Finite Group Databasebeta

← #30639104

Subgroups — #30639104

Order 16256.

370 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 89 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#22550359C211yesmaximal, characteristic, complemented
3#22549882C211yesmaximal, characteristic, complemented
4#22549778C211yesmaximal, characteristic, complemented
5#22549644C211yesmaximal, characteristic, complemented
6#22549636C211yesmaximal, characteristic, complemented
7C2×#104×C127C211yesmaximal, characteristic, complemented
8C22×D508C2211yescharacteristic
9#16861234—22no—
10#16861234C414yescomplemented
11C2×#14523235—21no—
12C2×#14523233—21no—
13C2×#14523232C2211yescharacteristic
14Q8×Dic127—21no—
15D4×Dic127—22no—
16D4×Dic127—21no—
17#16861168—21no—
18#16861168C2211yescharacteristic
19C2×C4⋊Dic127—21no—
20C2×C4⋊Dic127C2211yescharacteristic
21C2×(C2×D127)⋊C4—22no—
22#16861131—21no—
23#16861131—21no—
24C2×#41×C127C2211yescharacteristic
25C2×M16×C127C2211yescharacteristic
26C2×C42×C127C2211yescharacteristic
27#104×C127—22no—
28C23×D127—21no—
29C2×D508—21no—
30C2×D508—21no—
31C2×D508—21no—
32C2×D508—21no—
33C2×D508—21no—
34C2×D508C2×C412yescomplemented
35C22×Dic127—22no—
36C22×Dic127D412yes—
37#14523235—24no—
38#14523233—24no—
39#14523232—21no—
40#14523232C2×C412yescomplemented
41C2×C127⋊C8—21no—
42C4⋊Dic127—22no—
43C4⋊Dic127—21no—
44C4⋊Dic127—21no—
45C4⋊Dic127D411yescharacteristic
46C4⋊Dic127D411yescharacteristic
47C22⋊Dic127—22no—
48C22⋊Dic127—21no—
49(C2×D127)⋊C4—44no—
50C4×Dic127—22no—
51C4×Dic127—21no—
52#41×C127—21no—
53#41×C127—21no—
54#41×C127C2×C412yes—
55#41×C127D412yes—
56C2×Q8×C127D411yescharacteristic
57C2×D4×C127—21no—
58C2×D4×C127D411yescharacteristic
59C22×C4×C127—21no—
60C22×C4×C127—21no—
61C22×C4×C127C2311yescharacteristic
62M16×C127—21no—
63M16×C127—21no—
64C2×C8×C127—21no—
65C42×C127—21no—
66C42×C127—21no—
67C22×D127—22no—
68C22×D127—42no—
69C22×D127—42no—
70C22×D127—41no—
71C22×D127—41no—
72D508—22no—
73D508—22no—
74D508—41no—
75D508—41no—
76C2×Dic127—21no—
77C2×Dic127—22no—
78C2×Dic127—21no—
79C2×Dic127—42no—
80C2×Dic127—82no—
81C127⋊C8—22no—
82C23×C127—21no—
83C23×C127C2×D411yescharacteristic
84Q8×C127—21no—
85Q8×C127C22⋊C412yescomplemented
86D4×C127—22no—
87D4×C127—22no—
88D4×C127—21no—
89D4×C127C22⋊C412yescomplemented
90C2×C4×C127—21no—
91C2×C4×C127—22no—
92C2×C4×C127—22no—
93C2×C4×C127—22no—
94C2×C4×C127—41no—
95C2×C4×C127—41no—
96C2×C4×C127C22×C411yescharacteristic
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127C2×D411yescharacteristic
99C2×C4×C127C2×D411yescharacteristic
100C2×C4×C127#4111yescharacteristic
101C2×C4×C127#4111yescharacteristic
102C8×C127—41no—
103C22×C127—21no—
104C22×C127—21no—
105C22×C127—22no—
106C22×C127—41no—
107C22×C127—41no—
108C22×C127C4×D411yescharacteristic
109C22×C127#12011yescharacteristic
110C22×C127#12111yescharacteristic
111C2×D127—44no—
112C2×D127—42no—
113C2×D127—82no—
114C2×D127—81no—
115C2×D127—81no—
116C4×C127—22no—
117C4×C127—42no—
118C4×C127C2×C22⋊C411yescharacteristic
119C4×C127C4×D411yescharacteristic
120C4×C127#12111yescharacteristic
121C4×C127#12311yescharacteristic
122Dic127—42no—
123Dic127—41no—
124Dic127—41no—
125C2×C127—21no—
126C2×C127—21no—
127C2×C127—42no—
128C2×C127#38611yescharacteristic
129C2×C127#45311yescharacteristic
130C2×C127#45411yescharacteristic
131D127—82no—
132#1883—1271nomaximal, Sylow
133C127#188311yescharacteristic, complemented, Sylow
134C2×AGL(1,ℤ/8)—1271no—
135#520—1271no—
136C2×#104D12711yescharacteristic, complemented
137#418—1271no—
138#390—1271no—
139#343—1271no—
140#339—1271no—
141C2×#41C2×D12711yescharacteristic
142C22×D4—1271no—
143AGL(1,ℤ/8)—1274no—
144AGL(1,ℤ/8)—2542no—
145C2×SD16—2541no—
146C2×D8—2541no—
147C2×M16—1271no—
148C2×M16C2×D12711yescharacteristic
149C4×Q8—2541no—
150C4×D4—2542no—
151C4×D4—2541no—
152#117—1271no—
153#117—2541no—
154C2×C4⋊C4—1271no—
155C2×C4⋊C4—2541no—
156C2×C22⋊C4—2542no—
157C2×C42C2×D12711yescharacteristic
158#104—22no—
159#102—2541no—
160#102—2541no—
161C24—2541no—
162#41—21no—
163#41—21no—
164#41C4×D12712yes—
165#41C22⋊D12712yes—
166C2×Q8D50811yescharacteristic
167C2×D4—21no—
168C2×D4—1272no—
169C2×D4—2541no—
170C2×D4—2541no—
171C2×D4—2541no—
172C2×D4—2541no—
173C2×D4—2541no—
174C2×D4D50811yescharacteristic
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—1272no—
178C22×C4—2542no—
179C22×C4C22×D12711yescharacteristic
180SD16—2544no—
181D8—2544no—
182M16—21no—
183M16—21no—
184M16—1272no—
185M16—2541no—
186C2×C8—21no—
187C2×C8—2541no—
188C4⋊C4—1271no—
189C4⋊C4—1271no—
190C4⋊C4—2542no—
191C4⋊C4—2541no—
192C4⋊C4—2541no—
193C22⋊C4—5084no—
194C22⋊C4—2542no—
195C22⋊C4—2541no—
196C42—21no—
197C42—21no—
198C42—2542no—
199C42—2541no—
200C23—21no—
201C23—5082no—
202C23—5082no—
203C23—5081no—
204C23—5081no—
205C23—2542no—
206C23D4×D12711yescharacteristic
207Q8—21no—
208Q8(C2×D127)⋊C412yescomplemented
209D4—22no—
210D4—22no—
211D4—21no—
212D4—5081no—
213D4—5081no—
214D4—2542no—
215D4—2542no—
216D4(C2×D127)⋊C412yescomplemented
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—10162no—
224C2×C4—5082no—
225C2×C4—2541no—
226C2×C4—2541no—
227C2×C4—2542no—
228C2×C4C2×C4×D12711yescharacteristic
229C2×C4D4×D12711yescharacteristic
230C2×C4C2×C22⋊D12711yescharacteristic
231C2×C4C2×D50811yescharacteristic
232C2×C4#1452324711yescharacteristic
233C2×C4#1452324711yescharacteristic
234C8—41no—
235C8—2542no—
236C22—21no—
237C22—21no—
238C22—22no—
239C22—41no—
240C22—41no—
241C22—10162no—
242C22—10161no—
243C22—10161no—
244C22—5084no—
245C22—5082no—
246C22(C22⋊D127)⋊C411yescharacteristic
247C22#1686117911yescharacteristic
248C22#1686118111yescharacteristic
249C4—22no—
250C4—42no—
251C4—5081no—
252C4—5082no—
253C4—5081no—
254C4C2×(C2×D127)⋊C411yescharacteristic
255C4(C22⋊D127)⋊C411yescharacteristic
256C4#1686118111yescharacteristic
257C4#1686119311yescharacteristic
258C2—21no—
259C2—21no—
260C2—42no—
261C2—10162no—
262C2#2254975311yescharacteristic
263C2#2255004511yescharacteristic
264C2#2255005011yescharacteristic
subgroupG/H
centerC22#16861179
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#104×C127C2
socleC22×C127#120
prime radicalC2×#16861234C2
cosocleC22×C4×C127C23
hypercenterC2×#104D127
hypocenterC127#1883
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. #22550359C2semidirect
3. #22549882C2semidirect
4. #22549778C2semidirect
5. #22549644C2semidirect
6. #22549636C2semidirect
7. C2×#104×C127C2semidirect
10. #16861234C4semidirect
34. C2×D508C2×C4semidirect
40. #14523232C2×C4semidirect
85. Q8×C127C22⋊C4semidirect
89. D4×C127C22⋊C4semidirect
133. C127#1883semidirect
136. C2×#104D127semidirect
208. Q8(C2×D127)⋊C4semidirect
216. D4(C2×D127)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#168612342
#225503592
#225498822
#225497782
#225496442
#225496362
C2×#104×C1272
#1883127