Finite Group Databasebeta

← C3×#27706 #10646313

Subgroups — C3×#27706 #10646313

Order 768.

742 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2499C211yesmaximal, characteristic
2C3×#2440C211yesmaximal, characteristic
3C3×#2101C211yesmaximal, characteristic, complemented
4C3×#2099C211yesmaximal, characteristic, complemented
5C3×#2072C211yesmaximal, characteristic, complemented
6C3×#2024C211yesmaximal, characteristic, complemented
7C3×#1993C211yesmaximal, characteristic, complemented
8C3×#1935C211yesmaximal, characteristic, complemented
9C3×#1933C211yesmaximal, characteristic, complemented
10C3×#1894C211yesmaximal, characteristic
11C3×#1872C211yesmaximal, characteristic
12C3×#1834C211yesmaximal, characteristic, complemented
13C3×#1825C211yesmaximal, characteristic
14C3×#1753C211yesmaximal, characteristic
15C3×#1736C211yesmaximal, characteristic
16#27706C311yesmaximal, characteristic, direct product factor, Sylow
17C22×C3×M16C2211yescharacteristic
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×C22⋊Q8—21no—
20C2×C3×C22⋊Q8C2211yescharacteristic
21C2×C3×#117C2211yescharacteristic
22C22×C3×C4⋊C4C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×C22⋊C8C2211yescharacteristic
28C2×C3×#97C2211yescharacteristic
29C2×C3×C4×C8C2211yescharacteristic
30C3×#398—21no—
31C3×#397—21no—
32C3×#397—21no—
33C3×#395—21no—
34C3×#391C2211yescharacteristic
35C3×#391C2211yescharacteristic
36C3×#389C2211yescharacteristic
37C3×#389C2211yescharacteristic
38C3×#387—21no—
39C3×#387—21no—
40C3×#385—21no—
41C3×#384C2211yescharacteristic
42C3×#384C2211yescharacteristic
43C3×#382C2211yescharacteristic
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#2499C2×C311yescharacteristic
61#2440C2×C311yescharacteristic
62#2101C2×C311yescharacteristic, complemented
63#2099C2×C311yescharacteristic, complemented
64#2072C2×C311yescharacteristic, complemented
65#2024C2×C311yescharacteristic, complemented
66#1993C2×C311yescharacteristic, complemented
67#1935C2×C311yescharacteristic, complemented
68#1933C2×C311yescharacteristic, complemented
69#1894C2×C311yescharacteristic
70#1872C2×C311yescharacteristic
71#1834C2×C311yescharacteristic, complemented
72#1825C2×C311yescharacteristic
73#1753C2×C311yescharacteristic
74#1736C2×C311yescharacteristic
75C22×C3×Q8—21no—
76C22×C3×Q8C2311yescharacteristic
77C23×C3×C4—21no—
78C23×C3×C4C2311yescharacteristic
79C2×C3×M16—22no—
80C2×C3×M16—22no—
81C2×C3×M16—41no—
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×C4⋊Q8—22no—
87C3×C4⋊Q8—41no—
88C3×C22⋊Q8—22no—
89C3×C22⋊Q8—22no—
90C3×C22⋊Q8—44no—
91C3×#117—22no—
92C3×#117—22no—
93C2×C3×C4⋊C4—22no—
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—41no—
104C2×C3×C4⋊C4C2311yescharacteristic
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⋊C4D411yescharacteristic
115C2×C3×C22⋊C4D411yescharacteristic
116C2×C3×C42—21no—
117C2×C3×C42C2311yescharacteristic
118C2×C3×C42C2311yescharacteristic
119C3×#103—22no—
120C3×#103—22no—
121C3×#103—41no—
122C3×#103—41no—
123C3×C22⋊C8—22no—
124C3×#97—41no—
125C3×C4×C8—22no—
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×#95—21no—
135C3×#95D411yescharacteristic
136C3×#95D411yescharacteristic
137C22×M16C22×C311yescharacteristic
138C2×C4⋊Q8C22×C311yescharacteristic
139C2×C22⋊Q8—21no—
140C2×C22⋊Q8C22×C311yescharacteristic
141C2×#117C22×C311yescharacteristic
142C22×C4⋊C4C22×C311yescharacteristic
143C2×#103C22×C311yescharacteristic
144C2×#103C22×C311yescharacteristic
145C2×#103C22×C311yescharacteristic
146C2×#103C22×C311yescharacteristic
147C2×C22⋊C8C22×C311yescharacteristic
148C2×#97C22×C311yescharacteristic
149C2×C4×C8C22×C311yescharacteristic
150#398—21no—
151#397—21no—
152#397—21no—
153#395—21no—
154#391C22×C311yescharacteristic
155#391C22×C311yescharacteristic
156#389C22×C311yescharacteristic
157#389C22×C311yescharacteristic
158#387—21no—
159#387—21no—
160#385—21no—
161#384C22×C311yescharacteristic
162#384C22×C311yescharacteristic
163#382C22×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—81no—
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4#4112yes—
211C22×C3×C4#4112yes—
212C22×C3×C4C2411yescharacteristic
213C3×M16—44no—
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—21no—
224C2×C3×C8—21no—
225C2×C3×C8#4112yes—
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—21no—
239C3×C4⋊C4—42no—
240C3×C4⋊C4—42no—
241C3×C4⋊C4—42no—
242C3×C4⋊C4—42no—
243C3×C4⋊C4—42no—
244C3×C4⋊C4—42no—
245C3×C4⋊C4—42no—
246C3×C4⋊C4—42no—
247C3×C4⋊C4—81no—
248C3×C4⋊C4#4112yes—
249C3×C22⋊C4—22no—
250C3×C22⋊C4—22no—
251C3×C22⋊C4—42no—
252C3×C22⋊C4—42no—
253C3×C22⋊C4—42no—
254C3×C42—21no—
255C3×C42—21no—
256C3×C42—21no—
257C3×C42—21no—
258C3×C42—42no—
259C22×Q8—21no—
260C22×Q8C23×C311yescharacteristic
261C23×C4—21no—
262C23×C4C23×C311yescharacteristic
263C2×M16—22no—
264C2×M16—22no—
265C2×M16—41no—
266C22×C8C23×C311yescharacteristic
267C22×C8C23×C311yescharacteristic
268C22×C8C23×C311yescharacteristic
269C22×C8C23×C311yescharacteristic
270C4⋊Q8—22no—
271C4⋊Q8—41no—
272C22⋊Q8—22no—
273C22⋊Q8—22no—
274C22⋊Q8—44no—
275#117—22no—
276#117—22no—
277C2×C4⋊C4—22no—
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⋊C4—21no—
298C2×C22⋊C4C3×D411yescharacteristic
299C2×C22⋊C4C3×D411yescharacteristic
300C2×C42—21no—
301C2×C42C23×C311yescharacteristic
302C2×C42C23×C311yescharacteristic
303#103—22no—
304#103—22no—
305#103—41no—
306#103—41no—
307C22⋊C8—22no—
308#97—41no—
309C4×C8—22no—
310#95—21no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
318#95—21no—
319#95C3×D411yescharacteristic
320#95C3×D411yescharacteristic
321C23×C3—21no—
322C23×C3—21no—
323C23×C3—21no—
324C23×C3—21no—
325C23×C3—21no—
326C23×C3—21no—
327C23×C3#12412yes—
328C23×C3C22×D411yescharacteristic
329C3×Q8—41no—
330C3×Q8—41no—
331C3×Q8—41no—
332C3×Q8—41no—
333C3×Q8—84no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—22no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
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—41no—
374C2×C3×C4—84no—
375C2×C3×C4#12412yes—
376C2×C3×C4AGL(1,ℤ/8)12yes—
377C2×C3×C4#13712yes—
378C2×C3×C4C2×#4111yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380C2×C3×C4C2×#4111yescharacteristic
381C2×C3×C4C2×#4111yescharacteristic
382C2×C3×C4H(5,𝔽2)11yescharacteristic
383C2×C3×C42−1+411yescharacteristic
384C3×C8—22no—
385C3×C8—41no—
386C3×C8—41no—
387C3×C8—41no—
388C24C2×C3×D411yescharacteristic
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8—81no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—81no—
413C22×C4C2×C3×D411yescharacteristic
414C22×C4C2×C3×D411yescharacteristic
415C22×C4C2×C3×D411yescharacteristic
416C22×C4C2×C3×D411yescharacteristic
417C22×C4C2×C3×D411yescharacteristic
418C22×C4C3×#4112yes—
419C22×C4C3×#4112yes—
420C22×C4C24×C311yescharacteristic
421M16—44no—
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—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8C3×#4112yes—
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—21no—
446C4⋊C4—21no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—81no—
456C4⋊C4C3×#4112yes—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—21no—
466C42—42no—
467C22×C3—22no—
468C22×C3—22no—
469C22×C3—22no—
470C22×C3—41no—
471C22×C3—41no—
472C22×C3—41no—
473C22×C3—41no—
474C22×C3C2×#12411yescharacteristic
475C22×C3#54611yescharacteristic
476C22×C3#54711yescharacteristic
477C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
478C22×C3C2×#13711yescharacteristic
479C22×C3#57711yescharacteristic
480C22×C3#57811yescharacteristic
481C3×C4—22no—
482C3×C4—22no—
483C3×C4—22no—
484C3×C4—41no—
485C3×C4—41no—
486C3×C4—42no—
487C3×C4—81no—
488C3×C4—81no—
489C3×C4—81no—
490C3×C4—81no—
491C3×C4—81no—
492C3×C4—81no—
493C3×C4—81no—
494C3×C4#52911yescharacteristic
495C3×C4#53911yescharacteristic
496C3×C4#55111yescharacteristic
497C3×C4#55211yescharacteristic
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23C3×#12412yes—
505C23C22×C3×D411yescharacteristic
506Q8—41no—
507Q8—41no—
508Q8—41no—
509Q8—41no—
510Q8—84no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
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—41no—
551C2×C4—84no—
552C2×C4C3×#12412yes—
553C2×C4C3×AGL(1,ℤ/8)12yes—
554C2×C4C3×#13712yes—
555C2×C4C2×C3×#4111yescharacteristic
556C2×C4C2×C3×#4111yescharacteristic
557C2×C4C2×C3×#4111yescharacteristic
558C2×C4C2×C3×#4111yescharacteristic
559C2×C4C3×H(5,𝔽2)11yescharacteristic
560C2×C4C3×2−1+411yescharacteristic
561C8—22no—
562C8—41no—
563C8—41no—
564C8—41no—
565C2×C3—42no—
566C2×C3#248811yescharacteristic
567C2×C3#313711yescharacteristic
568C2×C3#314111yescharacteristic
569C2×C3#332411yescharacteristic
570C2×C3#332811yescharacteristic
571C2×C3#334311yescharacteristic
572C2×C3#334411yescharacteristic
573C22—22no—
574C22—22no—
575C22—22no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22C2×C3×#12411yescharacteristic
581C22C3×#54611yescharacteristic
582C22C3×#54711yescharacteristic
583C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
584C22C2×C3×#13711yescharacteristic
585C22C3×#57711yescharacteristic
586C22C3×#57811yescharacteristic
587C4—22no—
588C4—22no—
589C4—22no—
590C4—41no—
591C4—42no—
592C4—41no—
593C4—81no—
594C4—81no—
595C4—81no—
596C4—81no—
597C4—81no—
598C4—81no—
599C4—81no—
600C4C3×#52911yescharacteristic
601C4C3×#53911yescharacteristic
602C4C3×#55111yescharacteristic
603C4C3×#55211yescharacteristic
604C3#2770611yescharacteristic, direct product factor, Sylow
605C2—42no—
606C2C3×#248811yescharacteristic
607C2C3×#313711yescharacteristic
608C2C3×#314111yescharacteristic
609C2C3×#332411yescharacteristic
610C2C3×#332811yescharacteristic
611C2C3×#334311yescharacteristic
612C2C3×#334411yescharacteristic
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×#2101C2semidirect
4. C3×#2099C2semidirect
5. C3×#2072C2semidirect
6. C3×#2024C2semidirect
7. C3×#1993C2semidirect
8. C3×#1935C2semidirect
9. C3×#1933C2semidirect
12. C3×#1834C2semidirect
16. #27706C3direct (N × H)
62. #2101C2×C3semidirect
63. #2099C2×C3semidirect
64. #2072C2×C3semidirect
65. #2024C2×C3semidirect
66. #1993C2×C3semidirect
67. #1935C2×C3semidirect
68. #1933C2×C3semidirect
71. #1834C2×C3semidirect
604. C3#27706direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#24992
C3×#24402
C3×#21012
C3×#20992
C3×#20722
C3×#20242
C3×#19932
C3×#19352
C3×#19332
C3×#18942
C3×#18722
C3×#18342
C3×#18252
C3×#17532
C3×#17362
#277063