Finite Group Databasebeta

← C2×#72213856 #72226274

Subgroups — C2×#72213856 #72226274

Order 3072.

6060 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 6060 conjugacy classes of subgroups — too many to draw. Only its 92 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×#6342420C311yesmaximal, characteristic, complemented, Sylow
2#72213856C212yesmaximal, direct product factor
3C2×#11709197C211yesmaximal, characteristic, complemented
4C2×#85646—45nomaximal
5#11709197C2211yescharacteristic, complemented
6#6342420—310no—
7#6342420C2×C312yescomplemented
8C2×#14711—35no—
9C2×#14710—35no—
10C2×C44C2×C311yescharacteristic, complemented
11#85646—410no—
12C2×C22⋊(C42⋊C3)—45no—
13C23×#126A415yes—
14C22×#383—35no—
15C22×C43—35no—
16#14711—360no—
17#14711—310no—
18#14711—310no—
19#14710—310no—
20#14710—360no—
21#14710—310no—
22C2×#1437—310no—
23C2×#1436—330no—
24C44—35no—
25C44C22×C311yescharacteristic, complemented
26C22×C22⋊A4—161no—
27C22⋊(C42⋊C3)—45no—
28C2×#126⋊C3—1620no—
29C22×#126—310no—
30C22×#126—620no—
31C22×#126—620no—
32C22×#126C2×A4110yes—
33C23×C4⋊C4—35no—
34C23×C22⋊C4—35no—
35C23×C42—310no—
36C23×C42C2×A415yes—
37C2×#383—330no—
38C2×#383—310no—
39C2×#383—620no—
40C2×#383—620no—
41C22×#95—330no—
42C2×C43—330no—
43C2×C43—35no—
44C2×C43—35no—
45C2×C43—620no—
46C2×C43—620no—
47#1437—320no—
48#1437—360no—
49#1436—360no—
50#1436—360no—
51#1436—3120no—
52C2×C22⋊A4—161no—
53C2×C22⋊A4—162no—
54#126⋊C3—1640no—
55C2×C42⋊C3—1620no—
56C26C22⋊A411yescharacteristic
57C24×C4—35no—
58C24×C4—35no—
59C2×#126—420no—
60C2×#126—640no—
61C2×#126—6120no—
62C2×#126—1220no—
63C2×#126—1220no—
64C2×#126—1260no—
65C22×C4⋊C4—310no—
66C22×C4⋊C4—310no—
67C22×C4⋊C4—630no—
68C22×C4⋊C4—630no—
69C22×C22⋊C4—310no—
70C22×C22⋊C4—310no—
71C22×C22⋊C4—630no—
72C22×C22⋊C4—630no—
73C22×C42—35no—
74C22×C42—320no—
75C22×C42—320no—
76C22×C42—330no—
77C22×C42—310no—
78C22×C42—660no—
79C22×C42—660no—
80C22×C42C22×A415yes—
81#383—6120no—
82#383—640no—
83C2×#95—360no—
84C2×#95—360no—
85C2×#95—660no—
86C2×#95—660no—
87C43—6120no—
88C43—620no—
89C43—620no—
90C22⋊A4—161no—
91C22×A4—645no—
92C42⋊C3—1620no—
93C25—65no—
94C25—65no—
95C25C2×C22⋊A411yescharacteristic
96C25C2×C22⋊A412yes—
97C23×C4—330no—
98C23×C4—330no—
99C23×C4—35no—
100C23×C4—35no—
101C23×C4—310no—
102C23×C4—620no—
103C23×C4—620no—
104C23×C4—615no—
105C23×C4—615no—
106C23×C4—620no—
107C23×C4—620no—
108#126—440no—
109#126—1240no—
110#126—12120no—
111#126—1240no—
112C2×C4⋊C4—660no—
113C2×C4⋊C4—660no—
114C2×C4⋊C4—1220no—
115C2×C4⋊C4—1260no—
116C2×C22⋊C4—660no—
117C2×C22⋊C4—660no—
118C2×C22⋊C4—1220no—
119C2×C22⋊C4—1260no—
120C2×C42—360no—
121C2×C42—360no—
122C2×C42—320no—
123C2×C42—320no—
124C2×C42—320no—
125C2×C42—660no—
126C2×C42—660no—
127C2×C42—6120no—
128C2×C42—6240no—
129C2×C42—680no—
130C2×C42#126⋊C3120yescomplemented
131#95—6120no—
132#95—6120no—
133C2×A4—6410no—
134C2×A4—645no—
135C24—35no—
136C24—35no—
137C24—45no—
138C24—610no—
139C24—125no—
140C24—1210no—
141C24—1230no—
142C24C22×C22⋊A411yescharacteristic
143C22×C4—330no—
144C22×C4—330no—
145C22×C4—360no—
146C22×C4—320no—
147C22×C4—620no—
148C22×C4—620no—
149C22×C4—660no—
150C22×C4—6120no—
151C22×C4—660no—
152C22×C4—640no—
153C22×C4—630no—
154C22×C4—1215no—
155C22×C4—1230no—
156C22×C4—1260no—
157C22×C4—1230no—
158C22×C4—125no—
159C4⋊C4—1240no—
160C4⋊C4—1240no—
161C22⋊C4—1240no—
162C22⋊C4—1240no—
163C42—320no—
164C42—320no—
165C42—360no—
166C42—680no—
167C42—6240no—
168C42C2×#126⋊C3120yescomplemented
169C22×C3—2561no—
170A4—645no—
171C23—330no—
172C23—310no—
173C23—35no—
174C23—35no—
175C23—410no—
176C23—1220no—
177C23—245no—
178C23—245no—
179C23—2430no—
180C23#8564615yes—
181C2×C4—320no—
182C2×C4—320no—
183C2×C4—660no—
184C2×C4—660no—
185C2×C4—620no—
186C2×C4—6120no—
187C2×C4—620no—
188C2×C4—1260no—
189C2×C4—1210no—
190C2×C4—245no—
191C2×C4—2430no—
192C2×C4—245no—
193C2×C3—2562no—
194C2×C3—2561no—
195C22—35no—
196C22—310no—
197C22—330no—
198C22—35no—
199C22—161no—
200C22—2410no—
201C22—485no—
202C22C2×#8564615yes—
203C4—620no—
204C4—620no—
205C4—2410no—
206C3—2561noSylow
207C2—35no—
208C2—35no—
209C2—162no—
210C2#7221385611yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2#72213856
commutator, cosocle, hypocenterC44C22×C3
FrattiniC24C22×C22⋊A4
FittingC2×#6342420C3
socleC25C2×C22⋊A4
LCSG ▷ C44
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×#6342420C3semidirect
2. #72213856C2direct (N × H)
3. C2×#11709197C2semidirect
5. #11709197C22semidirect
7. #6342420C2×C3semidirect
10. C2×C44C2×C3semidirect
25. C44C22×C3semidirect
130. C2×C42#126⋊C3semidirect
168. C42C2×#126⋊C3semidirect
210. C2#72213856direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#63424203
#722138562
C2×#117091972
C2×#856464