Finite Group Databasebeta

← #30639828

Subgroups — #30639828

Order 16256.

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

normal subgroups only

This group has 370 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×#14523238C211yesmaximal, characteristic
2C2×#16861207C211yesmaximal, characteristic
3C2×#16861134C211yesmaximal, characteristic
4C2×#16861133C211yesmaximal, characteristic
5#22549783C211yesmaximal, characteristic
6#22549630C211yesmaximal, characteristic
7C22×Q16×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9C2×#14523238—22no—
10C2×#14523238—22no—
11C2×#14523238—22no—
12C22×C127⋊C8C2211yescharacteristic
13#16861207—22no—
14#16861207—22no—
15C2×C4⋊Dic127C2211yescharacteristic
16C2×Dic127⋊C4—22no—
17C2×C4×Dic127C2211yescharacteristic
18#16861134—22no—
19#16861133—22no—
20C127⋊#95—22no—
21C22×Q8×C127C2211yescharacteristic
22C22×Q8×C127C2211yescharacteristic
23C2×Q16×C127—22no—
24C2×Q16×C127—22no—
25C2×Q16×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#14523238—28no—
37C2×C127⋊C8—21no—
38C2×C127⋊C8—21no—
39C2×C127⋊C8D412yes—
40C4⋊Dic127—21no—
41C4⋊Dic127—21no—
42Dic127⋊C4—28no—
43C4×Dic127—21no—
44C4×Dic127—21no—
45C2×Q8×C127—21no—
46C2×Q8×C127—21no—
47C2×Q8×C127—21no—
48C2×Q8×C127—21no—
49C2×Q8×C127—21no—
50C2×Q8×C127—21no—
51C2×Q8×C127—21no—
52C2×Q8×C127—21no—
53C2×Q8×C127—21no—
54C2×Q8×C127—21no—
55C2×Q8×C127D412yes—
56C2×Q8×C127D412yes—
57C22×C4×C127—21no—
58C22×C4×C127—21no—
59C22×C4×C127C2311yescharacteristic
60Q16×C127—28no—
61C2×C8×C127—21no—
62C2×C8×C127—21no—
63C2×C8×C127D412yes—
64Dic254—22no—
65Dic254—22no—
66Dic254—41no—
67Dic254—41no—
68C2×Dic127—22no—
69C2×Dic127—22no—
70C2×Dic127—22no—
71C2×Dic127—41no—
72C2×Dic127—41no—
73C2×Dic127—41no—
74C2×Dic127—42no—
75C2×Dic127Q1614yescomplemented
76C127⋊C8—22no—
77C23×C127C2×D411yescharacteristic
78Q8×C127—22no—
79Q8×C127—22no—
80Q8×C127—22no—
81Q8×C127—22no—
82Q8×C127—41no—
83Q8×C127—41no—
84Q8×C127—41no—
85Q8×C127—41no—
86C2×C4×C127—22no—
87C2×C4×C127—22no—
88C2×C4×C127—41no—
89C2×C4×C127—41no—
90C2×C4×C127—41no—
91C2×C4×C127—41no—
92C2×C4×C127C2×D411yescharacteristic
93C2×C4×C127C2×D411yescharacteristic
94C2×C4×C127C2×D411yescharacteristic
95C2×C4×C127C2×D411yescharacteristic
96C2×C4×C127C2×D411yescharacteristic
97C2×C4×C127#4111yescharacteristic
98C8×C127—22no—
99C22×C127#12011yescharacteristic
100C22×C127#12111yescharacteristic
101C22×C127#12711yescharacteristic
102C22×C127C2×Q1612yes—
103C22×C127#13712yes—
104C4×C127—42no—
105C4×C127—42no—
106C4×C127#12011yescharacteristic
107C4×C127#12011yescharacteristic
108C4×C127#12111yescharacteristic
109C4×C127#12111yescharacteristic
110Dic127—24no—
111Dic127—42no—
112Dic127—81no—
113C2×C127#39211yescharacteristic
114C2×C127#45112yes—
115C2×C127#46212yes—
116C2×C127#49411yescharacteristic
117C2×C127#49711yescharacteristic
118#2016—1271nomaximal, Sylow
119C127#201611yescharacteristic, complemented, Sylow
120C22×Q16—1271no—
121C22×Q16D12711yescharacteristic
122C2×C4⋊Q8—1271no—
123C2×#103—1271no—
124C2×#103—1271no—
125#391—1271no—
126#336—1271no—
127C22×Q8—1271no—
128C22×Q8C2×D12711yescharacteristic
129C22×Q8C2×D12711yescharacteristic
130C2×Q16—22no—
131C2×Q16—22no—
132C2×Q16—22no—
133C2×Q16—2542no—
134C2×Q16—2542no—
135C2×Q16—2542no—
136C22×C8—1271no—
137C22×C8C2×D12711yescharacteristic
138C4⋊Q8—2542no—
139C4⋊Q8—2542no—
140C2×C4⋊C4—1271no—
141C2×C4⋊C4—2542no—
142C2×C42—1271no—
143#103—2542no—
144#103—2542no—
145#95—2542no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—1272no—
157C2×Q8—2541no—
158C2×Q8—2541no—
159C2×Q8—2541no—
160C2×Q8—2541no—
161C2×Q8—2541no—
162C2×Q8C22⋊D12712yes—
163C2×Q8C22⋊D12712yes—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—1272no—
167C22×C4—2541no—
168C22×C4—2541no—
169C22×C4C22×D12711yescharacteristic
170Q16—28no—
171Q16—2548no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—1272no—
175C2×C8—2541no—
176C2×C8—2541no—
177C2×C8C22⋊D12712yes—
178C4⋊C4—2548no—
179C4⋊C4—2541no—
180C4⋊C4—2541no—
181C42—2541no—
182C42—2541no—
183C23D4×D12711yescharacteristic
184Q8—22no—
185Q8—22no—
186Q8—22no—
187Q8—22no—
188Q8—41no—
189Q8—41no—
190Q8—41no—
191Q8—41no—
192Q8—5081no—
193Q8—5081no—
194Q8—2542no—
195Q8—2542no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—1274no—
203C2×C4—5081no—
204C2×C4—5082no—
205C2×C4—5081no—
206C2×C4—5081no—
207C2×C4—2542no—
208C2×C4—2542no—
209C2×C4—2542no—
210C2×C4D4×D12711yescharacteristic
211C2×C4D4×D12711yescharacteristic
212C2×C4C2×C22⋊D12711yescharacteristic
213C2×C4C2×C22⋊D12711yescharacteristic
214C2×C4C2×C22⋊D12711yescharacteristic
215C2×C4#1452324711yescharacteristic
216C8—22no—
217C8—2542no—
218C22#1686117711yescharacteristic
219C22#1686118111yescharacteristic
220C22#1686120511yescharacteristic
221C22Q16×D12712yes—
222C22#1686124112yes—
223C4—42no—
224C4—42no—
225C4—10161no—
226C4—5082no—
227C4—2544no—
228C4#1686117711yescharacteristic
229C4#1686117811yescharacteristic
230C4#1686118011yescharacteristic
231C4#1686118111yescharacteristic
232C2#2254978411yescharacteristic
233C2#2255003012yes—
234C2#2255008412yes—
235C2#2255025411yescharacteristic
236C2#2255026911yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC22×Q16×C127C2
socle, prime radicalC23×C127C2×D4
hypercenterC22×Q16D127
hypocenterC127#2016
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
75. C2×Dic127Q16semidirect
119. C127#2016semidirect

Maximal subgroups

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

subgroupindex
C22×#145232382
C2×#168612072
C2×#168611342
C2×#168611332
#225497832
#225496302
C22×Q16×C1272
#2016127