Finite Group Databasebeta

← C3×#32965 #10651572

Subgroups — C3×#32965 #10651572

Order 768.

1282 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#520C211yesmaximal, characteristic
2C2×C3×#475C211yesmaximal, characteristic, complemented
3C2×C3×#476C211yesmaximal, characteristic, complemented
4C3×#2628C211yesmaximal, characteristic, complemented
5C3×#1865C212yesmaximal, complemented
6C3×#1864C212yesmaximal, complemented
7C3×#1853C212yesmaximal, complemented
8C3×#1805C211yesmaximal, characteristic, complemented
9C3×#1804C211yesmaximal, characteristic, complemented
10C3×#1798C211yesmaximal, characteristic, complemented
11C3×#1788C212yesmaximal
12#32965C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C2×C3×C4⋊Q8C2211yescharacteristic
15C3×#520—24no—
16C3×#520—24no—
17C2×C3×C4×Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×D4—21no—
20C2×C3×C4×D4—21no—
21C2×C3×C4×D4C2211yescharacteristic
22C2×C3×C4×D4C2211yescharacteristic
23C2×C3×#117—21no—
24C2×C3×#117C2211yescharacteristic
25C22×C3×C4⋊C4—21no—
26C22×C3×C4⋊C4C2211yescharacteristic
27C3×#476—24no—
28C3×#475—24no—
29C2×C3×#106C2212yes—
30C2×C3×#106C2212yes—
31C2×C3×C4⋊C8C2211yescharacteristic
32C2×C3×C4⋊C8C2211yescharacteristic
33C2×C3×#103C2212yes—
34C2×C3×#103C2212yes—
35C2×C3×#102C2212yes—
36C2×C3×#102C2212yes—
37C2×C3×C22⋊C8C2212yes—
38C3×#400—21no—
39C3×#399—21no—
40C3×#395—21no—
41C3×#393—21no—
42C3×#389C2212yes—
43C3×#382C2211yescharacteristic
44C3×#380C2212yes—
45C3×#340C2212yes—
46C3×#340C2212yes—
47C3×#336C2212yes—
48C2×#520C2×C311yescharacteristic
49C2×#475C2×C311yescharacteristic, complemented
50C2×#476C2×C311yescharacteristic, complemented
51#2628C2×C311yescharacteristic, complemented
52#1865C2×C312yescomplemented
53#1864C2×C312yescomplemented
54#1853C2×C312yescomplemented
55#1805C2×C311yescharacteristic, complemented
56#1804C2×C311yescharacteristic, complemented
57#1798C2×C311yescharacteristic, complemented
58#1788C2×C312yes—
59C2×C3×#41—22no—
60C2×C3×#41—22no—
61C2×C3×#41—22no—
62C2×C3×#41—22no—
63C2×C3×#41—22no—
64C2×C3×#41Q814yescomplemented
65C22×C3×Q8—21no—
66C22×C3×Q8C2311yescharacteristic
67C22×C3×D4—21no—
68C22×C3×D4C2311yescharacteristic
69C23×C3×C4—21no—
70C23×C3×C4—21no—
71C23×C3×C4—21no—
72C23×C3×C4—21no—
73C23×C3×C4C2311yescharacteristic
74C22×C3×C8C2312yes—
75C22×C3×C8C2312yes—
76C3×C4⋊Q8—22no—
77C3×C4⋊Q8—22no—
78C3×C4×Q8—22no—
79C3×C4×Q8—22no—
80C3×C4×Q8—22no—
81C3×C4×Q8D414yes—
82C3×C4×D4—28no—
83C3×C4×D4—28no—
84C3×C4×D4—22no—
85C3×C4×D4—22no—
86C3×C4×D4—22no—
87C3×C4×D4D414yes—
88C3×#117—28no—
89C3×#117—22no—
90C3×#117—22no—
91C2×C3×C4⋊C4—28no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—22no—
96C2×C3×C4⋊C4—22no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—22no—
99C2×C3×C4⋊C4D411yescharacteristic
100C2×C3×C4⋊C4D411yescharacteristic
101C2×C3×C4⋊C4C2312yes—
102C2×C3×C4⋊C4C2312yes—
103C2×C3×C4⋊C4C2311yescharacteristic
104C2×C3×C4⋊C4C2311yescharacteristic
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C22⋊C4—22no—
108C2×C3×C22⋊C4D411yescharacteristic
109C2×C3×C22⋊C4D411yescharacteristic
110C2×C3×C42—21no—
111C2×C3×C42—21no—
112C2×C3×C42C2311yescharacteristic
113C2×C3×C42C2311yescharacteristic
114C3×#106—24no—
115C3×#106—24no—
116C3×C4⋊C8—22no—
117C3×C4⋊C8—22no—
118C3×#103—24no—
119C3×#103—24no—
120C3×#102—24no—
121C3×#102—24no—
122C3×C22⋊C8—24no—
123C3×#95—22no—
124C3×#95—21no—
125C3×#95—21no—
126C3×#95—22no—
127C22×#41C22×C311yescharacteristic
128C2×C4⋊Q8C22×C311yescharacteristic
129#520—24no—
130#520—24no—
131C2×C4×Q8C22×C311yescharacteristic
132C2×C4×Q8C22×C311yescharacteristic
133C2×C4×D4—21no—
134C2×C4×D4—21no—
135C2×C4×D4C22×C311yescharacteristic
136C2×C4×D4C22×C311yescharacteristic
137C2×#117—21no—
138C2×#117C22×C311yescharacteristic
139C22×C4⋊C4—21no—
140C22×C4⋊C4C22×C311yescharacteristic
141#476—24no—
142#475—24no—
143C2×#106C22×C312yes—
144C2×#106C22×C312yes—
145C2×C4⋊C8C22×C311yescharacteristic
146C2×C4⋊C8C22×C311yescharacteristic
147C2×#103C22×C312yes—
148C2×#103C22×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×C312yes—
157#382C22×C311yescharacteristic
158#380C22×C312yes—
159#340C22×C312yes—
160#340C22×C312yes—
161#336C22×C312yes—
162C24×C3—21no—
163C24×C3C2×D411yescharacteristic
164C3×#41—28no—
165C3×#41—28no—
166C3×#41—28no—
167C3×#41—28no—
168C2×C3×Q8—21no—
169C2×C3×Q8—21no—
170C2×C3×Q8—21no—
171C2×C3×Q8—42no—
172C2×C3×Q8—82no—
173C2×C3×Q8C2×D412yes—
174C2×C3×Q8C2×Q812yes—
175C2×C3×Q8#4112yes—
176C2×C3×D4—24no—
177C2×C3×D4—24no—
178C2×C3×D4—24no—
179C2×C3×D4—21no—
180C2×C3×D4—21no—
181C2×C3×D4—21no—
182C2×C3×D4C2×D412yes—
183C2×C3×D4C2×Q812yes—
184C2×C3×D4#4112yes—
185C22×C3×C4—24no—
186C22×C3×C4—24no—
187C22×C3×C4—21no—
188C22×C3×C4—24no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—24no—
196C22×C3×C4—24no—
197C22×C3×C4—24no—
198C22×C3×C4—22no—
199C22×C3×C4—22no—
200C22×C3×C4—82no—
201C22×C3×C4—82no—
202C22×C3×C4—82no—
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×Q812yes—
209C22×C3×C4#4112yes—
210C22×C3×C4C2411yescharacteristic
211C2×C3×C8—22no—
212C2×C3×C8—22no—
213C2×C3×C8—22no—
214C2×C3×C8—22no—
215C2×C3×C8—22no—
216C2×C3×C8—22no—
217C3×C4⋊C4—24no—
218C3×C4⋊C4—24no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—24no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—22no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4—22no—
227C3×C4⋊C4—22no—
228C3×C4⋊C4—42no—
229C3×C4⋊C4—82no—
230C3×C4⋊C4C2×D412yes—
231C3×C4⋊C4C2×D412yes—
232C3×C22⋊C4—24no—
233C3×C22⋊C4—24no—
234C3×C22⋊C4—22no—
235C3×C22⋊C4—22no—
236C3×C22⋊C4—82no—
237C3×C42—24no—
238C3×C42—24no—
239C3×C42—21no—
240C3×C42—21no—
241C3×C42C2×D412yes—
242C3×C42C2×D412yes—
243C2×#41—22no—
244C2×#41—22no—
245C2×#41—22no—
246C2×#41—22no—
247C2×#41—22no—
248C2×#41C3×Q814yescomplemented
249C22×Q8—21no—
250C22×Q8C23×C311yescharacteristic
251C22×D4—21no—
252C22×D4C23×C311yescharacteristic
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4—21no—
257C23×C4C23×C311yescharacteristic
258C22×C8C23×C312yes—
259C22×C8C23×C312yes—
260C4⋊Q8—22no—
261C4⋊Q8—22no—
262C4×Q8—22no—
263C4×Q8—22no—
264C4×Q8—22no—
265C4×Q8C3×D414yes—
266C4×D4—28no—
267C4×D4—28no—
268C4×D4—22no—
269C4×D4—22no—
270C4×D4—22no—
271C4×D4C3×D414yes—
272#117—28no—
273#117—22no—
274#117—22no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4C3×D411yescharacteristic
284C2×C4⋊C4C3×D411yescharacteristic
285C2×C4⋊C4C23×C312yes—
286C2×C4⋊C4C23×C312yes—
287C2×C4⋊C4C23×C311yescharacteristic
288C2×C4⋊C4C23×C311yescharacteristic
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4C3×D411yescharacteristic
293C2×C22⋊C4C3×D411yescharacteristic
294C2×C42—21no—
295C2×C42—21no—
296C2×C42C23×C311yescharacteristic
297C2×C42C23×C311yescharacteristic
298#106—24no—
299#106—24no—
300C4⋊C8—22no—
301C4⋊C8—22no—
302#103—24no—
303#103—24no—
304#102—24no—
305#102—24no—
306C22⋊C8—24no—
307#95—22no—
308#95—21no—
309#95—21no—
310#95—22no—
311C23×C3—21no—
312C23×C3—22no—
313C23×C3—22no—
314C23×C3—21no—
315C23×C3—22no—
316C23×C3—22no—
317C23×C3—22no—
318C23×C3—42no—
319C23×C3—42no—
320C23×C3C22⋊Q812yes—
321C23×C3C22×D411yescharacteristic
322C3×Q8—22no—
323C3×Q8—22no—
324C3×Q8—22no—
325C3×Q8—84no—
326C3×Q8C22⋊Q814yes—
327C3×D4—216no—
328C3×D4—22no—
329C3×D4—22no—
330C3×D4—22no—
331C3×D4C22⋊Q814yes—
332C2×C3×C4—216no—
333C2×C3×C4—22no—
334C2×C3×C4—21no—
335C2×C3×C4—22no—
336C2×C3×C4—21no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
357C2×C3×C4—42no—
358C2×C3×C4—84no—
359C2×C3×C4—84no—
360C2×C3×C4—84no—
361C2×C3×C4#12012yes—
362C2×C3×C4#12012yes—
363C2×C3×C4C22⋊Q812yes—
364C2×C3×C4C22×D411yescharacteristic
365C2×C3×C4C22×D411yescharacteristic
366C2×C3×C4C22×Q811yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368C2×C3×C4C2×#4111yescharacteristic
369C2×C3×C4C2×#4111yescharacteristic
370C3×C8—42no—
371C3×C8—42no—
372C24—21no—
373C24C2×C3×D411yescharacteristic
374#41—28no—
375#41—28no—
376#41—28no—
377#41—28no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—42no—
382C2×Q8—82no—
383C2×Q8C2×C3×D412yes—
384C2×Q8C2×C3×Q812yes—
385C2×Q8C3×#4112yes—
386C2×D4—24no—
387C2×D4—24no—
388C2×D4—24no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4C2×C3×D412yes—
393C2×D4C2×C3×Q812yes—
394C2×D4C3×#4112yes—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—21no—
398C22×C4—24no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—82no—
411C22×C4—82no—
412C22×C4—82no—
413C22×C4C2×C3×D411yescharacteristic
414C22×C4C2×C3×D411yescharacteristic
415C22×C4C2×C3×D411yescharacteristic
416C22×C4C2×C3×D411yescharacteristic
417C22×C4C2×C3×D411yescharacteristic
418C22×C4C2×C3×Q812yes—
419C22×C4C3×#4112yes—
420C22×C4C24×C311yescharacteristic
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—22no—
430C4⋊C4—24no—
431C4⋊C4—22no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—42no—
439C4⋊C4—82no—
440C4⋊C4C2×C3×D412yes—
441C4⋊C4C2×C3×D412yes—
442C22⋊C4—24no—
443C22⋊C4—24no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—82no—
447C42—24no—
448C42—24no—
449C42—21no—
450C42—21no—
451C42C2×C3×D412yes—
452C42C2×C3×D412yes—
453C22×C3—24no—
454C22×C3—22no—
455C22×C3—24no—
456C22×C3—44no—
457C22×C3—42no—
458C22×C3—44no—
459C22×C3—41no—
460C22×C3—42no—
461C22×C3—41no—
462C22×C3—42no—
463C22×C3C2×#12011yescharacteristic
464C22×C3C2×C22⋊Q811yescharacteristic
465C22×C3#52511yescharacteristic
466C22×C3#57712yes—
467C22×C3#57712yes—
468C3×C4—22no—
469C3×C4—24no—
470C3×C4—22no—
471C3×C4—22no—
472C3×C4—41no—
473C3×C4—42no—
474C3×C4—42no—
475C3×C4—41no—
476C3×C4—82no—
477C3×C4—82no—
478C3×C4C2×C22⋊Q811yescharacteristic
479C3×C4C2×C22⋊Q811yescharacteristic
480C3×C4#52511yescharacteristic
481C3×C4#52511yescharacteristic
482C23—22no—
483C23—22no—
484C23—21no—
485C23—22no—
486C23—22no—
487C23—21no—
488C23—22no—
489C23—42no—
490C23—42no—
491C23C3×C22⋊Q812yes—
492C23C22×C3×D411yescharacteristic
493Q8—22no—
494Q8—22no—
495Q8—22no—
496Q8—84no—
497Q8C3×C22⋊Q814yes—
498D4—216no—
499D4—22no—
500D4—22no—
501D4—22no—
502D4C3×C22⋊Q814yes—
503C2×C4—216no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—84no—
530C2×C4—84no—
531C2×C4—84no—
532C2×C4C3×#12012yes—
533C2×C4C3×#12012yes—
534C2×C4C3×C22⋊Q812yes—
535C2×C4C22×C3×D411yescharacteristic
536C2×C4C22×C3×D411yescharacteristic
537C2×C4C22×C3×Q811yescharacteristic
538C2×C4C2×C3×#4111yescharacteristic
539C2×C4C2×C3×#4111yescharacteristic
540C2×C4C2×C3×#4111yescharacteristic
541C8—42no—
542C8—42no—
543C2×C3—42no—
544C2×C3—44no—
545C2×C3#284811yescharacteristic
546C2×C3#300911yescharacteristic
547C2×C3#301011yescharacteristic
548C2×C3#308412yes—
549C2×C3#312512yes—
550C22—24no—
551C22—22no—
552C22—24no—
553C22—41no—
554C22—42no—
555C22—41no—
556C22—42no—
557C22—44no—
558C22—42no—
559C22—44no—
560C22C2×C3×#12011yescharacteristic
561C22C2×C3×C22⋊Q811yescharacteristic
562C22C3×#52511yescharacteristic
563C22C3×#57712yes—
564C22C3×#57712yes—
565C4—22no—
566C4—22no—
567C4—22no—
568C4—24no—
569C4—41no—
570C4—42no—
571C4—42no—
572C4—41no—
573C4—82no—
574C4—82no—
575C4C2×C3×C22⋊Q811yescharacteristic
576C4C2×C3×C22⋊Q811yescharacteristic
577C4C3×#52511yescharacteristic
578C4C3×#52511yescharacteristic
579C3#3296511yescharacteristic, direct product factor, Sylow
580C2—42no—
581C2—44no—
582C2C3×#284811yescharacteristic
583C2C3×#300911yescharacteristic
584C2C3×#301011yescharacteristic
585C2C3×#308412yes—
586C2C3×#312512yes—
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×#475C2semidirect
3. C2×C3×#476C2semidirect
4. C3×#2628C2semidirect
5. C3×#1865C2semidirect
6. C3×#1864C2semidirect
7. C3×#1853C2semidirect
8. C3×#1805C2semidirect
9. C3×#1804C2semidirect
10. C3×#1798C2semidirect
12. #32965C3direct (N × H)
49. C2×#475C2×C3semidirect
50. C2×#476C2×C3semidirect
51. #2628C2×C3semidirect
52. #1865C2×C3semidirect
53. #1864C2×C3semidirect
54. #1853C2×C3semidirect
55. #1805C2×C3semidirect
56. #1804C2×C3semidirect
57. #1798C2×C3semidirect
64. C2×C3×#41Q8semidirect
248. C2×#41C3×Q8semidirect
579. C3#32965direct (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×#4752
C2×C3×#4762
C3×#26282
C3×#18652
C3×#18642
C3×#18532
C3×#18052
C3×#18042
C3×#17982
C3×#17882
#329653