Finite Group Databasebeta

← C4⋊(#458⋊C3) #11709608

Subgroups — C4⋊(#458⋊C3) #11709608

Order 768.

322 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 26 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
1C4×#997C211yesmaximal, characteristic
2C2×#458⋊C3C212yesmaximal
3#32927C311yesmaximal, characteristic, complemented, Sylow
4C2×#997C2211yescharacteristic
5A4×C4⋊C4—41nomaximal
6#458⋊C3—22no—
7C4×H(5,𝔽2)C2×C311yescharacteristic
8C2×#458C2×C312yes—
9#2486—31no—
10#1890—32no—
11#1799—31no—
12#997D411yescharacteristic
13C2×C4×A4—41no—
14C2×C4×A4—42no—
15C2×C4×A4—81no—
16C4×SL(2,𝔽3)—81no—
17C2×H(5,𝔽2)C22×C311yescharacteristic
18C2×#123—32no—
19#520—61no—
20#518—31no—
21#518—61no—
22C4×#41—61no—
23C2×C4×D4—31no—
24C2×C4×D4—61no—
25C22×C4⋊C4A411yescharacteristic, complemented
26#458—22no—
27#458—62no—
28C2×#99—32no—
29#398—61no—
30#397—61no—
31#389—61no—
32#381—32no—
33#380—31no—
34#380—61no—
35#342—32no—
36C22×A4—41no—
37C22×A4—81no—
38C2×SL(2,𝔽3)—81no—
39C4×A4—42no—
40C4×A4—82no—
41C4×A4—82no—
42C3×C4⋊C4—161no—
43H(5,𝔽2)—31no—
44H(5,𝔽2)—61no—
45H(5,𝔽2)—61no—
46H(5,𝔽2)C3×D411yescharacteristic
47C2×#41—61no—
48C22×D4—31no—
49C22×D4—61no—
50C23×C4—21no—
51C23×C4—31no—
52C23×C4C2×A412yes—
53C23×C4C2×A411yescharacteristic
54#123—62no—
55#123—62no—
56#123—62no—
57#123—62no—
58C4×Q8—21no—
59C4×Q8—61no—
60C4×D4—62no—
61C4×D4—62no—
62C4×D4—62no—
63C4×D4—62no—
64C4×D4—62no—
65C4×D4—61no—
66C4×D4—61no—
67#117—61no—
68#117—62no—
69C2×C4⋊C4—31no—
70C2×C4⋊C4—61no—
71C2×C4⋊C4—61no—
72C2×C4⋊C4—62no—
73C2×C4⋊C4—61no—
74C2×C4⋊C4—61no—
75C2×C22⋊C4—31no—
76C2×C22⋊C4—31no—
77C2×C22⋊C4—32no—
78C2×C22⋊C4—61no—
79C2×C22⋊C4—61no—
80C2×C22⋊C4—62no—
81C2×C42—31no—
82C2×C42—61no—
83#99—62no—
84#99—62no—
85#95—61no—
86#95—62no—
87#95—62no—
88C2×A4—41no—
89C2×A4—41no—
90C2×A4—41no—
91C2×A4—81no—
92C2×A4—81no—
93C2×A4—81no—
94C2×C3×C4—162no—
95C2×C3×C4—161no—
96SL(2,𝔽3)—81no—
97C24—21no—
98C24—31no—
99C24C22×A411yescharacteristic
100#41—61no—
101#41—62no—
102#41—61no—
103#41—61no—
104#41—62no—
105#41—61no—
106C2×Q8—21no—
107C2×D4—31no—
108C2×D4—31no—
109C2×D4—31no—
110C2×D4—31no—
111C2×D4—61no—
112C2×D4—62no—
113C2×D4—62no—
114C2×D4—62no—
115C2×D4—61no—
116C2×D4—61no—
117C2×D4—62no—
118C2×D4—61no—
119C2×D4—61no—
120C22×C4—22no—
121C22×C4—31no—
122C22×C4—31no—
123C22×C4—31no—
124C22×C4—31no—
125C22×C4—31no—
126C22×C4—42no—
127C22×C4—61no—
128C22×C4—62no—
129C22×C4—61no—
130C22×C4—61no—
131C22×C4—61no—
132C22×C4—61no—
133C22×C4—62no—
134C22×C4—62no—
135C22×C4—82no—
136C22×C4—122no—
137C4⋊C4—41no—
138C4⋊C4—62no—
139C4⋊C4—61no—
140C4⋊C4—61no—
141C4⋊C4—121no—
142C4⋊C4—122no—
143C4⋊C4—122no—
144C4⋊C4—122no—
145C22⋊C4—62no—
146C22⋊C4—62no—
147C22⋊C4—62no—
148C22⋊C4—62no—
149C22⋊C4—62no—
150C22⋊C4—122no—
151C22⋊C4—122no—
152C42—61no—
153C42—62no—
154C42—61no—
155C22×C3—161no—
156A4—41no—
157A4—81no—
158C3×C4—162no—
159C3×C4—322no—
160C23—21no—
161C23—31no—
162C23—31no—
163C23—31no—
164C23—31no—
165C23—41no—
166C23—41no—
167C23—61no—
168C23—61no—
169C23—61no—
170C23—81no—
171C23—81no—
172C23—121no—
173C23—121no—
174C23D4×A411yescharacteristic
175Q8—21no—
176Q8—61no—
177D4—62no—
178D4—62no—
179D4—62no—
180D4—62no—
181D4—61no—
182D4—61no—
183D4—62no—
184C2×C4—31no—
185C2×C4—31no—
186C2×C4—31no—
187C2×C4—42no—
188C2×C4—61no—
189C2×C4—62no—
190C2×C4—61no—
191C2×C4—62no—
192C2×C4—61no—
193C2×C4—61no—
194C2×C4—61no—
195C2×C4—61no—
196C2×C4—121no—
197C2×C4—122no—
198C2×C4—122no—
199C2×C4—122no—
200C2×C4—121no—
201C2×C4—121no—
202C2×C4—122no—
203C2×C4—122no—
204C2×C4—122no—
205C2×C4—241no—
206C2×C4—241no—
207C2×C4—241no—
208C2×C4#120⋊C311yescharacteristic
209C2×C3—161no—
210C2×C3—161no—
211C2×C3—161no—
212C22—31no—
213C22—31no—
214C22—41no—
215C22—61no—
216C22—61no—
217C22—61no—
218C22—61no—
219C22—81no—
220C22—121no—
221C22—121no—
222C22—121no—
223C22—121no—
224C22—121no—
225C22—241no—
226C22—241no—
227C22C2×#120⋊C311yescharacteristic
228C4—61no—
229C4—61no—
230C4—61no—
231C4—82no—
232C4—121no—
233C4—121no—
234C4—242no—
235C4#458⋊C312yescomplemented
236C3—161noSylow
237C2—61no—
238C2—61no—
239C2—121no—
240C2—121no—
241C2C4⋊(#120⋊C3)11yescharacteristic
242C2C2×#458⋊C311yescharacteristic
243C2#3028⋊C311yescharacteristic
subgroupG/H
center, socleC22C2×#120⋊C3
commutator, cosocleC2×H(5,𝔽2)C22×C3
FrattiniC24C22×A4
Fitting#32927C3
prime radicalC2×#997C22
hypercenterC2×C4#120⋊C3
hypocenterH(5,𝔽2)C3×D4
LCSG ▷ C2×H(5,𝔽2) ▷ H(5,𝔽2)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. #32927C3semidirect
25. C22×C4⋊C4A4semidirect
235. C4#458⋊C3semidirect

Maximal subgroups

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

subgroupindex
C4×#9972
C2×#458⋊C32
#329273
A4×C4⋊C44