Finite Group Databasebeta

← #85472

Subgroups — #85472

Order 384.

248 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 32 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#5927C211yesmaximal, characteristic, complemented
2#5925C211yesmaximal, characteristic, complemented
3#577⋊C3C211yesmaximal, characteristic, complemented
4#5431C211yesmaximal, characteristic, complemented
5#5428C211yesmaximal, characteristic, complemented
6#5406C211yesmaximal, characteristic, complemented
7#5404C211yesmaximal, characteristic, complemented
8#3290—31nomaximal, Sylow
92−1+4⋊C3C2211yescharacteristic, complemented
10#994C2211yescharacteristic
11#986—21no—
12#986C2211yescharacteristic
13#985—21no—
14#985C2211yescharacteristic
15#984C2211yescharacteristic, complemented
16#983—21no—
17C2×#226—21no—
18#916—41nomaximal
19#131⋊C3C2211yescharacteristic, complemented
20GU(2,𝔽3)C2211yescharacteristic, complemented
21#578—31no—
22#577—31no—
23#577D311yescharacteristic, complemented
24#576—31no—
25#495—31no—
26#473—31no—
27#472—31no—
28#471—31no—
29#471—31no—
30#455—31no—
31#455—31no—
32#454—31no—
33#454—31no—
34#443—31no—
35#443—31no—
36#239—41no—
37#237—41no—
38#41⋊C3—21no—
39#41⋊C3C2311yescharacteristic
40C2×SL(2,𝔽3)—21no—
41GL(2,𝔽3)—21no—
42GL(2,𝔽3)D411yescharacteristic
43#226—21no—
44#226D411yescharacteristic
45C3×SD16—41no—
46#216—41no—
47#213—41no—
48#204—41no—
49D3×C8—41no—
502−1+4—31no—
512−1+4C2×D311yescharacteristic
52H(5,𝔽2)—31no—
53H(5,𝔽2)C2×D311yescharacteristic
54#137—31no—
55#137—31no—
56#137—31no—
57#137—31no—
58#137—61no—
59#137—61no—
60#137—61no—
61AGL(1,ℤ/8)—31no—
62AGL(1,ℤ/8)—31no—
63AGL(1,ℤ/8)—31no—
64AGL(1,ℤ/8)—31no—
65AGL(1,ℤ/8)—61no—
66AGL(1,ℤ/8)—61no—
67AGL(1,ℤ/8)—61no—
68#135—31no—
69#135—31no—
70#135—61no—
71#135—61no—
72#135—61no—
73C2×Q16—31no—
74C2×Q16—61no—
75C2×SD16—31no—
76C2×SD16—31no—
77C2×SD16—61no—
78C2×D8—31no—
79C2×D8—61no—
80#131—31no—
81#131—31no—
82#131—31no—
83#131C2×D311yescharacteristic
84#124—31no—
85#124—31no—
86#108—31no—
87#108—31no—
88#104—31no—
89#104—31no—
90#104—31no—
91#104—31no—
92#101—31no—
93#101—31no—
94#100—31no—
95#100—31no—
96C4×C8—31no—
97C3×Q8—41no—
98C3×D4—41no—
99#67—81no—
100C2×Dic3—81no—
101D12—41no—
102D12—81no—
103C4×D3—41no—
104C4×D3—81no—
105Dic6—41no—
106SL(2,𝔽3)C2×D411yescharacteristic
107C3×C8—41no—
108C3⋊C8—41no—
109#41—21no—
110#41—31no—
111#41—31no—
112#41—31no—
113#41—61no—
114#41—61no—
115#41—61no—
116#41—61no—
117#41—61no—
118#41—121no—
119#41C22×D311yescharacteristic
120C2×Q8—21no—
121C2×Q8—31no—
122C2×Q8—31no—
123C2×Q8—61no—
124C2×Q8—61no—
125C2×D4—31no—
126C2×D4—31no—
127C2×D4—61no—
128C2×D4—61no—
129C2×D4—61no—
130C2×D4—121no—
131Q16—31no—
132Q16—31no—
133Q16—61no—
134Q16—61no—
135Q16—61no—
136Q16—61no—
137Q16—61no—
138SD16—31no—
139SD16—31no—
140SD16—31no—
141SD16—61no—
142SD16—61no—
143SD16—61no—
144SD16—61no—
145SD16—61no—
146SD16—61no—
147SD16—61no—
148SD16—61no—
149SD16—61no—
150SD16S411yescharacteristic
151D8—31no—
152D8—31no—
153D8—61no—
154D8—61no—
155D8—61no—
156D8—61no—
157D8—61no—
158M16—31no—
159M16—31no—
160M16—31no—
161M16—31no—
162M16—61no—
163M16—61no—
164M16—61no—
165C2×C8—31no—
166C2×C8—31no—
167C2×C8—61no—
168C2×C8—61no—
169C2×C8—61no—
170C22⋊C4—121no—
171C22⋊C4—121no—
172C42—31no—
173C22×C3—81no—
174C2×D3—41no—
175C2×D3—81no—
176C3×C4—41no—
177C3×C4—81no—
178Dic3—41no—
179Dic3—81no—
180C23—61no—
181C23—61no—
182C23—121no—
183Q8—31no—
184Q8—31no—
185Q8—61no—
186Q8—61no—
187Q8—61no—
188Q8—121no—
189Q8D3×D411yescharacteristic
190Q8C2×S411yescharacteristic
191D4—31no—
192D4—31no—
193D4—31no—
194D4—61no—
195D4—61no—
196D4—61no—
197D4—61no—
198D4—61no—
199D4—61no—
200D4—121no—
201D4—121no—
202D4—121no—
203D4C2×S411yescharacteristic
204C2×C4—31no—
205C2×C4—61no—
206C2×C4—61no—
207C2×C4—61no—
208C2×C4—61no—
209C2×C4—61no—
210C2×C4—61no—
211C2×C4—121no—
212C2×C4—121no—
213C2×C4—121no—
214C8—31no—
215C8—31no—
216C8—31no—
217C8—61no—
218C8—61no—
219C8—61no—
220C8C2×S411yescharacteristic
221C2×C3—41no—
222C2×C3—161no—
223D3—81no—
224D3—161no—
225C22—21no—
226C22—31no—
227C22—61no—
228C22—61no—
229C22—121no—
230C22—241no—
231C22—241no—
232C22—481no—
233C4—21no—
234C4—31no—
235C4—61no—
236C4—61no—
237C4—121no—
238C4—121no—
239C4C22×S411yescharacteristic
240C3—41noSylow
241C2—41no—
242C2—61no—
243C2—121no—
244C2—121no—
245C2—241no—
246C2D4×S411yescharacteristic
subgroupG/H
center, socleC2D4×S4
commutator, cosocle#41⋊C3C23
FrattiniC4C22×S4
Fitting#577D3
hypercenterSD16S4
hypocenterSL(2,𝔽3)C2×D4
LCSG ▷ #41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ SD16

Complements

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

NH ≅ G/Ntype
1. #5927C2semidirect
2. #5925C2semidirect
3. #577⋊C3C2semidirect
4. #5431C2semidirect
5. #5428C2semidirect
6. #5406C2semidirect
7. #5404C2semidirect
9. 2−1+4⋊C3C22semidirect
15. #984C22semidirect
19. #131⋊C3C22semidirect
20. GU(2,𝔽3)C22semidirect
23. #577D3semidirect

Maximal subgroups

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

subgroupindex
#59272
#59252
#577⋊C32
#54312
#54282
#54062
#54042
#32903
#9164