Finite Group Databasebeta

← C3×#16604 #10635211

Subgroups — C3×#16604 #10635211

Order 768.

1578 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#125C211yesmaximal, characteristic, complemented
2C2×C3×#396C212yesmaximal
3C2×C3×#388C211yesmaximal, characteristic
4C3×#1444C212yesmaximal, complemented
5C3×#1444C211yesmaximal, characteristic, complemented
6C3×#1442C211yesmaximal, characteristic, complemented
7C3×#1441C212yesmaximal, complemented
8C3×#1439C211yesmaximal, characteristic
9C3×#1439C211yesmaximal, characteristic
10C3×#1439C212yesmaximal
11C3×C4⋊#95C211yesmaximal, characteristic, complemented
12#16604C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#125—28no—
14C2×C3×#125—24no—
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C4⋊C4C2212yes—
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C4⋊C4C2212yes—
19C22×C3×C4⋊C4C2212yes—
20C22×C3×C4⋊C4C2212yes—
21C22×C3×C4⋊C4C2211yescharacteristic
22C22×C3×C22⋊C4C2211yescharacteristic
23C22×C3×C22⋊C4C2212yes—
24C22×C3×C22⋊C4C2211yescharacteristic
25C22×C3×C22⋊C4C2212yes—
26C22×C3×C22⋊C4C2211yescharacteristic
27C22×C3×C42C2211yescharacteristic
28C3×#396—28no—
29C3×#388—24no—
30C2×C3×#95C2212yes—
31C2×C3×#95C2211yescharacteristic
32C2×C3×#95C2211yescharacteristic
33C2×C3×#95C2211yescharacteristic
34C2×C3×#95C2211yescharacteristic
35C2×C3×#95C2212yes—
36C2×C3×#95C2212yes—
37C2×C3×#95C2212yes—
38C2×C3×#95C2212yes—
39C2×C3×#95C2212yes—
40C22×#125C2×C311yescharacteristic, complemented
41C2×#396C2×C312yes—
42C2×#388C2×C311yescharacteristic
43#1444C2×C311yescharacteristic, complemented
44#1444C2×C312yescomplemented
45#1442C2×C311yescharacteristic, complemented
46#1441C2×C312yescomplemented
47#1439C2×C311yescharacteristic
48#1439C2×C312yes—
49#1439C2×C311yescharacteristic
50C4⋊#95C2×C311yescharacteristic, complemented
51C25×C3C2311yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2312yes—
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2312yes—
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2312yes—
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2312yes—
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C3×#125—232no—
63C2×C3×C4⋊C4—24no—
64C2×C3×C4⋊C4—24no—
65C2×C3×C4⋊C4—24no—
66C2×C3×C4⋊C4—24no—
67C2×C3×C4⋊C4—24no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—24no—
71C2×C3×C4⋊C4—24no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—24no—
74C2×C3×C4⋊C4—24no—
75C2×C3×C4⋊C4—24no—
76C2×C3×C4⋊C4—42no—
77C2×C3×C4⋊C4—42no—
78C2×C3×C4⋊C4—42no—
79C2×C3×C4⋊C4—42no—
80C2×C3×C4⋊C4D418yes—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—24no—
86C2×C3×C22⋊C4—24no—
87C2×C3×C22⋊C4—24no—
88C2×C3×C22⋊C4—24no—
89C2×C3×C22⋊C4—42no—
90C2×C3×C22⋊C4—42no—
91C2×C3×C22⋊C4—42no—
92C2×C3×C22⋊C4—42no—
93C2×C3×C22⋊C4Q814yes—
94C2×C3×C42—24no—
95C2×C3×C42D414yes—
96C3×#95—22no—
97C3×#95—22no—
98C3×#95—24no—
99C3×#95—24no—
100C3×#95—22no—
101C3×#95—22no—
102C3×#95—42no—
103C3×#95—42no—
104C3×#95—42no—
105C3×#95—42no—
106C2×#125—28no—
107C2×#125—24no—
108C22×C4⋊C4C22×C311yescharacteristic
109C22×C4⋊C4C22×C311yescharacteristic
110C22×C4⋊C4C22×C312yes—
111C22×C4⋊C4C22×C312yes—
112C22×C4⋊C4C22×C312yes—
113C22×C4⋊C4C22×C312yes—
114C22×C4⋊C4C22×C311yescharacteristic
115C22×C22⋊C4C22×C311yescharacteristic
116C22×C22⋊C4C22×C312yes—
117C22×C22⋊C4C22×C311yescharacteristic
118C22×C22⋊C4C22×C312yes—
119C22×C22⋊C4C22×C311yescharacteristic
120C22×C42C22×C311yescharacteristic
121#396—28no—
122#388—24no—
123C2×#95C22×C312yes—
124C2×#95C22×C311yescharacteristic
125C2×#95C22×C311yescharacteristic
126C2×#95C22×C312yes—
127C2×#95C22×C311yescharacteristic
128C2×#95C22×C311yescharacteristic
129C2×#95C22×C312yes—
130C2×#95C22×C312yes—
131C2×#95C22×C312yes—
132C2×#95C22×C312yes—
133C24×C3—21no—
134C24×C3—22no—
135C24×C3—21no—
136C24×C3—22no—
137C24×C3—21no—
138C24×C3—21no—
139C24×C3—22no—
140C24×C3—21no—
141C24×C3—21no—
142C24×C3—22no—
143C24×C3C2×Q812yes—
144C24×C3C2411yescharacteristic
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4—21no—
150C22×C3×C4—21no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—21no—
161C22×C3×C4—22no—
162C22×C3×C4—21no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—21no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—21no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—21no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—22no—
193C22×C3×C4C2×D414yes—
194C22×C3×C4C2×D412yes—
195C22×C3×C4C2×D414yes—
196C22×C3×C4C2×D412yes—
197C22×C3×C4C2×D414yes—
198C22×C3×C4C2×D412yes—
199C22×C3×C4C2×Q812yes—
200C22×C3×C4C2×Q812yes—
201C22×C3×C4#4114yes—
202C22×C3×C4#4112yes—
203C3×C4⋊C4—28no—
204C3×C4⋊C4—28no—
205C3×C4⋊C4—216no—
206C3×C4⋊C4—216no—
207C3×C4⋊C4—48no—
208C3×C4⋊C4—44no—
209C3×C4⋊C4—48no—
210C3×C22⋊C4—28no—
211C3×C22⋊C4—48no—
212C3×C22⋊C4—44no—
213C3×C22⋊C4—48no—
214C3×C22⋊C4—44no—
215C3×C42—28no—
216C25C23×C311yescharacteristic
217C23×C4C23×C311yescharacteristic
218C23×C4C23×C312yes—
219C23×C4C23×C311yescharacteristic
220C23×C4C23×C312yes—
221C23×C4C23×C312yes—
222C23×C4C23×C311yescharacteristic
223C23×C4C23×C311yescharacteristic
224C23×C4C23×C312yes—
225C23×C4C23×C311yescharacteristic
226C23×C4C23×C311yescharacteristic
227#125—232no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4C3×D418yes—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4C3×Q814yes—
259C2×C42—24no—
260C2×C42C3×D414yes—
261#95—22no—
262#95—22no—
263#95—24no—
264#95—24no—
265#95—22no—
266#95—22no—
267#95—42no—
268#95—42no—
269#95—42no—
270#95—42no—
271C23×C3—22no—
272C23×C3—22no—
273C23×C3—22no—
274C23×C3—24no—
275C23×C3—24no—
276C23×C3—41no—
277C23×C3—41no—
278C23×C3—41no—
279C23×C3—42no—
280C23×C3—41no—
281C23×C3—41no—
282C23×C3—41no—
283C23×C3—42no—
284C23×C3—42no—
285C23×C3—41no—
286C23×C3—41no—
287C23×C3—42no—
288C23×C3—42no—
289C23×C3—42no—
290C23×C3—42no—
291C23×C3—42no—
292C23×C3—42no—
293C23×C3—42no—
294C23×C3C22×D411yescharacteristic
295C23×C3C22×D412yes—
296C23×C3C22×Q811yescharacteristic
297C23×C3C2×#4111yescharacteristic
298C23×C3C2×#4112yes—
299C23×C3H(5,𝔽2)11yescharacteristic
300C23×C3H(5,𝔽2)12yes—
301C23×C3H(5,𝔽2)11yescharacteristic
302C23×C32−1+412yes—
303C23×C32−1+411yescharacteristic
304C23×C32−1+411yescharacteristic
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—24no—
310C2×C3×C4—24no—
311C2×C3×C4—24no—
312C2×C3×C4—24no—
313C2×C3×C4—22no—
314C2×C3×C4—24no—
315C2×C3×C4—24no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—24no—
319C2×C3×C4—22no—
320C2×C3×C4—24no—
321C2×C3×C4—24no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—41no—
328C2×C3×C4—42no—
329C2×C3×C4—41no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—42no—
347C2×C3×C4—41no—
348C2×C3×C4—42no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4—42no—
361C2×C3×C4#12014yes—
362C2×C3×C4C22⋊Q814yes—
363C24—21no—
364C24—22no—
365C24—21no—
366C24—22no—
367C24—21no—
368C24—21no—
369C24—22no—
370C24—21no—
371C24—21no—
372C24—22no—
373C24C2×C3×Q812yes—
374C24C24×C311yescharacteristic
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4C2×C3×D414yes—
424C22×C4C2×C3×D412yes—
425C22×C4C2×C3×D414yes—
426C22×C4C2×C3×D412yes—
427C22×C4C2×C3×D414yes—
428C22×C4C2×C3×D412yes—
429C22×C4C2×C3×Q812yes—
430C22×C4C2×C3×Q812yes—
431C22×C4C3×#4114yes—
432C22×C4C3×#4112yes—
433C4⋊C4—28no—
434C4⋊C4—28no—
435C4⋊C4—216no—
436C4⋊C4—216no—
437C4⋊C4—48no—
438C4⋊C4—44no—
439C4⋊C4—48no—
440C22⋊C4—28no—
441C22⋊C4—48no—
442C22⋊C4—44no—
443C22⋊C4—48no—
444C22⋊C4—44no—
445C42—28no—
446C22×C3—42no—
447C22×C3—42no—
448C22×C3—42no—
449C22×C3—44no—
450C22×C3—44no—
451C22×C3—81no—
452C22×C3—81no—
453C22×C3—81no—
454C22×C3—81no—
455C22×C3—82no—
456C22×C3—82no—
457C22×C3C2×#12011yescharacteristic
458C22×C3C2×C22⋊Q811yescharacteristic
459C22×C3#53211yescharacteristic
460C22×C3#53511yescharacteristic
461C22×C3#53512yes—
462C22×C3#53611yescharacteristic
463C22×C3#53711yescharacteristic
464C22×C3#53711yescharacteristic
465C22×C3#53712yes—
466C22×C3#53912yes—
467C22×C3#54011yescharacteristic
468C22×C3#54111yescharacteristic
469C22×C3#54112yes—
470C22×C3#54411yescharacteristic
471C22×C3#54411yescharacteristic
472C22×C3#54612yes—
473C22×C3#54712yes—
474C22×C3#54812yes—
475C22×C3D4×Q812yes—
476C22×C3#55312yes—
477C22×C3#55412yes—
478C22×C3#56212yes—
479C22×C3#56312yes—
480C3×C4—24no—
481C3×C4—24no—
482C3×C4—42no—
483C3×C4—42no—
484C3×C4—44no—
485C3×C4—44no—
486C3×C4—44no—
487C3×C4—81no—
488C3×C4—81no—
489C3×C4—82no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—24no—
494C23—24no—
495C23—42no—
496C23—42no—
497C23—41no—
498C23—41no—
499C23—42no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—42no—
505C23—42no—
506C23—41no—
507C23—41no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23C22×C3×D411yescharacteristic
514C23C22×C3×D412yes—
515C23C22×C3×Q811yescharacteristic
516C23C2×C3×#4111yescharacteristic
517C23C2×C3×#4112yes—
518C23C3×H(5,𝔽2)11yescharacteristic
519C23C3×H(5,𝔽2)12yes—
520C23C3×H(5,𝔽2)11yescharacteristic
521C23C3×2−1+412yes—
522C23C3×2−1+411yescharacteristic
523C23C3×2−1+411yescharacteristic
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4C3×#12014yes—
581C2×C4C3×C22⋊Q814yes—
582C2×C3—82no—
583C2×C3#240911yescharacteristic
584C2×C3#243011yescharacteristic
585C2×C3#256611yescharacteristic
586C2×C3#261511yescharacteristic
587C2×C3#262911yescharacteristic
588C2×C3#266411yescharacteristic
589C2×C3#270412yes—
590C2×C3#271112yes—
591C2×C3#272012yes—
592C2×C3#272512yes—
593C2×C3#282011yescharacteristic
594C22—42no—
595C22—42no—
596C22—44no—
597C22—44no—
598C22—42no—
599C22—81no—
600C22—81no—
601C22—82no—
602C22—82no—
603C22—81no—
604C22—81no—
605C22C2×C3×#12011yescharacteristic
606C22C2×C3×C22⋊Q811yescharacteristic
607C22C3×#53211yescharacteristic
608C22C3×#53512yes—
609C22C3×#53511yescharacteristic
610C22C3×#53611yescharacteristic
611C22C3×#53711yescharacteristic
612C22C3×#53711yescharacteristic
613C22C3×#53712yes—
614C22C3×#53912yes—
615C22C3×#54011yescharacteristic
616C22C3×#54111yescharacteristic
617C22C3×#54112yes—
618C22C3×#54411yescharacteristic
619C22C3×#54411yescharacteristic
620C22C3×#54612yes—
621C22C3×#54712yes—
622C22C3×#54812yes—
623C22C3×D4×Q812yes—
624C22C3×#55312yes—
625C22C3×#55412yes—
626C22C3×#56212yes—
627C22C3×#56312yes—
628C4—24no—
629C4—24no—
630C4—44no—
631C4—44no—
632C4—42no—
633C4—42no—
634C4—44no—
635C4—82no—
636C4—81no—
637C4—81no—
638C3#1660411yescharacteristic, direct product factor, Sylow
639C2—82no—
640C2C3×#240911yescharacteristic
641C2C3×#243011yescharacteristic
642C2C3×#256611yescharacteristic
643C2C3×#261511yescharacteristic
644C2C3×#262911yescharacteristic
645C2C3×#266411yescharacteristic
646C2C3×#270412yes—
647C2C3×#271112yes—
648C2C3×#272012yes—
649C2C3×#272512yes—
650C2C3×#282011yescharacteristic
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
1. C22×C3×#125C2semidirect
4. C3×#1444C2semidirect
5. C3×#1444C2semidirect
6. C3×#1442C2semidirect
7. C3×#1441C2semidirect
11. C3×C4⋊#95C2semidirect
12. #16604C3direct (N × H)
40. C22×#125C2×C3semidirect
43. #1444C2×C3semidirect
44. #1444C2×C3semidirect
45. #1442C2×C3semidirect
46. #1441C2×C3semidirect
50. C4⋊#95C2×C3semidirect
638. C3#16604direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1252
C2×C3×#3962
C2×C3×#3882
C3×#14442
C3×#14422
C3×#14412
C3×#14392
C3×C4⋊#952
#166043