Finite Group Databasebeta

← C2×#21451538 #29846926

Subgroups — C2×#21451538 #29846926

Order 15616.

5194 conjugacy classes of subgroups; only the 907 normal subgroups are shown.

normal subgroups only

This group has 5194 conjugacy classes of subgroups — too many to draw. Only its 907 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
1C22×#14519760C213yesmaximal
2#21451538C2116yesmaximal, direct product factor
3C2×#16839681C214yesmaximal, complemented
4C2×#16839673C214yesmaximal, complemented
5C2×#16839625C213yesmaximal, complemented
6C24×Q8×C61C211yesmaximal, characteristic, complemented
7C23×C22⋊D61C2211yescharacteristic
8C2×#14519760—224no—
9C2×#14519760C22124yes—
10C22×#11758196C2216yes—
11C22×C22⋊Dic61C2213yes—
12C22×#11758198C2216yes—
13#16839681C22132yescomplemented
14#16839673C22132yescomplemented
15#16839625C22124yescomplemented
16C2×#14519684C22112yes—
17C23×Q8×C61—224no—
18C23×Q8×C61C2214yes—
19C23×Q8×C61C2218yescomplemented
20C24×C4×C61C2213yes—
21C24×D61C2311yescharacteristic
22C22×C22⋊D61—212no—
23C22×C22⋊D61C2314yes—
24C23×Dic61C2316yes—
25C23×Dic61C2311yescharacteristic
26#14519760—296no—
27C2×#11758196—224no—
28C2×#11758196C23124yes—
29C2×C22⋊Dic61—212no—
30C2×C22⋊Dic61C23112yes—
31C2×#11758198—224no—
32C2×#11758198C23124yes—
33#14519684C23148yes—
34C25×C61C2311yescharacteristic
35C22×Q8×C61—2128no—
36C22×Q8×C61—248no—
37C22×Q8×C61—296no—
38C22×Q8×C61C23116yes—
39C23×C4×C61—224no—
40C23×C4×C61D4124yes—
41C23×C4×C61C2316yes—
42C23×C4×C61C23112yes—
43C24×C61—28no—
44C24×C61C2×D4112yes—
45C24×C61C2412yes—
46C24×C61C2411yescharacteristic
47C23×D61—26no—
48C23×D61—28no—
49C23×D61C2412yes—
50C2×Q8×C61—2128no—
51C2×Q8×C61—2256no—
52C2×Q8×C61—296no—
53C22×C4×C61—296no—
54C22×C4×C61—248no—
55C22×C4×C61—224no—
56C22×C4×C61C2×D4124yes—
57C22×C4×C61C2×D4148yes—
58C22×C4×C61C24112yes—
59C2×C22⋊D61—224no—
60C2×C22⋊D61—416no—
61C22×Dic61—26no—
62C22×Dic61—212no—
63C22×Dic61—224no—
64C22×Dic61C2412yes—
65C22×Dic61C24112yes—
66C22⋊Dic61—224no—
67#11758198—248no—
68#11758196—248no—
69C23×C61—248no—
70C23×C61—28no—
71C23×C61#120116yes—
72C23×C61C22×D4112yes—
73C23×C61C22×D416yes—
74C23×C612−1+418yes—
75C23×C61C2511yescharacteristic
76C22×D61—26no—
77C22×D61—28no—
78C22×D61—44no—
79C22×D61—424no—
80Q8×C61—2128no—
81C2×C4×C61—296no—
82C2×C4×C61—248no—
83C2×C4×C61—224no—
84C2×C4×C61C22×D4124yes—
85C22⋊D61—416no—
86C2×Dic61—26no—
87C2×Dic61—212no—
88C2×Dic61—224no—
89C2×Dic61—44no—
90C2×Dic61—424no—
91C2×#2614—611nomaximal, Sylow
92C22×C61—232no—
93C22×C61—224no—
94C22×C61C2×#12018yes—
95C22×C61C2×#12014yes—
96C22×C61#536124yes—
97C22×C61C23×D413yes—
98C22×C61C2×2−1+414yes—
99C2×D61—42no—
100C2×D61—412no—
101C2×D61—88no—
102C4×C61—224no—
103Dic61—42no—
104Dic61—412no—
105C24×Q8D6111yescharacteristic, complemented
106C22×#123—613no—
107#2614—6116no—
108C2×#395—614no—
109C2×#393—614no—
110C2×#381—613no—
111C2×C61—28no—
112C2×C61#261418yescomplemented
113C2×C61C22×#12011yescharacteristic
114C2×C61C2×#53616yes—
115D61—82no—
116C23×Q8—224no—
117C23×Q8C2×D6114yes—
118C23×Q8C2×D6118yescomplemented
119C23×D4—611no—
120C24×C4C2×D6113yes—
121C2×#123—6124no—
122C2×#123—12224no—
123C22×C4⋊C4—616no—
124C22×C22⋊C4—613no—
125C22×C22⋊C4—616no—
126#395—6132no—
127#393—6132no—
128#381—6124no—
129C2×#95—6112no—
130C61C2×#261411yescharacteristic, complemented, Sylow
131C25—611no—
132C25C22×D6111yescharacteristic
133C22×Q8—2128no—
134C22×Q8—248no—
135C22×Q8—296no—
136C22×Q8C22×D61116yes—
137C22×D4—614no—
138C22×D4—12212no—
139C23×C4—224no—
140C23×C4—616no—
141C23×C4—611no—
142C23×C4C22⋊D61124yes—
143C23×C4C22×D61112yes—
144C23×C4C22×D6116yes—
145#123—12296no—
146C2×C4⋊C4—6124no—
147C2×C4⋊C4—12224no—
148C2×C22⋊C4—6112no—
149C2×C22⋊C4—6124no—
150C2×C22⋊C4—12212no—
151C2×C22⋊C4—12224no—
152#95—6148no—
153C24—28no—
154C24—612no—
155C24—1226no—
156C24—1228no—
157C24C2×C22⋊D61112yes—
158C24C23×D6112yes—
159C24C23×D6111yescharacteristic
160C2×Q8—2128no—
161C2×Q8—2256no—
162C2×Q8—296no—
163C2×D4—12224no—
164C2×D4—24416no—
165C22×C4—248no—
166C22×C4—224no—
167C22×C4—296no—
168C22×C4—6112no—
169C22×C4—612no—
170C22×C4—12212no—
171C22×C4—1226no—
172C22×C4—12224no—
173C22×C4C2×C22⋊D61124yes—
174C22×C4C2×C22⋊D61148yes—
175C22×C4C23×D61112yes—
176C4⋊C4—12248no—
177C22⋊C4—12224no—
178C22⋊C4—12248no—
179C23—248no—
180C23—28no—
181C23—1226no—
182C23—1228no—
183C23—2444no—
184C23—24424no—
185C23#14519746116yes—
186C23C22×C22⋊D61112yes—
187C23C22×C22⋊D6116yes—
188C23#1451982518yes—
189C23C24×D6111yescharacteristic
190Q8—2128no—
191D4—24416no—
192C2×C4—296no—
193C2×C4—248no—
194C2×C4—224no—
195C2×C4—12212no—
196C2×C4—1226no—
197C2×C4—12224no—
198C2×C4—2444no—
199C2×C4—24424no—
200C2×C4C22×C22⋊D61124yes—
201C22—232no—
202C22—224no—
203C22—4888no—
204C22—2442no—
205C22—24412no—
206C22C2×#1451974618yes—
207C22C2×#1451974614yes—
208C22#16840336124yes—
209C22C23×C22⋊D6113yes—
210C22C2×#1451982514yes—
211C4—224no—
212C4—2442no—
213C4—24412no—
214C2—28no—
215C2—4882no—
216C2#2145153818yesdirect product factor
217C2C22×#1451974611yescharacteristic
218C2C2×#1684033616yes—
subgroupG/H
centerC24C23×D61
commutator, cosocleC23×C61C25
FrattiniC23C24×D61
FittingC24×Q8×C61C2
socleC24×C61C24
prime radicalC23×C22⋊D61C22
hypercenterC24×Q8D61
hypocenterC61C2×#2614
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C24×Q8

Complements

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

NH ≅ G/Ntype
2. #21451538C2direct (N × H)
3. C2×#16839681C2semidirect
4. C2×#16839673C2semidirect
5. C2×#16839625C2semidirect
6. C24×Q8×C61C2semidirect
13. #16839681C22semidirect
14. #16839673C22semidirect
15. #16839625C22semidirect
19. C23×Q8×C61C22semidirect
105. C24×Q8D61semidirect
112. C2×C61#2614semidirect
118. C23×Q8C2×D61semidirect
130. C61C2×#2614semidirect
216. C2#21451538direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145197602
#214515382
C2×#168396812
C2×#168396732
C2×#168396252
C24×Q8×C612
C2×#261461