Finite Group Databasebeta

← #30639814

Subgroups — #30639814

Order 16256.

406 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 93 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×#14523237C211yesmaximal, characteristic
2C22×#14523235C211yesmaximal, characteristic
3C2×#16861207C211yesmaximal, characteristic, complemented
4C2×#16861132C211yesmaximal, characteristic, complemented
5#22549777C211yesmaximal, characteristic
6#22549630C211yesmaximal, characteristic, complemented
7C2×#103×C127C211yesmaximal, characteristic, complemented
8C22×Dic254C2211yescharacteristic
9C22×D508C2211yescharacteristic
10C2×#14523237—22no—
11C2×#14523237—22no—
12C2×#14523237—22no—
13C2×#14523235—22no—
14C2×#14523235—22no—
15C2×#14523235—22no—
16C22×C127⋊C8C2211yescharacteristic
17#16861207—22no—
18#16861207—22no—
19C2×Dic127⋊C4—22no—
20C2×(C2×D127)⋊C4—22no—
21C2×C4×Dic127C2211yescharacteristic
22#16861132—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#14523237—28no—
43#14523235—28no—
44C2×C127⋊C8—21no—
45C2×C127⋊C8—21no—
46C2×C127⋊C8D412yes—
47Dic127⋊C4—28no—
48(C2×D127)⋊C4—44no—
49C4×Dic127—21no—
50C4×Dic127—21no—
51C2×Q8×C127—21no—
52C2×Q8×C127—21no—
53C2×Q8×C127—21no—
54C2×Q8×C127—21no—
55C2×Q8×C127—21no—
56C2×Q8×C127D412yes—
57C22×C4×C127—21no—
58C22×C4×C127—21no—
59C22×C4×C127C2311yescharacteristic
60C2×C8×C127—21no—
61C2×C8×C127—21no—
62C2×C8×C127D412yes—
63C4⋊C4×C127—21no—
64C4⋊C4×C127—21no—
65C22×D127—22no—
66C22×D127—41no—
67C22×D127—41no—
68C22×D127—42no—
69C22×D127—42no—
70Dic254—22no—
71Dic254—22no—
72Dic254—41no—
73Dic254—41no—
74D508—22no—
75D508—22no—
76D508—41no—
77D508—41no—
78C2×Dic127—22no—
79C2×Dic127—22no—
80C2×Dic127—22no—
81C2×Dic127—41no—
82C2×Dic127—41no—
83C2×Dic127SD1614yescomplemented
84C127⋊C8—22no—
85C23×C127C2×D411yescharacteristic
86Q8×C127—22no—
87Q8×C127—22no—
88Q8×C127—41no—
89Q8×C127—41no—
90C2×C4×C127—22no—
91C2×C4×C127—41no—
92C2×C4×C127—41no—
93C2×C4×C127—41no—
94C2×C4×C127—42no—
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#12011yescharacteristic
103C22×C127#12111yescharacteristic
104C22×C127#12711yescharacteristic
105C22×C127C2×SD1612yes—
106C22×C127#13712yes—
107C2×D127—42no—
108C2×D127—44no—
109C2×D127—82no—
110C2×D127—81no—
111C2×D127—81no—
112C4×C127—42no—
113C4×C127—81no—
114C4×C127#12011yescharacteristic
115C4×C127#12011yescharacteristic
116C4×C127#12111yescharacteristic
117C4×C127#12111yescharacteristic
118Dic127—24no—
119Dic127—42no—
120C2×C127#39211yescharacteristic
121C2×C127#44812yes—
122C2×C127#46112yes—
123C2×C127#49211yescharacteristic
124C2×C127#49711yescharacteristic
125D127—82no—
126#2013—1271nomaximal, Sylow
127C127#201311yescharacteristic, complemented, Sylow
128C22×SD16—1271no—
129C22×SD16—1271no—
130C2×C4⋊Q8—1271no—
131C2×#103—1271no—
132C2×#103D12711yescharacteristic, complemented
133#390—1271no—
134#336—1271no—
135C22×Q8—1271no—
136C22×Q8C2×D12711yescharacteristic
137C22×D4—1271no—
138C2×SD16—2542no—
139C2×SD16—2542no—
140C2×SD16—2542no—
141C2×SD16—2542no—
142C2×SD16—2542no—
143C2×SD16—2542no—
144C22×C8—1271no—
145C22×C8C2×D12711yescharacteristic
146C4⋊Q8—2542no—
147C4⋊Q8—2542no—
148C2×C4⋊C4—2542no—
149C2×C4⋊C4C2×D12711yescharacteristic
150C2×C22⋊C4—2542no—
151C2×C42—1271no—
152#103—22no—
153#103—2542no—
154C24—2541no—
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×Q8D50812yes—
167C2×D4—1272no—
168C2×D4—2541no—
169C2×D4—2541no—
170C2×D4—2541no—
171C2×D4—2541no—
172C2×D4—2541no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—1272no—
176C22×C4—2541no—
177C22×C4C22×D12711yescharacteristic
178SD16—2548no—
179SD16—2548no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—1272no—
183C2×C8—2541no—
184C2×C8—2541no—
185C2×C8C22⋊D12712yes—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—2548no—
189C22⋊C4—5084no—
190C42—2541no—
191C42—2541no—
192C23—5081no—
193C23—5082no—
194C23—5082no—
195C23—5081no—
196C23—2542no—
197C23D4×D12711yescharacteristic
198Q8—22no—
199Q8—22no—
200Q8—41no—
201Q8—41no—
202Q8—5081no—
203Q8—5081no—
204Q8—2542no—
205Q8—2542no—
206D4—5081no—
207D4—5081no—
208D4—2542no—
209D4—2542no—
210C2×C4—22no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—1274no—
216C2×C4—5081no—
217C2×C4—5081no—
218C2×C4—2542no—
219C2×C4—2542no—
220C2×C4—2542no—
221C2×C4D4×D12711yescharacteristic
222C2×C4D4×D12711yescharacteristic
223C2×C4D4×D12711yescharacteristic
224C2×C4C2×C22⋊D12711yescharacteristic
225C2×C4C2×D50811yescharacteristic
226C2×C4#1452324811yescharacteristic
227C8—22no—
228C8—2542no—
229C22—10162no—
230C22—10161no—
231C22—10161no—
232C22—5082no—
233C22—5084no—
234C22#1686117911yescharacteristic
235C22#1686118211yescharacteristic
236C22#1686120511yescharacteristic
237C22SD16×D12712yes—
238C22#1686124112yes—
239C4—42no—
240C4—81no—
241C4—5082no—
242C4—2544no—
243C4#1686117711yescharacteristic
244C4#1686117911yescharacteristic
245C4#1686118211yescharacteristic
246C4#1686118311yescharacteristic
247C2—10162no—
248C2#2254978511yescharacteristic
249C2#2255001612yes—
250C2#2255008012yes—
251C2#2255024711yescharacteristic
252C2#2255026911yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#103×C127C2
socleC23×C127C2×D4
prime radicalC22×D508C22
hypercenterC2×#103D127
hypocenterC127#2013
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
3. C2×#16861207C2semidirect
4. C2×#16861132C2semidirect
6. #22549630C2semidirect
7. C2×#103×C127C2semidirect
83. C2×Dic127SD16semidirect
127. C127#2013semidirect
132. C2×#103D127semidirect

Maximal subgroups

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

subgroupindex
C22×#145232372
C22×#145232352
C2×#168612072
C2×#168611322
#225497772
#225496302
C2×#103×C1272
#2013127