Finite Group Databasebeta

← C3×#10595209 #13174758

Subgroups — C3×#10595209 #13174758

Order 1728.

1176 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64147C211yesmaximal, characteristic
2C32×#935C212yesmaximal, complemented
3C3×#63975C212yesmaximal, complemented
4C3×#63964C212yesmaximal, complemented
5C3×#63956C212yesmaximal, complemented
6C3×#63952C212yesmaximal, complemented
7C3×#63883C211yesmaximal, characteristic, complemented
8C3×#63865C211yesmaximal, characteristic, complemented
9C3×#63845C212yesmaximal, complemented
10C3×#5660—32nomaximal
11#10595209C311yesmaximal, characteristic, direct product factor
12C2×C33×D4C2211yescharacteristic
13C2×C3×#638C2212yes—
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C22×C32×Dic3C2212yes—
16C2×C3×#627C2211yescharacteristic
17C2×C3×#624C2211yescharacteristic
18C2×C3×#624C2211yescharacteristic
19C3×#3757C2212yes—
20C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
21C32×#217C2212yes—
22C32×#217C2212yes—
23C32×#217C2212yes—
24C32×Dic3⋊C4C2212yes—
25C32×Dic3⋊C4C2212yes—
26C3×#3724C2212yes—
27C3×#3724C2211yescharacteristic
28C3×Dic3⋊Dic3C2212yes—
29C3×Dic3⋊Dic3C2212yes—
30C3×Dic3⋊Dic3C2212yes—
31C3×#3722C2212yes—
32C3×Dic32C2212yes—
33C32×#123—32no—
34#64147C2×C311yescharacteristic
35C3×#938—32no—
36C3×#938—61no—
37C3×#935—22no—
38C3×#935D312yescomplemented
39C3×#935C2×C312yescomplemented
40C3×C4×#67—32no—
41C3×#924—32no—
42C3×#896—32no—
43C3×D3×C4⋊C4—32no—
44C3×#890—32no—
45C3×#889—32no—
46C3×#883—32no—
47C3×#883—32no—
48C3×#67⋊C4—32no—
49C3×#879—32no—
50C3×#879—32no—
51C3×#878—32no—
52#63975C2×C312yescomplemented
53#63964C2×C312yescomplemented
54#63956C2×C312yescomplemented
55#63952C2×C312yescomplemented
56#63883C2×C311yescharacteristic, complemented
57#63865C2×C311yescharacteristic, complemented
58#63845C2×C312yescomplemented
59C23×C33C2312yes—
60C22×C3×C3⋊D3C2311yescharacteristic
61C33×D4—41no—
62C2×C33×C4C2311yescharacteristic
63C3×#638—24no—
64C2×C3×C3⋊Dic3C2312yes—
65C2×C3×C3⋊Dic3C2311yescharacteristic
66C3×C4×C3⋊D3—41no—
67C2×C32×Dic3—42no—
68C2×C32×Dic3C2312yes—
69C2×C32×Dic3C2312yes—
70C2×C32×Dic3C2312yes—
71C2×C32×Dic3C2312yes—
72C3×#627—41no—
73C3×#624—22no—
74C3×#624—41no—
75C3×#539—91no—
76#5660—32no—
77C2×C32×D4—21no—
78C2×C32×D4—22no—
79C2×C32×D4—41no—
80C2×C32×D4C2×D312yes—
81C2×C32×D4C22×C311yescharacteristic
82C22×C32×C4—32no—
83C2×#638C22×C312yes—
84C2×C4×C3⋊D3C22×C311yescharacteristic
85C2×C3×#67—32no—
86C2×C3×#67—32no—
87C2×C3×#67—62no—
88C22×C3×Dic3—22no—
89C22×C3×Dic3C2×D312yes—
90C22×C3×Dic3C22×C312yes—
91C2×C3×C4×D3—32no—
92C2×C3×C4×D3—32no—
93C2×C3×C4×D3—32no—
94C2×C3×C4×D3—61no—
95C2×C3×Dic6—32no—
96C2×#627C22×C311yescharacteristic
97C2×#624C22×C311yescharacteristic
98C2×#624C22×C311yescharacteristic
99C32×C4⋊C4—32no—
100C32×C4⋊C4—32no—
101C32×C22⋊C4—32no—
102C32×C22⋊C4—32no—
103C32×C22⋊C4—32no—
104#3757C22×C312yes—
105C4⋊(C3⋊Dic3)C22×C311yescharacteristic
106C3×#217—22no—
107C3×#217—22no—
108C3×#217—22no—
109C3×#217—32no—
110C3×#217—32no—
111C3×#217—62no—
112C3×#217C2×D312yes—
113C3×#217C2×D312yes—
114C3×#217C2×D312yes—
115C3×#217C22×C312yes—
116C3×#217C22×C312yes—
117C3×#217C22×C312yes—
118C3×(C2×D3)⋊C4—32no—
119C3×(C2×D3)⋊C4—32no—
120C3×C4⋊Dic3—32no—
121C3×C4⋊Dic3—32no—
122C3×C4⋊Dic3—32no—
123C3×C4⋊Dic3—32no—
124C3×C4⋊Dic3—61no—
125C3×Dic3⋊C4—22no—
126C3×Dic3⋊C4—22no—
127C3×Dic3⋊C4—32no—
128C3×Dic3⋊C4—32no—
129C3×Dic3⋊C4—32no—
130C3×Dic3⋊C4—32no—
131C3×Dic3⋊C4—32no—
132C3×Dic3⋊C4—32no—
133C3×Dic3⋊C4C2×D312yes—
134C3×Dic3⋊C4C2×D312yes—
135C3×Dic3⋊C4C22×C312yes—
136C3×Dic3⋊C4C22×C312yes—
137C3×C4×Dic3—32no—
138C3×C4×Dic3—32no—
139#3724C22×C312yes—
140#3724C22×C311yescharacteristic
141Dic3⋊Dic3C22×C312yes—
142Dic3⋊Dic3C22×C312yes—
143Dic3⋊Dic3C22×C312yes—
144#3722C22×C312yes—
145Dic32C22×C312yes—
146C22×C33—22no—
147C22×C33—22no—
148C22×C33—22no—
149C22×C33C2411yescharacteristic
150C2×C3×C3⋊D3—21no—
151C2×C3×C3⋊D3—21no—
152C2×C3×C3⋊D3#4112yes—
153C33×C4—21no—
154C3×C3⋊Dic3—22no—
155C3×C3⋊Dic3—21no—
156C32×Dic3—22no—
157C32×Dic3—22no—
158C32×Dic3—22no—
159C32×Dic3—22no—
160C3×#126—92no—
161C3×#125—92no—
162C3×#123—32no—
163C3×#123—32no—
164C3×#123—62no—
165C3×#123—92no—
166C3×C22⋊Q8—92no—
167C3×C22⋊Q8—91no—
168C3×#121—91no—
169C3×C4×D4—92no—
170C2×C3×C4⋊C4—91no—
171#938—32no—
172#938—61no—
173#935C3×D312yescomplemented
174C4×#67—32no—
175#924—32no—
176#896—32no—
177D3×C4⋊C4—32no—
178#890—32no—
179#889—32no—
180#883—32no—
181#883—32no—
182#67⋊C4—32no—
183#879—32no—
184#879—32no—
185#878—32no—
186C23×C32—22no—
187C23×C32—22no—
188C23×C32—22no—
189C23×C32—42no—
190C23×C32C22×D312yes—
191C23×C32C22×D312yes—
192C23×C32C23×C312yes—
193C22×C3⋊D3C23×C311yescharacteristic
194C22×C3×D3—32no—
195C22×C3×D3—61no—
196C32×D4—42no—
197C32×D4—41no—
198C32×D4—42no—
199C32×D4—44no—
200C32×D4—44no—
201C2×C32×C4—21no—
202C2×C32×C4—22no—
203C2×C32×C4—32no—
204C2×C32×C4—32no—
205C2×C32×C4—32no—
206C2×C32×C4—32no—
207C2×C32×C4—41no—
208C2×C32×C4—122no—
209C2×C32×C4C22×D312yes—
210C2×C32×C4C23×C311yescharacteristic
211#638—24no—
212C2×C3⋊Dic3C23×C312yes—
213C2×C3⋊Dic3C23×C311yescharacteristic
214C4×C3⋊D3—41no—
215C3×#67—64no—
216C3×#67—64no—
217C3×#67—124no—
218C2×C3×Dic3—22no—
219C2×C3×Dic3—22no—
220C2×C3×Dic3—22no—
221C2×C3×Dic3—22no—
222C2×C3×Dic3—32no—
223C2×C3×Dic3—32no—
224C2×C3×Dic3—32no—
225C2×C3×Dic3—42no—
226C2×C3×Dic3—42no—
227C2×C3×Dic3—44no—
228C2×C3×Dic3—62no—
229C2×C3×Dic3—61no—
230C2×C3×Dic3C22×D312yes—
231C2×C3×Dic3C22×D312yes—
232C2×C3×Dic3C22×D312yes—
233C2×C3×Dic3C22×D312yes—
234C2×C3×Dic3C23×C312yes—
235C2×C3×Dic3C23×C312yes—
236C2×C3×Dic3C23×C312yes—
237C2×C3×Dic3C23×C312yes—
238C3×C4×D3—64no—
239C3×C4×D3—122no—
240C3×C4×D3—122no—
241C3×C4×D3—122no—
242C3×Dic6—122no—
243#627—41no—
244#624—22no—
245#624—41no—
246#539—91noSylow
247C2×C33—42no—
248C2×C33C2×#4111yescharacteristic
249C2×C33H(5,𝔽2)11yescharacteristic
250C2×C332−1+411yescharacteristic
251C3×C3⋊D3—22no—
252C2×C3×Q8—91no—
253C2×C3×D4—21no—
254C2×C3×D4—22no—
255C2×C3×D4—41no—
256C2×C3×D4—92no—
257C2×C3×D4D3211yescharacteristic
258C2×C3×D4C2×C3×D312yes—
259C22×C3×C4—32no—
260C22×C3×C4—32no—
261C22×C3×C4—62no—
262C22×C3×C4—91no—
263C22×C3×C4—91no—
264C22×C3×C4—91no—
265C2×#67—32no—
266C2×#67—32no—
267C2×#67—62no—
268C22×Dic3C2×C3×D312yes—
269C2×C4×D3—32no—
270C2×C4×D3—32no—
271C2×C4×D3—32no—
272C2×C4×D3—61no—
273C2×Dic6—32no—
274C3×C4⋊C4—32no—
275C3×C4⋊C4—32no—
276C3×C4⋊C4—32no—
277C3×C4⋊C4—32no—
278C3×C4⋊C4—62no—
279C3×C4⋊C4—62no—
280C3×C4⋊C4—92no—
281C3×C4⋊C4—91no—
282C3×C4⋊C4—92no—
283C3×C4⋊C4—92no—
284C3×C4⋊C4—92no—
285C3×C4⋊C4—91no—
286C3×C22⋊C4—32no—
287C3×C22⋊C4—32no—
288C3×C22⋊C4—32no—
289C3×C22⋊C4—32no—
290C3×C22⋊C4—32no—
291C3×C22⋊C4—32no—
292C3×C22⋊C4—62no—
293C3×C22⋊C4—62no—
294C3×C22⋊C4—62no—
295C3×C22⋊C4—92no—
296C3×C22⋊C4—92no—
297C3×C42—92no—
298#217—32no—
299#217—32no—
300#217—62no—
301#217C2×C3×D312yes—
302#217C2×C3×D312yes—
303#217C2×C3×D312yes—
304(C2×D3)⋊C4—32no—
305(C2×D3)⋊C4—32no—
306C4⋊Dic3—32no—
307C4⋊Dic3—32no—
308C4⋊Dic3—32no—
309C4⋊Dic3—32no—
310C4⋊Dic3—61no—
311Dic3⋊C4—32no—
312Dic3⋊C4—32no—
313Dic3⋊C4—32no—
314Dic3⋊C4—32no—
315Dic3⋊C4—32no—
316Dic3⋊C4—32no—
317Dic3⋊C4C2×C3×D312yes—
318Dic3⋊C4C2×C3×D312yes—
319C4×Dic3—32no—
320C4×Dic3—32no—
321C22×C32—22no—
322C22×C32—24no—
323C22×C32—22no—
324C22×C32—22no—
325C22×C32—22no—
326C22×C32—22no—
327C22×C32—22no—
328C22×C32—22no—
329C22×C32—22no—
330C22×C32—22no—
331C22×C32—21no—
332C22×C32—22no—
333C22×C32—41no—
334C22×C32—42no—
335C22×C32—42no—
336C22×C32—44no—
337C22×C32—42no—
338C22×C32—42no—
339C22×C32—42no—
340C22×C32—42no—
341C22×C32—42no—
342C22×C32—42no—
343C22×C32—82no—
344C22×C32—82no—
345C22×C32C23×D312yes—
346C22×C32C24×C311yescharacteristic
347C2×C3⋊D3—21no—
348C2×C3⋊D3—21no—
349C2×C3⋊D3C3×#4112yes—
350C2×C3×D3—34no—
351C2×C3×D3—62no—
352C2×C3×D3—62no—
353C2×C3×D3—62no—
354C2×C3×D3—121no—
355C2×C3×D3—121no—
356C32×C4—21no—
357C32×C4—22no—
358C32×C4—22no—
359C32×C4—42no—
360C32×C4—42no—
361C32×C4—62no—
362C32×C4—62no—
363C32×C4—62no—
364C32×C4—62no—
365C3⋊Dic3—22no—
366C3⋊Dic3—21no—
367C3×Dic3—22no—
368C3×Dic3—22no—
369C3×Dic3—22no—
370C3×Dic3—22no—
371C3×Dic3—22no—
372C3×Dic3—22no—
373C3×Dic3—22no—
374C3×Dic3—22no—
375C3×Dic3—42no—
376C3×Dic3—42no—
377C3×Dic3—42no—
378C3×Dic3—42no—
379C3×Dic3—62no—
380C3×Dic3—62no—
381C3×Dic3—62no—
382C3×Dic3—121no—
383C3×Dic3—122no—
384#126—92no—
385#125—92no—
386#123—32no—
387#123—92no—
388C22⋊Q8—92no—
389C22⋊Q8—91no—
390#121—91no—
391C4×D4—92no—
392C2×C4⋊C4—91no—
393C33#53911yescharacteristic, complemented, Sylow
394C23×C3—22no—
395C23×C3—22no—
396C23×C3—22no—
397C23×C3—42no—
398C23×C3—91no—
399C23×C3C2×D3212yes—
400C23×C3C22×C3×D312yes—
401C23×C3C22×C3×D312yes—
402C22×D3—32no—
403C22×D3—61no—
404C3×Q8—361no—
405C3×D4—41no—
406C3×D4—42no—
407C3×D4—42no—
408C3×D4—44no—
409C3×D4—44no—
410C3×D4—184no—
411C2×C3×C4—21no—
412C2×C3×C4—22no—
413C2×C3×C4—32no—
414C2×C3×C4—32no—
415C2×C3×C4—32no—
416C2×C3×C4—32no—
417C2×C3×C4—32no—
418C2×C3×C4—32no—
419C2×C3×C4—32no—
420C2×C3×C4—32no—
421C2×C3×C4—41no—
422C2×C3×C4—62no—
423C2×C3×C4—62no—
424C2×C3×C4—62no—
425C2×C3×C4—62no—
426C2×C3×C4—92no—
427C2×C3×C4—91no—
428C2×C3×C4—122no—
429C2×C3×C4—122no—
430C2×C3×C4—124no—
431C2×C3×C4—182no—
432C2×C3×C4—361no—
433C2×C3×C4—361no—
434C2×C3×C4C2×D3211yescharacteristic
435C2×C3×C4C22×C3×D312yes—
436#67—64no—
437#67—64no—
438#67—124no—
439C2×Dic3—32no—
440C2×Dic3—32no—
441C2×Dic3—32no—
442C2×Dic3—42no—
443C2×Dic3—62no—
444C2×Dic3—61no—
445C2×Dic3C22×C3×D312yes—
446C2×Dic3C22×C3×D312yes—
447C2×Dic3C22×C3×D312yes—
448C2×Dic3C22×C3×D312yes—
449C4×D3—64no—
450C4×D3—122no—
451C4×D3—122no—
452C4×D3—122no—
453Dic6—122no—
454C2×C32—21no—
455C2×C32—21no—
456C2×C32—21no—
457C2×C32—22no—
458C2×C32—22no—
459C2×C32—22no—
460C2×C32—41no—
461C2×C32—41no—
462C2×C32—41no—
463C2×C32—42no—
464C2×C32—44no—
465C2×C32—42no—
466C2×C32—42no—
467C2×C32—82no—
468C2×C32—82no—
469C2×C32—84no—
470C2×C32#100412yes—
471C2×C32D3×#4112yes—
472C2×C32#101012yes—
473C2×C32C2×C3×#4111yescharacteristic
474C2×C32C3×H(5,𝔽2)11yescharacteristic
475C2×C32C3×2−1+411yescharacteristic
476C3⋊D3—22no—
477C3×D3—64no—
478C3×D3—122no—
479C2×Q8—91no—
480C2×D4—92no—
481C2×D4C3×D3211yescharacteristic
482C22×C4—32no—
483C22×C4—91no—
484C22×C4—91no—
485C22×C4—91no—
486C4⋊C4—32no—
487C4⋊C4—32no—
488C4⋊C4—92no—
489C4⋊C4—92no—
490C4⋊C4—92no—
491C4⋊C4—91no—
492C4⋊C4—92no—
493C4⋊C4—91no—
494C22⋊C4—32no—
495C22⋊C4—32no—
496C22⋊C4—32no—
497C22⋊C4—92no—
498C22⋊C4—92no—
499C42—92no—
500C22×C3—24no—
501C22×C3—22no—
502C22×C3—22no—
503C22×C3—22no—
504C22×C3—22no—
505C22×C3—22no—
506C22×C3—22no—
507C22×C3—22no—
508C22×C3—22no—
509C22×C3—22no—
510C22×C3—21no—
511C22×C3—22no—
512C22×C3—42no—
513C22×C3—42no—
514C22×C3—42no—
515C22×C3—42no—
516C22×C3—42no—
517C22×C3—42no—
518C22×C3—42no—
519C22×C3—42no—
520C22×C3—44no—
521C22×C3—41no—
522C22×C3—82no—
523C22×C3—82no—
524C22×C3—92no—
525C22×C3—181no—
526C22×C3—181no—
527C22×C3C22×D3211yescharacteristic
528C22×C3C23×C3×D312yes—
529C2×D3—34no—
530C2×D3—62no—
531C2×D3—62no—
532C2×D3—62no—
533C2×D3—121no—
534C2×D3—121no—
535C3×C4—22no—
536C3×C4—21no—
537C3×C4—22no—
538C3×C4—42no—
539C3×C4—42no—
540C3×C4—62no—
541C3×C4—62no—
542C3×C4—62no—
543C3×C4—62no—
544C3×C4—62no—
545C3×C4—62no—
546C3×C4—62no—
547C3×C4—62no—
548C3×C4—122no—
549C3×C4—122no—
550C3×C4—122no—
551C3×C4—122no—
552C3×C4—182no—
553C3×C4—181no—
554Dic3—22no—
555Dic3—22no—
556Dic3—22no—
557Dic3—22no—
558Dic3—62no—
559Dic3—62no—
560Dic3—62no—
561Dic3—121no—
562Dic3—122no—
563C32—21no—
564C32—22no—
565C32—41no—
566C32#566012yescomplemented
567C32C3×#53911yescharacteristic, complemented
568C23—91no—
569C23C2×C3×D3212yes—
570Q8—361no—
571D4—41no—
572D4—184no—
573C2×C4—32no—
574C2×C4—32no—
575C2×C4—32no—
576C2×C4—32no—
577C2×C4—92no—
578C2×C4—91no—
579C2×C4—122no—
580C2×C4—182no—
581C2×C4—361no—
582C2×C4—361no—
583C2×C4C2×C3×D3211yescharacteristic
584C2×C3—21no—
585C2×C3—22no—
586C2×C3—22no—
587C2×C3—21no—
588C2×C3—21no—
589C2×C3—22no—
590C2×C3—44no—
591C2×C3—42no—
592C2×C3—42no—
593C2×C3—42no—
594C2×C3—41no—
595C2×C3—41no—
596C2×C3—41no—
597C2×C3—82no—
598C2×C3—82no—
599C2×C3—84no—
600C2×C3—182no—
601C2×C3#6430911yescharacteristic
602C2×C3#6431211yescharacteristic
603C2×C3#6433011yescharacteristic
604C2×C3C3×#100412yes—
605C2×C3C3×D3×#4112yes—
606C2×C3C3×#101012yes—
607D3—64no—
608D3—122no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—92no—
613C22—181no—
614C22—181no—
615C22C22×C3×D3211yescharacteristic
616C4—21no—
617C4—62no—
618C4—62no—
619C4—62no—
620C4—62no—
621C4—181no—
622C4—182no—
623C3—21no—
624C3—22no—
625C3—41no—
626C3#1059520911yescharacteristic, direct product factor
627C3C3×#566012yescomplemented
628C2—42no—
629C2—182no—
630C2C3×#6430911yescharacteristic
631C2C3×#6431211yescharacteristic
632C2C3×#6433011yescharacteristic
subgroupG/H
centerC22×C3C22×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC2×C33×D4C22
socleC22×C33C24
prime radicalC3×#64147C2
hypercenterC2×C3×D4D32
hypocenterC32C3×#539
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. C32×#935C2semidirect
3. C3×#63975C2semidirect
4. C3×#63964C2semidirect
5. C3×#63956C2semidirect
6. C3×#63952C2semidirect
7. C3×#63883C2semidirect
8. C3×#63865C2semidirect
9. C3×#63845C2semidirect
11. #10595209C3direct (N × H)
38. C3×#935D3semidirect
39. C3×#935C2×C3semidirect
52. #63975C2×C3semidirect
53. #63964C2×C3semidirect
54. #63956C2×C3semidirect
55. #63952C2×C3semidirect
56. #63883C2×C3semidirect
57. #63865C2×C3semidirect
58. #63845C2×C3semidirect
173. #935C3×D3semidirect
393. C33#539semidirect
566. C32#5660semidirect
567. C32C3×#539semidirect
626. C3#10595209direct (N × H)
627. C3C3×#5660semidirect

Maximal subgroups

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

subgroupindex
C3×#641472
C32×#9352
C3×#639752
C3×#639642
C3×#639562
C3×#639522
C3×#638832
C3×#638652
C3×#638452
C3×#56603
#105952093