Finite Group Databasebeta

← #30647211

Subgroups — #30647211

Order 16256.

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

normal subgroups only

This group has 704 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×#16861256C211yesmaximal, characteristic
2#22550642C212yesmaximal, complemented
3#22549860C211yesmaximal, characteristic, complemented
4#22549857C214yesmaximal, complemented
5C2×#16861127C212yesmaximal, complemented
6#101×D127C214yesmaximal, complemented
7#413×C127C211yesmaximal, characteristic, complemented
8#16861256—24no—
9#16861256—24no—
10C2×#14523248C412yes—
11C2×#14523248C2211yescharacteristic
12C2×#14523247C414yes—
13C2×#14523247C2212yes—
14C22×Dic254C412yes—
15C22×Dic254C2211yescharacteristic
16C2×Q8×D127C2212yes—
17#16861217—24no—
18#16861215—24no—
19C2×#14523232C2212yes—
20M16×D127C2218yes—
21#16861128C2212yes—
22#16861127C2218yes—
23#16861126C2212yes—
24C2×#41×C127C2211yescharacteristic
25C2×M16×C127C2212yes—
26#101×C127C2212yes—
27#100×C127C2212yes—
28#14523248—24no—
29#14523248—24no—
30#14523248D414yes—
31#14523248D414yes—
32#14523247—216no—
33#14523247—28no—
34C2×Dic254—24no—
35C2×Dic254—24no—
36C2×Dic254—22no—
37C2×Dic254—22no—
38C2×Dic254C2×C414yes—
39C2×Dic254C2×C412yes—
40C2×Dic254C2×C411yescharacteristic
41C2×Dic254C2312yes—
42Q8×D127—28no—
43C2×D508C2×C411yescharacteristic
44C2×C22⋊D127C2×C414yes—
45C2×C22⋊D127C2×C412yes—
46C22×Dic127C2×C414yes—
47C22×Dic127C2×C412yes—
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×D127C22×C412yes—
66Dic254—22no—
67Dic254—22no—
68Dic254—24no—
69Dic254—28no—
70Dic254C22⋊C412yes—
71Dic254C22⋊C412yes—
72D508C22⋊C412yes—
73D508C22⋊C412yes—
74C22⋊D127—28no—
75C22⋊D127C22⋊C418yes—
76C2×Dic127—24no—
77C2×Dic127—28no—
78C2×Dic127C22×C414yes—
79C2×Dic127C22×C412yes—
80C4×D127—28no—
81C4×D127C2×D418yes—
82C127⋊C8—24no—
83C23×C127C22×C412yes—
84C23×C127C22×C411yescharacteristic
85Q8×C127—22no—
86D4×C127—24no—
87D4×C127—22no—
88C2×C4×C127—22no—
89C2×C4×C127—22no—
90C2×C4×C127C22×C412yes—
91C2×C4×C127C22×C411yescharacteristic
92C2×C4×C127C2×D412yes—
93C2×C4×C127C2×D412yes—
94C2×C4×C127C2411yescharacteristic
95C8×C127—24no—
96C22×C127—22no—
97C22×C127—41no—
98C22×C127—42no—
99C22×C127C2×C22⋊C412yes—
100C22×C127C23×C411yescharacteristic
101C2×D127—42no—
102C2×D127C2×C22⋊C414yes—
103C4×C127—22no—
104C4×C127C2×C22⋊C411yescharacteristic
105C4×C127C2×C22⋊C411yescharacteristic
106C4×C127C22×D412yes—
107Dic127—24no—
108Dic127C2×C22⋊C414yes—
109C2×C127—21no—
110C2×C127—22no—
111C2×C127—42no—
112C2×C127C22×C22⋊C411yescharacteristic
113D127—24no—
114#2889—1271nomaximal, Sylow
115C127#288911yescharacteristic, complemented, Sylow
116C2×2−1+4—1271no—
117#568—1272no—
118#413—1271no—
119#413—1274no—
120#413D12711yescharacteristic, complemented
121C2×#101—1272no—
122C2×#101—1274no—
1232−1+4—2544no—
1242−1+4—2544no—
125C2×#41—1274no—
126C2×#41—1272no—
127C2×#41—1272no—
128C2×#41—1271no—
129C2×#41C2×D12711yescharacteristic
130C22×Q8—1272no—
131C22×Q8—1272no—
132C22×Q8—1271no—
133#131—2544no—
134#131—2544no—
135C2×M16—1272no—
136C2×M16—1278no—
137C2×M16C2×D12712yes—
138#101—1272no—
139#101—1278no—
140#101C2×D12712yes—
141#100—1272no—
142#100C2×D12712yes—
143#41—22no—
144#41—22no—
145#41—1274no—
146#41—1274no—
147#41—2544no—
148#41—2544no—
149#41—2548no—
150#41—25416no—
151C2×Q8—1271no—
152C2×Q8—1274no—
153C2×Q8—1272no—
154C2×Q8—1272no—
155C2×Q8—2544no—
156C2×Q8—2544no—
157C2×Q8—2548no—
158C2×Q8—2542no—
159C2×Q8—2542no—
160C2×Q8C22×D12711yescharacteristic
161C2×D4—1272no—
162C2×D4—1274no—
163C2×D4—1271no—
164C2×D4C4×D12712yes—
165C2×D4C22×D12711yescharacteristic
166C22×C4—1272no—
167C22×C4—1274no—
168C22×C4—1272no—
169C22×C4—1274no—
170C22×C4C4×D12712yes—
171C22×C4C22×D12711yescharacteristic
172M16—22no—
173M16—1274no—
174M16—2548no—
175M16—2542no—
176M16C22×D12714yes—
177C2×C8—22no—
178C2×C8—2548no—
179C2×C8—2542no—
180C23—1272no—
181C23C2×C4×D12712yes—
182C23C2×C4×D12711yescharacteristic
183Q8—22no—
184Q8—1272no—
185Q8—1272no—
186Q8—2548no—
187Q8—2544no—
188Q8—2542no—
189Q8—2542no—
190D4—24no—
191D4—22no—
192D4—1278no—
193D4—1272no—
194D4—1272no—
195D4—2548no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—1272no—
199C2×C4—1278no—
200C2×C4—1274no—
201C2×C4—2548no—
202C2×C4—2548no—
203C2×C4—2544no—
204C2×C4C2×C4×D12712yes—
205C2×C4C2×C4×D12711yescharacteristic
206C2×C4D4×D12712yes—
207C2×C4D4×D12712yes—
208C2×C4C23×D12711yescharacteristic
209C8—24no—
210C8—2544no—
211C22—22no—
212C22—41no—
213C22—42no—
214C22—1274no—
215C22—5082no—
216C22C22⋊C4×D12712yes—
217C22C22×C4×D12711yescharacteristic
218C4—22no—
219C4—1274no—
220C4—2544no—
221C4C22⋊C4×D12711yescharacteristic
222C4C22⋊C4×D12711yescharacteristic
223C4C2×D4×D12712yes—
224C2—21no—
225C2—22no—
226C2—42no—
227C2—2544no—
228C2C2×C22⋊C4×D12711yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D127
commutatorC22×C127C23×C4
FrattiniC2×C4C23×D127
Fitting#413×C127C2
socleC2×C127C22×C22⋊C4
prime radicalC2×#16861256C2
cosocleC2×C4×C127C24
hypercenter#413D127
hypocenterC127#2889
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. #22549857C2semidirect
5. C2×#16861127C2semidirect
6. #101×D127C2semidirect
7. #413×C127C2semidirect
115. C127#2889semidirect
120. #413D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612562
#225506422
#225498602
#225498572
C2×#168611272
#101×D1272
#413×C1272
#2889127