Finite Group Databasebeta

← #30639235

Subgroups — #30639235

Order 16256.

332 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C2×#16861235C211yesmaximal, characteristic
2C2×#16861217C211yesmaximal, characteristic, complemented
3#22549878C211yesmaximal, characteristic, complemented
4C2×#16861130C211yesmaximal, characteristic, complemented
5#22549644C211yesmaximal, characteristic, complemented
6#22549640C211yesmaximal, characteristic, complemented
7C2×#100×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861235—22no—
10#16861235C414yescomplemented
11C2×#14523236—21no—
12C2×#14523233—21no—
13#16861217—22no—
14#16861217—22no—
15C2×#14523232C2211yescharacteristic
16C2×#14523231—21no—
17C22×C127⋊C8C2211yescharacteristic
18C2×C8×D127—21no—
19C2×C4⋊Dic127C2211yescharacteristic
20#16861134—21no—
21#16861131—21no—
22#16861130—21no—
23#16861130—21no—
24C22×D4×C127C2211yescharacteristic
25C2×M16×C127C2211yescharacteristic
26C2×M16×C127C2211yescharacteristic
27#100×C127—22no—
28#14523248—21no—
29#14523248—21no—
30#14523248C2×C412yes—
31#14523248D412yes—
32C2×Dic254D411yescharacteristic
33C2×D508D411yescharacteristic
34C2×C22⋊D127—21no—
35C22×Dic127—21no—
36C2×C4×D127—21no—
37#14523236—24no—
38#14523233—24no—
39#14523232—21no—
40#14523232C2×C412yes—
41#14523231—24no—
42C2×C127⋊C8—21no—
43C2×C127⋊C8—21no—
44C2×C127⋊C8—21no—
45C2×C127⋊C8D411yescharacteristic
46C2×C127⋊C8D411yescharacteristic
47C8×D127—24no—
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×C127C2×C412yes—
57C22×C4×C127C2311yescharacteristic
58M16×C127—21no—
59M16×C127—21no—
60M16×C127—21no—
61M16×C127D412yes—
62C2×C8×C127—21no—
63C2×C8×C127—21no—
64C22×D127—21no—
65Dic254—21no—
66Dic254C22⋊C412yes—
67D508—21no—
68D508C22⋊C412yes—
69C22⋊D127—22no—
70C22⋊D127—22no—
71C2×Dic127—21no—
72C2×Dic127—41no—
73C2×Dic127—81no—
74C4×D127—22no—
75C4×D127—22no—
76C127⋊C8—22no—
77C127⋊C8—21no—
78C127⋊C8—21no—
79C23×C127—22no—
80C23×C127—41no—
81C23×C127—41no—
82C23×C127—81no—
83C23×C127—81no—
84C23×C127#4111yescharacteristic
85D4×C127—22no—
86D4×C127—22no—
87D4×C127—41no—
88D4×C127—41no—
89C2×C4×C127C22×C411yescharacteristic
90C2×C4×C127C2×D411yescharacteristic
91C2×C4×C127C2×D411yescharacteristic
92C2×C4×C127C2×D411yescharacteristic
93C2×C4×C127C2×D411yescharacteristic
94C2×C4×C127#4111yescharacteristic
95C8×C127—22no—
96C8×C127—41no—
97C22×C127—21no—
98C22×C127—21no—
99C22×C127—42no—
100C22×C127—82no—
101C22×C127—82no—
102C22×C127—81no—
103C22×C127—81no—
104C22×C127C4×D411yescharacteristic
105C22×C127#12111yescharacteristic
106C22×C127#12311yescharacteristic
107C2×D127—22no—
108C2×D127—41no—
109C2×D127—41no—
110C4×C127C2×C22⋊C411yescharacteristic
111C4×C127C4×D411yescharacteristic
112C4×C127#12011yescharacteristic
113C4×C127#12111yescharacteristic
114Dic127—22no—
115Dic127—81no—
116C2×C127—21no—
117C2×C127—21no—
118C2×C127—82no—
119C2×C127#38611yescharacteristic
120C2×C127#47111yescharacteristic
121C2×C127#47211yescharacteristic
122D127—42no—
123#1907—1271nomaximal, Sylow
124C127#190711yescharacteristic, complemented, Sylow
125C2×AGL(1,ℤ/8)—1271no—
126C2×#131—1271no—
127#417—1271no—
128C2×#102—1271no—
129C2×#100D12711yescharacteristic, complemented
130#343—1271no—
131#341—1271no—
132C2×#41—1271no—
133C22×D4C2×D12711yescharacteristic
134AGL(1,ℤ/8)—1274no—
135AGL(1,ℤ/8)—2542no—
136C2×SD16—2541no—
137C2×D8—2541no—
138#131—2542no—
139#131—2542no—
140C2×M16—1271no—
141C2×M16—2541no—
142C2×M16C2×D12711yescharacteristic
143C2×M16C2×D12711yescharacteristic
144C22×C8—1271no—
145C22×C8—2541no—
146C2×C4⋊C4—1271no—
147#103—2541no—
148#102—2541no—
149#102—2541no—
150#102—2541no—
151#100—22no—
152C24—21no—
153#41—1272no—
154#41—1272no—
155#41—2541no—
156#41—2541no—
157C2×Q8—1271no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—1271no—
164C2×D4—2541no—
165C2×D4C4×D12712yes—
166C22×C4—2541no—
167C22×C4—2541no—
168C22×C4C22×D12711yescharacteristic
169SD16—2544no—
170D8—2544no—
171M16—21no—
172M16—21no—
173M16—21no—
174M16—1272no—
175M16—2544no—
176M16—2541no—
177M16C22⋊D12712yes—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—1271no—
181C2×C8—1271no—
182C2×C8—2544no—
183C2×C8—2541no—
184C2×C8—2541no—
185C2×C8—2541no—
186C4⋊C4—2541no—
187C4⋊C4—2541no—
188C23—22no—
189C23—41no—
190C23—41no—
191C23—81no—
192C23—81no—
193C23—2541no—
194C23#1452324811yescharacteristic
195Q8—1272no—
196Q8—2541no—
197D4—22no—
198D4—22no—
199D4—41no—
200D4—41no—
201D4—1272no—
202D4—2542no—
203D4—2542no—
204D4—2541no—
205C2×C4—10161no—
206C2×C4—5081no—
207C2×C4—2541no—
208C2×C4—2542no—
209C2×C4—2542no—
210C2×C4C2×C4×D12711yescharacteristic
211C2×C4D4×D12711yescharacteristic
212C2×C4D4×D12711yescharacteristic
213C2×C4D4×D12711yescharacteristic
214C2×C4C2×C22⋊D12711yescharacteristic
215C2×C4#1452324711yescharacteristic
216C8—22no—
217C8—41no—
218C8—2542no—
219C8—2541no—
220C8—2541no—
221C22—21no—
222C22—21no—
223C22—42no—
224C22—82no—
225C22—82no—
226C22—81no—
227C22—81no—
228C22—5081no—
229C22—5081no—
230C22—2542no—
231C22C4×C22⋊D12711yescharacteristic
232C22#1686118211yescharacteristic
233C22#1686119111yescharacteristic
234C4—10161no—
235C4—2542no—
236C4C22⋊C4×D12711yescharacteristic
237C4(C22⋊D127)⋊C411yescharacteristic
238C4#1686117711yescharacteristic
239C4#1686118011yescharacteristic
240C2—21no—
241C2—21no—
242C2—82no—
243C2—5082no—
244C2#2254975211yescharacteristic
245C2#2255013811yescharacteristic
246C2#2255014511yescharacteristic
subgroupG/H
centerC22#16861182
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#100×C127C2
socleC22×C127#121
prime radicalC2×#16861235C2
cosocleC22×C4×C127C23
hypercenterC2×#100D127
hypocenterC127#1907
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
2. C2×#16861217C2semidirect
3. #22549878C2semidirect
4. C2×#16861130C2semidirect
5. #22549644C2semidirect
6. #22549640C2semidirect
7. C2×#100×C127C2semidirect
10. #16861235C4semidirect
124. C127#1907semidirect
129. C2×#100D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612352
C2×#168612172
#225498782
C2×#168611302
#225496442
#225496402
C2×#100×C1272
#1907127