Finite Group Databasebeta

← C3×#32962 #10651569

Subgroups — C3×#32962 #10651569

Order 768.

1422 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#518C211yesmaximal, characteristic
2C2×C3×#476C211yesmaximal, characteristic, complemented
3C2×C3×#474C211yesmaximal, characteristic, complemented
4C3×#2628C211yesmaximal, characteristic, complemented
5C3×#1864C212yesmaximal, complemented
6C3×#1862C212yesmaximal, complemented
7C3×#1854C212yesmaximal, complemented
8C3×#1804C211yesmaximal, characteristic, complemented
9C3×#1804C211yesmaximal, characteristic, complemented
10C3×#1798C211yesmaximal, characteristic, complemented
11C3×#1786C212yesmaximal
12#32962C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×D4C2211yescharacteristic
14C2×C3×C4⋊Q8C2211yescharacteristic, complemented
15C3×#518—24no—
16C3×#518—24no—
17C2×C3×C4×D4—21no—
18C2×C3×C4×D4—21no—
19C2×C3×C4×D4C2211yescharacteristic
20C2×C3×C4×D4C2211yescharacteristic
21C2×C3×C4×D4C2211yescharacteristic
22C2×C3×C4×D4C2211yescharacteristic
23C2×C3×#117C2211yescharacteristic
24C2×C3×#117C2211yescharacteristic
25C22×C3×C22⋊C4—21no—
26C22×C3×C22⋊C4—21no—
27C3×#476—24no—
28C3×#474—24no—
29C2×C3×#107C2212yescomplemented
30C2×C3×#106C2212yescomplemented
31C2×C3×C4⋊C8C2211yescharacteristic, complemented
32C2×C3×C4⋊C8C2211yescharacteristic, complemented
33C2×C3×#102C2212yes—
34C2×C3×#102C2212yes—
35C2×C3×#102C2212yes—
36C2×C3×#102C2212yes—
37C2×C3×C22⋊C8C2212yes—
38C3×#400—21no—
39C3×#399—21no—
40C3×#395—21no—
41C3×#393—21no—
42C3×#389C2212yescomplemented
43C3×#382C2211yescharacteristic, complemented
44C3×#380C2212yes—
45C3×#340C2212yescomplemented
46C3×#340C2212yescomplemented
47C3×#336C2212yescomplemented
48C2×#518C2×C311yescharacteristic
49C2×#476C2×C311yescharacteristic, complemented
50C2×#474C2×C311yescharacteristic, complemented
51#2628C2×C311yescharacteristic, complemented
52#1864C2×C312yescomplemented
53#1862C2×C312yescomplemented
54#1854C2×C312yescomplemented
55#1804C2×C311yescharacteristic, complemented
56#1804C2×C311yescharacteristic, complemented
57#1798C2×C311yescharacteristic, complemented
58#1786C2×C312yes—
59C25×C3—21no—
60C22×C3×Q8—21no—
61C22×C3×D4—22no—
62C22×C3×D4—22no—
63C22×C3×D4—22no—
64C22×C3×D4—22no—
65C22×C3×D4—21no—
66C22×C3×D4—22no—
67C22×C3×D4Q814yescomplemented
68C22×C3×D4C2311yescharacteristic
69C22×C3×D4C2311yescharacteristic
70C23×C3×C4—21no—
71C23×C3×C4—21no—
72C23×C3×C4—21no—
73C23×C3×C4—21no—
74C23×C3×C4C2311yescharacteristic
75C22×C3×C8C2312yes—
76C22×C3×C8C2312yes—
77C3×C4⋊Q8—22no—
78C3×C4⋊Q8—22no—
79C3×C4×D4—28no—
80C3×C4×D4—28no—
81C3×C4×D4—22no—
82C3×C4×D4—22no—
83C3×C4×D4—22no—
84C3×C4×D4—22no—
85C3×C4×D4—22no—
86C3×C4×D4—22no—
87C3×C4×D4D414yes—
88C3×C4×D4D414yes—
89C3×#117—22no—
90C3×#117—22no—
91C3×#117—22no—
92C3×#117—22no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—22no—
95C2×C3×C4⋊C4D411yescharacteristic
96C2×C3×C4⋊C4D411yescharacteristic
97C2×C3×C4⋊C4C2312yes—
98C2×C3×C4⋊C4C2312yes—
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C22⋊C4—28no—
102C2×C3×C22⋊C4—28no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C22⋊C4—21no—
108C2×C3×C22⋊C4—22no—
109C2×C3×C22⋊C4D411yescharacteristic
110C2×C3×C22⋊C4D411yescharacteristic
111C2×C3×C42—21no—
112C2×C3×C42C2311yescharacteristic
113C2×C3×C42C2311yescharacteristic
114C3×#107—24no—
115C3×#106—24no—
116C3×C4⋊C8—22no—
117C3×C4⋊C8—22no—
118C3×#102—24no—
119C3×#102—24no—
120C3×#102—24no—
121C3×#102—24no—
122C3×C22⋊C8—24no—
123C3×#95—22no—
124C3×#95—21no—
125C3×#95—21no—
126C3×#95—22no—
127C23×D4C22×C311yescharacteristic
128C2×C4⋊Q8C22×C311yescharacteristic, complemented
129#518—24no—
130#518—24no—
131C2×C4×D4—21no—
132C2×C4×D4—21no—
133C2×C4×D4C22×C311yescharacteristic
134C2×C4×D4C22×C311yescharacteristic
135C2×C4×D4C22×C311yescharacteristic
136C2×C4×D4C22×C311yescharacteristic
137C2×#117C22×C311yescharacteristic
138C2×#117C22×C311yescharacteristic
139C22×C22⋊C4—21no—
140C22×C22⋊C4—21no—
141#476—24no—
142#474—24no—
143C2×#107C22×C312yescomplemented
144C2×#106C22×C312yescomplemented
145C2×C4⋊C8C22×C311yescharacteristic, complemented
146C2×C4⋊C8C22×C311yescharacteristic, complemented
147C2×#102C22×C312yes—
148C2×#102C22×C312yes—
149C2×#102C22×C312yes—
150C2×#102C22×C312yes—
151C2×C22⋊C8C22×C312yes—
152#400—21no—
153#399—21no—
154#395—21no—
155#393—21no—
156#389C22×C312yescomplemented
157#382C22×C311yescharacteristic, complemented
158#380C22×C312yes—
159#340C22×C312yescomplemented
160#340C22×C312yescomplemented
161#336C22×C312yescomplemented
162C24×C3—24no—
163C24×C3—24no—
164C24×C3—21no—
165C24×C3—21no—
166C24×C3—24no—
167C24×C3—82no—
168C24×C3—82no—
169C24×C3C2×D411yescharacteristic
170C2×C3×Q8—42no—
171C2×C3×Q8—82no—
172C2×C3×D4—24no—
173C2×C3×D4—24no—
174C2×C3×D4—28no—
175C2×C3×D4—28no—
176C2×C3×D4—28no—
177C2×C3×D4—28no—
178C2×C3×D4—24no—
179C2×C3×D4—21no—
180C2×C3×D4—21no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4C2×D412yes—
186C2×C3×D4C2×D412yes—
187C2×C3×D4C2×Q812yes—
188C2×C3×D4C2×Q812yes—
189C2×C3×D4#4112yes—
190C2×C3×D4#4112yes—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—24no—
198C22×C3×C4—24no—
199C22×C3×C4—24no—
200C22×C3×C4—24no—
201C22×C3×C4—22no—
202C22×C3×C4—22no—
203C22×C3×C4—82no—
204C22×C3×C4—82no—
205C22×C3×C4—82no—
206C22×C3×C4—82no—
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×Q812yes—
213C22×C3×C4#4112yes—
214C22×C3×C4C2411yescharacteristic
215C2×C3×C8—22no—
216C2×C3×C8—22no—
217C2×C3×C8—22no—
218C2×C3×C8—22no—
219C2×C3×C8—22no—
220C2×C3×C8—22no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—22no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4—22no—
227C3×C4⋊C4—22no—
228C3×C4⋊C4—22no—
229C3×C4⋊C4—42no—
230C3×C4⋊C4—82no—
231C3×C4⋊C4C2×D412yes—
232C3×C4⋊C4C2×D412yes—
233C3×C22⋊C4—24no—
234C3×C22⋊C4—24no—
235C3×C22⋊C4—24no—
236C3×C22⋊C4—22no—
237C3×C22⋊C4—22no—
238C3×C22⋊C4—24no—
239C3×C22⋊C4—24no—
240C3×C22⋊C4—82no—
241C3×C42—24no—
242C3×C42—21no—
243C3×C42—21no—
244C3×C42C2×D412yes—
245C3×C42C2×D412yes—
246C25—21no—
247C22×Q8—21no—
248C22×D4—22no—
249C22×D4—22no—
250C22×D4—22no—
251C22×D4—22no—
252C22×D4—21no—
253C22×D4—22no—
254C22×D4C3×Q814yescomplemented
255C22×D4C23×C311yescharacteristic
256C22×D4C23×C311yescharacteristic
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—21no—
261C23×C4C23×C311yescharacteristic
262C22×C8C23×C312yes—
263C22×C8C23×C312yes—
264C4⋊Q8—22no—
265C4⋊Q8—22no—
266C4×D4—28no—
267C4×D4—28no—
268C4×D4—22no—
269C4×D4—22no—
270C4×D4—22no—
271C4×D4—22no—
272C4×D4—22no—
273C4×D4—22no—
274C4×D4C3×D414yes—
275C4×D4C3×D414yes—
276#117—22no—
277#117—22no—
278#117—22no—
279#117—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4C3×D411yescharacteristic
283C2×C4⋊C4C3×D411yescharacteristic
284C2×C4⋊C4C23×C312yes—
285C2×C4⋊C4C23×C312yes—
286C2×C4⋊C4C23×C311yescharacteristic
287C2×C4⋊C4C23×C311yescharacteristic
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4C3×D411yescharacteristic
297C2×C22⋊C4C3×D411yescharacteristic
298C2×C42—21no—
299C2×C42C23×C311yescharacteristic
300C2×C42C23×C311yescharacteristic
301#107—24no—
302#106—24no—
303C4⋊C8—22no—
304C4⋊C8—22no—
305#102—24no—
306#102—24no—
307#102—24no—
308#102—24no—
309C22⋊C8—24no—
310#95—22no—
311#95—21no—
312#95—21no—
313#95—22no—
314C23×C3—216no—
315C23×C3—21no—
316C23×C3—22no—
317C23×C3—22no—
318C23×C3—22no—
319C23×C3—22no—
320C23×C3—21no—
321C23×C3—22no—
322C23×C3—22no—
323C23×C3—22no—
324C23×C3—22no—
325C23×C3—42no—
326C23×C3—42no—
327C23×C3—42no—
328C23×C3—42no—
329C23×C3—88no—
330C23×C3—88no—
331C23×C3—84no—
332C23×C3—84no—
333C23×C3C22⋊Q812yes—
334C23×C3C22×D411yescharacteristic
335C3×Q8—84no—
336C3×D4—216no—
337C3×D4—22no—
338C3×D4—22no—
339C3×D4—22no—
340C3×D4—22no—
341C3×D4—22no—
342C3×D4—22no—
343C3×D4C22⋊Q814yes—
344C3×D4C22⋊Q814yes—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—42no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
364C2×C3×C4—42no—
365C2×C3×C4—42no—
366C2×C3×C4—42no—
367C2×C3×C4—84no—
368C2×C3×C4—84no—
369C2×C3×C4—84no—
370C2×C3×C4—84no—
371C2×C3×C4#12012yes—
372C2×C3×C4#12012yes—
373C2×C3×C4C22⋊Q812yes—
374C2×C3×C4C22×D411yescharacteristic
375C2×C3×C4C22×D411yescharacteristic
376C2×C3×C4C22×Q811yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C2×C3×C4C2×#4111yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380C3×C8—42no—
381C3×C8—42no—
382C24—24no—
383C24—24no—
384C24—21no—
385C24—21no—
386C24—24no—
387C24—82no—
388C24—82no—
389C24C2×C3×D411yescharacteristic
390C2×Q8—42no—
391C2×Q8—82no—
392C2×D4—24no—
393C2×D4—24no—
394C2×D4—28no—
395C2×D4—28no—
396C2×D4—28no—
397C2×D4—28no—
398C2×D4—24no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4C2×C3×D412yes—
406C2×D4C2×C3×D412yes—
407C2×D4C2×C3×Q812yes—
408C2×D4C2×C3×Q812yes—
409C2×D4C3×#4112yes—
410C2×D4C3×#4112yes—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—82no—
424C22×C4—82no—
425C22×C4—82no—
426C22×C4—82no—
427C22×C4C2×C3×D411yescharacteristic
428C22×C4C2×C3×D411yescharacteristic
429C22×C4C2×C3×D411yescharacteristic
430C22×C4C2×C3×D411yescharacteristic
431C22×C4C2×C3×D411yescharacteristic
432C22×C4C2×C3×Q812yes—
433C22×C4C3×#4112yes—
434C22×C4C24×C311yescharacteristic
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—42no—
450C4⋊C4—82no—
451C4⋊C4C2×C3×D412yes—
452C4⋊C4C2×C3×D412yes—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C22⋊C4—24no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—24no—
459C22⋊C4—24no—
460C22⋊C4—82no—
461C42—24no—
462C42—21no—
463C42—21no—
464C42C2×C3×D412yes—
465C42C2×C3×D412yes—
466C22×C3—24no—
467C22×C3—22no—
468C22×C3—24no—
469C22×C3—24no—
470C22×C3—44no—
471C22×C3—44no—
472C22×C3—42no—
473C22×C3—44no—
474C22×C3—44no—
475C22×C3—42no—
476C22×C3—42no—
477C22×C3—41no—
478C22×C3—42no—
479C22×C3—42no—
480C22×C3—41no—
481C22×C3—816no—
482C22×C3C2×#12011yescharacteristic
483C22×C3C2×C22⋊Q811yescharacteristic
484C22×C3#52511yescharacteristic
485C22×C3#57612yes—
486C22×C3#57712yes—
487C3×C4—22no—
488C3×C4—22no—
489C3×C4—22no—
490C3×C4—41no—
491C3×C4—42no—
492C3×C4—41no—
493C3×C4—42no—
494C3×C4—82no—
495C3×C4—82no—
496C3×C4C2×C22⋊Q811yescharacteristic
497C3×C4C2×C22⋊Q811yescharacteristic
498C3×C4#52511yescharacteristic
499C3×C4#52511yescharacteristic
500C23—216no—
501C23—22no—
502C23—22no—
503C23—21no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—21no—
509C23—22no—
510C23—22no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—88no—
516C23—84no—
517C23—84no—
518C23—88no—
519C23C3×C22⋊Q812yes—
520C23C22×C3×D411yescharacteristic
521Q8—84no—
522D4—216no—
523D4—22no—
524D4—22no—
525D4—22no—
526D4—22no—
527D4—22no—
528D4—22no—
529D4C3×C22⋊Q814yes—
530D4C3×C22⋊Q814yes—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—84no—
554C2×C4—84no—
555C2×C4—84no—
556C2×C4—84no—
557C2×C4C3×#12012yes—
558C2×C4C3×#12012yes—
559C2×C4C3×C22⋊Q812yes—
560C2×C4C22×C3×D411yescharacteristic
561C2×C4C22×C3×D411yescharacteristic
562C2×C4C22×C3×Q811yescharacteristic
563C2×C4C2×C3×#4111yescharacteristic
564C2×C4C2×C3×#4111yescharacteristic
565C2×C4C2×C3×#4111yescharacteristic
566C8—42no—
567C8—42no—
568C2×C3—42no—
569C2×C3—44no—
570C2×C3—44no—
571C2×C3#284811yescharacteristic
572C2×C3#300811yescharacteristic
573C2×C3#300911yescharacteristic
574C2×C3#308112yes—
575C2×C3#311812yes—
576C22—24no—
577C22—22no—
578C22—24no—
579C22—24no—
580C22—42no—
581C22—42no—
582C22—41no—
583C22—42no—
584C22—42no—
585C22—41no—
586C22—44no—
587C22—44no—
588C22—42no—
589C22—44no—
590C22—44no—
591C22—816no—
592C22C2×C3×#12011yescharacteristic
593C22C2×C3×C22⋊Q811yescharacteristic
594C22C3×#52511yescharacteristic
595C22C3×#57612yes—
596C22C3×#57712yes—
597C4—22no—
598C4—22no—
599C4—22no—
600C4—41no—
601C4—42no—
602C4—41no—
603C4—42no—
604C4—82no—
605C4—82no—
606C4C2×C3×C22⋊Q811yescharacteristic
607C4C2×C3×C22⋊Q811yescharacteristic
608C4C3×#52511yescharacteristic
609C4C3×#52511yescharacteristic
610C3#3296211yescharacteristic, direct product factor, Sylow
611C2—42no—
612C2—44no—
613C2—44no—
614C2C3×#284811yescharacteristic
615C2C3×#300811yescharacteristic
616C2C3×#300911yescharacteristic
617C2C3×#308112yes—
618C2C3×#311812yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC23×C3×D4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#476C2semidirect
3. C2×C3×#474C2semidirect
4. C3×#2628C2semidirect
5. C3×#1864C2semidirect
6. C3×#1862C2semidirect
7. C3×#1854C2semidirect
8. C3×#1804C2semidirect
9. C3×#1804C2semidirect
10. C3×#1798C2semidirect
12. #32962C3direct (N × H)
14. C2×C3×C4⋊Q8C22semidirect
29. C2×C3×#107C22semidirect
30. C2×C3×#106C22semidirect
31. C2×C3×C4⋊C8C22semidirect
32. C2×C3×C4⋊C8C22semidirect
42. C3×#389C22semidirect
43. C3×#382C22semidirect
45. C3×#340C22semidirect
46. C3×#340C22semidirect
47. C3×#336C22semidirect
49. C2×#476C2×C3semidirect
50. C2×#474C2×C3semidirect
51. #2628C2×C3semidirect
52. #1864C2×C3semidirect
53. #1862C2×C3semidirect
54. #1854C2×C3semidirect
55. #1804C2×C3semidirect
56. #1804C2×C3semidirect
57. #1798C2×C3semidirect
67. C22×C3×D4Q8semidirect
128. C2×C4⋊Q8C22×C3semidirect
143. C2×#107C22×C3semidirect
144. C2×#106C22×C3semidirect
145. C2×C4⋊C8C22×C3semidirect
146. C2×C4⋊C8C22×C3semidirect
156. #389C22×C3semidirect
157. #382C22×C3semidirect
159. #340C22×C3semidirect
160. #340C22×C3semidirect
161. #336C22×C3semidirect
254. C22×D4C3×Q8semidirect
610. C3#32962direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5182
C2×C3×#4762
C2×C3×#4742
C3×#26282
C3×#18642
C3×#18622
C3×#18542
C3×#18042
C3×#17982
C3×#17862
#329623