Finite Group Databasebeta

← C2×C3×#1010⋊C3 #13181448

Subgroups — C2×C3×#1010⋊C3 #13181448

Order 1728.

1115 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 1115 conjugacy classes of subgroups — too many to draw. Only its 180 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
1C2×C32×#41⋊C3C211yesmaximal, characteristic, complemented
2C3×#1010⋊C3C214yesmaximal, direct product factor
3C2×C3×D3×SL(2,𝔽3)C212yesmaximal, complemented
4C2×C3×#1010C311yesmaximal, characteristic, complemented
5C2×C3×2−1+4⋊C3—31nomaximal
6C2×#1010⋊C3C313yesmaximal, direct product factor
7C2×C32×D12—41nomaximal
8C32×#41⋊C3C2212yescomplemented
9C2×C32×SL(2,𝔽3)C2211yescharacteristic
10C3×D3×SL(2,𝔽3)C2214yescomplemented
11C2×C32×#41C2×C311yescharacteristic, complemented
12C3×#1010—34no—
13C3×#1010C2×C314yescomplemented
14C2×C3×D3×Q8C2×C312yescomplemented
15C2×C3×#237—32no—
16C2×C3×#235—31no—
17C22×C3×Dic6—31no—
18C3×2−1+4⋊C3—34no—
19C2×C3×#41⋊C3—21no—
20C2×C3×#41⋊C3—23no—
21C2×C3×#41⋊C3D311yescharacteristic, complemented
22C2×C3×#41⋊C3C2×C313yescomplemented
23C22×C3×SL(2,𝔽3)—32no—
24#1010⋊C3C2×C3112yesdirect product factor
25C2×D3×SL(2,𝔽3)C2×C316yescomplemented
26C22×C32×D3—42no—
27C2×C33×C4—41no—
28C32×D12—44no—
29C32×SL(2,𝔽3)C2311yescharacteristic
30C2×C3×2−1+4—31no—
31C2×#1010C3211yescharacteristic, complemented
32C2×2−1+4⋊C3—33no—
33C32×#41—32no—
34C32×#41C22×C312yescomplemented
35C2×C32×Q8C22×C311yescharacteristic
36C2×C32×D4—31no—
37C2×C32×D4—121no—
38C22×C32×C4—31no—
39C2×C3×#67—32no—
40C22×C3×Dic3—32no—
41C3×D3×Q8—34no—
42C3×D3×Q8C22×C314yescomplemented
43C3×#237—38no—
44C3×#237—38no—
45C3×#235—34no—
46C3×#235—34no—
47C2×C3×D12—43no—
48C2×C3×D12A411yescharacteristic
49C2×C3×C4×D3—32no—
50C2×C3×Dic6—34no—
51C2×C3×Dic6—31no—
52C2×C3×Dic6—32no—
53C2×C3×Dic6—34no—
54C3×#41⋊C3—22no—
55C3×#41⋊C3—26no—
56C3×#41⋊C3C2×D312yescomplemented
57C3×#41⋊C3C22×C316yescomplemented
58C2×C3×SL(2,𝔽3)—21no—
59C2×C3×SL(2,𝔽3)—23no—
60C2×C3×SL(2,𝔽3)—34no—
61C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
62C2×C3×SL(2,𝔽3)C22×C313yes—
63D3×SL(2,𝔽3)C22×C3112yescomplemented
64C22×C33—41no—
65C2×C32×D3—44no—
66C2×C32×D3—84no—
67C33×C4—42no—
68C3×2−1+4—34no—
69C3×2−1+4—94no—
70C2×C3×#41—21no—
71C2×C3×#41—91no—
72C2×C3×#41—92no—
73C2×C3×#41C3×D311yescharacteristic, complemented
74C2×C3×#41C2×C3211yescharacteristic, complemented
75C22×C3×Q8—32no—
76C22×C3×Q8—91no—
77#1010—34no—
78#1010C2×C3214yescomplemented
79C2×D3×Q8C2×C3212yescomplemented
80C2×#237—32no—
81C2×#235—31no—
82C22×Dic6—31no—
832−1+4⋊C3—312no—
84C2×#41⋊C3—23no—
85C2×#41⋊C3C3×D313yescomplemented
86C22×SL(2,𝔽3)—36no—
87C23×C32—31no—
88C23×C32—122no—
89C22×C3×D3—46no—
90C22×C3×D3C2×A412yes—
91C32×Q8—31no—
92C32×Q8C23×C311yescharacteristic
93C32×D4—32no—
94C32×D4—32no—
95C32×D4—124no—
96C2×C32×C4—31no—
97C2×C32×C4—32no—
98C2×C32×C4—32no—
99C2×C32×C4—41no—
100C2×C32×C4—43no—
101C2×C32×C4—83no—
102C2×C32×C4—81no—
103C2×C32×C4C2×A411yescharacteristic
104C3×#67—38no—
105C2×C3×Dic3—38no—
106C2×C3×Dic3—32no—
107C3×D12—412no—
108C3×D12C2×A414yes—
109C3×C4×D3—34no—
110C3×C4×D3—34no—
111C3×Dic6—34no—
112C3×Dic6—34no—
113C3×Dic6—34no—
114C3×SL(2,𝔽3)—21no—
115C3×SL(2,𝔽3)—23no—
116C3×SL(2,𝔽3)C22×D311yescharacteristic
117C3×SL(2,𝔽3)C23×C313yes—
118C2×2−1+4—31noSylow
119C2×C33—42no—
120C2×C33—41no—
121C32×D3—84no—
122C3×#41—22no—
123C3×#41—32no—
124C3×#41—32no—
125C3×#41—62no—
126C3×#41—94no—
127C3×#41—94no—
128C3×#41—98no—
129C3×#41—98no—
130C3×#41C2×C3×D312yescomplemented
131C3×#41C22×C3212yescomplemented
132C2×C3×Q8—21no—
133C2×C3×Q8—34no—
134C2×C3×Q8—94no—
135C2×C3×Q8—94no—
136C2×C3×Q8—92no—
137C2×C3×Q8—91no—
138C2×C3×Q8—94no—
139C2×C3×Q8C2×C3×D311yescharacteristic
140C2×C3×Q8C22×C3211yescharacteristic
141C2×C3×D4—31no—
142C2×C3×D4—31no—
143C2×C3×D4—31no—
144C2×C3×D4—61no—
145C2×C3×D4—92no—
146C2×C3×D4—123no—
147C22×C3×C4—31no—
148C22×C3×C4—31no—
149C22×C3×C4—61no—
150C22×C3×C4—92no—
151C22×C3×C4—92no—
152C2×#67—32no—
153C22×Dic3—32no—
154D3×Q8—34no—
155D3×Q8C22×C3214yescomplemented
156#237—38no—
157#237—38no—
158#235—34no—
159#235—34no—
160C2×D12C3×A411yescharacteristic
161C2×C4×D3—32no—
162C2×Dic6—34no—
163C2×Dic6—31no—
164C2×Dic6—32no—
165C2×Dic6—34no—
166#41⋊C3—26no—
167#41⋊C3C2×C3×D316yescomplemented
168C2×SL(2,𝔽3)—23no—
169C2×SL(2,𝔽3)—312no—
170C2×SL(2,𝔽3)C2×C3×D313yes—
171C22×C32—32no—
172C22×C32—41no—
173C22×C32—43no—
174C22×C32—62no—
175C22×C32—83no—
176C22×C32—81no—
177C22×C32—124no—
178C22×C32—244no—
179C22×C32C22×A411yescharacteristic
180C2×C3×D3—24no—
181C2×C3×D3—412no—
182C2×C3×D3—812no—
183C2×C3×D3C22×A414yes—
184C32×C4—31no—
185C32×C4—31no—
186C32×C4—42no—
187C32×C4—46no—
188C32×C4—86no—
189C32×C4—82no—
190C32×C4C22×A412yes—
191C3×Dic3—34no—
1922−1+4—34no—
1932−1+4—94no—
194C2×#41—91no—
195C2×#41—92no—
196C2×#41C32×D311yescharacteristic, complemented
197C22×Q8—32no—
198C22×Q8—91no—
199C33—41noSylow
200C23×C3—32no—
201C23×C3—31no—
202C23×C3—31no—
203C23×C3—61no—
204C23×C3—126no—
205C22×D3C2×C3×A412yes—
206C3×Q8—21no—
207C3×Q8—31no—
208C3×Q8—31no—
209C3×Q8—61no—
210C3×Q8—94no—
211C3×Q8—94no—
212C3×Q8—94no—
213C3×Q8C22×C3×D311yescharacteristic
214C3×Q8C23×C3211yescharacteristic
215C3×D4—32no—
216C3×D4—32no—
217C3×D4—32no—
218C3×D4—32no—
219C3×D4—34no—
220C3×D4—62no—
221C3×D4—62no—
222C3×D4—98no—
223C3×D4—1212no—
224C2×C3×C4—21no—
225C2×C3×C4—31no—
226C2×C3×C4—31no—
227C2×C3×C4—32no—
228C2×C3×C4—32no—
229C2×C3×C4—32no—
230C2×C3×C4—32no—
231C2×C3×C4—43no—
232C2×C3×C4—62no—
233C2×C3×C4—62no—
234C2×C3×C4—61no—
235C2×C3×C4—83no—
236C2×C3×C4—94no—
237C2×C3×C4—94no—
238C2×C3×C4—98no—
239C2×C3×C4—92no—
240C2×C3×C4D3×A411yescharacteristic
241C2×C3×C4C2×C3×A411yescharacteristic
242#67—38no—
243C2×Dic3—38no—
244C2×Dic3—32no—
245D12C2×C3×A414yes—
246C4×D3—34no—
247C4×D3—34no—
248Dic6—34no—
249Dic6—34no—
250Dic6—34no—
251SL(2,𝔽3)—23no—
252SL(2,𝔽3)C22×C3×D313yes—
253C2×C32—42no—
254C2×C32—41no—
255C2×C32—46no—
256C2×C32—43no—
257C2×C32—62no—
258C2×C32—83no—
259C2×C32—86no—
260C2×C32—82no—
261C2×C32—81no—
262C2×C32—244no—
263C2×C322−1+4⋊C312yescomplemented
264C2×C32C23×A411yescharacteristic
265C3×D3—24no—
266C3×D3—812no—
267#41—32no—
268#41—94no—
269#41—94no—
270#41—98no—
271#41—98no—
272#41C2×C32×D312yescomplemented
273C2×Q8—34no—
274C2×Q8—94no—
275C2×Q8—94no—
276C2×Q8—92no—
277C2×Q8—91no—
278C2×Q8—94no—
279C2×Q8C2×C32×D311yescharacteristic
280C2×D4—31no—
281C2×D4—31no—
282C2×D4—92no—
283C22×C4—31no—
284C22×C4—92no—
285C22×C4—92no—
286C22×C3—21no—
287C22×C3—32no—
288C22×C3—34no—
289C22×C3—32no—
290C22×C3—43no—
291C22×C3—62no—
292C22×C3—64no—
293C22×C3—62no—
294C22×C3—62no—
295C22×C3—83no—
296C22×C3—1212no—
297C22×C3—122no—
298C22×C3—2412no—
299C22×C3C2×D3×A411yescharacteristic
300C22×C3C22×C3×A411yescharacteristic
301C2×D3—24no—
302C2×D3C22×C3×A414yes—
303C3×C4—22no—
304C3×C4—31no—
305C3×C4—31no—
306C3×C4—31no—
307C3×C4—31no—
308C3×C4—46no—
309C3×C4—61no—
310C3×C4—61no—
311C3×C4—86no—
312C3×C4—94no—
313C3×C4C2×D3×A412yes—
314C3×C4C22×C3×A412yes—
315Dic3—34no—
316C32—41no—
317C32—43no—
318C32—83no—
319C32—81no—
320C32C2×2−1+4⋊C311yescharacteristic, complemented
321C23—31no—
322C23—32no—
323Q8—31no—
324Q8—94no—
325Q8—94no—
326Q8—94no—
327Q8C22×C32×D311yescharacteristic
328D4—32no—
329D4—32no—
330D4—34no—
331D4—98no—
332C2×C4—31no—
333C2×C4—32no—
334C2×C4—32no—
335C2×C4—94no—
336C2×C4—94no—
337C2×C4—98no—
338C2×C4—92no—
339C2×C4C3×D3×A411yescharacteristic
340C2×C3—22no—
341C2×C3—21no—
342C2×C3—46no—
343C2×C3—43no—
344C2×C3—64no—
345C2×C3—62no—
346C2×C3—62no—
347C2×C3—86no—
348C2×C3—83no—
349C2×C3—122no—
350C2×C3—2412no—
351C2×C3#1010⋊C312yesdirect product factor
352C2×C3C3×2−1+4⋊C312yescomplemented
353C2×C3C22×D3×A411yescharacteristic
354C2×C3C23×C3×A411yescharacteristic
355D3—24no—
356C22—32no—
357C22—34no—
358C22—64no—
359C22—62no—
360C22C2×C3×D3×A411yescharacteristic
361C4—31no—
362C4—31no—
363C4—94no—
364C4C2×C3×D3×A412yes—
365C3—21no—
366C3—43no—
367C3—83no—
368C3C2×#1010⋊C311yescharacteristic, direct product factor
369C3C2×C3×2−1+4⋊C311yescharacteristic, complemented
370C2—64no—
371C2—62no—
372C2C3×#1010⋊C312yesdirect product factor
373C2C22×C3×D3×A411yescharacteristic
subgroupG/H
centerC22×C3C2×D3×A4
commutator, cosocle, hypocenterC3×Q8C23×C32
FrattiniC2C22×C3×D3×A4
FittingC2×C32×#41C2×C3
socleC22×C32C22×A4
hypercenterC2×C3×C4D3×A4
LCSG ▷ C3×Q8
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C32×#41⋊C3C2semidirect
2. C3×#1010⋊C3C2direct (N × H)
3. C2×C3×D3×SL(2,𝔽3)C2semidirect
4. C2×C3×#1010C3semidirect
6. C2×#1010⋊C3C3direct (N × H)
8. C32×#41⋊C3C22semidirect
10. C3×D3×SL(2,𝔽3)C22semidirect
11. C2×C32×#41C2×C3semidirect
13. C3×#1010C2×C3semidirect
14. C2×C3×D3×Q8C2×C3semidirect
21. C2×C3×#41⋊C3D3semidirect
22. C2×C3×#41⋊C3C2×C3semidirect
24. #1010⋊C3C2×C3direct (N × H)
25. C2×D3×SL(2,𝔽3)C2×C3semidirect
31. C2×#1010C32semidirect
34. C32×#41C22×C3semidirect
42. C3×D3×Q8C22×C3semidirect
56. C3×#41⋊C3C2×D3semidirect
57. C3×#41⋊C3C22×C3semidirect
63. D3×SL(2,𝔽3)C22×C3semidirect
73. C2×C3×#41C3×D3semidirect
74. C2×C3×#41C2×C32semidirect
78. #1010C2×C32semidirect
79. C2×D3×Q8C2×C32semidirect
85. C2×#41⋊C3C3×D3semidirect
130. C3×#41C2×C3×D3semidirect
131. C3×#41C22×C32semidirect
155. D3×Q8C22×C32semidirect
167. #41⋊C3C2×C3×D3semidirect
196. C2×#41C32×D3semidirect
263. C2×C322−1+4⋊C3semidirect
272. #41C2×C32×D3semidirect
320. C32C2×2−1+4⋊C3semidirect
351. C2×C3#1010⋊C3direct (N × H)
352. C2×C3C3×2−1+4⋊C3semidirect
368. C3C2×#1010⋊C3direct (N × H)
369. C3C2×C3×2−1+4⋊C3semidirect
372. C2C3×#1010⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#41⋊C32
C3×#1010⋊C32
C2×C3×D3×SL(2,𝔽3)2
C2×C3×#10103
C2×C3×2−1+4⋊C33
C2×#1010⋊C33
C2×C32×D124