Finite Group Databasebeta

← #30639826

Subgroups — #30639826

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×Dic508C211yesmaximal, characteristic, complemented
2C22×#14523236C211yesmaximal, characteristic
3C2×#16861196C211yesmaximal, characteristic
4C2×#16861132C211yesmaximal, characteristic, complemented
5#22549781C211yesmaximal, characteristic, complemented
6#22549630C211yesmaximal, characteristic, complemented
7C2×#102×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9C22×Dic254C2211yescharacteristic
10C2×Dic508—22no—
11C2×Dic508—22no—
12C2×Dic508—22no—
13C2×#14523236—22no—
14C2×#14523236—22no—
15C2×#14523236—22no—
16C22×C127⋊C8C2211yescharacteristic
17#16861196—22no—
18#16861196—22no—
19C2×C22⋊Dic127—22no—
20C2×C4×Dic127C2211yescharacteristic
21#16861132—22no—
22C127⋊#95—22no—
23C22×D4×C127C2211yescharacteristic
24C22×C8×C127C2211yescharacteristic
25C2×C4⋊C4×C127C2211yescharacteristic
26#102×C127—22no—
27C2×Dic254—21no—
28C2×Dic254—21no—
29C2×Dic254—21no—
30C2×Dic254—21no—
31C2×Dic254—21no—
32C2×Dic254—21no—
33C2×Dic254—21no—
34C2×Dic254—21no—
35C2×Dic254—21no—
36C2×Dic254—21no—
37C2×Dic254D412yes—
38C2×Dic254D412yes—
39C22×Dic127—21no—
40C22×Dic127—21no—
41C22×Dic127D412yes—
42Dic508—28no—
43#14523236—28no—
44C2×C127⋊C8—21no—
45C2×C127⋊C8—21no—
46C2×C127⋊C8D412yes—
47C22⋊Dic127—28no—
48C4×Dic127—21no—
49C4×Dic127—21no—
50C24×C127—21no—
51C2×D4×C127—21no—
52C2×D4×C127—21no—
53C2×D4×C127—21no—
54C2×D4×C127—21no—
55C2×D4×C127—21no—
56C2×D4×C127D412yes—
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—
64Dic254—22no—
65Dic254—22no—
66Dic254—22no—
67Dic254—22no—
68Dic254—41no—
69Dic254—41no—
70Dic254—41no—
71Dic254—41no—
72C2×Dic127—22no—
73C2×Dic127—22no—
74C2×Dic127—22no—
75C2×Dic127—22no—
76C2×Dic127—22no—
77C2×Dic127—41no—
78C2×Dic127—41no—
79C2×Dic127—41no—
80C2×Dic127—41no—
81C127⋊C8—22no—
82C23×C127—22no—
83C23×C127—41no—
84C23×C127—41no—
85C23×C127—42no—
86C23×C127—42no—
87C23×C127C2×D411yescharacteristic
88D4×C127—22no—
89D4×C127—22no—
90D4×C127—41no—
91D4×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—42no—
102C22×C127—44no—
103C22×C127—82no—
104C22×C127—81no—
105C22×C127—81no—
106C22×C127#12011yescharacteristic
107C22×C127#12111yescharacteristic
108C22×C127#12711yescharacteristic
109C22×C127#13512yes—
110C22×C127#13712yes—
111C4×C127—81no—
112C4×C127#12011yescharacteristic
113C4×C127#12011yescharacteristic
114C4×C127#12111yescharacteristic
115C4×C127#12111yescharacteristic
116Dic127—42no—
117Dic127—42no—
118Dic127—42no—
119C2×C127—82no—
120C2×C127#39211yescharacteristic
121C2×C127#45212yes—
122C2×C127#46412yes—
123C2×C127#49511yescharacteristic
124C2×C127#49711yescharacteristic
125#2015—1271nomaximal, Sylow
126C127#201511yescharacteristic, complemented, Sylow
127C22×Q16—1271no—
128C22×SD16—1271no—
129C2×#124—1271no—
130C2×#103—1271no—
131C2×#102D12711yescharacteristic
132#391—1271no—
133#336—1271no—
134C22×Q8—1271no—
135C22×Q8—1271no—
136C22×D4C2×D12711yescharacteristic
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—2542no—
151#102—22no—
152#95—2542no—
153C24—21no—
154C2×Q8—1272no—
155C2×Q8—1272no—
156C2×Q8—2541no—
157C2×Q8—2541no—
158C2×Q8—2541no—
159C2×Q8—2541no—
160C2×Q8—2541no—
161C2×Q8—2541no—
162C2×Q8—2541no—
163C2×Q8—2541no—
164C2×Q8—2541no—
165C2×Q8—2541no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4D50812yes—
172C22×C4—21no—
173C22×C4—1272no—
174C22×C4—2541no—
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—22no—
191C23—41no—
192C23—42no—
193C23—42no—
194C23—41no—
195C23D4×D12711yescharacteristic
196Q8—5081no—
197Q8—5081no—
198Q8—5081no—
199Q8—5081no—
200Q8—2542no—
201Q8—2542no—
202Q8—2542no—
203Q8—2542no—
204D4—22no—
205D4—22no—
206D4—41no—
207D4—41no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—5081no—
211C2×C4—5081no—
212C2×C4—5081no—
213C2×C4—5081no—
214C2×C4—2542no—
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—44no—
228C22—42no—
229C22—82no—
230C22—81no—
231C22—81no—
232C22#1686117911yescharacteristic
233C22#1686118211yescharacteristic
234C22#1686120511yescharacteristic
235C22#1686122812yes—
236C22#1686123812yes—
237C4—81no—
238C4—5082no—
239C4—5082no—
240C4—5082no—
241C4#1686117711yescharacteristic
242C4#1686117911yescharacteristic
243C4#1686118211yescharacteristic
244C4#1686118311yescharacteristic
245C2—82no—
246C2#2254978511yescharacteristic
247C2#2255003912yes—
248C2#2255009712yes—
249C2#2255025911yescharacteristic
250C2#2255026711yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#102×C127C2
socleC23×C127C2×D4
prime radicalC22×D4×C127C22
hypercenterC2×#102D127
hypocenterC127#2015
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C22×Dic508C2semidirect
4. C2×#16861132C2semidirect
5. #22549781C2semidirect
6. #22549630C2semidirect
126. C127#2015semidirect

Maximal subgroups

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

subgroupindex
C22×Dic5082
C22×#145232362
C2×#168611962
C2×#168611322
#225497812
#225496302
C2×#102×C1272
#2015127