Finite Group Databasebeta

← #85384

Subgroups — #85384

Order 384.

283 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 283 conjugacy classes of subgroups — too many to draw. Only its 34 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×#986C211yesmaximal, characteristic, complemented
2C2×#983C211yesmaximal, characteristic, complemented
3#548⋊C3C211yesmaximal, characteristic, complemented
4#5425C211yesmaximal, characteristic, complemented
5#5421C211yesmaximal, characteristic, complemented
6#5395C211yesmaximal, characteristic, complemented
7C4×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
8#3275—31nomaximal, Sylow
9C2×#41⋊C3C2211yescharacteristic
10C22×SL(2,𝔽3)C2211yescharacteristic, complemented
11#986—22no—
12#983—22no—
13C2×GL(2,𝔽3)C2211yescharacteristic
14C2×GL(2,𝔽3)C2211yescharacteristic
15C2×#226C2211yescharacteristic, complemented
16#884—41nomaximal
17C4×SL(2,𝔽3)C2211yescharacteristic, complemented
18#859C2211yescharacteristic, complemented
19C2×#137—31no—
20C2×AGL(1,ℤ/8)—31no—
21#548D311yescharacteristic, complemented
22#546—31no—
23#495—31no—
24#470—31no—
25#469—31no—
26#464—31no—
27#463—31no—
28#452—31no—
29#450—31no—
30#449—31no—
31#448—31no—
32C4×SD16—31no—
33#436—31no—
34C2×#67—41no—
35C2×#67—41no—
36C2×D12—41no—
37C2×C4×D3—41no—
38#41⋊C3—21no—
39C2×SL(2,𝔽3)—21no—
40C2×SL(2,𝔽3)C2311yescharacteristic
41GL(2,𝔽3)—21no—
42GL(2,𝔽3)D412yescomplemented
43#226—21no—
44C3×C22⋊C4—41no—
45(C2×D3)⋊C4—41no—
46Dic3⋊C4—41no—
47C2×#41—31no—
48C2×#41C2×D311yescharacteristic
49C22×Q8C2×D311yescharacteristic
50C22×D4—31no—
51#137—62no—
52#137—62no—
53AGL(1,ℤ/8)—62no—
54AGL(1,ℤ/8)—62no—
55C2×Q16—31no—
56C2×Q16—31no—
57C2×SD16—31no—
58C2×SD16—31no—
59C2×SD16—31no—
60C2×SD16—31no—
61C2×SD16—31no—
62C2×D8—31no—
63C2×D8—31no—
64C2×M16—31no—
65C2×M16—31no—
66#124—31no—
67#124—31no—
68#123—61no—
69C22⋊Q8—31no—
70C22⋊Q8—31no—
71#121—31no—
72#121—31no—
73#121—61no—
74#120—61no—
75C4×Q8C2×D311yescharacteristic
76C4×D4—31no—
77C4×D4—31no—
78C2×C22⋊C4—61no—
79#106—31no—
80C4⋊C8—31no—
81#103—31no—
82#103—31no—
83#103—31no—
84#102—31no—
85#102—31no—
86#102—31no—
87C22⋊C8—31no—
88C22⋊C8—31no—
89C4×C8—31no—
90C23×C3—41no—
91C22×D3—41no—
92C22×D3—41no—
93C2×C3×C4—41no—
94C2×C3×C4—41no—
95#67—82no—
96#67—82no—
97C2×Dic3—41no—
98C2×Dic3—41no—
99D12—82no—
100C4×D3—82no—
101SL(2,𝔽3)C2×D411yescharacteristic
102C24—61no—
103#41—21no—
104#41—61no—
105#41—62no—
106#41—121no—
107C2×Q8—21no—
108C2×Q8—31no—
109C2×Q8—31no—
110C2×Q8—61no—
111C2×Q8C22×D311yescharacteristic
112C2×D4—31no—
113C2×D4—31no—
114C2×D4—31no—
115C2×D4—31no—
116C2×D4—62no—
117C2×D4—61no—
118C2×D4—61no—
119C2×D4—61no—
120C2×D4—121no—
121C22×C4—31no—
122C22×C4—31no—
123C22×C4—61no—
124C22×C4—61no—
125Q16—62no—
126Q16—61no—
127Q16—61no—
128SD16—32no—
129SD16—32no—
130SD16—62no—
131SD16—61no—
132SD16—61no—
133SD16—121no—
134SD16—121no—
135D8—62no—
136D8—62no—
137M16—62no—
138M16—62no—
139C2×C8—31no—
140C2×C8—31no—
141C2×C8—31no—
142C2×C8—31no—
143C4⋊C4—31no—
144C4⋊C4—31no—
145C4⋊C4—31no—
146C4⋊C4—61no—
147C22⋊C4—31no—
148C22⋊C4—61no—
149C22⋊C4—61no—
150C22⋊C4—61no—
151C22⋊C4—61no—
152C22⋊C4—61no—
153C22⋊C4—61no—
154C22⋊C4S411yescharacteristic
155C42—31no—
156C22×C3—41no—
157C22×C3—81no—
158C22×C3—81no—
159C22×C3—81no—
160C2×D3—42no—
161C2×D3—81no—
162C2×D3—81no—
163C2×D3—81no—
164C2×D3—81no—
165C2×D3—81no—
166C3×C4—81no—
167C3×C4—81no—
168Dic3—42no—
169Dic3—81no—
170C23—31no—
171C23—61no—
172C23—61no—
173C23—122no—
174C23—122no—
175C23—241no—
176C23C2×S411yescharacteristic
177Q8—31no—
178Q8—61no—
179Q8—61no—
180Q8—61no—
181Q8—61no—
182Q8D3×D411yescharacteristic
183D4—32no—
184D4—61no—
185D4—61no—
186D4—61no—
187D4—61no—
188D4—61no—
189D4—122no—
190D4—121no—
191D4—122no—
192D4—241no—
193C2×C4—31no—
194C2×C4—31no—
195C2×C4—31no—
196C2×C4—61no—
197C2×C4—61no—
198C2×C4—61no—
199C2×C4—61no—
200C2×C4—61no—
201C2×C4—61no—
202C2×C4—122no—
203C2×C4—241no—
204C2×C4C2×S411yescharacteristic
205C2×C4C2×S411yescharacteristic
206C8—32no—
207C8—61no—
208C8—61no—
209C8—61no—
210C2×C3—41no—
211C2×C3—41no—
212C2×C3—41no—
213C2×C3—161no—
214D3—82no—
215D3—161no—
216C22—21no—
217C22—21no—
218C22—21no—
219C22—61no—
220C22—62no—
221C22—61no—
222C22—61no—
223C22—121no—
224C22—121no—
225C22—121no—
226C22—121no—
227C22—121no—
228C22—244no—
229C22C22×S411yescharacteristic
230C4—21no—
231C4—21no—
232C4—31no—
233C4—31no—
234C4—61no—
235C4—61no—
236C4—121no—
237C4—121no—
238C3—41noSylow
239C2—41no—
240C2—121no—
241C2—122no—
242C2—241no—
243C2D4×S411yescharacteristic
244C2#592311yescharacteristic
245C2#592611yescharacteristic
subgroupG/H
center, Frattini, socleC22C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23
Fitting#548D3
hypercenterC22⋊C4S4
hypocenterSL(2,𝔽3)C2×D4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#986C2semidirect
2. C2×#983C2semidirect
3. #548⋊C3C2semidirect
4. #5425C2semidirect
5. #5421C2semidirect
6. #5395C2semidirect
7. C4×GL(2,𝔽3)C2semidirect
10. C22×SL(2,𝔽3)C22semidirect
15. C2×#226C22semidirect
17. C4×SL(2,𝔽3)C22semidirect
18. #859C22semidirect
21. #548D3semidirect
42. GL(2,𝔽3)D4semidirect

Maximal subgroups

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

subgroupindex
C2×#9862
C2×#9832
#548⋊C32
#54252
#54212
#53952
C4×GL(2,𝔽3)2
#32753
#8844