Finite Group Databasebeta

← #13147302

Subgroups — #13147302

Order 1728.

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

normal subgroups only

This group has 374 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
12−1+4⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#11722436C211yesmaximal, characteristic, complemented
3#11720767C211yesmaximal, characteristic, complemented
4#10596308C311yesmaximal, characteristic, complemented
5#5837⋊C3—31nomaximal
6#41⋊H(3,𝔽3)C2211yescharacteristic
7C2×Q8⋊H(3,𝔽3)—21no—
8#88622—21no—
9#3754⋊C3—41nomaximal
10C32×2−1+4C2×C311yescharacteristic
11#64366C2×C311yescharacteristic, complemented
12C3×2−1+4⋊C3—21no—
13C3×2−1+4⋊C3D311yescharacteristic, complemented
14C3⋊#994—31no—
15#64159—31no—
16#64158—31no—
17#64157C2×C311yescharacteristic, complemented
18#64151—31no—
19C2×#3755—31no—
20C3⋊(#131⋊C3)—31no—
21D4×H(3,𝔽3)—41no—
22#634⋊C3—41no—
23Q8⋊H(3,𝔽3)D411yescharacteristic
24(C3⋊(C3⋊C8))⋊C3—41no—
25#5837—31no—
26#5837—91no—
27#577⋊C3—91no—
28C32×#41—31no—
29C32×#41—61no—
30C32×#41C22×C311yescharacteristic
31C2×C32×Q8—21no—
32C2×C32×Q8—31no—
33#3832—21no—
34D4×C3⋊D3—61no—
35#3828—31no—
36C2×C4⋊(C3⋊D3)—31no—
37C3×#41⋊C3—21no—
38C3×#41⋊C3C2×D311yescharacteristic
39C2×C3×SL(2,𝔽3)—22no—
40C2×C3×SL(2,𝔽3)—21no—
41C3⋊(#41⋊C3)—61no—
42#3756—31no—
43#3755—32no—
44#3755—31no—
45#3754A411yescharacteristic
46#3753—31no—
47#3748—31no—
48C2×C3⋊(C3⋊C8)—31no—
49C3×#214—121no—
50C22×H(3,𝔽3)—81no—
51C4×H(3,𝔽3)—41no—
52(C3⋊Dic3)⋊C3—81no—
53C3×2−1+4—31no—
54C3×2−1+4C3×D311yescharacteristic
55#1009—31no—
56#1009—91no—
572−1+4⋊C3—31no—
582−1+4⋊C3—61no—
59#994—91no—
60#950—91no—
61#950—93no—
62#949—93no—
63#949—91no—
64#948—31no—
65#948—91no—
66#942—93no—
67#942—91no—
68C2×#215—93no—
69C2×#215—91no—
70#131⋊C3—91no—
71C22×C3⋊D3—61no—
72C32×Q8—31no—
73C32×Q8—61no—
74C32×Q8C3×D411yescharacteristic
75C32×D4—31no—
76C32×D4—41no—
77C32×D4—61no—
78C32×D4—81no—
79C32×D4C2×A411yescharacteristic
80C2×C32×C4—31no—
81C2×C32×C4—61no—
82C2×C32×C4—61no—
83#638—61no—
84C4⋊(C3⋊D3)—31no—
85C4⋊(C3⋊D3)—61no—
86C4×C3⋊D3—61no—
87#634C2×A411yescharacteristic
88C3×Dic6—121no—
89C3×SL(2,𝔽3)—21no—
90C3×SL(2,𝔽3)#6711yescharacteristic
91C3⋊(C3⋊C8)—31no—
92C3⋊(C3⋊C8)C2×A411yescharacteristic
93C3×C3⋊C8—121no—
94#577—91noSylow
95C2×H(3,𝔽3)—41no—
96C2×H(3,𝔽3)—161no—
97C3×#41—33no—
98C3×#41—31no—
99C3×#41—31no—
100C3×#41—63no—
101C3×#41—61no—
102C3×#41C2×C3×D311yescharacteristic
103C2×C3×Q8—21no—
104C2×C3×Q8—33no—
105C2×C3×Q8—31no—
106C2×C3×Q8—61no—
107#239—61no—
108#239—181no—
109D3×D4—183no—
110D3×D4—181no—
111#235—91no—
112#235—93no—
113C2×D12—93no—
114C2×D12—91no—
115#41⋊C3—31no—
116#41⋊C3—61no—
117#41⋊C3—181no—
118C2×SL(2,𝔽3)—61no—
119C2×SL(2,𝔽3)—62no—
120C3×SD16—361no—
121#216—91no—
122#216—93no—
123#215—91no—
124#215—93no—
125#215—96no—
126#215—92no—
127#214—31no—
128#214—91no—
129#213—93no—
130#213—91no—
131#208—93no—
132#208—91no—
133C2×C3⋊C8—91no—
134C2×C3⋊C8—93no—
135C22×C32—21no—
136C22×C32—31no—
137C22×C32—81no—
138C22×C32—82no—
139C2×C3⋊D3—61no—
140C2×C3⋊D3—241no—
141C32×C4—31no—
142C32×C4—41no—
143C32×C4—61no—
144C32×C4—81no—
145C32×C4C22×A411yescharacteristic
146C3⋊Dic3—21no—
147C3×Dic3—241no—
1482−1+4(C3⋊D3)⋊C311yescharacteristic
149H(5,𝔽2)—91no—
150#137—271no—
151AGL(1,ℤ/8)—271no—
152#135—271no—
153C2×SD16—271no—
154#131—91no—
155H(3,𝔽3)—41noSylow
156C22×D3—183no—
157C22×D3—181no—
158C3×Q8—33no—
159C3×Q8—31no—
160C3×Q8—31no—
161C3×Q8—61no—
162C3×Q8—63no—
163C3×Q8—361no—
164C3×Q8C3×#6711yescharacteristic
165C3×D4—31no—
166C3×D4—33no—
167C3×D4—31no—
168C3×D4—61no—
169C3×D4—63no—
170C3×D4—121no—
171C3×D4—241no—
172C3×D4D3×A411yescharacteristic
173C2×C3×C4—33no—
174C2×C3×C4—31no—
175C2×C3×C4—61no—
176C2×C3×C4—63no—
177C2×C3×C4—63no—
178C2×C3×C4—61no—
179#67—181no—
180#67—183no—
181D12—93no—
182D12—91no—
183D12—181no—
184D12—183no—
185C4×D3—181no—
186C4×D3—183no—
187Dic6—31no—
188Dic6—91no—
189SL(2,𝔽3)—31no—
190SL(2,𝔽3)—61no—
191C3×C8—361no—
192C3⋊C8—31no—
193C3⋊C8—91no—
194C3⋊C8—93no—
195C3⋊C8—91no—
196C2×C32—41no—
197C2×C32—41no—
198C2×C32—61no—
199C2×C32—81no—
200C2×C32—161no—
201C2×C32—162no—
202C2×C32D4×A411yescharacteristic
203C3⋊D3—121no—
204#41—31no—
205#41—61no—
206#41—181no—
207#41—271no—
208#41C2×(C3⋊D3)⋊C311yescharacteristic
209C2×Q8—21no—
210C2×Q8—31no—
211C2×D4—271no—
212C2×D4—541no—
213Q16—271no—
214SD16—91no—
215SD16—271no—
216SD16—272no—
217D8—271no—
218M16—271no—
219C2×C8—271no—
220C22×C3—21no—
221C22×C3—33no—
222C22×C3—31no—
223C22×C3—61no—
224C22×C3—241no—
225C22×C3—242no—
226C2×D3—183no—
227C2×D3—181no—
228C2×D3—723no—
229C2×D3—721no—
230C3×C4—33no—
231C3×C4—31no—
232C3×C4—31no—
233C3×C4—63no—
234C3×C4—61no—
235C3×C4—121no—
236C3×C4—241no—
237C3×C4—721no—
238C3×C4C2×D3×A411yescharacteristic
239Dic3—61no—
240Dic3—181no—
241C32—41no—
242C32—81no—
243C32#577⋊C311yescharacteristic, complemented
244C23—541no—
245Q8—31no—
246Q8—61no—
247Q8—91no—
248Q8#638⋊C311yescharacteristic
249D4—31no—
250D4—61no—
251D4—271no—
252D4—541no—
253D4—541no—
254D4(C3⋊D3)⋊A411yescharacteristic
255C2×C4—31no—
256C2×C4—61no—
257C2×C4—61no—
258C2×C4—541no—
259C8—91no—
260C8—271no—
261C2×C3—31no—
262C2×C3—41no—
263C2×C3—63no—
264C2×C3—61no—
265C2×C3—121no—
266C2×C3—121no—
267C2×C3—241no—
268C2×C3—481no—
269C2×C3—482no—
270C2×C3A4×#6711yescharacteristic
271D3—363no—
272D3—361no—
273C22—21no—
274C22—31no—
275C22—541no—
276C22—2161no—
277C4—31no—
278C4—61no—
279C4—181no—
280C4C2×(C3⋊D3)⋊A411yescharacteristic
281C3—31no—
282C3—121no—
283C3—241no—
284C3#5837⋊C311yescharacteristic
285C2—41no—
286C2—61no—
287C2—1081no—
288C2#638⋊A411yescharacteristic
subgroupG/H
centerC2#638⋊A4
commutator, cosocleC32×#41C22×C3
FrattiniC3×C4C2×D3×A4
FittingC32×2−1+4C2×C3
socleC2×C3A4×#67
hypercenterD4(C3⋊D3)⋊A4
hypocenterC32×Q8C3×D4
LCSG ▷ C32×#41 ▷ C32×Q8
UCSC1 ◁ C2 ◁ C4 ◁ D4

Complements

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

NH ≅ G/Ntype
1. 2−1+4⋊H(3,𝔽3)C2semidirect
2. #11722436C2semidirect
3. #11720767C2semidirect
4. #10596308C3semidirect
11. #64366C2×C3semidirect
13. C3×2−1+4⋊C3D3semidirect
17. #64157C2×C3semidirect
243. C32#577⋊C3semidirect

Maximal subgroups

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

subgroupindex
2−1+4⋊H(3,𝔽3)2
#117224362
#117207672
#105963083
#5837⋊C33
#3754⋊C34