Finite Group Databasebeta

← C2×#11724490 #13181464

Subgroups — C2×#11724490 #13181464

Order 1728.

1392 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 1392 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C22×C32×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2#11724490C214yesmaximal, direct product factor
3C2×C3⋊D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C2×C3⋊(C3⋊(#41⋊C3))C211yesmaximal, characteristic, complemented
5C2×#64364C311yesmaximal, characteristic, complemented
6C2×#1007⋊C3—34nomaximal
7C2×C3×#638—41nomaximal
8C2×C32×SL(2,𝔽3)C2211yescharacteristic
9C2×C32×SL(2,𝔽3)C2212yescomplemented
10C3⋊D3×SL(2,𝔽3)C2212yescomplemented
11C3⋊(C3⋊(#41⋊C3))C2212yescomplemented
12C22×C32×Q8C2×C311yescharacteristic, complemented
13#64364—34no—
14#64364C2×C314yescomplemented
15C2×#3832—31no—
16C2×#3832C2×C311yescharacteristic, complemented
17C2×Q8×C3⋊D3—31no—
18C2×Q8×C3⋊D3C2×C311yescharacteristic, complemented
19C2×#3828—31no—
20C2×#3828—31no—
21C22×C3×SL(2,𝔽3)—24no—
22C22×C3×SL(2,𝔽3)D314yescomplemented
23#1007⋊C3—316no—
24C2×D3×SL(2,𝔽3)—34no—
25C2×C3⋊(#41⋊C3)—34no—
26C23×C33—41no—
27C22×C3×C3⋊D3—41no—
28C3×#638—44no—
29C2×C3×C3⋊Dic3—41no—
30C32×SL(2,𝔽3)C2311yescharacteristic
31C2×#1007—34no—
32C2×2−1+4⋊C3—91no—
33C2×C32×Q8—31no—
34C2×C32×Q8—32no—
35C2×C32×Q8C22×C311yescharacteristic
36C2×C32×Q8C22×C312yescomplemented
37C22×C32×C4—31no—
38C2×#638—31no—
39C2×#638A411yescharacteristic
40#3832—34no—
41#3832—34no—
42#3832—32no—
43#3832C22×C312yescomplemented
44Q8×C3⋊D3—34no—
45Q8×C3⋊D3—32no—
46Q8×C3⋊D3—34no—
47Q8×C3⋊D3C22×C312yescomplemented
48#3828—34no—
49#3828—34no—
50#3828—34no—
51#3828—34no—
52C2×C4⋊(C3⋊D3)—31no—
53C2×C4⋊(C3⋊D3)—31no—
54C2×C4×C3⋊D3—32no—
55C2×C4×C3⋊D3—31no—
56C2×C4×C3⋊D3—31no—
57C2×#634—31no—
58C2×#634—31no—
59C2×C3×#67—124no—
60C2×C3×SL(2,𝔽3)—24no—
61C2×C3×SL(2,𝔽3)—28no—
62C2×C3×SL(2,𝔽3)C2×D314yes—
63C2×C3×SL(2,𝔽3)C2×D318yescomplemented
64D3×SL(2,𝔽3)—38no—
65C3⋊(#41⋊C3)—38no—
66C22×C33—41no—
67C22×C33—42no—
68C22×C33—82no—
69C2×C3×C3⋊D3—42no—
70C2×C3×C3⋊D3—82no—
71C3×C3⋊Dic3—42no—
72C22×C3×Q8C3×D314yescomplemented
73#1007—316no—
74#1007—916no—
75C2×#239—34no—
76C2×#239—94no—
77C2×D3×Q8—34no—
78C2×D3×Q8—94no—
79C2×#235—94no—
80C2×#235—94no—
812−1+4⋊C3—94no—
82C2×#41⋊C3—91no—
83C22×SL(2,𝔽3)—24no—
84C22×SL(2,𝔽3)—91no—
85C22×SL(2,𝔽3)C3⋊D311yescharacteristic, complemented
86C23×C32—44no—
87C23×C32—84no—
88C23×C32C2×A411yescharacteristic
89C22×C3⋊D3—31no—
90C22×C3⋊D3C2×A411yescharacteristic
91C22×C3×D3—124no—
92C32×Q8—31no—
93C32×Q8—32no—
94C32×Q8—32no—
95C32×Q8C23×C311yescharacteristic
96C2×C32×C4—31no—
97C2×C32×C4—31no—
98C2×C32×C4—32no—
99C2×C32×C4—32no—
100#638—34no—
101#638C2×A414yes—
102C2×C3⋊Dic3—31no—
103C2×C3⋊Dic3C2×A411yescharacteristic
104C4⋊(C3⋊D3)—32no—
105C4⋊(C3⋊D3)—32no—
106C4⋊(C3⋊D3)—34no—
107C4×C3⋊D3—38no—
108C4×C3⋊D3—32no—
109C4×C3⋊D3—32no—
110C4×C3⋊D3—32no—
111C4×C3⋊D3—32no—
112#634—32no—
113#634—34no—
114#634—32no—
115C3×#67—1216no—
116C2×C3×Dic3—124no—
117C3×SL(2,𝔽3)—24no—
118C3×SL(2,𝔽3)C22×D314yes—
119C2×2−1+4—91noSylow
120C2×C33—42no—
121C2×C33—41no—
122C2×C33—82no—
123C3×C3⋊D3—82no—
124C2×C3×Q8—34no—
125C2×C3×Q8—38no—
126C2×C3×Q8C2×C3×D314yes—
127C2×C3×Q8C2×C3×D318yescomplemented
128C2×C3×D4—361no—
129C22×C3×C4—34no—
130C2×#67—34no—
131C2×#67—94no—
132#239—38no—
133#239—916no—
134#239—916no—
135#239—98no—
136D3×Q8—38no—
137D3×Q8—916no—
138D3×Q8—98no—
139D3×Q8—916no—
140#235—916no—
141#235—916no—
142#235—916no—
143#235—916no—
144C2×D12—94no—
145C2×D12—94no—
146C2×C4×D3—94no—
147C2×C4×D3—94no—
148C2×C4×D3—98no—
149C2×Dic6—94no—
150C2×Dic6—94no—
151#41⋊C3—92no—
152C2×SL(2,𝔽3)—24no—
153C2×SL(2,𝔽3)—28no—
154C2×SL(2,𝔽3)—92no—
155C2×SL(2,𝔽3)C2×C3⋊D311yescharacteristic
156C2×SL(2,𝔽3)C2×C3⋊D312yescomplemented
157C22×C32—22no—
158C22×C32—44no—
159C22×C32—48no—
160C22×C32—88no—
161C22×C32—816no—
162C22×C32—88no—
163C22×C32—84no—
164C22×C32C22×A412yes—
165C22×C32C22×A411yescharacteristic
166C2×C3⋊D3—22no—
167C2×C3⋊D3—32no—
168C2×C3⋊D3—62no—
169C2×C3⋊D3C22×A412yes—
170C2×C3×D3—128no—
171C2×C3×D3—248no—
172C32×C4—31no—
173C32×C4—32no—
174C32×C4—31no—
175C3⋊Dic3—32no—
176C3⋊Dic3C22×A412yes—
177C3×Dic3—128no—
1782−1+4—94no—
1792−1+4—274no—
180C2×#41—91no—
181C2×#41—271no—
182C2×#41—271no—
183C2×#41—271no—
184C22×Q8—91no—
185C22×Q8—271no—
186C22×Q8C3×C3⋊D311yescharacteristic, complemented
187C33—41noSylow
188C23×C3—41no—
189C23×C3—84no—
190C23×C3—361no—
191C23×C3D3×A414yes—
192C22×D3—34no—
193C22×D3—94no—
194C3×Q8—34no—
195C3×Q8—38no—
196C3×Q8—38no—
197C3×Q8C22×C3×D314yes—
198C3×D4—364no—
199C2×C3×C4—34no—
200C2×C3×C4—34no—
201C2×C3×C4—38no—
202C2×C3×C4—38no—
203C2×C3×C4—361no—
204#67—316no—
205#67—916no—
206C2×Dic3—34no—
207C2×Dic3—94no—
208D12—916no—
209D12—98no—
210D12—98no—
211C4×D3—98no—
212C4×D3—98no—
213C4×D3—98no—
214C4×D3—98no—
215C4×D3—932no—
216Dic6—916no—
217Dic6—98no—
218Dic6—98no—
219SL(2,𝔽3)—24no—
220SL(2,𝔽3)C22×C3⋊D311yescharacteristic
221C2×C32—22no—
222C2×C32—48no—
223C2×C32—44no—
224C2×C32—88no—
225C2×C32—816no—
226C2×C32—88no—
227C2×C32—84no—
228C2×C322−1+4⋊C312yescomplemented
229C2×C32C23×A411yescharacteristic
230C3⋊D3—22no—
231C3⋊D3—62no—
232C3×D3—248no—
233#41—92no—
234#41—274no—
235#41—274no—
236#41—274no—
237#41—274no—
238#41—274no—
239#41—274no—
240#41—272no—
241C2×Q8—31no—
242C2×Q8—32no—
243C2×Q8—92no—
244C2×Q8—271no—
245C2×Q8—271no—
246C2×Q8—274no—
247C2×Q8—274no—
248C2×Q8—272no—
249C2×Q8C2×C3×C3⋊D311yescharacteristic
250C2×Q8C2×C3×C3⋊D312yescomplemented
251C2×D4—91no—
252C2×D4—271no—
253C2×D4—271no—
254C2×D4—271no—
255C22×C4—31no—
256C22×C4—271no—
257C22×C4—271no—
258C22×C4—272no—
259C22×C3—28no—
260C22×C3—42no—
261C22×C3—41no—
262C22×C3—82no—
263C22×C3—816no—
264C22×C3—88no—
265C22×C3—84no—
266C22×C3—362no—
267C22×C3—722no—
268C22×C3C2×D3×A418yes—
269C22×C3C2×D3×A414yes—
270C2×D3—38no—
271C2×D3—68no—
272C2×D3—98no—
273C2×D3—188no—
274C3×C4—34no—
275C3×C4—38no—
276C3×C4—34no—
277C3×C4—362no—
278Dic3—38no—
279Dic3—98no—
280C32—44no—
281C32—84no—
282C32C2×2−1+4⋊C311yescharacteristic, complemented
283C23—91no—
284C23—271no—
285C23A4×C3⋊D311yescharacteristic
286Q8—31no—
287Q8—32no—
288Q8—32no—
289Q8—274no—
290Q8—272no—
291Q8—272no—
292Q8C22×C3×C3⋊D311yescharacteristic
293D4—94no—
294D4—274no—
295D4—272no—
296D4—272no—
297D4—274no—
298C2×C4—32no—
299C2×C4—32no—
300C2×C4—31no—
301C2×C4—31no—
302C2×C4—91no—
303C2×C4—272no—
304C2×C4—272no—
305C2×C4—272no—
306C2×C4—272no—
307C2×C4—278no—
308C2×C4—271no—
309C2×C3—28no—
310C2×C3—41no—
311C2×C3—42no—
312C2×C3—82no—
313C2×C3—816no—
314C2×C3—84no—
315C2×C3—88no—
316C2×C3—722no—
317C2×C3#1007⋊C318yescomplemented
318C2×C3C22×D3×A414yes—
319D3—68no—
320D3—188no—
321C22—22no—
322C22—92no—
323C22—182no—
324C22—272no—
325C22—542no—
326C22C2×A4×C3⋊D312yes—
327C22C2×A4×C3⋊D311yescharacteristic
328C4—31no—
329C4—32no—
330C4—31no—
331C4—92no—
332C4—272no—
333C3—41no—
334C3—84no—
335C3C2×#1007⋊C314yescomplemented
336C2—22no—
337C2—182no—
338C2—542no—
339C2#1172449012yesdirect product factor
340C2C22×A4×C3⋊D311yescharacteristic
subgroupG/H
centerC22C2×A4×C3⋊D3
commutator, cosocle, hypocenterC32×Q8C23×C3
FrattiniC2C22×A4×C3⋊D3
FittingC22×C32×Q8C2×C3
socleC22×C32C22×A4
hypercenterC23A4×C3⋊D3
LCSG ▷ C32×Q8
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×C32×SL(2,𝔽3)C2semidirect
2. #11724490C2direct (N × H)
3. C2×C3⋊D3×SL(2,𝔽3)C2semidirect
4. C2×C3⋊(C3⋊(#41⋊C3))C2semidirect
5. C2×#64364C3semidirect
9. C2×C32×SL(2,𝔽3)C22semidirect
10. C3⋊D3×SL(2,𝔽3)C22semidirect
11. C3⋊(C3⋊(#41⋊C3))C22semidirect
12. C22×C32×Q8C2×C3semidirect
14. #64364C2×C3semidirect
16. C2×#3832C2×C3semidirect
18. C2×Q8×C3⋊D3C2×C3semidirect
22. C22×C3×SL(2,𝔽3)D3semidirect
36. C2×C32×Q8C22×C3semidirect
43. #3832C22×C3semidirect
47. Q8×C3⋊D3C22×C3semidirect
63. C2×C3×SL(2,𝔽3)C2×D3semidirect
72. C22×C3×Q8C3×D3semidirect
85. C22×SL(2,𝔽3)C3⋊D3semidirect
127. C2×C3×Q8C2×C3×D3semidirect
156. C2×SL(2,𝔽3)C2×C3⋊D3semidirect
186. C22×Q8C3×C3⋊D3semidirect
228. C2×C322−1+4⋊C3semidirect
250. C2×Q8C2×C3×C3⋊D3semidirect
282. C32C2×2−1+4⋊C3semidirect
317. C2×C3#1007⋊C3semidirect
335. C3C2×#1007⋊C3semidirect
339. C2#11724490direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×SL(2,𝔽3)2
#117244902
C2×C3⋊D3×SL(2,𝔽3)2
C2×C3⋊(C3⋊(#41⋊C3))2
C2×#643643
C2×#1007⋊C33
C2×C3×#6384