Finite Group Databasebeta

← C3×#32916 #10651523

Subgroups — C3×#32916 #10651523

Order 768.

1406 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#520C211yesmaximal, characteristic
2C2×C3×#462C211yesmaximal, characteristic, complemented
3C2×C3×#461C211yesmaximal, characteristic, complemented
4C2×C3×#452C212yesmaximal
5C3×#2621C211yesmaximal, characteristic, complemented
6C3×#1878C212yesmaximal, complemented
7C3×#1877C212yesmaximal, complemented
8C3×#1872C212yesmaximal, complemented
9C3×#1805C211yesmaximal, characteristic, complemented
10C3×#1804C211yesmaximal, characteristic, complemented
11C3×#1798C211yesmaximal, characteristic, complemented
12#32916C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C22×C3×Q16C2212yes—
15C22×C3×SD16C2212yes—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×C22⋊Q8C2212yes—
18C3×#520—24no—
19C3×#520—24no—
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×D4—21no—
23C2×C3×C4×D4—21no—
24C2×C3×C4×D4C2211yescharacteristic
25C2×C3×C4×D4C2211yescharacteristic
26C2×C3×#117—21no—
27C2×C3×#117C2211yescharacteristic
28C22×C3×C4⋊C4—21no—
29C22×C3×C4⋊C4C2211yescharacteristic
30C3×#462—24no—
31C3×#461—24no—
32C3×#452—28no—
33C2×C3×C4⋊C8C2211yescharacteristic
34C2×C3×C4⋊C8C2211yescharacteristic
35C2×C3×#103C2212yes—
36C2×C3×#103C2212yes—
37C2×C3×#103C2212yes—
38C2×C3×#102C2212yes—
39C2×C3×C22⋊C8C2212yes—
40C3×#398—21no—
41C3×#398—21no—
42C3×#397—21no—
43C3×#397—21no—
44C3×#391C2212yes—
45C3×#382C2211yescharacteristic
46C3×#340C2212yes—
47C3×#340C2212yes—
48C3×#336C2212yes—
49C2×#520C2×C311yescharacteristic
50C2×#462C2×C311yescharacteristic, complemented
51C2×#461C2×C311yescharacteristic, complemented
52C2×#452C2×C312yes—
53#2621C2×C311yescharacteristic, complemented
54#1878C2×C312yescomplemented
55#1877C2×C312yescomplemented
56#1872C2×C312yescomplemented
57#1805C2×C311yescharacteristic, complemented
58#1804C2×C311yescharacteristic, complemented
59#1798C2×C311yescharacteristic, complemented
60C2×C3×#41—22no—
61C2×C3×#41—22no—
62C2×C3×#41—22no—
63C2×C3×#41—22no—
64C2×C3×#41—22no—
65C2×C3×#41D414yes—
66C22×C3×Q8C2312yes—
67C22×C3×Q8C2311yescharacteristic
68C22×C3×D4—21no—
69C22×C3×D4C2311yescharacteristic
70C23×C3×C4—21no—
71C23×C3×C4—21no—
72C23×C3×C4—21no—
73C23×C3×C4—21no—
74C23×C3×C4C2311yescharacteristic
75C2×C3×Q16—24no—
76C2×C3×Q16—24no—
77C2×C3×Q16—24no—
78C2×C3×SD16—24no—
79C2×C3×SD16—24no—
80C2×C3×SD16—24no—
81C22×C3×C8C2312yes—
82C22×C3×C8C2312yes—
83C3×C4⋊Q8—22no—
84C3×C4⋊Q8—22no—
85C3×C22⋊Q8—24no—
86C3×C22⋊Q8—24no—
87C3×C4×Q8—22no—
88C3×C4×Q8—22no—
89C3×C4×Q8—22no—
90C3×C4×Q8D414yes—
91C3×C4×D4—28no—
92C3×C4×D4—28no—
93C3×C4×D4—22no—
94C3×C4×D4—22no—
95C3×C4×D4—22no—
96C3×C4×D4D414yes—
97C3×#117—28no—
98C3×#117—22no—
99C3×#117—22no—
100C2×C3×C4⋊C4—28no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4—21no—
103C2×C3×C4⋊C4—21no—
104C2×C3×C4⋊C4—22no—
105C2×C3×C4⋊C4—22no—
106C2×C3×C4⋊C4—21no—
107C2×C3×C4⋊C4—21no—
108C2×C3×C4⋊C4—22no—
109C2×C3×C4⋊C4D411yescharacteristic
110C2×C3×C4⋊C4D411yescharacteristic
111C2×C3×C4⋊C4C2312yes—
112C2×C3×C4⋊C4C2311yescharacteristic
113C2×C3×C4⋊C4C2311yescharacteristic
114C2×C3×C22⋊C4—21no—
115C2×C3×C22⋊C4—21no—
116C2×C3×C22⋊C4—22no—
117C2×C3×C22⋊C4D411yescharacteristic
118C2×C3×C22⋊C4D411yescharacteristic
119C2×C3×C42—21no—
120C2×C3×C42—21no—
121C2×C3×C42C2311yescharacteristic
122C2×C3×C42C2311yescharacteristic
123C3×C4⋊C8—22no—
124C3×C4⋊C8—22no—
125C3×#103—24no—
126C3×#103—24no—
127C3×#103—24no—
128C3×#102—24no—
129C3×C22⋊C8—24no—
130C3×#95—21no—
131C3×#95—21no—
132C3×#95—22no—
133C3×#95—22no—
134C22×#41C22×C311yescharacteristic
135C22×Q16C22×C312yes—
136C22×SD16C22×C312yes—
137C2×C4⋊Q8C22×C311yescharacteristic
138C2×C22⋊Q8C22×C312yes—
139#520—24no—
140#520—24no—
141C2×C4×Q8C22×C311yescharacteristic
142C2×C4×Q8C22×C311yescharacteristic
143C2×C4×D4—21no—
144C2×C4×D4—21no—
145C2×C4×D4C22×C311yescharacteristic
146C2×C4×D4C22×C311yescharacteristic
147C2×#117—21no—
148C2×#117C22×C311yescharacteristic
149C22×C4⋊C4—21no—
150C22×C4⋊C4C22×C311yescharacteristic
151#462—24no—
152#461—24no—
153#452—28no—
154C2×C4⋊C8C22×C311yescharacteristic
155C2×C4⋊C8C22×C311yescharacteristic
156C2×#103C22×C312yes—
157C2×#103C22×C312yes—
158C2×#103C22×C312yes—
159C2×#102C22×C312yes—
160C2×C22⋊C8C22×C312yes—
161#398—21no—
162#398—21no—
163#397—21no—
164#397—21no—
165#391C22×C312yes—
166#382C22×C311yescharacteristic
167#340C22×C312yes—
168#340C22×C312yes—
169#336C22×C312yes—
170C24×C3—21no—
171C24×C3C2×D411yescharacteristic
172C3×#41—28no—
173C3×#41—28no—
174C3×#41—28no—
175C3×#41—28no—
176C2×C3×Q8—21no—
177C2×C3×Q8—21no—
178C2×C3×Q8—21no—
179C2×C3×Q8—22no—
180C2×C3×Q8—22no—
181C2×C3×Q8—22no—
182C2×C3×Q8—22no—
183C2×C3×Q8—22no—
184C2×C3×Q8—22no—
185C2×C3×Q8C2×D412yes—
186C2×C3×Q8C2×D412yes—
187C2×C3×Q8#4112yes—
188C2×C3×D4—24no—
189C2×C3×D4—24no—
190C2×C3×D4—24no—
191C2×C3×D4—21no—
192C2×C3×D4—21no—
193C2×C3×D4—21no—
194C2×C3×D4C2×D412yes—
195C2×C3×D4C2×D412yes—
196C2×C3×D4#4112yes—
197C22×C3×C4—24no—
198C22×C3×C4—24no—
199C22×C3×C4—21no—
200C22×C3×C4—24no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—24no—
208C22×C3×C4—24no—
209C22×C3×C4—24no—
210C22×C3×C4—22no—
211C22×C3×C4—22no—
212C22×C3×C4—82no—
213C22×C3×C4—82no—
214C22×C3×C4—82no—
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D412yes—
221C22×C3×C4#4112yes—
222C22×C3×C4C2411yescharacteristic
223C3×Q16—48no—
224C3×SD16—48no—
225C2×C3×C8—22no—
226C2×C3×C8—22no—
227C2×C3×C8—22no—
228C2×C3×C8—22no—
229C2×C3×C8—22no—
230C2×C3×C8—22no—
231C3×C4⋊C4—24no—
232C3×C4⋊C4—24no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—24no—
235C3×C4⋊C4—22no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—22no—
239C3×C4⋊C4—22no—
240C3×C4⋊C4—42no—
241C3×C4⋊C4—42no—
242C3×C4⋊C4—44no—
243C3×C4⋊C4C2×D412yes—
244C3×C4⋊C4C2×D412yes—
245C3×C22⋊C4—24no—
246C3×C22⋊C4—24no—
247C3×C22⋊C4—22no—
248C3×C22⋊C4—22no—
249C3×C22⋊C4—44no—
250C3×C42—24no—
251C3×C42—24no—
252C3×C42—21no—
253C3×C42—21no—
254C3×C42C2×D412yes—
255C3×C42C2×D412yes—
256C2×#41—22no—
257C2×#41—22no—
258C2×#41—22no—
259C2×#41—22no—
260C2×#41—22no—
261C2×#41C3×D414yes—
262C22×Q8C23×C312yes—
263C22×Q8C23×C311yescharacteristic
264C22×D4—21no—
265C22×D4C23×C311yescharacteristic
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4—21no—
270C23×C4C23×C311yescharacteristic
271C2×Q16—24no—
272C2×Q16—24no—
273C2×Q16—24no—
274C2×SD16—24no—
275C2×SD16—24no—
276C2×SD16—24no—
277C22×C8C23×C312yes—
278C22×C8C23×C312yes—
279C4⋊Q8—22no—
280C4⋊Q8—22no—
281C22⋊Q8—24no—
282C22⋊Q8—24no—
283C4×Q8—22no—
284C4×Q8—22no—
285C4×Q8—22no—
286C4×Q8C3×D414yes—
287C4×D4—28no—
288C4×D4—28no—
289C4×D4—22no—
290C4×D4—22no—
291C4×D4—22no—
292C4×D4C3×D414yes—
293#117—28no—
294#117—22no—
295#117—22no—
296C2×C4⋊C4—28no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4C3×D411yescharacteristic
306C2×C4⋊C4C3×D411yescharacteristic
307C2×C4⋊C4C23×C312yes—
308C2×C4⋊C4C23×C311yescharacteristic
309C2×C4⋊C4C23×C311yescharacteristic
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4C3×D411yescharacteristic
314C2×C22⋊C4C3×D411yescharacteristic
315C2×C42—21no—
316C2×C42—21no—
317C2×C42C23×C311yescharacteristic
318C2×C42C23×C311yescharacteristic
319C4⋊C8—22no—
320C4⋊C8—22no—
321#103—24no—
322#103—24no—
323#103—24no—
324#102—24no—
325C22⋊C8—24no—
326#95—21no—
327#95—21no—
328#95—22no—
329#95—22no—
330C23×C3—21no—
331C23×C3—22no—
332C23×C3—22no—
333C23×C3—21no—
334C23×C3—22no—
335C23×C3—22no—
336C23×C3—22no—
337C23×C3—42no—
338C23×C3—42no—
339C23×C3#12112yes—
340C23×C3C22×D411yescharacteristic
341C3×Q8—22no—
342C3×Q8—22no—
343C3×Q8—22no—
344C3×Q8—42no—
345C3×Q8—42no—
346C3×Q8—42no—
347C3×Q8—42no—
348C3×Q8#12114yes—
349C3×D4—216no—
350C3×D4—22no—
351C3×D4—22no—
352C3×D4—22no—
353C3×D4#12114yes—
354C2×C3×C4—216no—
355C2×C3×C4—22no—
356C2×C3×C4—21no—
357C2×C3×C4—22no—
358C2×C3×C4—21no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—22no—
372C2×C3×C4—42no—
373C2×C3×C4—42no—
374C2×C3×C4—42no—
375C2×C3×C4—42no—
376C2×C3×C4—42no—
377C2×C3×C4—42no—
378C2×C3×C4—42no—
379C2×C3×C4—42no—
380C2×C3×C4—84no—
381C2×C3×C4—84no—
382C2×C3×C4—84no—
383C2×C3×C4#12012yes—
384C2×C3×C4#12012yes—
385C2×C3×C4#12112yes—
386C2×C3×C4C22×D411yescharacteristic
387C2×C3×C4C22×D411yescharacteristic
388C2×C3×C4C22×D411yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392C3×C8—42no—
393C3×C8—42no—
394C24—21no—
395C24C2×C3×D411yescharacteristic
396#41—28no—
397#41—28no—
398#41—28no—
399#41—28no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8C2×C3×D412yes—
410C2×Q8C2×C3×D412yes—
411C2×Q8C3×#4112yes—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4C2×C3×D412yes—
419C2×D4C2×C3×D412yes—
420C2×D4C3×#4112yes—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—82no—
437C22×C4—82no—
438C22×C4—82no—
439C22×C4C2×C3×D411yescharacteristic
440C22×C4C2×C3×D411yescharacteristic
441C22×C4C2×C3×D411yescharacteristic
442C22×C4C2×C3×D411yescharacteristic
443C22×C4C2×C3×D411yescharacteristic
444C22×C4C2×C3×D412yes—
445C22×C4C3×#4112yes—
446C22×C4C24×C311yescharacteristic
447Q16—48no—
448SD16—48no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4C2×C3×D412yes—
468C4⋊C4C2×C3×D412yes—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—44no—
474C42—24no—
475C42—24no—
476C42—21no—
477C42—21no—
478C42C2×C3×D412yes—
479C42C2×C3×D412yes—
480C22×C3—22no—
481C22×C3—24no—
482C22×C3—24no—
483C22×C3—42no—
484C22×C3—44no—
485C22×C3—41no—
486C22×C3—42no—
487C22×C3—41no—
488C22×C3—42no—
489C22×C3—44no—
490C22×C3C2×#12011yescharacteristic
491C22×C3C2×#12111yescharacteristic
492C22×C3#52511yescharacteristic
493C22×C3#57712yes—
494C22×C3#57812yes—
495C3×C4—22no—
496C3×C4—24no—
497C3×C4—22no—
498C3×C4—22no—
499C3×C4—41no—
500C3×C4—42no—
501C3×C4—42no—
502C3×C4—41no—
503C3×C4—82no—
504C3×C4—82no—
505C3×C4C2×#12111yescharacteristic
506C3×C4C2×#12111yescharacteristic
507C3×C4#52511yescharacteristic
508C3×C4#52511yescharacteristic
509C23—22no—
510C23—22no—
511C23—21no—
512C23—22no—
513C23—22no—
514C23—21no—
515C23—22no—
516C23—42no—
517C23—42no—
518C23C3×#12112yes—
519C23C22×C3×D411yescharacteristic
520Q8—22no—
521Q8—22no—
522Q8—22no—
523Q8—42no—
524Q8—42no—
525Q8—42no—
526Q8—42no—
527Q8C3×#12114yes—
528D4—216no—
529D4—22no—
530D4—22no—
531D4—22no—
532D4C3×#12114yes—
533C2×C4—216no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—84no—
560C2×C4—84no—
561C2×C4—84no—
562C2×C4C3×#12012yes—
563C2×C4C3×#12012yes—
564C2×C4C3×#12112yes—
565C2×C4C22×C3×D411yescharacteristic
566C2×C4C22×C3×D411yescharacteristic
567C2×C4C22×C3×D411yescharacteristic
568C2×C4C2×C3×#4111yescharacteristic
569C2×C4C2×C3×#4111yescharacteristic
570C2×C4C2×C3×#4111yescharacteristic
571C8—42no—
572C8—42no—
573C2×C3—42no—
574C2×C3—44no—
575C2×C3#284711yescharacteristic
576C2×C3#300911yescharacteristic
577C2×C3#301111yescharacteristic
578C2×C3#304612yes—
579C2×C3#312012yes—
580C22—24no—
581C22—22no—
582C22—24no—
583C22—41no—
584C22—42no—
585C22—41no—
586C22—42no—
587C22—44no—
588C22—42no—
589C22—44no—
590C22C2×C3×#12011yescharacteristic
591C22C2×C3×#12111yescharacteristic
592C22C3×#52511yescharacteristic
593C22C3×#57712yes—
594C22C3×#57812yes—
595C4—22no—
596C4—22no—
597C4—22no—
598C4—24no—
599C4—41no—
600C4—42no—
601C4—42no—
602C4—41no—
603C4—82no—
604C4—82no—
605C4C2×C3×#12111yescharacteristic
606C4C2×C3×#12111yescharacteristic
607C4C3×#52511yescharacteristic
608C4C3×#52511yescharacteristic
609C3#3291611yescharacteristic, direct product factor, Sylow
610C2—42no—
611C2—44no—
612C2C3×#284711yescharacteristic
613C2C3×#300911yescharacteristic
614C2C3×#301111yescharacteristic
615C2C3×#304612yes—
616C2C3×#312012yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×#41C22
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×#462C2semidirect
3. C2×C3×#461C2semidirect
5. C3×#2621C2semidirect
6. C3×#1878C2semidirect
7. C3×#1877C2semidirect
8. C3×#1872C2semidirect
9. C3×#1805C2semidirect
10. C3×#1804C2semidirect
11. C3×#1798C2semidirect
12. #32916C3direct (N × H)
50. C2×#462C2×C3semidirect
51. C2×#461C2×C3semidirect
53. #2621C2×C3semidirect
54. #1878C2×C3semidirect
55. #1877C2×C3semidirect
56. #1872C2×C3semidirect
57. #1805C2×C3semidirect
58. #1804C2×C3semidirect
59. #1798C2×C3semidirect
609. C3#32916direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5202
C2×C3×#4622
C2×C3×#4612
C2×C3×#4522
C3×#26212
C3×#18782
C3×#18772
C3×#18722
C3×#18052
C3×#18042
C3×#17982
#329163