Finite Group Databasebeta

← #30639811

Subgroups — #30639811

Order 16256.

406 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 95 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×#14523236C211yesmaximal, characteristic
2C2×#16861207C211yesmaximal, characteristic, complemented
3C2×#16861134C211yesmaximal, characteristic, complemented
4C2×#16861133C211yesmaximal, characteristic, complemented
5#22549779C211yesmaximal, characteristic
6#22549630C211yesmaximal, characteristic, complemented
7C22×SD16×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9C2×#14523236—22no—
10C2×#14523236—22no—
11C2×#14523236—22no—
12C22×C127⋊C8C2211yescharacteristic
13#16861207—22no—
14#16861207—22no—
15C2×C4⋊Dic127C2211yescharacteristic
16C2×Dic127⋊C4—22no—
17C2×C22⋊Dic127—22no—
18C2×C4×Dic127C2211yescharacteristic
19#16861134—22no—
20#16861133—22no—
21C22×Q8×C127C2211yescharacteristic
22C22×D4×C127C2211yescharacteristic
23C2×SD16×C127—22no—
24C2×SD16×C127—22no—
25C2×SD16×C127—22no—
26C22×C8×C127C2211yescharacteristic
27C2×Dic254—21no—
28C2×Dic254—21no—
29C2×Dic254—21no—
30C2×Dic254—21no—
31C2×Dic254—21no—
32C2×Dic254D412yes—
33C22×Dic127—21no—
34C22×Dic127—21no—
35C22×Dic127D412yes—
36#14523236—28no—
37C2×C127⋊C8—21no—
38C2×C127⋊C8—21no—
39C2×C127⋊C8D412yes—
40C4⋊Dic127—21no—
41C4⋊Dic127—21no—
42Dic127⋊C4—28no—
43C22⋊Dic127—44no—
44C4×Dic127—21no—
45C4×Dic127—21no—
46C24×C127—21no—
47C2×Q8×C127—21no—
48C2×Q8×C127—21no—
49C2×Q8×C127—21no—
50C2×Q8×C127—21no—
51C2×Q8×C127—21no—
52C2×Q8×C127D412yes—
53C2×D4×C127—21no—
54C2×D4×C127—21no—
55C2×D4×C127—21no—
56C2×D4×C127—21no—
57C2×D4×C127—21no—
58C2×D4×C127D412yes—
59C22×C4×C127—21no—
60C22×C4×C127C2311yescharacteristic
61SD16×C127—28no—
62C2×C8×C127—21no—
63C2×C8×C127—21no—
64C2×C8×C127D412yes—
65Dic254—22no—
66Dic254—22no—
67Dic254—41no—
68Dic254—41no—
69C2×Dic127—22no—
70C2×Dic127—22no—
71C2×Dic127—22no—
72C2×Dic127—41no—
73C2×Dic127—41no—
74C2×Dic127—41no—
75C2×Dic127—42no—
76C2×Dic127SD1614yescomplemented
77C127⋊C8—22no—
78C23×C127—22no—
79C23×C127—41no—
80C23×C127—41no—
81C23×C127—42no—
82C23×C127—42no—
83C23×C127C2×D411yescharacteristic
84Q8×C127—22no—
85Q8×C127—22no—
86Q8×C127—41no—
87Q8×C127—41no—
88D4×C127—22no—
89D4×C127—22no—
90D4×C127—41no—
91D4×C127—41no—
92C2×C4×C127—22no—
93C2×C4×C127—41no—
94C2×C4×C127—41no—
95C2×C4×C127C2×D411yescharacteristic
96C2×C4×C127C2×D411yescharacteristic
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127C2×D411yescharacteristic
99C2×C4×C127C2×D411yescharacteristic
100C2×C4×C127#4111yescharacteristic
101C8×C127—22no—
102C22×C127—42no—
103C22×C127—44no—
104C22×C127—82no—
105C22×C127—81no—
106C22×C127—81no—
107C22×C127#12011yescharacteristic
108C22×C127#12111yescharacteristic
109C22×C127#12711yescharacteristic
110C22×C127C2×SD1612yes—
111C22×C127#13712yes—
112C4×C127—42no—
113C4×C127#12011yescharacteristic
114C4×C127#12011yescharacteristic
115C4×C127#12111yescharacteristic
116C4×C127#12111yescharacteristic
117Dic127—24no—
118Dic127—42no—
119Dic127—81no—
120C2×C127—82no—
121C2×C127#39211yescharacteristic
122C2×C127#44812yes—
123C2×C127#46112yes—
124C2×C127#49211yescharacteristic
125C2×C127#49711yescharacteristic
126#2013—1271nomaximal, Sylow
127C127#201311yescharacteristic, complemented, Sylow
128C22×SD16—1271no—
129C22×SD16D12711yescharacteristic
130C2×C4⋊Q8—1271no—
131C2×#103—1271no—
132C2×#103—1271no—
133#390—1271no—
134#336—1271no—
135C22×Q8—1271no—
136C22×Q8C2×D12711yescharacteristic
137C22×D4C2×D12711yescharacteristic
138C2×SD16—22no—
139C2×SD16—22no—
140C2×SD16—22no—
141C2×SD16—2542no—
142C2×SD16—2542no—
143C2×SD16—2542no—
144C22×C8—1271no—
145C22×C8C2×D12711yescharacteristic
146C4⋊Q8—2542no—
147C4⋊Q8—2542no—
148C2×C4⋊C4—1271no—
149C2×C4⋊C4—2542no—
150C2×C22⋊C4—2542no—
151C2×C42—1271no—
152#103—2542no—
153#103—2542no—
154C24—21no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—1272no—
161C2×Q8—2541no—
162C2×Q8—2541no—
163C2×Q8—2541no—
164C2×Q8—2541no—
165C2×Q8—2541no—
166C2×Q8C22⋊D12712yes—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4C22⋊D12712yes—
173C22×C4—21no—
174C22×C4—1272no—
175C22×C4—2541no—
176C22×C4—2541no—
177C22×C4C22×D12711yescharacteristic
178SD16—28no—
179SD16—2548no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—1272no—
183C2×C8—2541no—
184C2×C8—2541no—
185C2×C8C22⋊D12712yes—
186C4⋊C4—2548no—
187C4⋊C4—2541no—
188C4⋊C4—2541no—
189C22⋊C4—5084no—
190C42—2541no—
191C42—2541no—
192C23—22no—
193C23—41no—
194C23—42no—
195C23—42no—
196C23—41no—
197C23D4×D12711yescharacteristic
198Q8—22no—
199Q8—22no—
200Q8—41no—
201Q8—41no—
202Q8—5081no—
203Q8—5081no—
204Q8—2542no—
205Q8—2542no—
206D4—22no—
207D4—22no—
208D4—41no—
209D4—41no—
210C2×C4—22no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—1274no—
214C2×C4—5081no—
215C2×C4—5082no—
216C2×C4—5081no—
217C2×C4—5081no—
218C2×C4—2542no—
219C2×C4—2542no—
220C2×C4—2542no—
221C2×C4D4×D12711yescharacteristic
222C2×C4D4×D12711yescharacteristic
223C2×C4C2×C22⋊D12711yescharacteristic
224C2×C4C2×C22⋊D12711yescharacteristic
225C2×C4C2×C22⋊D12711yescharacteristic
226C2×C4#1452324711yescharacteristic
227C8—22no—
228C8—2542no—
229C22—42no—
230C22—44no—
231C22—81no—
232C22—81no—
233C22—82no—
234C22#1686117711yescharacteristic
235C22#1686118111yescharacteristic
236C22#1686120511yescharacteristic
237C22SD16×D12712yes—
238C22#1686123812yes—
239C4—42no—
240C4—10161no—
241C4—5082no—
242C4—2544no—
243C4#1686117711yescharacteristic
244C4#1686117811yescharacteristic
245C4#1686118011yescharacteristic
246C4#1686118111yescharacteristic
247C2—82no—
248C2#2254978411yescharacteristic
249C2#2255001312yes—
250C2#2255007912yes—
251C2#2255024711yescharacteristic
252C2#2255026711yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC22×SD16×C127C2
socleC23×C127C2×D4
prime radicalC22×D4×C127C22
hypercenterC22×SD16D127
hypocenterC127#2013
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
2. C2×#16861207C2semidirect
3. C2×#16861134C2semidirect
4. C2×#16861133C2semidirect
6. #22549630C2semidirect
76. C2×Dic127SD16semidirect
127. C127#2013semidirect

Maximal subgroups

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

subgroupindex
C22×#145232362
C2×#168612072
C2×#168611342
C2×#168611332
#225497792
#225496302
C22×SD16×C1272
#2013127