Finite Group Databasebeta

← C4×A4⋊Q8 #85282

Subgroups — C4×A4⋊Q8 #85282

Order 384.

325 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 325 conjugacy classes of subgroups — too many to draw. Only its 58 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×A4⋊Q8C211yesmaximal, characteristic
2C42×A4C211yesmaximal, characteristic
3C4⋊SL(2,ℤ/4)C211yesmaximal, characteristic
4SL(2,ℤ/4)⋊C4C212yesmaximal
5C4×SL(2,ℤ/4)C212yesmaximal
6C4×C22⋊Q8—31nomaximal, Sylow
7C2×C4×A4C2211yescharacteristic
8C2×C4×A4C2211yescharacteristic
9C2×C4×A4C2211yescharacteristic
10C2×SL(2,ℤ/4)C2212yes—
11C2×SL(2,ℤ/4)C2212yes—
12A4⋊Q8C414yesdirect product factor
13C4×Dic6—41nomaximal
14C2×C22⋊Q8—31no—
15C2×C4×Q8—31no—
16C22×C42D311yescharacteristic
17#391—31no—
18#389—31no—
19#387—32no—
20#385—32no—
21#380—31no—
22C4×C4⋊C4—32no—
23C4×C4⋊C4—31no—
24C4×C22⋊C4—32no—
25C22×A4C2311yescharacteristic
26C2×Dic6—41no—
27C4×A4—21no—
28C4×A4C2×C412yes—
29C4×A4Q812yescomplemented
30SL(2,ℤ/4)—22no—
31SL(2,ℤ/4)C2×C414yes—
32C3×C42—41no—
33C4⋊Dic3—41no—
34Dic3⋊C4—42no—
35C4×Dic3—42no—
36C22×Q8—31no—
37C23×C4C2×D311yescharacteristic
38C23×C4C2×D311yescharacteristic
39C23×C4C2×D311yescharacteristic
40C22⋊Q8—34no—
41C22⋊Q8—34no—
42C4×Q8—61no—
43C4×Q8—61no—
44C4×Q8—61no—
45C4×Q8—61no—
46C2×C4⋊C4—32no—
47C2×C4⋊C4—31no—
48C2×C4⋊C4—31no—
49C2×C4⋊C4—32no—
50C2×C4⋊C4—31no—
51C2×C4⋊C4—32no—
52C2×C22⋊C4—32no—
53C2×C22⋊C4—32no—
54C2×C42—31no—
55C2×C42—31no—
56C2×C42—31no—
57C2×C42—31no—
58C2×C42—32no—
59C2×C42—32no—
60#95—32no—
61#95—32no—
62#95—32no—
63#95—32no—
64#95—32no—
65C2×A4C22×C411yescharacteristic
66C2×A4C2×Q811yescharacteristic
67C2×A4#4111yescharacteristic
68C2×C3×C4—41no—
69C2×C3×C4—41no—
70C2×C3×C4—41no—
71C2×Dic3—42no—
72C2×Dic3—42no—
73Dic6—44no—
74C24C22×D311yescharacteristic
75C2×Q8—34no—
76C2×Q8—61no—
77C2×Q8—62no—
78C2×Q8—61no—
79C22×C4—21no—
80C22×C4—31no—
81C22×C4—31no—
82C22×C4—31no—
83C22×C4—31no—
84C22×C4—31no—
85C22×C4—32no—
86C22×C4—32no—
87C22×C4—32no—
88C22×C4—32no—
89C22×C4—31no—
90C22×C4—32no—
91C22×C4—32no—
92C22×C4—61no—
93C22×C4Dic612yescomplemented
94C22×C4C4×D312yes—
95C4⋊C4—34no—
96C4⋊C4—38no—
97C4⋊C4—62no—
98C4⋊C4—64no—
99C4⋊C4—61no—
100C4⋊C4—62no—
101C4⋊C4—62no—
102C4⋊C4—61no—
103C22⋊C4—34no—
104C22⋊C4—34no—
105C22⋊C4—62no—
106C22⋊C4—62no—
107C42—31no—
108C42—31no—
109C42—31no—
110C42—61no—
111C42—64no—
112C42—62no—
113C42—62no—
114C42S411yescharacteristic
115C22×C3—41no—
116A4C4×Q811yescharacteristic, complemented
117C3×C4—42no—
118C3×C4—42no—
119C3×C4—81no—
120Dic3—44no—
121Dic3—82no—
122C23—31no—
123C23—31no—
124C23—31no—
125C23—31no—
126C23C2×Dic611yescharacteristic
127C23C2×C4×D311yescharacteristic
128C23#23511yescharacteristic
129Q8—64no—
130Q8—64no—
131C2×C4—31no—
132C2×C4—31no—
133C2×C4—32no—
134C2×C4—32no—
135C2×C4—32no—
136C2×C4—32no—
137C2×C4—32no—
138C2×C4—32no—
139C2×C4—34no—
140C2×C4—34no—
141C2×C4—32no—
142C2×C4—31no—
143C2×C4—61no—
144C2×C4—61no—
145C2×C4—61no—
146C2×C4—61no—
147C2×C4—62no—
148C2×C4—62no—
149C2×C4—62no—
150C2×C4—62no—
151C2×C4—61no—
152C2×C4—62no—
153C2×C4—62no—
154C2×C4—62no—
155C2×C4—62no—
156C2×C4—62no—
157C2×C4—62no—
158C2×C4C2×S411yescharacteristic
159C2×C4C2×S411yescharacteristic
160C2×C4C2×S411yescharacteristic
161C2×C3—41no—
162C2×C3—41no—
163C2×C3—41no—
164C22—31no—
165C22—31no—
166C22—31no—
167C22—31no—
168C22—31no—
169C22—31no—
170C22—31no—
171C22—31no—
172C22—31no—
173C22—61no—
174C22C4×Dic611yescharacteristic, complemented
175C22C22×S411yescharacteristic
176C4—21no—
177C4—32no—
178C4—32no—
179C4—64no—
180C4—64no—
181C4—61no—
182C4—122no—
183C4—122no—
184C4A4⋊Q812yesdirect product factor
185C4C4×S412yes—
186C3—41noSylow
187C2—31no—
188C2—31no—
189C2—31no—
190C2—31no—
191C2C2×A4⋊Q811yescharacteristic
192C2C2×C4×S411yescharacteristic
193C2#591211yescharacteristic
subgroupG/H
centerC2×C4C2×S4
commutatorC2×A4C22×C4
FrattiniC22C22×S4
FittingC22×C42D3
socleC24C22×D3
prime radical, cosocleC22×A4C23
hypercenterC42S4
hypocenterA4C4×Q8
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
12. A4⋊Q8C4direct (N × H)
29. C4×A4Q8semidirect
93. C22×C4Dic6semidirect
116. A4C4×Q8semidirect
174. C22C4×Dic6semidirect
184. C4A4⋊Q8direct (N × H)

Maximal subgroups

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

subgroupindex
C2×A4⋊Q82
C42×A42
C4⋊SL(2,ℤ/4)2
SL(2,ℤ/4)⋊C42
C4×SL(2,ℤ/4)2
C4×C22⋊Q83
C4×Dic64