Finite Group Databasebeta

← #85474

Subgroups — #85474

Order 384.

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

normal subgroups only

This group has 250 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#5926C211yesmaximal, characteristic, complemented
3#578⋊C3C211yesmaximal, characteristic, complemented
4#5431C211yesmaximal, characteristic, complemented
5#5430C211yesmaximal, characteristic, complemented
6#5406C211yesmaximal, characteristic, complemented
7#5405C211yesmaximal, characteristic, complemented
8#3289—31nomaximal, Sylow
92−1+4⋊C3C2211yescharacteristic
102−1+4⋊C3C2211yescharacteristic, complemented
11#986C2211yescharacteristic
12#985C2211yescharacteristic
13#984C2211yescharacteristic, complemented
14#983—21no—
15#983—21no—
16C2×GL(2,𝔽3)—21no—
17C2×#226—21no—
18#911—41nomaximal
19#131⋊C3C2211yescharacteristic, complemented
20GU(2,𝔽3)C2211yescharacteristic, complemented
21#578—31no—
22#578D311yescharacteristic, complemented
23#577—31no—
24#577—31no—
25#497—31no—
26#473—31no—
27#473—31no—
28#471—31no—
29#471—31no—
30#456—31no—
31#456—31no—
32#455—31no—
33#454—31no—
34#444—31no—
35#444—31no—
36#237—41no—
37D3×D4—41no—
38#41⋊C3C2311yescharacteristic
39C2×SL(2,𝔽3)—21no—
40C2×SL(2,𝔽3)—21no—
41GL(2,𝔽3)—21no—
42GL(2,𝔽3)D411yescharacteristic
43#226—21no—
44#226D411yescharacteristic
45C3×D8—41no—
46#214—41no—
47#213—41no—
48#204—41no—
49ΓL(1,𝔽25)—41no—
502−1+4—31no—
512−1+4C2×D311yescharacteristic
522−1+4C2×D311yescharacteristic, complemented
53H(5,𝔽2)—31no—
54#137—31no—
55#137—31no—
56#137—31no—
57#137—31no—
58#137—31no—
59#137—31no—
60#137—61no—
61#137—61no—
62#137—61no—
63#137—61no—
64AGL(1,ℤ/8)—31no—
65AGL(1,ℤ/8)—31no—
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—31no—
75C2×Q16—61no—
76C2×SD16—31no—
77C2×SD16—31no—
78C2×SD16—61no—
79C2×SD16—61no—
80#131—31no—
81#131—31no—
82#131—31no—
83#131C2×D311yescharacteristic, complemented
84C4⋊Q8—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#101—31no—
95#100—31no—
96#97—31no—
97C22×D3—81no—
98C3×D4—41no—
99C3×D4—41no—
100#67—81no—
101#67—81no—
102C2×Dic3—81no—
103D12—41no—
104C4×D3—41no—
105Dic6—41no—
106SL(2,𝔽3)C2×D411yescharacteristic
107C3×C8—41no—
108C3⋊C8—41no—
109#41—31no—
110#41—31no—
111#41—31no—
112#41—61no—
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—21no—
122C2×Q8—31no—
123C2×Q8—31no—
124C2×Q8—31no—
125C2×Q8—61no—
126C2×Q8—61no—
127C2×D4—31no—
128C2×D4—61no—
129C2×D4—61no—
130C2×D4—121no—
131Q16—31no—
132Q16—31no—
133Q16—61no—
134Q16—61no—
135Q16—61no—
136Q16—61no—
137Q16—61no—
138Q16—61no—
139Q16—61no—
140SD16—31no—
141SD16—31no—
142SD16—31no—
143SD16—31no—
144SD16—61no—
145SD16—61no—
146SD16—61no—
147SD16—61no—
148SD16—61no—
149SD16—61no—
150SD16—61no—
151SD16—61no—
152SD16—61no—
153SD16—61no—
154D8—31no—
155D8—61no—
156D8—61no—
157D8S411yescharacteristic
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—
170C4⋊C4—121no—
171C22⋊C4—121no—
172C42—31no—
173C22×C3—81no—
174C22×C3—81no—
175C2×D3—41no—
176C2×D3—81no—
177C2×D3—162no—
178C3×C4—41no—
179Dic3—41no—
180Dic3—81no—
181C23—61no—
182C23—121no—
183Q8—31no—
184Q8—31no—
185Q8—31no—
186Q8—61no—
187Q8—61no—
188Q8—61no—
189Q8—61no—
190Q8—121no—
191Q8D3×D411yescharacteristic
192D4—31no—
193D4—31no—
194D4—61no—
195D4—61no—
196D4—61no—
197D4—61no—
198D4—61no—
199D4—121no—
200D4—121no—
201D4—121no—
202D4C2×S411yescharacteristic
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—61no—
212C2×C4—121no—
213C2×C4—121no—
214C2×C4—121no—
215C8—31no—
216C8—31no—
217C8—31no—
218C8—61no—
219C8—61no—
220C8—61no—
221C8C2×S411yescharacteristic
222C2×C3—41no—
223C2×C3—161no—
224C2×C3—161no—
225D3—81no—
226D3—161no—
227C22—21no—
228C22—21no—
229C22—31no—
230C22—61no—
231C22—121no—
232C22—241no—
233C22—481no—
234C4—31no—
235C4—61no—
236C4—61no—
237C4—61no—
238C4—121no—
239C4—121no—
240C4C22×S411yescharacteristic
241C3—41noSylow
242C2—41no—
243C2—41no—
244C2—61no—
245C2—121no—
246C2—241no—
247C2D4×S411yescharacteristic
subgroupG/H
center, socleC2D4×S4
commutator, cosocle#41⋊C3C23
FrattiniC4C22×S4
Fitting#578D3
hypercenterD8S4
hypocenterSL(2,𝔽3)C2×D4
LCSG ▷ #41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ D8

Complements

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

NH ≅ G/Ntype
1. #5927C2semidirect
2. #5926C2semidirect
3. #578⋊C3C2semidirect
4. #5431C2semidirect
5. #5430C2semidirect
6. #5406C2semidirect
7. #5405C2semidirect
10. 2−1+4⋊C3C22semidirect
13. #984C22semidirect
19. #131⋊C3C22semidirect
20. GU(2,𝔽3)C22semidirect
22. #578D3semidirect
52. 2−1+4C2×D3semidirect
83. #131C2×D3semidirect

Maximal subgroups

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

subgroupindex
#59272
#59262
#578⋊C32
#54312
#54302
#54062
#54052
#32893
#9114