Finite Group Databasebeta

← C2×C3×#3094 #10679116

Subgroups — C2×C3×#3094 #10679116

Order 768.

2670 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#121C211yesmaximal, characteristic
2C22×C3×#117C211yesmaximal, characteristic, complemented
3C3×#3094C2116yesmaximal, direct product factor
4C2×C3×#481C212yesmaximal, complemented
5C2×C3×#482C214yesmaximal, complemented
6C2×C3×#480C212yesmaximal, complemented
7C2×C3×#428C212yesmaximal, complemented
8C2×C3×#418C212yesmaximal, complemented
9C2×C3×#407C211yesmaximal, characteristic, complemented
10C2×#3094C311yesmaximal, characteristic, direct product factor, Sylow
11C23×C3×D4—21no—
12C23×C3×D4C2211yescharacteristic
13C24×C3×C4C2211yescharacteristic
14C22×C3×M16C2212yes—
15C2×C3×#121—21no—
16C2×C3×#121—24no—
17C2×C3×#121—21no—
18C2×C3×#121C2212yes—
19C2×C3×#121C2212yes—
20C2×C3×#121C2218yes—
21C2×C3×#117—21no—
22C2×C3×#117—21no—
23C2×C3×#117—24no—
24C2×C3×#117C2212yes—
25C2×C3×#117C2212yescomplemented
26C2×C3×#117C2218yescomplemented
27C22×C3×C4⋊C4C2211yescharacteristic
28C22×C3×C4⋊C4C2211yescharacteristic, complemented
29C22×C3×C4⋊C4C2211yescharacteristic, complemented
30C22×C3×C22⋊C4—21no—
31C22×C3×C22⋊C4—21no—
32C22×C3×C42—21no—
33C3×#482C22132yescomplemented
34C3×#481C22116yescomplemented
35C3×#480C22116yescomplemented
36C3×#428C22116yescomplemented
37C2×C3×#107C2214yescomplemented
38C2×C3×#106C2214yescomplemented
39C3×#418C22116yescomplemented
40C2×C3×#102C2214yescomplemented
41C2×C3×#102C2214yescomplemented
42C3×#407C2218yescomplemented
43C2×C3×C22⋊C8C2212yescomplemented
44C2×C3×C22⋊C8C2212yescomplemented
45C22×#121C2×C311yescharacteristic
46C22×#117C2×C311yescharacteristic, complemented
47#3094C2×C3116yesdirect product factor
48C2×#481C2×C312yescomplemented
49C2×#482C2×C314yescomplemented
50C2×#480C2×C312yescomplemented
51C2×#428C2×C312yescomplemented
52C2×#418C2×C312yescomplemented
53C2×#407C2×C311yescharacteristic, complemented
54C25×C3—21no—
55C25×C3D411yescharacteristic
56C22×C3×D4—28no—
57C22×C3×D4—22no—
58C22×C3×D4—22no—
59C22×C3×D4—22no—
60C22×C3×D4—21no—
61C22×C3×D4—28no—
62C22×C3×D4—48no—
63C22×C3×D4C2312yes—
64C22×C3×D4C2314yes—
65C23×C3×C4—28no—
66C23×C3×C4—21no—
67C23×C3×C4—21no—
68C23×C3×C4—21no—
69C23×C3×C4D411yescharacteristic
70C23×C3×C4D411yescharacteristic
71C23×C3×C4D414yes—
72C23×C3×C4D411yescharacteristic
73C23×C3×C4C2311yescharacteristic
74C23×C3×C4C2311yescharacteristic
75C23×C3×C4C2314yes—
76C23×C3×C4C2311yescharacteristic
77C2×C3×M16—28no—
78C2×C3×M16C2318yes—
79C22×C3×C8C2314yes—
80C3×#121—28no—
81C3×#121—24no—
82C3×#121—24no—
83C3×#121—28no—
84C3×#121C2318yes—
85C3×#121C2318yes—
86C3×#117—24no—
87C3×#117—28no—
88C3×#117—28no—
89C3×#117—24no—
90C3×#117C2318yes—
91C3×#117C2318yescomplemented
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—22no—
95C2×C3×C4⋊C4—22no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4C2312yes—
99C2×C3×C4⋊C4C2312yes—
100C2×C3×C4⋊C4C2312yes—
101C2×C3×C4⋊C4C2314yes—
102C2×C3×C4⋊C4C2314yescomplemented
103C2×C3×C4⋊C4C2314yescomplemented
104C2×C3×C22⋊C4—22no—
105C2×C3×C22⋊C4—28no—
106C2×C3×C22⋊C4—22no—
107C2×C3×C22⋊C4—28no—
108C2×C3×C22⋊C4—22no—
109C2×C3×C22⋊C4—22no—
110C2×C3×C42—22no—
111C2×C3×C42—28no—
112C2×C3×C42—22no—
113C3×#107C23116yescomplemented
114C3×#106C23116yescomplemented
115C3×#102C23116yescomplemented
116C3×#102C23116yescomplemented
117C3×C22⋊C8C2318yescomplemented
118C3×C22⋊C8C2318yescomplemented
119C23×D4—21no—
120C23×D4C22×C311yescharacteristic
121C24×C4C22×C311yescharacteristic
122C22×M16C22×C312yes—
123C2×#121—21no—
124C2×#121—24no—
125C2×#121—21no—
126C2×#121C22×C312yes—
127C2×#121C22×C312yes—
128C2×#121C22×C318yes—
129C2×#117—21no—
130C2×#117—21no—
131C2×#117—24no—
132C2×#117C22×C312yes—
133C2×#117C22×C312yescomplemented
134C2×#117C22×C318yescomplemented
135C22×C4⋊C4C22×C311yescharacteristic
136C22×C4⋊C4C22×C311yescharacteristic, complemented
137C22×C4⋊C4C22×C311yescharacteristic, complemented
138C22×C22⋊C4—21no—
139C22×C22⋊C4—21no—
140C22×C42—21no—
141#482C22×C3132yescomplemented
142#481C22×C3116yescomplemented
143#480C22×C3116yescomplemented
144#428C22×C3116yescomplemented
145C2×#107C22×C314yescomplemented
146C2×#106C22×C314yescomplemented
147#418C22×C3116yescomplemented
148C2×#102C22×C314yescomplemented
149C2×#102C22×C314yescomplemented
150#407C22×C318yescomplemented
151C2×C22⋊C8C22×C312yescomplemented
152C2×C22⋊C8C22×C312yescomplemented
153C24×C3—24no—
154C24×C3—24no—
155C24×C3—24no—
156C24×C3—22no—
157C24×C3—42no—
158C24×C3—44no—
159C24×C3—44no—
160C24×C3—44no—
161C24×C3C2×D411yescharacteristic
162C24×C3C2×D414yes—
163C24×C3C2×D411yescharacteristic
164C24×C3C2×D411yescharacteristic
165C2×C3×D4—28no—
166C2×C3×D4—28no—
167C2×C3×D4—24no—
168C2×C3×D4—22no—
169C2×C3×D4—28no—
170C2×C3×D4—216no—
171C2×C3×D4—432no—
172C2×C3×D4—48no—
173C2×C3×D4—48no—
174C2×C3×D4—48no—
175C2×C3×D4—44no—
176C2×C3×D4C2414yes—
177C22×C3×C4—24no—
178C22×C3×C4—24no—
179C22×C3×C4—216no—
180C22×C3×C4—24no—
181C22×C3×C4—24no—
182C22×C3×C4—24no—
183C22×C3×C4—216no—
184C22×C3×C4—24no—
185C22×C3×C4—24no—
186C22×C3×C4—24no—
187C22×C3×C4—22no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—44no—
191C22×C3×C4—44no—
192C22×C3×C4—44no—
193C22×C3×C4C2×D414yes—
194C22×C3×C4C2×D414yes—
195C22×C3×C4C2×D412yes—
196C22×C3×C4C2×D412yes—
197C22×C3×C4C2×D414yes—
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D412yes—
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4#4118yes—
203C22×C3×C4C2412yes—
204C22×C3×C4C2412yes—
205C22×C3×C4C2412yes—
206C22×C3×C4C2411yescharacteristic
207C3×M16—216no—
208C2×C3×C8—28no—
209C2×C3×C8C2418yes—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—24no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—24no—
215C3×C4⋊C4—24no—
216C3×C4⋊C4C2414yes—
217C3×C4⋊C4C2414yes—
218C3×C4⋊C4C2414yes—
219C3×C22⋊C4—28no—
220C3×C22⋊C4—28no—
221C3×C22⋊C4—28no—
222C3×C22⋊C4—28no—
223C3×C42—28no—
224C3×C42—28no—
225C25—21no—
226C25C3×D411yescharacteristic
227C22×D4—22no—
228C22×D4—28no—
229C22×D4—22no—
230C22×D4—21no—
231C22×D4—22no—
232C22×D4—28no—
233C22×D4—48no—
234C22×D4C23×C312yes—
235C22×D4C23×C314yes—
236C23×C4—28no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4C3×D411yescharacteristic
241C23×C4C3×D414yes—
242C23×C4C3×D411yescharacteristic
243C23×C4C3×D411yescharacteristic
244C23×C4C23×C311yescharacteristic
245C23×C4C23×C311yescharacteristic
246C23×C4C23×C314yes—
247C23×C4C23×C311yescharacteristic
248C2×M16—28no—
249C2×M16C23×C318yes—
250C22×C8C23×C314yes—
251#121—28no—
252#121—24no—
253#121—24no—
254#121—28no—
255#121C23×C318yes—
256#121C23×C318yes—
257#117—24no—
258#117—28no—
259#117—28no—
260#117—24no—
261#117C23×C318yes—
262#117C23×C318yescomplemented
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4C23×C312yes—
270C2×C4⋊C4C23×C314yes—
271C2×C4⋊C4C23×C312yes—
272C2×C4⋊C4C23×C312yes—
273C2×C4⋊C4C23×C314yescomplemented
274C2×C4⋊C4C23×C314yescomplemented
275C2×C22⋊C4—22no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—28no—
281C2×C42—22no—
282C2×C42—28no—
283C2×C42—22no—
284#107C23×C3116yescomplemented
285#106C23×C3116yescomplemented
286#102C23×C3116yescomplemented
287#102C23×C3116yescomplemented
288C22⋊C8C23×C318yescomplemented
289C22⋊C8C23×C318yescomplemented
290C23×C3—22no—
291C23×C3—22no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3—28no—
295C23×C3—22no—
296C23×C3—28no—
297C23×C3—22no—
298C23×C3—22no—
299C23×C3—24no—
300C23×C3—44no—
301C23×C3—416no—
302C23×C3—48no—
303C23×C3—48no—
304C23×C3—48no—
305C23×C3—44no—
306C23×C3—44no—
307C23×C3—44no—
308C23×C3—816no—
309C23×C3#12314yes—
310C23×C3AGL(1,ℤ/8)14yes—
311C23×C3C22×D412yes—
312C23×C3C22×D412yes—
313C23×C3C22×D412yes—
314C23×C3C22×D411yescharacteristic
315C3×D4—28no—
316C3×D4—416no—
317C3×D4—416no—
318C3×D4—48no—
319C3×D4—44no—
320C2×C3×C4—28no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—28no—
324C2×C3×C4—24no—
325C2×C3×C4—24no—
326C2×C3×C4—28no—
327C2×C3×C4—24no—
328C2×C3×C4—24no—
329C2×C3×C4—24no—
330C2×C3×C4—24no—
331C2×C3×C4—24no—
332C2×C3×C4—48no—
333C2×C3×C4—48no—
334C2×C3×C4—48no—
335C2×C3×C4—44no—
336C2×C3×C4—44no—
337C2×C3×C4—44no—
338C2×C3×C4#12318yes—
339C2×C3×C4#12314yes—
340C2×C3×C4C22×D412yes—
341C2×C3×C4C22×D412yes—
342C2×C3×C4C22×D412yes—
343C2×C3×C4C22×D411yescharacteristic
344C2×C3×C4C2×#4114yes—
345C2×C3×C4C2×#4118yes—
346C2×C3×C4C2511yescharacteristic
347C3×C8—28no—
348C24—24no—
349C24—24no—
350C24—24no—
351C24—22no—
352C24—42no—
353C24—44no—
354C24—44no—
355C24—44no—
356C24C2×C3×D411yescharacteristic
357C24C2×C3×D414yes—
358C24C2×C3×D411yescharacteristic
359C24C2×C3×D411yescharacteristic
360C2×D4—28no—
361C2×D4—28no—
362C2×D4—24no—
363C2×D4—22no—
364C2×D4—28no—
365C2×D4—216no—
366C2×D4—432no—
367C2×D4—48no—
368C2×D4—48no—
369C2×D4—48no—
370C2×D4—44no—
371C2×D4C24×C314yes—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—216no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—216no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4C2×C3×D411yescharacteristic
389C22×C4C2×C3×D414yes—
390C22×C4C2×C3×D411yescharacteristic
391C22×C4C2×C3×D414yes—
392C22×C4C2×C3×D412yes—
393C22×C4C2×C3×D412yes—
394C22×C4C2×C3×D414yes—
395C22×C4C2×C3×D411yescharacteristic
396C22×C4C2×C3×D412yes—
397C22×C4C3×#4118yes—
398C22×C4C24×C312yes—
399C22×C4C24×C312yes—
400C22×C4C24×C311yescharacteristic
401C22×C4C24×C312yes—
402M16—216no—
403C2×C8—28no—
404C2×C8C24×C318yes—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4C24×C314yes—
412C4⋊C4C24×C314yes—
413C4⋊C4C24×C314yes—
414C22⋊C4—28no—
415C22⋊C4—28no—
416C22⋊C4—28no—
417C22⋊C4—28no—
418C42—28no—
419C42—28no—
420C22×C3—28no—
421C22×C3—28no—
422C22×C3—22no—
423C22×C3—22no—
424C22×C3—22no—
425C22×C3—22no—
426C22×C3—22no—
427C22×C3—22no—
428C22×C3—22no—
429C22×C3—416no—
430C22×C3—44no—
431C22×C3—44no—
432C22×C3—44no—
433C22×C3—44no—
434C22×C3—816no—
435C22×C3—88no—
436C22×C3C2×#12314yes—
437C22×C3C2×#12312yes—
438C22×C3C2×AGL(1,ℤ/8)14yes—
439C22×C3C2×AGL(1,ℤ/8)12yes—
440C22×C3#57512yes—
441C22×C3C23×D411yescharacteristic
442C3×C4—22no—
443C3×C4—22no—
444C3×C4—44no—
445C3×C4—44no—
446C3×C4—44no—
447C3×C4C2×#12314yes—
448C3×C4C2×#12312yes—
449C3×C4C22×#4112yes—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—28no—
455C23—22no—
456C23—22no—
457C23—28no—
458C23—22no—
459C23—24no—
460C23—44no—
461C23—416no—
462C23—48no—
463C23—48no—
464C23—44no—
465C23—44no—
466C23—48no—
467C23—44no—
468C23—816no—
469C23C3×#12314yes—
470C23C3×AGL(1,ℤ/8)14yes—
471C23C22×C3×D412yes—
472C23C22×C3×D412yes—
473C23C22×C3×D412yes—
474C23C22×C3×D411yescharacteristic
475D4—28no—
476D4—416no—
477D4—416no—
478D4—48no—
479D4—44no—
480C2×C4—28no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—28no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—28no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—48no—
493C2×C4—48no—
494C2×C4—44no—
495C2×C4—48no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4C3×#12318yes—
499C2×C4C3×#12314yes—
500C2×C4C22×C3×D412yes—
501C2×C4C22×C3×D412yes—
502C2×C4C22×C3×D412yes—
503C2×C4C22×C3×D411yescharacteristic
504C2×C4C2×C3×#4114yes—
505C2×C4C2×C3×#4118yes—
506C2×C4C25×C311yescharacteristic
507C8—28no—
508C2×C3—24no—
509C2×C3—24no—
510C2×C3—42no—
511C2×C3—84no—
512C2×C3#309414yesdirect product factor
513C2×C3C22×#12311yescharacteristic
514C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
515C2×C3C2×#57511yescharacteristic
516C22—22no—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—28no—
521C22—28no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—416no—
526C22—44no—
527C22—44no—
528C22—44no—
529C22—44no—
530C22—816no—
531C22—88no—
532C22C2×C3×#12314yes—
533C22C2×C3×#12312yes—
534C22C2×C3×AGL(1,ℤ/8)14yes—
535C22C2×C3×AGL(1,ℤ/8)12yes—
536C22C3×#57512yes—
537C22C23×C3×D411yescharacteristic
538C4—22no—
539C4—22no—
540C4—44no—
541C4—44no—
542C4—44no—
543C4C2×C3×#12314yes—
544C4C2×C3×#12312yes—
545C4C22×C3×#4112yes—
546C3C2×#309411yescharacteristic, direct product factor, Sylow
547C2—24no—
548C2—24no—
549C2—42no—
550C2—84no—
551C2C3×#309414yesdirect product factor
552C2C22×C3×#12311yescharacteristic
553C2C22×C3×AGL(1,ℤ/8)11yescharacteristic
554C2C2×C3×#57511yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC2×C4C25×C3
prime radicalC22×C3×#121C2
LCSG ▷ C2×C4 ▷ C2 ▷ 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×#117C2semidirect
3. C3×#3094C2direct (N × H)
4. C2×C3×#481C2semidirect
5. C2×C3×#482C2semidirect
6. C2×C3×#480C2semidirect
7. C2×C3×#428C2semidirect
8. C2×C3×#418C2semidirect
9. C2×C3×#407C2semidirect
10. C2×#3094C3direct (N × H)
25. C2×C3×#117C22semidirect
26. C2×C3×#117C22semidirect
28. C22×C3×C4⋊C4C22semidirect
29. C22×C3×C4⋊C4C22semidirect
33. C3×#482C22semidirect
34. C3×#481C22semidirect
35. C3×#480C22semidirect
36. C3×#428C22semidirect
37. C2×C3×#107C22semidirect
38. C2×C3×#106C22semidirect
39. C3×#418C22semidirect
40. C2×C3×#102C22semidirect
41. C2×C3×#102C22semidirect
42. C3×#407C22semidirect
43. C2×C3×C22⋊C8C22semidirect
44. C2×C3×C22⋊C8C22semidirect
46. C22×#117C2×C3semidirect
47. #3094C2×C3direct (N × H)
48. C2×#481C2×C3semidirect
49. C2×#482C2×C3semidirect
50. C2×#480C2×C3semidirect
51. C2×#428C2×C3semidirect
52. C2×#418C2×C3semidirect
53. C2×#407C2×C3semidirect
91. C3×#117C23semidirect
102. C2×C3×C4⋊C4C23semidirect
103. C2×C3×C4⋊C4C23semidirect
113. C3×#107C23semidirect
114. C3×#106C23semidirect
115. C3×#102C23semidirect
116. C3×#102C23semidirect
117. C3×C22⋊C8C23semidirect
118. C3×C22⋊C8C23semidirect
133. C2×#117C22×C3semidirect
134. C2×#117C22×C3semidirect
136. C22×C4⋊C4C22×C3semidirect
137. C22×C4⋊C4C22×C3semidirect
141. #482C22×C3semidirect
142. #481C22×C3semidirect
143. #480C22×C3semidirect
144. #428C22×C3semidirect
145. C2×#107C22×C3semidirect
146. C2×#106C22×C3semidirect
147. #418C22×C3semidirect
148. C2×#102C22×C3semidirect
149. C2×#102C22×C3semidirect
150. #407C22×C3semidirect
151. C2×C22⋊C8C22×C3semidirect
152. C2×C22⋊C8C22×C3semidirect
262. #117C23×C3semidirect
273. C2×C4⋊C4C23×C3semidirect
274. C2×C4⋊C4C23×C3semidirect
284. #107C23×C3semidirect
285. #106C23×C3semidirect
286. #102C23×C3semidirect
287. #102C23×C3semidirect
288. C22⋊C8C23×C3semidirect
289. C22⋊C8C23×C3semidirect
512. C2×C3#3094direct (N × H)
546. C3C2×#3094direct (N × H)
551. C2C3×#3094direct (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×#1172
C3×#30942
C2×C3×#4812
C2×C3×#4822
C2×C3×#4802
C2×C3×#4282
C2×C3×#4182
C2×C3×#4072
C2×#30943