Finite Group Databasebeta

← C3×#27697 #10646304

Subgroups — C3×#27697 #10646304

Order 768.

790 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2502C211yesmaximal, characteristic
2C3×#2440C211yesmaximal, characteristic
3C3×#2100C211yesmaximal, characteristic, complemented
4C3×#2100C211yesmaximal, characteristic, complemented
5C3×#2072C211yesmaximal, characteristic, complemented
6C3×#2024C211yesmaximal, characteristic, complemented
7C3×#1990C211yesmaximal, characteristic, complemented
8C3×#1933C211yesmaximal, characteristic, complemented
9C3×#1933C211yesmaximal, characteristic, complemented
10C3×#1892C211yesmaximal, characteristic
11C3×#1872C211yesmaximal, characteristic
12C3×#1834C211yesmaximal, characteristic, complemented
13C3×#1824C211yesmaximal, characteristic
14C3×C8×C22⋊C4C211yesmaximal, characteristic
15C2×C3×#340C211yesmaximal, characteristic
16#27697C311yesmaximal, characteristic, direct product factor, Sylow
17C23×C3×C8C2211yescharacteristic
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×C22⋊Q8—21no—
20C2×C3×C22⋊Q8C2211yescharacteristic
21C22×C3×C4⋊C4C2211yescharacteristic
22C22×C3×C4⋊C4C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×C22⋊C8C2211yescharacteristic
28C2×C3×C4×C8C2211yescharacteristic
29C2×C3×C4×C8C2211yescharacteristic
30C3×#398—21no—
31C3×#398—21no—
32C3×#398—21no—
33C3×#395—21no—
34C3×#391C2211yescharacteristic
35C3×#391C2211yescharacteristic
36C3×#389C2211yescharacteristic
37C3×#389C2211yescharacteristic
38C3×#385—21no—
39C3×#385—21no—
40C3×#385—21no—
41C3×#384C2211yescharacteristic
42C3×#384C2211yescharacteristic
43C3×#384C2211yescharacteristic
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#2440C2×C311yescharacteristic
62#2100C2×C311yescharacteristic, complemented
63#2100C2×C311yescharacteristic, complemented
64#2072C2×C311yescharacteristic, complemented
65#2024C2×C311yescharacteristic, complemented
66#1990C2×C311yescharacteristic, complemented
67#1933C2×C311yescharacteristic, complemented
68#1933C2×C311yescharacteristic, complemented
69#1892C2×C311yescharacteristic
70#1872C2×C311yescharacteristic
71#1834C2×C311yescharacteristic, complemented
72#1824C2×C311yescharacteristic
73C8×C22⋊C4C2×C311yescharacteristic
74C2×#340C2×C311yescharacteristic
75C22×C3×Q8—21no—
76C22×C3×Q8C2311yescharacteristic
77C23×C3×C4—21no—
78C23×C3×C4—21no—
79C23×C3×C4C2311yescharacteristic
80C22×C3×C8—22no—
81C22×C3×C8—22no—
82C22×C3×C8—41no—
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C3×C4⋊Q8—22no—
88C3×C4⋊Q8—41no—
89C3×C22⋊Q8—22no—
90C3×C22⋊Q8—22no—
91C3×C22⋊Q8—44no—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4—22no—
94C2×C3×C4⋊C4—22no—
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—21no—
105C2×C3×C4⋊C4—21no—
106C2×C3×C4⋊C4—21no—
107C2×C3×C4⋊C4—41no—
108C2×C3×C4⋊C4C2311yescharacteristic
109C2×C3×C4⋊C4C2311yescharacteristic
110C2×C3×C4⋊C4C2311yescharacteristic
111C2×C3×C4⋊C4C2311yescharacteristic
112C2×C3×C4⋊C4C2311yescharacteristic
113C2×C3×C4⋊C4C2311yescharacteristic
114C2×C3×C4⋊C4C2311yescharacteristic
115C2×C3×C22⋊C4—21no—
116C2×C3×C22⋊C4—21no—
117C2×C3×C22⋊C4D411yescharacteristic
118C2×C3×C22⋊C4D411yescharacteristic
119C2×C3×C42C2311yescharacteristic
120C2×C3×C42C2311yescharacteristic
121C3×#103—22no—
122C3×#103—22no—
123C3×#103—41no—
124C3×#103—41no—
125C3×C22⋊C8—22no—
126C3×C4×C8—22no—
127C3×C4×C8—41no—
128C3×#95—21no—
129C3×#95—21no—
130C3×#95—21no—
131C3×#95—21no—
132C3×#95—21no—
133C3×#95—21no—
134C3×#95—21no—
135C3×#95D411yescharacteristic
136C3×#95D411yescharacteristic
137C23×C8C22×C311yescharacteristic
138C2×C4⋊Q8C22×C311yescharacteristic
139C2×C22⋊Q8—21no—
140C2×C22⋊Q8C22×C311yescharacteristic
141C22×C4⋊C4C22×C311yescharacteristic
142C22×C4⋊C4C22×C311yescharacteristic
143C2×#103C22×C311yescharacteristic
144C2×#103C22×C311yescharacteristic
145C2×#103C22×C311yescharacteristic
146C2×#103C22×C311yescharacteristic
147C2×C22⋊C8C22×C311yescharacteristic
148C2×C4×C8C22×C311yescharacteristic
149C2×C4×C8C22×C311yescharacteristic
150#398—21no—
151#398—21no—
152#398—21no—
153#395—21no—
154#391C22×C311yescharacteristic
155#391C22×C311yescharacteristic
156#389C22×C311yescharacteristic
157#389C22×C311yescharacteristic
158#385—21no—
159#385—21no—
160#385—21no—
161#384C22×C311yescharacteristic
162#384C22×C311yescharacteristic
163#384C22×C311yescharacteristic
164#380C22×C311yescharacteristic
165C4×C22⋊C4C22×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#340C22×C311yescharacteristic
178#336C22×C311yescharacteristic
179#336C22×C311yescharacteristic
180C24×C3C2×D411yescharacteristic
181C2×C3×Q8—21no—
182C2×C3×Q8—21no—
183C2×C3×Q8—21no—
184C2×C3×Q8—21no—
185C2×C3×Q8—21no—
186C2×C3×Q8—21no—
187C2×C3×Q8—42no—
188C2×C3×Q8—42no—
189C2×C3×Q8—81no—
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—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—42no—
203C22×C3×C4—42no—
204C22×C3×C4—42no—
205C22×C3×C4—81no—
206C22×C3×C4—81no—
207C22×C3×C4—81no—
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4#4112yes—
214C22×C3×C4#4112yes—
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—44no—
228C2×C3×C8#4112yes—
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—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—42no—
243C3×C4⋊C4—42no—
244C3×C4⋊C4—42no—
245C3×C4⋊C4—42no—
246C3×C4⋊C4—42no—
247C3×C4⋊C4—42no—
248C3×C4⋊C4—42no—
249C3×C4⋊C4—81no—
250C3×C4⋊C4—81no—
251C3×C4⋊C4—81no—
252C3×C4⋊C4—81no—
253C3×C4⋊C4—81no—
254C3×C4⋊C4#4112yes—
255C3×C22⋊C4—42no—
256C3×C22⋊C4—42no—
257C3×C22⋊C4SD1614yes—
258C3×C22⋊C4Q1614yes—
259C3×C42—21no—
260C3×C42—21no—
261C3×C42—21no—
262C3×C42—21no—
263C22×Q8—21no—
264C22×Q8C23×C311yescharacteristic
265C23×C4—21no—
266C23×C4—21no—
267C23×C4C23×C311yescharacteristic
268C22×C8—22no—
269C22×C8—22no—
270C22×C8—41no—
271C22×C8C23×C311yescharacteristic
272C22×C8C23×C311yescharacteristic
273C22×C8C23×C311yescharacteristic
274C22×C8C23×C311yescharacteristic
275C4⋊Q8—22no—
276C4⋊Q8—41no—
277C22⋊Q8—22no—
278C22⋊Q8—22no—
279C22⋊Q8—44no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
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⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—41no—
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×C4⋊C4C23×C311yescharacteristic
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4C3×D411yescharacteristic
306C2×C22⋊C4C3×D411yescharacteristic
307C2×C42C23×C311yescharacteristic
308C2×C42C23×C311yescharacteristic
309#103—22no—
310#103—22no—
311#103—41no—
312#103—41no—
313C22⋊C8—22no—
314C4×C8—22no—
315C4×C8—41no—
316#95—21no—
317#95—21no—
318#95—21no—
319#95—21no—
320#95—21no—
321#95—21no—
322#95—21no—
323#95C3×D411yescharacteristic
324#95C3×D411yescharacteristic
325C23×C3—21no—
326C23×C3—21no—
327C23×C3—21no—
328C23×C3—21no—
329C23×C3#12412yes—
330C23×C3C2×SD1612yes—
331C23×C3C2×Q1612yes—
332C23×C3C22×D411yescharacteristic
333C3×Q8—41no—
334C3×Q8—41no—
335C3×Q8—41no—
336C3×Q8—41no—
337C3×Q8—84no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
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—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
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—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—84no—
377C2×C3×C4—84no—
378C2×C3×C4#12412yes—
379C2×C3×C4C2×SD1612yes—
380C2×C3×C4C2×SD1612yes—
381C2×C3×C4C2×Q1612yes—
382C2×C3×C4C2×Q1612yes—
383C2×C3×C4C2×#4111yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387C2×C3×C4H(5,𝔽2)11yescharacteristic
388C2×C3×C42−1+411yescharacteristic
389C3×C8—22no—
390C3×C8—41no—
391C3×C8—41no—
392C3×C8—41no—
393C24C2×C3×D411yescharacteristic
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8—81no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
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—42no—
417C22×C4—42no—
418C22×C4—81no—
419C22×C4—81no—
420C22×C4—81no—
421C22×C4C2×C3×D411yescharacteristic
422C22×C4C2×C3×D411yescharacteristic
423C22×C4C2×C3×D411yescharacteristic
424C22×C4C2×C3×D411yescharacteristic
425C22×C4C2×C3×D411yescharacteristic
426C22×C4C3×#4112yes—
427C22×C4C3×#4112yes—
428C22×C4C24×C311yescharacteristic
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—44no—
441C2×C8C3×#4112yes—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
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—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—81no—
463C4⋊C4—81no—
464C4⋊C4—81no—
465C4⋊C4—81no—
466C4⋊C4—81no—
467C4⋊C4C3×#4112yes—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C22⋊C4C3×SD1614yes—
471C22⋊C4C3×Q1614yes—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—21no—
476C22×C3—22no—
477C22×C3—22no—
478C22×C3—22no—
479C22×C3—22no—
480C22×C3—22no—
481C22×C3—22no—
482C22×C3#48714yes—
483C22×C3C2×#12411yescharacteristic
484C22×C3#54611yescharacteristic
485C22×C3#54711yescharacteristic
486C22×C3C22×SD1611yescharacteristic
487C22×C3C22×Q1611yescharacteristic
488C22×C3#57611yescharacteristic
489C22×C3#57711yescharacteristic
490C3×C4—22no—
491C3×C4—22no—
492C3×C4—22no—
493C3×C4—22no—
494C3×C4—22no—
495C3×C4—42no—
496C3×C4—81no—
497C3×C4—81no—
498C3×C4—81no—
499C3×C4—81no—
500C3×C4—81no—
501C3×C4—81no—
502C3×C4—81no—
503C3×C4#52911yescharacteristic
504C3×C4#53911yescharacteristic
505C3×C4#55111yescharacteristic
506C3×C4#55211yescharacteristic
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23C3×#12412yes—
512C23C2×C3×SD1612yes—
513C23C2×C3×Q1612yes—
514C23C22×C3×D411yescharacteristic
515Q8—41no—
516Q8—41no—
517Q8—41no—
518Q8—41no—
519Q8—84no—
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—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
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—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—84no—
559C2×C4—84no—
560C2×C4C3×#12412yes—
561C2×C4C2×C3×SD1612yes—
562C2×C4C2×C3×SD1612yes—
563C2×C4C2×C3×Q1612yes—
564C2×C4C2×C3×Q1612yes—
565C2×C4C2×C3×#4111yescharacteristic
566C2×C4C2×C3×#4111yescharacteristic
567C2×C4C2×C3×#4111yescharacteristic
568C2×C4C2×C3×#4111yescharacteristic
569C2×C4C3×H(5,𝔽2)11yescharacteristic
570C2×C4C3×2−1+411yescharacteristic
571C8—22no—
572C8—41no—
573C8—41no—
574C8—41no—
575C2×C3—24no—
576C2×C3#248811yescharacteristic
577C2×C3C2×#48711yescharacteristic
578C2×C3#313611yescharacteristic
579C2×C3#329511yescharacteristic
580C2×C3#329911yescharacteristic
581C2×C3#333511yescharacteristic
582C2×C3#333811yescharacteristic
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22C3×#48714yes—
590C22C2×C3×#12411yescharacteristic
591C22C3×#54611yescharacteristic
592C22C3×#54711yescharacteristic
593C22C22×C3×SD1611yescharacteristic
594C22C22×C3×Q1611yescharacteristic
595C22C3×#57611yescharacteristic
596C22C3×#57711yescharacteristic
597C4—22no—
598C4—22no—
599C4—22no—
600C4—22no—
601C4—22no—
602C4—42no—
603C4—81no—
604C4—81no—
605C4—81no—
606C4—81no—
607C4—81no—
608C4—81no—
609C4—81no—
610C4C3×#52911yescharacteristic
611C4C3×#53911yescharacteristic
612C4C3×#55111yescharacteristic
613C4C3×#55211yescharacteristic
614C3#2769711yescharacteristic, direct product factor, Sylow
615C2—24no—
616C2C3×#248811yescharacteristic
617C2C2×C3×#48711yescharacteristic
618C2C3×#313611yescharacteristic
619C2C3×#329511yescharacteristic
620C2C3×#329911yescharacteristic
621C2C3×#333511yescharacteristic
622C2C3×#333811yescharacteristic
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×#2100C2semidirect
4. C3×#2100C2semidirect
5. C3×#2072C2semidirect
6. C3×#2024C2semidirect
7. C3×#1990C2semidirect
8. C3×#1933C2semidirect
9. C3×#1933C2semidirect
12. C3×#1834C2semidirect
16. #27697C3direct (N × H)
62. #2100C2×C3semidirect
63. #2100C2×C3semidirect
64. #2072C2×C3semidirect
65. #2024C2×C3semidirect
66. #1990C2×C3semidirect
67. #1933C2×C3semidirect
68. #1933C2×C3semidirect
71. #1834C2×C3semidirect
614. C3#27697direct (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×#24402
C3×#21002
C3×#20722
C3×#20242
C3×#19902
C3×#19332
C3×#18922
C3×#18722
C3×#18342
C3×#18242
C3×C8×C22⋊C42
C2×C3×#3402
#276973