Finite Group Databasebeta

← #87497

Subgroups — #87497

Order 384.

738 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 100 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#5927C213yesmaximal, complemented
3#5926C213yesmaximal, complemented
4#5925C211yesmaximal, characteristic, complemented
5#5924C211yesmaximal, characteristic, complemented
6#5923C213yesmaximal, complemented
7C2×#985C213yesmaximal, complemented
8#3584—31nomaximal, Sylow
9D3×#41—41nomaximal
102−1+4⋊C3C2213yescomplemented
11#994C2211yescharacteristic, complemented
12C2×#41⋊C3C2213yescomplemented
13#986C2213yescomplemented
14#985C2211yescharacteristic, complemented
15#985C2213yescomplemented
16#985C2213yescomplemented
17#985C2213yescomplemented
18#984C2213yescomplemented
19#983C2216yescomplemented
20C2×GL(2,𝔽3)C2213yescomplemented
21C2×#226C2213yescomplemented
22#585—31no—
23#585D311yescharacteristic, complemented
24#578—31no—
25#578—33no—
26#577—33no—
27#577—31no—
28#577—33no—
29#577—31no—
30#576—33no—
31#576—31no—
32#575—33no—
33#575—33no—
34C2×#135—33no—
35C2×#135—33no—
36C2×#131—31no—
37C3×#41—41no—
38#239—41no—
39D3×Q8—41no—
40#237—43no—
41D3×D4—43no—
42#235—43no—
43C2×C4×D3—43no—
44#41⋊C3C2313yes—
45#41⋊C3C2311yescharacteristic
46C2×SL(2,𝔽3)C2313yes—
47GL(2,𝔽3)C2311yescharacteristic
48GL(2,𝔽3)C2313yes—
49#226C2311yescharacteristic
50#226C2313yes—
512−1+4—31no—
522−1+4—33no—
532−1+4—31no—
542−1+4—61no—
552−1+4C2×D313yescomplemented
56H(5,𝔽2)—33no—
57H(5,𝔽2)—31no—
58H(5,𝔽2)—33no—
59H(5,𝔽2)—63no—
60H(5,𝔽2)C2×D311yescharacteristic, complemented
61C2×#41—31no—
62C2×#41—33no—
63C2×#41—33no—
64C2×#41—33no—
65C2×#41—63no—
66C2×#41—61no—
67C2×#41C2×D313yescomplemented
68#137—33no—
69#137—33no—
70#137—33no—
71#137—36no—
72#137—33no—
73#137—36no—
74AGL(1,ℤ/8)—33no—
75AGL(1,ℤ/8)—33no—
76AGL(1,ℤ/8)—36no—
77AGL(1,ℤ/8)—33no—
78AGL(1,ℤ/8)—36no—
79AGL(1,ℤ/8)—33no—
80#135—33no—
81#135—36no—
82#135—33no—
83#135—33no—
84#135—36no—
85#135—33no—
86#135—36no—
87#135—31no—
88#135—33no—
89#135—33no—
90#135—33no—
91C2×Q16—33no—
92C2×Q16—33no—
93C2×SD16—33no—
94C2×SD16—33no—
95C2×SD16—33no—
96C2×SD16—33no—
97C2×D8—33no—
98C2×D8—33no—
99#131—31no—
100#131—33no—
101#131—31no—
102#131—33no—
103C2×M16—33no—
104C22×C8—33no—
105C22×D3—43no—
106C3×Q8—41no—
107C3×D4—43no—
108C2×C3×C4—43no—
109#67—46no—
110C2×Dic3—43no—
111D12—43no—
112C4×D3—41no—
113C4×D3—43no—
114C4×D3—43no—
115C4×D3—43no—
116Dic6—43no—
117SL(2,𝔽3)C2411yescharacteristic
118#41—31no—
119#41—33no—
120#41—33no—
121#41—33no—
122#41—33no—
123#41—36no—
124#41—33no—
125#41—31no—
126#41—36no—
127#41—33no—
128#41—33no—
129#41—63no—
130#41—63no—
131#41—63no—
132#41—63no—
133#41—66no—
134#41—63no—
135#41—63no—
136#41—61no—
137#41—63no—
138#41—63no—
139#41S411yescharacteristic
140#41C22×D311yescharacteristic
141#41C22×D313yes—
142C2×Q8—31no—
143C2×Q8—33no—
144C2×Q8—33no—
145C2×Q8—33no—
146C2×Q8—63no—
147C2×Q8—61no—
148C2×Q8C22×D313yes—
149C2×D4—33no—
150C2×D4—33no—
151C2×D4—33no—
152C2×D4—33no—
153C2×D4—33no—
154C2×D4—63no—
155C2×D4—66no—
156C2×D4—63no—
157C2×D4—63no—
158C2×D4—63no—
159C2×D4—63no—
160C22×C4—33no—
161C22×C4—61no—
162C22×C4—63no—
163C22×C4—63no—
164Q16—33no—
165Q16—36no—
166Q16—33no—
167Q16—31no—
168Q16—33no—
169SD16—33no—
170SD16—31no—
171SD16—33no—
172SD16—36no—
173SD16—33no—
174SD16—33no—
175SD16—36no—
176SD16—33no—
177SD16—31no—
178SD16—33no—
179D8—33no—
180D8—31no—
181D8—36no—
182D8—33no—
183D8—33no—
184M16—36no—
185M16—33no—
186M16—33no—
187C2×C8—31no—
188C2×C8—33no—
189C2×C8—33no—
190C2×C8—33no—
191C2×C8—33no—
192C2×C8—33no—
193C22×C3—43no—
194C2×D3—41no—
195C2×D3—43no—
196C2×D3—86no—
197C3×C4—43no—
198C3×C4—41no—
199Dic3—43no—
200Dic3—41no—
201C23—33no—
202C23—61no—
203C23—63no—
204C23—63no—
205Q8—33no—
206Q8—33no—
207Q8—33no—
208Q8—31no—
209Q8—63no—
210Q8—63no—
211Q8C2×S411yescharacteristic
212Q8C23×D311yescharacteristic
213D4—31no—
214D4—33no—
215D4—36no—
216D4—33no—
217D4—31no—
218D4—33no—
219D4—63no—
220D4—63no—
221D4—63no—
222D4—61no—
223D4—63no—
224D4—63no—
225D4—66no—
226D4—63no—
227D4C2×S413yes—
228C2×C4—33no—
229C2×C4—33no—
230C2×C4—33no—
231C2×C4—31no—
232C2×C4—33no—
233C2×C4—63no—
234C2×C4—66no—
235C2×C4—63no—
236C2×C4—63no—
237C2×C4—63no—
238C2×C4—63no—
239C2×C4—63no—
240C2×C4—61no—
241C2×C4C2×S413yes—
242C8—31no—
243C8—31no—
244C8—33no—
245C8—33no—
246C2×C3—41no—
247C2×C3—83no—
248D3—42no—
249D3—83no—
250C22—31no—
251C22—33no—
252C22—61no—
253C22—63no—
254C22—123no—
255C22—243no—
256C22—241no—
257C22—243no—
258C22C22×S413yes—
259C4—31no—
260C4—33no—
261C4—63no—
262C4—61no—
263C4C22×S413yes—
264C4C22×S411yescharacteristic
265C3—41noSylow
266C2—23no—
267C2—61no—
268C2—63no—
269C2—121no—
270C2—123no—
271C2C23×S411yescharacteristic
subgroupG/H
centerC4C22×S4
commutator, cosocle, hypocenterSL(2,𝔽3)C24
Frattini, socleC2C23×S4
Fitting#585D3
hypercenter#41S4
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C4 ◁ #41

Complements

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

NH ≅ G/Ntype
1. #5945C2semidirect
2. #5927C2semidirect
3. #5926C2semidirect
4. #5925C2semidirect
5. #5924C2semidirect
6. #5923C2semidirect
7. C2×#985C2semidirect
10. 2−1+4⋊C3C22semidirect
11. #994C22semidirect
12. C2×#41⋊C3C22semidirect
13. #986C22semidirect
14. #985C22semidirect
15. #985C22semidirect
16. #985C22semidirect
17. #985C22semidirect
18. #984C22semidirect
19. #983C22semidirect
20. C2×GL(2,𝔽3)C22semidirect
21. C2×#226C22semidirect
23. #585D3semidirect
55. 2−1+4C2×D3semidirect
60. H(5,𝔽2)C2×D3semidirect
67. C2×#41C2×D3semidirect

Maximal subgroups

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

subgroupindex
#59452
#59272
#59262
#59252
#59242
#59232
C2×#9852
#35843
D3×#414