Finite Group Databasebeta

← #85444

Subgroups — #85444

Order 384.

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

normal subgroups only

This group has 265 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
1#5912C211yesmaximal, characteristic
2C2×A4⋊Q8C211yesmaximal, characteristic, complemented
3A4×M16C211yesmaximal, characteristic, complemented
4#5401C212yesmaximal, complemented
5#5398C212yesmaximal, complemented
6#3056—31nomaximal, Sylow
7C2×C4×A4C2211yescharacteristic
8GL(2,ℤ/4)—21no—
9C2×SL(2,ℤ/4)—21no—
10C4⋊S4C2211yescharacteristic
11C4×S4—21no—
12A4⋊Q8—21no—
13A4⋊Q8C2212yes—
14A4⋊Q8C2211yescharacteristic
15#909—41nomaximal
16C8×A4C2212yes—
17C2×#137—31no—
18C22×M16D311yescharacteristic, complemented
19#525—31no—
20C2×C22⋊Q8—31no—
21#470—32no—
22#468—32no—
23#467—32no—
24#465—32no—
25#428—31no—
26#419—31no—
27#417—31no—
28C22×A4D411yescharacteristic
29C2×S4—21no—
30#235—41no—
31C2×Dic6—41no—
32C4×A4D411yescharacteristic
33C4×A4C2311yescharacteristic
34SL(2,ℤ/4)—22no—
35SL(2,ℤ/4)—21no—
36C3×M16—41no—
37Dic12—42no—
38#204—42no—
39C2×#41—31no—
40C22×Q8—31no—
41C23×C4C2×D311yescharacteristic
42#137—61no—
43#137—62no—
44#137—61no—
45C2×Q16—32no—
46C2×SD16—32no—
47C2×M16—31no—
48C2×M16—31no—
49C2×M16—32no—
50C22×C8C2×D312yes—
51#123—61no—
52C22⋊Q8—32no—
53C22⋊Q8—31no—
54C22⋊Q8—32no—
55C22⋊Q8—31no—
56C22⋊Q8—61no—
57C22⋊Q8—61no—
58#121—31no—
59#121—31no—
60#120—61no—
61C4×D4—61no—
62C4×D4—61no—
63#117—31no—
64C2×C4⋊C4—31no—
65C2×C4⋊C4—61no—
66C2×C22⋊C4—61no—
67#107—32no—
68#106—32no—
69#103—32no—
70#103—32no—
71#103—32no—
72#102—32no—
73C2×A4—21no—
74C2×A4C2×D411yescharacteristic
75S4—41no—
76C2×C3×C4—41no—
77#67—81no—
78C2×Dic3—81no—
79D12—41no—
80C4×D3—81no—
81Dic6—42no—
82Dic6—41no—
83Dic6—81no—
84C3×C8—42no—
85C24D1211yescharacteristic
86#41—61no—
87#41—61no—
88#41—61no—
89#41—61no—
90C2×Q8—32no—
91C2×Q8—31no—
92C2×Q8—61no—
93C2×Q8—61no—
94C2×Q8—61no—
95C2×Q8—61no—
96C2×Q8—61no—
97C2×D4—31no—
98C2×D4—61no—
99C2×D4—61no—
100C2×D4—61no—
101C22×C4—31no—
102C22×C4—31no—
103C22×C4—31no—
104C22×C4—31no—
105C22×C4—61no—
106C22×C4—61no—
107C22×C4—61no—
108C22×C4D1211yescharacteristic
109C22×C4C22×D311yescharacteristic
110Q16—62no—
111Q16—62no—
112SD16—62no—
113SD16—62no—
114M16—31no—
115M16—32no—
116M16—61no—
117M16S411yescharacteristic, complemented
118C2×C8—32no—
119C2×C8—32no—
120C4⋊C4—32no—
121C4⋊C4—31no—
122C4⋊C4—31no—
123C4⋊C4—61no—
124C4⋊C4—62no—
125C4⋊C4—62no—
126C4⋊C4—61no—
127C4⋊C4—61no—
128C22⋊C4—61no—
129C22⋊C4—61no—
130C22⋊C4—62no—
131C22⋊C4—61no—
132C22⋊C4—61no—
133C22⋊C4—62no—
134C22⋊C4—61no—
135C42—61no—
136C22×C3—41no—
137C2×D3—81no—
138A4#13711yescharacteristic, complemented
139C3×C4—41no—
140C3×C4—41no—
141Dic3—82no—
142Dic3—81no—
143C23—21no—
144C23—31no—
145C23—31no—
146C23—61no—
147C23—61no—
148C23C2×D1211yescharacteristic
149Q8—62no—
150Q8—61no—
151Q8—62no—
152Q8—61no—
153Q8—121no—
154Q8—121no—
155D4—61no—
156D4—61no—
157D4—121no—
158D4—121no—
159D4—122no—
160D4—121no—
161D4—121no—
162C2×C4—31no—
163C2×C4—31no—
164C2×C4—31no—
165C2×C4—31no—
166C2×C4—31no—
167C2×C4—31no—
168C2×C4—31no—
169C2×C4—61no—
170C2×C4—61no—
171C2×C4—61no—
172C2×C4—62no—
173C2×C4—61no—
174C2×C4—62no—
175C2×C4—121no—
176C2×C4—121no—
177C2×C4—122no—
178C2×C4—121no—
179C2×C4—121no—
180C2×C4C2×S411yescharacteristic
181C8—32no—
182C8C2×S412yes—
183C2×C3—41no—
184C2×C3—81no—
185D3—161no—
186C22—31no—
187C22—31no—
188C22—31no—
189C22—61no—
190C22—61no—
191C22—61no—
192C22—61no—
193C22—121no—
194C22—121no—
195C22—121no—
196C22#90911yescharacteristic, complemented
197C22C4⋊S411yescharacteristic
198C4—31no—
199C4—31no—
200C4—122no—
201C4—121no—
202C4—121no—
203C4—121no—
204C4—122no—
205C4C4⋊S411yescharacteristic
206C4C22×S411yescharacteristic
207C3—41noSylow
208C2—21no—
209C2—31no—
210C2—31no—
211C2—61no—
212C2—241no—
213C2C2×C4⋊S411yescharacteristic
subgroupG/H
centerC2C2×C4⋊S4
commutator, cosocleC4×A4C23
FrattiniC4C22×S4
FittingC22×M16D3
socleC23C2×D12
prime radical#5912C2
hypercenterM16S4
hypocenterA4#137
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
2. C2×A4⋊Q8C2semidirect
3. A4×M16C2semidirect
4. #5401C2semidirect
5. #5398C2semidirect
18. C22×M16D3semidirect
117. M16S4semidirect
138. A4#137semidirect
196. C22#909semidirect

Maximal subgroups

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

subgroupindex
#59122
C2×A4⋊Q82
A4×M162
#54012
#53982
#30563
#9094