Finite Group Databasebeta

← C3×#10594866 #13174404

Subgroups — C3×#10594866 #13174404

Order 1728.

1530 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3828C211yesmaximal, characteristic
2C22×C32×Dic6C211yesmaximal, characteristic, complemented
3C32×#926C211yesmaximal, characteristic, complemented
4C3×#63966C214yesmaximal, complemented
5C3×#63871C214yesmaximal, complemented
6C3×#63866C212yesmaximal, complemented
7C3×#63864C212yesmaximal, complemented
8C3×#5816—31nomaximal
9C3×#5526—31nomaximal
10#10594866C311yesmaximal, characteristic, direct product factor
11C22×C33×C4C2211yescharacteristic
12C2×C3×#638C2212yes—
13C3×#3828—24no—
14C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C3×C4×C3⋊D3C2212yes—
16C2×C3×#634C2211yescharacteristic
17C22×C32×Dic3C2212yes—
18C2×C32×Dic6—24no—
19C2×C32×Dic6C2212yes—
20C2×C32×Dic6C2212yes—
21C32×#217C2212yes—
22C32×C4⋊Dic3C2212yes—
23C32×C4×Dic3C2212yes—
24C3×Dic3⋊Dic3C2218yes—
25C3×#3722C2218yes—
26C22×C32×Q8—31no—
27C2×#3828C2×C311yescharacteristic
28C2×C3×#235—31no—
29C2×C3×#235—31no—
30C2×C3×#235—61no—
31C22×C3×Dic6—21no—
32C22×C3×Dic6D311yescharacteristic, complemented
33C22×C3×Dic6C2×C311yescharacteristic, complemented
34C32×#117—31no—
35C3×#947—32no—
36C3×#946—34no—
37C3×#944—32no—
38C3×#929—34no—
39C3×#926—21no—
40C3×#926D311yescharacteristic, complemented
41C3×#926C2×C311yescharacteristic, complemented
42C3×#897—34no—
43C3×#886—34no—
44C3×#875—32no—
45C3×#869—32no—
46#63966C2×C314yescomplemented
47#63871C2×C314yescomplemented
48#63866C2×C312yescomplemented
49#63864C2×C312yescomplemented
50C23×C33C2311yescharacteristic
51C22×C3×C3⋊D3C2312yes—
52C2×C33×C4D414yes—
53C2×C33×C4C2311yescharacteristic
54C2×C33×C4C2311yescharacteristic
55C3×#638—24no—
56C2×C3×C3⋊Dic3C2312yes—
57C3×C4⋊(C3⋊D3)—22no—
58C3×C4×C3⋊D3—24no—
59C3×#634—22no—
60C2×C32×Dic3—24no—
61C2×C32×Dic3C2314yes—
62C2×C32×Dic3C2314yes—
63C32×Dic6—24no—
64C32×Dic6—24no—
65C3×#536—91no—
66#5816—31no—
67#5526—31no—
68C2×C32×Q8—32no—
69C2×C32×Q8—32no—
70C2×C32×Q8—64no—
71C22×C32×C4—21no—
72C22×C32×C4—21no—
73C22×C32×C4—21no—
74C22×C32×C4—32no—
75C22×C32×C4—41no—
76C22×C32×C4C2×D311yescharacteristic
77C22×C32×C4C2×D311yescharacteristic
78C22×C32×C4C22×C311yescharacteristic
79C2×#638C22×C312yes—
80#3828—24no—
81C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
82C2×C4×C3⋊D3C22×C312yes—
83C2×#634C22×C311yescharacteristic
84C2×C3×#67—32no—
85C2×C3×#67—32no—
86C2×C3×#67—62no—
87C22×C3×Dic3—22no—
88C22×C3×Dic3C2×D312yes—
89C22×C3×Dic3C22×C312yes—
90C3×#235—68no—
91C3×#235—64no—
92C3×#235—64no—
93C2×C3×D12—31no—
94C2×C3×D12—31no—
95C2×C3×D12—61no—
96C2×C3×C4×D3—32no—
97C2×C3×C4×D3—32no—
98C2×C3×C4×D3—62no—
99C2×C3×Dic6—28no—
100C2×C3×Dic6—22no—
101C2×C3×Dic6—22no—
102C2×C3×Dic6—24no—
103C2×C3×Dic6—24no—
104C2×C3×Dic6—31no—
105C2×C3×Dic6—31no—
106C2×C3×Dic6—61no—
107C2×C3×Dic6C2×D312yes—
108C2×C3×Dic6C2×D312yes—
109C2×C3×Dic6C22×C312yes—
110C2×C3×Dic6C22×C312yes—
111C32×C4⋊C4—32no—
112C32×C22⋊C4—32no—
113C32×C42—32no—
114C3×#217—22no—
115C3×#217C2×D312yes—
116C3×#217C22×C312yes—
117C3×(C2×D3)⋊C4—38no—
118C3×(C2×D3)⋊C4—38no—
119C3×C4⋊Dic3—22no—
120C3×C4⋊Dic3—38no—
121C3×C4⋊Dic3C2×D312yes—
122C3×C4⋊Dic3C22×C312yes—
123C3×Dic3⋊C4—38no—
124C3×C4×Dic3—22no—
125C3×C4×Dic3C2×D312yes—
126C3×C4×Dic3C22×C312yes—
127Dic3⋊Dic3C22×C318yes—
128#3722C22×C318yes—
129C22×C33—22no—
130C22×C33C2×D412yes—
131C22×C33C2411yescharacteristic
132C2×C3×C3⋊D3—22no—
133C2×C3×C3⋊D3—24no—
134C33×C4C2×D412yes—
135C33×C4C2×D412yes—
136C3×C3⋊Dic3—22no—
137C32×Dic3—24no—
138C32×Dic3—24no—
139C2×C3×#41—91no—
140C22×C3×Q8—31no—
141C22×C3×Q8—31no—
142C22×C3×Q8—61no—
143C2×#235—31no—
144C2×#235—31no—
145C2×#235—61no—
146C22×Dic6C3×D311yescharacteristic, complemented
147C3×C4⋊Q8—92no—
148C3×#124—92no—
149C3×#123—94no—
150C3×C22⋊Q8—94no—
151C3×#117—31no—
152C3×#117—31no—
153C3×#117—61no—
154#947—32no—
155#946—34no—
156#944—32no—
157#929—34no—
158#926C3×D311yescharacteristic, complemented
159#897—34no—
160#886—34no—
161#875—32no—
162#869—32no—
163C23×C32—21no—
164C23×C32—21no—
165C23×C32—21no—
166C23×C32—41no—
167C23×C32C22×D311yescharacteristic
168C23×C32C22×D311yescharacteristic
169C23×C32C23×C311yescharacteristic
170C22×C3⋊D3C23×C312yes—
171C22×C3×D3—32no—
172C22×C3×D3—32no—
173C22×C3×D3—62no—
174C32×Q8—64no—
175C32×Q8—64no—
176C2×C32×C4—24no—
177C2×C32×C4—24no—
178C2×C32×C4—24no—
179C2×C32×C4—21no—
180C2×C32×C4—21no—
181C2×C32×C4—21no—
182C2×C32×C4—21no—
183C2×C32×C4—21no—
184C2×C32×C4—21no—
185C2×C32×C4—34no—
186C2×C32×C4—34no—
187C2×C32×C4—44no—
188C2×C32×C4—41no—
189C2×C32×C4—41no—
190C2×C32×C4—64no—
191C2×C32×C4D1214yes—
192C2×C32×C4#6714yes—
193C2×C32×C4C3×D414yes—
194C2×C32×C4C22×D311yescharacteristic
195C2×C32×C4C22×D311yescharacteristic
196C2×C32×C4C22×D311yescharacteristic
197C2×C32×C4C22×D311yescharacteristic
198C2×C32×C4C23×C311yescharacteristic
199C2×C32×C4C23×C311yescharacteristic
200#638—24no—
201C2×C3⋊Dic3C23×C312yes—
202C4⋊(C3⋊D3)—22no—
203C4×C3⋊D3—24no—
204#634—22no—
205C3×#67—68no—
206C3×#67—64no—
207C3×#67—64no—
208C2×C3×Dic3—28no—
209C2×C3×Dic3—24no—
210C2×C3×Dic3—24no—
211C2×C3×Dic3—24no—
212C2×C3×Dic3—24no—
213C2×C3×Dic3—32no—
214C2×C3×Dic3—32no—
215C2×C3×Dic3—62no—
216C2×C3×Dic3C22×D314yes—
217C2×C3×Dic3C22×D314yes—
218C2×C3×Dic3C23×C314yes—
219C2×C3×Dic3C23×C314yes—
220C3×D12—64no—
221C3×D12—62no—
222C3×D12—62no—
223C3×C4×D3—68no—
224C3×C4×D3—64no—
225C3×C4×D3—64no—
226C3×Dic6—28no—
227C3×Dic6—28no—
228C3×Dic6—24no—
229C3×Dic6—24no—
230C3×Dic6—24no—
231C3×Dic6—24no—
232C3×Dic6—64no—
233C3×Dic6—62no—
234C3×Dic6—62no—
235#536—91noSylow
236C2×C33—22no—
237C2×C33C22×D411yescharacteristic
238C2×C332−1+412yes—
239C3×C3⋊D3—42no—
240C3×#41—184no—
241C2×C3×Q8—32no—
242C2×C3×Q8—32no—
243C2×C3×Q8—32no—
244C2×C3×Q8—32no—
245C2×C3×Q8—62no—
246C2×C3×Q8—62no—
247C2×C3×Q8—68no—
248C2×C3×Q8—64no—
249C2×C3×Q8—64no—
250C2×C3×Q8—91no—
251C2×C3×D4—92no—
252C2×C3×D4—91no—
253C22×C3×C4—21no—
254C22×C3×C4—21no—
255C22×C3×C4—21no—
256C22×C3×C4—32no—
257C22×C3×C4—32no—
258C22×C3×C4—41no—
259C22×C3×C4—62no—
260C22×C3×C4—92no—
261C22×C3×C4D3211yescharacteristic
262C22×C3×C4C2×C3×D311yescharacteristic
263C22×C3×C4C2×C3×D311yescharacteristic
264C2×#67—32no—
265C2×#67—32no—
266C2×#67—62no—
267C22×Dic3C2×C3×D312yes—
268#235—68no—
269#235—64no—
270#235—64no—
271C2×D12—31no—
272C2×D12—31no—
273C2×D12—61no—
274C2×C4×D3—32no—
275C2×C4×D3—32no—
276C2×C4×D3—62no—
277C2×Dic6—24no—
278C2×Dic6—31no—
279C2×Dic6—31no—
280C2×Dic6—61no—
281C2×Dic6C2×C3×D312yes—
282C2×Dic6C2×C3×D312yes—
283C3×C4⋊C4—32no—
284C3×C4⋊C4—32no—
285C3×C4⋊C4—62no—
286C3×C4⋊C4—98no—
287C3×C22⋊C4—32no—
288C3×C22⋊C4—32no—
289C3×C22⋊C4—62no—
290C3×C22⋊C4—98no—
291C3×C42—32no—
292C3×C42—32no—
293C3×C42—62no—
294#217C2×C3×D312yes—
295(C2×D3)⋊C4—38no—
296(C2×D3)⋊C4—38no—
297C4⋊Dic3—38no—
298C4⋊Dic3C2×C3×D312yes—
299Dic3⋊C4—38no—
300C4×Dic3C2×C3×D312yes—
301C22×C32—22no—
302C22×C32—24no—
303C22×C32—22no—
304C22×C32—22no—
305C22×C32—21no—
306C22×C32—21no—
307C22×C32—21no—
308C22×C32—22no—
309C22×C32—22no—
310C22×C32—22no—
311C22×C32—42no—
312C22×C32—41no—
313C22×C32—42no—
314C22×C32—42no—
315C22×C32—44no—
316C22×C32C2×D1212yes—
317C22×C32C2×#6712yes—
318C22×C32C2×C3×D412yes—
319C22×C32C23×D311yescharacteristic
320C22×C32C23×D311yescharacteristic
321C22×C32C24×C311yescharacteristic
322C2×C3⋊D3—22no—
323C2×C3⋊D3—24no—
324C2×C3×D3—64no—
325C2×C3×D3—62no—
326C2×C3×D3—64no—
327C2×C3×D3—64no—
328C2×C3×D3—62no—
329C2×C3×D3—124no—
330C32×C4—22no—
331C32×C4—22no—
332C32×C4—22no—
333C32×C4—22no—
334C32×C4—22no—
335C32×C4—22no—
336C32×C4—42no—
337C32×C4—42no—
338C32×C4—64no—
339C32×C4—64no—
340C32×C4C2×D1212yes—
341C32×C4C2×D1212yes—
342C32×C4C2×#6712yes—
343C32×C4C2×#6712yes—
344C32×C4C2×C3×D412yes—
345C32×C4C2×C3×D412yes—
346C3⋊Dic3—22no—
347C3×Dic3—28no—
348C3×Dic3—24no—
349C3×Dic3—24no—
350C3×Dic3—24no—
351C3×Dic3—24no—
352C3×Dic3—44no—
353C3×Dic3—64no—
354C3×Dic3—62no—
355C3×Dic3—62no—
356C2×#41—91no—
357C22×Q8—31no—
358C4⋊Q8—92no—
359#124—92no—
360#123—94no—
361C22⋊Q8—94no—
362#117—31no—
363C33#53611yescharacteristic, complemented, Sylow
364C23×C3—21no—
365C23×C3—21no—
366C23×C3—21no—
367C23×C3—41no—
368C23×C3—92no—
369C23×C3C2×D3211yescharacteristic
370C23×C3C22×C3×D311yescharacteristic
371C23×C3C22×C3×D311yescharacteristic
372C22×D3—32no—
373C22×D3—32no—
374C22×D3—62no—
375C3×Q8—68no—
376C3×Q8—68no—
377C3×Q8—64no—
378C3×Q8—64no—
379C3×Q8—64no—
380C3×Q8—64no—
381C3×Q8—182no—
382C3×D4—184no—
383C3×D4—182no—
384C2×C3×C4—24no—
385C2×C3×C4—24no—
386C2×C3×C4—24no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—34no—
394C2×C3×C4—34no—
395C2×C3×C4—34no—
396C2×C3×C4—34no—
397C2×C3×C4—44no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—64no—
401C2×C3×C4—64no—
402C2×C3×C4—68no—
403C2×C3×C4—64no—
404C2×C3×C4—64no—
405C2×C3×C4—92no—
406C2×C3×C4—184no—
407C2×C3×C4#62614yes—
408C2×C3×C4C3×D1214yes—
409C2×C3×C4C3×#6714yes—
410C2×C3×C4C2×D3211yescharacteristic
411C2×C3×C4C2×D3211yescharacteristic
412C2×C3×C4C22×C3×D311yescharacteristic
413C2×C3×C4C22×C3×D311yescharacteristic
414C2×C3×C4C22×C3×D311yescharacteristic
415C2×C3×C4C22×C3×D311yescharacteristic
416#67—68no—
417#67—64no—
418#67—64no—
419C2×Dic3—24no—
420C2×Dic3—32no—
421C2×Dic3—32no—
422C2×Dic3—62no—
423C2×Dic3C22×C3×D314yes—
424C2×Dic3C22×C3×D314yes—
425D12—64no—
426D12—62no—
427D12—62no—
428C4×D3—68no—
429C4×D3—64no—
430C4×D3—64no—
431Dic6—24no—
432Dic6—24no—
433Dic6—64no—
434Dic6—62no—
435Dic6—62no—
436C2×C32—22no—
437C2×C32—24no—
438C2×C32—22no—
439C2×C32—22no—
440C2×C32—22no—
441C2×C32—21no—
442C2×C32—21no—
443C2×C32—22no—
444C2×C32—22no—
445C2×C32—21no—
446C2×C32—41no—
447C2×C32—42no—
448C2×C32—42no—
449C2×C32—42no—
450C2×C32—44no—
451C2×C32C22×D1211yescharacteristic
452C2×C32#100712yes—
453C2×C32#101012yes—
454C2×C32C22×#6711yescharacteristic
455C2×C32C22×C3×D411yescharacteristic
456C2×C32C3×2−1+412yes—
457C3⋊D3—42no—
458C3×D3—124no—
459C3×D3—122no—
460C3×D3—122no—
461#41—184no—
462C2×Q8—32no—
463C2×Q8—32no—
464C2×Q8—64no—
465C2×Q8—91no—
466C2×D4—92no—
467C2×D4—91no—
468C22×C4—32no—
469C22×C4—92no—
470C22×C4C3×D3211yescharacteristic
471C4⋊C4—32no—
472C4⋊C4—98no—
473C22⋊C4—32no—
474C22⋊C4—98no—
475C42—32no—
476C22×C3—24no—
477C22×C3—22no—
478C22×C3—22no—
479C22×C3—22no—
480C22×C3—21no—
481C22×C3—21no—
482C22×C3—21no—
483C22×C3—22no—
484C22×C3—22no—
485C22×C3—22no—
486C22×C3—42no—
487C22×C3—42no—
488C22×C3—44no—
489C22×C3—41no—
490C22×C3—42no—
491C22×C3—184no—
492C22×C3—182no—
493C22×C3C2×#62612yes—
494C22×C3C2×C3×D1212yes—
495C22×C3C2×C3×#6712yes—
496C22×C3C22×D3211yescharacteristic
497C22×C3C23×C3×D311yescharacteristic
498C22×C3C23×C3×D311yescharacteristic
499C2×D3—64no—
500C2×D3—62no—
501C2×D3—64no—
502C2×D3—64no—
503C2×D3—62no—
504C2×D3—124no—
505C3×C4—22no—
506C3×C4—22no—
507C3×C4—22no—
508C3×C4—22no—
509C3×C4—22no—
510C3×C4—22no—
511C3×C4—42no—
512C3×C4—42no—
513C3×C4—68no—
514C3×C4—64no—
515C3×C4—64no—
516C3×C4—64no—
517C3×C4—64no—
518C3×C4—124no—
519C3×C4—182no—
520C3×C4C2×#62612yes—
521C3×C4C2×#62612yes—
522C3×C4C2×C3×D1212yes—
523C3×C4C2×C3×D1212yes—
524C3×C4C2×C3×#6712yes—
525C3×C4C2×C3×#6712yes—
526Dic3—24no—
527Dic3—24no—
528Dic3—64no—
529Dic3—62no—
530Dic3—62no—
531C32—21no—
532C32—21no—
533C32—21no—
534C32—41no—
535C32#552611yescharacteristic, complemented
536C32#581611yescharacteristic, complemented
537C32C3×#53611yescharacteristic, complemented
538C23—92no—
539C23C2×C3×D3211yescharacteristic
540Q8—64no—
541Q8—64no—
542Q8—182no—
543D4—184no—
544D4—182no—
545C2×C4—34no—
546C2×C4—34no—
547C2×C4—64no—
548C2×C4—92no—
549C2×C4—184no—
550C2×C4C3×#62614yes—
551C2×C4C2×C3×D3211yescharacteristic
552C2×C4C2×C3×D3211yescharacteristic
553C2×C3—24no—
554C2×C3—22no—
555C2×C3—22no—
556C2×C3—22no—
557C2×C3—21no—
558C2×C3—21no—
559C2×C3—21no—
560C2×C3—22no—
561C2×C3—22no—
562C2×C3—22no—
563C2×C3—42no—
564C2×C3—42no—
565C2×C3—44no—
566C2×C3—41no—
567C2×C3—42no—
568C2×C3—362no—
569C2×C3#6429712yes—
570C2×C3C22×#62611yescharacteristic
571C2×C3C22×C3×D1211yescharacteristic
572C2×C3C3×#100712yes—
573C2×C3C3×#101012yes—
574C2×C3C22×C3×#6711yescharacteristic
575D3—124no—
576D3—122no—
577D3—122no—
578C22—22no—
579C22—182no—
580C22—184no—
581C22C2×C3×#62612yes—
582C22C22×C3×D3211yescharacteristic
583C4—64no—
584C4—64no—
585C4—182no—
586C4C2×C3×#62612yes—
587C4C2×C3×#62612yes—
588C3—21no—
589C3—21no—
590C3—21no—
591C3—41no—
592C3#1059486611yescharacteristic, direct product factor
593C3C3×#552611yescharacteristic, complemented
594C3C3×#581611yescharacteristic, complemented
595C2—22no—
596C2—362no—
597C2C3×#6429712yes—
598C2C22×C3×#62611yescharacteristic
subgroupG/H
centerC22×C3C22×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC22×C33×C4C22
socleC22×C33C24
prime radicalC2×C3×#3828C2
hypercenterC22×C3×C4D32
hypocenterC32C3×#536
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
2. C22×C32×Dic6C2semidirect
3. C32×#926C2semidirect
4. C3×#63966C2semidirect
5. C3×#63871C2semidirect
6. C3×#63866C2semidirect
7. C3×#63864C2semidirect
10. #10594866C3direct (N × H)
32. C22×C3×Dic6D3semidirect
33. C22×C3×Dic6C2×C3semidirect
40. C3×#926D3semidirect
41. C3×#926C2×C3semidirect
46. #63966C2×C3semidirect
47. #63871C2×C3semidirect
48. #63866C2×C3semidirect
49. #63864C2×C3semidirect
146. C22×Dic6C3×D3semidirect
158. #926C3×D3semidirect
363. C33#536semidirect
535. C32#5526semidirect
536. C32#5816semidirect
537. C32C3×#536semidirect
592. C3#10594866direct (N × H)
593. C3C3×#5526semidirect
594. C3C3×#5816semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38282
C22×C32×Dic62
C32×#9262
C3×#639662
C3×#638712
C3×#638662
C3×#638642
C3×#58163
C3×#55263
#105948663