Finite Group Databasebeta

← #3586⋊C3 #85641

Subgroups — #3586⋊C3 #85641

Order 384.

337 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 337 conjugacy classes of subgroups — too many to draw. Only its 58 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#5945C211yesmaximal, characteristic, complemented
2C2×2−1+4⋊C3C211yesmaximal, characteristic, complemented
3#578⋊C3C212yesmaximal, complemented
4#577⋊C3C212yesmaximal, complemented
5#568⋊C3C211yesmaximal, characteristic, complemented
6#3586C311yesmaximal, characteristic, complemented, Sylow
72−1+4⋊C3—21no—
82−1+4⋊C3—21no—
92−1+4⋊C3C2212yescomplemented
102−1+4⋊C3C2211yescharacteristic, complemented
11#994C2211yescharacteristic, complemented
12C2×#41⋊C3—21no—
13C2×#41⋊C3C2211yescharacteristic
14C22×SL(2,𝔽3)—21no—
15C3×AGL(1,ℤ/8)—41nomaximal
16#131⋊C3C2212yescomplemented
17#585C2×C311yescharacteristic, complemented
18C2×2−1+4C2×C311yescharacteristic, complemented
19#578—32no—
20#578C2×C312yescomplemented
21#577—32no—
22#577C2×C312yescomplemented
23#575—31no—
24#575—31no—
25C2×#137—31no—
26C2×#137—31no—
27#568C2×C311yescharacteristic, complemented
28C3×#41—41no—
29C2×C3×D4—41no—
30#41⋊C3—21no—
31#41⋊C3D411yescharacteristic
32#41⋊C3C2311yescharacteristic
33C2×SL(2,𝔽3)—22no—
34C2×SL(2,𝔽3)—21no—
35C2×SL(2,𝔽3)D411yescharacteristic
36C3×SD16—42no—
37C3×D8—42no—
38C3×M16—41no—
392−1+4—21no—
402−1+4—21no—
412−1+4—32no—
422−1+4—31no—
432−1+4—61no—
442−1+4C22×C312yes—
452−1+4C22×C311yescharacteristic, complemented
46H(5,𝔽2)—31no—
47H(5,𝔽2)—61no—
48H(5,𝔽2)C22×C311yescharacteristic, complemented
49C2×#41—21no—
50C2×#41—31no—
51C2×#41—31no—
52C2×#41—31no—
53C2×#41—61no—
54C2×#41—61no—
55C2×#41C22×C311yescharacteristic
56C22×Q8—21no—
57C22×Q8—31no—
58#137—32no—
59#137—34no—
60#137—31no—
61#137—32no—
62#137—32no—
63#137—32no—
64#137—32no—
65#137—31no—
66AGL(1,ℤ/8)—31no—
67AGL(1,ℤ/8)—32no—
68AGL(1,ℤ/8)—32no—
69AGL(1,ℤ/8)A411yescharacteristic
70#135—34no—
71#135—32no—
72#135—32no—
73C2×Q16—32no—
74C2×Q16—32no—
75C2×SD16—32no—
76C2×SD16—32no—
77#131—32no—
78#131C22×C312yescomplemented
79C2×M16—31no—
80C2×M16—31no—
81C23×C3—81no—
82C3×Q8—41no—
83C3×D4—41no—
84C3×D4—42no—
85C3×D4—81no—
86C3×D4—81no—
87C2×C3×C4—41no—
88C2×C3×C4—81no—
89SL(2,𝔽3)C2×D411yescharacteristic
90C3×C8—42no—
91#41—21no—
92#41—31no—
93#41—31no—
94#41—32no—
95#41—31no—
96#41—32no—
97#41—32no—
98#41—31no—
99#41—31no—
100#41—61no—
101#41—61no—
102#41—61no—
103#41—61no—
104#41—61no—
105#41—61no—
106#41—62no—
107#41—64no—
108#41—62no—
109#41—61no—
110#41—61no—
111#41—61no—
112#41—62no—
113#41C3×D411yescharacteristic
114#41C2×A411yescharacteristic
115#41C23×C311yescharacteristic
116C2×Q8—22no—
117C2×Q8—21no—
118C2×Q8—32no—
119C2×Q8—32no—
120C2×Q8—31no—
121C2×Q8—31no—
122C2×Q8—31no—
123C2×Q8—61no—
124C2×Q8—61no—
125C2×Q8—61no—
126C2×Q8—61no—
127C2×Q8—62no—
128C2×Q8C3×D411yescharacteristic
129C2×D4—31no—
130C2×D4—31no—
131C2×D4—31no—
132C2×D4—61no—
133C2×D4—61no—
134C2×D4—62no—
135C2×D4—61no—
136C2×D4—61no—
137C2×D4—61no—
138C2×D4C2×A411yescharacteristic
139C22×C4—31no—
140C22×C4—61no—
141C22×C4—61no—
142C22×C4—61no—
143C22×C4—61no—
144Q16—34no—
145Q16—32no—
146Q16—32no—
147SD16—34no—
148SD16—32no—
149SD16—32no—
150SD16—32no—
151SD16C2×A412yes—
152D8—32no—
153D8C2×A412yes—
154M16—31no—
155M16—32no—
156M16—32no—
157M16C2×A411yescharacteristic
158C2×C8—32no—
159C2×C8—32no—
160C22×C3—41no—
161C22×C3—81no—
162C22×C3—82no—
163C22×C3—162no—
164C3×C4—41no—
165C3×C4—41no—
166C3×C4—81no—
167C23—21no—
168C23—31no—
169C23—61no—
170C23—61no—
171Q8—31no—
172Q8—32no—
173Q8—31no—
174Q8—61no—
175Q8—62no—
176Q8—61no—
177Q8—62no—
178Q8—61no—
179Q8C2×C3×D411yescharacteristic
180Q8C22×A411yescharacteristic
181D4—21no—
182D4—21no—
183D4—31no—
184D4—31no—
185D4—32no—
186D4—31no—
187D4—61no—
188D4—61no—
189D4—61no—
190D4—61no—
191D4—62no—
192D4—62no—
193D4—62no—
194D4—61no—
195D4C22×A412yes—
196D4C22×A411yescharacteristic
197C2×C4—21no—
198C2×C4—31no—
199C2×C4—31no—
200C2×C4—31no—
201C2×C4—31no—
202C2×C4—31no—
203C2×C4—61no—
204C2×C4—61no—
205C2×C4—61no—
206C2×C4—62no—
207C2×C4—62no—
208C2×C4—62no—
209C2×C4—62no—
210C2×C4—61no—
211C2×C4—61no—
212C2×C4—61no—
213C2×C4—62no—
214C2×C4C22×A411yescharacteristic
215C8—32no—
216C8C22×A412yes—
217C2×C3—41no—
218C2×C3—81no—
219C2×C3—161no—
220C2×C3—162no—
221C22—21no—
222C22—22no—
223C22—31no—
224C22—31no—
225C22—42no—
226C22—61no—
227C22—121no—
228C22—241no—
229C22—241no—
230C22D4×A411yescharacteristic
231C4—21no—
232C4—31no—
233C4—31no—
234C4—61no—
235C4—62no—
236C4D4×A411yescharacteristic
237C4C23×A411yescharacteristic
238C3—41noSylow
239C2—21no—
240C2—42no—
241C2—41no—
242C2—61no—
243C2—61no—
244C2—121no—
245C2C2×D4×A411yescharacteristic
subgroupG/H
center, socleC2C2×D4×A4
commutator, cosocle#41C23×C3
FrattiniC4C23×A4
Fitting#3586C3
hypercenterAGL(1,ℤ/8)A4
hypocenterQ8C2×C3×D4
LCSG ▷ #41 ▷ Q8
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #5945C2semidirect
2. C2×2−1+4⋊C3C2semidirect
3. #578⋊C3C2semidirect
4. #577⋊C3C2semidirect
5. #568⋊C3C2semidirect
6. #3586C3semidirect
9. 2−1+4⋊C3C22semidirect
10. 2−1+4⋊C3C22semidirect
11. #994C22semidirect
16. #131⋊C3C22semidirect
17. #585C2×C3semidirect
18. C2×2−1+4C2×C3semidirect
20. #578C2×C3semidirect
22. #577C2×C3semidirect
27. #568C2×C3semidirect
45. 2−1+4C22×C3semidirect
48. H(5,𝔽2)C22×C3semidirect
78. #131C22×C3semidirect

Maximal subgroups

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

subgroupindex
#59452
C2×2−1+4⋊C32
#578⋊C32
#577⋊C32
#568⋊C32
#35863
C3×AGL(1,ℤ/8)4