Finite Group Databasebeta

← C3×#29603 #10648210

Subgroups — C3×#29603 #10648210

Order 768.

1326 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×#536C211yesmaximal, characteristic, complemented
2C2×C3×#482C211yesmaximal, characteristic, complemented
3C2×C3×#448C211yesmaximal, characteristic, complemented
4C3×#2615C211yesmaximal, characteristic
5C3×#2036C212yesmaximal, complemented
6C3×#2035C212yesmaximal, complemented
7C3×#2000C212yesmaximal, complemented
8C3×#2000C212yesmaximal, complemented
9C3×#1825C211yesmaximal, characteristic, complemented
10C3×#1787C211yesmaximal, characteristic, complemented
11C3×#1780C211yesmaximal, characteristic, complemented
12#29603C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×C3×Q8C2211yescharacteristic
15C22×C3×SD16C2212yes—
16C3×#536—24no—
17C3×#536—24no—
18C2×C3×C4⋊Q8—21no—
19C2×C3×#124—21no—
20C2×C3×#123—22no—
21C2×C3×C22⋊Q8C2212yes—
22C2×C3×C22⋊Q8C2212yes—
23C2×C3×#121C2211yescharacteristic
24C2×C3×#121C2212yes—
25C2×C3×#117C2211yescharacteristic
26C3×#482—24no—
27C3×#448—24no—
28C2×C3×#107C2211yescharacteristic
29C2×C3×#106C2211yescharacteristic
30C2×C3×#103C2212yes—
31C2×C3×#103C2212yes—
32C2×C3×#103C2212yes—
33C2×C3×#102C2211yescharacteristic
34C2×C3×#102C2211yescharacteristic
35C2×C3×C22⋊C8C2211yescharacteristic
36C2×C3×C22⋊C8C2211yescharacteristic
37C2×C3×C22⋊C8C2212yes—
38C2×C3×C22⋊C8C2212yes—
39C3×#393—21no—
40C3×#392—22no—
41C3×#392—21no—
42C3×#380C2212yes—
43C3×#380C2211yescharacteristic
44C3×#340C2212yes—
45C3×#340C2212yes—
46C2×#536C2×C311yescharacteristic, complemented
47C2×#482C2×C311yescharacteristic, complemented
48C2×#448C2×C311yescharacteristic, complemented
49#2615C2×C311yescharacteristic
50#2036C2×C312yescomplemented
51#2035C2×C312yescomplemented
52#2000C2×C312yescomplemented
53#2000C2×C312yescomplemented
54#1825C2×C311yescharacteristic, complemented
55#1787C2×C311yescharacteristic, complemented
56#1780C2×C311yescharacteristic, complemented
57C2×C3×#41—22no—
58C2×C3×#41—22no—
59C2×C3×#41—42no—
60C2×C3×#41—42no—
61C22×C3×Q8—21no—
62C22×C3×Q8—22no—
63C22×C3×Q8—42no—
64C22×C3×Q8—42no—
65C22×C3×Q8D411yescharacteristic
66C22×C3×Q8D414yescomplemented
67C22×C3×Q8C2312yes—
68C22×C3×D4—22no—
69C22×C3×D4—21no—
70C22×C3×D4D412yes—
71C22×C3×D4D411yescharacteristic
72C22×C3×D4C2311yescharacteristic
73C23×C3×C4—21no—
74C23×C3×C4C2312yes—
75C23×C3×C4C2311yescharacteristic
76C2×C3×SD16—24no—
77C2×C3×SD16—42no—
78C2×C3×SD16—42no—
79C22×C3×C8C2312yes—
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×C4⋊Q8—28no—
83C3×#124—28no—
84C3×#123—216no—
85C3×C22⋊Q8—24no—
86C3×C22⋊Q8—24no—
87C3×C22⋊Q8—24no—
88C3×C22⋊Q8—24no—
89C3×#121—22no—
90C3×#121—22no—
91C3×#121—42no—
92C3×#121—42no—
93C3×#117—22no—
94C3×#117D414yescomplemented
95C2×C3×C4⋊C4—22no—
96C2×C3×C4⋊C4—22no—
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2312yes—
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—22no—
103C2×C3×C22⋊C4—22no—
104C2×C3×C22⋊C4—22no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—22no—
107C2×C3×C22⋊C4—21no—
108C2×C3×C42—21no—
109C3×#107—22no—
110C3×#106—22no—
111C3×#103—24no—
112C3×#103—42no—
113C3×#103—42no—
114C3×#102—22no—
115C3×#102—22no—
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
123C23×Q8C22×C311yescharacteristic
124C22×SD16C22×C312yes—
125#536—24no—
126#536—24no—
127C2×C4⋊Q8—21no—
128C2×#124—21no—
129C2×#123—22no—
130C2×C22⋊Q8C22×C312yes—
131C2×C22⋊Q8C22×C312yes—
132C2×#121C22×C311yescharacteristic
133C2×#121C22×C312yes—
134C2×#117C22×C311yescharacteristic
135#482—24no—
136#448—24no—
137C2×#107C22×C311yescharacteristic
138C2×#106C22×C311yescharacteristic
139C2×#103C22×C312yes—
140C2×#103C22×C312yes—
141C2×#103C22×C312yes—
142C2×#102C22×C311yescharacteristic
143C2×#102C22×C311yescharacteristic
144C2×C22⋊C8C22×C311yescharacteristic
145C2×C22⋊C8C22×C311yescharacteristic
146C2×C22⋊C8C22×C312yes—
147C2×C22⋊C8C22×C312yes—
148#393—21no—
149#392—22no—
150#392—21no—
151#380C22×C311yescharacteristic
152#380C22×C312yes—
153#340C22×C312yes—
154#340C22×C312yes—
155C24×C3—21no—
156C24×C3C2×D412yes—
157C24×C3C2×D411yescharacteristic
158C3×#41—48no—
159C3×#41—44no—
160C3×#41—44no—
161C2×C3×Q8—24no—
162C2×C3×Q8—22no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8—22no—
168C2×C3×Q8—42no—
169C2×C3×Q8—48no—
170C2×C3×Q8—44no—
171C2×C3×Q8—44no—
172C2×C3×Q8—44no—
173C2×C3×Q8C2×D414yes—
174C2×C3×D4—24no—
175C2×C3×D4—22no—
176C2×C3×D4—21no—
177C2×C3×D4—21no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—42no—
181C2×C3×D4—42no—
182C2×C3×D4—42no—
183C2×C3×D4—44no—
184C2×C3×D4—42no—
185C2×C3×D4—82no—
186C2×C3×D4—82no—
187C2×C3×D4—82no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—24no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—22no—
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—44no—
203C22×C3×C4C2×D412yes—
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D412yes—
206C22×C3×C4C2×D412yes—
207C22×C3×C4C2411yescharacteristic
208C3×SD16—48no—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8—22no—
212C2×C3×C8—22no—
213C2×C3×C8—22no—
214C2×C3×C8—22no—
215C2×C3×C8—22no—
216C3×C4⋊C4—28no—
217C3×C4⋊C4—28no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—22no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4C2×D412yes—
225C3×C4⋊C4C2×D412yes—
226C3×C22⋊C4—28no—
227C3×C22⋊C4—28no—
228C3×C22⋊C4—24no—
229C3×C22⋊C4—44no—
230C3×C22⋊C4—42no—
231C3×C22⋊C4—81no—
232C3×C22⋊C4—82no—
233C3×C42—24no—
234C2×#41—22no—
235C2×#41—22no—
236C2×#41—42no—
237C2×#41—42no—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8—42no—
241C22×Q8—42no—
242C22×Q8C3×D411yescharacteristic
243C22×Q8C3×D414yescomplemented
244C22×Q8C23×C312yes—
245C22×D4—22no—
246C22×D4—21no—
247C22×D4C3×D412yes—
248C22×D4C3×D411yescharacteristic
249C22×D4C23×C311yescharacteristic
250C23×C4—21no—
251C23×C4C23×C312yes—
252C23×C4C23×C311yescharacteristic
253C2×SD16—24no—
254C2×SD16—42no—
255C2×SD16—42no—
256C22×C8C23×C312yes—
257C22×C8C23×C311yescharacteristic
258C22×C8C23×C311yescharacteristic
259C4⋊Q8—28no—
260#124—28no—
261#123—216no—
262C22⋊Q8—24no—
263C22⋊Q8—24no—
264C22⋊Q8—24no—
265C22⋊Q8—24no—
266#121—22no—
267#121—22no—
268#121—42no—
269#121—42no—
270#117—22no—
271#117C3×D414yescomplemented
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4C23×C312yes—
275C2×C4⋊C4C23×C311yescharacteristic
276C2×C4⋊C4C23×C311yescharacteristic
277C2×C4⋊C4C23×C311yescharacteristic
278C2×C22⋊C4—21no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—21no—
285C2×C42—21no—
286#107—22no—
287#106—22no—
288#103—24no—
289#103—42no—
290#103—42no—
291#102—22no—
292#102—22no—
293C22⋊C8—22no—
294C22⋊C8—22no—
295C22⋊C8—42no—
296C22⋊C8—42no—
297#95—22no—
298#95—21no—
299C23×C3—22no—
300C23×C3—21no—
301C23×C3—22no—
302C23×C3—21no—
303C23×C3—22no—
304C23×C3—22no—
305C23×C3—22no—
306C23×C3—22no—
307C23×C3—22no—
308C23×C3—22no—
309C23×C3—22no—
310C23×C3—41no—
311C23×C3—42no—
312C23×C3—41no—
313C23×C3—42no—
314C23×C3—41no—
315C23×C3#12012yes—
316C23×C3C22×D411yescharacteristic
317C3×Q8—24no—
318C3×Q8—24no—
319C3×Q8—48no—
320C3×Q8—44no—
321C3×Q8—42no—
322C3×Q8—42no—
323C3×D4—44no—
324C3×D4—48no—
325C3×D4—42no—
326C3×D4—41no—
327C3×D4—41no—
328C3×D4—84no—
329C3×D4—88no—
330C2×C3×C4—22no—
331C2×C3×C4—24no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—24no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—48no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—41no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4—42no—
355C2×C3×C4#12012yes—
356C2×C3×C4C22×D411yescharacteristic
357C2×C3×C4C22×D411yescharacteristic
358C2×C3×C4H(5,𝔽2)12yes—
359C2×C3×C4H(5,𝔽2)12yes—
360C3×C8—42no—
361C3×C8—41no—
362C3×C8—41no—
363C24—21no—
364C24C2×C3×D412yes—
365C24C2×C3×D411yescharacteristic
366#41—48no—
367#41—44no—
368#41—44no—
369C2×Q8—24no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—42no—
377C2×Q8—48no—
378C2×Q8—44no—
379C2×Q8—44no—
380C2×Q8—44no—
381C2×Q8C2×C3×D414yes—
382C2×D4—24no—
383C2×D4—22no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—42no—
391C2×D4—44no—
392C2×D4—42no—
393C2×D4—82no—
394C2×D4—82no—
395C2×D4—82no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—44no—
411C22×C4C2×C3×D412yes—
412C22×C4C2×C3×D411yescharacteristic
413C22×C4C2×C3×D412yes—
414C22×C4C2×C3×D412yes—
415C22×C4C24×C311yescharacteristic
416SD16—48no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C4⋊C4—28no—
425C4⋊C4—28no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4C2×C3×D412yes—
433C4⋊C4C2×C3×D412yes—
434C22⋊C4—28no—
435C22⋊C4—28no—
436C22⋊C4—24no—
437C22⋊C4—44no—
438C22⋊C4—42no—
439C22⋊C4—81no—
440C22⋊C4—82no—
441C42—24no—
442C22×C3—22no—
443C22×C3—22no—
444C22×C3—22no—
445C22×C3—42no—
446C22×C3—42no—
447C22×C3—42no—
448C22×C3—42no—
449C22×C3—42no—
450C22×C3—42no—
451C22×C3—42no—
452C22×C3—42no—
453C22×C3—42no—
454C22×C3—81no—
455C22×C3—82no—
456C22×C3—82no—
457C22×C3—81no—
458C22×C3—81no—
459C22×C3C2×#12011yescharacteristic
460C22×C3#53412yes—
461C22×C3#57711yescharacteristic
462C22×C3#57712yes—
463C22×C3#57811yescharacteristic
464C3×C4—22no—
465C3×C4—42no—
466C3×C4—44no—
467C3×C4—42no—
468C3×C4—42no—
469C3×C4—82no—
470C3×C4—81no—
471C3×C4#53412yes—
472C3×C4#53511yescharacteristic
473C3×C4#53511yescharacteristic
474C23—21no—
475C23—22no—
476C23—21no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—41no—
486C23—42no—
487C23—41no—
488C23—41no—
489C23—42no—
490C23C3×#12012yes—
491C23C22×C3×D411yescharacteristic
492Q8—24no—
493Q8—24no—
494Q8—48no—
495Q8—44no—
496Q8—42no—
497Q8—42no—
498D4—44no—
499D4—48no—
500D4—42no—
501D4—41no—
502D4—41no—
503D4—84no—
504D4—88no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—48no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
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×C4C3×#12012yes—
531C2×C4C22×C3×D411yescharacteristic
532C2×C4C22×C3×D411yescharacteristic
533C2×C4C3×H(5,𝔽2)12yes—
534C2×C4C3×H(5,𝔽2)12yes—
535C8—42no—
536C8—41no—
537C8—41no—
538C2×C3—42no—
539C2×C3—82no—
540C2×C3—161no—
541C2×C3#261311yescharacteristic
542C2×C3#301011yescharacteristic
543C2×C3#301311yescharacteristic
544C2×C3#320012yes—
545C2×C3#320112yes—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—82no—
559C22—82no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22C2×C3×#12011yescharacteristic
564C22C3×#53412yes—
565C22C3×#57711yescharacteristic
566C22C3×#57712yes—
567C22C3×#57811yescharacteristic
568C4—22no—
569C4—42no—
570C4—42no—
571C4—42no—
572C4—44no—
573C4—82no—
574C4—81no—
575C4C3×#53412yes—
576C4C3×#53511yescharacteristic
577C4C3×#53511yescharacteristic
578C3#2960311yescharacteristic, direct product factor, Sylow
579C2—42no—
580C2—82no—
581C2—161no—
582C2C3×#261311yescharacteristic
583C2C3×#301011yescharacteristic
584C2C3×#301311yescharacteristic
585C2C3×#320012yes—
586C2C3×#320112yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#2615C2
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
1. C2×C3×#536C2semidirect
2. C2×C3×#482C2semidirect
3. C2×C3×#448C2semidirect
5. C3×#2036C2semidirect
6. C3×#2035C2semidirect
7. C3×#2000C2semidirect
8. C3×#2000C2semidirect
9. C3×#1825C2semidirect
10. C3×#1787C2semidirect
11. C3×#1780C2semidirect
12. #29603C3direct (N × H)
46. C2×#536C2×C3semidirect
47. C2×#482C2×C3semidirect
48. C2×#448C2×C3semidirect
50. #2036C2×C3semidirect
51. #2035C2×C3semidirect
52. #2000C2×C3semidirect
53. #2000C2×C3semidirect
54. #1825C2×C3semidirect
55. #1787C2×C3semidirect
56. #1780C2×C3semidirect
66. C22×C3×Q8D4semidirect
94. C3×#117D4semidirect
243. C22×Q8C3×D4semidirect
271. #117C3×D4semidirect
578. C3#29603direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5362
C2×C3×#4822
C2×C3×#4482
C3×#26152
C3×#20362
C3×#20352
C3×#20002
C3×#18252
C3×#17872
C3×#17802
#296033