Finite Group Databasebeta

← C2×#22550048 #30648065

Subgroups — C2×#22550048 #30648065

Order 16256.

728 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 149 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, complemented
2C2×#16861235C212yesmaximal, complemented
3#22550048C218yesmaximal, direct product factor
4C2×#16861137C212yesmaximal, complemented
5C2×#16861128C211yesmaximal, characteristic, complemented
6C2×#124×C127C211yesmaximal, characteristic, complemented
7#16861257—24no—
8#16861257—22no—
9#16861257C2214yescomplemented
10C2×#14523249—22no—
11C2×#14523248—22no—
12C2×#14523248C2212yes—
13C2×Q8×D127—22no—
14#16861235—24no—
15#16861235C2218yescomplemented
16C2×#14523236—22no—
17C2×#14523233—22no—
18C2×#14523232C2212yes—
19#16861137C2218yescomplemented
20#16861128C2214yescomplemented
21C22×Q8×C127C2211yescharacteristic
22C22×D4×C127C2211yescharacteristic
23#124×C127—22no—
24#124×C127C2214yescomplemented
25C2×C22⋊C4×C127—22no—
26C2×C42×C127C2211yescharacteristic
27#14523249—28no—
28#14523249—28no—
29#14523248—28no—
30#14523248—28no—
31#14523248—22no—
32#14523248—22no—
33#14523248D414yes—
34#14523248C2314yes—
35C2×Dic254—22no—
36C2×Dic254D412yes—
37Q8×D127—28no—
38Q8×D127—28no—
39C2×D508—22no—
40C2×D508D412yes—
41C2×C22⋊D127—22no—
42C2×C4×D127—24no—
43C2×C4×D127—22no—
44#14523236—28no—
45#14523233—28no—
46#14523232—22no—
47#14523232C2314yes—
48C2×C127⋊C8—22no—
49C24×C127—21no—
50C2×Q8×C127—22no—
51C2×Q8×C127—22no—
52C2×Q8×C127D412yes—
53C2×Q8×C127C2312yes—
54C2×D4×C127—22no—
55C2×D4×C127—22no—
56C2×D4×C127—21no—
57C2×D4×C127C2312yes—
58C22×C4×C127—21no—
59C22×C4×C127D412yes—
60C22×C4×C127C2311yescharacteristic
61C22⋊C4×C127—28no—
62C42×C127—21no—
63C42×C127C2312yes—
64C22×D127—22no—
65Dic254—28no—
66Dic254—22no—
67Dic254C2×D414yes—
68D508—28no—
69D508—22no—
70D508C2×D414yes—
71C22⋊D127—24no—
72C22⋊D127—24no—
73C2×Dic127—22no—
74C4×D127—216no—
75C4×D127—24no—
76C4×D127—24no—
77C127⋊C8—24no—
78C23×C127—22no—
79C23×C127—44no—
80C23×C127—41no—
81C23×C127—41no—
82C23×C127C2×D411yescharacteristic
83Q8×C127—24no—
84Q8×C127—24no—
85D4×C127—24no—
86D4×C127—42no—
87C2×C4×C127—22no—
88C2×C4×C127—22no—
89C2×C4×C127—22no—
90C2×C4×C127—42no—
91C2×C4×C127C2×D412yes—
92C2×C4×C127C2×D412yes—
93C2×C4×C127C2×D414yes—
94C2×C4×C127C2×D411yescharacteristic
95C2×C4×C127C2411yescharacteristic
96C22×C127—22no—
97C22×C127—44no—
98C22×C127—42no—
99C22×C127—82no—
100C22×C127—82no—
101C22×C127#12011yescharacteristic
102C22×C127#12011yescharacteristic
103C22×C127C22×D411yescharacteristic
104C2×D127—24no—
105C2×D127—44no—
106C4×C127—24no—
107C4×C127—42no—
108C4×C127#12012yes—
109C4×C127C22×D412yes—
110Dic127—24no—
111C2×C127—21no—
112C2×C127—21no—
113C2×C127—82no—
114C2×C127#45412yescomplemented
115C2×C127C2×#12011yescharacteristic
116D127—44no—
117C2×#454—1271nomaximal, Sylow
118C127C2×#45411yescharacteristic, complemented, Sylow
119C2×2−1+4—1271no—
120C2×AGL(1,ℤ/8)—1272no—
121C2×#124D12711yescharacteristic, complemented
122#454—1278no—
123C2×#104—1272no—
124C2×#101—1271no—
1252−1+4—1274no—
1262−1+4—2544no—
1272−1+4—2542no—
128C2×#41—1272no—
129C2×#41—2542no—
130C2×#41—2542no—
131C22×Q8—2542no—
132C22×Q8C2×D12711yescharacteristic
133C22×D4C2×D12711yescharacteristic
134AGL(1,ℤ/8)—1278no—
135AGL(1,ℤ/8)—2544no—
136C2×SD16—2542no—
137C2×D8—2542no—
138C2×M16—1272no—
139#124—22no—
140#124C2×D12714yescomplemented
141C2×C22⋊C4—22no—
142C2×C42C2×D12711yescharacteristic
143#104—1278no—
144#101—1274no—
145C24—21no—
146#41—1274no—
147#41—1274no—
148#41—2548no—
149#41—2548no—
150#41—2548no—
151#41—2542no—
152#41—2542no—
153#41—2548no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—1272no—
157C2×Q8—2542no—
158C2×Q8—2548no—
159C2×Q8—2548no—
160C2×Q8C22⋊D12712yes—
161C2×Q8C22×D12712yes—
162C2×D4—22no—
163C2×D4—22no—
164C2×D4—21no—
165C2×D4—1272no—
166C2×D4—2542no—
167C2×D4—2542no—
168C2×D4C22×D12712yes—
169C22×C4—21no—
170C22×C4—2544no—
171C22×C4—2542no—
172C22×C4C22⋊D12712yes—
173C22×C4C22×D12711yescharacteristic
174SD16—2548no—
175D8—2548no—
176M16—1274no—
177M16—2542no—
178C2×C8—2542no—
179C22⋊C4—28no—
180C42—21no—
181C42C22×D12712yes—
182C23—22no—
183C23—44no—
184C23—41no—
185C23—41no—
186C23—2542no—
187C23C2×C22⋊D12711yescharacteristic
188Q8—24no—
189Q8—24no—
190Q8—1274no—
191Q8—2548no—
192Q8—2542no—
193D4—24no—
194D4—42no—
195D4—1274no—
196D4—2548no—
197D4—2544no—
198D4—2544no—
199D4—2542no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—42no—
204C2×C4—25416no—
205C2×C4—2542no—
206C2×C4—2544no—
207C2×C4—2544no—
208C2×C4D4×D12712yes—
209C2×C4D4×D12712yes—
210C2×C4C2×C22⋊D12714yes—
211C2×C4C2×C22⋊D12711yescharacteristic
212C2×C4C23×D12711yescharacteristic
213C8—2544no—
214C22—22no—
215C22—44no—
216C22—42no—
217C22—82no—
218C22—82no—
219C22—5084no—
220C22—2544no—
221C22#1686117711yescharacteristic
222C22#1686117711yescharacteristic
223C22C22×C22⋊D12711yescharacteristic
224C4—24no—
225C4—42no—
226C4—2544no—
227C4#1686117712yes—
228C4C2×D4×D12712yes—
229C2—21no—
230C2—21no—
231C2—82no—
232C2—5084no—
233C2#2255004812yesdirect product factor
234C2C2×#1686117711yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#124×C127C2
socleC22×C127#120
hypercenterC2×#124D127
hypocenterC127C2×#454
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#16861257C2semidirect
2. C2×#16861235C2semidirect
3. #22550048C2direct (N × H)
4. C2×#16861137C2semidirect
5. C2×#16861128C2semidirect
6. C2×#124×C127C2semidirect
9. #16861257C22semidirect
15. #16861235C22semidirect
19. #16861137C22semidirect
20. #16861128C22semidirect
24. #124×C127C22semidirect
114. C2×C127#454semidirect
118. C127C2×#454semidirect
121. C2×#124D127semidirect
140. #124C2×D127semidirect
233. C2#22550048direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168612572
C2×#168612352
#225500482
C2×#168611372
C2×#168611282
C2×#124×C1272
C2×#454127