Finite Group Databasebeta

← C2×#22550058 #30648071

Subgroups — C2×#22550058 #30648071

Order 16256.

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

normal subgroups only

This group has 708 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
2C2×#16861240C212yesmaximal, complemented
3#22550058C218yesmaximal, direct product factor
4C2×#16861137C212yesmaximal, complemented
5C2×#16861128C211yesmaximal, characteristic, complemented
6C2×C4⋊Q8×C127C211yesmaximal, characteristic, complemented
7#16861257—24no—
8#16861257—22no—
9#16861257C2214yes—
10C2×#14523249—22no—
11C2×#14523248—22no—
12C2×#14523248C2212yes—
13C2×Q8×D127—22no—
14#16861240—24no—
15#16861240C2218yescomplemented
16C2×#14523238—22no—
17C2×#14523235—22no—
18C2×#14523232C2212yes—
19#16861137C2218yescomplemented
20#16861128C2214yescomplemented
21C22×Q8×C127C2211yescharacteristic
22C22×Q8×C127C2211yescharacteristic
23C4⋊Q8×C127—22no—
24C4⋊Q8×C127C2214yescomplemented
25C2×C4⋊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#14523238—28no—
45#14523235—28no—
46#14523232—22no—
47#14523232C2314yes—
48C2×C127⋊C8—22no—
49C2×Q8×C127—22no—
50C2×Q8×C127—22no—
51C2×Q8×C127—21no—
52C2×Q8×C127—22no—
53C2×Q8×C127—22no—
54C2×Q8×C127D412yes—
55C2×Q8×C127C2312yes—
56C2×Q8×C127C2312yes—
57C22×C4×C127—21no—
58C22×C4×C127—21no—
59C22×C4×C127D412yes—
60C22×C4×C127C2311yescharacteristic
61C4⋊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×C127C2×D411yescharacteristic
79Q8×C127—24no—
80Q8×C127—24no—
81Q8×C127—24no—
82Q8×C127—42no—
83C2×C4×C127—24no—
84C2×C4×C127—22no—
85C2×C4×C127—22no—
86C2×C4×C127—22no—
87C2×C4×C127—22no—
88C2×C4×C127—41no—
89C2×C4×C127—41no—
90C2×C4×C127C2×D412yes—
91C2×C4×C127C2×D412yes—
92C2×C4×C127C2×D414yes—
93C2×C4×C127C2×D411yescharacteristic
94C2×C4×C127C2411yescharacteristic
95C22×C127—22no—
96C22×C127#12011yescharacteristic
97C22×C127#12011yescharacteristic
98C22×C127C22×D411yescharacteristic
99C2×D127—24no—
100C2×D127—44no—
101C4×C127—24no—
102C4×C127—24no—
103C4×C127—42no—
104C4×C127#12012yes—
105C4×C127C22×D412yes—
106Dic127—24no—
107C2×C127—21no—
108C2×C127—21no—
109C2×C127#45612yescomplemented
110C2×C127C2×#12011yescharacteristic
111D127—44no—
112C2×#456—1271nomaximal, Sylow
113C127C2×#45611yescharacteristic, complemented, Sylow
114C2×2−1+4—1271no—
115C2×#137—1272no—
116C2×C4⋊Q8D12711yescharacteristic, complemented
117#456—1278no—
118C2×#104—1272no—
119C2×#101—1271no—
1202−1+4—1274no—
1212−1+4—2544no—
1222−1+4—2542no—
123C2×#41—1272no—
124C2×#41—2542no—
125C2×#41—2542no—
126C22×Q8—2542no—
127C22×Q8C2×D12711yescharacteristic
128C22×Q8C2×D12711yescharacteristic
129#137—1278no—
130#137—2544no—
131C2×Q16—2542no—
132C2×SD16—2542no—
133C2×M16—1272no—
134C4⋊Q8—22no—
135C4⋊Q8C2×D12714yescomplemented
136C2×C4⋊C4—22no—
137C2×C42C2×D12711yescharacteristic
138#104—1278no—
139#101—1274no—
140#41—1274no—
141#41—1274no—
142#41—2548no—
143#41—2548no—
144#41—2548no—
145#41—2542no—
146#41—2542no—
147#41—2548no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—21no—
151C2×Q8—22no—
152C2×Q8—22no—
153C2×Q8—1272no—
154C2×Q8—2542no—
155C2×Q8—2548no—
156C2×Q8—2548no—
157C2×Q8C22⋊D12712yes—
158C2×Q8C22×D12712yes—
159C2×Q8C22×D12712yes—
160C2×D4—1272no—
161C2×D4—2542no—
162C2×D4—2542no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—2544no—
166C22×C4—2542no—
167C22×C4C22⋊D12712yes—
168C22×C4C22×D12711yescharacteristic
169Q16—2548no—
170SD16—2548no—
171M16—1274no—
172M16—2542no—
173C2×C8—2542no—
174C4⋊C4—28no—
175C42—21no—
176C42C22×D12712yes—
177C23—2542no—
178C23C2×C22⋊D12711yescharacteristic
179Q8—24no—
180Q8—24no—
181Q8—24no—
182Q8—42no—
183Q8—1274no—
184Q8—2548no—
185Q8—2542no—
186D4—1274no—
187D4—2548no—
188D4—2544no—
189D4—2544no—
190D4—2542no—
191C2×C4—24no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—25416no—
199C2×C4—2542no—
200C2×C4—2544no—
201C2×C4—2544no—
202C2×C4D4×D12712yes—
203C2×C4D4×D12712yes—
204C2×C4C2×C22⋊D12714yes—
205C2×C4C2×C22⋊D12711yescharacteristic
206C2×C4C23×D12711yescharacteristic
207C8—2544no—
208C22—22no—
209C22—5084no—
210C22—2544no—
211C22#1686117711yescharacteristic
212C22#1686117711yescharacteristic
213C22C22×C22⋊D12711yescharacteristic
214C4—24no—
215C4—24no—
216C4—42no—
217C4—2544no—
218C4#1686117712yes—
219C4C2×D4×D12712yes—
220C2—21no—
221C2—21no—
222C2—5084no—
223C2#2255005812yesdirect product factor
224C2C2×#1686117711yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×C4⋊Q8×C127C2
socleC22×C127#120
prime radicalC2×#16861257C2
hypercenterC2×C4⋊Q8D127
hypocenterC127C2×#456
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#16861240C2semidirect
3. #22550058C2direct (N × H)
4. C2×#16861137C2semidirect
5. C2×#16861128C2semidirect
6. C2×C4⋊Q8×C127C2semidirect
15. #16861240C22semidirect
19. #16861137C22semidirect
20. #16861128C22semidirect
24. C4⋊Q8×C127C22semidirect
109. C2×C127#456semidirect
113. C127C2×#456semidirect
116. C2×C4⋊Q8D127semidirect
135. C4⋊Q8C2×D127semidirect
223. C2#22550058direct (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×#168612402
#225500582
C2×#168611372
C2×#168611282
C2×C4⋊Q8×C1272
C2×#456127