Finite Group Databasebeta

← #30647210

Subgroups — #30647210

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×#16861257C211yesmaximal, characteristic
2#22550642C212yesmaximal, complemented
3#22549859C214yesmaximal, complemented
4#22549858C212yesmaximal, complemented
5C2×#16861128C211yesmaximal, characteristic, complemented
6#101×D127C214yesmaximal, complemented
7C2×#101×C127C211yesmaximal, characteristic, complemented
8#16861257—24no—
9#16861257—24no—
10C2×#14523249C412yes—
11C2×#14523249C2212yes—
12C2×#14523248C414yes—
13C2×#14523248C2211yescharacteristic
14C2×#14523248C2211yescharacteristic
15C2×Q8×D127C412yes—
16C2×Q8×D127C2212yes—
17#16861217—24no—
18#16861215—24no—
19C2×#14523232C2212yes—
20M16×D127C2218yes—
21#16861128C2214yes—
22#16861127C2214yes—
23#16861125C2214yes—
24C22×Q8×C127C2211yescharacteristic
25C2×M16×C127C2212yes—
26#101×C127C2214yes—
27#14523249—28no—
28#14523249—28no—
29#14523248—28no—
30#14523248—28no—
31#14523248—22no—
32#14523248—22no—
33#14523248D414yes—
34#14523248D414yes—
35C2×Dic254C2×C414yes—
36C2×Dic254C2×C411yescharacteristic
37C2×Dic254C2311yescharacteristic
38Q8×D127—28no—
39Q8×D127—28no—
40C2×D508C2×C414yes—
41C2×D508C2×C411yescharacteristic
42C2×D508C2311yescharacteristic
43C2×C22⋊D127C2×C412yes—
44C2×C22⋊D127C2×C412yes—
45C2×C4×D127C2×C414yes—
46C2×C4×D127C2×C412yes—
47C2×C4×D127C2×C414yes—
48C2×C4×D127C2312yes—
49#14523232—22no—
50#14523232C2314yes—
51#14523231—28no—
52C2×C127⋊C8—22no—
53C8×D127—28no—
54C2×Q8×C127—22no—
55C2×Q8×C127—22no—
56C2×Q8×C127C2×C412yes—
57C2×Q8×C127C2312yes—
58C22×C4×C127C2×C412yes—
59C22×C4×C127C2311yescharacteristic
60M16×C127—22no—
61M16×C127C2314yes—
62C2×C8×C127—22no—
63C22×D127C22×C412yes—
64C22×D127C22×C412yes—
65Dic254—22no—
66Dic254—28no—
67Dic254C22⋊C414yes—
68D508—28no—
69D508—22no—
70D508C22⋊C414yes—
71C22⋊D127—22no—
72C22⋊D127—22no—
73C22⋊D127C22⋊C414yes—
74C22⋊D127C22⋊C414yes—
75C2×Dic127C22×C412yes—
76C2×Dic127C22×C412yes—
77C4×D127—28no—
78C4×D127—24no—
79C4×D127—28no—
80C4×D127C2×D418yes—
81C127⋊C8—24no—
82C23×C127C22×C411yescharacteristic
83Q8×C127—24no—
84Q8×C127—24no—
85C2×C4×C127—22no—
86C2×C4×C127—22no—
87C2×C4×C127C22×C414yes—
88C2×C4×C127C22×C411yescharacteristic
89C2×C4×C127C2×D412yes—
90C2×C4×C127C2×D412yes—
91C2×C4×C127C2411yescharacteristic
92C8×C127—24no—
93C22×C127—41no—
94C22×C127C2×C22⋊C411yescharacteristic
95C22×C127C2×C22⋊C411yescharacteristic
96C22×C127C23×C411yescharacteristic
97C2×D127—22no—
98C2×D127—42no—
99C2×D127—42no—
100C2×D127C2×C22⋊C414yes—
101C4×C127—24no—
102C4×C127C2×C22⋊C412yes—
103C4×C127C22×D412yes—
104Dic127—22no—
105Dic127C2×C22⋊C414yes—
106C2×C127—21no—
107C2×C127—21no—
108C2×C127—21no—
109C2×C127C22×C22⋊C411yescharacteristic
110D127—24no—
111D127—42no—
112#2889—1271nomaximal, Sylow
113C127#288911yescharacteristic, complemented, Sylow
114C2×2−1+4—1271no—
115#568—1272no—
116#413—1272no—
117#413—1274no—
118C2×#101—1271no—
119C2×#101—1274no—
120C2×#101D12711yescharacteristic, complemented
1212−1+4—2544no—
1222−1+4—2544no—
123C2×#41—1272no—
124C2×#41—1272no—
125C2×#41—1274no—
126C2×#41—1271no—
127C2×#41—1271no—
128C22×Q8—1272no—
129C22×Q8—1272no—
130C22×Q8C2×D12711yescharacteristic
131#131—2544no—
132#131—2544no—
133C2×M16—1272no—
134C2×M16—1278no—
135C2×M16C2×D12712yes—
136#101—1274no—
137#101—1274no—
138#101C2×D12714yes—
139#100—1274no—
140#41—1274no—
141#41—1274no—
142#41—2548no—
143#41—2548no—
144#41—2548no—
145#41—2548no—
146#41—2542no—
147#41—2542no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—1271no—
151C2×Q8—1274no—
152C2×Q8—1271no—
153C2×Q8—2548no—
154C2×Q8—2548no—
155C2×Q8C4×D12712yes—
156C2×Q8C22×D12712yes—
157C2×D4—1272no—
158C2×D4—1272no—
159C2×D4—1271no—
160C2×D4—1274no—
161C2×D4—1271no—
162C22×C4—1272no—
163C22×C4—1274no—
164C22×C4—1274no—
165C22×C4—1272no—
166C22×C4C4×D12712yes—
167C22×C4C22×D12711yescharacteristic
168M16—22no—
169M16—1274no—
170M16—2548no—
171M16—2542no—
172M16C22×D12714yes—
173C2×C8—22no—
174C2×C8—2548no—
175C2×C8—2542no—
176C23—1272no—
177C23—1272no—
178C23C2×C4×D12711yescharacteristic
179Q8—24no—
180Q8—24no—
181Q8—1274no—
182Q8—2548no—
183Q8—2542no—
184D4—1274no—
185D4—1274no—
186D4—1274no—
187D4—2548no—
188D4—2542no—
189D4—2542no—
190D4—2542no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—1272no—
194C2×C4—1278no—
195C2×C4—1272no—
196C2×C4—2548no—
197C2×C4—2548no—
198C2×C4—2544no—
199C2×C4C2×C4×D12714yes—
200C2×C4C2×C4×D12711yescharacteristic
201C2×C4D4×D12712yes—
202C2×C4D4×D12712yes—
203C2×C4C23×D12711yescharacteristic
204C8—24no—
205C8—2544no—
206C22—41no—
207C22—1274no—
208C22—5082no—
209C22—5082no—
210C22—2542no—
211C22C22⋊C4×D12711yescharacteristic
212C22C22⋊C4×D12711yescharacteristic
213C22C22×C4×D12711yescharacteristic
214C4—24no—
215C4—1274no—
216C4—2542no—
217C4C22⋊C4×D12712yes—
218C4C2×D4×D12712yes—
219C2—21no—
220C2—21no—
221C2—21no—
222C2—5082no—
223C2—2544no—
224C2C2×C22⋊C4×D12711yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D127
commutatorC22×C127C23×C4
FrattiniC2×C4C23×D127
FittingC2×#101×C127C2
socleC2×C127C22×C22⋊C4
prime radicalC2×#16861257C2
cosocleC2×C4×C127C24
hypercenterC2×#101D127
hypocenterC127#2889
LCSG ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
2. #22550642C2semidirect
3. #22549859C2semidirect
4. #22549858C2semidirect
5. C2×#16861128C2semidirect
6. #101×D127C2semidirect
7. C2×#101×C127C2semidirect
113. C127#2889semidirect
120. C2×#101D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612572
#225506422
#225498592
#225498582
C2×#168611282
#101×D1272
C2×#101×C1272
#2889127