Finite Group Databasebeta

← C3×#10590757 #13157743

Subgroups — C3×#10590757 #13157743

Order 1728.

578 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#63918C211yesmaximal, characteristic, complemented
2C3×Dic3×D12C211yesmaximal, characteristic
3C3×#63644C211yesmaximal, characteristic, complemented
4C32×#821C211yesmaximal, characteristic
5C32×Dic3⋊C8C211yesmaximal, characteristic, complemented
6C3×#63573C211yesmaximal, characteristic, complemented
7C3×#63563C211yesmaximal, characteristic
8C3×#4842—31nomaximal
9C3×#4761—31nomaximal
10#10590757C311yesmaximal, characteristic, direct product factor
11C2×C32×D12C2211yescharacteristic
12C2×C3×#627—21no—
13C2×C3×D3×Dic3—21no—
14C2×C33×C8C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4×Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C3×Dic3⋊Dic3—21no—
21C3×(C2×D3)⋊Dic3—21no—
22C3×D4×Dic3—31no—
23C3×#888—31no—
24C3×C4×D12—31no—
25C3×#869—31no—
26#63918C2×C311yescharacteristic, complemented
27Dic3×D12C2×C311yescharacteristic
28C32×C4⋊C8—31no—
29C32×#102—31no—
30#63644C2×C311yescharacteristic, complemented
31C3×#832—31no—
32C3×#821—21no—
33C3×#821—31no—
34C3×#821D311yescharacteristic
35C3×#821C2×C311yescharacteristic
36C3×#817—31no—
37C3×#817—31no—
38C3×#817—31no—
39C3×#817—61no—
40C3×Dic3⋊C8—21no—
41C3×Dic3⋊C8D311yescharacteristic, complemented
42C3×Dic3⋊C8C2×C311yescharacteristic, complemented
43C3×#808—31no—
44#63573C2×C311yescharacteristic, complemented
45#63563C2×C311yescharacteristic
46C22×C32×D3—21no—
47C2×C33×C4C2311yescharacteristic
48C2×C3×C3⋊Dic3—21no—
49C2×C3×C3⋊Dic3—21no—
50C2×C32×Dic3—21no—
51C2×C32×Dic3D411yescharacteristic
52C2×C32×Dic3D411yescharacteristic
53C32×D12—21no—
54C32×D12Q812yescomplemented
55C3×#627—42no—
56C3×D3×Dic3—42no—
57C33×C8—21no—
58C32×C3⋊C8—21no—
59C3×#476—91no—
60#4842—31no—
61#4761—31no—
62C2×C32×D4—31no—
63C22×C3×Dic3—61no—
64C2×C3×D12—21no—
65C2×C3×D12C2×D311yescharacteristic
66C2×C3×D12C22×C311yescharacteristic
67C2×C3×C4×D3—61no—
68C2×C3×Dic6—61no—
69C2×C3×Dic6—61no—
70C2×#627—21no—
71C2×D3×Dic3—21no—
72C2×C32×C8—21no—
73C2×C32×C8—21no—
74C2×C32×C8—21no—
75C2×C32×C8—31no—
76C2×C32×C8—41no—
77C2×C32×C8C2×D311yescharacteristic
78C2×C32×C8C2×D311yescharacteristic
79C2×C32×C8C22×C311yescharacteristic
80C32×C4⋊C4—31no—
81C32×C42—31no—
82C4⋊(C3⋊Dic3)C22×C311yescharacteristic
83C4⋊(C3⋊Dic3)C22×C311yescharacteristic
84C3×#217—61no—
85C3×(C2×D3)⋊C4—61no—
86C3×C4⋊Dic3—21no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—31no—
89C3×C4⋊Dic3—31no—
90C3×C4⋊Dic3—31no—
91C3×C4⋊Dic3—61no—
92C3×C4⋊Dic3—61no—
93C3×C4⋊Dic3—61no—
94C3×C4⋊Dic3C2×D311yescharacteristic
95C3×C4⋊Dic3C22×C311yescharacteristic
96C3×Dic3⋊C4—61no—
97C3×C4×Dic3—21no—
98C3×C4×Dic3C2×D311yescharacteristic
99C3×C4×Dic3C22×C311yescharacteristic
100C2×C3×C3⋊C8—21no—
101C2×C3×C3⋊C8C2×D311yescharacteristic
102C2×C3×C3⋊C8C22×C311yescharacteristic
103Dic3⋊Dic3—21no—
104(C2×D3)⋊Dic3—21no—
105C22×C33C2×D411yescharacteristic
106C2×C32×D3—22no—
107C2×C32×D3—41no—
108C2×C32×D3—41no—
109C33×C4C2×Q811yescharacteristic
110C33×C4#4111yescharacteristic
111C3×C3⋊Dic3—41no—
112C3×C3⋊Dic3—41no—
113C32×Dic3—21no—
114C32×Dic3—41no—
115C32×Dic3SD1612yes—
116C3×C4⋊Q8—91no—
117C3×C4×D4—91no—
118D4×Dic3—31no—
119#888—31no—
120C4×D12—31no—
121#869—31no—
122C3×#106—91no—
123C3×#106—91no—
124C3×C4⋊C8—31no—
125C3×C4⋊C8—31no—
126C3×C4⋊C8—61no—
127C3×#102—31no—
128C3×#102—31no—
129C3×#102—61no—
130C3×#102—91no—
131#832—31no—
132#821—31no—
133#821C3×D311yescharacteristic
134#817—31no—
135#817—31no—
136#817—31no—
137#817—61no—
138Dic3⋊C8C3×D311yescharacteristic, complemented
139#808—31no—
140C23×C32—61no—
141C22×C3×D3—21no—
142C22×C3×D3—21no—
143C22×C3×D3—41no—
144C32×D4—32no—
145C32×D4—61no—
146C2×C32×C4—21no—
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4—31no—
150C2×C32×C4—31no—
151C2×C32×C4—41no—
152C2×C32×C4—61no—
153C2×C32×C4C22×D311yescharacteristic
154C2×C32×C4C22×D311yescharacteristic
155C2×C32×C4C23×C311yescharacteristic
156C2×C3⋊Dic3—21no—
157C2×C3⋊Dic3—21no—
158C2×C3×Dic3—21no—
159C2×C3×Dic3—21no—
160C2×C3×Dic3—21no—
161C2×C3×Dic3—21no—
162C2×C3×Dic3—41no—
163C2×C3×Dic3—61no—
164C2×C3×Dic3—61no—
165C2×C3×Dic3—61no—
166C2×C3×Dic3—61no—
167C2×C3×Dic3—121no—
168C2×C3×Dic3—121no—
169C2×C3×Dic3—122no—
170C2×C3×Dic3D1211yescharacteristic
171C2×C3×Dic3D1211yescharacteristic
172C2×C3×Dic3C3×D411yescharacteristic
173C2×C3×Dic3C3×D411yescharacteristic
174C3×D12—21no—
175C3×D12—21no—
176C3×D12—22no—
177C3×D12—41no—
178C3×D12Dic612yescomplemented
179C3×D12C3×Q812yescomplemented
180C3×C4×D3—122no—
181C3×Dic6—122no—
182C3×Dic6—122no—
183#627—42no—
184D3×Dic3—42no—
185C32×C8—21no—
186C32×C8—22no—
187C32×C8—21no—
188C32×C8—21no—
189C32×C8—41no—
190C32×C8—41no—
191C32×C8—42no—
192C32×C8—61no—
193C3×C3⋊C8—21no—
194C3×C3⋊C8—21no—
195C3×C3⋊C8—41no—
196#476—91noSylow
197C2×C33C22⋊Q811yescharacteristic
198C2×C33C2×SD1611yescharacteristic
199C2×C33AGL(1,ℤ/8)11yescharacteristic
200C32×D3—42no—
201C2×C3×Q8—181no—
202C2×C3×D4—31no—
203C2×C3×D4—31no—
204C2×C3×D4—61no—
205C22×C3×C4—181no—
206C22×Dic3—61no—
207C2×D12C2×C3×D311yescharacteristic
208C2×C4×D3—61no—
209C2×Dic6—61no—
210C2×Dic6—61no—
211C2×C3×C8—21no—
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8—31no—
215C2×C3×C8—31no—
216C2×C3×C8—41no—
217C2×C3×C8—61no—
218C2×C3×C8D3211yescharacteristic
219C2×C3×C8C2×C3×D311yescharacteristic
220C2×C3×C8C2×C3×D311yescharacteristic
221C3×C4⋊C4—31no—
222C3×C4⋊C4—31no—
223C3×C4⋊C4—61no—
224C3×C4⋊C4—91no—
225C3×C4⋊C4—91no—
226C3×C4⋊C4—181no—
227C3×C22⋊C4—181no—
228C3×C42—31no—
229C3×C42—31no—
230C3×C42—61no—
231#217—61no—
232(C2×D3)⋊C4—61no—
233C4⋊Dic3—31no—
234C4⋊Dic3—31no—
235C4⋊Dic3—31no—
236C4⋊Dic3—31no—
237C4⋊Dic3—61no—
238C4⋊Dic3—61no—
239C4⋊Dic3—61no—
240C4⋊Dic3C2×C3×D311yescharacteristic
241Dic3⋊C4—61no—
242C4×Dic3C2×C3×D311yescharacteristic
243C2×C3⋊C8C2×C3×D311yescharacteristic
244C22×C32—21no—
245C22×C32—21no—
246C22×C32—21no—
247C22×C32—41no—
248C22×C32—62no—
249C22×C32—121no—
250C22×C32—121no—
251C22×C32C2×D1211yescharacteristic
252C22×C32D3×D411yescharacteristic
253C22×C32C2×C3×D411yescharacteristic
254C2×C3×D3—22no—
255C2×C3×D3—22no—
256C2×C3×D3—42no—
257C2×C3×D3—41no—
258C2×C3×D3—41no—
259C2×C3×D3—41no—
260C2×C3×D3—41no—
261C2×C3×D3—81no—
262C2×C3×D3—81no—
263C32×C4—21no—
264C32×C4—21no—
265C32×C4—21no—
266C32×C4—21no—
267C32×C4—21no—
268C32×C4—21no—
269C32×C4—32no—
270C32×C4—41no—
271C32×C4—41no—
272C32×C4—61no—
273C32×C4—121no—
274C32×C4C2×Dic611yescharacteristic
275C32×C4#23711yescharacteristic
276C32×C4#23711yescharacteristic
277C32×C4D3×Q811yescharacteristic
278C32×C4C2×C3×Q811yescharacteristic
279C32×C4C3×#4111yescharacteristic
280C3⋊Dic3—41no—
281C3⋊Dic3—41no—
282C3×Dic3—21no—
283C3×Dic3—21no—
284C3×Dic3—22no—
285C3×Dic3—41no—
286C3×Dic3—41no—
287C3×Dic3—41no—
288C3×Dic3—81no—
289C3×Dic3—121no—
290C3×Dic3—121no—
291C3×Dic3—121no—
292C3×Dic3—121no—
293C3×Dic3—241no—
294C3×Dic3—241no—
295C3×Dic3#20412yes—
296C3×Dic3C3×SD1612yes—
297C4⋊Q8—91no—
298C4×D4—91no—
299#106—91no—
300#106—91no—
301C4⋊C8—31no—
302#102—31no—
303#102—91no—
304C33#47611yescharacteristic, complemented, Sylow
305C23×C3—61no—
306C23×C3—61no—
307C23×C3—121no—
308C22×D3—21no—
309C3×Q8—362no—
310C3×D4—32no—
311C3×D4—32no—
312C3×D4—62no—
313C3×D4—61no—
314C3×D4—61no—
315C3×D4—121no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—31no—
320C2×C3×C4—31no—
321C2×C3×C4—31no—
322C2×C3×C4—31no—
323C2×C3×C4—41no—
324C2×C3×C4—61no—
325C2×C3×C4—61no—
326C2×C3×C4—61no—
327C2×C3×C4—61no—
328C2×C3×C4—121no—
329C2×C3×C4—181no—
330C2×C3×C4—181no—
331C2×C3×C4—362no—
332C2×C3×C4C2×D3211yescharacteristic
333C2×C3×C4C22×C3×D311yescharacteristic
334C2×C3×C4C22×C3×D311yescharacteristic
335C2×Dic3—21no—
336C2×Dic3—61no—
337C2×Dic3—61no—
338C2×Dic3—61no—
339C2×Dic3—61no—
340C2×Dic3—121no—
341C2×Dic3—121no—
342C2×Dic3—122no—
343C2×Dic3C3×D1211yescharacteristic
344C2×Dic3C3×D1211yescharacteristic
345D12—21no—
346D12C3×Dic612yescomplemented
347C4×D3—122no—
348Dic6—122no—
349Dic6—122no—
350C3×C8—22no—
351C3×C8—21no—
352C3×C8—21no—
353C3×C8—21no—
354C3×C8—41no—
355C3×C8—41no—
356C3×C8—42no—
357C3×C8—61no—
358C3×C8—61no—
359C3×C8—121no—
360C3⋊C8—21no—
361C2×C32—21no—
362C2×C32—21no—
363C2×C32—21no—
364C2×C32—21no—
365C2×C32—21no—
366C2×C32—21no—
367C2×C32—21no—
368C2×C32—21no—
369C2×C32—21no—
370C2×C32—41no—
371C2×C32—41no—
372C2×C32—41no—
373C2×C32—122no—
374C2×C32#87811yescharacteristic
375C2×C32#89711yescharacteristic
376C2×C32C2×#20411yescharacteristic
377C2×C32#90811yescharacteristic
378C2×C32#91111yescharacteristic
379C2×C32D3×SD1611yescharacteristic
380C2×C32C3×C22⋊Q811yescharacteristic
381C2×C32C2×C3×SD1611yescharacteristic
382C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
383C3×D3—42no—
384C3×D3—42no—
385C3×D3—82no—
386C2×Q8—181no—
387C2×D4—31no—
388C22×C4—181no—
389C2×C8—31no—
390C2×C8C3×D3211yescharacteristic
391C4⋊C4—31no—
392C4⋊C4—91no—
393C4⋊C4—91no—
394C4⋊C4—181no—
395C22⋊C4—181no—
396C42—31no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—41no—
401C22×C3—62no—
402C22×C3—62no—
403C22×C3—122no—
404C22×C3—121no—
405C22×C3—121no—
406C22×C3—121no—
407C22×C3—121no—
408C22×C3—241no—
409C22×C3—241no—
410C22×C3D3×D1211yescharacteristic
411C22×C3C2×C3×D1211yescharacteristic
412C22×C3C3×D3×D411yescharacteristic
413C2×D3—22no—
414C2×D3—41no—
415C2×D3—41no—
416C3×C4—21no—
417C3×C4—21no—
418C3×C4—21no—
419C3×C4—21no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—32no—
423C3×C4—32no—
424C3×C4—41no—
425C3×C4—41no—
426C3×C4—62no—
427C3×C4—61no—
428C3×C4—61no—
429C3×C4—121no—
430C3×C4—121no—
431C3×C4—121no—
432C3×C4—241no—
433C3×C4—361no—
434C3×C4—361no—
435C3×C4D3×Dic611yescharacteristic
436C3×C4#380511yescharacteristic
437C3×C4C2×C3×Dic611yescharacteristic
438C3×C4C3×#23711yescharacteristic
439C3×C4C3×#23711yescharacteristic
440C3×C4C3×D3×Q811yescharacteristic
441Dic3—21no—
442Dic3—41no—
443Dic3—121no—
444Dic3—121no—
445Dic3—121no—
446Dic3—121no—
447Dic3—241no—
448Dic3—241no—
449Dic3C3×#20412yes—
450C32—21no—
451C32—21no—
452C32—21no—
453C32—41no—
454C32#476111yescharacteristic, complemented
455C32#484211yescharacteristic, complemented
456C32C3×#47611yescharacteristic, complemented
457C23—61no—
458Q8—362no—
459D4—32no—
460D4—61no—
461C2×C4—31no—
462C2×C4—31no—
463C2×C4—61no—
464C2×C4—181no—
465C2×C4—181no—
466C2×C4—362no—
467C2×C4C2×C3×D3211yescharacteristic
468C8—21no—
469C8—61no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—41no—
480C2×C3—41no—
481C2×C3—41no—
482C2×C3—122no—
483C2×C3—122no—
484C2×C3—242no—
485C2×C3D3×#20411yescharacteristic
486C2×C3#6379511yescharacteristic
487C2×C3#6389311yescharacteristic
488C2×C3C3×#87811yescharacteristic
489C2×C3C3×#89711yescharacteristic
490C2×C3C2×C3×#20411yescharacteristic
491C2×C3C3×#90811yescharacteristic
492C2×C3C3×#91111yescharacteristic
493C2×C3C3×D3×SD1611yescharacteristic
494D3—42no—
495C22—62no—
496C22—121no—
497C22—121no—
498C22C3×D3×D1211yescharacteristic
499C4—32no—
500C4—61no—
501C4—121no—
502C4—361no—
503C4—361no—
504C4C3×D3×Dic611yescharacteristic
505C4C3×#380511yescharacteristic
506C3—21no—
507C3—21no—
508C3—21no—
509C3—41no—
510C3#1059075711yescharacteristic, direct product factor
511C3C3×#476111yescharacteristic, complemented
512C3C3×#484211yescharacteristic, complemented
513C2—122no—
514C2C3×D3×#20411yescharacteristic
515C2C3×#6379511yescharacteristic
516C2C3×#6389311yescharacteristic
subgroupG/H
centerC22×C3D3×D12
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypercenterC2×C3×C8D32
hypocenterC32C3×#476
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
1. C3×#63918C2semidirect
3. C3×#63644C2semidirect
5. C32×Dic3⋊C8C2semidirect
6. C3×#63573C2semidirect
10. #10590757C3direct (N × H)
26. #63918C2×C3semidirect
30. #63644C2×C3semidirect
41. C3×Dic3⋊C8D3semidirect
42. C3×Dic3⋊C8C2×C3semidirect
44. #63573C2×C3semidirect
54. C32×D12Q8semidirect
138. Dic3⋊C8C3×D3semidirect
178. C3×D12Dic6semidirect
179. C3×D12C3×Q8semidirect
304. C33#476semidirect
346. D12C3×Dic6semidirect
454. C32#4761semidirect
455. C32#4842semidirect
456. C32C3×#476semidirect
510. C3#10590757direct (N × H)
511. C3C3×#4761semidirect
512. C3C3×#4842semidirect

Maximal subgroups

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

subgroupindex
C3×#639182
C3×Dic3×D122
C3×#636442
C32×#8212
C32×Dic3⋊C82
C3×#635732
C3×#635632
C3×#48423
C3×#47613
#105907573