Finite Group Databasebeta

← #13147303

Subgroups — #13147303

Order 1728.

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

normal subgroups only

This group has 369 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#11722438C211yesmaximal, characteristic, complemented
3#11720767C211yesmaximal, characteristic, complemented
4#10596309C311yesmaximal, characteristic, complemented
5#5838⋊C3—31nomaximal
6#41⋊H(3,𝔽3)C2211yescharacteristic
7C2×Q8⋊H(3,𝔽3)—21no—
8#88623—21no—
9#3753⋊C3—41nomaximal
10C32×2−1+4C2×C311yescharacteristic, complemented
11#64367C2×C311yescharacteristic, complemented
12C3×2−1+4⋊C3—21no—
13C3×2−1+4⋊C3D311yescharacteristic, complemented
14#1010⋊C3—31no—
15#64160—31no—
16#64159—31no—
17#64157C2×C311yescharacteristic, complemented
18C2×#3756—31no—
19#64151—31no—
20C3⋊(#131⋊C3)—31no—
21D4×H(3,𝔽3)—41no—
22C4⋊((C3⋊D3)⋊C3)—41no—
23Q8⋊H(3,𝔽3)D411yescharacteristic
24(C3⋊(C3⋊C8))⋊C3—41no—
25#5838—31no—
26#5838—91no—
27#578⋊C3—91no—
28C32×#41—31no—
29C32×#41—61no—
30C32×#41C22×C311yescharacteristic
31C2×C32×Q8—21no—
32C2×C32×Q8—31no—
33Q8×C3⋊D3—21no—
34#3830—61no—
35#3828—31no—
36C2×#634—31no—
37C3×#41⋊C3—21no—
38C3×#41⋊C3C2×D311yescharacteristic
39C2×C3×SL(2,𝔽3)—22no—
40C2×C3×SL(2,𝔽3)—21no—
41D3×SL(2,𝔽3)—61no—
42#3756—32no—
43#3756—31no—
44#3755—31no—
45#3754—31no—
46#3753A411yescharacteristic
47#3748—31no—
48C2×C3⋊(C3⋊C8)—31no—
49C3×#213—121no—
50C22×H(3,𝔽3)—81no—
51C2×(C3⋊D3)⋊C3—81no—
52C4×H(3,𝔽3)—41no—
53C3×2−1+4—31no—
54C3×2−1+4C3×D311yescharacteristic
55#1010—31no—
56#1010—91no—
572−1+4⋊C3—31no—
582−1+4⋊C3—61no—
592−1+4⋊C3—91no—
60#951—93no—
61#951—91no—
62#950—91no—
63#950—93no—
64#948—31no—
65#948—91no—
66C2×#216—93no—
67C2×#216—91no—
68#942—93no—
69#942—91no—
70#131⋊C3—91no—
71C32×Q8—31no—
72C32×Q8—61no—
73C32×Q8C3×D411yescharacteristic
74C32×D4—31no—
75C32×D4—41no—
76C32×D4—61no—
77C32×D4—81no—
78C32×D4C2×A411yescharacteristic
79C2×C32×C4—31no—
80C2×C32×C4—61no—
81C2×C32×C4—61no—
82#638—61no—
83C2×C3⋊Dic3—61no—
84C4⋊(C3⋊D3)C2×A411yescharacteristic
85C4×C3⋊D3—61no—
86#634—31no—
87#634—61no—
88C3×D12—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#578—91noSylow
95C2×H(3,𝔽3)—41no—
96C2×H(3,𝔽3)—161no—
97(C3⋊D3)⋊C3—161no—
98C3×#41—33no—
99C3×#41—31no—
100C3×#41—31no—
101C3×#41—63no—
102C3×#41—61no—
103C3×#41C2×C3×D311yescharacteristic
104C2×C3×Q8—21no—
105C2×C3×Q8—33no—
106C2×C3×Q8—31no—
107C2×C3×Q8—61no—
108D3×Q8—61no—
109D3×Q8—181no—
110#237—183no—
111#237—181no—
112#235—91no—
113#235—93no—
114C2×Dic6—93no—
115C2×Dic6—91no—
116#41⋊C3—31no—
117#41⋊C3—61no—
118C2×SL(2,𝔽3)—61no—
119C2×SL(2,𝔽3)—62no—
120C2×SL(2,𝔽3)—181no—
121C3×D8—361no—
122#216—91no—
123#216—96no—
124#216—92no—
125#216—93no—
126#215—91no—
127#215—93no—
128#214—93no—
129#214—91no—
130#213—31no—
131#213—91no—
132#208—93no—
133#208—91no—
134C2×C3⋊C8—91no—
135C2×C3⋊C8—93no—
136C22×C32—21no—
137C22×C32—31no—
138C22×C32—81no—
139C22×C32—82no—
140C2×C3⋊D3—21no—
141C2×C3×D3—241no—
142C32×C4—31no—
143C32×C4—41no—
144C32×C4—61no—
145C32×C4—81no—
146C32×C4C22×A411yescharacteristic
147C3⋊Dic3—61no—
1482−1+4—91no—
1492−1+4(C3⋊D3)⋊C311yescharacteristic, complemented
150#137—271no—
151#137—271no—
152#135—271no—
153C2×Q16—271no—
154#131—91no—
155H(3,𝔽3)—41noSylow
156C3×Q8—33no—
157C3×Q8—31no—
158C3×Q8—31no—
159C3×Q8—61no—
160C3×Q8—63no—
161C3×Q8C3×#6711yescharacteristic
162C3×D4—31no—
163C3×D4—33no—
164C3×D4—31no—
165C3×D4—61no—
166C3×D4—63no—
167C3×D4—121no—
168C3×D4—241no—
169C3×D4—361no—
170C3×D4D3×A411yescharacteristic
171C2×C3×C4—33no—
172C2×C3×C4—31no—
173C2×C3×C4—61no—
174C2×C3×C4—63no—
175C2×C3×C4—63no—
176C2×C3×C4—61no—
177#67—181no—
178#67—183no—
179C2×Dic3—183no—
180C2×Dic3—181no—
181D12—31no—
182D12—91no—
183C4×D3—181no—
184C4×D3—183no—
185Dic6—93no—
186Dic6—91no—
187Dic6—183no—
188Dic6—181no—
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—41no—
204C3×D3—481no—
205#41—31no—
206#41—61no—
207#41—271no—
208#41—541no—
209#41C2×(C3⋊D3)⋊C311yescharacteristic
210C2×Q8—21no—
211C2×Q8—31no—
212C2×Q8—181no—
213C2×Q8—271no—
214Q16—271no—
215Q16—272no—
216SD16—271no—
217SD16—271no—
218D8—91no—
219M16—271no—
220C2×C8—271no—
221C22×C3—21no—
222C22×C3—33no—
223C22×C3—31no—
224C22×C3—61no—
225C22×C3—241no—
226C22×C3—242no—
227C22×C3—721no—
228C2×D3—61no—
229C2×D3—181no—
230C3×C4—33no—
231C3×C4—31no—
232C3×C4—31no—
233C3×C4—63no—
234C3×C4—61no—
235C3×C4—121no—
236C3×C4—241no—
237C3×C4C2×D3×A411yescharacteristic
238Dic3—183no—
239Dic3—181no—
240C32—41no—
241C32—81no—
242C32#578⋊C311yescharacteristic, complemented
243Q8—31no—
244Q8—61no—
245Q8—271no—
246Q8—541no—
247Q8#638⋊C311yescharacteristic
248D4—31no—
249D4—61no—
250D4—91no—
251D4—541no—
252D4(C3⋊D3)⋊A411yescharacteristic
253C2×C4—31no—
254C2×C4—61no—
255C2×C4—61no—
256C2×C4—541no—
257C2×C4—541no—
258C8—91no—
259C8—271no—
260C2×C3—31no—
261C2×C3—41no—
262C2×C3—63no—
263C2×C3—61no—
264C2×C3—121no—
265C2×C3—121no—
266C2×C3—241no—
267C2×C3—481no—
268C2×C3—482no—
269C2×C3—1441no—
270C2×C3A4×#6711yescharacteristic
271D3—121no—
272D3—361no—
273C22—21no—
274C22—31no—
275C22—181no—
276C4—31no—
277C4—61no—
278C4—541no—
279C4C2×(C3⋊D3)⋊A411yescharacteristic
280C3—31no—
281C3—121no—
282C3—241no—
283C3#5838⋊C311yescharacteristic
284C2—41no—
285C2—61no—
286C2—361no—
287C2#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. #11722438C2semidirect
3. #11720767C2semidirect
4. #10596309C3semidirect
10. C32×2−1+4C2×C3semidirect
11. #64367C2×C3semidirect
13. C3×2−1+4⋊C3D3semidirect
17. #64157C2×C3semidirect
149. 2−1+4(C3⋊D3)⋊C3semidirect
242. C32#578⋊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
#117224382
#117207672
#105963093
#5838⋊C33
#3753⋊C34