Finite Group Databasebeta

← #85512

Subgroups — #85512

Order 384.

302 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 38 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
1A4×#41C211yesmaximal, characteristic
2C2×A4⋊Q8C211yesmaximal, characteristic, complemented
3#5416C212yesmaximal, complemented
4#5414C212yesmaximal, complemented
5#5409C211yesmaximal, characteristic, complemented
6#3007—31nomaximal, Sylow
7Q8×A4C2211yescharacteristic
8D4×A4—21no—
9D4×A4C2211yescharacteristic
10C2×C4×A4—21no—
11C2×C4×A4C2211yescharacteristic
12C2×SL(2,ℤ/4)—21no—
13A4⋊Q8—21no—
14A4⋊Q8C2212yes—
15#951—41nomaximal
16A4⋊C8C2212yes—
17C22×#41D311yescharacteristic
18C2×#137—31no—
19C2×#137—31no—
20C2×C22⋊Q8—31no—
21#452—32no—
22#452—32no—
23#451—32no—
24#450—32no—
25#417—31no—
26#417—31no—
27#407—31no—
28C22×A4—21no—
29C22×A4D411yescharacteristic
30C3×#41—41no—
31C2×Dic6—41no—
32C4×A4—21no—
33C4×A4D411yescharacteristic
34C4×A4C2311yescharacteristic
35SL(2,ℤ/4)—22no—
36#216—42no—
37#214—42no—
38#208—41no—
39C2×#41—31no—
40C2×#41—31no—
41C2×#41—31no—
42C2×#41—31no—
43C2×#41—61no—
44C2×#41—61no—
45C22×Q8—31no—
46C22×Q8C2×D311yescharacteristic
47C22×D4—21no—
48C22×D4C2×D311yescharacteristic
49C23×C4—21no—
50C23×C4C2×D311yescharacteristic
51#137—62no—
52#137—62no—
53#137—61no—
54#137—61no—
55#137—61no—
56#137—61no—
57C2×Q16—32no—
58C2×Q16—32no—
59C2×SD16—32no—
60C2×SD16—32no—
61C2×M16—31no—
62C2×M16—31no—
63C22⋊Q8—32no—
64C22⋊Q8—32no—
65C22⋊Q8—61no—
66C22⋊Q8—61no—
67C2×C4⋊C4—31no—
68C2×C4⋊C4—61no—
69C2×C22⋊C4—61no—
70#103—32no—
71#103—32no—
72#102—32no—
73#102—32no—
74C22⋊C8—32no—
75C22⋊C8—32no—
76C2×A4—21no—
77C2×A4—41no—
78C2×A4C2×D411yescharacteristic
79C3×Q8—41no—
80C3×D4—41no—
81C3×D4—81no—
82C2×C3×C4—41no—
83C2×C3×C4—81no—
84C2×Dic3—81no—
85Dic6—42no—
86Dic6—81no—
87C3⋊C8—42no—
88C24—21no—
89C24#6711yescharacteristic
90#41—32no—
91#41—32no—
92#41—31no—
93#41—31no—
94#41—31no—
95#41—61no—
96#41—61no—
97#41—61no—
98#41—62no—
99#41—62no—
100#41S411yescharacteristic
101C2×Q8—31no—
102C2×Q8—31no—
103C2×Q8—32no—
104C2×Q8—61no—
105C2×Q8—61no—
106C2×Q8—61no—
107C2×Q8—61no—
108C2×Q8—61no—
109C2×Q8—61no—
110C2×D4—31no—
111C2×D4—31no—
112C2×D4—61no—
113C2×D4—61no—
114C2×D4—61no—
115C2×D4—61no—
116C2×D4—61no—
117C22×C4—21no—
118C22×C4—31no—
119C22×C4—31no—
120C22×C4—31no—
121C22×C4—31no—
122C22×C4—61no—
123C22×C4—61no—
124C22×C4—61no—
125C22×C4—61no—
126C22×C4—61no—
127C22×C4—61no—
128C22×C4#6711yescharacteristic
129C22×C4C22×D311yescharacteristic
130Q16—64no—
131Q16—62no—
132Q16—62no—
133SD16—64no—
134SD16—62no—
135SD16—62no—
136M16—61no—
137M16—61no—
138M16—61no—
139M16—61no—
140C2×C8—32no—
141C2×C8—32no—
142C4⋊C4—32no—
143C4⋊C4—61no—
144C4⋊C4—62no—
145C4⋊C4—62no—
146C22⋊C4—62no—
147C22⋊C4—62no—
148C22×C3—41no—
149C22×C3—81no—
150A4#13711yescharacteristic, complemented
151C3×C4—41no—
152C3×C4—41no—
153C3×C4—81no—
154Dic3—82no—
155C23—21no—
156C23—31no—
157C23—31no—
158C23—41no—
159C23—61no—
160C23—61no—
161C23—61no—
162C23—121no—
163C23C2×#6711yescharacteristic
164Q8—32no—
165Q8—31no—
166Q8—61no—
167Q8—62no—
168Q8—62no—
169Q8—121no—
170Q8—121no—
171Q8C2×S411yescharacteristic
172D4—21no—
173D4—32no—
174D4—31no—
175D4—61no—
176D4—61no—
177D4—61no—
178D4—62no—
179D4—61no—
180D4C2×S411yescharacteristic
181C2×C4—21no—
182C2×C4—31no—
183C2×C4—31no—
184C2×C4—31no—
185C2×C4—31no—
186C2×C4—31no—
187C2×C4—31no—
188C2×C4—31no—
189C2×C4—61no—
190C2×C4—61no—
191C2×C4—61no—
192C2×C4—61no—
193C2×C4—62no—
194C2×C4—61no—
195C2×C4—61no—
196C2×C4—62no—
197C2×C4—62no—
198C2×C4—121no—
199C2×C4—122no—
200C2×C4—121no—
201C2×C4C2×S411yescharacteristic
202C8—62no—
203C8—62no—
204C2×C3—41no—
205C2×C3—81no—
206C2×C3—161no—
207C22—21no—
208C22—31no—
209C22—31no—
210C22—31no—
211C22—61no—
212C22—61no—
213C22—61no—
214C22—61no—
215C22—61no—
216C22—121no—
217C22—121no—
218C22—121no—
219C22—121no—
220C22#95111yescharacteristic, complemented
221C22GL(2,ℤ/4)11yescharacteristic
222C4—21no—
223C4—31no—
224C4—31no—
225C4—61no—
226C4—122no—
227C4—122no—
228C4GL(2,ℤ/4)11yescharacteristic
229C4C22×S411yescharacteristic
230C3—41noSylow
231C2—21no—
232C2—31no—
233C2—31no—
234C2—41no—
235C2—61no—
236C2—121no—
237C2C2×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC2C2×GL(2,ℤ/4)
commutator, cosocleC4×A4C23
FrattiniC4C22×S4
FittingC22×#41D3
socleC23C2×#67
prime radicalA4×#41C2
hypercenter#41S4
hypocenterA4#137
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
2. C2×A4⋊Q8C2semidirect
3. #5416C2semidirect
4. #5414C2semidirect
5. #5409C2semidirect
150. A4#137semidirect
220. C22#951semidirect

Maximal subgroups

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

subgroupindex
A4×#412
C2×A4⋊Q82
#54162
#54142
#54092
#30073
#9514