Finite Group Databasebeta

← C3×#10591056 #13158084

Subgroups — C3×#10591056 #13158084

Order 1728.

614 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#926C211yesmaximal, characteristic
2C3×#63657C211yesmaximal, characteristic
3C32×#830C211yesmaximal, characteristic
4C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
5C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6C3×#63554C211yesmaximal, characteristic, complemented
7C3×Dic3×C3⋊C8C211yesmaximal, characteristic, complemented
8C3×#5000—31nomaximal
9C3×#4720—31nomaximal
10#10591056C311yesmaximal, characteristic, direct product factor
11C22×C33×C4C2211yescharacteristic
12C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
13C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
14C32×#217C411yescharacteristic
15C32×#217C411yescharacteristic
16C32×C4⋊Dic3C411yescharacteristic
17C32×C4⋊Dic3C411yescharacteristic
18C32×C4×Dic3C2211yescharacteristic
19C32×C4×Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22C32×#117—31no—
23C3×#926—21no—
24C3×#926D311yescharacteristic
25C3×#926C2×C311yescharacteristic
26C32×C22⋊C8—31no—
27#63657C2×C311yescharacteristic
28C3×#830—21no—
29C3×#830—31no—
30C3×#830—31no—
31C3×#830—61no—
32C3×#830D311yescharacteristic
33C3×#830C2×C311yescharacteristic
34C3×#815—31no—
35C3×Dic3⋊C8—31no—
36C3×Dic3⋊C8—31no—
37C3×C8×Dic3—31no—
38C3×C4⋊(C3⋊C8)—31no—
39C3×C4⋊(C3⋊C8)—31no—
40C3×#803—31no—
41C3×C4×C3⋊C8—31no—
42Dic3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
43Dic3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
44#63554C2×C311yescharacteristic, complemented
45Dic3×C3⋊C8C2×C311yescharacteristic, complemented
46C23×C33C2×C411yescharacteristic
47C2×C33×C4—21no—
48C2×C33×C4—21no—
49C2×C33×C4C2×C411yescharacteristic
50C2×C33×C4C2311yescharacteristic
51C2×C32×Dic3C2×C411yescharacteristic
52C2×C32×Dic3C2×C411yescharacteristic
53C2×C32×Dic3C2×C411yescharacteristic
54C2×C32×Dic3C2×C411yescharacteristic
55C3×C3⋊(C3⋊C8)—21no—
56C3×C3⋊(C3⋊C8)—21no—
57C32×C3⋊C8—21no—
58C32×C3⋊C8—21no—
59C3×#433—91no—
60#5000—31no—
61#4720—31no—
62C22×C32×C4—21no—
63C22×C32×C4—21no—
64C22×C32×C4—21no—
65C22×C32×C4—41no—
66C22×C32×C4C2×D311yescharacteristic
67C22×C32×C4C2×D311yescharacteristic
68C22×C32×C4C22×C311yescharacteristic
69C2×C32×C8—31no—
70C2×C32×C8—31no—
71C32×C4⋊C4—31no—
72C32×C4⋊C4—31no—
73C32×C22⋊C4—31no—
74C32×C22⋊C4—31no—
75C32×C42—31no—
76C32×C42—31no—
77C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
78C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
79C3×#217—21no—
80C3×#217—21no—
81C3×#217Dic311yescharacteristic
82C3×#217Dic311yescharacteristic
83C3×#217C3×C411yescharacteristic
84C3×#217C3×C411yescharacteristic
85C3×C4⋊Dic3—21no—
86C3×C4⋊Dic3—21no—
87C3×C4⋊Dic3Dic311yescharacteristic
88C3×C4⋊Dic3Dic311yescharacteristic
89C3×C4⋊Dic3C3×C411yescharacteristic
90C3×C4⋊Dic3C3×C411yescharacteristic
91C3×C4×Dic3—21no—
92C3×C4×Dic3—21no—
93C3×C4×Dic3C2×D311yescharacteristic
94C3×C4×Dic3C2×D311yescharacteristic
95C3×C4×Dic3C22×C311yescharacteristic
96C3×C4×Dic3C22×C311yescharacteristic
97C2×C3×C3⋊C8—21no—
98C2×C3×C3⋊C8—21no—
99C2×C3×C3⋊C8—31no—
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—31no—
102C2×C3×C3⋊C8—31no—
103C2×C3×C3⋊C8—61no—
104C2×C3×C3⋊C8—61no—
105C2×C3×C3⋊C8C2×D311yescharacteristic
106C2×C3×C3⋊C8C2×D311yescharacteristic
107C2×C3×C3⋊C8C22×C311yescharacteristic
108C2×C3×C3⋊C8C22×C311yescharacteristic
109C22×C33—21no—
110C22×C33—21no—
111C22×C33—21no—
112C22×C33C22×C411yescharacteristic
113C33×C4—21no—
114C33×C4#4111yescharacteristic
115C33×C4#4111yescharacteristic
116C32×Dic3—21no—
117C32×Dic3—21no—
118C32×Dic3—21no—
119C32×Dic3—21no—
120C3×#117—31no—
121C3×#117—31no—
122C3×#117—61no—
123#926C3×D311yescharacteristic
124C3×C4⋊C8—91no—
125C3×C4⋊C8—91no—
126C3×C22⋊C8—31no—
127C3×C22⋊C8—31no—
128C3×C22⋊C8—61no—
129C3×C22⋊C8—91no—
130C3×#97—91no—
131C3×C4×C8—91no—
132#830—31no—
133#830—31no—
134#830—61no—
135#830C3×D311yescharacteristic
136#815—31no—
137Dic3⋊C8—31no—
138Dic3⋊C8—31no—
139C8×Dic3—31no—
140C4⋊(C3⋊C8)—31no—
141C4⋊(C3⋊C8)—31no—
142#803—31no—
143C4×C3⋊C8—31no—
144C23×C32—21no—
145C23×C32—21no—
146C23×C32—21no—
147C23×C32—41no—
148C23×C32C4×D311yescharacteristic
149C23×C32C2×Dic311yescharacteristic
150C23×C32C2×C3×C411yescharacteristic
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—22no—
154C2×C32×C4—22no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—31no—
166C2×C32×C4—31no—
167C2×C32×C4—31no—
168C2×C32×C4—31no—
169C2×C32×C4—41no—
170C2×C32×C4—41no—
171C2×C32×C4—41no—
172C2×C32×C4—41no—
173C2×C32×C4—41no—
174C2×C32×C4—41no—
175C2×C32×C4—42no—
176C2×C32×C4—42no—
177C2×C32×C4C4×D311yescharacteristic
178C2×C32×C4C2×Dic311yescharacteristic
179C2×C32×C4C2×C3×C411yescharacteristic
180C2×C32×C4C22×D311yescharacteristic
181C2×C32×C4C22×D311yescharacteristic
182C2×C32×C4C23×C311yescharacteristic
183C2×C3×Dic3—21no—
184C2×C3×Dic3—21no—
185C2×C3×Dic3—21no—
186C2×C3×Dic3—21no—
187C2×C3×Dic3C2×Dic311yescharacteristic
188C2×C3×Dic3C2×Dic311yescharacteristic
189C2×C3×Dic3C2×Dic311yescharacteristic
190C2×C3×Dic3C2×Dic311yescharacteristic
191C2×C3×Dic3C2×C3×C411yescharacteristic
192C2×C3×Dic3C2×C3×C411yescharacteristic
193C2×C3×Dic3C2×C3×C411yescharacteristic
194C2×C3×Dic3C2×C3×C411yescharacteristic
195C32×C8—61no—
196C32×C8—61no—
197C3⋊(C3⋊C8)—21no—
198C3⋊(C3⋊C8)—21no—
199C3×C3⋊C8—21no—
200C3×C3⋊C8—21no—
201C3×C3⋊C8—21no—
202C3×C3⋊C8—21no—
203C3×C3⋊C8—41no—
204C3×C3⋊C8—41no—
205C3×C3⋊C8—61no—
206C3×C3⋊C8—61no—
207C3×C3⋊C8—61no—
208C3×C3⋊C8—61no—
209C3×C3⋊C8—121no—
210C3×C3⋊C8—121no—
211#433—91noSylow
212C2×C33—41no—
213C2×C33#11711yescharacteristic
214C2×C33#13111yescharacteristic
215C2×C33#13111yescharacteristic
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4—41no—
220C22×C3×C4D3211yescharacteristic
221C22×C3×C4C2×C3×D311yescharacteristic
222C22×C3×C4C2×C3×D311yescharacteristic
223C2×C3×C8—31no—
224C2×C3×C8—31no—
225C2×C3×C8—31no—
226C2×C3×C8—31no—
227C2×C3×C8—61no—
228C2×C3×C8—61no—
229C2×C3×C8—91no—
230C2×C3×C8—91no—
231C3×C4⋊C4—31no—
232C3×C4⋊C4—31no—
233C3×C4⋊C4—31no—
234C3×C4⋊C4—31no—
235C3×C4⋊C4—61no—
236C3×C4⋊C4—61no—
237C3×C22⋊C4—31no—
238C3×C22⋊C4—31no—
239C3×C22⋊C4—31no—
240C3×C22⋊C4—31no—
241C3×C22⋊C4—61no—
242C3×C22⋊C4—61no—
243C3×C42—31no—
244C3×C42—31no—
245C3×C42—31no—
246C3×C42—31no—
247C3×C42—61no—
248C3×C42—61no—
249#217C3×Dic311yescharacteristic
250#217C3×Dic311yescharacteristic
251C4⋊Dic3C3×Dic311yescharacteristic
252C4⋊Dic3C3×Dic311yescharacteristic
253C4×Dic3C2×C3×D311yescharacteristic
254C4×Dic3C2×C3×D311yescharacteristic
255C2×C3⋊C8—31no—
256C2×C3⋊C8—31no—
257C2×C3⋊C8—31no—
258C2×C3⋊C8—31no—
259C2×C3⋊C8—61no—
260C2×C3⋊C8—61no—
261C2×C3⋊C8C2×C3×D311yescharacteristic
262C2×C3⋊C8C2×C3×D311yescharacteristic
263C22×C32—21no—
264C22×C32—21no—
265C22×C32—21no—
266C22×C32—22no—
267C22×C32—22no—
268C22×C32—22no—
269C22×C32—21no—
270C22×C32—21no—
271C22×C32—21no—
272C22×C32—21no—
273C22×C32—21no—
274C22×C32—21no—
275C22×C32—21no—
276C22×C32—21no—
277C22×C32—21no—
278C22×C32—41no—
279C22×C32—41no—
280C22×C32—41no—
281C22×C32—41no—
282C22×C32—41no—
283C22×C32—41no—
284C22×C32—41no—
285C22×C32—42no—
286C22×C32—42no—
287C22×C32—42no—
288C22×C32C2×C4×D311yescharacteristic
289C22×C32C22×Dic311yescharacteristic
290C22×C32C22×C3×C411yescharacteristic
291C32×C4—21no—
292C32×C4—22no—
293C32×C4—21no—
294C32×C4—21no—
295C32×C4—21no—
296C32×C4—21no—
297C32×C4—21no—
298C32×C4—21no—
299C32×C4—21no—
300C32×C4—21no—
301C32×C4—41no—
302C32×C4—41no—
303C32×C4—41no—
304C32×C4—41no—
305C32×C4—42no—
306C32×C4—61no—
307C32×C4—61no—
308C32×C4—61no—
309C32×C4—61no—
310C32×C4#23511yescharacteristic
311C32×C4#23511yescharacteristic
312C32×C4#23711yescharacteristic
313C32×C4#23711yescharacteristic
314C32×C4C3×#4111yescharacteristic
315C32×C4C3×#4111yescharacteristic
316C3×Dic3—21no—
317C3×Dic3—21no—
318C3×Dic3—21no—
319C3×Dic3—21no—
320C3×Dic3—21no—
321C3×Dic3—21no—
322C3×Dic3—21no—
323C3×Dic3—21no—
324C3×Dic3—41no—
325C3×Dic3—41no—
326C3×Dic3—41no—
327C3×Dic3—41no—
328#117—31no—
329C4⋊C8—91no—
330C4⋊C8—91no—
331C22⋊C8—31no—
332C22⋊C8—91no—
333#97—91no—
334C4×C8—91no—
335C33#43311yescharacteristic, complemented, Sylow
336C23×C3—21no—
337C23×C3—21no—
338C23×C3—21no—
339C23×C3—41no—
340C23×C3D3×Dic311yescharacteristic
341C23×C3C3×C4×D311yescharacteristic
342C23×C3C2×C3×Dic311yescharacteristic
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—31no—
358C2×C3×C4—31no—
359C2×C3×C4—31no—
360C2×C3×C4—31no—
361C2×C3×C4—31no—
362C2×C3×C4—31no—
363C2×C3×C4—31no—
364C2×C3×C4—31no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—42no—
370C2×C3×C4—42no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—61no—
374C2×C3×C4—61no—
375C2×C3×C4—61no—
376C2×C3×C4—61no—
377C2×C3×C4D3×Dic311yescharacteristic
378C2×C3×C4C3×C4×D311yescharacteristic
379C2×C3×C4C2×C3×Dic311yescharacteristic
380C2×C3×C4C2×D3211yescharacteristic
381C2×C3×C4C22×C3×D311yescharacteristic
382C2×C3×C4C22×C3×D311yescharacteristic
383C2×Dic3C2×C3×Dic311yescharacteristic
384C2×Dic3C2×C3×Dic311yescharacteristic
385C2×Dic3C2×C3×Dic311yescharacteristic
386C2×Dic3C2×C3×Dic311yescharacteristic
387C3×C8—61no—
388C3×C8—61no—
389C3×C8—61no—
390C3×C8—61no—
391C3×C8—121no—
392C3×C8—121no—
393C3×C8—181no—
394C3×C8—181no—
395C3⋊C8—21no—
396C3⋊C8—21no—
397C3⋊C8—61no—
398C3⋊C8—61no—
399C3⋊C8—61no—
400C3⋊C8—61no—
401C3⋊C8—121no—
402C3⋊C8—121no—
403C2×C32—21no—
404C2×C32—21no—
405C2×C32—21no—
406C2×C32—21no—
407C2×C32—21no—
408C2×C32—21no—
409C2×C32—21no—
410C2×C32—21no—
411C2×C32—21no—
412C2×C32—41no—
413C2×C32—41no—
414C2×C32—41no—
415C2×C32—42no—
416C2×C32—42no—
417C2×C32—42no—
418C2×C32—44no—
419C2×C32—41no—
420C2×C32—41no—
421C2×C32—41no—
422C2×C32#87711yescharacteristic
423C2×C32#90111yescharacteristic
424C2×C32#90711yescharacteristic
425C2×C32#92611yescharacteristic
426C2×C32#94811yescharacteristic
427C2×C32#94811yescharacteristic
428C2×C32C3×#11711yescharacteristic
429C2×C32C3×#13111yescharacteristic
430C2×C32C3×#13111yescharacteristic
431C22×C4C3×D3211yescharacteristic
432C2×C8—31no—
433C2×C8—31no—
434C2×C8—91no—
435C2×C8—91no—
436C4⋊C4—31no—
437C4⋊C4—31no—
438C22⋊C4—31no—
439C22⋊C4—31no—
440C42—31no—
441C42—31no—
442C22×C3—22no—
443C22×C3—22no—
444C22×C3—22no—
445C22×C3—21no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—21no—
451C22×C3—21no—
452C22×C3—21no—
453C22×C3—21no—
454C22×C3—21no—
455C22×C3—21no—
456C22×C3—21no—
457C22×C3—41no—
458C22×C3—41no—
459C22×C3—41no—
460C22×C3—41no—
461C22×C3—41no—
462C22×C3—41no—
463C22×C3—42no—
464C22×C3—42no—
465C22×C3—42no—
466C22×C3—41no—
467C22×C3C2×D3×Dic311yescharacteristic
468C22×C3C2×C3×C4×D311yescharacteristic
469C22×C3C22×C3×Dic311yescharacteristic
470C3×C4—22no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—21no—
475C3×C4—21no—
476C3×C4—21no—
477C3×C4—21no—
478C3×C4—21no—
479C3×C4—21no—
480C3×C4—41no—
481C3×C4—41no—
482C3×C4—42no—
483C3×C4—41no—
484C3×C4—41no—
485C3×C4—61no—
486C3×C4—61no—
487C3×C4—61no—
488C3×C4—61no—
489C3×C4—61no—
490C3×C4—61no—
491C3×C4—61no—
492C3×C4—61no—
493C3×C4—121no—
494C3×C4—121no—
495C3×C4—121no—
496C3×C4—121no—
497C3×C4#380411yescharacteristic
498C3×C4#380411yescharacteristic
499C3×C4C3×#23511yescharacteristic
500C3×C4C3×#23511yescharacteristic
501C3×C4C3×#23711yescharacteristic
502C3×C4C3×#23711yescharacteristic
503Dic3—21no—
504Dic3—21no—
505Dic3—21no—
506Dic3—21no—
507C32—21no—
508C32—21no—
509C32—21no—
510C32—41no—
511C32#472011yescharacteristic, complemented
512C32#500011yescharacteristic, complemented
513C32C3×#43311yescharacteristic, complemented
514C23C3×D3×Dic311yescharacteristic
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—31no—
518C2×C4—31no—
519C2×C4—31no—
520C2×C4—31no—
521C2×C4C3×D3×Dic311yescharacteristic
522C2×C4C2×C3×D3211yescharacteristic
523C8—61no—
524C8—61no—
525C8—181no—
526C8—181no—
527C2×C3—21no—
528C2×C3—21no—
529C2×C3—21no—
530C2×C3—21no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—21no—
535C2×C3—21no—
536C2×C3—42no—
537C2×C3—42no—
538C2×C3—42no—
539C2×C3—44no—
540C2×C3—41no—
541C2×C3—41no—
542C2×C3—41no—
543C2×C3—41no—
544C2×C3—41no—
545C2×C3—41no—
546C2×C3#6381411yescharacteristic
547C2×C3#6381511yescharacteristic
548C2×C3#6395511yescharacteristic
549C2×C3C3×#87711yescharacteristic
550C2×C3C3×#90111yescharacteristic
551C2×C3C3×#90711yescharacteristic
552C2×C3C3×#92611yescharacteristic
553C2×C3C3×#94811yescharacteristic
554C2×C3C3×#94811yescharacteristic
555C22—21no—
556C22—21no—
557C22—21no—
558C22C2×C3×D3×Dic311yescharacteristic
559C4—21no—
560C4—61no—
561C4—61no—
562C4—61no—
563C4—61no—
564C4C3×#380411yescharacteristic
565C4C3×#380411yescharacteristic
566C3—21no—
567C3—21no—
568C3—21no—
569C3—41no—
570C3#1059105611yescharacteristic, direct product factor
571C3C3×#472011yescharacteristic, complemented
572C3C3×#500011yescharacteristic, complemented
573C2—41no—
574C2C3×#6381411yescharacteristic
575C2C3×#6381511yescharacteristic
576C2C3×#6395511yescharacteristic
subgroupG/H
centerC2×C3×C4C2×D32
commutatorC22×C32C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socleC22×C33C22×C4
prime radicalC23×C33C2×C4
cosocleC2×C32×C4C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C3×#433
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
4. C3×Dic3⋊(C3⋊C8)C2semidirect
5. C3×Dic3⋊(C3⋊C8)C2semidirect
6. C3×#63554C2semidirect
7. C3×Dic3×C3⋊C8C2semidirect
10. #10591056C3direct (N × H)
42. Dic3⋊(C3⋊C8)C2×C3semidirect
43. Dic3⋊(C3⋊C8)C2×C3semidirect
44. #63554C2×C3semidirect
45. Dic3×C3⋊C8C2×C3semidirect
335. C33#433semidirect
511. C32#4720semidirect
512. C32#5000semidirect
513. C32C3×#433semidirect
570. C3#10591056direct (N × H)
571. C3C3×#4720semidirect
572. C3C3×#5000semidirect

Maximal subgroups

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

subgroupindex
C32×#9262
C3×#636572
C32×#8302
C3×Dic3⋊(C3⋊C8)2
C3×#635542
C3×Dic3×C3⋊C82
C3×#50003
C3×#47203
#105910563