Finite Group Databasebeta

← #85307

Subgroups — #85307

Order 384.

345 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 345 conjugacy classes of subgroups — too many to draw. Only its 52 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×C4×S4C211yesmaximal, characteristic
2A4×C4⋊C4C211yesmaximal, characteristic, complemented
3(C2×S4)⋊C4C212yesmaximal
4C4⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
5C4×SL(2,ℤ/4)C212yesmaximal, complemented
6#2352—31nomaximal, Sylow
7C22×S4C2211yescharacteristic
8C2×C4×A4C2212yes—
9C2×C4×A4C2211yescharacteristic
10C2×SL(2,ℤ/4)C2212yes—
11C2×SL(2,ℤ/4)C2211yescharacteristic
12C4×S4C414yescomplemented
13#892—41nomaximal
14C2×C4×D4—31no—
15C2×#117—31no—
16C22×C4⋊C4D311yescharacteristic, complemented
17#388—32no—
18#387—32no—
19#386—32no—
20#382—31no—
21#380—31no—
22#380—31no—
23C4×C4⋊C4—31no—
24C4×C22⋊C4—32no—
25C22×A4C2311yescharacteristic
26C2×S4—21no—
27C2×S4—21no—
28C2×S4C2×C412yes—
29C2×C4×D3—41no—
30C4×A4—22no—
31C4×A4C2×C412yes—
32SL(2,ℤ/4)—22no—
33SL(2,ℤ/4)C2×C412yes—
34C3×C4⋊C4—41no—
35(C2×D3)⋊C4—42no—
36C4⋊Dic3—41no—
37C4×Dic3—42no—
38C22×D4—31no—
39C23×C4—31no—
40C23×C4C2×D312yes—
41C23×C4C2×D311yescharacteristic
42C4×D4—34no—
43C4×D4—34no—
44#117—61no—
45#117—62no—
46#117—61no—
47C2×C4⋊C4—31no—
48C2×C4⋊C4—31no—
49C2×C4⋊C4—31no—
50C2×C4⋊C4—32no—
51C2×C4⋊C4—31no—
52C2×C4⋊C4—31no—
53C2×C22⋊C4—31no—
54C2×C22⋊C4—31no—
55C2×C22⋊C4—32no—
56C2×C22⋊C4—32no—
57C2×C22⋊C4—32no—
58C2×C42—31no—
59C2×C42—32no—
60C2×C42—32no—
61#95—32no—
62#95—32no—
63#95—32no—
64#95—32no—
65#95—32no—
66C22×D3—41no—
67C2×A4C22×C411yescharacteristic
68C2×A4#4111yescharacteristic
69C2×A4#4111yescharacteristic
70S4—22no—
71C2×C3×C4—41no—
72C2×C3×C4—42no—
73C2×Dic3—41no—
74C2×Dic3—42no—
75C4×D3—44no—
76C24—31no—
77C24C22×D311yescharacteristic
78C2×D4—32no—
79C2×D4—32no—
80C2×D4—61no—
81C2×D4—61no—
82C2×D4—61no—
83C2×D4—61no—
84C22×C4—22no—
85C22×C4—31no—
86C22×C4—31no—
87C22×C4—32no—
88C22×C4—31no—
89C22×C4—31no—
90C22×C4—31no—
91C22×C4—34no—
92C22×C4—32no—
93C22×C4—32no—
94C22×C4—32no—
95C22×C4—32no—
96C22×C4—61no—
97C22×C4—62no—
98C22×C4—61no—
99C22×C4—62no—
100C22×C4C4×D312yes—
101C4⋊C4—34no—
102C4⋊C4—32no—
103C4⋊C4—31no—
104C4⋊C4—61no—
105C4⋊C4—62no—
106C4⋊C4—61no—
107C4⋊C4—61no—
108C4⋊C4S411yescharacteristic, complemented
109C22⋊C4—34no—
110C22⋊C4—32no—
111C22⋊C4—32no—
112C22⋊C4—62no—
113C22⋊C4—62no—
114C22⋊C4—62no—
115C22⋊C4—62no—
116C22⋊C4—122no—
117C42—34no—
118C42—62no—
119C42—62no—
120C42—122no—
121C22×C3—41no—
122C2×D3—42no—
123C2×D3—81no—
124C2×D3—81no—
125A4#11711yescharacteristic, complemented
126C3×C4—42no—
127C3×C4—82no—
128Dic3—42no—
129Dic3—82no—
130C23—31no—
131C23—31no—
132C23—31no—
133C23—31no—
134C23—32no—
135C23—61no—
136C23—61no—
137C23—61no—
138C23—61no—
139C23—61no—
140C23—61no—
141C23C2×C4×D311yescharacteristic
142C23#23711yescharacteristic
143C23#23911yescharacteristic
144D4—62no—
145D4—62no—
146D4—62no—
147D4—62no—
148C2×C4—31no—
149C2×C4—32no—
150C2×C4—32no—
151C2×C4—32no—
152C2×C4—32no—
153C2×C4—32no—
154C2×C4—32no—
155C2×C4—32no—
156C2×C4—62no—
157C2×C4—62no—
158C2×C4—61no—
159C2×C4—64no—
160C2×C4—61no—
161C2×C4—61no—
162C2×C4—61no—
163C2×C4—61no—
164C2×C4—61no—
165C2×C4—61no—
166C2×C4—64no—
167C2×C4—62no—
168C2×C4—62no—
169C2×C4—62no—
170C2×C4—62no—
171C2×C4—62no—
172C2×C4—62no—
173C2×C4—62no—
174C2×C4—62no—
175C2×C4C2×S412yes—
176C2×C4C2×S411yescharacteristic
177C2×C3—41no—
178C2×C3—41no—
179C2×C3—41no—
180D3—82no—
181C22—31no—
182C22—31no—
183C22—31no—
184C22—31no—
185C22—31no—
186C22—31no—
187C22—31no—
188C22—31no—
189C22—31no—
190C22—61no—
191C22—62no—
192C22—62no—
193C22—62no—
194C22—121no—
195C22—121no—
196C22—121no—
197C22—121no—
198C22#89211yescharacteristic, complemented
199C22C22×S411yescharacteristic
200C4—22no—
201C4—32no—
202C4—62no—
203C4—62no—
204C4—62no—
205C4—62no—
206C4—122no—
207C4—122no—
208C4C4×S412yes—
209C3—41noSylow
210C2—31no—
211C2—31no—
212C2—31no—
213C2—31no—
214C2—122no—
215C2C2×C4×S411yescharacteristic
216C2#591411yescharacteristic
217C2#591911yescharacteristic
subgroupG/H
center, FrattiniC22C22×S4
commutatorC2×A4C22×C4
FittingC22×C4⋊C4D3
socleC24C22×D3
prime radicalC22×S4C22
cosocleC22×A4C23
hypercenterC4⋊C4S4
hypocenterA4#117
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. A4×C4⋊C4C2semidirect
4. C4⋊SL(2,ℤ/4)C2semidirect
5. C4×SL(2,ℤ/4)C2semidirect
12. C4×S4C4semidirect
16. C22×C4⋊C4D3semidirect
108. C4⋊C4S4semidirect
125. A4#117semidirect
198. C22#892semidirect

Maximal subgroups

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

subgroupindex
C2×C4×S42
A4×C4⋊C42
(C2×S4)⋊C42
C4⋊SL(2,ℤ/4)2
C4×SL(2,ℤ/4)2
#23523
#8924