Finite Group Databasebeta

← C3×D3×2−1+4⋊C3 #13181452

Subgroups — C3×D3×2−1+4⋊C3 #13181452

Order 1728.

1112 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×2−1+4⋊C3C211yesmaximal, characteristic, complemented
2C3×#1010⋊C3C211yesmaximal, characteristic, complemented
3C3×D3×#41⋊C3C211yesmaximal, characteristic, complemented
4C3×#1007⋊C3C212yesmaximal, complemented
5C2×C3×D3×SL(2,𝔽3)C212yesmaximal, complemented
6C3×D3×2−1+4C311yesmaximal, characteristic, complemented
7C2×C3×2−1+4⋊C3—31nomaximal
8D3×2−1+4⋊C3C313yesmaximal, direct product factor
9C32×D3×D4—41nomaximal
10C32×#41⋊C3C2211yescharacteristic, complemented
11C2×C32×SL(2,𝔽3)C2212yescomplemented
12C3×D3×SL(2,𝔽3)C2211yescharacteristic
13C3×D3×SL(2,𝔽3)C2212yescomplemented
14C3×C3⋊(#41⋊C3)C2211yescharacteristic, complemented
15C32×2−1+4C2×C311yescharacteristic, complemented
16C3×#1010—32no—
17C3×#1010—31no—
18C3×#1010C2×C311yescharacteristic, complemented
19C3×D3×#41—31no—
20C3×D3×#41—32no—
21C3×D3×#41C2×C311yescharacteristic, complemented
22C3×#1007—31no—
23C3×#1007C2×C312yescomplemented
24C2×C3×D3×Q8—31no—
25C2×C3×D3×Q8C2×C312yescomplemented
26C3×2−1+4⋊C3—21no—
27C3×2−1+4⋊C3—23no—
28C3×2−1+4⋊C3—31no—
29C3×2−1+4⋊C3—32no—
30C3×2−1+4⋊C3D311yescharacteristic, direct product factor
31C3×2−1+4⋊C3C2×C313yescomplemented
32C2×C3×#41⋊C3—31no—
33C22×C3×SL(2,𝔽3)—32no—
34#1010⋊C3C2×C313yescomplemented
35D3×#41⋊C3C2×C313yescomplemented
36#1007⋊C3C2×C316yescomplemented
37C2×D3×SL(2,𝔽3)C2×C316yescomplemented
38C22×C32×D3—42no—
39C33×D4—41no—
40C32×#67—42no—
41C32×D12—41no—
42C32×C4×D3—41no—
43C32×SL(2,𝔽3)C2311yescharacteristic
44C2×C3×2−1+4—31no—
45D3×2−1+4C3211yescharacteristic, complemented
46C2×2−1+4⋊C3—33no—
47C32×#41—31no—
48C32×#41—32no—
49C32×#41C22×C311yescharacteristic, complemented
50C2×C32×Q8—31no—
51C2×C32×Q8C22×C312yescomplemented
52C2×C32×D4—121no—
53C3×#239—31no—
54C3×#239—32no—
55C3×#239C22×C311yescharacteristic, complemented
56C3×D3×Q8—32no—
57C3×D3×Q8—31no—
58C3×D3×Q8—32no—
59C3×D3×Q8—32no—
60C3×D3×Q8—32no—
61C3×D3×Q8C22×C311yescharacteristic
62C3×D3×Q8C22×C312yescomplemented
63C3×#237—34no—
64C3×#237—31no—
65C3×#237—32no—
66C3×#237—32no—
67C3×#237—31no—
68C3×D3×D4—32no—
69C3×D3×D4—31no—
70C3×D3×D4—43no—
71C3×D3×D4A411yescharacteristic
72C3×#235—34no—
73C3×#235—32no—
74C3×#235—32no—
75C3×#235—31no—
76C3×#235—31no—
77C2×C3×C4×D3—32no—
78C2×C3×C4×D3—32no—
79C2×C3×C4×D3—31no—
80C2×C3×Dic6—32no—
81C2×C3×Dic6—32no—
82C2×C3×Dic6—31no—
83C3×#41⋊C3—21no—
84C3×#41⋊C3—23no—
85C3×#41⋊C3—31no—
86C3×#41⋊C3C2×D311yescharacteristic, complemented
87C3×#41⋊C3C22×C313yescomplemented
88C2×C3×SL(2,𝔽3)—22no—
89C2×C3×SL(2,𝔽3)—26no—
90C2×C3×SL(2,𝔽3)—31no—
91C2×C3×SL(2,𝔽3)—32no—
92C2×C3×SL(2,𝔽3)C2×D312yescomplemented
93C2×C3×SL(2,𝔽3)C22×C316yescomplemented
94D3×SL(2,𝔽3)C22×C313yes—
95D3×SL(2,𝔽3)C22×C316yescomplemented
96C3⋊(#41⋊C3)C22×C313yescomplemented
97C22×C33—42no—
98C2×C32×D3—41no—
99C2×C32×D3—42no—
100C2×C32×D3—84no—
101C33×C4—41no—
102C32×Dic3—41no—
103C3×2−1+4—21no—
104C3×2−1+4—31no—
105C3×2−1+4—32no—
106C3×2−1+4—91no—
107C3×2−1+4—91no—
108C3×2−1+4—92no—
109C3×2−1+4C3×D311yescharacteristic, complemented
110C3×2−1+4C2×C3211yescharacteristic, complemented
111C2×C3×#41—31no—
112C2×C3×#41—91no—
113C2×C3×#41—92no—
114C22×C3×Q8—32no—
115C22×C3×Q8—91no—
116#1010—32no—
117#1010—31no—
118#1010C2×C3211yescharacteristic, complemented
119D3×#41—31no—
120D3×#41—32no—
121D3×#41C2×C3211yescharacteristic, complemented
122#1007—31no—
123#1007C2×C3212yescomplemented
124C2×D3×Q8—31no—
125C2×D3×Q8C2×C3212yescomplemented
1262−1+4⋊C3—23no—
1272−1+4⋊C3—33no—
1282−1+4⋊C3—36no—
1292−1+4⋊C3C3×D313yesdirect product factor
130C2×#41⋊C3—33no—
131C22×SL(2,𝔽3)—36no—
132C23×C32—122no—
133C22×C3×D3—31no—
134C22×C3×D3—46no—
135C22×C3×D3C2×A412yes—
136C32×Q8—32no—
137C32×Q8—31no—
138C32×Q8C23×C311yescharacteristic
139C32×D4—31no—
140C32×D4—32no—
141C32×D4—41no—
142C32×D4—43no—
143C32×D4—83no—
144C32×D4—81no—
145C32×D4—121no—
146C32×D4—122no—
147C32×D4C2×A411yescharacteristic
148C2×C32×C4—32no—
149C2×C32×C4—32no—
150C2×C32×C4—31no—
151C2×C32×C4—121no—
152C3×#67—32no—
153C3×#67—32no—
154C3×#67—32no—
155C3×#67—46no—
156C3×#67C2×A412yes—
157C2×C3×Dic3—32no—
158C2×C3×Dic3—32no—
159C2×C3×Dic3—31no—
160C3×D12—31no—
161C3×D12—32no—
162C3×D12—43no—
163C3×D12C2×A411yescharacteristic
164C3×C4×D3—34no—
165C3×C4×D3—32no—
166C3×C4×D3—31no—
167C3×C4×D3—32no—
168C3×C4×D3—32no—
169C3×C4×D3—31no—
170C3×C4×D3—31no—
171C3×C4×D3—43no—
172C3×C4×D3C2×A411yescharacteristic
173C3×Dic6—34no—
174C3×Dic6—31no—
175C3×Dic6—32no—
176C3×Dic6—31no—
177C3×Dic6—32no—
178C3×SL(2,𝔽3)—21no—
179C3×SL(2,𝔽3)—23no—
180C3×SL(2,𝔽3)C22×D311yescharacteristic
181C3×SL(2,𝔽3)C23×C313yes—
182C2×2−1+4—31noSylow
183C2×C33—41no—
184C2×C33—82no—
185C32×D3—42no—
186C32×D3—82no—
187C3×#41—21no—
188C3×#41—31no—
189C3×#41—31no—
190C3×#41—32no—
191C3×#41—32no—
192C3×#41—31no—
193C3×#41—61no—
194C3×#41—62no—
195C3×#41—92no—
196C3×#41—91no—
197C3×#41—94no—
198C3×#41—92no—
199C3×#41—91no—
200C3×#41—92no—
201C3×#41—91no—
202C3×#41—94no—
203C3×#41—92no—
204C3×#41—91no—
205C3×#41—92no—
206C3×#41—91no—
207C3×#41C2×C3×D311yescharacteristic, complemented
208C3×#41C22×C3211yescharacteristic, complemented
209C2×C3×Q8—22no—
210C2×C3×Q8—31no—
211C2×C3×Q8—31no—
212C2×C3×Q8—31no—
213C2×C3×Q8—32no—
214C2×C3×Q8—61no—
215C2×C3×Q8—91no—
216C2×C3×Q8—91no—
217C2×C3×Q8—92no—
218C2×C3×Q8—92no—
219C2×C3×Q8—92no—
220C2×C3×Q8—92no—
221C2×C3×Q8—92no—
222C2×C3×Q8—92no—
223C2×C3×Q8C2×C3×D312yescomplemented
224C2×C3×Q8C22×C3212yescomplemented
225C2×C3×D4—31no—
226C2×C3×D4—92no—
227C2×C3×D4—91no—
228C2×C3×D4—123no—
229C22×C3×C4—91no—
230C22×C3×C4—92no—
231C22×C3×C4—92no—
232#239—31no—
233#239—32no—
234#239C22×C3211yescharacteristic, complemented
235D3×Q8—32no—
236D3×Q8—31no—
237D3×Q8—32no—
238D3×Q8—32no—
239D3×Q8—32no—
240D3×Q8C22×C3211yescharacteristic
241D3×Q8C22×C3212yescomplemented
242#237—34no—
243#237—31no—
244#237—32no—
245#237—32no—
246#237—31no—
247D3×D4—31no—
248D3×D4—32no—
249D3×D4C3×A411yescharacteristic
250#235—34no—
251#235—32no—
252#235—32no—
253#235—31no—
254#235—31no—
255C2×C4×D3—32no—
256C2×C4×D3—32no—
257C2×C4×D3—31no—
258C2×Dic6—32no—
259C2×Dic6—32no—
260C2×Dic6—31no—
261#41⋊C3—23no—
262#41⋊C3—33no—
263#41⋊C3C2×C3×D313yescomplemented
264C2×SL(2,𝔽3)—26no—
265C2×SL(2,𝔽3)—33no—
266C2×SL(2,𝔽3)—36no—
267C2×SL(2,𝔽3)C2×C3×D316yescomplemented
268C22×C32—31no—
269C22×C32—42no—
270C22×C32—46no—
271C22×C32—86no—
272C22×C32—82no—
273C22×C32—121no—
274C22×C32—122no—
275C22×C32—244no—
276C22×C32C22×A412yes—
277C2×C3×D3—24no—
278C2×C3×D3—31no—
279C2×C3×D3—43no—
280C2×C3×D3—46no—
281C2×C3×D3—62no—
282C2×C3×D3—812no—
283C2×C3×D3C22×A412yes—
284C2×C3×D3C22×A411yescharacteristic
285C32×C4—32no—
286C32×C4—31no—
287C32×C4—41no—
288C32×C4—43no—
289C32×C4—83no—
290C32×C4—81no—
291C32×C4—121no—
292C32×C4C22×A411yescharacteristic
293C3×Dic3—31no—
294C3×Dic3—32no—
295C3×Dic3—43no—
296C3×Dic3C22×A411yescharacteristic
2972−1+4—31no—
2982−1+4—32no—
2992−1+4—91no—
3002−1+4—91no—
3012−1+4—92no—
3022−1+4C32×D311yescharacteristic, complemented
303C2×#41—31no—
304C2×#41—91no—
305C2×#41—92no—
306C22×Q8—32no—
307C22×Q8—91no—
308C33—41noSylow
309C23×C3—32no—
310C23×C3—91no—
311C23×C3—126no—
312C22×D3—31no—
313C22×D3C2×C3×A412yes—
314C3×Q8—21no—
315C3×Q8—32no—
316C3×Q8—32no—
317C3×Q8—31no—
318C3×Q8—31no—
319C3×Q8—61no—
320C3×Q8—62no—
321C3×Q8—92no—
322C3×Q8—91no—
323C3×Q8—92no—
324C3×Q8—94no—
325C3×Q8—91no—
326C3×Q8C22×C3×D311yescharacteristic
327C3×Q8C23×C3211yescharacteristic
328C3×D4—21no—
329C3×D4—31no—
330C3×D4—31no—
331C3×D4—32no—
332C3×D4—32no—
333C3×D4—32no—
334C3×D4—31no—
335C3×D4—43no—
336C3×D4—62no—
337C3×D4—61no—
338C3×D4—83no—
339C3×D4—92no—
340C3×D4—91no—
341C3×D4—92no—
342C3×D4—92no—
343C3×D4—92no—
344C3×D4—123no—
345C3×D4—126no—
346C3×D4D3×A411yescharacteristic
347C3×D4C2×C3×A411yescharacteristic
348C2×C3×C4—32no—
349C2×C3×C4—32no—
350C2×C3×C4—32no—
351C2×C3×C4—32no—
352C2×C3×C4—31no—
353C2×C3×C4—31no—
354C2×C3×C4—31no—
355C2×C3×C4—61no—
356C2×C3×C4—62no—
357C2×C3×C4—62no—
358C2×C3×C4—91no—
359C2×C3×C4—91no—
360C2×C3×C4—92no—
361C2×C3×C4—92no—
362C2×C3×C4—92no—
363C2×C3×C4—91no—
364C2×C3×C4—94no—
365C2×C3×C4—92no—
366C2×C3×C4—92no—
367C2×C3×C4—91no—
368C2×C3×C4—123no—
369#67—32no—
370#67—32no—
371#67—32no—
372#67C2×C3×A412yes—
373C2×Dic3—32no—
374C2×Dic3—32no—
375C2×Dic3—31no—
376D12—31no—
377D12—32no—
378D12C2×C3×A411yescharacteristic
379C4×D3—34no—
380C4×D3—32no—
381C4×D3—31no—
382C4×D3—32no—
383C4×D3—32no—
384C4×D3—31no—
385C4×D3—31no—
386C4×D3C2×C3×A411yescharacteristic
387Dic6—34no—
388Dic6—31no—
389Dic6—32no—
390Dic6—31no—
391Dic6—32no—
392SL(2,𝔽3)—23no—
393SL(2,𝔽3)C22×C3×D313yes—
394C2×C32—22no—
395C2×C32—41no—
396C2×C32—43no—
397C2×C32—61no—
398C2×C32—83no—
399C2×C32—82no—
400C2×C32—86no—
401C2×C32—81no—
402C2×C32—122no—
403C2×C32—162no—
404C2×C32—166no—
405C2×C32—242no—
406C2×C32C23×A411yescharacteristic
407C3×D3—22no—
408C3×D3—46no—
409C3×D3—61no—
410C3×D3—86no—
411C3×D32−1+4⋊C312yesdirect product factor
412#41—31no—
413#41—32no—
414#41—31no—
415#41—92no—
416#41—91no—
417#41—94no—
418#41—92no—
419#41—91no—
420#41—92no—
421#41—91no—
422#41—94no—
423#41—92no—
424#41—91no—
425#41—92no—
426#41—91no—
427#41C2×C32×D311yescharacteristic, complemented
428C2×Q8—31no—
429C2×Q8—31no—
430C2×Q8—32no—
431C2×Q8—91no—
432C2×Q8—91no—
433C2×Q8—92no—
434C2×Q8—92no—
435C2×Q8—92no—
436C2×Q8—92no—
437C2×Q8—92no—
438C2×Q8—92no—
439C2×Q8C2×C32×D312yescomplemented
440C2×D4—31no—
441C2×D4—92no—
442C2×D4—91no—
443C22×C4—91no—
444C22×C4—92no—
445C22×C4—92no—
446C22×C3—22no—
447C22×C3—31no—
448C22×C3—31no—
449C22×C3—32no—
450C22×C3—31no—
451C22×C3—46no—
452C22×C3—61no—
453C22×C3—64no—
454C22×C3—86no—
455C22×C3—91no—
456C22×C3—123no—
457C22×C3—126no—
458C22×C3—182no—
459C22×C3—2412no—
460C22×C3C2×D3×A412yes—
461C22×C3C22×C3×A412yes—
462C2×D3—24no—
463C2×D3—31no—
464C2×D3—62no—
465C2×D3C22×C3×A412yes—
466C2×D3C22×C3×A411yescharacteristic
467C3×C4—21no—
468C3×C4—31no—
469C3×C4—32no—
470C3×C4—31no—
471C3×C4—31no—
472C3×C4—32no—
473C3×C4—43no—
474C3×C4—61no—
475C3×C4—62no—
476C3×C4—83no—
477C3×C4—91no—
478C3×C4—92no—
479C3×C4—123no—
480C3×C4C2×D3×A411yescharacteristic
481C3×C4C22×C3×A411yescharacteristic
482Dic3—31no—
483Dic3—32no—
484Dic3C22×C3×A411yescharacteristic
485C32—41no—
486C32—43no—
487C32—83no—
488C32—81no—
489C32C2×2−1+4⋊C311yescharacteristic, complemented
490C23—32no—
491C23—91no—
492Q8—32no—
493Q8—31no—
494Q8—92no—
495Q8—91no—
496Q8—92no—
497Q8—94no—
498Q8—91no—
499Q8C22×C32×D311yescharacteristic
500D4—31no—
501D4—32no—
502D4—32no—
503D4—31no—
504D4—92no—
505D4—91no—
506D4—92no—
507D4—92no—
508D4—92no—
509D4C3×D3×A411yescharacteristic
510C2×C4—32no—
511C2×C4—32no—
512C2×C4—31no—
513C2×C4—31no—
514C2×C4—91no—
515C2×C4—91no—
516C2×C4—92no—
517C2×C4—92no—
518C2×C4—91no—
519C2×C4—92no—
520C2×C4—94no—
521C2×C4—92no—
522C2×C4—92no—
523C2×C4—91no—
524C2×C3—22no—
525C2×C3—22no—
526C2×C3—21no—
527C2×C3—32no—
528C2×C3—42no—
529C2×C3—43no—
530C2×C3—62no—
531C2×C3—61no—
532C2×C3—61no—
533C2×C3—86no—
534C2×C3—83no—
535C2×C3—126no—
536C2×C3—121no—
537C2×C3—166no—
538C2×C3—181no—
539C2×C3—246no—
540C2×C3C22×D3×A411yescharacteristic
541C2×C3C23×C3×A411yescharacteristic
542D3—22no—
543D3—61no—
544D3C3×2−1+4⋊C312yesdirect product factor
545C22—31no—
546C22—31no—
547C22—32no—
548C22—64no—
549C22—91no—
550C22—182no—
551C22C2×C3×D3×A412yes—
552C4—31no—
553C4—32no—
554C4—31no—
555C4—91no—
556C4—92no—
557C4C2×C3×D3×A411yescharacteristic
558C3—21no—
559C3—43no—
560C3—83no—
561C3D3×2−1+4⋊C311yescharacteristic, direct product factor
562C3C2×C3×2−1+4⋊C311yescharacteristic, complemented
563C2—22no—
564C2—32no—
565C2—62no—
566C2—61no—
567C2—181no—
568C2C22×C3×D3×A411yescharacteristic
subgroupG/H
centerC2×C3C22×D3×A4
commutator, cosocle, hypocenterC3×Q8C23×C32
FrattiniC2C22×C3×D3×A4
FittingC32×2−1+4C2×C3
socleC2×C32C23×A4
hypercenterC3×D4D3×A4
LCSG ▷ C3×Q8
UCSC1 ◁ C2×C3 ◁ C3×D4

Complements

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

NH ≅ G/Ntype
1. C32×2−1+4⋊C3C2semidirect
2. C3×#1010⋊C3C2semidirect
3. C3×D3×#41⋊C3C2semidirect
4. C3×#1007⋊C3C2semidirect
5. C2×C3×D3×SL(2,𝔽3)C2semidirect
6. C3×D3×2−1+4C3semidirect
8. D3×2−1+4⋊C3C3direct (N × H)
10. C32×#41⋊C3C22semidirect
11. C2×C32×SL(2,𝔽3)C22semidirect
13. C3×D3×SL(2,𝔽3)C22semidirect
14. C3×C3⋊(#41⋊C3)C22semidirect
15. C32×2−1+4C2×C3semidirect
18. C3×#1010C2×C3semidirect
21. C3×D3×#41C2×C3semidirect
23. C3×#1007C2×C3semidirect
25. C2×C3×D3×Q8C2×C3semidirect
30. C3×2−1+4⋊C3D3direct (N × H)
31. C3×2−1+4⋊C3C2×C3semidirect
34. #1010⋊C3C2×C3semidirect
35. D3×#41⋊C3C2×C3semidirect
36. #1007⋊C3C2×C3semidirect
37. C2×D3×SL(2,𝔽3)C2×C3semidirect
45. D3×2−1+4C32semidirect
49. C32×#41C22×C3semidirect
51. C2×C32×Q8C22×C3semidirect
55. C3×#239C22×C3semidirect
62. C3×D3×Q8C22×C3semidirect
86. C3×#41⋊C3C2×D3semidirect
87. C3×#41⋊C3C22×C3semidirect
92. C2×C3×SL(2,𝔽3)C2×D3semidirect
93. C2×C3×SL(2,𝔽3)C22×C3semidirect
95. D3×SL(2,𝔽3)C22×C3semidirect
96. C3⋊(#41⋊C3)C22×C3semidirect
109. C3×2−1+4C3×D3semidirect
110. C3×2−1+4C2×C32semidirect
118. #1010C2×C32semidirect
121. D3×#41C2×C32semidirect
123. #1007C2×C32semidirect
125. C2×D3×Q8C2×C32semidirect
129. 2−1+4⋊C3C3×D3direct (N × H)
207. C3×#41C2×C3×D3semidirect
208. C3×#41C22×C32semidirect
223. C2×C3×Q8C2×C3×D3semidirect
224. C2×C3×Q8C22×C32semidirect
234. #239C22×C32semidirect
241. D3×Q8C22×C32semidirect
263. #41⋊C3C2×C3×D3semidirect
267. C2×SL(2,𝔽3)C2×C3×D3semidirect
302. 2−1+4C32×D3semidirect
411. C3×D32−1+4⋊C3direct (N × H)
427. #41C2×C32×D3semidirect
439. C2×Q8C2×C32×D3semidirect
489. C32C2×2−1+4⋊C3semidirect
544. D3C3×2−1+4⋊C3direct (N × H)
561. C3D3×2−1+4⋊C3direct (N × H)
562. C3C2×C3×2−1+4⋊C3semidirect

Maximal subgroups

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

subgroupindex
C32×2−1+4⋊C32
C3×#1010⋊C32
C3×D3×#41⋊C32
C3×#1007⋊C32
C2×C3×D3×SL(2,𝔽3)2
C3×D3×2−1+43
C2×C3×2−1+4⋊C33
D3×2−1+4⋊C33
C32×D3×D44