Finite Group Databasebeta

← C3×#10590780 #13157765

Subgroups — C3×#10590780 #13157765

Order 1728.

554 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#63898C211yesmaximal, characteristic
2C3×#63872C211yesmaximal, characteristic, complemented
3C3×#63641C211yesmaximal, characteristic, complemented
4C32×#821C211yesmaximal, characteristic
5C32×#818C211yesmaximal, characteristic, complemented
6C3×#63574C211yesmaximal, characteristic, complemented
7C3×#63562C211yesmaximal, characteristic
8C3×#4891—31nomaximal
9C3×#4766—31nomaximal
10#10590780C311yesmaximal, characteristic, direct product factor
11C2×C32×D12C2211yescharacteristic
12C2×C3×D3×Dic3—21no—
13C2×C33×C8C2211yescharacteristic
14C3×C4×C3⋊Dic3C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C3×Dic3⋊Dic3—21no—
20C3×Dic3⋊Dic3—21no—
21C3×(C2×D3)⋊Dic3—21no—
22C3×D4×Dic3—31no—
23C3×#893—31no—
24C3×#890—31no—
25C3×#890—31no—
26#63898C2×C311yescharacteristic
27#63872C2×C311yescharacteristic, complemented
28C32×#107—31no—
29C32×#102—31no—
30#63641C2×C311yescharacteristic, complemented
31C3×#832—31no—
32C3×#821—21no—
33C3×#821D311yescharacteristic
34C3×#821C2×C311yescharacteristic
35C3×#818—21no—
36C3×#818D311yescharacteristic, complemented
37C3×#818C2×C311yescharacteristic, complemented
38C3×Dic3⋊C8—31no—
39C3×Dic3⋊C8—31no—
40C3×Dic3⋊C8—61no—
41C3×#809—31no—
42C3×#808—31no—
43C3×#808—31no—
44#63574C2×C311yescharacteristic, complemented
45#63562C2×C311yescharacteristic
46C22×C32×D3—21no—
47C2×C33×C4C2311yescharacteristic
48C2×C3×C3⋊Dic3D411yescharacteristic
49C2×C3×C3⋊Dic3D411yescharacteristic
50C2×C32×Dic3—21no—
51C2×C32×Dic3—21no—
52C2×C32×Dic3—21no—
53C32×D12—21no—
54C32×D12Q812yes—
55C3×D3×Dic3—42no—
56C33×C8—21no—
57C3×C3⋊(C3⋊C8)—21no—
58C3×#478—91no—
59#4891—31no—
60#4766—31no—
61C2×C32×D4—31no—
62C22×C3×Dic3—61no—
63C2×C3×D12—21no—
64C2×C3×D12C2×D311yescharacteristic
65C2×C3×D12C22×C311yescharacteristic
66C2×C3×C4×D3—61no—
67C2×D3×Dic3—21no—
68C2×C32×C8—21no—
69C2×C32×C8—21no—
70C2×C32×C8—21no—
71C2×C32×C8—41no—
72C2×C32×C8C2×D311yescharacteristic
73C2×C32×C8C2×D311yescharacteristic
74C2×C32×C8C22×C311yescharacteristic
75C32×C4⋊C4—31no—
76C32×C4⋊C4—31no—
77C32×C4⋊C4—31no—
78C4×C3⋊Dic3C22×C311yescharacteristic
79C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
80C3×#217—61no—
81C3×(C2×D3)⋊C4—61no—
82C3×C4⋊Dic3—21no—
83C3×C4⋊Dic3—21no—
84C3×C4⋊Dic3—21no—
85C3×C4⋊Dic3—61no—
86C3×C4⋊Dic3—61no—
87C3×C4⋊Dic3C2×D311yescharacteristic
88C3×C4⋊Dic3C2×D311yescharacteristic
89C3×C4⋊Dic3C2×D311yescharacteristic
90C3×C4⋊Dic3C22×C311yescharacteristic
91C3×C4⋊Dic3C22×C311yescharacteristic
92C3×C4⋊Dic3C22×C311yescharacteristic
93C3×Dic3⋊C4—61no—
94C3×Dic3⋊C4—61no—
95C3×C4×Dic3—31no—
96C3×C4×Dic3—31no—
97C3×C4×Dic3—61no—
98C2×C3×C3⋊C8—31no—
99C2×C3×C3⋊C8—31no—
100C2×C3×C3⋊C8—61no—
101Dic3⋊Dic3—21no—
102Dic3⋊Dic3—21no—
103(C2×D3)⋊Dic3—21no—
104C22×C33C2×D411yescharacteristic
105C2×C32×D3—22no—
106C2×C32×D3—41no—
107C2×C32×D3—41no—
108C33×C4C2×Q811yescharacteristic
109C33×C4#4111yescharacteristic
110C3×C3⋊Dic3—21no—
111C3×C3⋊Dic3—21no—
112C32×Dic3—41no—
113C32×Dic3—41no—
114C32×Dic3—41no—
115C3×#125—91no—
116C3×C4×D4—91no—
117D4×Dic3—31no—
118#893—31no—
119#890—31no—
120#890—31no—
121C3×#107—31no—
122C3×#107—31no—
123C3×#107—61no—
124C3×#106—91no—
125C3×C4⋊C8—91no—
126C3×#102—31no—
127C3×#102—31no—
128C3×#102—61no—
129C3×#102—91no—
130#832—31no—
131#821C3×D311yescharacteristic
132#818C3×D311yescharacteristic, complemented
133Dic3⋊C8—31no—
134Dic3⋊C8—31no—
135Dic3⋊C8—61no—
136#809—31no—
137#808—31no—
138#808—31no—
139C23×C32—61no—
140C22×C3×D3—21no—
141C22×C3×D3—21no—
142C22×C3×D3—41no—
143C32×D4—32no—
144C32×D4—61no—
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—21no—
148C2×C32×C4—41no—
149C2×C32×C4—61no—
150C2×C32×C4—61no—
151C2×C32×C4—61no—
152C2×C32×C4C22×D311yescharacteristic
153C2×C32×C4C22×D311yescharacteristic
154C2×C32×C4C23×C311yescharacteristic
155C2×C3⋊Dic3C3×D411yescharacteristic
156C2×C3⋊Dic3C3×D411yescharacteristic
157C2×C3×Dic3—21no—
158C2×C3×Dic3—21no—
159C2×C3×Dic3—21no—
160C2×C3×Dic3—21no—
161C2×C3×Dic3—21no—
162C2×C3×Dic3—21no—
163C2×C3×Dic3—31no—
164C2×C3×Dic3—31no—
165C2×C3×Dic3—31no—
166C2×C3×Dic3—31no—
167C2×C3×Dic3—41no—
168C2×C3×Dic3—41no—
169C2×C3×Dic3—41no—
170C2×C3×Dic3—61no—
171C2×C3×Dic3—61no—
172C2×C3×Dic3—122no—
173C3×D12—21no—
174C3×D12—21no—
175C3×D12—22no—
176C3×D12—41no—
177C3×D12Dic612yes—
178C3×D12C3×Q812yes—
179C3×C4×D3—122no—
180D3×Dic3—42no—
181C32×C8—21no—
182C32×C8—22no—
183C32×C8—21no—
184C32×C8—21no—
185C32×C8—41no—
186C32×C8—41no—
187C32×C8—42no—
188C3⋊(C3⋊C8)—21no—
189C3×C3⋊C8—61no—
190C3×C3⋊C8—61no—
191C3×C3⋊C8—121no—
192#478—91noSylow
193C2×C33C22⋊Q811yescharacteristic
194C2×C33#13511yescharacteristic
195C2×C33AGL(1,ℤ/8)11yescharacteristic
196C32×D3—42no—
197C2×C3×D4—31no—
198C2×C3×D4—31no—
199C2×C3×D4—61no—
200C22×C3×C4—181no—
201C22×Dic3—61no—
202C2×D12C2×C3×D311yescharacteristic
203C2×C4×D3—61no—
204C2×C3×C8—21no—
205C2×C3×C8—21no—
206C2×C3×C8—21no—
207C2×C3×C8—41no—
208C2×C3×C8—91no—
209C2×C3×C8D3211yescharacteristic
210C2×C3×C8C2×C3×D311yescharacteristic
211C2×C3×C8C2×C3×D311yescharacteristic
212C3×C4⋊C4—31no—
213C3×C4⋊C4—31no—
214C3×C4⋊C4—31no—
215C3×C4⋊C4—31no—
216C3×C4⋊C4—31no—
217C3×C4⋊C4—31no—
218C3×C4⋊C4—61no—
219C3×C4⋊C4—61no—
220C3×C4⋊C4—61no—
221C3×C4⋊C4—181no—
222C3×C4⋊C4—181no—
223C3×C22⋊C4—181no—
224C3×C42—91no—
225#217—61no—
226(C2×D3)⋊C4—61no—
227C4⋊Dic3—61no—
228C4⋊Dic3—61no—
229C4⋊Dic3C2×C3×D311yescharacteristic
230C4⋊Dic3C2×C3×D311yescharacteristic
231C4⋊Dic3C2×C3×D311yescharacteristic
232Dic3⋊C4—61no—
233Dic3⋊C4—61no—
234C4×Dic3—31no—
235C4×Dic3—31no—
236C4×Dic3—61no—
237C2×C3⋊C8—31no—
238C2×C3⋊C8—31no—
239C2×C3⋊C8—61no—
240C22×C32—21no—
241C22×C32—21no—
242C22×C32—21no—
243C22×C32—41no—
244C22×C32—62no—
245C22×C32—121no—
246C22×C32—121no—
247C22×C32D3×D411yescharacteristic
248C22×C32D3×D411yescharacteristic
249C22×C32C2×C3×D411yescharacteristic
250C2×C3×D3—22no—
251C2×C3×D3—22no—
252C2×C3×D3—42no—
253C2×C3×D3—41no—
254C2×C3×D3—41no—
255C2×C3×D3—41no—
256C2×C3×D3—41no—
257C2×C3×D3—81no—
258C2×C3×D3—81no—
259C32×C4—21no—
260C32×C4—21no—
261C32×C4—21no—
262C32×C4—21no—
263C32×C4—21no—
264C32×C4—21no—
265C32×C4—41no—
266C32×C4—41no—
267C32×C4—121no—
268C32×C4—121no—
269C32×C4—121no—
270C32×C4C2×Dic611yescharacteristic
271C32×C4#23511yescharacteristic
272C32×C4#23711yescharacteristic
273C32×C4D3×Q811yescharacteristic
274C32×C4C2×C3×Q811yescharacteristic
275C32×C4C3×#4111yescharacteristic
276C3⋊Dic3—21no—
277C3⋊Dic3—21no—
278C3×Dic3—41no—
279C3×Dic3—41no—
280C3×Dic3—41no—
281C3×Dic3—41no—
282C3×Dic3—41no—
283C3×Dic3—41no—
284C3×Dic3—62no—
285C3×Dic3—61no—
286C3×Dic3—61no—
287C3×Dic3—61no—
288C3×Dic3—61no—
289C3×Dic3—81no—
290C3×Dic3—81no—
291C3×Dic3—81no—
292C3×Dic3—121no—
293#125—91no—
294C4×D4—91no—
295#107—31no—
296#106—91no—
297C4⋊C8—91no—
298#102—31no—
299#102—91no—
300C33#47811yescharacteristic, complemented, Sylow
301C23×C3—61no—
302C23×C3—61no—
303C23×C3—121no—
304C22×D3—21no—
305C3×D4—32no—
306C3×D4—32no—
307C3×D4—62no—
308C3×D4—61no—
309C3×D4—61no—
310C3×D4—121no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—41no—
315C2×C3×C4—61no—
316C2×C3×C4—61no—
317C2×C3×C4—61no—
318C2×C3×C4—61no—
319C2×C3×C4—61no—
320C2×C3×C4—61no—
321C2×C3×C4—91no—
322C2×C3×C4—91no—
323C2×C3×C4—121no—
324C2×C3×C4—121no—
325C2×C3×C4—121no—
326C2×C3×C4—362no—
327C2×C3×C4C2×D3211yescharacteristic
328C2×C3×C4C22×C3×D311yescharacteristic
329C2×C3×C4C22×C3×D311yescharacteristic
330C2×Dic3—21no—
331C2×Dic3—21no—
332C2×Dic3—21no—
333C2×Dic3—31no—
334C2×Dic3—31no—
335C2×Dic3—31no—
336C2×Dic3—31no—
337C2×Dic3—61no—
338C2×Dic3—61no—
339C2×Dic3—122no—
340D12—21no—
341D12C3×Dic612yes—
342C4×D3—122no—
343C3×C8—22no—
344C3×C8—21no—
345C3×C8—21no—
346C3×C8—21no—
347C3×C8—41no—
348C3×C8—41no—
349C3×C8—42no—
350C3×C8—181no—
351C3⋊C8—61no—
352C3⋊C8—61no—
353C3⋊C8—121no—
354C2×C32—21no—
355C2×C32—21no—
356C2×C32—21no—
357C2×C32—21no—
358C2×C32—21no—
359C2×C32—21no—
360C2×C32—21no—
361C2×C32—21no—
362C2×C32—21no—
363C2×C32—41no—
364C2×C32—41no—
365C2×C32—41no—
366C2×C32—122no—
367C2×C32#87811yescharacteristic
368C2×C32#89611yescharacteristic
369C2×C32#91111yescharacteristic
370C2×C32#91211yescharacteristic
371C2×C32#91411yescharacteristic
372C2×C32#91911yescharacteristic
373C2×C32C3×C22⋊Q811yescharacteristic
374C2×C32C3×#13511yescharacteristic
375C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
376C3×D3—42no—
377C3×D3—42no—
378C3×D3—82no—
379C2×D4—31no—
380C22×C4—181no—
381C2×C8—91no—
382C2×C8C3×D3211yescharacteristic
383C4⋊C4—31no—
384C4⋊C4—31no—
385C4⋊C4—31no—
386C4⋊C4—181no—
387C4⋊C4—181no—
388C22⋊C4—181no—
389C42—91no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—41no—
394C22×C3—62no—
395C22×C3—62no—
396C22×C3—122no—
397C22×C3—121no—
398C22×C3—121no—
399C22×C3—121no—
400C22×C3—121no—
401C22×C3—241no—
402C22×C3—241no—
403C22×C3#380211yescharacteristic
404C22×C3C3×D3×D411yescharacteristic
405C22×C3C3×D3×D411yescharacteristic
406C2×D3—22no—
407C2×D3—41no—
408C2×D3—41no—
409C3×C4—21no—
410C3×C4—21no—
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—21no—
415C3×C4—41no—
416C3×C4—41no—
417C3×C4—121no—
418C3×C4—121no—
419C3×C4—121no—
420C3×C4—121no—
421C3×C4—121no—
422C3×C4—121no—
423C3×C4—181no—
424C3×C4—181no—
425C3×C4—241no—
426C3×C4—241no—
427C3×C4—241no—
428C3×C4D3×Dic611yescharacteristic
429C3×C4#380411yescharacteristic
430C3×C4C2×C3×Dic611yescharacteristic
431C3×C4C3×#23511yescharacteristic
432C3×C4C3×#23711yescharacteristic
433C3×C4C3×D3×Q811yescharacteristic
434Dic3—41no—
435Dic3—41no—
436Dic3—41no—
437Dic3—62no—
438Dic3—61no—
439Dic3—61no—
440Dic3—61no—
441Dic3—61no—
442Dic3—121no—
443C32—21no—
444C32—21no—
445C32—21no—
446C32—41no—
447C32#476611yescharacteristic, complemented
448C32#489111yescharacteristic, complemented
449C32C3×#47811yescharacteristic, complemented
450C23—61no—
451D4—32no—
452D4—61no—
453C2×C4—61no—
454C2×C4—61no—
455C2×C4—61no—
456C2×C4—91no—
457C2×C4—91no—
458C2×C4—362no—
459C2×C4C2×C3×D3211yescharacteristic
460C8—21no—
461C8—181no—
462C2×C3—21no—
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—41no—
472C2×C3—41no—
473C2×C3—41no—
474C2×C3—122no—
475C2×C3—122no—
476C2×C3—242no—
477C2×C3#6379811yescharacteristic
478C2×C3#6381011yescharacteristic
479C2×C3#6389911yescharacteristic
480C2×C3C3×#87811yescharacteristic
481C2×C3C3×#89611yescharacteristic
482C2×C3C3×#91111yescharacteristic
483C2×C3C3×#91211yescharacteristic
484C2×C3C3×#91411yescharacteristic
485C2×C3C3×#91911yescharacteristic
486D3—42no—
487C22—62no—
488C22—121no—
489C22—121no—
490C22C3×#380211yescharacteristic
491C4—121no—
492C4—121no—
493C4—121no—
494C4—181no—
495C4—181no—
496C4C3×D3×Dic611yescharacteristic
497C4C3×#380411yescharacteristic
498C3—21no—
499C3—21no—
500C3—21no—
501C3—41no—
502C3#1059078011yescharacteristic, direct product factor
503C3C3×#476611yescharacteristic, complemented
504C3C3×#489111yescharacteristic, complemented
505C2—122no—
506C2C3×#6379811yescharacteristic
507C2C3×#6381011yescharacteristic
508C2C3×#6389911yescharacteristic
subgroupG/H
centerC22×C3#3802
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypercenterC2×C3×C8D32
hypocenterC32C3×#478
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
2. C3×#63872C2semidirect
3. C3×#63641C2semidirect
5. C32×#818C2semidirect
6. C3×#63574C2semidirect
10. #10590780C3direct (N × H)
27. #63872C2×C3semidirect
30. #63641C2×C3semidirect
36. C3×#818D3semidirect
37. C3×#818C2×C3semidirect
44. #63574C2×C3semidirect
132. #818C3×D3semidirect
300. C33#478semidirect
447. C32#4766semidirect
448. C32#4891semidirect
449. C32C3×#478semidirect
502. C3#10590780direct (N × H)
503. C3C3×#4766semidirect
504. C3C3×#4891semidirect

Maximal subgroups

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

subgroupindex
C3×#638982
C3×#638722
C3×#636412
C32×#8212
C32×#8182
C3×#635742
C3×#635622
C3×#48913
C3×#47663
#105907803