Finite Group Databasebeta

← C3×#32665 #10651272

Subgroups — C3×#32665 #10651272

Order 768.

734 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2582C211yesmaximal, characteristic, complemented
2C3×#2538C211yesmaximal, characteristic
3C3×#2095C211yesmaximal, characteristic, complemented
4C3×#2093C211yesmaximal, characteristic
5C3×#2083C211yesmaximal, characteristic, complemented
6C3×#2083C211yesmaximal, characteristic, complemented
7C3×#2068C211yesmaximal, characteristic, complemented
8C3×#2067C211yesmaximal, characteristic
9C3×#1985C211yesmaximal, characteristic, complemented
10C3×#1973C211yesmaximal, characteristic
11C3×#1943C211yesmaximal, characteristic, complemented
12C3×#1880C211yesmaximal, characteristic
13C3×#1877C211yesmaximal, characteristic
14C3×#1865C211yesmaximal, characteristic, complemented
15C3×#1864C211yesmaximal, characteristic
16#32665C311yesmaximal, characteristic, direct product factor, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×C3×C22⋊Q8—21no—
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×D4C2211yescharacteristic
22C2×C3×#107C2211yescharacteristic
23C2×C3×#107C2211yescharacteristic
24C2×C3×#107C2211yescharacteristic
25C2×C3×#107C2211yescharacteristic
26C2×C3×#106C2211yescharacteristic
27C2×C3×C4⋊C8C2211yescharacteristic
28C2×C3×#103C2211yescharacteristic
29C2×C3×#103C2211yescharacteristic
30C2×C3×#103C2211yescharacteristic
31C2×C3×#102C2211yescharacteristic
32C2×C3×#102C2211yescharacteristic
33C2×C3×#102C2211yescharacteristic
34C2×C3×#97C2211yescharacteristic
35C3×#398—21no—
36C3×#398—21no—
37C3×#396—21no—
38C3×#396—21no—
39C3×#391—21no—
40C3×#391C2211yescharacteristic
41C3×#391C2211yescharacteristic
42C3×#390C2211yescharacteristic
43C3×#390C2211yescharacteristic
44C3×#389—21no—
45C3×#389C2211yescharacteristic
46C3×#389C2211yescharacteristic
47C3×#389C2211yescharacteristic
48C3×#389C2211yescharacteristic
49C3×#384C2211yescharacteristic
50C3×#384C2211yescharacteristic
51C3×#380—21no—
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#336C2211yescharacteristic
57C3×#336C2211yescharacteristic
58C3×#336C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#2582C2×C311yescharacteristic, complemented
61#2538C2×C311yescharacteristic
62#2095C2×C311yescharacteristic, complemented
63#2093C2×C311yescharacteristic
64#2083C2×C311yescharacteristic, complemented
65#2083C2×C311yescharacteristic, complemented
66#2068C2×C311yescharacteristic, complemented
67#2067C2×C311yescharacteristic
68#1985C2×C311yescharacteristic, complemented
69#1973C2×C311yescharacteristic
70#1943C2×C311yescharacteristic, complemented
71#1880C2×C311yescharacteristic
72#1877C2×C311yescharacteristic
73#1865C2×C311yescharacteristic, complemented
74#1864C2×C311yescharacteristic
75C22×C3×Q8D411yescharacteristic
76C22×C3×Q8C2311yescharacteristic
77C22×C3×D4C2311yescharacteristic
78C23×C3×C4—21no—
79C2×C3×SD16—41no—
80C2×C3×SD16—41no—
81C2×C3×SD16Q814yescomplemented
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×C22⋊Q8—44no—
87C3×C4×Q8—41no—
88C3×C4×Q8—41no—
89C3×C4×Q8—41no—
90C3×C4×Q8—41no—
91C3×C4×D4—41no—
92C3×C4×D4—41no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4D411yescharacteristic
103C2×C3×C4⋊C4D411yescharacteristic
104C2×C3×C4⋊C4D411yescharacteristic
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C4⋊C4C2311yescharacteristic
109C2×C3×C4⋊C4C2311yescharacteristic
110C2×C3×C4⋊C4C2311yescharacteristic
111C2×C3×C22⋊C4—21no—
112C2×C3×C22⋊C4—21no—
113C2×C3×C22⋊C4—21no—
114C2×C3×C22⋊C4—21no—
115C2×C3×C22⋊C4—21no—
116C2×C3×C42—21no—
117C2×C3×C42C2311yescharacteristic
118C2×C3×C42C2311yescharacteristic
119C2×C3×C42C2311yescharacteristic
120C3×#107—22no—
121C3×#107—41no—
122C3×#107—41no—
123C3×#107—41no—
124C3×#106—41no—
125C3×C4⋊C8—41no—
126C3×#103—22no—
127C3×#103—41no—
128C3×#103—41no—
129C3×#102—22no—
130C3×#102—41no—
131C3×#102—41no—
132C3×#97—41no—
133C3×#95—21no—
134C3×#95—21no—
135C3×#95—21no—
136C3×#95—21no—
137C3×#95—21no—
138C3×#95—21no—
139C3×#95—21no—
140C3×#95—21no—
141C22×SD16C22×C311yescharacteristic
142C2×C22⋊Q8—21no—
143C2×C4×Q8C22×C311yescharacteristic
144C2×C4×Q8C22×C311yescharacteristic
145C2×C4×D4C22×C311yescharacteristic
146C2×#107C22×C311yescharacteristic
147C2×#107C22×C311yescharacteristic
148C2×#107C22×C311yescharacteristic
149C2×#107C22×C311yescharacteristic
150C2×#106C22×C311yescharacteristic
151C2×C4⋊C8C22×C311yescharacteristic
152C2×#103C22×C311yescharacteristic
153C2×#103C22×C311yescharacteristic
154C2×#103C22×C311yescharacteristic
155C2×#102C22×C311yescharacteristic
156C2×#102C22×C311yescharacteristic
157C2×#102C22×C311yescharacteristic
158C2×#97C22×C311yescharacteristic
159#398—21no—
160#398—21no—
161#396—21no—
162#396—21no—
163#391—21no—
164#391C22×C311yescharacteristic
165#391C22×C311yescharacteristic
166#390C22×C311yescharacteristic
167#390C22×C311yescharacteristic
168#389—21no—
169#389C22×C311yescharacteristic
170#389C22×C311yescharacteristic
171#389C22×C311yescharacteristic
172#389C22×C311yescharacteristic
173#384C22×C311yescharacteristic
174#384C22×C311yescharacteristic
175#380—21no—
176#340C22×C311yescharacteristic
177#340C22×C311yescharacteristic
178#340C22×C311yescharacteristic
179#340C22×C311yescharacteristic
180#336C22×C311yescharacteristic
181#336C22×C311yescharacteristic
182#336C22×C311yescharacteristic
183#336C22×C311yescharacteristic
184C24×C3—21no—
185C2×C3×Q8—22no—
186C2×C3×Q8—22no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8—21no—
191C2×C3×Q8—21no—
192C2×C3×Q8—41no—
193C2×C3×Q8C2×Q812yes—
194C2×C3×D4—21no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4C2×Q812yes—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—42no—
208C22×C3×C4—42no—
209C22×C3×C4—42no—
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2411yescharacteristic
217C3×SD16—44no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8C2×Q812yes—
229C2×C3×C8#4112yes—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—41no—
243C3×C4⋊C4—41no—
244C3×C4⋊C4—41no—
245C3×C4⋊C4—42no—
246C3×C4⋊C4—42no—
247C3×C4⋊C4—42no—
248C3×C4⋊C4—42no—
249C3×C4⋊C4—42no—
250C3×C4⋊C4—81no—
251C3×C4⋊C4—81no—
252C3×C4⋊C4—81no—
253C3×C4⋊C4—81no—
254C3×C22⋊C4—42no—
255C3×C22⋊C4—42no—
256C3×C22⋊C4—42no—
257C3×C22⋊C4—81no—
258C3×C22⋊C4—81no—
259C3×C42—21no—
260C3×C42—21no—
261C3×C42—21no—
262C3×C42—21no—
263C3×C42—21no—
264C3×C42—21no—
265C3×C42—42no—
266C22×Q8C3×D411yescharacteristic
267C22×Q8C23×C311yescharacteristic
268C22×D4C23×C311yescharacteristic
269C23×C4—21no—
270C2×SD16—41no—
271C2×SD16—41no—
272C2×SD16C3×Q814yescomplemented
273C22×C8C23×C311yescharacteristic
274C22×C8C23×C311yescharacteristic
275C22×C8C23×C311yescharacteristic
276C22×C8C23×C311yescharacteristic
277C22⋊Q8—44no—
278C4×Q8—41no—
279C4×Q8—41no—
280C4×Q8—41no—
281C4×Q8—41no—
282C4×D4—41no—
283C4×D4—41no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4C3×D411yescharacteristic
294C2×C4⋊C4C3×D411yescharacteristic
295C2×C4⋊C4C3×D411yescharacteristic
296C2×C4⋊C4C23×C311yescharacteristic
297C2×C4⋊C4C23×C311yescharacteristic
298C2×C4⋊C4C23×C311yescharacteristic
299C2×C4⋊C4C23×C311yescharacteristic
300C2×C4⋊C4C23×C311yescharacteristic
301C2×C4⋊C4C23×C311yescharacteristic
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C42—21no—
308C2×C42C23×C311yescharacteristic
309C2×C42C23×C311yescharacteristic
310C2×C42C23×C311yescharacteristic
311#107—22no—
312#107—41no—
313#107—41no—
314#107—41no—
315#106—41no—
316C4⋊C8—41no—
317#103—22no—
318#103—41no—
319#103—41no—
320#102—22no—
321#102—41no—
322#102—41no—
323#97—41no—
324#95—21no—
325#95—21no—
326#95—21no—
327#95—21no—
328#95—21no—
329#95—21no—
330#95—21no—
331#95—21no—
332C23×C3—22no—
333C23×C3—41no—
334C23×C3—41no—
335C23×C3—41no—
336C23×C3—41no—
337C23×C3—41no—
338C23×C3—41no—
339C23×C3C22×D411yescharacteristic
340C3×Q8—22no—
341C3×Q8—22no—
342C3×Q8—44no—
343C3×Q8—41no—
344C3×Q8—41no—
345C3×D4—22no—
346C3×D4—22no—
347C3×D4—41no—
348C3×D4—41no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—22no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—84no—
387C2×C3×C4AGL(1,ℤ/8)12yes—
388C2×C3×C4#13712yes—
389C2×C3×C4C22×Q811yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4H(5,𝔽2)11yescharacteristic
392C2×C3×C4H(5,𝔽2)11yescharacteristic
393C2×C3×C4H(5,𝔽2)11yescharacteristic
394C2×C3×C42−1+411yescharacteristic
395C3×C8—22no—
396C3×C8—22no—
397C3×C8—41no—
398C3×C8—41no—
399C24—21no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—41no—
408C2×Q8C2×C3×Q812yes—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4C2×C3×Q812yes—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4C2×C3×D411yescharacteristic
426C22×C4C2×C3×D411yescharacteristic
427C22×C4C2×C3×D411yescharacteristic
428C22×C4C2×C3×D411yescharacteristic
429C22×C4C2×C3×D411yescharacteristic
430C22×C4C2×C3×D411yescharacteristic
431C22×C4C24×C311yescharacteristic
432SD16—44no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8C2×C3×Q812yes—
444C2×C8C3×#4112yes—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—81no—
466C4⋊C4—81no—
467C4⋊C4—81no—
468C4⋊C4—81no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—81no—
473C22⋊C4—81no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—42no—
481C22×C3—42no—
482C22×C3—42no—
483C22×C3—42no—
484C22×C3—81no—
485C22×C3—81no—
486C22×C3—81no—
487C22×C3—81no—
488C22×C3#53411yescharacteristic
489C22×C3D4×Q811yescharacteristic
490C22×C3#55011yescharacteristic
491C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
492C22×C3C2×#13711yescharacteristic
493C22×C3#57611yescharacteristic
494C22×C3#57811yescharacteristic
495C3×C4—22no—
496C3×C4—22no—
497C3×C4—41no—
498C3×C4—41no—
499C3×C4—41no—
500C3×C4—41no—
501C3×C4—42no—
502C3×C4—81no—
503C3×C4—81no—
504C3×C4—81no—
505C3×C4—81no—
506C3×C4—81no—
507C3×C4—81no—
508C3×C4#53911yescharacteristic
509C3×C4#54311yescharacteristic
510C3×C4#55411yescharacteristic
511C3×C4#56211yescharacteristic
512C23—22no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23C22×C3×D411yescharacteristic
520Q8—22no—
521Q8—22no—
522Q8—44no—
523Q8—41no—
524Q8—41no—
525D4—22no—
526D4—22no—
527D4—41no—
528D4—41no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—84no—
567C2×C4C3×AGL(1,ℤ/8)12yes—
568C2×C4C3×#13712yes—
569C2×C4C22×C3×Q811yescharacteristic
570C2×C4C2×C3×#4111yescharacteristic
571C2×C4C3×H(5,𝔽2)11yescharacteristic
572C2×C4C3×H(5,𝔽2)11yescharacteristic
573C2×C4C3×H(5,𝔽2)11yescharacteristic
574C2×C4C3×2−1+411yescharacteristic
575C8—22no—
576C8—22no—
577C8—41no—
578C8—41no—
579C2×C3—82no—
580C2×C3#273211yescharacteristic
581C2×C3#319311yescharacteristic
582C2×C3#320311yescharacteristic
583C2×C3#338511yescharacteristic
584C2×C3#338611yescharacteristic
585C2×C3#339811yescharacteristic
586C2×C3#339911yescharacteristic
587C22—42no—
588C22—42no—
589C22—42no—
590C22—81no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22C3×#53411yescharacteristic
595C22C3×D4×Q811yescharacteristic
596C22C3×#55011yescharacteristic
597C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
598C22C2×C3×#13711yescharacteristic
599C22C3×#57611yescharacteristic
600C22C3×#57811yescharacteristic
601C4—22no—
602C4—22no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—41no—
607C4—42no—
608C4—81no—
609C4—81no—
610C4—81no—
611C4—81no—
612C4—81no—
613C4—81no—
614C4C3×#53911yescharacteristic
615C4C3×#54311yescharacteristic
616C4C3×#55411yescharacteristic
617C4C3×#56211yescharacteristic
618C3#3266511yescharacteristic, direct product factor, Sylow
619C2—82no—
620C2C3×#273211yescharacteristic
621C2C3×#319311yescharacteristic
622C2C3×#320311yescharacteristic
623C2C3×#338511yescharacteristic
624C2C3×#338611yescharacteristic
625C2C3×#339811yescharacteristic
626C2C3×#339911yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#2582C2semidirect
3. C3×#2095C2semidirect
5. C3×#2083C2semidirect
6. C3×#2083C2semidirect
7. C3×#2068C2semidirect
9. C3×#1985C2semidirect
11. C3×#1943C2semidirect
14. C3×#1865C2semidirect
16. #32665C3direct (N × H)
60. #2582C2×C3semidirect
62. #2095C2×C3semidirect
64. #2083C2×C3semidirect
65. #2083C2×C3semidirect
66. #2068C2×C3semidirect
68. #1985C2×C3semidirect
70. #1943C2×C3semidirect
73. #1865C2×C3semidirect
81. C2×C3×SD16Q8semidirect
272. C2×SD16C3×Q8semidirect
618. C3#32665direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#25822
C3×#25382
C3×#20952
C3×#20932
C3×#20832
C3×#20682
C3×#20672
C3×#19852
C3×#19732
C3×#19432
C3×#18802
C3×#18772
C3×#18652
C3×#18642
#326653