21no—
36C2×D508—21no—
37C2×D508—21no—
38C2×D508—21no—
39C2×D508D412yes—
40C22×Dic127—21no—
41C22×Dic127D412yes—
42#14523238—28no—
43#14523237—28no—
44C2×C127⋊C8—21no—
45C2×C127⋊C8—21no—
46C2×C127⋊C8D412yes—
47(C2×D127)⋊C4—28no—
48C4×Dic127—21no—
49C4×Dic127—21no—
50C2×Q8×C127—21no—
51C2×Q8×C127—21no—
52C2×Q8×C127—21no—
53C2×Q8×C127—21no—
54C2×Q8×C127—21no—
55C2×Q8×C127D412yes—
56C22×C4×C127—21no—
57C22×C4×C127—21no—
58C22×C4×C127C2311yescharacteristic
59C2×C8×C127—21no—
60C2×C8×C127—21no—
61C2×C8×C127D412yes—
62C4⋊C4×C127—21no—
63C4⋊C4×C127—21no—
64C22×D127—22no—
65C22×D127—41no—
66C22×D127—41no—
67C22×D127—42no—
68C22×D127—42no—
69Dic254—22no—
70Dic254—22no—
71Dic254—41no—
72Dic254—41no—
73D508—22no—
74D508—22no—
75D508—41no—
76D508—41no—
77C2×Dic127—22no—
78C2×Dic127—22no—
79C2×Dic127—22no—
80C2×Dic127—22no—
81C2×Dic127—41no—
82C2×Dic127—41no—
83C127⋊C8—22no—
84C23×C127C2×D411yescharacteristic
85Q8×C127—22no—
86Q8×C127—22no—
87Q8×C127—41no—
88Q8×C127—41no—
89C2×C4×C127—22no—
90C2×C4×C127—41no—
91C2×C4×C127—41no—
92C2×C4×C127—41no—
93C2×C4×C127—42no—
94C2×C4×C127C2×D411yescharacteristic
95C2×C4×C127C2×D411yescharacteristic
96C2×C4×C127C2×D411yescharacteristic
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127C2×D411yescharacteristic
99C2×C4×C127#4111yescharacteristic
100C8×C127—22no—
101C22×C127#12011yescharacteristic
102C22×C127#12111yescharacteristic
103C22×C127#12711yescharacteristic
104C22×C127#13512yes—
105C22×C127#13712yes—
106C2×D127—42no—
107C2×D127—44no—
108C2×D127—82no—
109C2×D127—81no—
110C2×D127—81no—
111C4×C127—42no—
112C4×C127—81no—
113C4×C127#12011yescharacteristic
114C4×C127#12011yescharacteristic
115C4×C127#12111yescharacteristic
116C4×C127#12111yescharacteristic
117Dic127—42no—
118Dic127—42no—
119C2×C127#39211yescharacteristic
120C2×C127#45212yes—
121C2×C127#46412yes—
122C2×C127#49511yescharacteristic
123C2×C127#49711yescharacteristic
124D127—82no—
125#2015—1271nomaximal, Sylow
126C127#201511yescharacteristic, complemented, Sylow
127C22×Q16—1271no—
128C22×SD16—1271no—
129C2×#124—1271no—
130C2×#103D12711yescharacteristic, complemented
131C2×#102—1271no—
132#391—1271no—
133#336—1271no—
134C22×Q8—1271no—
135C22×Q8C2×D12711yescharacteristic
136C22×D4—1271no—
137C2×Q16—2542no—
138C2×Q16—2542no—
139C2×Q16—2542no—
140C2×SD16—2542no—
141C2×SD16—2542no—
142C2×SD16—2542no—
143C22×C8—1271no—
144C22×C8C2×D12711yescharacteristic
145#124—2542no—
146#124—2542no—
147C2×C4⋊C4C2×D12711yescharacteristic
148C2×C22⋊C4—2542no—
149C2×C42—1271no—
150#103—22no—
151#102—2542no—
152#95—2542no—
153C24—2541no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—1272no—
160C2×Q8—2541no—
161C2×Q8—2541no—
162C2×Q8—2541no—
163C2×Q8—2541no—
164C2×Q8—2541no—
165C2×Q8D50812yes—
166C2×D4—1272no—
167C2×D4—2541no—
168C2×D4—2541no—
169C2×D4—2541no—
170C2×D4—2541no—
171C2×D4—2541no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—1272no—
175C22×C4—2541no—
176C22×C4C22×D12711yescharacteristic
177Q16—2548no—
178SD16—2548no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—1272no—
182C2×C8—2541no—
183C2×C8—2541no—
184C2×C8C22⋊D12712yes—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C22⋊C4—2548no—
188C42—2541no—
189C42—2541no—
190C23—5081no—
191C23—5082no—
192C23—5082no—
193C23—5081no—
194C23—2542no—
195C23D4×D12711yescharacteristic
196Q8—22no—
197Q8—22no—
198Q8—41no—
199Q8—41no—
200Q8—5081no—
201Q8—5081no—
202Q8—2542no—
203Q8—2542no—
204D4—5081no—
205D4—5081no—
206D4—2542no—
207D4—2542no—
208C2×C4—22no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—42no—
213C2×C4—5081no—
214C2×C4—5081no—
215C2×C4—2542no—
216C2×C4—2542no—
217C2×C4—2542no—
218C2×C4—2542no—
219C2×C4D4×D12711yescharacteristic
220C2×C4D4×D12711yescharacteristic
221C2×C4D4×D12711yescharacteristic
222C2×C4C2×C22⋊D12711yescharacteristic
223C2×C4C2×D50811yescharacteristic
224C2×C4#1452324811yescharacteristic
225C8—22no—
226C8—2542no—
227C22—10162no—
228C22—10161no—
229C22—10161no—
230C22—5082no—
231C22—5084no—
232C22#1686117911yescharacteristic
233C22#1686118211yescharacteristic
234C22#1686120511yescharacteristic
235C22#1686123012yes—
236C22#1686124112yes—
237C4—42no—
238C4—81no—
239C4—5082no—
240C4—5082no—
241C4#1686117711yescharacteristic
242C4#1686117911yescharacteristic
243C4#1686118211yescharacteristic
244C4#1686118311yescharacteristic
245C2—10162no—
246C2#2254978511yescharacteristic
247C2#2255004112yes—
248C2#2255010112yes—
249C2#2255025711yescharacteristic
250C2#2255026911yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#103×C127C2
socleC23×C127C2×D4
prime radicalC22×D508C22
hypercenterC2×#103D127
hypocenterC127#2015
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
1. C22×#14523238C2semidirect
5. #22549781C2semidirect
6. #22549630C2semidirect
7. C2×#103×C127C2semidirect
126. C127#2015semidirect
130. C2×#103D127semidirect

Maximal subgroups

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

subgroupindex
C22×#145232382
C22×#145232372
C2×#168611982
C2×#168611292
#225497812
#225496302
C2×#103×C1272
#2015127