Finite Group Databasebeta

← C2×#14521349 #16855468

Subgroups — C2×#14521349 #16855468

Order 3968.

448 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 165 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×#11758447C211yesmaximal, characteristic, complemented
2#14521349C218yesmaximal, direct product factor
3C2×#11758395C211yesmaximal, characteristic, complemented
4C2×#11758396C211yesmaximal, characteristic, complemented
5C2×C4⋊(C31⋊C8)C211yesmaximal, characteristic, complemented
6C2×#11758420C211yesmaximal, characteristic
7C2×#11758420C211yesmaximal, characteristic
8C2×C4×D4×C31C211yesmaximal, characteristic
9C22×D4×C31C2211yescharacteristic
10C23×C4×C31—21no—
11C4×D4×C31—22no—
12C4×D4×C31C2214yes—
13C2×C4⋊C4×C31C2211yescharacteristic
14C2×C22⋊C4×C31—21no—
15C2×C42×C31C2211yescharacteristic
16C2×C4⋊Dic31C2211yescharacteristic
17C2×C4⋊Dic31C2211yescharacteristic
18C2×Dic31⋊C4—21no—
19C2×Dic31⋊C4—21no—
20C22×C31⋊C8C2211yescharacteristic
21C22×C31⋊C8C2211yescharacteristic
22#11758447—22no—
23#11758447C2214yescomplemented
24#11758420C2214yes—
25#11758420C2214yes—
26#11758396C2214yescomplemented
27#11758395C2214yescomplemented
28C4⋊(C31⋊C8)C2214yescomplemented
29C24×C31—21no—
30C2×D4×C31—21no—
31C2×D4×C31—22no—
32C2×D4×C31Q814yes—
33C2×D4×C31C2312yes—
34C22×C4×C31—21no—
35C22×C4×C31—24no—
36C22×C4×C31—44no—
37C22×C4×C31D411yescharacteristic
38C22×C4×C31D411yescharacteristic
39C22×C4×C31C2311yescharacteristic
40C22×Dic31—21no—
41C22×Dic31—21no—
42C4⋊C4×C31—21no—
43C4⋊C4×C31C2312yes—
44C22⋊C4×C31—24no—
45C42×C31D412yes—
46C42×C31C2312yes—
47C4⋊Dic31—21no—
48C4⋊Dic31—21no—
49C4⋊Dic31C2312yes—
50C4⋊Dic31C2312yes—
51Dic31⋊C4—24no—
52Dic31⋊C4—24no—
53C2×C31⋊C8—22no—
54C2×C31⋊C8—22no—
55C2×C31⋊C8C2312yes—
56C2×C31⋊C8C2312yes—
57C23×C31—24no—
58C23×C31—22no—
59C23×C31—42no—
60C23×C31—42no—
61C23×C31C2×D411yescharacteristic
62D4×C31—24no—
63D4×C31—22no—
64D4×C31C2×Q814yes—
65C2×C4×C31—22no—
66C2×C4×C31—22no—
67C2×C4×C31—22no—
68C2×C4×C31—48no—
69C2×C4×C31—42no—
70C2×C4×C31C2×D412yes—
71C2×C4×C31C2×D412yes—
72C2×C4×C31C2×D411yescharacteristic
73C2×C4×C31C2×Q812yes—
74C2×C4×C31#4112yes—
75C2×C4×C31C2411yescharacteristic
76C2×Dic31—22no—
77C2×Dic31—22no—
78C2×Dic31—42no—
79C2×Dic31—42no—
80C31⋊C8—22no—
81C31⋊C8—22no—
82C2×#478—311nomaximal, Sylow
83C22×C31—24no—
84C22×C31—42no—
85C22×C31—42no—
86C22×C31—48no—
87C22×C31C22⋊Q812yes—
88C22×C31#13512yes—
89C22×C31AGL(1,ℤ/8)12yes—
90C22×C31C22×D411yescharacteristic
91C4×C31—22no—
92C4×C31—22no—
93C4×C31—42no—
94C4×C31C22⋊Q812yes—
95C4×C31C22×Q811yescharacteristic
96C4×C31C2×#4111yescharacteristic
97Dic31—42no—
98Dic31—42no—
99C2×#125—311no—
100C2×C4×D4D3111yescharacteristic
101#478—318no—
102C2×#107—311no—
103C2×#106—311no—
104C2×C4⋊C8—311no—
105C2×#102—311no—
106C2×#102—311no—
107C2×C31—44no—
108C2×C31#47814yescomplemented
109C2×C31C2×C22⋊Q811yescharacteristic
110C2×C31C2×#13511yescharacteristic
111C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
112C22×D4C2×D3111yescharacteristic
113C23×C4—21no—
114C22×C8—311no—
115C22×C8—311no—
116#125—314no—
117#125—622no—
118C4×D4—22no—
119C4×D4C2×D3114yes—
120C2×C4⋊C4—311no—
121C2×C4⋊C4—311no—
122C2×C4⋊C4—621no—
123C2×C4⋊C4—621no—
124C2×C4⋊C4C2×D3111yescharacteristic
125C2×C22⋊C4—21no—
126C2×C42C2×D3111yescharacteristic
127#107—314no—
128#106—314no—
129C4⋊C8—314no—
130#102—314no—
131#102—314no—
132C31C2×#47811yescharacteristic, complemented, Sylow
133C24—21no—
134C2×D4—22no—
135C2×D4—21no—
136C2×D4Dic6214yes—
137C2×D4C22×D3112yes—
138C22×C4—21no—
139C22×C4—24no—
140C22×C4—44no—
141C22×C4—621no—
142C22×C4—621no—
143C22×C4C22⋊D3111yescharacteristic
144C22×C4C22⋊D3111yescharacteristic
145C22×C4C22×D3111yescharacteristic
146C2×C8—312no—
147C2×C8—312no—
148C2×C8—622no—
149C2×C8—622no—
150C4⋊C4—21no—
151C4⋊C4—312no—
152C4⋊C4—312no—
153C4⋊C4—624no—
154C4⋊C4—624no—
155C4⋊C4—621no—
156C4⋊C4—621no—
157C4⋊C4C22×D3112yes—
158C22⋊C4—24no—
159C42C22⋊D3112yes—
160C42C22×D3112yes—
161C23—24no—
162C23—22no—
163C23—42no—
164C23—42no—
165C23C2×C22⋊D3111yescharacteristic
166D4—24no—
167D4—22no—
168D4C2×Dic6214yes—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—48no—
173C2×C4—42no—
174C2×C4—622no—
175C2×C4—622no—
176C2×C4—1242no—
177C2×C4—1242no—
178C2×C4C2×Dic6212yes—
179C2×C4#9249512yes—
180C2×C4C2×C22⋊D3112yes—
181C2×C4C2×C22⋊D3112yes—
182C2×C4C2×C22⋊D3111yescharacteristic
183C2×C4C23×D3111yescharacteristic
184C8—622no—
185C8—622no—
186C22—24no—
187C22—48no—
188C22—42no—
189C22—42no—
190C22#1175850112yes—
191C22#1175850912yes—
192C22#1175852712yes—
193C22C22×C22⋊D3111yescharacteristic
194C4—22no—
195C4—22no—
196C4—42no—
197C4—1242no—
198C4—1242no—
199C4#1175850112yes—
200C4C22×Dic6211yescharacteristic
201C4C2×#9249511yescharacteristic
202C2—44no—
203C2#1452134914yesdirect product factor
204C2C2×#1175850111yescharacteristic
205C2C2×#1175852711yescharacteristic
206C2C2×#1175850911yescharacteristic
subgroupG/H
centerC23C2×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC2×C4×D4×C31C2
socleC23×C31C2×D4
prime radicalC22×D4×C31C22
hypercenterC2×C4×D4D31
hypocenterC31C2×#478
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#11758447C2semidirect
2. #14521349C2direct (N × H)
3. C2×#11758395C2semidirect
4. C2×#11758396C2semidirect
5. C2×C4⋊(C31⋊C8)C2semidirect
23. #11758447C22semidirect
26. #11758396C22semidirect
27. #11758395C22semidirect
28. C4⋊(C31⋊C8)C22semidirect
108. C2×C31#478semidirect
132. C31C2×#478semidirect
203. C2#14521349direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#117584472
#145213492
C2×#117583952
C2×#117583962
C2×C4⋊(C31⋊C8)2
C2×#117584202
C2×C4×D4×C312
C2×#47831