Finite Group Databasebeta

← C3×#15135 #10633742

Subgroups — C3×#15135 #10633742

Order 768.

1814 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#399C212yesmaximal
2C2×C3×#389C212yesmaximal, complemented
3C2×C3×#384C211yesmaximal, characteristic, complemented
4C2×C3×#380C212yesmaximal
5C2×C3×C4×C22⋊C4C211yesmaximal, characteristic
6C3×#1445C212yesmaximal, complemented
7C3×#1444C212yesmaximal, complemented
8C3×#1439C212yesmaximal
9C3×C4⋊#95C211yesmaximal, characteristic, complemented
10#15135C311yesmaximal, characteristic, direct product factor, Sylow
11C24×C3×C4C2211yescharacteristic
12C22×C3×C4⋊C4C2212yes—
13C22×C3×C4⋊C4C2212yes—
14C22×C3×C4⋊C4C2214yes—
15C22×C3×C4⋊C4C2212yes—
16C22×C3×C4⋊C4C2212yes—
17C22×C3×C22⋊C4C2214yes—
18C22×C3×C22⋊C4C2211yescharacteristic
19C22×C3×C22⋊C4C2211yescharacteristic
20C22×C3×C42C2211yescharacteristic
21C22×C3×C42C2211yescharacteristic
22C3×#399—28no—
23C3×#389—28no—
24C3×#384—24no—
25C3×#380—28no—
26C3×C4×C22⋊C4—24no—
27C2×C3×#95C2212yes—
28C2×C3×#95C2211yescharacteristic
29C2×C3×#95C2211yescharacteristic
30C2×C3×#95C2214yes—
31C2×C3×#95C2214yes—
32C2×C3×#95C2212yes—
33C2×#399C2×C312yes—
34C2×#389C2×C312yescomplemented
35C2×#384C2×C311yescharacteristic, complemented
36C2×#380C2×C312yes—
37C2×C4×C22⋊C4C2×C311yescharacteristic
38#1445C2×C312yescomplemented
39#1444C2×C312yescomplemented
40#1439C2×C312yes—
41C4⋊#95C2×C311yescharacteristic, complemented
42C25×C3C2311yescharacteristic
43C23×C3×C4—22no—
44C23×C3×C4—24no—
45C23×C3×C4—24no—
46C23×C3×C4—22no—
47C23×C3×C4Q814yes—
48C23×C3×C4C2314yes—
49C23×C3×C4C2314yes—
50C23×C3×C4C2311yescharacteristic
51C23×C3×C4C2311yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C2×C3×C4⋊C4—24no—
57C2×C3×C4⋊C4—24no—
58C2×C3×C4⋊C4—24no—
59C2×C3×C4⋊C4—24no—
60C2×C3×C4⋊C4—24no—
61C2×C3×C4⋊C4—24no—
62C2×C3×C4⋊C4—28no—
63C2×C3×C4⋊C4—24no—
64C2×C3×C4⋊C4—28no—
65C2×C3×C4⋊C4—24no—
66C2×C3×C4⋊C4—24no—
67C2×C3×C4⋊C4—44no—
68C2×C3×C4⋊C4—44no—
69C2×C3×C4⋊C4D418yes—
70C2×C3×C22⋊C4—22no—
71C2×C3×C22⋊C4—22no—
72C2×C3×C22⋊C4—22no—
73C2×C3×C22⋊C4—22no—
74C2×C3×C22⋊C4—28no—
75C2×C3×C22⋊C4—28no—
76C2×C3×C22⋊C4—44no—
77C2×C3×C22⋊C4D414yes—
78C2×C3×C22⋊C4Q814yes—
79C2×C3×C42—24no—
80C2×C3×C42—22no—
81C2×C3×C42—24no—
82C2×C3×C42D414yes—
83C3×#95—22no—
84C3×#95—22no—
85C3×#95—28no—
86C3×#95—28no—
87C3×#95—24no—
88C3×#95—24no—
89C24×C4C22×C311yescharacteristic
90C22×C4⋊C4C22×C312yes—
91C22×C4⋊C4C22×C312yes—
92C22×C4⋊C4C22×C314yes—
93C22×C4⋊C4C22×C312yes—
94C22×C4⋊C4C22×C312yes—
95C22×C22⋊C4C22×C314yes—
96C22×C22⋊C4C22×C311yescharacteristic
97C22×C22⋊C4C22×C311yescharacteristic
98C22×C42C22×C311yescharacteristic
99C22×C42C22×C311yescharacteristic
100#399—28no—
101#389—28no—
102#384—24no—
103#380—28no—
104C4×C22⋊C4—24no—
105C2×#95C22×C312yes—
106C2×#95C22×C312yes—
107C2×#95C22×C314yes—
108C2×#95C22×C311yescharacteristic
109C2×#95C22×C314yes—
110C2×#95C22×C311yescharacteristic
111C24×C3—21no—
112C24×C3—22no—
113C24×C3—21no—
114C24×C3—21no—
115C24×C3—21no—
116C24×C3—22no—
117C24×C3—22no—
118C24×C3—22no—
119C24×C3C2×D412yes—
120C24×C3C2×Q812yes—
121C24×C3C2×Q812yes—
122C24×C3C2411yescharacteristic
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—21no—
129C22×C3×C4—22no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—24no—
133C22×C3×C4—22no—
134C22×C3×C4—24no—
135C22×C3×C4—21no—
136C22×C3×C4—21no—
137C22×C3×C4—24no—
138C22×C3×C4—24no—
139C22×C3×C4—21no—
140C22×C3×C4—22no—
141C22×C3×C4—24no—
142C22×C3×C4—22no—
143C22×C3×C4—24no—
144C22×C3×C4—22no—
145C22×C3×C4—28no—
146C22×C3×C4—216no—
147C22×C3×C4—24no—
148C22×C3×C4—24no—
149C22×C3×C4—24no—
150C22×C3×C4—24no—
151C22×C3×C4—24no—
152C22×C3×C4—24no—
153C22×C3×C4—24no—
154C22×C3×C4—44no—
155C22×C3×C4—48no—
156C22×C3×C4—44no—
157C22×C3×C4C2×D418yes—
158C22×C3×C4C2×D412yes—
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D412yes—
163C22×C3×C4C2×D414yes—
164C22×C3×C4C2×Q812yes—
165C22×C3×C4C2×Q812yes—
166C22×C3×C4C2×Q812yes—
167C22×C3×C4C2×Q812yes—
168C22×C3×C4#4112yes—
169C22×C3×C4#4112yes—
170C22×C3×C4#4114yes—
171C22×C3×C4#4112yes—
172C3×C4⋊C4—216no—
173C3×C4⋊C4—416no—
174C3×C4⋊C4—48no—
175C3×C4⋊C4—48no—
176C3×C4⋊C4—48no—
177C3×C22⋊C4—28no—
178C3×C22⋊C4—28no—
179C3×C22⋊C4—416no—
180C3×C42—28no—
181C3×C42—44no—
182C25C23×C311yescharacteristic
183C23×C4—22no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—22no—
187C23×C4C3×Q814yes—
188C23×C4C23×C314yes—
189C23×C4C23×C311yescharacteristic
190C23×C4C23×C314yes—
191C23×C4C23×C311yescharacteristic
192C23×C4C23×C311yescharacteristic
193C23×C4C23×C311yescharacteristic
194C23×C4C23×C311yescharacteristic
195C23×C4C23×C311yescharacteristic
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4C3×D418yes—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4C3×D414yes—
218C2×C22⋊C4C3×Q814yes—
219C2×C42—24no—
220C2×C42—22no—
221C2×C42—24no—
222C2×C42C3×D414yes—
223#95—22no—
224#95—22no—
225#95—28no—
226#95—28no—
227#95—24no—
228#95—24no—
229C23×C3—22no—
230C23×C3—22no—
231C23×C3—22no—
232C23×C3—24no—
233C23×C3—22no—
234C23×C3—22no—
235C23×C3—22no—
236C23×C3—24no—
237C23×C3—24no—
238C23×C3—24no—
239C23×C3—24no—
240C23×C3—24no—
241C23×C3—42no—
242C23×C3—41no—
243C23×C3—41no—
244C23×C3—41no—
245C23×C3—41no—
246C23×C3—42no—
247C23×C3—42no—
248C23×C3—42no—
249C23×C3—42no—
250C23×C3—42no—
251C23×C3C4⋊Q814yes—
252C23×C3C22×D411yescharacteristic
253C23×C3C22×D411yescharacteristic
254C23×C3C22×D412yes—
255C23×C3C22×Q811yescharacteristic
256C23×C3C22×Q811yescharacteristic
257C23×C3C2×#4111yescharacteristic
258C23×C3C2×#4111yescharacteristic
259C23×C3C2×#4112yes—
260C23×C3C2×#4111yescharacteristic
261C23×C3H(5,𝔽2)12yes—
262C23×C32−1+412yes—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—24no—
268C2×C3×C4—24no—
269C2×C3×C4—24no—
270C2×C3×C4—24no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—24no—
275C2×C3×C4—28no—
276C2×C3×C4—28no—
277C2×C3×C4—24no—
278C2×C3×C4—28no—
279C2×C3×C4—24no—
280C2×C3×C4—42no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—44no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4—44no—
289C2×C3×C4—44no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—44no—
293C2×C3×C4—416no—
294C2×C3×C4—44no—
295C2×C3×C4—44no—
296C2×C3×C4—44no—
297C2×C3×C4—44no—
298C2×C3×C4—44no—
299C2×C3×C4—44no—
300C2×C3×C4—44no—
301C2×C3×C4#12114yes—
302C2×C3×C4C22⋊Q814yes—
303C2×C3×C4C22⋊Q814yes—
304C2×C3×C4C22⋊Q818yes—
305C2×C3×C4#12314yes—
306C2×C3×C4#12318yes—
307C2×C3×C4#12714yes—
308C24—21no—
309C24—22no—
310C24—21no—
311C24—21no—
312C24—21no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24C2×C3×D412yes—
317C24C2×C3×Q812yes—
318C24C2×C3×Q812yes—
319C24C24×C311yescharacteristic
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—28no—
343C22×C4—216no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—44no—
352C22×C4—48no—
353C22×C4—44no—
354C22×C4C2×C3×D418yes—
355C22×C4C2×C3×D412yes—
356C22×C4C2×C3×D412yes—
357C22×C4C2×C3×D412yes—
358C22×C4C2×C3×D412yes—
359C22×C4C2×C3×D412yes—
360C22×C4C2×C3×D414yes—
361C22×C4C2×C3×Q812yes—
362C22×C4C2×C3×Q812yes—
363C22×C4C2×C3×Q812yes—
364C22×C4C2×C3×Q812yes—
365C22×C4C3×#4112yes—
366C22×C4C3×#4112yes—
367C22×C4C3×#4114yes—
368C22×C4C3×#4112yes—
369C4⋊C4—216no—
370C4⋊C4—416no—
371C4⋊C4—48no—
372C4⋊C4—48no—
373C4⋊C4—48no—
374C22⋊C4—28no—
375C22⋊C4—28no—
376C22⋊C4—416no—
377C42—28no—
378C42—44no—
379C22×C3—24no—
380C22×C3—28no—
381C22×C3—42no—
382C22×C3—42no—
383C22×C3—44no—
384C22×C3—44no—
385C22×C3—44no—
386C22×C3—42no—
387C22×C3—42no—
388C22×C3—42no—
389C22×C3—42no—
390C22×C3C2×#12111yescharacteristic
391C22×C3C2×C22⋊Q811yescharacteristic
392C22×C3C2×C22⋊Q811yescharacteristic
393C22×C3C2×C22⋊Q812yes—
394C22×C3C2×#12312yes—
395C22×C3C2×#12311yescharacteristic
396C22×C3C2×#12711yescharacteristic
397C22×C3C2×C4⋊Q811yescharacteristic
398C22×C3#53211yescharacteristic
399C22×C3#53311yescharacteristic
400C22×C3#53311yescharacteristic
401C22×C3#53511yescharacteristic
402C22×C3#53611yescharacteristic
403C22×C3#54212yes—
404C22×C3#54412yes—
405C22×C3D4212yes—
406C22×C3#54712yes—
407C22×C3#54712yes—
408C22×C3D4×Q812yes—
409C22×C3#55012yes—
410C22×C3#55212yes—
411C22×C3#55312yes—
412C22×C3#55412yes—
413C3×C4—24no—
414C3×C4—24no—
415C3×C4—42no—
416C3×C4—48no—
417C3×C4—42no—
418C3×C4—42no—
419C3×C4—84no—
420C3×C4#39918yescomplemented
421C23—22no—
422C23—22no—
423C23—24no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—24no—
429C23—24no—
430C23—24no—
431C23—24no—
432C23—24no—
433C23—42no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23C3×C4⋊Q814yes—
444C23C22×C3×D411yescharacteristic
445C23C22×C3×D411yescharacteristic
446C23C22×C3×D412yes—
447C23C22×C3×Q811yescharacteristic
448C23C22×C3×Q811yescharacteristic
449C23C2×C3×#4111yescharacteristic
450C23C2×C3×#4111yescharacteristic
451C23C2×C3×#4112yes—
452C23C2×C3×#4111yescharacteristic
453C23C3×H(5,𝔽2)12yes—
454C23C3×2−1+412yes—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—28no—
468C2×C4—28no—
469C2×C4—24no—
470C2×C4—28no—
471C2×C4—24no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—44no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—416no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4C3×#12114yes—
494C2×C4C3×C22⋊Q814yes—
495C2×C4C3×C22⋊Q814yes—
496C2×C4C3×C22⋊Q818yes—
497C2×C4C3×#12314yes—
498C2×C4C3×#12318yes—
499C2×C4C3×#12714yes—
500C2×C3—44no—
501C2×C3C2×#39911yescharacteristic
502C2×C3#242212yes—
503C2×C3#242312yes—
504C2×C3#244012yes—
505C2×C3#244112yes—
506C2×C3#248611yescharacteristic
507C2×C3#248711yescharacteristic
508C2×C3#254111yescharacteristic
509C2×C3#254211yescharacteristic
510C2×C3#262811yescharacteristic
511C2×C3#285711yescharacteristic
512C22—28no—
513C22—24no—
514C22—42no—
515C22—42no—
516C22—44no—
517C22—44no—
518C22—44no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22C2×C3×#12111yescharacteristic
524C22C2×C3×C22⋊Q811yescharacteristic
525C22C2×C3×C22⋊Q812yes—
526C22C2×C3×C22⋊Q811yescharacteristic
527C22C2×C3×#12312yes—
528C22C2×C3×#12311yescharacteristic
529C22C2×C3×#12711yescharacteristic
530C22C2×C3×C4⋊Q811yescharacteristic
531C22C3×#53211yescharacteristic
532C22C3×#53311yescharacteristic
533C22C3×#53311yescharacteristic
534C22C3×#53511yescharacteristic
535C22C3×#53611yescharacteristic
536C22C3×#54212yes—
537C22C3×#54412yes—
538C22C3×D4212yes—
539C22C3×#54712yes—
540C22C3×#54712yes—
541C22C3×D4×Q812yes—
542C22C3×#55012yes—
543C22C3×#55212yes—
544C22C3×#55312yes—
545C22C3×#55412yes—
546C4—24no—
547C4—24no—
548C4—42no—
549C4—48no—
550C4—42no—
551C4—42no—
552C4—84no—
553C4C3×#39918yescomplemented
554C3#1513511yescharacteristic, direct product factor, Sylow
555C2—44no—
556C2C2×C3×#39911yescharacteristic
557C2C3×#242212yes—
558C2C3×#242312yes—
559C2C3×#244012yes—
560C2C3×#244112yes—
561C2C3×#248611yescharacteristic
562C2C3×#248711yescharacteristic
563C2C3×#254111yescharacteristic
564C2C3×#254211yescharacteristic
565C2C3×#262811yescharacteristic
566C2C3×#285711yescharacteristic
subgroupG/H
center, socleC24×C3C24
commutator, Frattini, cosocleC24C24×C3
prime radicalC25×C3C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#389C2semidirect
3. C2×C3×#384C2semidirect
6. C3×#1445C2semidirect
7. C3×#1444C2semidirect
9. C3×C4⋊#95C2semidirect
10. #15135C3direct (N × H)
34. C2×#389C2×C3semidirect
35. C2×#384C2×C3semidirect
38. #1445C2×C3semidirect
39. #1444C2×C3semidirect
41. C4⋊#95C2×C3semidirect
420. C3×C4#399semidirect
553. C4C3×#399semidirect
554. C3#15135direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3992
C2×C3×#3892
C2×C3×#3842
C2×C3×#3802
C2×C3×C4×C22⋊C42
C3×#14452
C3×#14442
C3×#14392
C3×C4⋊#952
#151353