Finite Group Databasebeta

← C2×C3×#1519 #10636171

Subgroups — C2×C3×#1519 #10636171

Order 768.

1866 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#121C211yesmaximal, characteristic
2C22×C3×C22⋊C8C211yesmaximal, characteristic, complemented
3C2×C3×#380C211yesmaximal, characteristic, complemented
4C3×#1519C218yesmaximal, direct product factor
5C2×C3×#327C214yesmaximal, complemented
6C2×#1519C311yesmaximal, characteristic, direct product factor, Sylow
7C23×C3×D4—21no—
8C23×C3×D4C411yescharacteristic, complemented
9C24×C3×C4C2211yescharacteristic
10C23×C3×C8—21no—
11C2×C3×#121—21no—
12C2×C3×#121—22no—
13C2×C3×#121—21no—
14C2×C3×#121—22no—
15C2×C3×#121C412yescomplemented
16C2×C3×#121C414yescomplemented
17C2×C3×#121C2212yes—
18C2×C3×#121C2214yes—
19C22×C3×C4⋊C4—21no—
20C22×C3×C4⋊C4C411yescharacteristic, complemented
21C22×C3×C22⋊C4—21no—
22C22×C3×C22⋊C4—21no—
23C2×C3×C22⋊C8—21no—
24C2×C3×C22⋊C8—22no—
25C2×C3×C22⋊C8C2212yescomplemented
26C2×C3×C22⋊C8C2214yescomplemented
27C3×#380—22no—
28C3×#380C2214yescomplemented
29C2×C3×#95C2212yescomplemented
30C3×#327C22116yescomplemented
31C22×#121C2×C311yescharacteristic
32C22×C22⋊C8C2×C311yescharacteristic, complemented
33C2×#380C2×C311yescharacteristic, complemented
34#1519C2×C318yesdirect product factor
35C2×#327C2×C314yescomplemented
36C25×C3—21no—
37C25×C3D411yescharacteristic
38C22×C3×D4—24no—
39C22×C3×D4—22no—
40C22×C3×D4—24no—
41C22×C3×D4—22no—
42C22×C3×D4—22no—
43C22×C3×D4—21no—
44C22×C3×D4—44no—
45C22×C3×D4—44no—
46C22×C3×D4—44no—
47C22×C3×D4C2×C412yes—
48C22×C3×D4C2×C414yescomplemented
49C23×C3×C4—21no—
50C23×C3×C4—22no—
51C23×C3×C4—44no—
52C23×C3×C4C2×C411yescharacteristic
53C23×C3×C4C2×C412yes—
54C23×C3×C4C2×C411yescharacteristic
55C23×C3×C4D411yescharacteristic
56C23×C3×C4D411yescharacteristic
57C23×C3×C4D412yes—
58C23×C3×C4D411yescharacteristic
59C23×C3×C4D412yes—
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2312yes—
62C22×C3×C8—24no—
63C22×C3×C8—22no—
64C22×C3×C8—44no—
65C3×#121—24no—
66C3×#121—24no—
67C3×#121—24no—
68C3×#121—22no—
69C3×#121—24no—
70C3×#121—22no—
71C3×#121—24no—
72C3×#121C2×C414yes—
73C3×#121C2×C418yescomplemented
74C3×#121C2314yes—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—24no—
78C2×C3×C4⋊C4—44no—
79C2×C3×C4⋊C4C2×C412yes—
80C2×C3×C4⋊C4C2×C414yescomplemented
81C2×C3×C22⋊C4—24no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—24no—
85C2×C3×C22⋊C4—24no—
86C2×C3×C22⋊C4—44no—
87C3×C22⋊C8—24no—
88C3×C22⋊C8—22no—
89C3×C22⋊C8C2314yescomplemented
90C3×#95—22no—
91C3×#95C2314yescomplemented
92C23×D4—21no—
93C23×D4C3×C411yescharacteristic, complemented
94C24×C4C22×C311yescharacteristic
95C23×C8—21no—
96C2×#121—21no—
97C2×#121—22no—
98C2×#121—21no—
99C2×#121—22no—
100C2×#121C3×C412yescomplemented
101C2×#121C3×C414yescomplemented
102C2×#121C22×C312yes—
103C2×#121C22×C314yes—
104C22×C4⋊C4—21no—
105C22×C4⋊C4C3×C411yescharacteristic, complemented
106C22×C22⋊C4—21no—
107C22×C22⋊C4—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—22no—
110C2×C22⋊C8C22×C312yescomplemented
111C2×C22⋊C8C22×C314yescomplemented
112#380—22no—
113#380C22×C314yescomplemented
114C2×#95C22×C312yescomplemented
115#327C22×C3116yescomplemented
116C24×C3—24no—
117C24×C3—22no—
118C24×C3—42no—
119C24×C3—42no—
120C24×C3—44no—
121C24×C3—44no—
122C24×C3—44no—
123C24×C3C22⋊C411yescharacteristic
124C24×C3C22⋊C411yescharacteristic
125C24×C3C22⋊C412yes—
126C24×C3D814yes—
127C24×C3SD1614yes—
128C24×C3C2×D411yescharacteristic
129C24×C3C2×D412yes—
130C2×C3×D4—24no—
131C2×C3×D4—28no—
132C2×C3×D4—24no—
133C2×C3×D4—22no—
134C2×C3×D4—24no—
135C2×C3×D4—44no—
136C2×C3×D4—48no—
137C2×C3×D4—48no—
138C2×C3×D4—48no—
139C2×C3×D4—416no—
140C2×C3×D4—44no—
141C2×C3×D4—48no—
142C2×C3×D4—44no—
143C2×C3×D4—48no—
144C2×C3×D4—44no—
145C2×C3×D4C22×C414yes—
146C22×C3×C4—24no—
147C22×C3×C4—24no—
148C22×C3×C4—24no—
149C22×C3×C4—24no—
150C22×C3×C4—22no—
151C22×C3×C4—24no—
152C22×C3×C4—44no—
153C22×C3×C4—44no—
154C22×C3×C4—44no—
155C22×C3×C4—44no—
156C22×C3×C4—42no—
157C22×C3×C4—42no—
158C22×C3×C4—42no—
159C22×C3×C4—44no—
160C22×C3×C4—44no—
161C22×C3×C4—44no—
162C22×C3×C4—44no—
163C22×C3×C4C22⋊C412yes—
164C22×C3×C4C22⋊C412yes—
165C22×C3×C4C22⋊C412yes—
166C22×C3×C4C22⋊C411yescharacteristic
167C22×C3×C4C22⋊C412yes—
168C22×C3×C4C22⋊C412yes—
169C22×C3×C4C22⋊C411yescharacteristic
170C22×C3×C4C22×C412yes—
171C22×C3×C4C22×C411yescharacteristic
172C22×C3×C4C22×C411yescharacteristic
173C22×C3×C4C22×C412yes—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2411yescharacteristic
181C2×C3×C8—24no—
182C2×C3×C8—44no—
183C2×C3×C8—48no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—24no—
186C3×C4⋊C4—48no—
187C3×C4⋊C4C22×C414yes—
188C3×C22⋊C4—24no—
189C3×C22⋊C4—28no—
190C3×C22⋊C4—24no—
191C3×C22⋊C4—48no—
192C25—21no—
193C25C3×D411yescharacteristic
194C22×D4—22no—
195C22×D4—24no—
196C22×D4—22no—
197C22×D4—24no—
198C22×D4—22no—
199C22×D4—21no—
200C22×D4—44no—
201C22×D4—44no—
202C22×D4—44no—
203C22×D4C2×C3×C412yes—
204C22×D4C2×C3×C414yescomplemented
205C23×C4—21no—
206C23×C4—22no—
207C23×C4—44no—
208C23×C4C2×C3×C411yescharacteristic
209C23×C4C2×C3×C412yes—
210C23×C4C2×C3×C411yescharacteristic
211C23×C4C3×D411yescharacteristic
212C23×C4C3×D411yescharacteristic
213C23×C4C3×D412yes—
214C23×C4C3×D411yescharacteristic
215C23×C4C3×D412yes—
216C23×C4C23×C311yescharacteristic
217C23×C4C23×C312yes—
218C22×C8—22no—
219C22×C8—24no—
220C22×C8—44no—
221#121—24no—
222#121—24no—
223#121—24no—
224#121—22no—
225#121—24no—
226#121—22no—
227#121—24no—
228#121C2×C3×C414yes—
229#121C2×C3×C418yescomplemented
230#121C23×C314yes—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4C2×C3×C412yes—
236C2×C4⋊C4C2×C3×C414yescomplemented
237C2×C22⋊C4—24no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—44no—
243C22⋊C8—24no—
244C22⋊C8—22no—
245C22⋊C8C23×C314yescomplemented
246#95—22no—
247#95C23×C314yescomplemented
248C23×C3—22no—
249C23×C3—22no—
250C23×C3—22no—
251C23×C3—22no—
252C23×C3—22no—
253C23×C3—22no—
254C23×C3—24no—
255C23×C3—42no—
256C23×C3—42no—
257C23×C3—42no—
258C23×C3—48no—
259C23×C3—48no—
260C23×C3—42no—
261C23×C3—48no—
262C23×C3—48no—
263C23×C3—44no—
264C23×C3—48no—
265C23×C3—44no—
266C23×C3—44no—
267C23×C3—816no—
268C23×C3#10214yes—
269C23×C3#10214yes—
270C23×C3#10218yes—
271C23×C3C2×C22⋊C412yes—
272C23×C3C2×C22⋊C411yescharacteristic
273C23×C3C2×C22⋊C412yes—
274C23×C3C2×C22⋊C411yescharacteristic
275C23×C3C2×D812yes—
276C23×C3C2×D814yes—
277C23×C3C2×SD1612yes—
278C23×C3C2×SD1614yes—
279C23×C3C22×D411yescharacteristic
280C3×D4—44no—
281C3×D4—44no—
282C3×D4—44no—
283C3×D4—44no—
284C3×D4—48no—
285C3×D4—416no—
286C3×D4—48no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—24no—
291C2×C3×C4—24no—
292C2×C3×C4—24no—
293C2×C3×C4—24no—
294C2×C3×C4—24no—
295C2×C3×C4—24no—
296C2×C3×C4—24no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—41no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—48no—
306C2×C3×C4—44no—
307C2×C3×C4—44no—
308C2×C3×C4—44no—
309C2×C3×C4—44no—
310C2×C3×C4—88no—
311C2×C3×C4#9914yes—
312C2×C3×C4C2×C22⋊C411yescharacteristic
313C2×C3×C4C2×C22⋊C412yes—
314C2×C3×C4C2×C22⋊C411yescharacteristic
315C2×C3×C4C2×C22⋊C412yes—
316C2×C3×C4C23×C411yescharacteristic
317C2×C3×C4C22×D411yescharacteristic
318C3×C8—44no—
319C24—22no—
320C24—24no—
321C24—42no—
322C24—42no—
323C24—44no—
324C24—44no—
325C24—44no—
326C24C3×C22⋊C411yescharacteristic
327C24C3×C22⋊C411yescharacteristic
328C24C3×C22⋊C412yes—
329C24C3×D814yes—
330C24C3×SD1614yes—
331C24C2×C3×D411yescharacteristic
332C24C2×C3×D412yes—
333C2×D4—24no—
334C2×D4—28no—
335C2×D4—24no—
336C2×D4—22no—
337C2×D4—24no—
338C2×D4—44no—
339C2×D4—48no—
340C2×D4—48no—
341C2×D4—48no—
342C2×D4—416no—
343C2×D4—44no—
344C2×D4—48no—
345C2×D4—44no—
346C2×D4—48no—
347C2×D4—44no—
348C2×D4C22×C3×C414yes—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4C3×C22⋊C411yescharacteristic
367C22×C4C3×C22⋊C412yes—
368C22×C4C3×C22⋊C412yes—
369C22×C4C3×C22⋊C412yes—
370C22×C4C3×C22⋊C412yes—
371C22×C4C3×C22⋊C412yes—
372C22×C4C3×C22⋊C411yescharacteristic
373C22×C4C22×C3×C411yescharacteristic
374C22×C4C22×C3×C412yes—
375C22×C4C22×C3×C411yescharacteristic
376C22×C4C22×C3×C412yes—
377C22×C4C2×C3×D412yes—
378C22×C4C2×C3×D411yescharacteristic
379C22×C4C2×C3×D412yes—
380C22×C4C2×C3×D411yescharacteristic
381C22×C4C2×C3×D412yes—
382C22×C4C2×C3×D411yescharacteristic
383C22×C4C24×C311yescharacteristic
384C2×C8—24no—
385C2×C8—44no—
386C2×C8—48no—
387C4⋊C4—22no—
388C4⋊C4—24no—
389C4⋊C4—48no—
390C4⋊C4C22×C3×C414yes—
391C22⋊C4—24no—
392C22⋊C4—28no—
393C22⋊C4—24no—
394C22⋊C4—48no—
395C22×C3—21no—
396C22×C3—22no—
397C22×C3—22no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—22no—
401C22×C3—21no—
402C22×C3—22no—
403C22×C3—24no—
404C22×C3—28no—
405C22×C3—28no—
406C22×C3—24no—
407C22×C3—48no—
408C22×C3—44no—
409C22×C3—44no—
410C22×C3—44no—
411C22×C3—44no—
412C22×C3—44no—
413C22×C3—816no—
414C22×C3—88no—
415C22×C3C2×#9912yes—
416C22×C3C2×#10214yes—
417C22×C3C2×#10214yes—
418C22×C3C2×#10212yes—
419C22×C3C2×#10212yes—
420C22×C3#42112yes—
421C22×C3C22×C22⋊C411yescharacteristic
422C22×C3C22×D811yescharacteristic
423C22×C3C22×SD1611yescharacteristic
424C3×C4—21no—
425C3×C4—21no—
426C3×C4—22no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—44no—
430C3×C4—84no—
431C3×C4C2×#9914yes—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—24no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—48no—
443C23—48no—
444C23—42no—
445C23—48no—
446C23—48no—
447C23—44no—
448C23—48no—
449C23—44no—
450C23—44no—
451C23—816no—
452C23C3×#10214yes—
453C23C3×#10214yes—
454C23C3×#10218yes—
455C23C2×C3×C22⋊C412yes—
456C23C2×C3×C22⋊C411yescharacteristic
457C23C2×C3×C22⋊C412yes—
458C23C2×C3×C22⋊C411yescharacteristic
459C23C2×C3×D812yes—
460C23C2×C3×D814yes—
461C23C2×C3×SD1612yes—
462C23C2×C3×SD1614yes—
463C23C22×C3×D411yescharacteristic
464D4—44no—
465D4—44no—
466D4—44no—
467D4—44no—
468D4—48no—
469D4—416no—
470D4—48no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—48no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—88no—
495C2×C4C3×#9914yes—
496C2×C4C2×C3×C22⋊C411yescharacteristic
497C2×C4C2×C3×C22⋊C412yes—
498C2×C4C2×C3×C22⋊C411yescharacteristic
499C2×C4C2×C3×C22⋊C412yes—
500C2×C4C23×C3×C411yescharacteristic
501C2×C4C22×C3×D411yescharacteristic
502C8—44no—
503C2×C3—21no—
504C2×C3—24no—
505C2×C3—21no—
506C2×C3—22no—
507C2×C3—24no—
508C2×C3—84no—
509C2×C3#151914yesdirect product factor
510C2×C3C22×#9911yescharacteristic
511C2×C3C22×#10211yescharacteristic
512C2×C3C2×#42111yescharacteristic
513C22—21no—
514C22—22no—
515C22—22no—
516C22—21no—
517C22—21no—
518C22—22no—
519C22—21no—
520C22—22no—
521C22—24no—
522C22—28no—
523C22—28no—
524C22—24no—
525C22—48no—
526C22—44no—
527C22—44no—
528C22—44no—
529C22—44no—
530C22—44no—
531C22—816no—
532C22—88no—
533C22C2×C3×#9912yes—
534C22C2×C3×#10214yes—
535C22C2×C3×#10214yes—
536C22C2×C3×#10212yes—
537C22C2×C3×#10212yes—
538C22C3×#42112yes—
539C22C22×C3×C22⋊C411yescharacteristic
540C22C22×C3×D811yescharacteristic
541C22C22×C3×SD1611yescharacteristic
542C4—21no—
543C4—21no—
544C4—21no—
545C4—22no—
546C4—21no—
547C4—44no—
548C4—84no—
549C4C2×C3×#9914yes—
550C3C2×#151911yescharacteristic, direct product factor, Sylow
551C2—21no—
552C2—24no—
553C2—21no—
554C2—22no—
555C2—24no—
556C2—84no—
557C2C3×#151914yesdirect product factor
558C2C22×C3×#9911yescharacteristic
559C2C22×C3×#10211yescharacteristic
560C2C2×C3×#42111yescharacteristic
subgroupG/H
center, socleC23×C3C2×C22⋊C4
commutatorC2×C4C23×C3×C4
Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×#121C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C3 ◁ C24×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C3×C22⋊C8C2semidirect
3. C2×C3×#380C2semidirect
4. C3×#1519C2direct (N × H)
5. C2×C3×#327C2semidirect
6. C2×#1519C3direct (N × H)
8. C23×C3×D4C4semidirect
15. C2×C3×#121C4semidirect
16. C2×C3×#121C4semidirect
20. C22×C3×C4⋊C4C4semidirect
25. C2×C3×C22⋊C8C22semidirect
26. C2×C3×C22⋊C8C22semidirect
28. C3×#380C22semidirect
29. C2×C3×#95C22semidirect
30. C3×#327C22semidirect
32. C22×C22⋊C8C2×C3semidirect
33. C2×#380C2×C3semidirect
34. #1519C2×C3direct (N × H)
35. C2×#327C2×C3semidirect
48. C22×C3×D4C2×C4semidirect
73. C3×#121C2×C4semidirect
80. C2×C3×C4⋊C4C2×C4semidirect
89. C3×C22⋊C8C23semidirect
91. C3×#95C23semidirect
93. C23×D4C3×C4semidirect
100. C2×#121C3×C4semidirect
101. C2×#121C3×C4semidirect
105. C22×C4⋊C4C3×C4semidirect
110. C2×C22⋊C8C22×C3semidirect
111. C2×C22⋊C8C22×C3semidirect
113. #380C22×C3semidirect
114. C2×#95C22×C3semidirect
115. #327C22×C3semidirect
204. C22×D4C2×C3×C4semidirect
229. #121C2×C3×C4semidirect
236. C2×C4⋊C4C2×C3×C4semidirect
245. C22⋊C8C23×C3semidirect
247. #95C23×C3semidirect
509. C2×C3#1519direct (N × H)
550. C3C2×#1519direct (N × H)
557. C2C3×#1519direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1212
C22×C3×C22⋊C82
C2×C3×#3802
C3×#15192
C2×C3×#3272
C2×#15193