Finite Group Databasebeta

← C2×C3×#1007⋊C3 #13181443

Subgroups — C2×C3×#1007⋊C3 #13181443

Order 1728.

1175 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 1175 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
1C22×C32×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C3×#1007⋊C3C214yesmaximal, direct product factor
3C2×C3×D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C2×C3×C3⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
5C2×C3×#1007C311yesmaximal, characteristic, complemented
6C2×C3×2−1+4⋊C3—31nomaximal
7C2×#1007⋊C3C313yesmaximal, direct product factor
8C2×C32×#67—41nomaximal
9C2×C32×SL(2,𝔽3)C2211yescharacteristic
10C2×C32×SL(2,𝔽3)C2212yescomplemented
11C3×D3×SL(2,𝔽3)C2212yescomplemented
12C3×C3⋊(#41⋊C3)C2212yescomplemented
13C22×C32×Q8C2×C311yescharacteristic, complemented
14C3×#1007—34no—
15C3×#1007C2×C314yescomplemented
16C2×C3×#239—31no—
17C2×C3×#239C2×C311yescharacteristic, complemented
18C2×C3×D3×Q8—31no—
19C2×C3×D3×Q8C2×C311yescharacteristic, complemented
20C2×C3×#235—31no—
21C2×C3×#235—31no—
22C3×2−1+4⋊C3—34no—
23C2×C3×#41⋊C3—31no—
24C22×C3×SL(2,𝔽3)—21no—
25C22×C3×SL(2,𝔽3)—23no—
26C22×C3×SL(2,𝔽3)—31no—
27C22×C3×SL(2,𝔽3)D311yescharacteristic, complemented
28C22×C3×SL(2,𝔽3)C2×C313yescomplemented
29#1007⋊C3C2×C3112yesdirect product factor
30C2×D3×SL(2,𝔽3)C2×C313yescomplemented
31C2×C3⋊(#41⋊C3)C2×C313yescomplemented
32C23×C33—41no—
33C22×C32×D3—41no—
34C32×#67—44no—
35C2×C32×Dic3—41no—
36C32×SL(2,𝔽3)C2311yescharacteristic
37C2×C3×2−1+4—31no—
38C2×#1007C3211yescharacteristic, complemented
39C2×2−1+4⋊C3—33no—
40C2×C32×Q8—31no—
41C2×C32×Q8—32no—
42C2×C32×Q8C22×C311yescharacteristic
43C2×C32×Q8C22×C312yescomplemented
44C2×C32×D4—121no—
45C22×C32×C4—31no—
46C2×C3×#67—31no—
47C2×C3×#67—43no—
48C2×C3×#67A411yescharacteristic
49C3×#239—34no—
50C3×#239—34no—
51C3×#239—32no—
52C3×#239C22×C312yescomplemented
53C3×D3×Q8—34no—
54C3×D3×Q8—32no—
55C3×D3×Q8—34no—
56C3×D3×Q8C22×C312yescomplemented
57C3×#235—34no—
58C3×#235—34no—
59C3×#235—34no—
60C3×#235—34no—
61C2×C3×D12—31no—
62C2×C3×D12—31no—
63C2×C3×C4×D3—32no—
64C2×C3×C4×D3—31no—
65C2×C3×C4×D3—31no—
66C2×C3×Dic6—31no—
67C2×C3×Dic6—31no—
68C3×#41⋊C3—32no—
69C2×C3×SL(2,𝔽3)—21no—
70C2×C3×SL(2,𝔽3)—22no—
71C2×C3×SL(2,𝔽3)—23no—
72C2×C3×SL(2,𝔽3)—26no—
73C2×C3×SL(2,𝔽3)—32no—
74C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
75C2×C3×SL(2,𝔽3)C2×D312yescomplemented
76C2×C3×SL(2,𝔽3)C22×C313yes—
77C2×C3×SL(2,𝔽3)C22×C316yescomplemented
78D3×SL(2,𝔽3)C22×C316yescomplemented
79C3⋊(#41⋊C3)C22×C316yescomplemented
80C22×C33—41no—
81C22×C33—42no—
82C22×C33—82no—
83C2×C32×D3—42no—
84C2×C32×D3—82no—
85C32×Dic3—42no—
86C3×2−1+4—34no—
87C3×2−1+4—94no—
88C2×C3×#41—31no—
89C2×C3×#41—91no—
90C2×C3×#41—91no—
91C2×C3×#41—91no—
92C22×C3×Q8—21no—
93C22×C3×Q8—31no—
94C22×C3×Q8—91no—
95C22×C3×Q8C3×D311yescharacteristic, complemented
96C22×C3×Q8C2×C3211yescharacteristic, complemented
97#1007—34no—
98#1007C2×C3214yescomplemented
99C2×#239—31no—
100C2×#239C2×C3211yescharacteristic, complemented
101C2×D3×Q8—31no—
102C2×D3×Q8C2×C3211yescharacteristic, complemented
103C2×#235—31no—
104C2×#235—31no—
1052−1+4⋊C3—312no—
106C2×#41⋊C3—33no—
107C22×SL(2,𝔽3)—23no—
108C22×SL(2,𝔽3)—33no—
109C22×SL(2,𝔽3)C3×D313yescomplemented
110C23×C32—41no—
111C23×C32—43no—
112C23×C32—83no—
113C23×C32—81no—
114C23×C32—121no—
115C23×C32C2×A411yescharacteristic
116C22×C3×D3—31no—
117C22×C3×D3—43no—
118C22×C3×D3C2×A411yescharacteristic
119C32×Q8—31no—
120C32×Q8—32no—
121C32×Q8—32no—
122C32×Q8C23×C311yescharacteristic
123C32×D4—124no—
124C2×C32×C4—31no—
125C2×C32×C4—31no—
126C2×C32×C4—32no—
127C2×C32×C4—32no—
128C2×C32×C4—121no—
129C3×#67—34no—
130C3×#67—412no—
131C3×#67C2×A414yes—
132C2×C3×Dic3—31no—
133C2×C3×Dic3—43no—
134C2×C3×Dic3C2×A411yescharacteristic
135C3×D12—32no—
136C3×D12—32no—
137C3×D12—34no—
138C3×C4×D3—38no—
139C3×C4×D3—32no—
140C3×C4×D3—32no—
141C3×C4×D3—32no—
142C3×C4×D3—32no—
143C3×Dic6—32no—
144C3×Dic6—34no—
145C3×Dic6—32no—
146C3×SL(2,𝔽3)—21no—
147C3×SL(2,𝔽3)—23no—
148C3×SL(2,𝔽3)C22×D311yescharacteristic
149C3×SL(2,𝔽3)C23×C313yes—
150C2×2−1+4—31noSylow
151C2×C33—42no—
152C2×C33—41no—
153C2×C33—82no—
154C32×D3—82no—
155C3×#41—32no—
156C3×#41—94no—
157C3×#41—94no—
158C3×#41—94no—
159C3×#41—94no—
160C3×#41—94no—
161C3×#41—94no—
162C3×#41—92no—
163C2×C3×Q8—21no—
164C2×C3×Q8—22no—
165C2×C3×Q8—31no—
166C2×C3×Q8—32no—
167C2×C3×Q8—31no—
168C2×C3×Q8—32no—
169C2×C3×Q8—32no—
170C2×C3×Q8—61no—
171C2×C3×Q8—62no—
172C2×C3×Q8—91no—
173C2×C3×Q8—91no—
174C2×C3×Q8—94no—
175C2×C3×Q8—94no—
176C2×C3×Q8—92no—
177C2×C3×Q8C2×C3×D311yescharacteristic
178C2×C3×Q8C2×C3×D312yescomplemented
179C2×C3×Q8C22×C3211yescharacteristic
180C2×C3×Q8C22×C3212yescomplemented
181C2×C3×D4—31no—
182C2×C3×D4—91no—
183C2×C3×D4—91no—
184C2×C3×D4—91no—
185C2×C3×D4—123no—
186C22×C3×C4—31no—
187C22×C3×C4—31no—
188C22×C3×C4—61no—
189C22×C3×C4—91no—
190C22×C3×C4—91no—
191C22×C3×C4—92no—
192C2×#67—31no—
193C2×#67C3×A411yescharacteristic
194#239—34no—
195#239—34no—
196#239—32no—
197#239C22×C3212yescomplemented
198D3×Q8—34no—
199D3×Q8—32no—
200D3×Q8—34no—
201D3×Q8C22×C3212yescomplemented
202#235—34no—
203#235—34no—
204#235—34no—
205#235—34no—
206C2×D12—31no—
207C2×D12—31no—
208C2×C4×D3—32no—
209C2×C4×D3—31no—
210C2×C4×D3—31no—
211C2×Dic6—31no—
212C2×Dic6—31no—
213#41⋊C3—36no—
214C2×SL(2,𝔽3)—23no—
215C2×SL(2,𝔽3)—26no—
216C2×SL(2,𝔽3)—36no—
217C2×SL(2,𝔽3)C2×C3×D313yes—
218C2×SL(2,𝔽3)C2×C3×D316yescomplemented
219C22×C32—22no—
220C22×C32—42no—
221C22×C32—41no—
222C22×C32—43no—
223C22×C32—46no—
224C22×C32—83no—
225C22×C32—86no—
226C22×C32—82no—
227C22×C32—86no—
228C22×C32—812no—
229C22×C32—84no—
230C22×C32—82no—
231C22×C32—81no—
232C22×C32—122no—
233C22×C32—242no—
234C22×C32C22×A412yes—
235C22×C32C22×A411yescharacteristic
236C2×C3×D3—22no—
237C2×C3×D3—32no—
238C2×C3×D3—46no—
239C2×C3×D3—62no—
240C2×C3×D3—86no—
241C2×C3×D3C22×A412yes—
242C32×C4—31no—
243C32×C4—32no—
244C32×C4—31no—
245C32×C4—122no—
246C3×Dic3—32no—
247C3×Dic3—46no—
248C3×Dic3C22×A412yes—
2492−1+4—34no—
2502−1+4—94no—
251C2×#41—31no—
252C2×#41—91no—
253C2×#41—91no—
254C2×#41—91no—
255C22×Q8—31no—
256C22×Q8—91no—
257C22×Q8C32×D311yescharacteristic, complemented
258C33—41noSylow
259C23×C3—21no—
260C23×C3—31no—
261C23×C3—43no—
262C23×C3—83no—
263C23×C3—91no—
264C23×C3—123no—
265C23×C3D3×A411yescharacteristic
266C23×C3C2×C3×A411yescharacteristic
267C22×D3—31no—
268C22×D3C2×C3×A411yescharacteristic
269C3×Q8—21no—
270C3×Q8—31no—
271C3×Q8—32no—
272C3×Q8—31no—
273C3×Q8—32no—
274C3×Q8—32no—
275C3×Q8—32no—
276C3×Q8—61no—
277C3×Q8—62no—
278C3×Q8—62no—
279C3×Q8—94no—
280C3×Q8—92no—
281C3×Q8—92no—
282C3×Q8C22×C3×D311yescharacteristic
283C3×Q8C23×C3211yescharacteristic
284C3×D4—34no—
285C3×D4—94no—
286C3×D4—92no—
287C3×D4—92no—
288C3×D4—94no—
289C3×D4—1212no—
290C2×C3×C4—32no—
291C2×C3×C4—32no—
292C2×C3×C4—31no—
293C2×C3×C4—31no—
294C2×C3×C4—31no—
295C2×C3×C4—31no—
296C2×C3×C4—31no—
297C2×C3×C4—32no—
298C2×C3×C4—32no—
299C2×C3×C4—61no—
300C2×C3×C4—62no—
301C2×C3×C4—61no—
302C2×C3×C4—62no—
303C2×C3×C4—92no—
304C2×C3×C4—92no—
305C2×C3×C4—92no—
306C2×C3×C4—92no—
307C2×C3×C4—98no—
308C2×C3×C4—91no—
309C2×C3×C4—123no—
310#67—34no—
311#67C2×C3×A414yes—
312C2×Dic3—31no—
313C2×Dic3C2×C3×A411yescharacteristic
314D12—32no—
315D12—32no—
316D12—34no—
317C4×D3—38no—
318C4×D3—32no—
319C4×D3—32no—
320C4×D3—32no—
321C4×D3—32no—
322Dic6—32no—
323Dic6—34no—
324Dic6—32no—
325SL(2,𝔽3)—23no—
326SL(2,𝔽3)C22×C3×D313yes—
327C2×C32—22no—
328C2×C32—42no—
329C2×C32—41no—
330C2×C32—46no—
331C2×C32—43no—
332C2×C32—83no—
333C2×C32—86no—
334C2×C32—82no—
335C2×C32—86no—
336C2×C32—812no—
337C2×C32—84no—
338C2×C32—82no—
339C2×C32—81no—
340C2×C32—242no—
341C2×C322−1+4⋊C312yescomplemented
342C2×C32C23×A411yescharacteristic
343C3×D3—22no—
344C3×D3—62no—
345C3×D3—86no—
346#41—32no—
347#41—94no—
348#41—94no—
349#41—94no—
350#41—94no—
351#41—94no—
352#41—94no—
353#41—92no—
354C2×Q8—31no—
355C2×Q8—32no—
356C2×Q8—32no—
357C2×Q8—91no—
358C2×Q8—91no—
359C2×Q8—94no—
360C2×Q8—94no—
361C2×Q8—92no—
362C2×Q8C2×C32×D311yescharacteristic
363C2×Q8C2×C32×D312yescomplemented
364C2×D4—31no—
365C2×D4—91no—
366C2×D4—91no—
367C2×D4—91no—
368C22×C4—31no—
369C22×C4—91no—
370C22×C4—91no—
371C22×C4—92no—
372C22×C3—24no—
373C22×C3—22no—
374C22×C3—22no—
375C22×C3—22no—
376C22×C3—21no—
377C22×C3—32no—
378C22×C3—43no—
379C22×C3—46no—
380C22×C3—62no—
381C22×C3—86no—
382C22×C3—812no—
383C22×C3—86no—
384C22×C3—83no—
385C22×C3—92no—
386C22×C3—126no—
387C22×C3—182no—
388C22×C3—246no—
389C22×C3C2×D3×A412yes—
390C22×C3C2×D3×A411yescharacteristic
391C22×C3C22×C3×A412yes—
392C22×C3C22×C3×A411yescharacteristic
393C2×D3—22no—
394C2×D3—32no—
395C2×D3—62no—
396C2×D3C22×C3×A412yes—
397C3×C4—31no—
398C3×C4—32no—
399C3×C4—31no—
400C3×C4—32no—
401C3×C4—31no—
402C3×C4—31no—
403C3×C4—32no—
404C3×C4—61no—
405C3×C4—61no—
406C3×C4—62no—
407C3×C4—92no—
408C3×C4—126no—
409Dic3—32no—
410Dic3C22×C3×A412yes—
411C32—41no—
412C32—43no—
413C32—83no—
414C32—81no—
415C32C2×2−1+4⋊C311yescharacteristic, complemented
416C23—31no—
417C23—91no—
418C23C3×D3×A411yescharacteristic
419Q8—31no—
420Q8—32no—
421Q8—32no—
422Q8—94no—
423Q8—92no—
424Q8—92no—
425Q8C22×C32×D311yescharacteristic
426D4—34no—
427D4—94no—
428D4—92no—
429D4—92no—
430D4—94no—
431C2×C4—31no—
432C2×C4—31no—
433C2×C4—32no—
434C2×C4—32no—
435C2×C4—31no—
436C2×C4—92no—
437C2×C4—92no—
438C2×C4—92no—
439C2×C4—92no—
440C2×C4—98no—
441C2×C4—91no—
442C2×C3—24no—
443C2×C3—22no—
444C2×C3—22no—
445C2×C3—22no—
446C2×C3—21no—
447C2×C3—46no—
448C2×C3—43no—
449C2×C3—62no—
450C2×C3—86no—
451C2×C3—812no—
452C2×C3—86no—
453C2×C3—83no—
454C2×C3—182no—
455C2×C3—246no—
456C2×C3#1007⋊C312yesdirect product factor
457C2×C3C3×2−1+4⋊C312yescomplemented
458C2×C3C22×D3×A411yescharacteristic
459C2×C3C23×C3×A411yescharacteristic
460D3—22no—
461D3—62no—
462C22—22no—
463C22—32no—
464C22—62no—
465C22—92no—
466C22—182no—
467C22C2×C3×D3×A412yes—
468C22C2×C3×D3×A411yescharacteristic
469C4—31no—
470C4—31no—
471C4—32no—
472C4—32no—
473C4—92no—
474C3—21no—
475C3—43no—
476C3—83no—
477C3C2×#1007⋊C311yescharacteristic, direct product factor
478C3C2×C3×2−1+4⋊C311yescharacteristic, complemented
479C2—22no—
480C2—62no—
481C2—182no—
482C2C3×#1007⋊C312yesdirect product factor
483C2C22×C3×D3×A411yescharacteristic
subgroupG/H
centerC22×C3C2×D3×A4
commutator, cosocle, hypocenterC3×Q8C23×C32
FrattiniC2C22×C3×D3×A4
FittingC22×C32×Q8C2×C3
socleC22×C32C22×A4
hypercenterC23×C3D3×A4
LCSG ▷ C3×Q8
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C22×C32×SL(2,𝔽3)C2semidirect
2. C3×#1007⋊C3C2direct (N × H)
3. C2×C3×D3×SL(2,𝔽3)C2semidirect
4. C2×C3×C3⋊(#41⋊C3)C2semidirect
5. C2×C3×#1007C3semidirect
7. C2×#1007⋊C3C3direct (N × H)
10. C2×C32×SL(2,𝔽3)C22semidirect
11. C3×D3×SL(2,𝔽3)C22semidirect
12. C3×C3⋊(#41⋊C3)C22semidirect
13. C22×C32×Q8C2×C3semidirect
15. C3×#1007C2×C3semidirect
17. C2×C3×#239C2×C3semidirect
19. C2×C3×D3×Q8C2×C3semidirect
27. C22×C3×SL(2,𝔽3)D3semidirect
28. C22×C3×SL(2,𝔽3)C2×C3semidirect
29. #1007⋊C3C2×C3direct (N × H)
30. C2×D3×SL(2,𝔽3)C2×C3semidirect
31. C2×C3⋊(#41⋊C3)C2×C3semidirect
38. C2×#1007C32semidirect
43. C2×C32×Q8C22×C3semidirect
52. C3×#239C22×C3semidirect
56. C3×D3×Q8C22×C3semidirect
75. C2×C3×SL(2,𝔽3)C2×D3semidirect
77. C2×C3×SL(2,𝔽3)C22×C3semidirect
78. D3×SL(2,𝔽3)C22×C3semidirect
79. C3⋊(#41⋊C3)C22×C3semidirect
95. C22×C3×Q8C3×D3semidirect
96. C22×C3×Q8C2×C32semidirect
98. #1007C2×C32semidirect
100. C2×#239C2×C32semidirect
102. C2×D3×Q8C2×C32semidirect
109. C22×SL(2,𝔽3)C3×D3semidirect
178. C2×C3×Q8C2×C3×D3semidirect
180. C2×C3×Q8C22×C32semidirect
197. #239C22×C32semidirect
201. D3×Q8C22×C32semidirect
218. C2×SL(2,𝔽3)C2×C3×D3semidirect
257. C22×Q8C32×D3semidirect
341. C2×C322−1+4⋊C3semidirect
363. C2×Q8C2×C32×D3semidirect
415. C32C2×2−1+4⋊C3semidirect
456. C2×C3#1007⋊C3direct (N × H)
457. C2×C3C3×2−1+4⋊C3semidirect
477. C3C2×#1007⋊C3direct (N × H)
478. C3C2×C3×2−1+4⋊C3semidirect
482. C2C3×#1007⋊C3direct (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
C3×#1007⋊C32
C2×C3×D3×SL(2,𝔽3)2
C2×C3×C3⋊(#41⋊C3)2
C2×C3×#10073
C2×C3×2−1+4⋊C33
C2×#1007⋊C33
C2×C32×#674