Finite Group Databasebeta

← C3×#24157 #10642764

Subgroups — C3×#24157 #10642764

Order 768.

1114 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#533C211yesmaximal, characteristic
2C2×C3×#475C211yesmaximal, characteristic, complemented
3C2×C3×#462C211yesmaximal, characteristic, complemented
4C3×C4×C22⋊Q8C211yesmaximal, characteristic
5C3×#2035C212yesmaximal
6C3×#2001C212yesmaximal
7C3×#1988C211yesmaximal, characteristic, complemented
8C3×#1981C211yesmaximal, characteristic, complemented
9C3×#1878C212yesmaximal, complemented
10C3×#1865C212yesmaximal, complemented
11C3×#1797C211yesmaximal, characteristic
12#24157C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×Q16C2212yes—
14C3×#533—24no—
15C3×#533—24no—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#125—21no—
19C2×C3×C22⋊Q8—21no—
20C2×C3×C22⋊Q8C2211yescharacteristic
21C2×C3×C22⋊Q8C2212yes—
22C2×C3×C4×Q8—21no—
23C2×C3×C4×Q8C2212yes—
24C2×C3×C4×Q8C2211yescharacteristic
25C2×C3×#117—21no—
26C22×C3×C42C2211yescharacteristic
27C3×#475—24no—
28C3×#462—24no—
29C2×C3×#106C2212yes—
30C2×C3×C4⋊C8C2211yescharacteristic
31C2×C3×C4⋊C8C2211yescharacteristic
32C2×C3×#103C2212yes—
33C2×C3×#103C2212yes—
34C2×C3×#103C2212yes—
35C2×C3×#103C2212yes—
36C2×C3×C22⋊C8C2212yes—
37C3×#391C2211yescharacteristic
38C3×#389C2211yescharacteristic
39C3×#387—21no—
40C3×#385—21no—
41C3×#380C2211yescharacteristic
42C3×C4×C4⋊C4—21no—
43C3×C4×C4⋊C4C2211yescharacteristic
44C3×C4×C22⋊C4—21no—
45C3×#340C2212yes—
46C3×#340C2212yes—
47C3×#336C2212yes—
48C2×#533C2×C311yescharacteristic
49C2×#475C2×C311yescharacteristic, complemented
50C2×#462C2×C311yescharacteristic, complemented
51C4×C22⋊Q8C2×C311yescharacteristic
52#2035C2×C312yes—
53#2001C2×C312yes—
54#1988C2×C311yescharacteristic, complemented
55#1981C2×C311yescharacteristic, complemented
56#1878C2×C312yescomplemented
57#1865C2×C312yescomplemented
58#1797C2×C311yescharacteristic
59C22×C3×Q8—21no—
60C22×C3×Q8C2312yes—
61C22×C3×Q8C2311yescharacteristic
62C23×C3×C4D411yescharacteristic
63C23×C3×C4D411yescharacteristic
64C23×C3×C4C2311yescharacteristic
65C2×C3×Q16—24no—
66C2×C3×Q16—24no—
67C2×C3×Q16—42no—
68C22×C3×C8C2312yes—
69C22×C3×C8C2312yes—
70C3×C4⋊Q8—22no—
71C3×C4⋊Q8—22no—
72C3×C4⋊Q8D414yes—
73C3×C4⋊Q8D414yes—
74C3×#125—28no—
75C3×C22⋊Q8—28no—
76C3×C22⋊Q8—24no—
77C3×C22⋊Q8—22no—
78C3×C22⋊Q8—24no—
79C3×C22⋊Q8—22no—
80C3×C4×Q8—28no—
81C3×C4×Q8—24no—
82C3×C4×Q8—24no—
83C3×C4×Q8—22no—
84C3×C4×Q8—22no—
85C3×#117—28no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—22no—
91C2×C3×C4⋊C4—22no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2312yes—
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4—22no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C42—22no—
102C2×C3×C42—22no—
103C2×C3×C42—22no—
104C2×C3×C42—21no—
105C2×C3×C42—21no—
106C2×C3×C42D411yescharacteristic
107C2×C3×C42D411yescharacteristic
108C2×C3×C42C2311yescharacteristic
109C2×C3×C42C2311yescharacteristic
110C3×#106—24no—
111C3×C4⋊C8—22no—
112C3×C4⋊C8—22no—
113C3×#103—24no—
114C3×#103—24no—
115C3×#103—24no—
116C3×#103—42no—
117C3×C22⋊C8—42no—
118C3×#95—21no—
119C3×#95—21no—
120C3×#95—21no—
121C3×#95—21no—
122C3×#95—21no—
123C22×Q16C22×C312yes—
124#533—24no—
125#533—24no—
126C2×C4⋊Q8C22×C311yescharacteristic
127C2×C4⋊Q8C22×C311yescharacteristic
128C2×#125—21no—
129C2×C22⋊Q8—21no—
130C2×C22⋊Q8C22×C312yes—
131C2×C22⋊Q8C22×C311yescharacteristic
132C2×C4×Q8—21no—
133C2×C4×Q8C22×C311yescharacteristic
134C2×C4×Q8C22×C312yes—
135C2×#117—21no—
136C22×C42C22×C311yescharacteristic
137#475—24no—
138#462—24no—
139C2×#106C22×C312yes—
140C2×C4⋊C8C22×C311yescharacteristic
141C2×C4⋊C8C22×C311yescharacteristic
142C2×#103C22×C312yes—
143C2×#103C22×C312yes—
144C2×#103C22×C312yes—
145C2×#103C22×C312yes—
146C2×C22⋊C8C22×C312yes—
147#391C22×C311yescharacteristic
148#389C22×C311yescharacteristic
149#387—21no—
150#385—21no—
151#380C22×C311yescharacteristic
152C4×C4⋊C4—21no—
153C4×C4⋊C4C22×C311yescharacteristic
154C4×C22⋊C4—21no—
155#340C22×C312yes—
156#340C22×C312yes—
157#336C22×C312yes—
158C24×C3C2×D411yescharacteristic
159C2×C3×Q8—24no—
160C2×C3×Q8—22no—
161C2×C3×Q8—21no—
162C2×C3×Q8—21no—
163C2×C3×Q8—21no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8—21no—
168C2×C3×Q8—22no—
169C2×C3×Q8—21no—
170C2×C3×Q8—42no—
171C2×C3×Q8—42no—
172C2×C3×Q8C2×D414yes—
173C2×C3×Q8#4112yes—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2411yescharacteristic
198C3×Q16—48no—
199C2×C3×C8—22no—
200C2×C3×C8—22no—
201C2×C3×C8—22no—
202C2×C3×C8—22no—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C3×C4⋊C4—28no—
206C3×C4⋊C4—28no—
207C3×C4⋊C4—24no—
208C3×C4⋊C4—24no—
209C3×C4⋊C4—24no—
210C3×C4⋊C4—24no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—42no—
218C3×C4⋊C4—42no—
219C3×C4⋊C4C2×D414yes—
220C3×C4⋊C4#4112yes—
221C3×C22⋊C4—28no—
222C3×C22⋊C4—24no—
223C3×C22⋊C4—42no—
224C3×C42—28no—
225C3×C42—22no—
226C3×C42—22no—
227C3×C42—21no—
228C3×C42—21no—
229C3×C42—42no—
230C3×C42—42no—
231C3×C42C2×D412yes—
232C3×C42C2×D412yes—
233C22×Q8—21no—
234C22×Q8C23×C312yes—
235C22×Q8C23×C311yescharacteristic
236C23×C4C3×D411yescharacteristic
237C23×C4C3×D411yescharacteristic
238C23×C4C23×C311yescharacteristic
239C2×Q16—24no—
240C2×Q16—24no—
241C2×Q16—42no—
242C22×C8C23×C312yes—
243C22×C8C23×C312yes—
244C4⋊Q8—22no—
245C4⋊Q8—22no—
246C4⋊Q8C3×D414yes—
247C4⋊Q8C3×D414yes—
248#125—28no—
249C22⋊Q8—28no—
250C22⋊Q8—24no—
251C22⋊Q8—22no—
252C22⋊Q8—24no—
253C22⋊Q8—22no—
254C4×Q8—28no—
255C4×Q8—24no—
256C4×Q8—24no—
257C4×Q8—22no—
258C4×Q8—22no—
259#117—28no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4C23×C311yescharacteristic
269C2×C4⋊C4C23×C311yescharacteristic
270C2×C4⋊C4C23×C312yes—
271C2×C4⋊C4C23×C311yescharacteristic
272C2×C22⋊C4—22no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42C3×D411yescharacteristic
281C2×C42C3×D411yescharacteristic
282C2×C42C23×C311yescharacteristic
283C2×C42C23×C311yescharacteristic
284#106—24no—
285C4⋊C8—22no—
286C4⋊C8—22no—
287#103—24no—
288#103—24no—
289#103—24no—
290#103—42no—
291C22⋊C8—42no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296#95—21no—
297C23×C3—21no—
298C23×C3—21no—
299C23×C3—21no—
300C23×C3—21no—
301C23×C3—21no—
302C23×C3—21no—
303C23×C3#13712yes—
304C23×C3C22×D411yescharacteristic
305C3×Q8—22no—
306C3×Q8—24no—
307C3×Q8—24no—
308C3×Q8—22no—
309C3×Q8—41no—
310C3×Q8—42no—
311C3×Q8—42no—
312C3×Q8—41no—
313C3×Q8—48no—
314C2×C3×C4—28no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—24no—
322C2×C3×C4—24no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—41no—
330C2×C3×C4—42no—
331C2×C3×C4—41no—
332C2×C3×C4—42no—
333C2×C3×C4—41no—
334C2×C3×C4—42no—
335C2×C3×C4—41no—
336C2×C3×C4—42no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—44no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4#12012yes—
349C2×C3×C4#12012yes—
350C2×C3×C4#13512yes—
351C2×C3×C4#13712yes—
352C2×C3×C4#13712yes—
353C2×C3×C4C22×D411yescharacteristic
354C2×C3×C4C22×D411yescharacteristic
355C2×C3×C4C2×#4111yescharacteristic
356C2×C3×C4C2×#4111yescharacteristic
357C2×C3×C4H(5,𝔽2)11yescharacteristic
358C2×C3×C42−1+411yescharacteristic
359C3×C8—42no—
360C3×C8—42no—
361C24C2×C3×D411yescharacteristic
362C2×Q8—24no—
363C2×Q8—22no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—21no—
371C2×Q8—22no—
372C2×Q8—21no—
373C2×Q8—42no—
374C2×Q8—42no—
375C2×Q8C2×C3×D414yes—
376C2×Q8C3×#4112yes—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4C2×C3×D411yescharacteristic
396C22×C4C2×C3×D411yescharacteristic
397C22×C4C2×C3×D411yescharacteristic
398C22×C4C2×C3×D411yescharacteristic
399C22×C4C2×C3×D411yescharacteristic
400C22×C4C24×C311yescharacteristic
401Q16—48no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C4⋊C4—28no—
409C4⋊C4—28no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4C2×C3×D414yes—
423C4⋊C4C3×#4112yes—
424C22⋊C4—28no—
425C22⋊C4—24no—
426C22⋊C4—42no—
427C42—28no—
428C42—22no—
429C42—22no—
430C42—21no—
431C42—21no—
432C42—42no—
433C42—42no—
434C42C2×C3×D412yes—
435C42C2×C3×D412yes—
436C22×C3—22no—
437C22×C3—22no—
438C22×C3—22no—
439C22×C3—41no—
440C22×C3—41no—
441C22×C3—41no—
442C22×C3—41no—
443C22×C3C2×#12011yescharacteristic
444C22×C3#52511yescharacteristic
445C22×C3#53711yescharacteristic
446C22×C3C2×#13511yescharacteristic
447C22×C3C2×#13711yescharacteristic
448C22×C3C2×#13711yescharacteristic
449C22×C3C2×#13711yescharacteristic
450C3×C4—22no—
451C3×C4—22no—
452C3×C4—42no—
453C3×C4—41no—
454C3×C4—44no—
455C3×C4—42no—
456C3×C4—41no—
457C3×C4—44no—
458C3×C4—81no—
459C3×C4—81no—
460C3×C4#45214yes—
461C3×C4#54611yescharacteristic
462C3×C4#54711yescharacteristic
463C3×C4#54811yescharacteristic
464C3×C4#55011yescharacteristic
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23C3×#13712yes—
472C23C22×C3×D411yescharacteristic
473Q8—22no—
474Q8—24no—
475Q8—24no—
476Q8—22no—
477Q8—48no—
478Q8—41no—
479Q8—42no—
480Q8—42no—
481Q8—41no—
482C2×C4—28no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4C3×#12012yes—
517C2×C4C3×#12012yes—
518C2×C4C3×#13512yes—
519C2×C4C3×#13712yes—
520C2×C4C3×#13712yes—
521C2×C4C22×C3×D411yescharacteristic
522C2×C4C22×C3×D411yescharacteristic
523C2×C4C2×C3×#4111yescharacteristic
524C2×C4C2×C3×#4111yescharacteristic
525C2×C4C3×H(5,𝔽2)11yescharacteristic
526C2×C4C3×2−1+411yescharacteristic
527C8—42no—
528C8—42no—
529C2×C3—42no—
530C2×C3#240711yescharacteristic
531C2×C3C2×#45211yescharacteristic
532C2×C3#300411yescharacteristic
533C2×C3#310711yescharacteristic
534C2×C3#311311yescharacteristic
535C2×C3#324811yescharacteristic
536C2×C3#325511yescharacteristic
537C22—22no—
538C22—22no—
539C22—22no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22C2×C3×#12011yescharacteristic
545C22C3×#52511yescharacteristic
546C22C3×#53711yescharacteristic
547C22C2×C3×#13511yescharacteristic
548C22C2×C3×#13711yescharacteristic
549C22C2×C3×#13711yescharacteristic
550C22C2×C3×#13711yescharacteristic
551C4—22no—
552C4—22no—
553C4—41no—
554C4—44no—
555C4—42no—
556C4—41no—
557C4—44no—
558C4—42no—
559C4—81no—
560C4—81no—
561C4C3×#45214yes—
562C4C3×#54611yescharacteristic
563C4C3×#54711yescharacteristic
564C4C3×#54811yescharacteristic
565C4C3×#55011yescharacteristic
566C3#2415711yescharacteristic, direct product factor, Sylow
567C2—42no—
568C2C3×#240711yescharacteristic
569C2C2×C3×#45211yescharacteristic
570C2C3×#300411yescharacteristic
571C2C3×#310711yescharacteristic
572C2C3×#311311yescharacteristic
573C2C3×#324811yescharacteristic
574C2C3×#325511yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
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
2. C2×C3×#475C2semidirect
3. C2×C3×#462C2semidirect
7. C3×#1988C2semidirect
8. C3×#1981C2semidirect
9. C3×#1878C2semidirect
10. C3×#1865C2semidirect
12. #24157C3direct (N × H)
49. C2×#475C2×C3semidirect
50. C2×#462C2×C3semidirect
54. #1988C2×C3semidirect
55. #1981C2×C3semidirect
56. #1878C2×C3semidirect
57. #1865C2×C3semidirect
566. C3#24157direct (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×#4752
C2×C3×#4622
C3×C4×C22⋊Q82
C3×#20352
C3×#20012
C3×#19882
C3×#19812
C3×#18782
C3×#18652
C3×#17972
#241573