Finite Group Databasebeta

← C3×#10590555 #13157538

Subgroups — C3×#10590555 #13157538

Order 1728.

624 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#893C211yesmaximal, characteristic
2C3×#63918C211yesmaximal, characteristic, complemented
3C3×#63634C211yesmaximal, characteristic, complemented
4C32×#807C211yesmaximal, characteristic, complemented
5C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6C3×#63563C211yesmaximal, characteristic
7C3×#63562C211yesmaximal, characteristic
8C3×#5007—31nomaximal
9C3×#4890—31nomaximal
10#10590555C311yesmaximal, characteristic, direct product factor
11C2×C32×D12C2211yescharacteristic
12C2×C32×C4×D3—21no—
13C2×C3×#627—21no—
14C33×C4⋊C4C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×(C2×D3)⋊C4—21no—
18C32×C4⋊Dic3C2211yescharacteristic
19C32×C4×Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C3×Dic3⋊Dic3—21no—
22C32×C4×D4—31no—
23C3×#893—21no—
24C3×#893D311yescharacteristic
25C3×#893C2×C311yescharacteristic
26C3×#888—31no—
27C3×#869—31no—
28#63918C2×C311yescharacteristic, complemented
29C32×#107—31no—
30#63634C2×C311yescharacteristic, complemented
31C3×#832—31no—
32C3×#832—31no—
33C3×#821—31no—
34C3×Dic3⋊C8—31no—
35C3×#809—31no—
36C3×#807—21no—
37C3×#807—31no—
38C3×#807—31no—
39C3×#807—61no—
40C3×#807D311yescharacteristic, complemented
41C3×#807C2×C311yescharacteristic, complemented
42C3×C4⋊(C3⋊C8)—31no—
43Dic3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
44#63563C2×C311yescharacteristic
45#63562C2×C311yescharacteristic
46C22×C32×D3—21no—
47C2×C33×C4—21no—
48C2×C33×C4C2311yescharacteristic
49C2×C3×C3⋊Dic3—21no—
50C2×C32×Dic3—21no—
51C2×C32×Dic3D411yescharacteristic
52C2×C32×Dic3D411yescharacteristic
53C32×D12—21no—
54C32×D12Q812yescomplemented
55C32×C4×D3—42no—
56C3×#627—42no—
57C3×C3⋊(C3⋊C8)—21no—
58C32×C3⋊C8—21no—
59C3×#474—91no—
60#5007—31no—
61#4890—31no—
62C2×C32×D4—31no—
63C22×C32×C4—61no—
64C2×C3×D12—21no—
65C2×C3×D12C2×D311yescharacteristic
66C2×C3×D12C22×C311yescharacteristic
67C2×C3×C4×D3—22no—
68C2×C3×C4×D3—21no—
69C2×C3×C4×D3—21no—
70C2×C3×Dic6—61no—
71C2×C3×Dic6—61no—
72C2×#627—21no—
73C2×C32×C8—31no—
74C32×C4⋊C4—21no—
75C32×C4⋊C4—21no—
76C32×C4⋊C4—21no—
77C32×C4⋊C4—31no—
78C32×C4⋊C4—41no—
79C32×C4⋊C4C2×D311yescharacteristic
80C32×C4⋊C4C2×D311yescharacteristic
81C32×C4⋊C4C22×C311yescharacteristic
82C32×C22⋊C4—61no—
83C32×C42—31no—
84C4⋊(C3⋊Dic3)C22×C311yescharacteristic
85C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
86C3×(C2×D3)⋊C4—22no—
87C3×(C2×D3)⋊C4—21no—
88C3×(C2×D3)⋊C4—21no—
89C3×C4⋊Dic3—21no—
90C3×C4⋊Dic3—31no—
91C3×C4⋊Dic3—31no—
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⋊C8—31no—
102C2×C3×C3⋊C8—31no—
103C2×C3×C3⋊C8—61no—
104C2×C3×C3⋊C8C2×D311yescharacteristic
105C2×C3×C3⋊C8C22×C311yescharacteristic
106Dic3⋊Dic3—21no—
107C22×C33C2×D411yescharacteristic
108C2×C32×D3—22no—
109C2×C32×D3—41no—
110C2×C32×D3—41no—
111C33×C4—41no—
112C33×C4C2×Q811yescharacteristic
113C33×C4#4111yescharacteristic
114C3×C3⋊Dic3—41no—
115C32×Dic3—21no—
116C32×Dic3—41no—
117C32×Dic3D812yes—
118C3×C4⋊Q8—91no—
119C3×C4×D4—31no—
120C3×C4×D4—31no—
121C3×C4×D4—61no—
122#893C3×D311yescharacteristic
123#888—31no—
124#869—31no—
125C3×#107—31no—
126C3×#107—31no—
127C3×#107—61no—
128C3×#107—91no—
129C3×C4⋊C8—91no—
130C3×#102—91no—
131C3×#102—91no—
132#832—31no—
133#832—31no—
134#821—31no—
135Dic3⋊C8—31no—
136#809—31no—
137#807—31no—
138#807—31no—
139#807—61no—
140#807C3×D311yescharacteristic, complemented
141C4⋊(C3⋊C8)—31no—
142C23×C32—61no—
143C22×C3×D3—22no—
144C22×C3×D3—21no—
145C22×C3×D3—21no—
146C32×D4—32no—
147C32×D4—61no—
148C2×C32×C4—21no—
149C2×C32×C4—22no—
150C2×C32×C4—21no—
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—31no—
156C2×C32×C4—31no—
157C2×C32×C4—41no—
158C2×C32×C4—41no—
159C2×C32×C4—41no—
160C2×C32×C4—42no—
161C2×C32×C4—61no—
162C2×C32×C4—122no—
163C2×C32×C4C22×D311yescharacteristic
164C2×C32×C4C22×D311yescharacteristic
165C2×C32×C4C23×C311yescharacteristic
166C2×C3⋊Dic3—21no—
167C2×C3×Dic3—21no—
168C2×C3×Dic3—21no—
169C2×C3×Dic3—21no—
170C2×C3×Dic3—21no—
171C2×C3×Dic3—41no—
172C2×C3×Dic3—61no—
173C2×C3×Dic3—61no—
174C2×C3×Dic3—121no—
175C2×C3×Dic3#6711yescharacteristic
176C2×C3×Dic3#6711yescharacteristic
177C2×C3×Dic3C3×D411yescharacteristic
178C2×C3×Dic3C3×D411yescharacteristic
179C3×D12—22no—
180C3×D12—21no—
181C3×D12—21no—
182C3×D12—22no—
183C3×D12Dic612yescomplemented
184C3×D12C3×Q812yescomplemented
185C3×C4×D3—44no—
186C3×C4×D3—42no—
187C3×C4×D3—42no—
188C3×Dic6—122no—
189C3×Dic6—122no—
190#627—42no—
191C32×C8—61no—
192C3⋊(C3⋊C8)—21no—
193C3×C3⋊C8—22no—
194C3×C3⋊C8—21no—
195C3×C3⋊C8—21no—
196C3×C3⋊C8—62no—
197C3×C3⋊C8—61no—
198C3×C3⋊C8—61no—
199#474—91noSylow
200C2×C33C22⋊Q811yescharacteristic
201C2×C33C2×D811yescharacteristic
202C2×C33#13711yescharacteristic
203C32×D3—42no—
204C2×C3×Q8—181no—
205C2×C3×D4—31no—
206C2×C3×D4—31no—
207C2×C3×D4—61no—
208C22×C3×C4—61no—
209C22×C3×C4—61no—
210C22×C3×C4—62no—
211C2×D12C2×C3×D311yescharacteristic
212C2×C4×D3—21no—
213C2×Dic6—61no—
214C2×Dic6—61no—
215C2×C3×C8—31no—
216C2×C3×C8—31no—
217C2×C3×C8—61no—
218C2×C3×C8—91no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—31no—
223C3×C4⋊C4—31no—
224C3×C4⋊C4—41no—
225C3×C4⋊C4—61no—
226C3×C4⋊C4—91no—
227C3×C4⋊C4—181no—
228C3×C4⋊C4D3211yescharacteristic
229C3×C4⋊C4C2×C3×D311yescharacteristic
230C3×C4⋊C4C2×C3×D311yescharacteristic
231C3×C22⋊C4—61no—
232C3×C22⋊C4—61no—
233C3×C22⋊C4—62no—
234C3×C42—31no—
235C3×C42—31no—
236C3×C42—61no—
237(C2×D3)⋊C4—21no—
238C4⋊Dic3—31no—
239C4⋊Dic3—31no—
240C4⋊Dic3—61no—
241C4⋊Dic3—61no—
242C4⋊Dic3C2×C3×D311yescharacteristic
243Dic3⋊C4—61no—
244C4×Dic3C2×C3×D311yescharacteristic
245C2×C3⋊C8—31no—
246C2×C3⋊C8—31no—
247C2×C3⋊C8—61no—
248C2×C3⋊C8C2×C3×D311yescharacteristic
249C22×C32—21no—
250C22×C32—21no—
251C22×C32—21no—
252C22×C32—41no—
253C22×C32—62no—
254C22×C32—121no—
255C22×C32—121no—
256C22×C32D3×D411yescharacteristic
257C22×C32C2×#6711yescharacteristic
258C22×C32C2×C3×D411yescharacteristic
259C2×C3×D3—24no—
260C2×C3×D3—22no—
261C2×C3×D3—22no—
262C2×C3×D3—42no—
263C2×C3×D3—42no—
264C2×C3×D3—41no—
265C2×C3×D3—41no—
266C2×C3×D3—41no—
267C2×C3×D3—41no—
268C32×C4—21no—
269C32×C4—21no—
270C32×C4—21no—
271C32×C4—21no—
272C32×C4—21no—
273C32×C4—21no—
274C32×C4—32no—
275C32×C4—41no—
276C32×C4—41no—
277C32×C4—41no—
278C32×C4—42no—
279C32×C4—41no—
280C32×C4—41no—
281C32×C4—61no—
282C32×C4—81no—
283C32×C4—81no—
284C32×C4—82no—
285C32×C4—121no—
286C32×C4C2×Dic611yescharacteristic
287C32×C4#23511yescharacteristic
288C32×C4#23511yescharacteristic
289C32×C4D3×Q811yescharacteristic
290C32×C4C2×C3×Q811yescharacteristic
291C32×C4C3×#4111yescharacteristic
292C3⋊Dic3—41no—
293C3×Dic3—21no—
294C3×Dic3—21no—
295C3×Dic3—22no—
296C3×Dic3—41no—
297C3×Dic3—41no—
298C3×Dic3—41no—
299C3×Dic3—81no—
300C3×Dic3—121no—
301C3×Dic3—121no—
302C3×Dic3—241no—
303C3×Dic3#21312yes—
304C3×Dic3C3×D812yes—
305C4⋊Q8—91no—
306C4×D4—31no—
307#107—31no—
308#107—91no—
309C4⋊C8—91no—
310#102—91no—
311#102—91no—
312C33#47411yescharacteristic, complemented, Sylow
313C23×C3—62no—
314C23×C3—61no—
315C23×C3—61no—
316C22×D3—21no—
317C3×Q8—362no—
318C3×D4—32no—
319C3×D4—32no—
320C3×D4—62no—
321C3×D4—62no—
322C3×D4—61no—
323C3×D4—61no—
324C2×C3×C4—22no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—31no—
332C2×C3×C4—31no—
333C2×C3×C4—31no—
334C2×C3×C4—31no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—42no—
338C2×C3×C4—41no—
339C2×C3×C4—61no—
340C2×C3×C4—61no—
341C2×C3×C4—61no—
342C2×C3×C4—61no—
343C2×C3×C4—121no—
344C2×C3×C4—122no—
345C2×C3×C4—122no—
346C2×C3×C4—124no—
347C2×C3×C4—181no—
348C2×C3×C4C2×D3211yescharacteristic
349C2×C3×C4C22×C3×D311yescharacteristic
350C2×C3×C4C22×C3×D311yescharacteristic
351C2×Dic3—21no—
352C2×Dic3—61no—
353C2×Dic3—61no—
354C2×Dic3—121no—
355C2×Dic3C3×#6711yescharacteristic
356C2×Dic3C3×#6711yescharacteristic
357D12—21no—
358D12C3×Dic612yescomplemented
359C4×D3—42no—
360Dic6—122no—
361Dic6—122no—
362C3×C8—62no—
363C3×C8—61no—
364C3×C8—61no—
365C3×C8—181no—
366C3⋊C8—21no—
367C3⋊C8—62no—
368C3⋊C8—61no—
369C3⋊C8—61no—
370C2×C32—21no—
371C2×C32—21no—
372C2×C32—21no—
373C2×C32—21no—
374C2×C32—21no—
375C2×C32—21no—
376C2×C32—21no—
377C2×C32—21no—
378C2×C32—21no—
379C2×C32—41no—
380C2×C32—41no—
381C2×C32—41no—
382C2×C32—122no—
383C2×C32#89611yescharacteristic
384C2×C32D3×D811yescharacteristic
385C2×C32#91811yescharacteristic
386C2×C32#92411yescharacteristic
387C2×C32C2×#21311yescharacteristic
388C2×C32#95111yescharacteristic
389C2×C32C3×C22⋊Q811yescharacteristic
390C2×C32C2×C3×D811yescharacteristic
391C2×C32C3×#13711yescharacteristic
392C3×D3—44no—
393C3×D3—42no—
394C3×D3—42no—
395C2×Q8—181no—
396C2×D4—31no—
397C22×C4—61no—
398C2×C8—31no—
399C2×C8—91no—
400C4⋊C4—31no—
401C4⋊C4—91no—
402C4⋊C4—181no—
403C4⋊C4C3×D3211yescharacteristic
404C22⋊C4—61no—
405C42—31no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—41no—
410C22×C3—64no—
411C22×C3—62no—
412C22×C3—62no—
413C22×C3—122no—
414C22×C3—122no—
415C22×C3—121no—
416C22×C3—121no—
417C22×C3—121no—
418C22×C3—121no—
419C22×C3D3×#6711yescharacteristic
420C22×C3C3×D3×D411yescharacteristic
421C22×C3C2×C3×#6711yescharacteristic
422C2×D3—22no—
423C2×D3—41no—
424C2×D3—41no—
425C3×C4—21no—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—32no—
432C3×C4—32no—
433C3×C4—42no—
434C3×C4—41no—
435C3×C4—41no—
436C3×C4—41no—
437C3×C4—41no—
438C3×C4—41no—
439C3×C4—62no—
440C3×C4—61no—
441C3×C4—61no—
442C3×C4—81no—
443C3×C4—81no—
444C3×C4—82no—
445C3×C4—121no—
446C3×C4—121no—
447C3×C4—121no—
448C3×C4—241no—
449C3×C4—361no—
450C3×C4D3×Dic611yescharacteristic
451C3×C4#379811yescharacteristic
452C3×C4C2×C3×Dic611yescharacteristic
453C3×C4C3×#23511yescharacteristic
454C3×C4C3×#23511yescharacteristic
455C3×C4C3×D3×Q811yescharacteristic
456Dic3—21no—
457Dic3—41no—
458Dic3—121no—
459Dic3—121no—
460Dic3—241no—
461Dic3C3×#21312yes—
462C32—21no—
463C32—21no—
464C32—21no—
465C32—41no—
466C32#489011yescharacteristic, complemented
467C32#500711yescharacteristic, complemented
468C32C3×#47411yescharacteristic, complemented
469C23—61no—
470Q8—362no—
471D4—32no—
472D4—61no—
473C2×C4—21no—
474C2×C4—31no—
475C2×C4—31no—
476C2×C4—61no—
477C2×C4—122no—
478C2×C4—181no—
479C2×C4C2×C3×D3211yescharacteristic
480C8—61no—
481C8—181no—
482C2×C3—21no—
483C2×C3—21no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—41no—
492C2×C3—41no—
493C2×C3—41no—
494C2×C3—124no—
495C2×C3—122no—
496C2×C3—122no—
497C2×C3#6385111yescharacteristic
498C2×C3D3×#21311yescharacteristic
499C2×C3#6394311yescharacteristic
500C2×C3C3×#89611yescharacteristic
501C2×C3C3×D3×D811yescharacteristic
502C2×C3C3×#91811yescharacteristic
503C2×C3C3×#92411yescharacteristic
504C2×C3C2×C3×#21311yescharacteristic
505C2×C3C3×#95111yescharacteristic
506D3—42no—
507C22—62no—
508C22—121no—
509C22—121no—
510C22C3×D3×#6711yescharacteristic
511C4—32no—
512C4—41no—
513C4—61no—
514C4—121no—
515C4—361no—
516C4C3×D3×Dic611yescharacteristic
517C4C3×#379811yescharacteristic
518C3—21no—
519C3—21no—
520C3—21no—
521C3—41no—
522C3#1059055511yescharacteristic, direct product factor
523C3C3×#489011yescharacteristic, complemented
524C3C3×#500711yescharacteristic, complemented
525C2—122no—
526C2C3×#6385111yescharacteristic
527C2C3×D3×#21311yescharacteristic
528C2C3×#6394311yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#474
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#63918C2semidirect
3. C3×#63634C2semidirect
4. C32×#807C2semidirect
5. C3×Dic3⋊(C3⋊C8)C2semidirect
10. #10590555C3direct (N × H)
28. #63918C2×C3semidirect
30. #63634C2×C3semidirect
40. C3×#807D3semidirect
41. C3×#807C2×C3semidirect
43. Dic3⋊(C3⋊C8)C2×C3semidirect
54. C32×D12Q8semidirect
140. #807C3×D3semidirect
183. C3×D12Dic6semidirect
184. C3×D12C3×Q8semidirect
312. C33#474semidirect
358. D12C3×Dic6semidirect
466. C32#4890semidirect
467. C32#5007semidirect
468. C32C3×#474semidirect
522. C3#10590555direct (N × H)
523. C3C3×#4890semidirect
524. C3C3×#5007semidirect

Maximal subgroups

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

subgroupindex
C32×#8932
C3×#639182
C3×#636342
C32×#8072
C3×Dic3⋊(C3⋊C8)2
C3×#635632
C3×#635622
C3×#50073
C3×#48903
#105905553