Finite Group Databasebeta

← C3×#23574 #10642181

Subgroups — C3×#23574 #10642181

Order 768.

1228 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#533C211yesmaximal, characteristic, complemented
2C2×C3×#440C212yesmaximal, complemented
3C3×C4×#121C211yesmaximal, characteristic
4C3×#2057C211yesmaximal, characteristic, complemented
5C3×#2043C211yesmaximal, characteristic, complemented
6C3×#2036C212yesmaximal, complemented
7C3×#2023C211yesmaximal, characteristic, complemented
8C3×#2013C211yesmaximal, characteristic, complemented
9C3×#2000C212yesmaximal
10C3×#1763C212yesmaximal, complemented
11C3×#1749C211yesmaximal, characteristic, complemented
12#23574C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2212yes—
14C3×#533—28no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#125—21no—
18C2×C3×C22⋊Q8—21no—
19C2×C3×C22⋊Q8C2212yes—
20C2×C3×#121C2211yescharacteristic
21C2×C3×C4×Q8—21no—
22C2×C3×C4×Q8C2212yes—
23C2×C3×C4×D4—21no—
24C2×C3×C4×D4C2211yescharacteristic
25C2×C3×#117—21no—
26C22×C3×C42C2211yescharacteristic
27C3×#440—28no—
28C2×C3×#107C2212yes—
29C2×C3×#103C2212yes—
30C2×C3×#103C2212yes—
31C2×C3×#103C2212yes—
32C2×C3×#102C2212yes—
33C2×C3×C22⋊C8C2212yes—
34C2×C3×#97C2212yes—
35C3×#390C2211yescharacteristic
36C3×#389C2211yescharacteristic
37C3×#388—21no—
38C3×#386—21no—
39C3×#380C2211yescharacteristic
40C3×C4×C4⋊C4C2211yescharacteristic
41C3×C4×C22⋊C4—21no—
42C3×#340C2212yes—
43C3×#340C2212yes—
44C3×#336C2211yescharacteristic
45C3×#336C2211yescharacteristic
46C2×#533C2×C311yescharacteristic, complemented
47C2×#440C2×C312yescomplemented
48C4×#121C2×C311yescharacteristic
49#2057C2×C311yescharacteristic, complemented
50#2043C2×C311yescharacteristic, complemented
51#2036C2×C312yescomplemented
52#2023C2×C311yescharacteristic, complemented
53#2013C2×C311yescharacteristic, complemented
54#2000C2×C312yes—
55#1763C2×C312yescomplemented
56#1749C2×C311yescharacteristic, complemented
57C22×C3×Q8—21no—
58C22×C3×Q8C2312yes—
59C22×C3×D4—21no—
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4—21no—
62C23×C3×C4D411yescharacteristic
63C23×C3×C4D411yescharacteristic
64C23×C3×C4C2311yescharacteristic
65C2×C3×SD16—24no—
66C2×C3×SD16—24no—
67C2×C3×SD16—24no—
68C22×C3×C8C2312yes—
69C22×C3×C8C2312yes—
70C3×C4⋊Q8—24no—
71C3×C4⋊Q8—24no—
72C3×#125—28no—
73C3×C22⋊Q8—28no—
74C3×C22⋊Q8—24no—
75C3×C22⋊Q8—24no—
76C3×#121—22no—
77C3×#121—22no—
78C3×C4×Q8—28no—
79C3×C4×Q8—24no—
80C3×C4×Q8D418yescomplemented
81C3×C4×D4—24no—
82C3×C4×D4—44no—
83C3×#117—28no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—22no—
88C2×C3×C4⋊C4—22no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2312yes—
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C22⋊C4—22no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C42—22no—
101C2×C3×C42—24no—
102C2×C3×C42—21no—
103C2×C3×C42D411yescharacteristic
104C2×C3×C42D411yescharacteristic
105C2×C3×C42C2311yescharacteristic
106C2×C3×C42C2311yescharacteristic
107C3×#107—24no—
108C3×#103—24no—
109C3×#103—24no—
110C3×#103—24no—
111C3×#102—24no—
112C3×C22⋊C8—24no—
113C3×#97—24no—
114C3×#95—21no—
115C3×#95—21no—
116C3×#95—21no—
117C22×SD16C22×C312yes—
118#533—28no—
119C2×C4⋊Q8C22×C311yescharacteristic
120C2×C4⋊Q8C22×C311yescharacteristic
121C2×#125—21no—
122C2×C22⋊Q8—21no—
123C2×C22⋊Q8C22×C312yes—
124C2×#121C22×C311yescharacteristic
125C2×C4×Q8—21no—
126C2×C4×Q8C22×C312yes—
127C2×C4×D4—21no—
128C2×C4×D4C22×C311yescharacteristic
129C2×#117—21no—
130C22×C42C22×C311yescharacteristic
131#440—28no—
132C2×#107C22×C312yes—
133C2×#103C22×C312yes—
134C2×#103C22×C312yes—
135C2×#103C22×C312yes—
136C2×#102C22×C312yes—
137C2×C22⋊C8C22×C312yes—
138C2×#97C22×C312yes—
139#390C22×C311yescharacteristic
140#389C22×C311yescharacteristic
141#388—21no—
142#386—21no—
143#380C22×C311yescharacteristic
144C4×C4⋊C4C22×C311yescharacteristic
145C4×C22⋊C4—21no—
146#340C22×C312yes—
147#340C22×C312yes—
148#336C22×C311yescharacteristic
149#336C22×C311yescharacteristic
150C24×C3—21no—
151C24×C3C2×D411yescharacteristic
152C2×C3×Q8—24no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8—22no—
157C2×C3×Q8—22no—
158C2×C3×Q8—42no—
159C2×C3×Q8—42no—
160C2×C3×Q8C2×D414yes—
161C2×C3×D4—24no—
162C2×C3×D4—22no—
163C2×C3×D4—22no—
164C2×C3×D4—21no—
165C2×C3×D4—42no—
166C2×C3×D4—42no—
167C2×C3×D4#4112yes—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—21no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—41no—
183C22×C3×C4—42no—
184C22×C3×C4—42no—
185C22×C3×C4—42no—
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2411yescharacteristic
192C3×SD16—216no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8—22no—
197C2×C3×C8—22no—
198C2×C3×C8#4114yes—
199C3×C4⋊C4—24no—
200C3×C4⋊C4—24no—
201C3×C4⋊C4—28no—
202C3×C4⋊C4—24no—
203C3×C4⋊C4—28no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—42no—
209C3×C4⋊C4C2×D414yes—
210C3×C4⋊C4#4112yes—
211C3×C22⋊C4—28no—
212C3×C22⋊C4—24no—
213C3×C22⋊C4—42no—
214C3×C22⋊C4—42no—
215C3×C22⋊C4—42no—
216C3×C22⋊C4—42no—
217C3×C42—22no—
218C3×C42—22no—
219C3×C42—28no—
220C3×C42—22no—
221C3×C42—42no—
222C3×C42C2×D414yes—
223C22×Q8—21no—
224C22×Q8C23×C312yes—
225C22×D4—21no—
226C22×D4C23×C311yescharacteristic
227C23×C4—21no—
228C23×C4C3×D411yescharacteristic
229C23×C4C3×D411yescharacteristic
230C23×C4C23×C311yescharacteristic
231C2×SD16—24no—
232C2×SD16—24no—
233C2×SD16—24no—
234C22×C8C23×C312yes—
235C22×C8C23×C312yes—
236C4⋊Q8—24no—
237C4⋊Q8—24no—
238#125—28no—
239C22⋊Q8—28no—
240C22⋊Q8—24no—
241C22⋊Q8—24no—
242#121—22no—
243#121—22no—
244C4×Q8—28no—
245C4×Q8—24no—
246C4×Q8C3×D418yescomplemented
247C4×D4—24no—
248C4×D4—44no—
249#117—28no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4C23×C311yescharacteristic
257C2×C4⋊C4C23×C311yescharacteristic
258C2×C4⋊C4C23×C312yes—
259C2×C4⋊C4C23×C311yescharacteristic
260C2×C22⋊C4—22no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C42—22no—
267C2×C42—24no—
268C2×C42—21no—
269C2×C42C3×D411yescharacteristic
270C2×C42C3×D411yescharacteristic
271C2×C42C23×C311yescharacteristic
272C2×C42C23×C311yescharacteristic
273#107—24no—
274#103—24no—
275#103—24no—
276#103—24no—
277#102—24no—
278C22⋊C8—24no—
279#97—24no—
280#95—21no—
281#95—21no—
282#95—21no—
283C23×C3—21no—
284C23×C3—21no—
285C23×C3—21no—
286C23×C3—21no—
287C23×C3—21no—
288C23×C3—21no—
289C23×C3—22no—
290C23×C3—41no—
291C23×C3—41no—
292C23×C3—41no—
293C23×C3—41no—
294C23×C3—41no—
295C23×C3—41no—
296C23×C3#13712yes—
297C23×C3C22×D411yescharacteristic
298C3×Q8—24no—
299C3×Q8—24no—
300C3×Q8—48no—
301C3×Q8—42no—
302C3×Q8—42no—
303C3×D4—24no—
304C3×D4—48no—
305C3×D4—42no—
306C2×C3×C4—28no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—24no—
317C2×C3×C4—24no—
318C2×C3×C4—21no—
319C2×C3×C4—22no—
320C2×C3×C4—21no—
321C2×C3×C4—44no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—41no—
325C2×C3×C4—42no—
326C2×C3×C4—41no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—84no—
337C2×C3×C4#12012yes—
338C2×C3×C4#12012yes—
339C2×C3×C4AGL(1,ℤ/8)12yes—
340C2×C3×C4#13712yes—
341C2×C3×C4C22×D412yes—
342C2×C3×C4C2×#4111yescharacteristic
343C2×C3×C4C2×#4112yes—
344C2×C3×C4C2×#4111yescharacteristic
345C3×C8—24no—
346C3×C8—42no—
347C24—21no—
348C24C2×C3×D411yescharacteristic
349C2×Q8—24no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—42no—
356C2×Q8—42no—
357C2×Q8C2×C3×D414yes—
358C2×D4—24no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—21no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4C3×#4112yes—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—41no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4C2×C3×D411yescharacteristic
384C22×C4C2×C3×D411yescharacteristic
385C22×C4C2×C3×D411yescharacteristic
386C22×C4C2×C3×D411yescharacteristic
387C22×C4C2×C3×D411yescharacteristic
388C22×C4C24×C311yescharacteristic
389SD16—216no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8C3×#4114yes—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—28no—
399C4⋊C4—24no—
400C4⋊C4—28no—
401C4⋊C4—22no—
402C4⋊C4—21no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—42no—
406C4⋊C4C2×C3×D414yes—
407C4⋊C4C3×#4112yes—
408C22⋊C4—28no—
409C22⋊C4—24no—
410C22⋊C4—42no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C22⋊C4—42no—
414C42—22no—
415C42—22no—
416C42—28no—
417C42—22no—
418C42—42no—
419C42C2×C3×D414yes—
420C22×C3—22no—
421C22×C3—22no—
422C22×C3—22no—
423C22×C3—41no—
424C22×C3—41no—
425C22×C3—41no—
426C22×C3—41no—
427C22×C3—42no—
428C22×C3—42no—
429C22×C3—42no—
430C22×C3—81no—
431C22×C3—81no—
432C22×C3—81no—
433C22×C3—81no—
434C22×C3C2×#12011yescharacteristic
435C22×C3#52511yescharacteristic
436C22×C3#53211yescharacteristic
437C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
438C22×C3C2×#13711yescharacteristic
439C22×C3C2×#13711yescharacteristic
440C22×C3#57511yescharacteristic
441C3×C4—22no—
442C3×C4—22no—
443C3×C4—22no—
444C3×C4—22no—
445C3×C4—42no—
446C3×C4—41no—
447C3×C4—44no—
448C3×C4—44no—
449C3×C4—81no—
450C3×C4—81no—
451C3×C4#52512yes—
452C3×C4#53212yes—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—22no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23C3×#13712yes—
467C23C22×C3×D411yescharacteristic
468Q8—24no—
469Q8—24no—
470Q8—48no—
471Q8—42no—
472Q8—42no—
473D4—24no—
474D4—48no—
475D4—42no—
476C2×C4—28no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—44no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—84no—
507C2×C4C3×#12012yes—
508C2×C4C3×#12012yes—
509C2×C4C3×AGL(1,ℤ/8)12yes—
510C2×C4C3×#13712yes—
511C2×C4C22×C3×D412yes—
512C2×C4C2×C3×#4111yescharacteristic
513C2×C4C2×C3×#4112yes—
514C2×C4C2×C3×#4111yescharacteristic
515C8—24no—
516C8—42no—
517C2×C3—42no—
518C2×C3—82no—
519C2×C3#238811yescharacteristic
520C2×C3#300411yescharacteristic
521C2×C3#300611yescharacteristic
522C2×C3#311011yescharacteristic
523C2×C3#311611yescharacteristic
524C2×C3#318111yescharacteristic
525C2×C3#318211yescharacteristic
526C22—22no—
527C22—22no—
528C22—22no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22—41no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22C2×C3×#12011yescharacteristic
541C22C3×#52511yescharacteristic
542C22C3×#53211yescharacteristic
543C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
544C22C2×C3×#13711yescharacteristic
545C22C2×C3×#13711yescharacteristic
546C22C3×#57511yescharacteristic
547C4—22no—
548C4—22no—
549C4—22no—
550C4—22no—
551C4—41no—
552C4—44no—
553C4—44no—
554C4—42no—
555C4—81no—
556C4—81no—
557C4C3×#52512yes—
558C4C3×#53212yes—
559C3#2357411yescharacteristic, direct product factor, Sylow
560C2—42no—
561C2—82no—
562C2C3×#238811yescharacteristic
563C2C3×#300411yescharacteristic
564C2C3×#300611yescharacteristic
565C2C3×#311011yescharacteristic
566C2C3×#311611yescharacteristic
567C2C3×#318111yescharacteristic
568C2C3×#318211yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#533C2semidirect
2. C2×C3×#440C2semidirect
4. C3×#2057C2semidirect
5. C3×#2043C2semidirect
6. C3×#2036C2semidirect
7. C3×#2023C2semidirect
8. C3×#2013C2semidirect
10. C3×#1763C2semidirect
11. C3×#1749C2semidirect
12. #23574C3direct (N × H)
46. C2×#533C2×C3semidirect
47. C2×#440C2×C3semidirect
49. #2057C2×C3semidirect
50. #2043C2×C3semidirect
51. #2036C2×C3semidirect
52. #2023C2×C3semidirect
53. #2013C2×C3semidirect
55. #1763C2×C3semidirect
56. #1749C2×C3semidirect
80. C3×C4×Q8D4semidirect
246. C4×Q8C3×D4semidirect
559. C3#23574direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5332
C2×C3×#4402
C3×C4×#1212
C3×#20572
C3×#20432
C3×#20362
C3×#20232
C3×#20132
C3×#20002
C3×#17632
C3×#17492
#235743