Finite Group Databasebeta

← #30647207

Subgroups — #30647207

Order 16256.

756 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 209 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×#16861255C211yesmaximal, characteristic
2#22550642C212yesmaximal, complemented
3#22549860C211yesmaximal, characteristic, complemented
4#22549859C214yesmaximal, complemented
5C2×#16861125C212yesmaximal, complemented
6#100×D127C214yesmaximal, complemented
7#413×C127C211yesmaximal, characteristic, complemented
8#16861255—24no—
9#16861255—24no—
10C2×#14523249C2212yes—
11C2×#14523248C412yes—
12C2×#14523248C2211yescharacteristic
13C22×D508C412yes—
14C22×D508C2211yescharacteristic
15C2×D4×D127C414yes—
16C2×D4×D127C2212yes—
17#16861217—24no—
18#16861215—24no—
19C2×#14523232C2212yescomplemented
20M16×D127C2218yescomplemented
21#16861128C2212yescomplemented
22#16861126C2212yescomplemented
23#16861125C2218yescomplemented
24C2×#41×C127C2211yescharacteristic
25C2×M16×C127C2212yescomplemented
26#101×C127C2212yescomplemented
27#100×C127C2212yescomplemented
28C23×D127C2×C412yes—
29C23×D127C2×C414yes—
30#14523249—28no—
31#14523248—24no—
32#14523248—24no—
33#14523248D414yes—
34#14523248D414yes—
35C2×Dic254C2×C411yescharacteristic
36C2×D508—24no—
37C2×D508—24no—
38C2×D508—22no—
39C2×D508—22no—
40C2×D508C2×C414yes—
41C2×D508C2×C412yes—
42C2×D508C2×C411yescharacteristic
43C2×D508C2312yes—
44C2×C22⋊D127C2×C414yes—
45C2×C22⋊D127C2×C412yes—
46D4×D127—216no—
47D4×D127—28no—
48C2×C4×D127C2×C414yes—
49C2×C4×D127C2312yes—
50#14523232—22no—
51#14523232C2314yes—
52#14523231—28no—
53C2×C127⋊C8—22no—
54C8×D127—28no—
55#41×C127—22no—
56#41×C127—22no—
57C2×Q8×C127C2311yescharacteristic
58C2×D4×C127C2×C412yes—
59C2×D4×C127C2311yescharacteristic
60C22×C4×C127C2×C412yes—
61C22×C4×C127C2311yescharacteristic
62M16×C127—22no—
63M16×C127C2314yes—
64C2×C8×C127—22no—
65C22×D127—24no—
66C22×D127—28no—
67C22×D127—84no—
68C22×D127—82no—
69C22×D127C22×C414yes—
70C22×D127C22×C412yes—
71Dic254C22⋊C412yes—
72Dic254C22⋊C412yes—
73D508—28no—
74D508—22no—
75D508—22no—
76D508—24no—
77D508C22⋊C412yes—
78D508C22⋊C412yes—
79C22⋊D127—28no—
80C22⋊D127C22⋊C418yes—
81C2×Dic127C22×C412yes—
82C4×D127—28no—
83C4×D127C2×D418yes—
84C127⋊C8—24no—
85C23×C127C22×C412yes—
86C23×C127C22×C411yescharacteristic
87Q8×C127—22no—
88D4×C127—24no—
89D4×C127—22no—
90C2×C4×C127—22no—
91C2×C4×C127—22no—
92C2×C4×C127C22×C412yes—
93C2×C4×C127C22×C411yescharacteristic
94C2×C4×C127C2×D412yes—
95C2×C4×C127C2×D412yes—
96C2×C4×C127C2411yescharacteristic
97C8×C127—24no—
98C22×C127—22no—
99C22×C127—41no—
100C22×C127—42no—
101C22×C127C2×C22⋊C412yes—
102C22×C127C23×C411yescharacteristic
103C2×D127—24no—
104C2×D127—42no—
105C2×D127—44no—
106C2×D127—88no—
107C2×D127—84no—
108C2×D127C2×C22⋊C414yes—
109C4×C127—22no—
110C4×C127C2×C22⋊C411yescharacteristic
111C4×C127C2×C22⋊C411yescharacteristic
112C4×C127C22×D412yes—
113Dic127C2×C22⋊C414yes—
114C2×C127—21no—
115C2×C127—22no—
116C2×C127—42no—
117C2×C127C22×C22⋊C411yescharacteristic
118D127—24no—
119D127—44no—
120#2888—1271nomaximal, Sylow
121C127#288811yescharacteristic, complemented, Sylow
122C2×H(5,𝔽2)—1271no—
123#568—1272no—
124#413—1271no—
125#413—1274no—
126#413D12711yescharacteristic, complemented
127C2×#100—1272no—
128C2×#100—1274no—
129H(5,𝔽2)—2544no—
130H(5,𝔽2)—2544no—
131C2×#41—1272no—
132C2×#41—1272no—
133C2×#41—1271no—
134C2×#41C2×D12711yescharacteristic
135C22×D4—1274no—
136C22×D4—1272no—
137C22×D4—1272no—
138C22×D4—1271no—
139#131—2544no—
140#131—2544no—
141C2×M16—1272no—
142C2×M16—1278no—
143C2×M16C2×D12712yescomplemented
144#101—1272no—
145#101C2×D12712yescomplemented
146#100—1272no—
147#100—1278no—
148#100C2×D12712yescomplemented
149C24—1274no—
150C24—1272no—
151#41—22no—
152#41—22no—
153#41—1274no—
154#41—1274no—
155#41—2544no—
156#41—2544no—
157#41—2548no—
158C2×Q8—1271no—
159C2×Q8C22×D12711yescharacteristic
160C2×D4—1272no—
161C2×D4—1274no—
162C2×D4—1271no—
163C2×D4—1274no—
164C2×D4—1272no—
165C2×D4—1272no—
166C2×D4—2544no—
167C2×D4—2544no—
168C2×D4—2548no—
169C2×D4—25416no—
170C2×D4—2542no—
171C2×D4—2542no—
172C2×D4C4×D12712yes—
173C2×D4C22×D12711yescharacteristic
174C22×C4—1272no—
175C22×C4—1274no—
176C22×C4C4×D12712yes—
177C22×C4C22×D12711yescharacteristic
178M16—22no—
179M16—1274no—
180M16—2548no—
181M16—2542no—
182M16C22×D12714yes—
183C2×C8—22no—
184C2×C8—2548no—
185C2×C8—2542no—
186C23—1272no—
187C23—1274no—
188C23—10162no—
189C23—10164no—
190C23—2548no—
191C23—2544no—
192C23C2×C4×D12712yes—
193C23C2×C4×D12711yescharacteristic
194Q8—22no—
195Q8—1272no—
196Q8—1272no—
197D4—24no—
198D4—22no—
199D4—1278no—
200D4—1272no—
201D4—1272no—
202D4—2548no—
203D4—2548no—
204D4—2544no—
205D4—2542no—
206D4—2542no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—1272no—
210C2×C4—1278no—
211C2×C4—2548no—
212C2×C4C2×C4×D12712yes—
213C2×C4C2×C4×D12711yescharacteristic
214C2×C4D4×D12712yes—
215C2×C4D4×D12712yes—
216C2×C4C23×D12711yescharacteristic
217C8—24no—
218C8—2544no—
219C22—22no—
220C22—41no—
221C22—42no—
222C22—1274no—
223C22—10164no—
224C22—10168no—
225C22—5084no—
226C22—5082no—
227C22—2544no—
228C22C22⋊C4×D12712yes—
229C22C22×C4×D12711yescharacteristic
230C4—22no—
231C4—1274no—
232C4C22⋊C4×D12711yescharacteristic
233C4C22⋊C4×D12711yescharacteristic
234C4C2×D4×D12712yes—
235C2—21no—
236C2—22no—
237C2—42no—
238C2—5084no—
239C2—2544no—
240C2C2×C22⋊C4×D12711yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D127
commutatorC22×C127C23×C4
FrattiniC2×C4C23×D127
Fitting#413×C127C2
socleC2×C127C22×C22⋊C4
prime radicalC2×#16861255C2
cosocleC2×C4×C127C24
hypercenter#413D127
hypocenterC127#2888
LCSG ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #22550642C2semidirect
3. #22549860C2semidirect
4. #22549859C2semidirect
5. C2×#16861125C2semidirect
6. #100×D127C2semidirect
7. #413×C127C2semidirect
19. C2×#14523232C22semidirect
20. M16×D127C22semidirect
21. #16861128C22semidirect
22. #16861126C22semidirect
23. #16861125C22semidirect
25. C2×M16×C127C22semidirect
26. #101×C127C22semidirect
27. #100×C127C22semidirect
121. C127#2888semidirect
126. #413D127semidirect
143. C2×M16C2×D127semidirect
145. #101C2×D127semidirect
148. #100C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612552
#225506422
#225498602
#225498592
C2×#168611252
#100×D1272
#413×C1272
#2888127