Finite Group Databasebeta

← C3×#10595213 #13174759

Subgroups — C3×#10595213 #13174759

Order 1728.

1198 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64145C211yesmaximal, characteristic, complemented
2C32×#939C212yesmaximal, complemented
3C3×#63970C212yesmaximal, complemented
4C3×#63969C212yesmaximal, complemented
5C3×#63964C212yesmaximal, complemented
6C3×#63961C212yesmaximal, complemented
7C3×#63915C211yesmaximal, characteristic, complemented
8C3×#63896C212yesmaximal, complemented
9C3×#63870C211yesmaximal, characteristic, complemented
10C3×#5621—32nomaximal
11#10595213C311yesmaximal, characteristic, direct product factor
12C2×C33×D4C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C2×C32×#67C2212yes—
15C2×C32×#67C2212yes—
16C22×C32×Dic3C2212yes—
17C2×C3×#627C2211yescharacteristic
18C2×C3×#625C2211yescharacteristic
19C2×C3×D3×Dic3C2212yes—
20C3×#3757C2212yes—
21C3×#3757C2212yes—
22C3×C4×C3⋊Dic3C2211yescharacteristic
23C32×#217C2212yes—
24C32×(C2×D3)⋊C4C2212yes—
25C32×Dic3⋊C4C2212yes—
26C3×#3724C2211yescharacteristic
27C3×#3724C2212yes—
28C3×#3724C2211yescharacteristic
29C3×Dic3⋊Dic3C2212yes—
30C3×Dic3⋊Dic3C2212yes—
31C3×(C2×D3)⋊Dic3C2212yes—
32C3×(C2×D3)⋊Dic3C2212yes—
33C32×#121—32no—
34#64145C2×C311yescharacteristic, complemented
35C3×#939—22no—
36C3×#939—32no—
37C3×#939D312yescomplemented
38C3×#939C2×C312yescomplemented
39C3×#938—32no—
40C3×#936—32no—
41C3×#936—61no—
42C3×#935—32no—
43C3×#935—32no—
44C3×#929—32no—
45C3×#924—32no—
46C3×#896—32no—
47C3×#889—32no—
48C3×#885—32no—
49C3×#883—32no—
50C3×#878—32no—
51C3×#878—32no—
52#63970C2×C312yescomplemented
53#63969C2×C312yescomplemented
54#63964C2×C312yescomplemented
55#63961C2×C312yescomplemented
56#63915C2×C311yescharacteristic, complemented
57#63896C2×C312yescomplemented
58#63870C2×C311yescharacteristic, complemented
59C23×C33C2312yes—
60C22×C32×D3C2312yes—
61C33×D4—41no—
62C2×C33×C4C2311yescharacteristic
63C2×C3×C3⋊Dic3C2311yescharacteristic
64C2×C3×C3⋊Dic3C2311yescharacteristic
65C2×C3×C3⋊Dic3C2312yes—
66C3×#634—41no—
67C32×#67—42no—
68C32×#67—42no—
69C2×C32×Dic3—42no—
70C2×C32×Dic3C2312yes—
71C2×C32×Dic3C2312yes—
72C2×C32×Dic3C2312yes—
73C3×#627—41no—
74C3×#625—41no—
75C3×D3×Dic3—42no—
76C3×#562—91no—
77#5621—32no—
78C2×C32×D4—21no—
79C2×C32×D4—22no—
80C2×C32×D4—32no—
81C2×C32×D4—32no—
82C2×C32×D4—41no—
83C2×C32×D4C2×D312yes—
84C2×C32×D4C22×C311yescharacteristic
85C22×C32×C4—32no—
86C2×#634C22×C311yescharacteristic
87C2×C3×#67—22no—
88C2×C3×#67—22no—
89C2×C3×#67—32no—
90C2×C3×#67C2×D312yes—
91C2×C3×#67C2×D312yes—
92C2×C3×#67C22×C312yes—
93C2×C3×#67C22×C312yes—
94C22×C3×Dic3—22no—
95C22×C3×Dic3—32no—
96C22×C3×Dic3C2×D312yes—
97C22×C3×Dic3C22×C312yes—
98C2×C3×C4×D3—32no—
99C2×C3×Dic6—32no—
100C2×C3×Dic6—32no—
101C2×C3×Dic6—61no—
102C2×#627C22×C311yescharacteristic
103C2×#625C22×C311yescharacteristic
104C2×D3×Dic3C22×C312yes—
105C32×C4⋊C4—32no—
106C32×C22⋊C4—32no—
107C32×C22⋊C4—32no—
108#3757C22×C312yes—
109#3757C22×C312yes—
110C4×C3⋊Dic3C22×C311yescharacteristic
111C3×#217—22no—
112C3×#217—32no—
113C3×#217—32no—
114C3×#217—32no—
115C3×#217—32no—
116C3×#217—32no—
117C3×#217—32no—
118C3×#217—62no—
119C3×#217—62no—
120C3×#217C2×D312yes—
121C3×#217C22×C312yes—
122C3×(C2×D3)⋊C4—22no—
123C3×(C2×D3)⋊C4—32no—
124C3×(C2×D3)⋊C4—32no—
125C3×(C2×D3)⋊C4C2×D312yes—
126C3×(C2×D3)⋊C4C22×C312yes—
127C3×C4⋊Dic3—32no—
128C3×C4⋊Dic3—32no—
129C3×Dic3⋊C4—22no—
130C3×Dic3⋊C4—32no—
131C3×Dic3⋊C4—32no—
132C3×Dic3⋊C4—32no—
133C3×Dic3⋊C4—32no—
134C3×Dic3⋊C4—32no—
135C3×Dic3⋊C4—32no—
136C3×Dic3⋊C4C2×D312yes—
137C3×Dic3⋊C4C22×C312yes—
138C3×C4×Dic3—32no—
139C3×C4×Dic3—61no—
140#3724C22×C311yescharacteristic
141#3724C22×C311yescharacteristic
142#3724C22×C312yes—
143Dic3⋊Dic3C22×C312yes—
144Dic3⋊Dic3C22×C312yes—
145(C2×D3)⋊Dic3C22×C312yes—
146(C2×D3)⋊Dic3C22×C312yes—
147C22×C33—22no—
148C22×C33—22no—
149C22×C33—22no—
150C22×C33C2411yescharacteristic
151C2×C32×D3—22no—
152C2×C32×D3—22no—
153C2×C32×D3—22no—
154C33×C4—21no—
155C3×C3⋊Dic3—22no—
156C3×C3⋊Dic3—21no—
157C3×C3⋊Dic3—21no—
158C32×Dic3—22no—
159C32×Dic3—22no—
160C32×Dic3—22no—
161C3×#125—91no—
162C3×#124—91no—
163C3×#124—91no—
164C3×#123—92no—
165C3×#123—92no—
166C3×C22⋊Q8—92no—
167C3×C22⋊Q8—92no—
168C3×#121—32no—
169C3×#121—32no—
170C3×#121—62no—
171C3×#121—92no—
172#939—32no—
173#939C3×D312yescomplemented
174#938—32no—
175#936—32no—
176#936—61no—
177#935—32no—
178#935—32no—
179#929—32no—
180#924—32no—
181#896—32no—
182#889—32no—
183#885—32no—
184#883—32no—
185#878—32no—
186#878—32no—
187C23×C32—22no—
188C23×C32—22no—
189C23×C32—22no—
190C23×C32—32no—
191C23×C32—42no—
192C23×C32C22×D312yes—
193C23×C32C22×D312yes—
194C23×C32C23×C312yes—
195C22×C3×D3—22no—
196C22×C3×D3C22×D312yes—
197C22×C3×D3C23×C312yes—
198C32×D4—42no—
199C32×D4—41no—
200C32×D4—42no—
201C32×D4—44no—
202C32×D4—44no—
203C32×D4—122no—
204C32×D4—122no—
205C2×C32×C4—21no—
206C2×C32×C4—22no—
207C2×C32×C4—32no—
208C2×C32×C4—32no—
209C2×C32×C4—32no—
210C2×C32×C4—41no—
211C2×C32×C4—122no—
212C2×C32×C4C22×D312yes—
213C2×C32×C4C23×C311yescharacteristic
214C2×C3⋊Dic3C23×C311yescharacteristic
215C2×C3⋊Dic3C23×C312yes—
216C2×C3⋊Dic3C23×C311yescharacteristic
217#634—41no—
218C3×#67—42no—
219C3×#67—42no—
220C3×#67—42no—
221C3×#67—42no—
222C3×#67—44no—
223C3×#67—44no—
224C3×#67—122no—
225C2×C3×Dic3—22no—
226C2×C3×Dic3—22no—
227C2×C3×Dic3—22no—
228C2×C3×Dic3—32no—
229C2×C3×Dic3—32no—
230C2×C3×Dic3—32no—
231C2×C3×Dic3—32no—
232C2×C3×Dic3—42no—
233C2×C3×Dic3—42no—
234C2×C3×Dic3—44no—
235C2×C3×Dic3—61no—
236C2×C3×Dic3—61no—
237C2×C3×Dic3—62no—
238C2×C3×Dic3—122no—
239C2×C3×Dic3C22×D312yes—
240C2×C3×Dic3C22×D312yes—
241C2×C3×Dic3C22×D312yes—
242C2×C3×Dic3C23×C312yes—
243C2×C3×Dic3C23×C312yes—
244C2×C3×Dic3C23×C312yes—
245C3×C4×D3—122no—
246C3×Dic6—122no—
247C3×Dic6—122no—
248C3×Dic6—122no—
249#627—41no—
250#625—41no—
251D3×Dic3—42no—
252#562—91noSylow
253C2×C33—42no—
254C2×C33H(5,𝔽2)11yescharacteristic
255C2×C33H(5,𝔽2)11yescharacteristic
256C2×C332−1+411yescharacteristic
257C32×D3—42no—
258C2×C3×Q8—91no—
259C2×C3×Q8—91no—
260C2×C3×D4—21no—
261C2×C3×D4—22no—
262C2×C3×D4—32no—
263C2×C3×D4—32no—
264C2×C3×D4—32no—
265C2×C3×D4—32no—
266C2×C3×D4—41no—
267C2×C3×D4—62no—
268C2×C3×D4—62no—
269C2×C3×D4—91no—
270C2×C3×D4D3211yescharacteristic
271C2×C3×D4C2×C3×D312yes—
272C22×C3×C4—32no—
273C22×C3×C4—32no—
274C22×C3×C4—62no—
275C22×C3×C4—92no—
276C2×#67—32no—
277C2×#67C2×C3×D312yes—
278C2×#67C2×C3×D312yes—
279C22×Dic3—32no—
280C22×Dic3C2×C3×D312yes—
281C2×C4×D3—32no—
282C2×Dic6—32no—
283C2×Dic6—32no—
284C2×Dic6—61no—
285C3×C4⋊C4—32no—
286C3×C4⋊C4—32no—
287C3×C4⋊C4—62no—
288C3×C4⋊C4—92no—
289C3×C4⋊C4—91no—
290C3×C4⋊C4—91no—
291C3×C4⋊C4—92no—
292C3×C4⋊C4—92no—
293C3×C22⋊C4—32no—
294C3×C22⋊C4—32no—
295C3×C22⋊C4—32no—
296C3×C22⋊C4—32no—
297C3×C22⋊C4—62no—
298C3×C22⋊C4—62no—
299C3×C22⋊C4—92no—
300C3×C22⋊C4—92no—
301C3×C22⋊C4—92no—
302C3×C22⋊C4—92no—
303C3×C42—91no—
304#217—32no—
305#217—32no—
306#217—32no—
307#217—32no—
308#217—32no—
309#217—32no—
310#217—62no—
311#217—62no—
312#217C2×C3×D312yes—
313(C2×D3)⋊C4—32no—
314(C2×D3)⋊C4—32no—
315(C2×D3)⋊C4C2×C3×D312yes—
316C4⋊Dic3—32no—
317C4⋊Dic3—32no—
318Dic3⋊C4—32no—
319Dic3⋊C4—32no—
320Dic3⋊C4—32no—
321Dic3⋊C4—32no—
322Dic3⋊C4—32no—
323Dic3⋊C4—32no—
324Dic3⋊C4C2×C3×D312yes—
325C4×Dic3—32no—
326C4×Dic3—61no—
327C22×C32—22no—
328C22×C32—24no—
329C22×C32—22no—
330C22×C32—22no—
331C22×C32—22no—
332C22×C32—22no—
333C22×C32—22no—
334C22×C32—22no—
335C22×C32—22no—
336C22×C32—22no—
337C22×C32—21no—
338C22×C32—22no—
339C22×C32—41no—
340C22×C32—42no—
341C22×C32—42no—
342C22×C32—44no—
343C22×C32—42no—
344C22×C32—42no—
345C22×C32—42no—
346C22×C32—42no—
347C22×C32—42no—
348C22×C32—42no—
349C22×C32—62no—
350C22×C32—62no—
351C22×C32—62no—
352C22×C32—82no—
353C22×C32—82no—
354C22×C32C23×D312yes—
355C22×C32C24×C311yescharacteristic
356C2×C3×D3—22no—
357C2×C3×D3—22no—
358C2×C3×D3—22no—
359C2×C3×D3—22no—
360C2×C3×D3—22no—
361C2×C3×D3—22no—
362C2×C3×D3—42no—
363C2×C3×D3—42no—
364C2×C3×D3—42no—
365C32×C4—21no—
366C32×C4—22no—
367C32×C4—22no—
368C32×C4—42no—
369C32×C4—42no—
370C32×C4—62no—
371C32×C4—62no—
372C32×C4—62no—
373C3⋊Dic3—22no—
374C3⋊Dic3—21no—
375C3⋊Dic3—21no—
376C3×Dic3—24no—
377C3×Dic3—22no—
378C3×Dic3—22no—
379C3×Dic3—22no—
380C3×Dic3—22no—
381C3×Dic3—22no—
382C3×Dic3—22no—
383C3×Dic3—42no—
384C3×Dic3—42no—
385C3×Dic3—62no—
386C3×Dic3—62no—
387C3×Dic3—62no—
388C3×Dic3—62no—
389C3×Dic3—122no—
390C3×Dic3—121no—
391C3×Dic3—121no—
392#125—91no—
393#124—91no—
394#124—91no—
395#123—92no—
396#123—92no—
397C22⋊Q8—92no—
398C22⋊Q8—92no—
399#121—32no—
400#121—92no—
401C33#56211yescharacteristic, complemented, Sylow
402C23×C3—22no—
403C23×C3—22no—
404C23×C3—22no—
405C23×C3—32no—
406C23×C3—32no—
407C23×C3—42no—
408C23×C3—62no—
409C23×C3C2×D3212yes—
410C23×C3C22×C3×D312yes—
411C23×C3C22×C3×D312yes—
412C22×D3C22×C3×D312yes—
413C3×Q8—361no—
414C3×Q8—361no—
415C3×D4—41no—
416C3×D4—42no—
417C3×D4—42no—
418C3×D4—44no—
419C3×D4—44no—
420C3×D4—122no—
421C3×D4—122no—
422C3×D4—122no—
423C3×D4—122no—
424C3×D4—124no—
425C3×D4—124no—
426C3×D4—361no—
427C2×C3×C4—21no—
428C2×C3×C4—22no—
429C2×C3×C4—32no—
430C2×C3×C4—32no—
431C2×C3×C4—32no—
432C2×C3×C4—32no—
433C2×C3×C4—32no—
434C2×C3×C4—32no—
435C2×C3×C4—41no—
436C2×C3×C4—62no—
437C2×C3×C4—62no—
438C2×C3×C4—62no—
439C2×C3×C4—91no—
440C2×C3×C4—91no—
441C2×C3×C4—92no—
442C2×C3×C4—122no—
443C2×C3×C4—122no—
444C2×C3×C4—124no—
445C2×C3×C4—362no—
446C2×C3×C4C2×D3211yescharacteristic
447C2×C3×C4C22×C3×D312yes—
448#67—42no—
449#67—42no—
450#67—122no—
451C2×Dic3—32no—
452C2×Dic3—32no—
453C2×Dic3—32no—
454C2×Dic3—32no—
455C2×Dic3—42no—
456C2×Dic3—61no—
457C2×Dic3—61no—
458C2×Dic3—62no—
459C2×Dic3—122no—
460C2×Dic3C22×C3×D312yes—
461C2×Dic3C22×C3×D312yes—
462C2×Dic3C22×C3×D312yes—
463C4×D3—122no—
464Dic6—122no—
465Dic6—122no—
466Dic6—122no—
467C2×C32—21no—
468C2×C32—21no—
469C2×C32—21no—
470C2×C32—22no—
471C2×C32—22no—
472C2×C32—22no—
473C2×C32—41no—
474C2×C32—41no—
475C2×C32—41no—
476C2×C32—42no—
477C2×C32—44no—
478C2×C32—42no—
479C2×C32—42no—
480C2×C32—82no—
481C2×C32—82no—
482C2×C32—84no—
483C2×C32—122no—
484C2×C32#100412yes—
485C2×C32#100412yes—
486C2×C32#101012yes—
487C2×C32C3×H(5,𝔽2)11yescharacteristic
488C2×C32C3×H(5,𝔽2)11yescharacteristic
489C2×C32C3×2−1+411yescharacteristic
490C3×D3—42no—
491C3×D3—42no—
492C3×D3—82no—
493C2×Q8—91no—
494C2×Q8—91no—
495C2×D4—32no—
496C2×D4—32no—
497C2×D4—91no—
498C2×D4C3×D3211yescharacteristic
499C22×C4—32no—
500C22×C4—92no—
501C4⋊C4—32no—
502C4⋊C4—92no—
503C4⋊C4—92no—
504C4⋊C4—92no—
505C4⋊C4—91no—
506C4⋊C4—91no—
507C22⋊C4—32no—
508C22⋊C4—32no—
509C22⋊C4—92no—
510C22⋊C4—92no—
511C22⋊C4—92no—
512C22⋊C4—92no—
513C42—91no—
514C22×C3—24no—
515C22×C3—22no—
516C22×C3—22no—
517C22×C3—22no—
518C22×C3—22no—
519C22×C3—22no—
520C22×C3—22no—
521C22×C3—22no—
522C22×C3—22no—
523C22×C3—22no—
524C22×C3—21no—
525C22×C3—22no—
526C22×C3—42no—
527C22×C3—42no—
528C22×C3—42no—
529C22×C3—42no—
530C22×C3—42no—
531C22×C3—42no—
532C22×C3—42no—
533C22×C3—42no—
534C22×C3—44no—
535C22×C3—41no—
536C22×C3—62no—
537C22×C3—62no—
538C22×C3—62no—
539C22×C3—62no—
540C22×C3—62no—
541C22×C3—62no—
542C22×C3—82no—
543C22×C3—82no—
544C22×C3—122no—
545C22×C3—122no—
546C22×C3—122no—
547C22×C3C22×D3211yescharacteristic
548C22×C3C23×C3×D312yes—
549C2×D3—22no—
550C2×D3—22no—
551C2×D3—22no—
552C3×C4—22no—
553C3×C4—21no—
554C3×C4—22no—
555C3×C4—42no—
556C3×C4—42no—
557C3×C4—64no—
558C3×C4—62no—
559C3×C4—62no—
560C3×C4—62no—
561C3×C4—62no—
562C3×C4—62no—
563C3×C4—62no—
564C3×C4—122no—
565C3×C4—122no—
566C3×C4—181no—
567C3×C4—181no—
568C3×C4—182no—
569Dic3—22no—
570Dic3—22no—
571Dic3—22no—
572Dic3—62no—
573Dic3—62no—
574Dic3—62no—
575Dic3—62no—
576Dic3—122no—
577Dic3—121no—
578Dic3—121no—
579C32—22no—
580C32—21no—
581C32—41no—
582C32#562112yescomplemented
583C32C3×#56211yescharacteristic, complemented
584C23—32no—
585C23C2×C3×D3212yes—
586Q8—361no—
587Q8—361no—
588D4—41no—
589D4—122no—
590D4—122no—
591D4—361no—
592C2×C4—32no—
593C2×C4—32no—
594C2×C4—32no—
595C2×C4—91no—
596C2×C4—91no—
597C2×C4—92no—
598C2×C4—122no—
599C2×C4—362no—
600C2×C4C2×C3×D3211yescharacteristic
601C2×C3—21no—
602C2×C3—22no—
603C2×C3—21no—
604C2×C3—22no—
605C2×C3—21no—
606C2×C3—22no—
607C2×C3—44no—
608C2×C3—42no—
609C2×C3—42no—
610C2×C3—42no—
611C2×C3—41no—
612C2×C3—41no—
613C2×C3—41no—
614C2×C3—82no—
615C2×C3—82no—
616C2×C3—84no—
617C2×C3—122no—
618C2×C3—122no—
619C2×C3—242no—
620C2×C3#6430911yescharacteristic
621C2×C3#6431311yescharacteristic
622C2×C3#6433011yescharacteristic
623C2×C3C3×#100412yes—
624C2×C3C3×#100412yes—
625C2×C3C3×#101012yes—
626D3—42no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—62no—
631C22—62no—
632C22—62no—
633C22C22×C3×D3211yescharacteristic
634C4—21no—
635C4—62no—
636C4—62no—
637C4—62no—
638C4—182no—
639C4—181no—
640C4—181no—
641C3—22no—
642C3—21no—
643C3—41no—
644C3#1059521311yescharacteristic, direct product factor
645C3C3×#562112yescomplemented
646C2—42no—
647C2—122no—
648C2C3×#6430911yescharacteristic
649C2C3×#6431311yescharacteristic
650C2C3×#6433011yescharacteristic
subgroupG/H
centerC22×C3C22×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC2×C33×D4C22
socleC22×C33C24
hypercenterC2×C3×D4D32
hypocenterC32C3×#562
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
1. C3×#64145C2semidirect
2. C32×#939C2semidirect
3. C3×#63970C2semidirect
4. C3×#63969C2semidirect
5. C3×#63964C2semidirect
6. C3×#63961C2semidirect
7. C3×#63915C2semidirect
8. C3×#63896C2semidirect
9. C3×#63870C2semidirect
11. #10595213C3direct (N × H)
34. #64145C2×C3semidirect
37. C3×#939D3semidirect
38. C3×#939C2×C3semidirect
52. #63970C2×C3semidirect
53. #63969C2×C3semidirect
54. #63964C2×C3semidirect
55. #63961C2×C3semidirect
56. #63915C2×C3semidirect
57. #63896C2×C3semidirect
58. #63870C2×C3semidirect
173. #939C3×D3semidirect
401. C33#562semidirect
582. C32#5621semidirect
583. C32C3×#562semidirect
644. C3#10595213direct (N × H)
645. C3C3×#5621semidirect

Maximal subgroups

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

subgroupindex
C3×#641452
C32×#9392
C3×#639702
C3×#639692
C3×#639642
C3×#639612
C3×#639152
C3×#638962
C3×#638702
C3×#56213
#105952133