Finite Group Databasebeta

← #30639238

Subgroups — #30639238

Order 16256.

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

normal subgroups only

This group has 332 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
1C2×#16861234C211yesmaximal, characteristic
2#22549877C211yesmaximal, characteristic, complemented
3C2×#16861131C211yesmaximal, characteristic, complemented
4C2×#16861125C211yesmaximal, characteristic, complemented
5#22549644C211yesmaximal, characteristic, complemented
6#22549641C211yesmaximal, characteristic, complemented
7C2×#131×C127C211yesmaximal, characteristic, complemented
8C22×D508C2211yescharacteristic
9#16861234—22no—
10#16861234C414yescomplemented
11C2×#14523235—21no—
12C2×#14523233—21no—
13C2×#14523232C2211yescharacteristic
14C2×#14523232C2211yescharacteristic
15C2×C4⋊Dic127C2211yescharacteristic
16#16861133—21no—
17#16861131—21no—
18#16861131—21no—
19#16861130—21no—
20#16861125—22no—
21C2×#41×C127C2211yescharacteristic
22#131×C127—22no—
23#131×C127—22no—
24C2×M16×C127—21no—
25C2×M16×C127C2211yescharacteristic
26C22×C8×C127—21no—
27C22×C8×C127C2211yescharacteristic
28C23×D127—21no—
29C2×D508—21no—
30C2×D508—21no—
31C2×D508—21no—
32C2×D508—21no—
33C2×D508—21no—
34C2×D508C2×C412yes—
35C22×Dic127—21no—
36#14523235—24no—
37#14523233—24no—
38#14523232—21no—
39#14523232—21no—
40#14523232—21no—
41#14523232C2×C412yes—
42C2×C127⋊C8—21no—
43C2×C127⋊C8—21no—
44C4⋊Dic127—21no—
45C4⋊Dic127—21no—
46#41×C127—21no—
47#41×C127—21no—
48#41×C127C2×C412yes—
49#41×C127D412yes—
50C2×Q8×C127D411yescharacteristic
51C2×D4×C127—21no—
52C2×D4×C127D411yescharacteristic
53C22×C4×C127—21no—
54C22×C4×C127C2311yescharacteristic
55M16×C127—24no—
56M16×C127—21no—
57M16×C127D412yes—
58C2×C8×C127—24no—
59C2×C8×C127—21no—
60C2×C8×C127—21no—
61C2×C8×C127—21no—
62C2×C8×C127D411yescharacteristic
63C2×C8×C127D411yescharacteristic
64C22×D127—22no—
65C22×D127—41no—
66C22×D127—41no—
67C22×D127—81no—
68C22×D127—81no—
69D508—22no—
70D508—22no—
71D508—41no—
72D508—41no—
73C2×Dic127—41no—
74C2×Dic127—81no—
75C127⋊C8—22no—
76C127⋊C8—41no—
77C23×C127—21no—
78C23×C127#4111yescharacteristic
79Q8×C127—21no—
80Q8×C127C22⋊C412yes—
81D4×C127—22no—
82D4×C127—22no—
83D4×C127—21no—
84D4×C127C22⋊C412yes—
85C2×C4×C127—21no—
86C2×C4×C127—22no—
87C2×C4×C127—22no—
88C2×C4×C127C22×C411yescharacteristic
89C2×C4×C127C2×D411yescharacteristic
90C2×C4×C127C2×D411yescharacteristic
91C2×C4×C127C2×D411yescharacteristic
92C2×C4×C127C2×D411yescharacteristic
93C2×C4×C127#4111yescharacteristic
94C8×C127—21no—
95C8×C127—22no—
96C8×C127—21no—
97C22×C127—21no—
98C22×C127—21no—
99C22×C127—22no—
100C22×C127—41no—
101C22×C127—41no—
102C22×C127C4×D411yescharacteristic
103C22×C127#12111yescharacteristic
104C22×C127#12311yescharacteristic
105C2×D127—42no—
106C2×D127—82no—
107C2×D127—81no—
108C2×D127—81no—
109C2×D127—82no—
110C4×C127—22no—
111C4×C127C2×C22⋊C411yescharacteristic
112C4×C127C4×D411yescharacteristic
113C4×C127#12011yescharacteristic
114C4×C127#12111yescharacteristic
115Dic127—81no—
116C2×C127—21no—
117C2×C127—21no—
118C2×C127—42no—
119C2×C127#38611yescharacteristic
120C2×C127#47111yescharacteristic
121C2×C127#47211yescharacteristic
122D127—82no—
123#1907—1271nomaximal, Sylow
124C127#190711yescharacteristic, complemented, Sylow
125C2×AGL(1,ℤ/8)—1271no—
126C2×#131D12711yescharacteristic, complemented
127#417—1271no—
128C2×#102—1271no—
129C2×#100—1271no—
130#343—1271no—
131#341—1271no—
132C2×#41C2×D12711yescharacteristic
133C22×D4—1271no—
134AGL(1,ℤ/8)—1274no—
135AGL(1,ℤ/8)—2542no—
136C2×SD16—2541no—
137C2×D8—2541no—
138#131—22no—
139#131—22no—
140C2×M16—21no—
141C2×M16—1271no—
142C2×M16—1271no—
143C2×M16C2×D12711yescharacteristic
144C22×C8—21no—
145C22×C8C2×D12711yescharacteristic
146C2×C4⋊C4—1271no—
147#103—2541no—
148#102—2541no—
149#102—2541no—
150#102—2541no—
151#100—2542no—
152C24—2541no—
153#41—21no—
154#41—21no—
155#41C4×D12712yes—
156#41C22⋊D12712yes—
157C2×Q8D50811yescharacteristic
158C2×D4—21no—
159C2×D4—1272no—
160C2×D4—2541no—
161C2×D4—2541no—
162C2×D4—2541no—
163C2×D4—2541no—
164C2×D4—2541no—
165C2×D4D50811yescharacteristic
166C22×C4—21no—
167C22×C4—2541no—
168C22×C4C22×D12711yescharacteristic
169SD16—2544no—
170D8—2544no—
171M16—24no—
172M16—21no—
173M16—1272no—
174M16—2541no—
175M16—2541no—
176M16—2541no—
177M16C22⋊D12712yes—
178C2×C8—24no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—2541no—
183C2×C8—2541no—
184C2×C8C22⋊D12711yescharacteristic
185C2×C8C22⋊D12711yescharacteristic
186C4⋊C4—2541no—
187C4⋊C4—2541no—
188C23—21no—
189C23—10161no—
190C23—10161no—
191C23—5081no—
192C23—5081no—
193C23—2542no—
194C23#1452324811yescharacteristic
195Q8—21no—
196Q8(C2×D127)⋊C412yes—
197D4—22no—
198D4—22no—
199D4—21no—
200D4—5081no—
201D4—5081no—
202D4—2542no—
203D4—2542no—
204D4(C2×D127)⋊C412yes—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—10161no—
209C2×C4—5081no—
210C2×C4C2×C4×D12711yescharacteristic
211C2×C4C2×C22⋊D12711yescharacteristic
212C2×C4C2×C22⋊D12711yescharacteristic
213C2×C4C2×C22⋊D12711yescharacteristic
214C2×C4C2×D50811yescharacteristic
215C2×C4#1452324811yescharacteristic
216C8—21no—
217C8—22no—
218C8—21no—
219C8—5081no—
220C8—2542no—
221C22—21no—
222C22—21no—
223C22—22no—
224C22—41no—
225C22—41no—
226C22—10162no—
227C22—10162no—
228C22—10161no—
229C22—10161no—
230C22—5082no—
231C22C4×C22⋊D12711yescharacteristic
232C22#1686118311yescharacteristic
233C22#1686119211yescharacteristic
234C4—22no—
235C4—10161no—
236C4C2×(C2×D127)⋊C411yescharacteristic
237C4C4×C22⋊D12711yescharacteristic
238C4#1686117811yescharacteristic
239C4#1686118311yescharacteristic
240C2—21no—
241C2—21no—
242C2—42no—
243C2—10162no—
244C2#2254975511yescharacteristic
245C2#2255013911yescharacteristic
246C2#2255014611yescharacteristic
subgroupG/H
centerC22#16861183
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#131×C127C2
socleC22×C127#121
prime radicalC2×#16861234C2
cosocleC22×C4×C127C23
hypercenterC2×#131D127
hypocenterC127#1907
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#131

Complements

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

NH ≅ G/Ntype
2. #22549877C2semidirect
3. C2×#16861131C2semidirect
4. C2×#16861125C2semidirect
5. #22549644C2semidirect
6. #22549641C2semidirect
7. C2×#131×C127C2semidirect
10. #16861234C4semidirect
124. C127#1907semidirect
126. C2×#131D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612342
#225498772
C2×#168611312
C2×#168611252
#225496442
#225496412
C2×#131×C1272
#1907127