Finite Group Databasebeta

← #73049

Subgroups — #73049

Order 384.

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

normal subgroups only

This group has 250 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
1C2×GL(2,ℤ/4)C211yesmaximal, characteristic
2A4×C22⋊C4C211yesmaximal, characteristic, complemented
3#5432C211yesmaximal, characteristic, complemented
4#1898—31nomaximal, Sylow
5C23×A4C2211yescharacteristic
6C22×S4C411yescharacteristic, complemented
7C2×C4×A4—21no—
8GL(2,ℤ/4)—21no—
9GL(2,ℤ/4)—21no—
10C2×SL(2,ℤ/4)—21no—
11C2×SL(2,ℤ/4)C411yescharacteristic, complemented
12#806—41nomaximal
13C2×#120—31no—
14C22×C22⋊C4D311yescharacteristic, complemented
15C2×#99—31no—
16C2×#99—31no—
17#379—31no—
18#342—31no—
19#342—31no—
20C22×A4—41no—
21C22×A4C2×C411yescharacteristic
22C22×A4D411yescharacteristic
23C22×A4D411yescharacteristic
24C2×S4—21no—
25C2×S4—41no—
26C2×#67—41no—
27C4×A4—41no—
28SL(2,ℤ/4)—21no—
29SL(2,ℤ/4)—41no—
30C3×C22⋊C4—41no—
31#217—41no—
32C25C2×D311yescharacteristic
33C22×D4—31no—
34C22×D4—31no—
35C22×D4—31no—
36C23×C4—21no—
37#120—61no—
38#120—61no—
39#120—61no—
40#120—61no—
41C2×C22⋊C4—31no—
42C2×C22⋊C4—31no—
43C2×C22⋊C4—31no—
44C2×C22⋊C4—31no—
45C2×C22⋊C4—31no—
46C2×C22⋊C4—31no—
47C2×C22⋊C4—31no—
48C2×C22⋊C4—61no—
49C2×C22⋊C4—61no—
50C2×C22⋊C4—61no—
51#99—62no—
52#99—61no—
53#99—61no—
54C23×C3—41no—
55C22×D3—41no—
56C2×A4—21no—
57C2×A4—21no—
58C2×A4—21no—
59C2×A4C22⋊C411yescharacteristic
60S4—41no—
61C2×C3×C4—81no—
62#67—81no—
63#67—81no—
64C2×Dic3—41no—
65C2×Dic3—81no—
66C24—31no—
67C24—31no—
68C24—31no—
69C24—31no—
70C24—31no—
71C24—41no—
72C24—121no—
73C24C4×D311yescharacteristic
74C24D1211yescharacteristic
75C24#6711yescharacteristic
76C2×D4—61no—
77C2×D4—61no—
78C2×D4—61no—
79C2×D4—61no—
80C2×D4—61no—
81C2×D4—61no—
82C2×D4—61no—
83C2×D4—61no—
84C2×D4—61no—
85C2×D4—61no—
86C2×D4—121no—
87C2×D4—121no—
88C2×D4—122no—
89C22×C4—31no—
90C22×C4—31no—
91C22×C4—31no—
92C22×C4—41no—
93C22×C4—61no—
94C22×C4—61no—
95C22×C4—61no—
96C22×C4—61no—
97C22×C4—121no—
98C22⋊C4—32no—
99C22⋊C4—31no—
100C22⋊C4—61no—
101C22⋊C4—61no—
102C22⋊C4—61no—
103C22⋊C4—61no—
104C22⋊C4—61no—
105C22⋊C4—61no—
106C22⋊C4—61no—
107C22⋊C4—61no—
108C22⋊C4—61no—
109C22⋊C4—61no—
110C22⋊C4—61no—
111C22⋊C4—121no—
112C22⋊C4—121no—
113C22⋊C4—122no—
114C22⋊C4S411yescharacteristic, complemented
115C22×C3—41no—
116C22×C3—41no—
117C22×C3—41no—
118C22×C3—161no—
119C2×D3—81no—
120C2×D3—161no—
121A4#9911yescharacteristic, complemented
122C3×C4—161no—
123Dic3—81no—
124Dic3—161no—
125C23—21no—
126C23—21no—
127C23—21no—
128C23—31no—
129C23—31no—
130C23—31no—
131C23—31no—
132C23—31no—
133C23—31no—
134C23—31no—
135C23—31no—
136C23—31no—
137C23—61no—
138C23—61no—
139C23—61no—
140C23—61no—
141C23—62no—
142C23—62no—
143C23—61no—
144C23—61no—
145C23—61no—
146C23—121no—
147C23—121no—
148C23—121no—
149C23—121no—
150C23—121no—
151C23—121no—
152C23—121no—
153C23—121no—
154C23(C2×D3)⋊C411yescharacteristic
155C23C2×S411yescharacteristic
156D4—121no—
157D4—121no—
158D4—121no—
159D4—122no—
160D4—122no—
161D4—122no—
162D4—121no—
163D4—121no—
164D4—121no—
165C2×C4—21no—
166C2×C4—61no—
167C2×C4—61no—
168C2×C4—61no—
169C2×C4—61no—
170C2×C4—61no—
171C2×C4—61no—
172C2×C4—121no—
173C2×C4—121no—
174C2×C4—121no—
175C2×C4—121no—
176C2×C4—121no—
177C2×C4—121no—
178C2×C4—121no—
179C2×C4—121no—
180C2×C4—121no—
181C2×C4—121no—
182C2×C4—121no—
183C2×C4—121no—
184C2×C3—41no—
185C2×C3—81no—
186C2×C3—81no—
187C2×C3—81no—
188D3—161no—
189C22—31no—
190C22—31no—
191C22—32no—
192C22—31no—
193C22—31no—
194C22—31no—
195C22—41no—
196C22—61no—
197C22—61no—
198C22—61no—
199C22—61no—
200C22—61no—
201C22—61no—
202C22—61no—
203C22—61no—
204C22—61no—
205C22—61no—
206C22—61no—
207C22—121no—
208C22—121no—
209C22—121no—
210C22—121no—
211C22—121no—
212C22—122no—
213C22—121no—
214C22—241no—
215C22—241no—
216C22#80611yescharacteristic, complemented
217C22C4×S411yescharacteristic
218C22C4⋊S411yescharacteristic
219C22GL(2,ℤ/4)11yescharacteristic
220C4—41no—
221C4—121no—
222C4—121no—
223C4—121no—
224C4—121no—
225C4—241no—
226C4—241no—
227C3—41noSylow
228C2—21no—
229C2—21no—
230C2—21no—
231C2—31no—
232C2—31no—
233C2—61no—
234C2—61no—
235C2—61no—
236C2—241no—
237C2(C2×S4)⋊C411yescharacteristic
subgroupG/H
centerC2(C2×S4)⋊C4
commutatorC22×A4C2×C4
FrattiniC23C2×S4
FittingC22×C22⋊C4D3
socleC23(C2×D3)⋊C4
prime radicalC2×GL(2,ℤ/4)C2
cosocleC23×A4C22
hypercenterC22⋊C4S4
hypocenterA4#99
LCSG ▷ C22×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C2 ◁ C23 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. A4×C22⋊C4C2semidirect
3. #5432C2semidirect
6. C22×S4C4semidirect
11. C2×SL(2,ℤ/4)C4semidirect
14. C22×C22⋊C4D3semidirect
114. C22⋊C4S4semidirect
121. A4#99semidirect
216. C22#806semidirect

Maximal subgroups

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

subgroupindex
C2×GL(2,ℤ/4)2
A4×C22⋊C42
#54322
#18983
#8064