Finite Group Databasebeta

← C3×#29669 #10648276

Subgroups — C3×#29669 #10648276

Order 768.

1196 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#542C211yesmaximal, characteristic, complemented
2C2×C3×#460C211yesmaximal, characteristic, complemented
3C2×C3×#459C211yesmaximal, characteristic, complemented
4C3×#2619C211yesmaximal, characteristic, complemented
5C3×#2023C212yesmaximal, complemented
6C3×#2012C212yesmaximal, complemented
7C3×#2000C212yesmaximal, complemented
8C3×#1999C212yesmaximal, complemented
9C3×#1805C211yesmaximal, characteristic, complemented
10C3×#1804C211yesmaximal, characteristic, complemented
11C3×#1798C211yesmaximal, characteristic, complemented
12#29669C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2212yes—
14C22×C3×D8C2212yes—
15C3×#542—24no—
16C3×#542—24no—
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#127C2211yescharacteristic
19C2×C3×#126—21no—
20C2×C3×#124—21no—
21C2×C3×#124C2211yescharacteristic
22C2×C3×#123—21no—
23C2×C3×C22⋊Q8—21no—
24C2×C3×C22⋊Q8C2211yescharacteristic
25C2×C3×#121C2211yescharacteristic
26C2×C3×#121C2212yes—
27C2×C3×C4×Q8C2211yescharacteristic
28C2×C3×C4×D4C2211yescharacteristic
29C2×C3×#117C2211yescharacteristic
30C3×#460—24no—
31C3×#459—24no—
32C2×C3×C4⋊C8C2211yescharacteristic
33C2×C3×C4⋊C8C2211yescharacteristic
34C2×C3×#103C2212yes—
35C2×C3×#102C2212yes—
36C2×C3×#102C2212yes—
37C2×C3×#102C2212yes—
38C2×C3×C22⋊C8C2212yes—
39C3×#394—21no—
40C3×#394—21no—
41C3×#392—21no—
42C3×#392—21no—
43C3×#390C2212yes—
44C3×#382C2211yescharacteristic
45C3×#340C2212yes—
46C3×#340C2212yes—
47C3×#336C2212yes—
48C2×#542C2×C311yescharacteristic, complemented
49C2×#460C2×C311yescharacteristic, complemented
50C2×#459C2×C311yescharacteristic, complemented
51#2619C2×C311yescharacteristic, complemented
52#2023C2×C312yescomplemented
53#2012C2×C312yescomplemented
54#2000C2×C312yescomplemented
55#1999C2×C312yescomplemented
56#1805C2×C311yescharacteristic, complemented
57#1804C2×C311yescharacteristic, complemented
58#1798C2×C311yescharacteristic, complemented
59C22×C3×Q8—21no—
60C22×C3×Q8C2311yescharacteristic
61C22×C3×D4—21no—
62C22×C3×D4—22no—
63C22×C3×D4—21no—
64C22×C3×D4—21no—
65C22×C3×D4C2312yes—
66C22×C3×D4C2311yescharacteristic
67C23×C3×C4—21no—
68C23×C3×C4C2311yescharacteristic
69C2×C3×SD16—24no—
70C2×C3×SD16—42no—
71C2×C3×SD16—42no—
72C2×C3×D8—24no—
73C2×C3×D8—42no—
74C2×C3×D8—42no—
75C22×C3×C8C2312yes—
76C22×C3×C8C2312yes—
77C3×C4⋊Q8—22no—
78C3×C4⋊Q8—22no—
79C3×#127—22no—
80C3×#127—22no—
81C3×#126—28no—
82C3×#124—28no—
83C3×#124—22no—
84C3×#124—22no—
85C3×#123—28no—
86C3×C22⋊Q8—28no—
87C3×C22⋊Q8—22no—
88C3×C22⋊Q8—22no—
89C3×#121—22no—
90C3×#121—22no—
91C3×#121—42no—
92C3×#121—42no—
93C3×C4×Q8—22no—
94C3×C4×Q8D414yescomplemented
95C3×C4×D4—22no—
96C3×C4×D4D414yescomplemented
97C3×#117—22no—
98C3×#117—22no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4D411yescharacteristic
103C2×C3×C4⋊C4D411yescharacteristic
104C2×C3×C4⋊C4C2312yes—
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C22⋊C4—21no—
108C2×C3×C22⋊C4—21no—
109C2×C3×C22⋊C4—21no—
110C2×C3×C22⋊C4—21no—
111C2×C3×C22⋊C4—21no—
112C2×C3×C22⋊C4—22no—
113C2×C3×C22⋊C4—22no—
114C2×C3×C22⋊C4—22no—
115C2×C3×C22⋊C4D411yescharacteristic
116C2×C3×C22⋊C4D411yescharacteristic
117C2×C3×C42—21no—
118C2×C3×C42C2311yescharacteristic
119C2×C3×C42C2311yescharacteristic
120C3×C4⋊C8—22no—
121C3×C4⋊C8—22no—
122C3×#103—42no—
123C3×#102—24no—
124C3×#102—24no—
125C3×#102—42no—
126C3×C22⋊C8—42no—
127C3×#95—21no—
128C3×#95—21no—
129C22×SD16C22×C312yes—
130C22×D8C22×C312yes—
131#542—24no—
132#542—24no—
133C2×C4⋊Q8C22×C311yescharacteristic
134C2×#127C22×C311yescharacteristic
135C2×#126—21no—
136C2×#124—21no—
137C2×#124C22×C311yescharacteristic
138C2×#123—21no—
139C2×C22⋊Q8—21no—
140C2×C22⋊Q8C22×C311yescharacteristic
141C2×#121C22×C311yescharacteristic
142C2×#121C22×C312yes—
143C2×C4×Q8C22×C311yescharacteristic
144C2×C4×D4C22×C311yescharacteristic
145C2×#117C22×C311yescharacteristic
146#460—24no—
147#459—24no—
148C2×C4⋊C8C22×C311yescharacteristic
149C2×C4⋊C8C22×C311yescharacteristic
150C2×#103C22×C312yes—
151C2×#102C22×C312yes—
152C2×#102C22×C312yes—
153C2×#102C22×C312yes—
154C2×C22⋊C8C22×C312yes—
155#394—21no—
156#394—21no—
157#392—21no—
158#392—21no—
159#390C22×C312yes—
160#382C22×C311yescharacteristic
161#340C22×C312yes—
162#340C22×C312yes—
163#336C22×C312yes—
164C24×C3—21no—
165C24×C3—22no—
166C24×C3C2×D411yescharacteristic
167C2×C3×Q8—24no—
168C2×C3×Q8—21no—
169C2×C3×Q8—21no—
170C2×C3×Q8—21no—
171C2×C3×Q8—21no—
172C2×C3×Q8—21no—
173C2×C3×Q8—82no—
174C2×C3×Q8C2×D412yes—
175C2×C3×D4—24no—
176C2×C3×D4—21no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—21no—
180C2×C3×D4—21no—
181C2×C3×D4—21no—
182C2×C3×D4—22no—
183C2×C3×D4—22no—
184C2×C3×D4—22no—
185C2×C3×D4—21no—
186C2×C3×D4—22no—
187C2×C3×D4—44no—
188C2×C3×D4—42no—
189C2×C3×D4—42no—
190C2×C3×D4—82no—
191C2×C3×D4—82no—
192C2×C3×D4—82no—
193C2×C3×D4—82no—
194C2×C3×D4—82no—
195C2×C3×D4C2×D412yes—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—24no—
206C22×C3×C4—22no—
207C22×C3×C4—82no—
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2411yescharacteristic
214C3×SD16—48no—
215C3×D8—48no—
216C2×C3×C8—22no—
217C2×C3×C8—22no—
218C2×C3×C8—22no—
219C2×C3×C8—22no—
220C2×C3×C8—22no—
221C2×C3×C8—22no—
222C3×C4⋊C4—24no—
223C3×C4⋊C4—24no—
224C3×C4⋊C4—24no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4—22no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4—22no—
231C3×C4⋊C4C2×D412yes—
232C3×C4⋊C4C2×D412yes—
233C3×C22⋊C4—24no—
234C3×C22⋊C4—24no—
235C3×C22⋊C4—24no—
236C3×C22⋊C4—24no—
237C3×C22⋊C4—24no—
238C3×C22⋊C4—22no—
239C3×C22⋊C4—22no—
240C3×C22⋊C4—44no—
241C3×C22⋊C4—82no—
242C3×C22⋊C4—82no—
243C3×C42—24no—
244C3×C42—21no—
245C3×C42—21no—
246C3×C42C2×D412yes—
247C3×C42C2×D412yes—
248C22×Q8—21no—
249C22×Q8C23×C311yescharacteristic
250C22×D4—21no—
251C22×D4—22no—
252C22×D4—21no—
253C22×D4—21no—
254C22×D4C23×C312yes—
255C22×D4C23×C311yescharacteristic
256C23×C4—21no—
257C23×C4C23×C311yescharacteristic
258C2×SD16—24no—
259C2×SD16—42no—
260C2×SD16—42no—
261C2×D8—24no—
262C2×D8—42no—
263C2×D8—42no—
264C22×C8C23×C312yes—
265C22×C8C23×C312yes—
266C4⋊Q8—22no—
267C4⋊Q8—22no—
268#127—22no—
269#127—22no—
270#126—28no—
271#124—28no—
272#124—22no—
273#124—22no—
274#123—28no—
275C22⋊Q8—28no—
276C22⋊Q8—22no—
277C22⋊Q8—22no—
278#121—22no—
279#121—22no—
280#121—42no—
281#121—42no—
282C4×Q8—22no—
283C4×Q8C3×D414yescomplemented
284C4×D4—22no—
285C4×D4C3×D414yescomplemented
286#117—22no—
287#117—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4C3×D411yescharacteristic
292C2×C4⋊C4C3×D411yescharacteristic
293C2×C4⋊C4C23×C312yes—
294C2×C4⋊C4C23×C311yescharacteristic
295C2×C4⋊C4C23×C311yescharacteristic
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4C3×D411yescharacteristic
305C2×C22⋊C4C3×D411yescharacteristic
306C2×C42—21no—
307C2×C42C23×C311yescharacteristic
308C2×C42C23×C311yescharacteristic
309C4⋊C8—22no—
310C4⋊C8—22no—
311#103—42no—
312#102—24no—
313#102—24no—
314#102—42no—
315C22⋊C8—42no—
316#95—21no—
317#95—21no—
318C23×C3—21no—
319C23×C3—22no—
320C23×C3—21no—
321C23×C3—21no—
322C23×C3—22no—
323C23×C3—22no—
324C23×C3—41no—
325C23×C3—41no—
326C23×C3—42no—
327C23×C3—42no—
328C23×C3—42no—
329C23×C3—42no—
330C23×C3—42no—
331C23×C3—42no—
332C23×C3—42no—
333C23×C3—42no—
334C23×C3—42no—
335C23×C3C22×D411yescharacteristic
336C3×Q8—22no—
337C3×Q8—22no—
338C3×Q8—41no—
339C3×Q8—41no—
340C3×Q8—84no—
341C3×D4—22no—
342C3×D4—22no—
343C3×D4—41no—
344C3×D4—42no—
345C3×D4—42no—
346C3×D4—41no—
347C3×D4—42no—
348C3×D4—42no—
349C3×D4—84no—
350C3×D4—84no—
351C3×D4—84no—
352C3×D4—88no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—21no—
356C2×C3×C4—22no—
357C2×C3×C4—21no—
358C2×C3×C4—22no—
359C2×C3×C4—21no—
360C2×C3×C4—22no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—42no—
368C2×C3×C4—42no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—42no—
374C2×C3×C4—42no—
375C2×C3×C4—42no—
376C2×C3×C4—84no—
377C2×C3×C4#12012yes—
378C2×C3×C4#12012yes—
379C2×C3×C4C22×D411yescharacteristic
380C2×C3×C4C22×D411yescharacteristic
381C2×C3×C4H(5,𝔽2)11yescharacteristic
382C2×C3×C4H(5,𝔽2)11yescharacteristic
383C2×C3×C4H(5,𝔽2)11yescharacteristic
384C2×C3×C42−1+411yescharacteristic
385C3×C8—42no—
386C3×C8—42no—
387C24—21no—
388C24—22no—
389C24C2×C3×D411yescharacteristic
390C2×Q8—24no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—82no—
397C2×Q8C2×C3×D412yes—
398C2×D4—24no—
399C2×D4—21no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—21no—
409C2×D4—22no—
410C2×D4—44no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—82no—
414C2×D4—82no—
415C2×D4—82no—
416C2×D4—82no—
417C2×D4—82no—
418C2×D4C2×C3×D412yes—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—82no—
431C22×C4C2×C3×D411yescharacteristic
432C22×C4C2×C3×D411yescharacteristic
433C22×C4C2×C3×D411yescharacteristic
434C22×C4C2×C3×D411yescharacteristic
435C22×C4C2×C3×D411yescharacteristic
436C22×C4C24×C311yescharacteristic
437SD16—48no—
438D8—48no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4C2×C3×D412yes—
455C4⋊C4C2×C3×D412yes—
456C22⋊C4—24no—
457C22⋊C4—24no—
458C22⋊C4—24no—
459C22⋊C4—24no—
460C22⋊C4—24no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—44no—
464C22⋊C4—82no—
465C22⋊C4—82no—
466C42—24no—
467C42—21no—
468C42—21no—
469C42C2×C3×D412yes—
470C42C2×C3×D412yes—
471C22×C3—41no—
472C22×C3—42no—
473C22×C3—42no—
474C22×C3—42no—
475C22×C3—42no—
476C22×C3—41no—
477C22×C3—41no—
478C22×C3—42no—
479C22×C3—82no—
480C22×C3—82no—
481C22×C3—82no—
482C22×C3—82no—
483C22×C3—82no—
484C22×C3—82no—
485C22×C3—82no—
486C22×C3—82no—
487C22×C3—82no—
488C22×C3C2×#12011yescharacteristic
489C22×C3#53511yescharacteristic
490C22×C3#53711yescharacteristic
491C22×C3#57612yes—
492C22×C3#57712yes—
493C3×C4—22no—
494C3×C4—22no—
495C3×C4—41no—
496C3×C4—42no—
497C3×C4—41no—
498C3×C4—42no—
499C3×C4—42no—
500C3×C4—41no—
501C3×C4—82no—
502C3×C4#53511yescharacteristic
503C3×C4#53511yescharacteristic
504C3×C4#53711yescharacteristic
505C3×C4#53711yescharacteristic
506C23—22no—
507C23—22no—
508C23—21no—
509C23—22no—
510C23—21no—
511C23—21no—
512C23—42no—
513C23—42no—
514C23—41no—
515C23—41no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23C22×C3×D411yescharacteristic
524Q8—22no—
525Q8—22no—
526Q8—41no—
527Q8—41no—
528Q8—84no—
529D4—22no—
530D4—22no—
531D4—41no—
532D4—42no—
533D4—42no—
534D4—41no—
535D4—42no—
536D4—42no—
537D4—84no—
538D4—84no—
539D4—84no—
540D4—88no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—84no—
565C2×C4C3×#12012yes—
566C2×C4C3×#12012yes—
567C2×C4C22×C3×D411yescharacteristic
568C2×C4C22×C3×D411yescharacteristic
569C2×C4C3×H(5,𝔽2)11yescharacteristic
570C2×C4C3×H(5,𝔽2)11yescharacteristic
571C2×C4C3×H(5,𝔽2)11yescharacteristic
572C2×C4C3×2−1+411yescharacteristic
573C8—42no—
574C8—42no—
575C2×C3—81no—
576C2×C3—82no—
577C2×C3—162no—
578C2×C3#261511yescharacteristic
579C2×C3#300811yescharacteristic
580C2×C3#301011yescharacteristic
581C2×C3#322412yes—
582C2×C3#325712yes—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—41no—
587C22—41no—
588C22—42no—
589C22—41no—
590C22—42no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22C2×C3×#12011yescharacteristic
601C22C3×#53511yescharacteristic
602C22C3×#53711yescharacteristic
603C22C3×#57612yes—
604C22C3×#57712yes—
605C4—22no—
606C4—22no—
607C4—41no—
608C4—42no—
609C4—42no—
610C4—41no—
611C4—41no—
612C4—42no—
613C4—82no—
614C4C3×#53511yescharacteristic
615C4C3×#53511yescharacteristic
616C4C3×#53711yescharacteristic
617C4C3×#53711yescharacteristic
618C3#2966911yescharacteristic, direct product factor, Sylow
619C2—81no—
620C2—82no—
621C2—162no—
622C2C3×#261511yescharacteristic
623C2C3×#300811yescharacteristic
624C2C3×#301011yescharacteristic
625C2C3×#322412yes—
626C2C3×#325712yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
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
1. C2×C3×#542C2semidirect
2. C2×C3×#460C2semidirect
3. C2×C3×#459C2semidirect
4. C3×#2619C2semidirect
5. C3×#2023C2semidirect
6. C3×#2012C2semidirect
7. C3×#2000C2semidirect
8. C3×#1999C2semidirect
9. C3×#1805C2semidirect
10. C3×#1804C2semidirect
11. C3×#1798C2semidirect
12. #29669C3direct (N × H)
48. C2×#542C2×C3semidirect
49. C2×#460C2×C3semidirect
50. C2×#459C2×C3semidirect
51. #2619C2×C3semidirect
52. #2023C2×C3semidirect
53. #2012C2×C3semidirect
54. #2000C2×C3semidirect
55. #1999C2×C3semidirect
56. #1805C2×C3semidirect
57. #1804C2×C3semidirect
58. #1798C2×C3semidirect
94. C3×C4×Q8D4semidirect
96. C3×C4×D4D4semidirect
283. C4×Q8C3×D4semidirect
285. C4×D4C3×D4semidirect
618. C3#29669direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5422
C2×C3×#4602
C2×C3×#4592
C3×#26192
C3×#20232
C3×#20122
C3×#20002
C3×#19992
C3×#18052
C3×#18042
C3×#17982
#296693