Finite Group Databasebeta

← C3×#35723 #10654330

Subgroups — C3×#35723 #10654330

Order 768.

2284 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×Q8C211yesmaximal, characteristic
2C2×C3×#519C211yesmaximal, characteristic, complemented
3C3×#2625C213yesmaximal, complemented
4C3×#2499C213yesmaximal, complemented
5C3×#2485C213yesmaximal
6C3×#2446C213yesmaximal, complemented
7C3×#2446C213yesmaximal, complemented
8C3×#2423C213yesmaximal, complemented
9C3×#2421C213yesmaximal, complemented
10C3×#2395C213yesmaximal
11C3×#2351C211yesmaximal, characteristic, complemented
12C3×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
13C3×Q8×C22⋊C4C211yesmaximal, characteristic, complemented
14C3×Q8×C22⋊C4C211yesmaximal, characteristic, complemented
15C3×Q8×C22⋊C4C211yesmaximal, characteristic, complemented
16#35723C311yesmaximal, characteristic, direct product factor, Sylow
17C23×C3×Q8C2211yescharacteristic
18C23×C3×Q8C2211yescharacteristic
19C3×D4×Q8—28no—
20C2×C3×C4⋊Q8C2213yes—
21C2×C3×C22⋊Q8C2213yes—
22C2×C3×C22⋊Q8C2213yes—
23C3×#519—28no—
24C2×C3×C4×Q8C2213yes—
25C2×C3×C4×Q8C2213yes—
26C2×C3×C4×Q8C2211yescharacteristic
27C2×C3×C4×Q8C2211yescharacteristic
28C2×C3×C4×Q8C2211yescharacteristic
29C2×C3×C4×Q8C2211yescharacteristic
30C2×C3×C4×Q8C2211yescharacteristic
31C2×C3×C4×D4C2213yes—
32C2×C3×#117C2213yes—
33C2×C3×#117C2213yes—
34C3×#400C2216yes—
35C3×#400C2213yes—
36C3×#399C2213yes—
37C3×#398C2216yes—
38C3×#398C2213yes—
39C3×#397C2213yes—
40C3×#397C2216yes—
41C3×#397C2213yes—
42C3×#397C2213yes—
43C3×#394C2211yescharacteristic
44C3×#394C2213yes—
45C3×#391C2213yes—
46C3×#391C2213yes—
47C3×#391C2213yes—
48C3×#391C2213yes—
49C3×#391C2213yes—
50C3×#391C2213yes—
51C3×#389C2213yes—
52C3×#389C2213yes—
53C3×#389C2213yes—
54C3×#389C2213yes—
55C3×#388C2216yes—
56C3×#387C2216yes—
57C3×#387C2216yes—
58C3×#387C2216yes—
59C3×#384C2213yes—
60C3×#382C2213yes—
61C3×#380C2213yes—
62C3×#380C2213yes—
63C3×#380C2213yes—
64C3×C4×C4⋊C4C2213yes—
65C3×C4×C4⋊C4C2213yes—
66C3×C4×C22⋊C4C2213yes—
67C3×C4×C22⋊C4C2213yes—
68C3×C4×C22⋊C4C2213yes—
69C2×D4×Q8C2×C311yescharacteristic
70C2×#519C2×C311yescharacteristic, complemented
71#2625C2×C313yescomplemented
72#2499C2×C313yescomplemented
73#2485C2×C313yes—
74#2446C2×C313yescomplemented
75#2446C2×C313yescomplemented
76#2423C2×C313yescomplemented
77#2421C2×C313yescomplemented
78#2395C2×C313yes—
79#2351C2×C311yescharacteristic, complemented
80Q8×C4⋊C4C2×C311yescharacteristic, complemented
81Q8×C22⋊C4C2×C311yescharacteristic, complemented
82Q8×C22⋊C4C2×C311yescharacteristic, complemented
83Q8×C22⋊C4C2×C311yescharacteristic, complemented
84C22×C3×Q8—24no—
85C22×C3×Q8—24no—
86C22×C3×Q8—24no—
87C22×C3×Q8—24no—
88C22×C3×Q8—24no—
89C22×C3×Q8D418yes—
90C22×C3×Q8C2313yes—
91C22×C3×Q8C2313yes—
92C22×C3×Q8C2311yescharacteristic
93C22×C3×D4C2311yescharacteristic
94C23×C3×C4C2313yes—
95C23×C3×C4C2313yes—
96C3×C4⋊Q8—212no—
97C3×C22⋊Q8—212no—
98C3×C22⋊Q8—212no—
99C3×C4×Q8—212no—
100C3×C4×Q8—212no—
101C3×C4×Q8—24no—
102C3×C4×Q8—24no—
103C3×C4×Q8—24no—
104C3×C4×Q8—24no—
105C3×C4×Q8—24no—
106C3×C4×D4—212no—
107C3×#117—212no—
108C3×#117—212no—
109C2×C3×C4⋊C4C2311yescharacteristic
110C2×C3×C4⋊C4C2313yes—
111C2×C3×C4⋊C4C2313yes—
112C2×C3×C4⋊C4C2313yes—
113C2×C3×C4⋊C4C2313yes—
114C2×C3×C4⋊C4C2313yes—
115C2×C3×C4⋊C4C2313yes—
116C2×C3×C4⋊C4C2313yes—
117C2×C3×C4⋊C4C2313yes—
118C2×C3×C4⋊C4C2316yes—
119C2×C3×C4⋊C4C2313yes—
120C2×C3×C4⋊C4C2313yes—
121C2×C3×C4⋊C4C2313yes—
122C2×C3×C4⋊C4C2316yes—
123C2×C3×C22⋊C4C2311yescharacteristic
124C2×C3×C22⋊C4C2311yescharacteristic
125C2×C3×C22⋊C4C2311yescharacteristic
126C2×C3×C22⋊C4C2313yes—
127C2×C3×C22⋊C4C2313yes—
128C2×C3×C22⋊C4C2311yescharacteristic
129C2×C3×C22⋊C4C2313yes—
130C2×C3×C22⋊C4C2313yes—
131C2×C3×C22⋊C4C2313yes—
132C2×C3×C22⋊C4C2313yes—
133C2×C3×C42C2313yes—
134C2×C3×C42C2313yes—
135C2×C3×C42C2313yes—
136C2×C3×C42C2313yes—
137C2×C3×C42C2313yes—
138C2×C3×C42C2316yes—
139C3×#95C2313yes—
140C3×#95C2313yes—
141C3×#95C2313yes—
142C3×#95C2313yes—
143C3×#95C2313yes—
144C3×#95C2313yes—
145C3×#95C2311yescharacteristic
146C3×#95C2313yes—
147C3×#95C2313yes—
148C3×#95C2313yes—
149C3×#95C2316yes—
150C3×#95C2316yes—
151C3×#95C2316yes—
152C3×#95C2316yes—
153C23×Q8C22×C311yescharacteristic
154C23×Q8C22×C311yescharacteristic
155D4×Q8—28no—
156C2×C4⋊Q8C22×C313yes—
157C2×C22⋊Q8C22×C313yes—
158C2×C22⋊Q8C22×C313yes—
159#519—28no—
160C2×C4×Q8C22×C311yescharacteristic
161C2×C4×Q8C22×C311yescharacteristic
162C2×C4×Q8C22×C311yescharacteristic
163C2×C4×Q8C22×C311yescharacteristic
164C2×C4×Q8C22×C311yescharacteristic
165C2×C4×Q8C22×C313yes—
166C2×C4×Q8C22×C313yes—
167C2×C4×D4C22×C313yes—
168C2×#117C22×C313yes—
169C2×#117C22×C313yes—
170#400C22×C316yes—
171#400C22×C313yes—
172#399C22×C313yes—
173#398C22×C316yes—
174#398C22×C313yes—
175#397C22×C313yes—
176#397C22×C316yes—
177#397C22×C313yes—
178#397C22×C313yes—
179#394C22×C311yescharacteristic
180#394C22×C313yes—
181#391C22×C313yes—
182#391C22×C313yes—
183#391C22×C313yes—
184#391C22×C313yes—
185#391C22×C313yes—
186#391C22×C313yes—
187#389C22×C313yes—
188#389C22×C313yes—
189#389C22×C313yes—
190#389C22×C313yes—
191#388C22×C316yes—
192#387C22×C316yes—
193#387C22×C316yes—
194#387C22×C316yes—
195#384C22×C313yes—
196#382C22×C313yes—
197#380C22×C313yes—
198#380C22×C313yes—
199#380C22×C313yes—
200C4×C4⋊C4C22×C313yes—
201C4×C4⋊C4C22×C313yes—
202C4×C22⋊C4C22×C313yes—
203C4×C22⋊C4C22×C313yes—
204C4×C22⋊C4C22×C313yes—
205C24×C3C2411yescharacteristic
206C24×C3C2411yescharacteristic
207C2×C3×Q8—232no—
208C2×C3×Q8—26no—
209C2×C3×Q8—26no—
210C2×C3×Q8—26no—
211C2×C3×Q8—26no—
212C2×C3×Q8—26no—
213C2×C3×Q8—416no—
214C2×C3×Q8C2×D4112yes—
215C2×C3×Q8C2×D414yes—
216C2×C3×Q8#4114yes—
217C2×C3×Q8#4114yes—
218C2×C3×D4—22no—
219C2×C3×D4—22no—
220C2×C3×D4—22no—
221C22×C3×C4—26no—
222C22×C3×C4—26no—
223C22×C3×C4—26no—
224C22×C3×C4—26no—
225C22×C3×C4—26no—
226C22×C3×C4C2×D4112yes—
227C22×C3×C4C2411yescharacteristic
228C22×C3×C4C2413yes—
229C22×C3×C4C2413yes—
230C22×C3×C4C2413yes—
231C22×C3×C4C2413yes—
232C22×C3×C4C2411yescharacteristic
233C22×C3×C4C2411yescharacteristic
234C22×C3×C4C2413yes—
235C22×C3×C4C2413yes—
236C22×C3×C4C2413yes—
237C22×C3×C4C2411yescharacteristic
238C22×C3×C4C2411yescharacteristic
239C22×C3×C4C2413yes—
240C3×C4⋊C4—26no—
241C3×C4⋊C4—26no—
242C3×C4⋊C4—26no—
243C3×C4⋊C4—26no—
244C3×C4⋊C4—26no—
245C3×C4⋊C4—26no—
246C3×C4⋊C4—212no—
247C3×C4⋊C4—26no—
248C3×C4⋊C4—26no—
249C3×C4⋊C4—22no—
250C3×C4⋊C4—26no—
251C3×C4⋊C4—26no—
252C3×C4⋊C4—43no—
253C3×C4⋊C4—46no—
254C3×C22⋊C4—22no—
255C3×C22⋊C4—26no—
256C3×C22⋊C4—26no—
257C3×C22⋊C4—22no—
258C3×C22⋊C4—26no—
259C3×C22⋊C4—26no—
260C3×C22⋊C4—22no—
261C3×C22⋊C4—22no—
262C3×C22⋊C4—43no—
263C3×C22⋊C4—43no—
264C3×C42—212no—
265C3×C42—26no—
266C3×C42—26no—
267C3×C42—26no—
268C3×C42—26no—
269C3×C42—26no—
270C22×Q8—24no—
271C22×Q8—24no—
272C22×Q8—24no—
273C22×Q8—24no—
274C22×Q8—24no—
275C22×Q8C3×D418yes—
276C22×Q8C23×C313yes—
277C22×Q8C23×C313yes—
278C22×Q8C23×C311yescharacteristic
279C22×D4C23×C311yescharacteristic
280C23×C4C23×C313yes—
281C23×C4C23×C313yes—
282C4⋊Q8—212no—
283C22⋊Q8—212no—
284C22⋊Q8—212no—
285C4×Q8—212no—
286C4×Q8—212no—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—24no—
290C4×Q8—24no—
291C4×Q8—24no—
292C4×D4—212no—
293#117—212no—
294#117—212no—
295C2×C4⋊C4C23×C313yes—
296C2×C4⋊C4C23×C313yes—
297C2×C4⋊C4C23×C313yes—
298C2×C4⋊C4C23×C313yes—
299C2×C4⋊C4C23×C313yes—
300C2×C4⋊C4C23×C313yes—
301C2×C4⋊C4C23×C313yes—
302C2×C4⋊C4C23×C313yes—
303C2×C4⋊C4C23×C316yes—
304C2×C4⋊C4C23×C311yescharacteristic
305C2×C4⋊C4C23×C313yes—
306C2×C4⋊C4C23×C313yes—
307C2×C4⋊C4C23×C313yes—
308C2×C4⋊C4C23×C316yes—
309C2×C22⋊C4C23×C311yescharacteristic
310C2×C22⋊C4C23×C313yes—
311C2×C22⋊C4C23×C313yes—
312C2×C22⋊C4C23×C313yes—
313C2×C22⋊C4C23×C311yescharacteristic
314C2×C22⋊C4C23×C311yescharacteristic
315C2×C22⋊C4C23×C311yescharacteristic
316C2×C22⋊C4C23×C313yes—
317C2×C22⋊C4C23×C313yes—
318C2×C22⋊C4C23×C313yes—
319C2×C42C23×C313yes—
320C2×C42C23×C313yes—
321C2×C42C23×C316yes—
322C2×C42C23×C313yes—
323C2×C42C23×C313yes—
324C2×C42C23×C313yes—
325#95C23×C313yes—
326#95C23×C313yes—
327#95C23×C313yes—
328#95C23×C313yes—
329#95C23×C313yes—
330#95C23×C313yes—
331#95C23×C311yescharacteristic
332#95C23×C313yes—
333#95C23×C313yes—
334#95C23×C313yes—
335#95C23×C316yes—
336#95C23×C316yes—
337#95C23×C316yes—
338#95C23×C316yes—
339C23×C3—21no—
340C23×C3—21no—
341C23×C3—21no—
342C23×C3—21no—
343C23×C3—21no—
344C23×C3—21no—
345C23×C3—21no—
346C23×C3—21no—
347C23×C3—21no—
348C23×C3—21no—
349C23×C3—21no—
350C23×C3—21no—
351C23×C3C22×D412yes—
352C23×C32−1+412yes—
353C23×C3C2511yescharacteristic
354C3×Q8—224no—
355C3×Q8—412no—
356C3×Q8#123116yes—
357C3×D4—28no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—23no—
361C2×C3×C4—23no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—224no—
365C2×C3×C4—23no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—23no—
369C2×C3×C4—23no—
370C2×C3×C4—23no—
371C2×C3×C4—21no—
372C2×C3×C4—23no—
373C2×C3×C4—23no—
374C2×C3×C4—23no—
375C2×C3×C4—23no—
376C2×C3×C4—21no—
377C2×C3×C4—23no—
378C2×C3×C4—21no—
379C2×C3×C4—23no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—23no—
384C2×C3×C4—23no—
385C2×C3×C4—23no—
386C2×C3×C4—23no—
387C2×C3×C4—23no—
388C2×C3×C4—23no—
389C2×C3×C4—23no—
390C2×C3×C4—21no—
391C2×C3×C4—23no—
392C2×C3×C4—412no—
393C2×C3×C4C22×D416yes—
394C2×C3×C4C22×D416yes—
395C2×C3×C4C2×#4116yes—
396C2×C3×C4C2×#4116yes—
397C2×C3×C42−1+412yes—
398C2×C3×C42−1+412yes—
399C24C24×C311yescharacteristic
400C24C24×C311yescharacteristic
401C2×Q8—232no—
402C2×Q8—26no—
403C2×Q8—26no—
404C2×Q8—26no—
405C2×Q8—26no—
406C2×Q8—26no—
407C2×Q8—416no—
408C2×Q8C2×C3×D4112yes—
409C2×Q8C2×C3×D414yes—
410C2×Q8C3×#4114yes—
411C2×Q8C3×#4114yes—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C22×C4—26no—
416C22×C4—26no—
417C22×C4—26no—
418C22×C4—26no—
419C22×C4—26no—
420C22×C4C2×C3×D4112yes—
421C22×C4C24×C311yescharacteristic
422C22×C4C24×C313yes—
423C22×C4C24×C313yes—
424C22×C4C24×C313yes—
425C22×C4C24×C313yes—
426C22×C4C24×C311yescharacteristic
427C22×C4C24×C311yescharacteristic
428C22×C4C24×C313yes—
429C22×C4C24×C313yes—
430C22×C4C24×C313yes—
431C22×C4C24×C311yescharacteristic
432C22×C4C24×C311yescharacteristic
433C22×C4C24×C313yes—
434C4⋊C4—26no—
435C4⋊C4—26no—
436C4⋊C4—26no—
437C4⋊C4—26no—
438C4⋊C4—26no—
439C4⋊C4—26no—
440C4⋊C4—212no—
441C4⋊C4—26no—
442C4⋊C4—26no—
443C4⋊C4—22no—
444C4⋊C4—26no—
445C4⋊C4—26no—
446C4⋊C4—43no—
447C4⋊C4—46no—
448C22⋊C4—22no—
449C22⋊C4—26no—
450C22⋊C4—26no—
451C22⋊C4—22no—
452C22⋊C4—26no—
453C22⋊C4—26no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—43no—
457C22⋊C4—43no—
458C42—212no—
459C42—26no—
460C42—26no—
461C42—26no—
462C42—26no—
463C42—26no—
464C22×C3—22no—
465C22×C3—22no—
466C22×C3—22no—
467C22×C3—22no—
468C22×C3—22no—
469C22×C3—22no—
470C22×C3—41no—
471C22×C3—41no—
472C22×C3—41no—
473C22×C3—41no—
474C22×C3—41no—
475C22×C3—41no—
476C22×C3—41no—
477C22×C3—41no—
478C22×C3C23×D411yescharacteristic
479C22×C3C22×#4111yescharacteristic
480C22×C3C22×#4111yescharacteristic
481C22×C3C2×2−1+411yescharacteristic
482C22×C3C2×2−1+411yescharacteristic
483C22×C3C2×2−1+411yescharacteristic
484C22×C3#58511yescharacteristic
485C3×C4—26no—
486C3×C4—22no—
487C3×C4—22no—
488C3×C4—43no—
489C3×C4—43no—
490C3×C4—41no—
491C3×C4—41no—
492C3×C4—43no—
493C3×C4—43no—
494C3×C4—43no—
495C3×C4—43no—
496C3×C4—41no—
497C3×C4C2×#123112yes—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23C22×C3×D412yes—
511C23C3×2−1+412yes—
512C23C25×C311yescharacteristic
513Q8—224no—
514Q8—412no—
515Q8C3×#123116yes—
516D4—28no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—23no—
520C2×C4—23no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—224no—
524C2×C4—23no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—23no—
528C2×C4—23no—
529C2×C4—23no—
530C2×C4—21no—
531C2×C4—23no—
532C2×C4—23no—
533C2×C4—23no—
534C2×C4—23no—
535C2×C4—21no—
536C2×C4—23no—
537C2×C4—21no—
538C2×C4—23no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—23no—
543C2×C4—23no—
544C2×C4—23no—
545C2×C4—23no—
546C2×C4—23no—
547C2×C4—23no—
548C2×C4—23no—
549C2×C4—21no—
550C2×C4—23no—
551C2×C4—412no—
552C2×C4C22×C3×D416yes—
553C2×C4C22×C3×D416yes—
554C2×C4C2×C3×#4116yes—
555C2×C4C2×C3×#4116yes—
556C2×C4C3×2−1+412yes—
557C2×C4C3×2−1+412yes—
558C2×C3—42no—
559C2×C3—42no—
560C2×C3C22×#12311yescharacteristic
561C2×C3#348611yescharacteristic
562C2×C3#349711yescharacteristic
563C2×C3#351611yescharacteristic
564C2×C3#351811yescharacteristic
565C2×C3#352311yescharacteristic
566C2×C3#352311yescharacteristic
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22C23×C3×D411yescharacteristic
582C22C22×C3×#4111yescharacteristic
583C22C22×C3×#4111yescharacteristic
584C22C2×C3×2−1+411yescharacteristic
585C22C2×C3×2−1+411yescharacteristic
586C22C2×C3×2−1+411yescharacteristic
587C22C3×#58511yescharacteristic
588C4—26no—
589C4—22no—
590C4—22no—
591C4—43no—
592C4—43no—
593C4—41no—
594C4—41no—
595C4—43no—
596C4—43no—
597C4—43no—
598C4—43no—
599C4—41no—
600C4C2×C3×#123112yes—
601C3#3572311yescharacteristic, direct product factor, Sylow
602C2—42no—
603C2—42no—
604C2C22×C3×#12311yescharacteristic
605C2C3×#348611yescharacteristic
606C2C3×#349711yescharacteristic
607C2C3×#351611yescharacteristic
608C2C3×#351811yescharacteristic
609C2C3×#352311yescharacteristic
610C2C3×#352311yescharacteristic
subgroupG/H
center, socleC23×C3C25
commutator, Frattini, cosocleC23C25×C3
prime radicalC22×C3×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C3 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#519C2semidirect
3. C3×#2625C2semidirect
4. C3×#2499C2semidirect
6. C3×#2446C2semidirect
7. C3×#2446C2semidirect
8. C3×#2423C2semidirect
9. C3×#2421C2semidirect
11. C3×#2351C2semidirect
12. C3×Q8×C4⋊C4C2semidirect
13. C3×Q8×C22⋊C4C2semidirect
14. C3×Q8×C22⋊C4C2semidirect
15. C3×Q8×C22⋊C4C2semidirect
16. #35723C3direct (N × H)
70. C2×#519C2×C3semidirect
71. #2625C2×C3semidirect
72. #2499C2×C3semidirect
74. #2446C2×C3semidirect
75. #2446C2×C3semidirect
76. #2423C2×C3semidirect
77. #2421C2×C3semidirect
79. #2351C2×C3semidirect
80. Q8×C4⋊C4C2×C3semidirect
81. Q8×C22⋊C4C2×C3semidirect
82. Q8×C22⋊C4C2×C3semidirect
83. Q8×C22⋊C4C2×C3semidirect
601. C3#35723direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D4×Q82
C2×C3×#5192
C3×#26252
C3×#24992
C3×#24852
C3×#24462
C3×#24232
C3×#24212
C3×#23952
C3×#23512
C3×Q8×C4⋊C42
C3×Q8×C22⋊C42
#357233