Finite Group Databasebeta

← C3×#29672 #10648279

Subgroups — C3×#29672 #10648279

Order 768.

1096 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×#542C211yesmaximal, characteristic
2C2×C3×#462C211yesmaximal, characteristic, complemented
3C2×C3×#461C211yesmaximal, characteristic, complemented
4C3×#2621C211yesmaximal, characteristic, complemented
5C3×#2024C212yesmaximal, complemented
6C3×#2015C212yesmaximal, complemented
7C3×#2001C212yesmaximal, complemented
8C3×#2000C212yesmaximal
9C3×#1805C211yesmaximal, characteristic, complemented
10C3×#1804C211yesmaximal, characteristic, complemented
11C3×#1798C211yesmaximal, characteristic, complemented
12#29672C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×Q16C2212yes—
14C22×C3×SD16C2212yes—
15C3×#542—24no—
16C3×#542—24no—
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×C4⋊Q8C2211yescharacteristic
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×C22⋊Q8C2212yes—
26C2×C3×#121C2211yescharacteristic
27C2×C3×C4×Q8C2211yescharacteristic
28C2×C3×C4×D4C2211yescharacteristic
29C2×C3×#117C2211yescharacteristic
30C3×#462—24no—
31C3×#461—24no—
32C2×C3×C4⋊C8C2211yescharacteristic
33C2×C3×C4⋊C8C2211yescharacteristic
34C2×C3×#103C2212yes—
35C2×C3×#103C2212yes—
36C2×C3×#103C2212yes—
37C2×C3×#102C2212yes—
38C2×C3×C22⋊C8C2212yes—
39C3×#398—21no—
40C3×#398—21no—
41C3×#397—21no—
42C3×#397—21no—
43C3×#391C2212yes—
44C3×#382C2211yescharacteristic
45C3×#340C2212yes—
46C3×#340C2212yes—
47C3×#336C2212yes—
48C2×#542C2×C311yescharacteristic
49C2×#462C2×C311yescharacteristic, complemented
50C2×#461C2×C311yescharacteristic, complemented
51#2621C2×C311yescharacteristic, complemented
52#2024C2×C312yescomplemented
53#2015C2×C312yescomplemented
54#2001C2×C312yescomplemented
55#2000C2×C312yes—
56#1805C2×C311yescharacteristic, complemented
57#1804C2×C311yescharacteristic, complemented
58#1798C2×C311yescharacteristic, complemented
59C22×C3×Q8—21no—
60C22×C3×Q8C2312yes—
61C22×C3×Q8C2311yescharacteristic
62C22×C3×D4—21no—
63C22×C3×D4C2311yescharacteristic
64C23×C3×C4—21no—
65C23×C3×C4C2311yescharacteristic
66C2×C3×Q16—24no—
67C2×C3×Q16—42no—
68C2×C3×Q16—42no—
69C2×C3×SD16—24no—
70C2×C3×SD16—42no—
71C2×C3×SD16—42no—
72C22×C3×C8C2312yes—
73C22×C3×C8C2312yes—
74C3×C4⋊Q8—22no—
75C3×C4⋊Q8—22no—
76C3×C4⋊Q8—22no—
77C3×C4⋊Q8—22no—
78C3×#126—28no—
79C3×#124—28no—
80C3×#124—22no—
81C3×#124—22no—
82C3×#123—28no—
83C3×C22⋊Q8—28no—
84C3×C22⋊Q8—22no—
85C3×C22⋊Q8—22no—
86C3×C22⋊Q8—42no—
87C3×C22⋊Q8—42no—
88C3×#121—22no—
89C3×#121—22no—
90C3×C4×Q8—22no—
91C3×C4×Q8D414yes—
92C3×C4×D4—22no—
93C3×C4×D4D414yes—
94C3×#117—22no—
95C3×#117—22no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—22no—
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⋊C4D411yescharacteristic
114C2×C3×C22⋊C4D411yescharacteristic
115C2×C3×C42—21no—
116C2×C3×C42C2311yescharacteristic
117C2×C3×C42C2311yescharacteristic
118C3×C4⋊C8—22no—
119C3×C4⋊C8—22no—
120C3×#103—24no—
121C3×#103—24no—
122C3×#103—42no—
123C3×#102—42no—
124C3×C22⋊C8—42no—
125C3×#95—21no—
126C3×#95—21no—
127C3×#95—22no—
128C3×#95—22no—
129C22×Q16C22×C312yes—
130C22×SD16C22×C312yes—
131#542—24no—
132#542—24no—
133C2×C4⋊Q8C22×C311yescharacteristic
134C2×C4⋊Q8C22×C311yescharacteristic
135C2×#126—21no—
136C2×#124—21no—
137C2×#124C22×C311yescharacteristic
138C2×#123—21no—
139C2×C22⋊Q8—21no—
140C2×C22⋊Q8C22×C311yescharacteristic
141C2×C22⋊Q8C22×C312yes—
142C2×#121C22×C311yescharacteristic
143C2×C4×Q8C22×C311yescharacteristic
144C2×C4×D4C22×C311yescharacteristic
145C2×#117C22×C311yescharacteristic
146#462—24no—
147#461—24no—
148C2×C4⋊C8C22×C311yescharacteristic
149C2×C4⋊C8C22×C311yescharacteristic
150C2×#103C22×C312yes—
151C2×#103C22×C312yes—
152C2×#103C22×C312yes—
153C2×#102C22×C312yes—
154C2×C22⋊C8C22×C312yes—
155#398—21no—
156#398—21no—
157#397—21no—
158#397—21no—
159#391C22×C312yes—
160#382C22×C311yescharacteristic
161#340C22×C312yes—
162#340C22×C312yes—
163#336C22×C312yes—
164C24×C3—21no—
165C24×C3C2×D411yescharacteristic
166C2×C3×Q8—24no—
167C2×C3×Q8—21no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8—21no—
171C2×C3×Q8—21no—
172C2×C3×Q8—21no—
173C2×C3×Q8—22no—
174C2×C3×Q8—22no—
175C2×C3×Q8—22no—
176C2×C3×Q8—21no—
177C2×C3×Q8—22no—
178C2×C3×Q8—82no—
179C2×C3×Q8C2×D412yes—
180C2×C3×D4—24no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4—82no—
187C2×C3×D4C2×D412yes—
188C22×C3×C4—21no—
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—21no—
196C22×C3×C4—21no—
197C22×C3×C4—24no—
198C22×C3×C4—22no—
199C22×C3×C4—22no—
200C22×C3×C4—82no—
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2411yescharacteristic
207C3×Q16—48no—
208C3×SD16—48no—
209C2×C3×C8—22no—
210C2×C3×C8—22no—
211C2×C3×C8—22no—
212C2×C3×C8—22no—
213C2×C3×C8—22no—
214C2×C3×C8—22no—
215C3×C4⋊C4—24no—
216C3×C4⋊C4—24no—
217C3×C4⋊C4—24no—
218C3×C4⋊C4—22no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—22no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—44no—
225C3×C4⋊C4—42no—
226C3×C4⋊C4—42no—
227C3×C4⋊C4C2×D412yes—
228C3×C4⋊C4C2×D412yes—
229C3×C22⋊C4—24no—
230C3×C22⋊C4—24no—
231C3×C22⋊C4—24no—
232C3×C22⋊C4—24no—
233C3×C22⋊C4—24no—
234C3×C22⋊C4—22no—
235C3×C22⋊C4—22no—
236C3×C22⋊C4—44no—
237C3×C42—24no—
238C3×C42—21no—
239C3×C42—21no—
240C3×C42C2×D412yes—
241C3×C42C2×D412yes—
242C22×Q8—21no—
243C22×Q8C23×C312yes—
244C22×Q8C23×C311yescharacteristic
245C22×D4—21no—
246C22×D4C23×C311yescharacteristic
247C23×C4—21no—
248C23×C4C23×C311yescharacteristic
249C2×Q16—24no—
250C2×Q16—42no—
251C2×Q16—42no—
252C2×SD16—24no—
253C2×SD16—42no—
254C2×SD16—42no—
255C22×C8C23×C312yes—
256C22×C8C23×C312yes—
257C4⋊Q8—22no—
258C4⋊Q8—22no—
259C4⋊Q8—22no—
260C4⋊Q8—22no—
261#126—28no—
262#124—28no—
263#124—22no—
264#124—22no—
265#123—28no—
266C22⋊Q8—28no—
267C22⋊Q8—22no—
268C22⋊Q8—22no—
269C22⋊Q8—42no—
270C22⋊Q8—42no—
271#121—22no—
272#121—22no—
273C4×Q8—22no—
274C4×Q8C3×D414yes—
275C4×D4—22no—
276C4×D4C3×D414yes—
277#117—22no—
278#117—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4C3×D411yescharacteristic
286C2×C4⋊C4C3×D411yescharacteristic
287C2×C4⋊C4C23×C312yes—
288C2×C4⋊C4C23×C311yescharacteristic
289C2×C4⋊C4C23×C311yescharacteristic
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
301C4⋊C8—22no—
302C4⋊C8—22no—
303#103—24no—
304#103—24no—
305#103—42no—
306#102—42no—
307C22⋊C8—42no—
308#95—21no—
309#95—21no—
310#95—22no—
311#95—22no—
312C23×C3—21no—
313C23×C3—22no—
314C23×C3—21no—
315C23×C3—21no—
316C23×C3—22no—
317C23×C3—22no—
318C23×C3—41no—
319C23×C3—41no—
320C23×C3—42no—
321C23×C3—42no—
322C23×C3C22×D411yescharacteristic
323C3×Q8—22no—
324C3×Q8—22no—
325C3×Q8—41no—
326C3×Q8—41no—
327C3×Q8—42no—
328C3×Q8—42no—
329C3×Q8—42no—
330C3×Q8—42no—
331C3×Q8—84no—
332C3×D4—22no—
333C3×D4—22no—
334C3×D4—41no—
335C3×D4—41no—
336C3×D4—84no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—21no—
340C2×C3×C4—22no—
341C2×C3×C4—21no—
342C2×C3×C4—22no—
343C2×C3×C4—21no—
344C2×C3×C4—22no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4—84no—
364C2×C3×C4#12012yes—
365C2×C3×C4#12012yes—
366C2×C3×C4C22×D411yescharacteristic
367C2×C3×C4C22×D411yescharacteristic
368C2×C3×C4H(5,𝔽2)11yescharacteristic
369C2×C3×C4H(5,𝔽2)11yescharacteristic
370C2×C3×C4H(5,𝔽2)11yescharacteristic
371C2×C3×C42−1+411yescharacteristic
372C3×C8—42no—
373C3×C8—42no—
374C24—21no—
375C24C2×C3×D411yescharacteristic
376C2×Q8—24no—
377C2×Q8—21no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—21no—
387C2×Q8—22no—
388C2×Q8—82no—
389C2×Q8C2×C3×D412yes—
390C2×D4—24no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—82no—
397C2×D4C2×C3×D412yes—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—82no—
411C22×C4C2×C3×D411yescharacteristic
412C22×C4C2×C3×D411yescharacteristic
413C22×C4C2×C3×D411yescharacteristic
414C22×C4C2×C3×D411yescharacteristic
415C22×C4C2×C3×D411yescharacteristic
416C22×C4C24×C311yescharacteristic
417Q16—48no—
418SD16—48no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—44no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4C2×C3×D412yes—
438C4⋊C4C2×C3×D412yes—
439C22⋊C4—24no—
440C22⋊C4—24no—
441C22⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—24no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—44no—
447C42—24no—
448C42—21no—
449C42—21no—
450C42C2×C3×D412yes—
451C42C2×C3×D412yes—
452C22×C3—41no—
453C22×C3—42no—
454C22×C3—42no—
455C22×C3—42no—
456C22×C3—42no—
457C22×C3—41no—
458C22×C3—41no—
459C22×C3—42no—
460C22×C3—82no—
461C22×C3—82no—
462C22×C3C2×#12011yescharacteristic
463C22×C3#53511yescharacteristic
464C22×C3#53711yescharacteristic
465C22×C3#57712yes—
466C22×C3#57812yes—
467C3×C4—22no—
468C3×C4—22no—
469C3×C4—41no—
470C3×C4—42no—
471C3×C4—41no—
472C3×C4—42no—
473C3×C4—42no—
474C3×C4—41no—
475C3×C4—82no—
476C3×C4—82no—
477C3×C4#53511yescharacteristic
478C3×C4#53511yescharacteristic
479C3×C4#53711yescharacteristic
480C3×C4#53711yescharacteristic
481C23—22no—
482C23—22no—
483C23—21no—
484C23—22no—
485C23—21no—
486C23—21no—
487C23—42no—
488C23—42no—
489C23—41no—
490C23—41no—
491C23C22×C3×D411yescharacteristic
492Q8—22no—
493Q8—22no—
494Q8—41no—
495Q8—41no—
496Q8—42no—
497Q8—42no—
498Q8—42no—
499Q8—42no—
500Q8—84no—
501D4—22no—
502D4—22no—
503D4—41no—
504D4—41no—
505D4—84no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—84no—
533C2×C4C3×#12012yes—
534C2×C4C3×#12012yes—
535C2×C4C22×C3×D411yescharacteristic
536C2×C4C22×C3×D411yescharacteristic
537C2×C4C3×H(5,𝔽2)11yescharacteristic
538C2×C4C3×H(5,𝔽2)11yescharacteristic
539C2×C4C3×H(5,𝔽2)11yescharacteristic
540C2×C4C3×2−1+411yescharacteristic
541C8—42no—
542C8—42no—
543C2×C3—81no—
544C2×C3—82no—
545C2×C3#261511yescharacteristic
546C2×C3#300911yescharacteristic
547C2×C3#301111yescharacteristic
548C2×C3#322512yes—
549C2×C3#325812yes—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—41no—
554C22—41no—
555C22—42no—
556C22—41no—
557C22—42no—
558C22—82no—
559C22—82no—
560C22C2×C3×#12011yescharacteristic
561C22C3×#53511yescharacteristic
562C22C3×#53711yescharacteristic
563C22C3×#57712yes—
564C22C3×#57812yes—
565C4—22no—
566C4—22no—
567C4—41no—
568C4—42no—
569C4—42no—
570C4—41no—
571C4—41no—
572C4—42no—
573C4—82no—
574C4—82no—
575C4C3×#53511yescharacteristic
576C4C3×#53511yescharacteristic
577C4C3×#53711yescharacteristic
578C4C3×#53711yescharacteristic
579C3#2967211yescharacteristic, direct product factor, Sylow
580C2—81no—
581C2—82no—
582C2C3×#261511yescharacteristic
583C2C3×#300911yescharacteristic
584C2C3×#301111yescharacteristic
585C2C3×#322512yes—
586C2C3×#325812yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
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
4. C3×#2621C2semidirect
5. C3×#2024C2semidirect
6. C3×#2015C2semidirect
7. C3×#2001C2semidirect
9. C3×#1805C2semidirect
10. C3×#1804C2semidirect
11. C3×#1798C2semidirect
12. #29672C3direct (N × H)
49. C2×#462C2×C3semidirect
50. C2×#461C2×C3semidirect
51. #2621C2×C3semidirect
52. #2024C2×C3semidirect
53. #2015C2×C3semidirect
54. #2001C2×C3semidirect
56. #1805C2×C3semidirect
57. #1804C2×C3semidirect
58. #1798C2×C3semidirect
579. C3#29672direct (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×#4622
C2×C3×#4612
C3×#26212
C3×#20242
C3×#20152
C3×#20012
C3×#20002
C3×#18052
C3×#18042
C3×#17982
#296723