Finite Group Databasebeta

← C3×#29609 #10648216

Subgroups — C3×#29609 #10648216

Order 768.

1332 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#537C211yesmaximal, characteristic
2C2×C3×#483C211yesmaximal, characteristic, complemented
3C2×C3×#452C211yesmaximal, characteristic, complemented
4C3×#2616C211yesmaximal, characteristic, complemented
5C3×#2035C212yesmaximal, complemented
6C3×#2034C212yesmaximal, complemented
7C3×#2001C212yesmaximal, complemented
8C3×#2000C212yesmaximal, complemented
9C3×#1824C211yesmaximal, characteristic, complemented
10C3×#1788C211yesmaximal, characteristic, complemented
11C3×#1780C211yesmaximal, characteristic, complemented
12#29609C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C22×C3×#41C2211yescharacteristic
15C22×C3×Q16C2211yescharacteristic
16C22×C3×SD16C2211yescharacteristic
17C3×#537—24no—
18C3×#537—24no—
19C2×C3×C22⋊Q8—21no—
20C2×C3×C22⋊Q8C2211yescharacteristic
21C2×C3×C22⋊Q8C2212yes—
22C2×C3×C22⋊Q8C2212yes—
23C2×C3×#121—21no—
24C2×C3×#121—22no—
25C2×C3×#121C2212yes—
26C22×C3×C4⋊C4C2211yescharacteristic
27C3×#483—24no—
28C3×#452—24no—
29C2×C3×#106C2212yes—
30C2×C3×#103C2212yes—
31C2×C3×#103C2211yescharacteristic
32C2×C3×#103C2212yes—
33C2×C3×#102C2211yescharacteristic
34C2×C3×#102C2212yes—
35C2×C3×C22⋊C8C2211yescharacteristic
36C2×C3×C22⋊C8C2211yescharacteristic
37C2×C3×C22⋊C8C2212yes—
38C2×C3×C22⋊C8C2212yes—
39C3×#395—21no—
40C3×#394—22no—
41C3×#394—21no—
42C3×#380C2212yes—
43C3×#380C2211yescharacteristic
44C3×#340C2212yes—
45C3×#340C2212yes—
46C2×#537C2×C311yescharacteristic
47C2×#483C2×C311yescharacteristic, complemented
48C2×#452C2×C311yescharacteristic, complemented
49#2616C2×C311yescharacteristic, complemented
50#2035C2×C312yescomplemented
51#2034C2×C312yescomplemented
52#2001C2×C312yescomplemented
53#2000C2×C312yescomplemented
54#1824C2×C311yescharacteristic, complemented
55#1788C2×C311yescharacteristic, complemented
56#1780C2×C311yescharacteristic, complemented
57C2×C3×#41—22no—
58C2×C3×#41—22no—
59C2×C3×#41—22no—
60C2×C3×#41—42no—
61C2×C3×#41—42no—
62C2×C3×#41—42no—
63C2×C3×#41—42no—
64C2×C3×#41D414yes—
65C22×C3×Q8D411yescharacteristic
66C22×C3×Q8C2311yescharacteristic
67C22×C3×Q8C2311yescharacteristic
68C22×C3×D4—21no—
69C22×C3×D4—22no—
70C22×C3×D4D412yes—
71C22×C3×D4D411yescharacteristic
72C22×C3×D4C2311yescharacteristic
73C23×C3×C4—21no—
74C23×C3×C4—21no—
75C23×C3×C4C2312yes—
76C23×C3×C4C2311yescharacteristic
77C2×C3×Q16—22no—
78C2×C3×Q16—42no—
79C2×C3×SD16—22no—
80C2×C3×SD16—42no—
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2312yes—
84C3×C22⋊Q8—28no—
85C3×C22⋊Q8—24no—
86C3×C22⋊Q8—24no—
87C3×C22⋊Q8—22no—
88C3×C22⋊Q8—22no—
89C3×C22⋊Q8—42no—
90C3×C22⋊Q8—42no—
91C3×#121—216no—
92C3×#121—28no—
93C3×#121—24no—
94C3×#121—24no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—22no—
97C2×C3×C4⋊C4—22no—
98C2×C3×C4⋊C4—22no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4D414yes—
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C4⋊C4C2312yes—
103C2×C3×C4⋊C4C2312yes—
104C2×C3×C22⋊C4—22no—
105C2×C3×C22⋊C4—22no—
106C2×C3×C22⋊C4—22no—
107C2×C3×C22⋊C4—21no—
108C2×C3×C22⋊C4—22no—
109C2×C3×C22⋊C4—21no—
110C3×#106—24no—
111C3×#103—22no—
112C3×#103—24no—
113C3×#103—42no—
114C3×#102—22no—
115C3×#102—42no—
116C3×C22⋊C8—22no—
117C3×C22⋊C8—22no—
118C3×C22⋊C8—42no—
119C3×C22⋊C8—42no—
120C3×#95—22no—
121C3×#95—21no—
122C22×#41C22×C311yescharacteristic
123C22×#41C22×C311yescharacteristic
124C22×Q16C22×C311yescharacteristic
125C22×SD16C22×C311yescharacteristic
126#537—24no—
127#537—24no—
128C2×C22⋊Q8—21no—
129C2×C22⋊Q8C22×C311yescharacteristic
130C2×C22⋊Q8C22×C312yes—
131C2×C22⋊Q8C22×C312yes—
132C2×#121—21no—
133C2×#121—22no—
134C2×#121C22×C312yes—
135C22×C4⋊C4C22×C311yescharacteristic
136#483—24no—
137#452—24no—
138C2×#106C22×C312yes—
139C2×#103C22×C312yes—
140C2×#103C22×C311yescharacteristic
141C2×#103C22×C312yes—
142C2×#102C22×C311yescharacteristic
143C2×#102C22×C312yes—
144C2×C22⋊C8C22×C311yescharacteristic
145C2×C22⋊C8C22×C311yescharacteristic
146C2×C22⋊C8C22×C312yes—
147C2×C22⋊C8C22×C312yes—
148#395—21no—
149#394—22no—
150#394—21no—
151#380C22×C311yescharacteristic
152#380C22×C312yes—
153#340C22×C312yes—
154#340C22×C312yes—
155C24×C3—21no—
156C24×C3C2×D412yes—
157C24×C3C2×D411yescharacteristic
158C3×#41—44no—
159C3×#41—44no—
160C3×#41—44no—
161C3×#41—44no—
162C3×#41—48no—
163C3×#41—48no—
164C2×C3×Q8—22no—
165C2×C3×Q8—21no—
166C2×C3×Q8—21no—
167C2×C3×Q8—21no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—42no—
173C2×C3×Q8C2×D412yes—
174C2×C3×D4—24no—
175C2×C3×D4—24no—
176C2×C3×D4—28no—
177C2×C3×D4—22no—
178C2×C3×D4—21no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—42no—
182C2×C3×D4—42no—
183C2×C3×D4—42no—
184C2×C3×D4—44no—
185C2×C3×D4—82no—
186C2×C3×D4—82no—
187C2×C3×D4C2×D412yes—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—22no—
191C22×C3×C4—21no—
192C22×C3×C4—24no—
193C22×C3×C4—24no—
194C22×C3×C4—22no—
195C22×C3×C4—22no—
196C22×C3×C4—22no—
197C22×C3×C4—22no—
198C22×C3×C4—22no—
199C22×C3×C4—22no—
200C22×C3×C4—21no—
201C22×C3×C4—22no—
202C22×C3×C4—21no—
203C22×C3×C4—44no—
204C22×C3×C4—82no—
205C22×C3×C4C2×D412yes—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D412yes—
208C22×C3×C4C2×D412yes—
209C22×C3×C4C2411yescharacteristic
210C3×Q16—44no—
211C3×SD16—44no—
212C2×C3×C8—21no—
213C2×C3×C8—22no—
214C2×C3×C8—21no—
215C2×C3×C8—22no—
216C2×C3×C8—22no—
217C2×C3×C8—22no—
218C2×C3×C8—22no—
219C3×C4⋊C4—28no—
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—44no—
227C3×C4⋊C4—42no—
228C3×C4⋊C4C2×D414yes—
229C3×C22⋊C4—28no—
230C3×C22⋊C4—28no—
231C3×C22⋊C4—44no—
232C3×C22⋊C4—42no—
233C3×C22⋊C4—81no—
234C3×C22⋊C4—82no—
235C2×#41—22no—
236C2×#41—22no—
237C2×#41—22no—
238C2×#41—42no—
239C2×#41—42no—
240C2×#41—42no—
241C2×#41—42no—
242C2×#41C3×D414yes—
243C22×Q8C3×D411yescharacteristic
244C22×Q8C23×C311yescharacteristic
245C22×Q8C23×C311yescharacteristic
246C22×D4—21no—
247C22×D4—22no—
248C22×D4C3×D412yes—
249C22×D4C3×D411yescharacteristic
250C22×D4C23×C311yescharacteristic
251C23×C4—21no—
252C23×C4—21no—
253C23×C4C23×C312yes—
254C23×C4C23×C311yescharacteristic
255C2×Q16—22no—
256C2×Q16—42no—
257C2×SD16—22no—
258C2×SD16—42no—
259C22×C8C23×C311yescharacteristic
260C22×C8C23×C311yescharacteristic
261C22×C8C23×C312yes—
262C22⋊Q8—28no—
263C22⋊Q8—24no—
264C22⋊Q8—24no—
265C22⋊Q8—22no—
266C22⋊Q8—22no—
267C22⋊Q8—42no—
268C22⋊Q8—42no—
269#121—216no—
270#121—28no—
271#121—24no—
272#121—24no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4C3×D414yes—
279C2×C4⋊C4C23×C312yes—
280C2×C4⋊C4C23×C311yescharacteristic
281C2×C4⋊C4C23×C312yes—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—21no—
288#106—24no—
289#103—22no—
290#103—24no—
291#103—42no—
292#102—22no—
293#102—42no—
294C22⋊C8—22no—
295C22⋊C8—22no—
296C22⋊C8—42no—
297C22⋊C8—42no—
298#95—22no—
299#95—21no—
300C23×C3—22no—
301C23×C3—21no—
302C23×C3—22no—
303C23×C3—21no—
304C23×C3—22no—
305C23×C3—22no—
306C23×C3—22no—
307C23×C3—22no—
308C23×C3—22no—
309C23×C3—22no—
310C23×C3—22no—
311C23×C3—22no—
312C23×C3—42no—
313C23×C3—42no—
314C23×C3—41no—
315C23×C3—41no—
316C23×C3#12012yes—
317C23×C3C22×D411yescharacteristic
318C3×Q8—22no—
319C3×Q8—22no—
320C3×Q8—44no—
321C3×Q8—42no—
322C3×Q8—41no—
323C3×Q8—41no—
324C3×Q8—42no—
325C3×D4—22no—
326C3×D4—22no—
327C3×D4—44no—
328C3×D4—48no—
329C3×D4—48no—
330C3×D4—42no—
331C3×D4—88no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—24no—
336C2×C3×C4—24no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—48no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4—41no—
355C2×C3×C4—42no—
356C2×C3×C4—41no—
357C2×C3×C4—84no—
358C2×C3×C4#12012yes—
359C2×C3×C4C22×D411yescharacteristic
360C2×C3×C4C22×D411yescharacteristic
361C2×C3×C4H(5,𝔽2)12yes—
362C2×C3×C4H(5,𝔽2)12yes—
363C3×C8—41no—
364C3×C8—41no—
365C3×C8—42no—
366C24—21no—
367C24C2×C3×D412yes—
368C24C2×C3×D411yescharacteristic
369#41—44no—
370#41—44no—
371#41—44no—
372#41—44no—
373#41—48no—
374#41—48no—
375C2×Q8—22no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—42no—
384C2×Q8C2×C3×D412yes—
385C2×D4—24no—
386C2×D4—24no—
387C2×D4—28no—
388C2×D4—22no—
389C2×D4—21no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—42no—
393C2×D4—42no—
394C2×D4—42no—
395C2×D4—44no—
396C2×D4—82no—
397C2×D4—82no—
398C2×D4C2×C3×D412yes—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—44no—
415C22×C4—82no—
416C22×C4C2×C3×D412yes—
417C22×C4C2×C3×D411yescharacteristic
418C22×C4C2×C3×D412yes—
419C22×C4C2×C3×D412yes—
420C22×C4C24×C311yescharacteristic
421Q16—44no—
422SD16—44no—
423C2×C8—21no—
424C2×C8—22no—
425C2×C8—21no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C4⋊C4—28no—
431C4⋊C4—24no—
432C4⋊C4—22no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—44no—
438C4⋊C4—42no—
439C4⋊C4C2×C3×D414yes—
440C22⋊C4—28no—
441C22⋊C4—28no—
442C22⋊C4—44no—
443C22⋊C4—42no—
444C22⋊C4—81no—
445C22⋊C4—82no—
446C22×C3—22no—
447C22×C3—22no—
448C22×C3—22no—
449C22×C3—42no—
450C22×C3—42no—
451C22×C3—42no—
452C22×C3—42no—
453C22×C3—42no—
454C22×C3—42no—
455C22×C3—42no—
456C22×C3—42no—
457C22×C3—42no—
458C22×C3—42no—
459C22×C3—42no—
460C22×C3—42no—
461C22×C3—82no—
462C22×C3—82no—
463C22×C3C2×#12011yescharacteristic
464C22×C3#53412yes—
465C22×C3#57712yes—
466C22×C3#57711yescharacteristic
467C22×C3#57811yescharacteristic
468C3×C4—22no—
469C3×C4—42no—
470C3×C4—42no—
471C3×C4—44no—
472C3×C4—81no—
473C3×C4—82no—
474C3×C4—81no—
475C3×C4#53411yescharacteristic
476C3×C4#53411yescharacteristic
477C3×C4#53512yes—
478C23—21no—
479C23—22no—
480C23—21no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—42no—
491C23—41no—
492C23—41no—
493C23—42no—
494C23C3×#12012yes—
495C23C22×C3×D411yescharacteristic
496Q8—22no—
497Q8—22no—
498Q8—44no—
499Q8—42no—
500Q8—41no—
501Q8—41no—
502Q8—42no—
503D4—22no—
504D4—22no—
505D4—44no—
506D4—48no—
507D4—48no—
508D4—42no—
509D4—88no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—48no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—84no—
536C2×C4C3×#12012yes—
537C2×C4C22×C3×D411yescharacteristic
538C2×C4C22×C3×D411yescharacteristic
539C2×C4C3×H(5,𝔽2)12yes—
540C2×C4C3×H(5,𝔽2)12yes—
541C8—41no—
542C8—41no—
543C8—42no—
544C2×C3—42no—
545C2×C3—82no—
546C2×C3—82no—
547C2×C3#261311yescharacteristic
548C2×C3#300911yescharacteristic
549C2×C3#301311yescharacteristic
550C2×C3#320112yes—
551C2×C3#320412yes—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—82no—
568C22—82no—
569C22C2×C3×#12011yescharacteristic
570C22C3×#53412yes—
571C22C3×#57712yes—
572C22C3×#57711yescharacteristic
573C22C3×#57811yescharacteristic
574C4—22no—
575C4—44no—
576C4—42no—
577C4—42no—
578C4—82no—
579C4—81no—
580C4—81no—
581C4C3×#53411yescharacteristic
582C4C3×#53411yescharacteristic
583C4C3×#53512yes—
584C3#2960911yescharacteristic, direct product factor, Sylow
585C2—42no—
586C2—82no—
587C2—82no—
588C2C3×#261311yescharacteristic
589C2C3×#300911yescharacteristic
590C2C3×#301311yescharacteristic
591C2C3×#320112yes—
592C2C3×#320412yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#537C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×#41 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#483C2semidirect
3. C2×C3×#452C2semidirect
4. C3×#2616C2semidirect
5. C3×#2035C2semidirect
6. C3×#2034C2semidirect
7. C3×#2001C2semidirect
8. C3×#2000C2semidirect
9. C3×#1824C2semidirect
10. C3×#1788C2semidirect
11. C3×#1780C2semidirect
12. #29609C3direct (N × H)
47. C2×#483C2×C3semidirect
48. C2×#452C2×C3semidirect
49. #2616C2×C3semidirect
50. #2035C2×C3semidirect
51. #2034C2×C3semidirect
52. #2001C2×C3semidirect
53. #2000C2×C3semidirect
54. #1824C2×C3semidirect
55. #1788C2×C3semidirect
56. #1780C2×C3semidirect
584. C3#29609direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5372
C2×C3×#4832
C2×C3×#4522
C3×#26162
C3×#20352
C3×#20342
C3×#20012
C3×#20002
C3×#18242
C3×#17882
C3×#17802
#296093