Finite Group Databasebeta

← C3×#21006 #10639613

Subgroups — C3×#21006 #10639613

Order 768.

1654 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#137C211yesmaximal, characteristic
2C2×C3×#568C211yesmaximal, characteristic
3C3×#2896C211yesmaximal, characteristic
4C2×C3×#419C211yesmaximal, characteristic, complemented
5C22×C3×#101C211yesmaximal, characteristic, complemented
6C3×#1908C218yesmaximal, complemented
7C3×#1742C211yesmaximal, characteristic, complemented
8C3×#1738C211yesmaximal, characteristic, complemented
9#21006C311yesmaximal, characteristic, direct product factor, Sylow
10C22×C3×#41C2211yescharacteristic
11C23×C3×Q8C2211yescharacteristic
12C2×C3×#137—24no—
13C2×C3×#137—22no—
14C2×C3×#137C418yescomplemented
15C2×C3×#137C2214yes—
16C22×C3×Q16—21no—
17C22×C3×SD16—21no—
18C3×#568—24no—
19C3×#568—24no—
20C2×C3×#131—22no—
21C2×C3×#131C2214yes—
22C22×C3×M16—22no—
23C22×C3×M16C2211yescharacteristic
24C22×C3×M16C2211yescharacteristic
25C22×C3×M16C2211yescharacteristic
26C22×C3×M16C2211yescharacteristic
27C2×C3×#117C2211yescharacteristic
28C3×#419—21no—
29C3×#419—22no—
30C3×#419—22no—
31C3×#418—21no—
32C3×#417C2214yes—
33C3×#416—22no—
34C2×C3×#103C2214yescomplemented
35C2×C3×#101—24no—
36C2×C3×#101C2214yescomplemented
37C3×#343C2214yescomplemented
38C3×#341C2214yescomplemented
39C22×#137C2×C311yescharacteristic
40C2×#568C2×C311yescharacteristic
41#2896C2×C311yescharacteristic
42C2×#419C2×C311yescharacteristic, complemented
43C22×#101C2×C311yescharacteristic, complemented
44#1908C2×C318yescomplemented
45#1742C2×C311yescharacteristic, complemented
46#1738C2×C311yescharacteristic, complemented
47C2×C3×#41—21no—
48C2×C3×#41—22no—
49C2×C3×#41—22no—
50C2×C3×#41—21no—
51C2×C3×#41C2×C414yes—
52C2×C3×#41D414yes—
53C2×C3×#41D412yes—
54C2×C3×#41C2312yes—
55C22×C3×Q8—22no—
56C22×C3×Q8—21no—
57C22×C3×Q8—21no—
58C22×C3×Q8—21no—
59C22×C3×Q8—22no—
60C22×C3×Q8—21no—
61C22×C3×Q8—21no—
62C22×C3×Q8—22no—
63C22×C3×Q8C2×C414yes—
64C22×C3×Q8D411yescharacteristic
65C22×C3×Q8C2312yes—
66C22×C3×D4—21no—
67C22×C3×D4D411yescharacteristic
68C23×C3×C4—21no—
69C23×C3×C4—21no—
70C23×C3×C4C2311yescharacteristic
71C3×#137—216no—
72C3×#137—28no—
73C3×#137C2×C4116yes—
74C2×C3×Q16—28no—
75C2×C3×Q16—24no—
76C2×C3×SD16—28no—
77C2×C3×SD16—24no—
78C3×#131—216no—
79C3×#131—28no—
80C3×#131—28no—
81C2×C3×M16—216no—
82C2×C3×M16—21no—
83C2×C3×M16—22no—
84C2×C3×M16—22no—
85C2×C3×M16—22no—
86C2×C3×M16—22no—
87C2×C3×M16—21no—
88C2×C3×M16—21no—
89C2×C3×M16—24no—
90C2×C3×M16—22no—
91C2×C3×M16—22no—
92C2×C3×M16C2×C414yes—
93C2×C3×M16D414yes—
94C2×C3×M16D412yes—
95C2×C3×M16D412yes—
96C2×C3×M16C2312yes—
97C2×C3×M16C2312yes—
98C2×C3×M16C2312yes—
99C22×C3×C8—21no—
100C22×C3×C8—21no—
101C22×C3×C8—24no—
102C22×C3×C8—21no—
103C22×C3×C8C2312yes—
104C3×#117—21no—
105C3×#117—21no—
106C3×#117—21no—
107C3×#117—21no—
108C2×C3×C4⋊C4C2312yes—
109C2×C3×C22⋊C4—21no—
110C2×C3×C42—21no—
111C3×#103—24no—
112C3×#103—28no—
113C3×#102—24no—
114C3×#101—28no—
115C22×#41C22×C311yescharacteristic
116C23×Q8C22×C311yescharacteristic
117C2×#137—24no—
118C2×#137—22no—
119C2×#137C3×C418yescomplemented
120C2×#137C22×C314yes—
121C22×Q16—21no—
122C22×SD16—21no—
123#568—24no—
124#568—24no—
125C2×#131—22no—
126C2×#131C22×C314yes—
127C22×M16—22no—
128C22×M16C22×C311yescharacteristic
129C22×M16C22×C311yescharacteristic
130C22×M16C22×C311yescharacteristic
131C22×M16C22×C311yescharacteristic
132C2×#117C22×C311yescharacteristic
133#419—21no—
134#419—22no—
135#419—22no—
136#418—21no—
137#417C22×C314yes—
138#416—22no—
139C2×#103C22×C314yescomplemented
140C2×#101—24no—
141C2×#101C22×C314yescomplemented
142#343C22×C314yescomplemented
143#341C22×C314yescomplemented
144C24×C3—21no—
145C24×C3#4111yescharacteristic
146C3×#41—24no—
147C3×#41—24no—
148C3×#41—24no—
149C3×#41—24no—
150C3×#41—24no—
151C3×#41—22no—
152C3×#41—22no—
153C3×#41C22⋊C418yes—
154C3×#41C22×C414yes—
155C3×#41C2×D414yes—
156C2×C3×Q8—24no—
157C2×C3×Q8—21no—
158C2×C3×Q8—22no—
159C2×C3×Q8—22no—
160C2×C3×Q8—24no—
161C2×C3×Q8—24no—
162C2×C3×Q8—24no—
163C2×C3×Q8—24no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8—22no—
168C2×C3×Q8—42no—
169C2×C3×Q8—42no—
170C2×C3×Q8—41no—
171C2×C3×Q8—41no—
172C2×C3×Q8—42no—
173C2×C3×Q8—42no—
174C2×C3×Q8—42no—
175C2×C3×Q8C22⋊C414yes—
176C2×C3×Q8C22×C414yes—
177C2×C3×Q8C2×D412yes—
178C2×C3×D4—24no—
179C2×C3×D4—22no—
180C2×C3×D4—24no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—21no—
184C2×C3×D4C22⋊C414yes—
185C2×C3×D4C2×D412yes—
186C22×C3×C4—21no—
187C22×C3×C4—24no—
188C22×C3×C4—24no—
189C22×C3×C4—22no—
190C22×C3×C4—22no—
191C22×C3×C4—24no—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4—41no—
195C22×C3×C4—41no—
196C22×C3×C4—42no—
197C22×C3×C4C22×C412yes—
198C22×C3×C4C2×D412yes—
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D412yes—
204C22×C3×C4#4111yescharacteristic
205C22×C3×C4#4112yes—
206C22×C3×C4C2411yescharacteristic
207C3×Q16—216no—
208C3×SD16—216no—
209C3×M16—216no—
210C3×M16—24no—
211C3×M16—24no—
212C3×M16—24no—
213C3×M16—24no—
214C3×M16—22no—
215C3×M16—22no—
216C3×M16—22no—
217C3×M16—22no—
218C3×M16—42no—
219C3×M16C22×C414yes—
220C3×M16C2×D414yes—
221C2×C3×C8—216no—
222C2×C3×C8—24no—
223C2×C3×C8—24no—
224C2×C3×C8—22no—
225C2×C3×C8—22no—
226C2×C3×C8—22no—
227C2×C3×C8—22no—
228C2×C3×C8—22no—
229C2×C3×C8—42no—
230C2×C3×C8C2×D414yes—
231C3×C4⋊C4—22no—
232C3×C4⋊C4—22no—
233C3×C22⋊C4—41no—
234C3×C22⋊C4—41no—
235C3×C42—41no—
236C3×C42—41no—
237C2×#41—21no—
238C2×#41—22no—
239C2×#41—22no—
240C2×#41—21no—
241C2×#41C2×C3×C414yes—
242C2×#41C3×D414yes—
243C2×#41C3×D412yes—
244C2×#41C23×C312yes—
245C22×Q8—22no—
246C22×Q8—21no—
247C22×Q8—21no—
248C22×Q8—21no—
249C22×Q8—22no—
250C22×Q8—21no—
251C22×Q8—21no—
252C22×Q8—22no—
253C22×Q8C2×C3×C414yes—
254C22×Q8C3×D411yescharacteristic
255C22×Q8C23×C312yes—
256C22×D4—21no—
257C22×D4C3×D411yescharacteristic
258C23×C4—21no—
259C23×C4—21no—
260C23×C4C23×C311yescharacteristic
261#137—216no—
262#137—28no—
263#137C2×C3×C4116yes—
264C2×Q16—28no—
265C2×Q16—24no—
266C2×SD16—28no—
267C2×SD16—24no—
268#131—216no—
269#131—28no—
270#131—28no—
271C2×M16—216no—
272C2×M16—21no—
273C2×M16—22no—
274C2×M16—22no—
275C2×M16—22no—
276C2×M16—22no—
277C2×M16—21no—
278C2×M16—21no—
279C2×M16—24no—
280C2×M16—22no—
281C2×M16—22no—
282C2×M16C2×C3×C414yes—
283C2×M16C3×D414yes—
284C2×M16C3×D412yes—
285C2×M16C3×D412yes—
286C2×M16C23×C312yes—
287C2×M16C23×C312yes—
288C2×M16C23×C312yes—
289C22×C8—21no—
290C22×C8—21no—
291C22×C8—24no—
292C22×C8—21no—
293C22×C8C23×C312yes—
294#117—21no—
295#117—21no—
296#117—21no—
297#117—21no—
298C2×C4⋊C4C23×C312yes—
299C2×C22⋊C4—21no—
300C2×C42—21no—
301#103—24no—
302#103—28no—
303#102—24no—
304#101—28no—
305C23×C3—22no—
306C23×C3—24no—
307C23×C3—42no—
308C23×C3—82no—
309C23×C3C4×D412yes—
310C23×C3#12111yescharacteristic
311C23×C3#12111yescharacteristic
312C23×C3#12312yes—
313C23×C3C2×#4111yescharacteristic
314C3×Q8—22no—
315C3×Q8—24no—
316C3×Q8—24no—
317C3×Q8—24no—
318C3×Q8—42no—
319C3×Q8—42no—
320C3×Q8—44no—
321C3×Q8—44no—
322C3×Q8C2×C22⋊C414yes—
323C3×D4—24no—
324C3×D4—24no—
325C3×D4—28no—
326C3×D4—22no—
327C3×D4—24no—
328C3×D4C2×C22⋊C414yes—
329C2×C3×C4—22no—
330C2×C3×C4—24no—
331C2×C3×C4—24no—
332C2×C3×C4—24no—
333C2×C3×C4—28no—
334C2×C3×C4—24no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—44no—
338C2×C3×C4—44no—
339C2×C3×C4—42no—
340C2×C3×C4—82no—
341C2×C3×C4C2×C22⋊C412yes—
342C2×C3×C4C2×C22⋊C412yes—
343C2×C3×C4C4×D412yes—
344C2×C3×C4C4×D412yes—
345C2×C3×C4C4×D412yes—
346C2×C3×C4#12012yes—
347C2×C3×C4#12012yes—
348C2×C3×C4#12112yes—
349C2×C3×C4#12112yes—
350C2×C3×C4#12111yescharacteristic
351C2×C3×C4#12111yescharacteristic
352C2×C3×C4#12312yes—
353C2×C3×C4C23×C411yescharacteristic
354C2×C3×C4C22×D411yescharacteristic
355C2×C3×C4C22×D411yescharacteristic
356C2×C3×C4C22×D411yescharacteristic
357C2×C3×C4C22×D411yescharacteristic
358C2×C3×C4C2×#4111yescharacteristic
359C3×C8—24no—
360C3×C8—24no—
361C3×C8—24no—
362C3×C8—42no—
363C24—21no—
364C24C3×#4111yescharacteristic
365#41—24no—
366#41—24no—
367#41—24no—
368#41—24no—
369#41—24no—
370#41—22no—
371#41—22no—
372#41C3×C22⋊C418yes—
373#41C22×C3×C414yes—
374#41C2×C3×D414yes—
375C2×Q8—24no—
376C2×Q8—22no—
377C2×Q8—21no—
378C2×Q8—22no—
379C2×Q8—24no—
380C2×Q8—24no—
381C2×Q8—24no—
382C2×Q8—24no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—41no—
392C2×Q8—41no—
393C2×Q8—42no—
394C2×Q8C3×C22⋊C414yes—
395C2×Q8C22×C3×C414yes—
396C2×Q8C2×C3×D412yes—
397C2×D4—24no—
398C2×D4—22no—
399C2×D4—24no—
400C2×D4—22no—
401C2×D4—21no—
402C2×D4—22no—
403C2×D4C3×C22⋊C414yes—
404C2×D4C2×C3×D412yes—
405C22×C4—21no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—42no—
416C22×C4C22×C3×C412yes—
417C22×C4C2×C3×D412yes—
418C22×C4C2×C3×D411yescharacteristic
419C22×C4C2×C3×D411yescharacteristic
420C22×C4C2×C3×D411yescharacteristic
421C22×C4C2×C3×D411yescharacteristic
422C22×C4C2×C3×D412yes—
423C22×C4C3×#4111yescharacteristic
424C22×C4C3×#4112yes—
425C22×C4C24×C311yescharacteristic
426Q16—216no—
427SD16—216no—
428M16—216no—
429M16—24no—
430M16—24no—
431M16—24no—
432M16—24no—
433M16—22no—
434M16—22no—
435M16—22no—
436M16—22no—
437M16—42no—
438M16C22×C3×C414yes—
439M16C2×C3×D414yes—
440C2×C8—216no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—42no—
449C2×C8C2×C3×D414yes—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C22⋊C4—41no—
453C22⋊C4—41no—
454C42—41no—
455C42—41no—
456C22×C3—22no—
457C22×C3—22no—
458C22×C3—22no—
459C22×C3—24no—
460C22×C3—42no—
461C22×C3—42no—
462C22×C3—44no—
463C22×C3—84no—
464C22×C3#38612yes—
465C22×C3#38612yes—
466C22×C3C2×C4×D411yescharacteristic
467C22×C3C2×#12111yescharacteristic
468C22×C3C2×#12311yescharacteristic
469C3×C4—24no—
470C3×C4—44no—
471C3×C4—82no—
472C3×C4#38612yes—
473C3×C4#38612yes—
474C3×C4C22×C22⋊C411yescharacteristic
475C3×C4C2×C4×D411yescharacteristic
476C3×C4C2×#12011yescharacteristic
477C3×C4C2×#12111yescharacteristic
478C23—22no—
479C23—24no—
480C23—42no—
481C23—82no—
482C23C3×C4×D412yes—
483C23C3×#12111yescharacteristic
484C23C3×#12111yescharacteristic
485C23C3×#12312yes—
486C23C2×C3×#4111yescharacteristic
487Q8—24no—
488Q8—22no—
489Q8—24no—
490Q8—24no—
491Q8—44no—
492Q8—44no—
493Q8—42no—
494Q8—42no—
495Q8C2×C3×C22⋊C414yes—
496D4—28no—
497D4—24no—
498D4—24no—
499D4—24no—
500D4—22no—
501D4C2×C3×C22⋊C414yes—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—28no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—82no—
514C2×C4C2×C3×C22⋊C412yes—
515C2×C4C2×C3×C22⋊C412yes—
516C2×C4C3×C4×D412yes—
517C2×C4C3×C4×D412yes—
518C2×C4C3×C4×D412yes—
519C2×C4C3×#12012yes—
520C2×C4C3×#12012yes—
521C2×C4C3×#12111yescharacteristic
522C2×C4C3×#12111yescharacteristic
523C2×C4C3×#12112yes—
524C2×C4C3×#12112yes—
525C2×C4C3×#12312yes—
526C2×C4C23×C3×C411yescharacteristic
527C2×C4C22×C3×D411yescharacteristic
528C2×C4C22×C3×D411yescharacteristic
529C2×C4C22×C3×D411yescharacteristic
530C2×C4C22×C3×D411yescharacteristic
531C2×C4C2×C3×#4111yescharacteristic
532C8—24no—
533C8—24no—
534C8—24no—
535C8—42no—
536C2×C3—22no—
537C2×C3—21no—
538C2×C3—21no—
539C2×C3—22no—
540C2×C3—44no—
541C2×C3C2×#38611yescharacteristic
542C2×C3#306912yes—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—24no—
547C22—42no—
548C22—42no—
549C22—44no—
550C22—84no—
551C22C3×#38612yes—
552C22C3×#38612yes—
553C22C2×C3×C4×D411yescharacteristic
554C22C2×C3×#12111yescharacteristic
555C22C2×C3×#12311yescharacteristic
556C4—24no—
557C4—44no—
558C4—82no—
559C4C3×#38612yes—
560C4C3×#38612yes—
561C4C22×C3×C22⋊C411yescharacteristic
562C4C2×C3×C4×D411yescharacteristic
563C4C2×C3×#12011yescharacteristic
564C4C2×C3×#12111yescharacteristic
565C3#2100611yescharacteristic, direct product factor, Sylow
566C2—22no—
567C2—22no—
568C2—21no—
569C2—21no—
570C2—44no—
571C2C2×C3×#38611yescharacteristic
572C2C3×#306912yes—
subgroupG/H
center, socleC22×C3C2×#121
commutatorC2×C4C23×C3×C4
Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×#41C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
4. C2×C3×#419C2semidirect
5. C22×C3×#101C2semidirect
6. C3×#1908C2semidirect
7. C3×#1742C2semidirect
8. C3×#1738C2semidirect
9. #21006C3direct (N × H)
14. C2×C3×#137C4semidirect
34. C2×C3×#103C22semidirect
36. C2×C3×#101C22semidirect
37. C3×#343C22semidirect
38. C3×#341C22semidirect
42. C2×#419C2×C3semidirect
43. C22×#101C2×C3semidirect
44. #1908C2×C3semidirect
45. #1742C2×C3semidirect
46. #1738C2×C3semidirect
119. C2×#137C3×C4semidirect
139. C2×#103C22×C3semidirect
141. C2×#101C22×C3semidirect
142. #343C22×C3semidirect
143. #341C22×C3semidirect
565. C3#21006direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1372
C2×C3×#5682
C3×#28962
C2×C3×#4192
C22×C3×#1012
C3×#19082
C3×#17422
C3×#17382
#210063