Finite Group Databasebeta

← C3×#29267 #10647874

Subgroups — C3×#29267 #10647874

Order 768.

734 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2331C211yesmaximal, characteristic, complemented
2C3×#2326C211yesmaximal, characteristic
3C3×#2097C211yesmaximal, characteristic, complemented
4C3×#2092C211yesmaximal, characteristic
5C3×#2024C211yesmaximal, characteristic, complemented
6C3×#2023C211yesmaximal, characteristic
7C3×#1983C211yesmaximal, characteristic
8C3×#1983C211yesmaximal, characteristic
9C3×#1948C211yesmaximal, characteristic, complemented
10C3×#1947C211yesmaximal, characteristic, complemented
11C3×#1865C211yesmaximal, characteristic, complemented
12C3×#1864C211yesmaximal, characteristic
13C3×#1863C211yesmaximal, characteristic, complemented
14C3×#1862C211yesmaximal, characteristic
15C3×#1832C211yesmaximal, characteristic, complemented
16#29267C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#124C2211yescharacteristic
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×D4C2211yescharacteristic
21C2×C3×#107C2211yescharacteristic
22C2×C3×#107C2211yescharacteristic
23C2×C3×#106C2211yescharacteristic
24C2×C3×#106C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#103C2211yescharacteristic
28C2×C3×#103C2211yescharacteristic
29C2×C3×#103C2211yescharacteristic
30C2×C3×#103C2211yescharacteristic
31C2×C3×#102C2211yescharacteristic
32C2×C3×#102C2211yescharacteristic
33C2×C3×#102C2211yescharacteristic
34C2×C3×#102C2211yescharacteristic
35C3×#391—21no—
36C3×#391—21no—
37C3×#391C2211yescharacteristic
38C3×#390C2211yescharacteristic
39C3×#389C2211yescharacteristic
40C3×#389C2211yescharacteristic
41C3×#388—21no—
42C3×#388—21no—
43C3×#387—21no—
44C3×#387—21no—
45C3×#385—21no—
46C3×#385—21no—
47C3×#384C2211yescharacteristic
48C3×#382C2211yescharacteristic
49C3×C4×C4⋊C4C2211yescharacteristic
50C3×C4×C4⋊C4C2211yescharacteristic
51C3×#376C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#336C2211yescharacteristic
57C3×#336C2211yescharacteristic
58C3×#336C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#2331C2×C311yescharacteristic, complemented
61#2326C2×C311yescharacteristic
62#2097C2×C311yescharacteristic, complemented
63#2092C2×C311yescharacteristic
64#2024C2×C311yescharacteristic, complemented
65#2023C2×C311yescharacteristic
66#1983C2×C311yescharacteristic
67#1983C2×C311yescharacteristic
68#1948C2×C311yescharacteristic, complemented
69#1947C2×C311yescharacteristic, complemented
70#1865C2×C311yescharacteristic, complemented
71#1864C2×C311yescharacteristic
72#1863C2×C311yescharacteristic, complemented
73#1862C2×C311yescharacteristic
74#1832C2×C311yescharacteristic, complemented
75C22×C3×Q8—21no—
76C22×C3×Q8C2311yescharacteristic
77C22×C3×D4C2311yescharacteristic
78C23×C3×C4—21no—
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C3×C4⋊Q8—22no—
84C3×C4⋊Q8—22no—
85C3×#124—22no—
86C3×#124Q814yescomplemented
87C3×C4×Q8—41no—
88C3×C4×Q8—41no—
89C3×C4×D4—41no—
90C3×C4×D4—41no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C22⋊C4—21no—
108C2×C3×C42—21no—
109C2×C3×C42—21no—
110C2×C3×C42—21no—
111C2×C3×C42C2311yescharacteristic
112C2×C3×C42C2311yescharacteristic
113C2×C3×C42C2311yescharacteristic
114C3×#107—41no—
115C3×#107—41no—
116C3×#106—41no—
117C3×#106—41no—
118C3×C4⋊C8—22no—
119C3×C4⋊C8—22no—
120C3×#103—22no—
121C3×#103—22no—
122C3×#103—41no—
123C3×#103—41no—
124C3×#102—22no—
125C3×#102—22no—
126C3×#102—41no—
127C3×#102—41no—
128C3×#95—21no—
129C3×#95—21no—
130C3×#95—21no—
131C3×#95—21no—
132C3×#95—21no—
133C3×#95—21no—
134C3×#95—21no—
135C3×#95—21no—
136C3×#95D411yescharacteristic
137C3×#95D411yescharacteristic
138C3×#95D411yescharacteristic
139C3×#95D411yescharacteristic
140C2×C4⋊Q8C22×C311yescharacteristic
141C2×#124C22×C311yescharacteristic
142C2×C4×Q8C22×C311yescharacteristic
143C2×C4×D4C22×C311yescharacteristic
144C2×#107C22×C311yescharacteristic
145C2×#107C22×C311yescharacteristic
146C2×#106C22×C311yescharacteristic
147C2×#106C22×C311yescharacteristic
148C2×C4⋊C8C22×C311yescharacteristic
149C2×C4⋊C8C22×C311yescharacteristic
150C2×#103C22×C311yescharacteristic
151C2×#103C22×C311yescharacteristic
152C2×#103C22×C311yescharacteristic
153C2×#103C22×C311yescharacteristic
154C2×#102C22×C311yescharacteristic
155C2×#102C22×C311yescharacteristic
156C2×#102C22×C311yescharacteristic
157C2×#102C22×C311yescharacteristic
158#391—21no—
159#391—21no—
160#391C22×C311yescharacteristic
161#390C22×C311yescharacteristic
162#389C22×C311yescharacteristic
163#389C22×C311yescharacteristic
164#388—21no—
165#388—21no—
166#387—21no—
167#387—21no—
168#385—21no—
169#385—21no—
170#384C22×C311yescharacteristic
171#382C22×C311yescharacteristic
172C4×C4⋊C4C22×C311yescharacteristic
173C4×C4⋊C4C22×C311yescharacteristic
174#376C22×C311yescharacteristic
175#340C22×C311yescharacteristic
176#340C22×C311yescharacteristic
177#340C22×C311yescharacteristic
178#340C22×C311yescharacteristic
179#336C22×C311yescharacteristic
180#336C22×C311yescharacteristic
181#336C22×C311yescharacteristic
182#336C22×C311yescharacteristic
183C24×C3—21no—
184C2×C3×Q8—24no—
185C2×C3×Q8—21no—
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8—81no—
191C2×C3×Q8—81no—
192C2×C3×Q8C2×Q812yes—
193C2×C3×D4—21no—
194C2×C3×D4—21no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4C2×Q812yes—
199C22×C3×C4—21no—
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—42no—
207C22×C3×C4—81no—
208C22×C3×C4—81no—
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2411yescharacteristic
216C2×C3×C8—21no—
217C2×C3×C8—21no—
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—
228C3×C4⋊C4—24no—
229C3×C4⋊C4—21no—
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—42no—
240C3×C4⋊C4—81no—
241C3×C4⋊C4—81no—
242C3×C4⋊C4—81no—
243C3×C4⋊C4—81no—
244C3×C4⋊C4#4112yes—
245C3×C4⋊C4#4112yes—
246C3×C22⋊C4—24no—
247C3×C22⋊C4—24no—
248C3×C22⋊C4—42no—
249C3×C22⋊C4—81no—
250C3×C22⋊C4—81no—
251C3×C42—21no—
252C3×C42—21no—
253C3×C42—21no—
254C3×C42—21no—
255C3×C42—42no—
256C3×C42—81no—
257C3×C42—81no—
258C3×C42C2×Q812yes—
259C3×C42#4112yes—
260C22×Q8—21no—
261C22×Q8C23×C311yescharacteristic
262C22×D4C23×C311yescharacteristic
263C23×C4—21no—
264C22×C8C23×C311yescharacteristic
265C22×C8C23×C311yescharacteristic
266C22×C8C23×C311yescharacteristic
267C22×C8C23×C311yescharacteristic
268C4⋊Q8—22no—
269C4⋊Q8—22no—
270#124—22no—
271#124C3×Q814yescomplemented
272C4×Q8—41no—
273C4×Q8—41no—
274C4×D4—41no—
275C4×D4—41no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4C23×C311yescharacteristic
283C2×C4⋊C4C23×C311yescharacteristic
284C2×C4⋊C4C23×C311yescharacteristic
285C2×C4⋊C4C23×C311yescharacteristic
286C2×C4⋊C4C23×C311yescharacteristic
287C2×C4⋊C4C23×C311yescharacteristic
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42C23×C311yescharacteristic
297C2×C42C23×C311yescharacteristic
298C2×C42C23×C311yescharacteristic
299#107—41no—
300#107—41no—
301#106—41no—
302#106—41no—
303C4⋊C8—22no—
304C4⋊C8—22no—
305#103—22no—
306#103—22no—
307#103—41no—
308#103—41no—
309#102—22no—
310#102—22no—
311#102—41no—
312#102—41no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
318#95—21no—
319#95—21no—
320#95—21no—
321#95C3×D411yescharacteristic
322#95C3×D411yescharacteristic
323#95C3×D411yescharacteristic
324#95C3×D411yescharacteristic
325C23×C3—22no—
326C23×C3—41no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3—41no—
330C23×C3—41no—
331C23×C3—41no—
332C23×C3C22×D411yescharacteristic
333C3×Q8—22no—
334C3×Q8—22no—
335C3×Q8—41no—
336C3×Q8—41no—
337C3×Q8—84no—
338C3×D4—22no—
339C3×D4—22no—
340C3×D4—41no—
341C3×D4—41no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—22no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
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—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
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—84no—
380C2×C3×C4C22⋊Q812yes—
381C2×C3×C4C22⋊Q812yes—
382C2×C3×C4C22×Q811yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C2×C3×C4H(5,𝔽2)11yescharacteristic
387C2×C3×C4H(5,𝔽2)11yescharacteristic
388C3×C8—41no—
389C3×C8—41no—
390C3×C8—41no—
391C3×C8—41no—
392C24—21no—
393C2×Q8—24no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—81no—
400C2×Q8—81no—
401C2×Q8C2×C3×Q812yes—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4C2×C3×Q812yes—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—42no—
416C22×C4—81no—
417C22×C4—81no—
418C22×C4C2×C3×D411yescharacteristic
419C22×C4C2×C3×D411yescharacteristic
420C22×C4C2×C3×D411yescharacteristic
421C22×C4C2×C3×D411yescharacteristic
422C22×C4C2×C3×D411yescharacteristic
423C22×C4C2×C3×D411yescharacteristic
424C22×C4C24×C311yescharacteristic
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C4⋊C4—24no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—42no—
449C4⋊C4—81no—
450C4⋊C4—81no—
451C4⋊C4—81no—
452C4⋊C4—81no—
453C4⋊C4C3×#4112yes—
454C4⋊C4C3×#4112yes—
455C22⋊C4—24no—
456C22⋊C4—24no—
457C22⋊C4—42no—
458C22⋊C4—81no—
459C22⋊C4—81no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—42no—
465C42—81no—
466C42—81no—
467C42C2×C3×Q812yes—
468C42C3×#4112yes—
469C22×C3—42no—
470C22×C3—42no—
471C22×C3—42no—
472C22×C3—81no—
473C22×C3—81no—
474C22×C3—81no—
475C22×C3—81no—
476C22×C3C2×C22⋊Q811yescharacteristic
477C22×C3#52511yescharacteristic
478C22×C3#53511yescharacteristic
479C22×C3#57611yescharacteristic
480C22×C3#57711yescharacteristic
481C22×C3#57711yescharacteristic
482C22×C3#57811yescharacteristic
483C3×C4—22no—
484C3×C4—22no—
485C3×C4—42no—
486C3×C4—41no—
487C3×C4—42no—
488C3×C4—42no—
489C3×C4—41no—
490C3×C4—41no—
491C3×C4—41no—
492C3×C4—81no—
493C3×C4—81no—
494C3×C4—81no—
495C3×C4—81no—
496C3×C4#55411yescharacteristic
497C3×C4#55411yescharacteristic
498C3×C4#55911yescharacteristic
499C3×C4#55911yescharacteristic
500C23—22no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23C22×C3×D411yescharacteristic
508Q8—22no—
509Q8—22no—
510Q8—41no—
511Q8—41no—
512Q8—84no—
513D4—22no—
514D4—22no—
515D4—41no—
516D4—41no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
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—84no—
555C2×C4C3×C22⋊Q812yes—
556C2×C4C3×C22⋊Q812yes—
557C2×C4C22×C3×Q811yescharacteristic
558C2×C4C2×C3×#4111yescharacteristic
559C2×C4C2×C3×#4111yescharacteristic
560C2×C4C2×C3×#4111yescharacteristic
561C2×C4C3×H(5,𝔽2)11yescharacteristic
562C2×C4C3×H(5,𝔽2)11yescharacteristic
563C8—41no—
564C8—41no—
565C8—41no—
566C8—41no—
567C2×C3—82no—
568C2×C3#258011yescharacteristic
569C2×C3#308311yescharacteristic
570C2×C3#308411yescharacteristic
571C2×C3#311811yescharacteristic
572C2×C3#312111yescharacteristic
573C2×C3#322411yescharacteristic
574C2×C3#322511yescharacteristic
575C22—42no—
576C22—42no—
577C22—42no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22C2×C3×C22⋊Q811yescharacteristic
583C22C3×#52511yescharacteristic
584C22C3×#53511yescharacteristic
585C22C3×#57611yescharacteristic
586C22C3×#57711yescharacteristic
587C22C3×#57711yescharacteristic
588C22C3×#57811yescharacteristic
589C4—22no—
590C4—22no—
591C4—41no—
592C4—41no—
593C4—42no—
594C4—41no—
595C4—42no—
596C4—42no—
597C4—41no—
598C4—81no—
599C4—81no—
600C4—81no—
601C4—81no—
602C4C3×#55411yescharacteristic
603C4C3×#55411yescharacteristic
604C4C3×#55911yescharacteristic
605C4C3×#55911yescharacteristic
606C3#2926711yescharacteristic, direct product factor, Sylow
607C2—82no—
608C2C3×#258011yescharacteristic
609C2C3×#308311yescharacteristic
610C2C3×#308411yescharacteristic
611C2C3×#311811yescharacteristic
612C2C3×#312111yescharacteristic
613C2C3×#322411yescharacteristic
614C2C3×#322511yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×#376 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#2331C2semidirect
3. C3×#2097C2semidirect
5. C3×#2024C2semidirect
9. C3×#1948C2semidirect
10. C3×#1947C2semidirect
11. C3×#1865C2semidirect
13. C3×#1863C2semidirect
15. C3×#1832C2semidirect
16. #29267C3direct (N × H)
60. #2331C2×C3semidirect
62. #2097C2×C3semidirect
64. #2024C2×C3semidirect
68. #1948C2×C3semidirect
69. #1947C2×C3semidirect
70. #1865C2×C3semidirect
72. #1863C2×C3semidirect
74. #1832C2×C3semidirect
86. C3×#124Q8semidirect
271. #124C3×Q8semidirect
606. C3#29267direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#23312
C3×#23262
C3×#20972
C3×#20922
C3×#20242
C3×#20232
C3×#19832
C3×#19482
C3×#19472
C3×#18652
C3×#18642
C3×#18632
C3×#18622
C3×#18322
#292673