Finite Group Databasebeta

← #85544

Subgroups — #85544

Order 384.

297 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 297 conjugacy classes of subgroups — too many to draw. Only its 36 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
2#5923C211yesmaximal, characteristic, complemented
3#5431C211yesmaximal, characteristic, complemented
4#5430C211yesmaximal, characteristic, complemented
5#5429C211yesmaximal, characteristic, complemented
6#5428C211yesmaximal, characteristic, complemented
7C2×GU(2,𝔽3)C211yesmaximal, characteristic, complemented
8#3018—31nomaximal, Sylow
92−1+4⋊C3—21no—
102−1+4⋊C3C2211yescharacteristic
11#994C2211yescharacteristic
12C2×#41⋊C3—21no—
13C2×#41⋊C3C2211yescharacteristic
14#986C2211yescharacteristic, complemented
15#985—21no—
16#984C2211yescharacteristic, complemented
17#983—21no—
18#950—41nomaximal
19GU(2,𝔽3)C2211yescharacteristic, complemented
20GU(2,𝔽3)C2211yescharacteristic, complemented
21#585D311yescharacteristic, complemented
22#575—31no—
23#575—31no—
24#532—31no—
25#456—31no—
26#456—31no—
27#455—31no—
28#455—31no—
29#454—31no—
30#454—31no—
31#453—31no—
32#453—31no—
33C2×#104—31no—
34C2×#104—31no—
35#413—31no—
36C3×#41—41no—
37#235—41no—
38#41⋊C3—21no—
39#41⋊C3D411yescharacteristic, complemented
40#41⋊C3C2311yescharacteristic
41C2×SL(2,𝔽3)—21no—
42C2×SL(2,𝔽3)D411yescharacteristic, complemented
43GL(2,𝔽3)—21no—
44#226—21no—
45#216—41no—
46#215—41no—
47#214—41no—
48#213—41no—
49C2×C3⋊C8—41no—
502−1+4—21no—
512−1+4—31no—
522−1+4C2×D311yescharacteristic
53H(5,𝔽2)—31no—
54H(5,𝔽2)—61no—
55H(5,𝔽2)C2×D311yescharacteristic
56C2×#41—21no—
57C2×#41—31no—
58C2×#41—31no—
59C2×#41—31no—
60C2×#41—61no—
61C2×#41C2×D311yescharacteristic
62#137—31no—
63#137—31no—
64#137—31no—
65#137—31no—
66#137—61no—
67#137—61no—
68AGL(1,ℤ/8)—31no—
69AGL(1,ℤ/8)—31no—
70AGL(1,ℤ/8)—31no—
71AGL(1,ℤ/8)—31no—
72AGL(1,ℤ/8)—61no—
73AGL(1,ℤ/8)—61no—
74#135—61no—
75#135—61no—
76#135—61no—
77#135—61no—
78C2×M16—31no—
79C2×M16—31no—
80C4⋊Q8—31no—
81#127—31no—
82#124—31no—
83#124—31no—
84#121—62no—
85C4×D4—62no—
86C2×C42—31no—
87#104—31no—
88#104—31no—
89#104—31no—
90#104—31no—
91#104—31no—
92#104—31no—
93#104—31no—
94#104—31no—
95#101—31no—
96#101—31no—
97#100—31no—
98#100—31no—
99C3×Q8—41no—
100C3×D4—41no—
101C3×D4—81no—
102C2×C3×C4—41no—
103C2×C3×C4—81no—
104#67—81no—
105D12—41no—
106C4×D3—81no—
107Dic6—41no—
108SL(2,𝔽3)C2×D411yescharacteristic
109C3⋊C8—41no—
110C3⋊C8—41no—
111#41—21no—
112#41—31no—
113#41—31no—
114#41—31no—
115#41—31no—
116#41—31no—
117#41—61no—
118#41—61no—
119#41—61no—
120#41—61no—
121#41—62no—
122#41—61no—
123#41—61no—
124#41—61no—
125#41—61no—
126#41—61no—
127#41—61no—
128#41—61no—
129#41—61no—
130#41#6711yescharacteristic
131#41S411yescharacteristic
132#41C22×D311yescharacteristic
133C2×Q8—21no—
134C2×Q8—31no—
135C2×Q8—31no—
136C2×Q8—31no—
137C2×Q8—61no—
138C2×Q8#6711yescharacteristic
139C2×D4—31no—
140C2×D4—31no—
141C2×D4—31no—
142C2×D4—32no—
143C2×D4—31no—
144C2×D4—61no—
145C2×D4—61no—
146C2×D4—61no—
147C2×D4—62no—
148C2×D4—62no—
149C2×D4—61no—
150C22×C4—31no—
151C22×C4—32no—
152C22×C4—61no—
153C22×C4—61no—
154C22×C4—61no—
155Q16—61no—
156Q16—61no—
157Q16—61no—
158Q16—61no—
159SD16—61no—
160SD16—61no—
161SD16—61no—
162SD16—61no—
163SD16—61no—
164SD16—61no—
165SD16—61no—
166SD16—61no—
167D8—61no—
168D8—61no—
169D8—61no—
170D8—61no—
171M16—31no—
172M16—31no—
173M16—31no—
174M16—31no—
175M16—61no—
176M16—61no—
177C2×C8—61no—
178C2×C8—61no—
179C4⋊C4—62no—
180C22⋊C4—62no—
181C22⋊C4—62no—
182C42—31no—
183C42—31no—
184C42—61no—
185C22×C3—41no—
186C22×C3—81no—
187C2×D3—81no—
188C3×C4—41no—
189C3×C4—41no—
190C3×C4—81no—
191Dic3—81no—
192C23—31no—
193C23—32no—
194C23—61no—
195C23—61no—
196Q8—31no—
197Q8—31no—
198Q8—61no—
199Q8—61no—
200Q8—61no—
201Q8—61no—
202Q8C2×#6711yescharacteristic
203Q8C2×S411yescharacteristic
204D4—21no—
205D4—31no—
206D4—31no—
207D4—31no—
208D4—61no—
209D4—61no—
210D4—61no—
211D4—61no—
212D4—61no—
213D4—61no—
214D4—62no—
215D4—61no—
216D4—61no—
217D4—124no—
218D4—121no—
219D4—121no—
220D4C2×S411yescharacteristic
221C2×C4—21no—
222C2×C4—31no—
223C2×C4—31no—
224C2×C4—31no—
225C2×C4—31no—
226C2×C4—31no—
227C2×C4—32no—
228C2×C4—61no—
229C2×C4—61no—
230C2×C4—61no—
231C2×C4—62no—
232C2×C4—61no—
233C2×C4—61no—
234C2×C4—61no—
235C2×C4—61no—
236C2×C4—121no—
237C2×C4—122no—
238C2×C4—121no—
239C2×C4C2×S411yescharacteristic
240C8—61no—
241C8—61no—
242C8—61no—
243C8—61no—
244C2×C3—41no—
245C2×C3—81no—
246C2×C3—161no—
247D3—161no—
248C22—21no—
249C22—31no—
250C22—31no—
251C22—61no—
252C22—121no—
253C22—122no—
254C22—122no—
255C22—121no—
256C22—241no—
257C22GL(2,ℤ/4)11yescharacteristic
258C4—21no—
259C4—31no—
260C4—31no—
261C4—62no—
262C4—61no—
263C4—121no—
264C4—121no—
265C4GL(2,ℤ/4)11yescharacteristic
266C4C22×S411yescharacteristic
267C3—41noSylow
268C2—21no—
269C2—41no—
270C2—61no—
271C2—61no—
272C2—121no—
273C2—241no—
274C2C2×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC4GL(2,ℤ/4)
commutator, cosocle#41⋊C3C23
FrattiniC4C22×S4
Fitting#585D3
socleC2C2×GL(2,ℤ/4)
hypercenter#41S4
hypocenterSL(2,𝔽3)C2×D4
LCSG ▷ #41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C4 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
1. #5945C2semidirect
2. #5923C2semidirect
3. #5431C2semidirect
4. #5430C2semidirect
5. #5429C2semidirect
6. #5428C2semidirect
7. C2×GU(2,𝔽3)C2semidirect
14. #986C22semidirect
16. #984C22semidirect
19. GU(2,𝔽3)C22semidirect
20. GU(2,𝔽3)C22semidirect
21. #585D3semidirect
39. #41⋊C3D4semidirect
42. C2×SL(2,𝔽3)D4semidirect

Maximal subgroups

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

subgroupindex
#59452
#59232
#54312
#54302
#54292
#54282
C2×GU(2,𝔽3)2
#30183
#9504