Finite Group Databasebeta

← C3×#27715 #10646322

Subgroups — C3×#27715 #10646322

Order 768.

734 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
1C3×#2502C211yesmaximal, characteristic
2C3×#2442C211yesmaximal, characteristic
3C3×#2099C211yesmaximal, characteristic, complemented
4C3×#2099C211yesmaximal, characteristic, complemented
5C3×#2074C211yesmaximal, characteristic, complemented
6C3×#2026C211yesmaximal, characteristic, complemented
7C3×#1992C211yesmaximal, characteristic, complemented
8C3×#1935C211yesmaximal, characteristic, complemented
9C3×#1935C211yesmaximal, characteristic, complemented
10C3×#1892C211yesmaximal, characteristic
11C3×#1872C211yesmaximal, characteristic
12C3×#1836C211yesmaximal, characteristic, complemented
13C3×#1824C211yesmaximal, characteristic
14C3×#1752C211yesmaximal, characteristic
15C2×C3×#340C211yesmaximal, characteristic
16#27715C311yesmaximal, characteristic, direct product factor, Sylow
17C23×C3×C8C2211yescharacteristic
18C2×C3×#125C2211yescharacteristic
19C2×C3×C22⋊Q8C2211yescharacteristic
20C22×C3×C4⋊C4C2211yescharacteristic
21C22×C3×C4⋊C4C2211yescharacteristic
22C2×C3×#103C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×C22⋊C8C2211yescharacteristic
27C2×C3×#97C2211yescharacteristic
28C2×C3×#97C2211yescharacteristic
29C3×#400—21no—
30C3×#398—21no—
31C3×#398—21no—
32C3×#398—21no—
33C3×#391C2211yescharacteristic
34C3×#391C2211yescharacteristic
35C3×#389C2211yescharacteristic
36C3×#389C2211yescharacteristic
37C3×#385—21no—
38C3×#385—21no—
39C3×#385—21no—
40C3×#384C2211yescharacteristic
41C3×#384C2211yescharacteristic
42C3×#382C2211yescharacteristic
43C3×#380—21no—
44C3×#380C2211yescharacteristic
45C3×C4×C22⋊C4C2211yescharacteristic
46C3×#340C2211yescharacteristic
47C3×#340C2211yescharacteristic
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58C3×#336C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#2502C2×C311yescharacteristic
61#2442C2×C311yescharacteristic
62#2099C2×C311yescharacteristic, complemented
63#2099C2×C311yescharacteristic, complemented
64#2074C2×C311yescharacteristic, complemented
65#2026C2×C311yescharacteristic, complemented
66#1992C2×C311yescharacteristic, complemented
67#1935C2×C311yescharacteristic, complemented
68#1935C2×C311yescharacteristic, complemented
69#1892C2×C311yescharacteristic
70#1872C2×C311yescharacteristic
71#1836C2×C311yescharacteristic, complemented
72#1824C2×C311yescharacteristic
73#1752C2×C311yescharacteristic
74C2×#340C2×C311yescharacteristic
75C22×C3×Q8C2311yescharacteristic
76C23×C3×C4—21no—
77C23×C3×C4—21no—
78C23×C3×C4C2311yescharacteristic
79C22×C3×C8—22no—
80C22×C3×C8—22no—
81C22×C3×C8—41no—
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×#125—22no—
87C3×#125—41no—
88C3×C22⋊Q8—22no—
89C3×C22⋊Q8—22no—
90C2×C3×C4⋊C4—22no—
91C2×C3×C4⋊C4—22no—
92C2×C3×C4⋊C4—22no—
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⋊C4—21no—
103C2×C3×C4⋊C4—21no—
104C2×C3×C4⋊C4—41no—
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×C4⋊C4C2311yescharacteristic
112C2×C3×C22⋊C4—21no—
113C2×C3×C22⋊C4—21no—
114C2×C3×C22⋊C4D411yescharacteristic
115C2×C3×C22⋊C4D411yescharacteristic
116C2×C3×C42C2311yescharacteristic
117C2×C3×C42C2311yescharacteristic
118C3×#103—22no—
119C3×#103—22no—
120C3×#103—41no—
121C3×#103—41no—
122C3×C22⋊C8—22no—
123C3×#97—22no—
124C3×#97—41no—
125C3×#95—21no—
126C3×#95—21no—
127C3×#95—21no—
128C3×#95—21no—
129C3×#95—21no—
130C3×#95—21no—
131C3×#95—21no—
132C3×#95—21no—
133C3×#95—21no—
134C3×#95D411yescharacteristic
135C3×#95D411yescharacteristic
136C23×C8C22×C311yescharacteristic
137C2×#125C22×C311yescharacteristic
138C2×C22⋊Q8C22×C311yescharacteristic
139C22×C4⋊C4C22×C311yescharacteristic
140C22×C4⋊C4C22×C311yescharacteristic
141C2×#103C22×C311yescharacteristic
142C2×#103C22×C311yescharacteristic
143C2×#103C22×C311yescharacteristic
144C2×#103C22×C311yescharacteristic
145C2×C22⋊C8C22×C311yescharacteristic
146C2×#97C22×C311yescharacteristic
147C2×#97C22×C311yescharacteristic
148#400—21no—
149#398—21no—
150#398—21no—
151#398—21no—
152#391C22×C311yescharacteristic
153#391C22×C311yescharacteristic
154#389C22×C311yescharacteristic
155#389C22×C311yescharacteristic
156#385—21no—
157#385—21no—
158#385—21no—
159#384C22×C311yescharacteristic
160#384C22×C311yescharacteristic
161#382C22×C311yescharacteristic
162#380—21no—
163#380C22×C311yescharacteristic
164C4×C22⋊C4C22×C311yescharacteristic
165#340C22×C311yescharacteristic
166#340C22×C311yescharacteristic
167#340C22×C311yescharacteristic
168#340C22×C311yescharacteristic
169#340C22×C311yescharacteristic
170#340C22×C311yescharacteristic
171#340C22×C311yescharacteristic
172#340C22×C311yescharacteristic
173#340C22×C311yescharacteristic
174#340C22×C311yescharacteristic
175#340C22×C311yescharacteristic
176#340C22×C311yescharacteristic
177#336C22×C311yescharacteristic
178#336C22×C311yescharacteristic
179C24×C3C2×D411yescharacteristic
180C2×C3×Q8—21no—
181C2×C3×Q8—21no—
182C2×C3×Q8—21no—
183C2×C3×Q8—21no—
184C2×C3×Q8—21no—
185C2×C3×Q8—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×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—42no—
199C22×C3×C4—42no—
200C22×C3×C4—42no—
201C22×C3×C4—81no—
202C22×C3×C4—81no—
203C22×C3×C4—81no—
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4#4112yes—
210C22×C3×C4#4112yes—
211C22×C3×C4C2411yescharacteristic
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
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—44no—
224C2×C3×C8#4112yes—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
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—42no—
239C3×C4⋊C4—42no—
240C3×C4⋊C4—42no—
241C3×C4⋊C4—42no—
242C3×C4⋊C4—42no—
243C3×C4⋊C4—81no—
244C3×C4⋊C4—81no—
245C3×C4⋊C4—81no—
246C3×C4⋊C4—81no—
247C3×C4⋊C4—81no—
248C3×C4⋊C4—81no—
249C3×C4⋊C4#4112yes—
250C3×C22⋊C4—22no—
251C3×C22⋊C4—22no—
252C3×C22⋊C4—42no—
253C3×C22⋊C4—42no—
254C3×C42—21no—
255C3×C42—21no—
256C3×C42—21no—
257C3×C42—21no—
258C22×Q8C23×C311yescharacteristic
259C23×C4—21no—
260C23×C4—21no—
261C23×C4C23×C311yescharacteristic
262C22×C8—22no—
263C22×C8—22no—
264C22×C8—41no—
265C22×C8C23×C311yescharacteristic
266C22×C8C23×C311yescharacteristic
267C22×C8C23×C311yescharacteristic
268C22×C8C23×C311yescharacteristic
269#125—22no—
270#125—41no—
271C22⋊Q8—22no—
272C22⋊Q8—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
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⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—41no—
288C2×C4⋊C4C23×C311yescharacteristic
289C2×C4⋊C4C23×C311yescharacteristic
290C2×C4⋊C4C23×C311yescharacteristic
291C2×C4⋊C4C23×C311yescharacteristic
292C2×C4⋊C4C23×C311yescharacteristic
293C2×C4⋊C4C23×C311yescharacteristic
294C2×C4⋊C4C23×C311yescharacteristic
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4C3×D411yescharacteristic
298C2×C22⋊C4C3×D411yescharacteristic
299C2×C42C23×C311yescharacteristic
300C2×C42C23×C311yescharacteristic
301#103—22no—
302#103—22no—
303#103—41no—
304#103—41no—
305C22⋊C8—22no—
306#97—22no—
307#97—41no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95C3×D411yescharacteristic
318#95C3×D411yescharacteristic
319C23×C3—21no—
320C23×C3—21no—
321C23×C3—21no—
322C23×C3—21no—
323C23×C3#12412yes—
324C23×C3#13712yes—
325C23×C3#13712yes—
326C23×C3C22×D411yescharacteristic
327C3×Q8—41no—
328C3×Q8—41no—
329C3×Q8—41no—
330C3×Q8—41no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—22no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
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—84no—
374C2×C3×C4—84no—
375C2×C3×C4#12412yes—
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C2×C3×C4C2×#4111yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380C2×C3×C4H(5,𝔽2)11yescharacteristic
381C2×C3×C42−1+411yescharacteristic
382C3×C8—22no—
383C3×C8—41no—
384C3×C8—41no—
385C3×C8—41no—
386C24C2×C3×D411yescharacteristic
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—81no—
409C22×C4—81no—
410C22×C4—81no—
411C22×C4C2×C3×D411yescharacteristic
412C22×C4C2×C3×D411yescharacteristic
413C22×C4C2×C3×D411yescharacteristic
414C22×C4C2×C3×D411yescharacteristic
415C22×C4C2×C3×D411yescharacteristic
416C22×C4C3×#4112yes—
417C22×C4C3×#4112yes—
418C22×C4C24×C311yescharacteristic
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—44no—
431C2×C8C3×#4112yes—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—81no—
451C4⋊C4—81no—
452C4⋊C4—81no—
453C4⋊C4—81no—
454C4⋊C4—81no—
455C4⋊C4—81no—
456C4⋊C4C3×#4112yes—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—21no—
465C22×C3—22no—
466C22×C3—22no—
467C22×C3—22no—
468C22×C3—22no—
469C22×C3—22no—
470C22×C3—22no—
471C22×C3#49114yes—
472C22×C3C2×#12411yescharacteristic
473C22×C3#54611yescharacteristic
474C22×C3#54711yescharacteristic
475C22×C3C2×#13711yescharacteristic
476C22×C3C2×#13711yescharacteristic
477C22×C3#57611yescharacteristic
478C22×C3#57711yescharacteristic
479C3×C4—22no—
480C3×C4—41no—
481C3×C4—41no—
482C3×C4—41no—
483C3×C4—42no—
484C3×C4—41no—
485C3×C4—81no—
486C3×C4—81no—
487C3×C4—81no—
488C3×C4—81no—
489C3×C4—81no—
490C3×C4—81no—
491C3×C4—81no—
492C3×C4#52911yescharacteristic
493C3×C4#53911yescharacteristic
494C3×C4#55111yescharacteristic
495C3×C4#55211yescharacteristic
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23C3×#12412yes—
501C23C3×#13712yes—
502C23C3×#13712yes—
503C23C22×C3×D411yescharacteristic
504Q8—41no—
505Q8—41no—
506Q8—41no—
507Q8—41no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
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—84no—
551C2×C4—84no—
552C2×C4C3×#12412yes—
553C2×C4C2×C3×#4111yescharacteristic
554C2×C4C2×C3×#4111yescharacteristic
555C2×C4C2×C3×#4111yescharacteristic
556C2×C4C2×C3×#4111yescharacteristic
557C2×C4C3×H(5,𝔽2)11yescharacteristic
558C2×C4C3×2−1+411yescharacteristic
559C8—22no—
560C8—41no—
561C8—41no—
562C8—41no—
563C2×C3—24no—
564C2×C3#248811yescharacteristic
565C2×C3C2×#49111yescharacteristic
566C2×C3#313611yescharacteristic
567C2×C3#331411yescharacteristic
568C2×C3#331611yescharacteristic
569C2×C3#334411yescharacteristic
570C2×C3#334611yescharacteristic
571C22—22no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22C3×#49114yes—
578C22C2×C3×#12411yescharacteristic
579C22C3×#54611yescharacteristic
580C22C3×#54711yescharacteristic
581C22C2×C3×#13711yescharacteristic
582C22C2×C3×#13711yescharacteristic
583C22C3×#57611yescharacteristic
584C22C3×#57711yescharacteristic
585C4—22no—
586C4—41no—
587C4—42no—
588C4—41no—
589C4—41no—
590C4—41no—
591C4—81no—
592C4—81no—
593C4—81no—
594C4—81no—
595C4—81no—
596C4—81no—
597C4—81no—
598C4C3×#52911yescharacteristic
599C4C3×#53911yescharacteristic
600C4C3×#55111yescharacteristic
601C4C3×#55211yescharacteristic
602C3#2771511yescharacteristic, direct product factor, Sylow
603C2—24no—
604C2C3×#248811yescharacteristic
605C2C2×C3×#49111yescharacteristic
606C2C3×#313611yescharacteristic
607C2C3×#331411yescharacteristic
608C2C3×#331611yescharacteristic
609C2C3×#334411yescharacteristic
610C2C3×#334611yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C4×C22⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×#2099C2semidirect
4. C3×#2099C2semidirect
5. C3×#2074C2semidirect
6. C3×#2026C2semidirect
7. C3×#1992C2semidirect
8. C3×#1935C2semidirect
9. C3×#1935C2semidirect
12. C3×#1836C2semidirect
16. #27715C3direct (N × H)
62. #2099C2×C3semidirect
63. #2099C2×C3semidirect
64. #2074C2×C3semidirect
65. #2026C2×C3semidirect
66. #1992C2×C3semidirect
67. #1935C2×C3semidirect
68. #1935C2×C3semidirect
71. #1836C2×C3semidirect
602. C3#27715direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#25022
C3×#24422
C3×#20992
C3×#20742
C3×#20262
C3×#19922
C3×#19352
C3×#18922
C3×#18722
C3×#18362
C3×#18242
C3×#17522
C2×C3×#3402
#277153