Finite Group Databasebeta

← C3×#27739 #10646346

Subgroups — C3×#27739 #10646346

Order 768.

690 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2566C211yesmaximal, characteristic
2C3×#2552C211yesmaximal, characteristic, complemented
3C3×#2101C211yesmaximal, characteristic, complemented
4C3×#2100C211yesmaximal, characteristic, complemented
5C3×#2071C211yesmaximal, characteristic
6C3×#2071C211yesmaximal, characteristic
7C3×#1992C211yesmaximal, characteristic, complemented
8C3×#1989C211yesmaximal, characteristic, complemented
9C3×#1936C211yesmaximal, characteristic
10C3×#1934C211yesmaximal, characteristic
11C3×#1836C211yesmaximal, characteristic, complemented
12C3×#1834C211yesmaximal, characteristic, complemented
13C3×#1802C211yesmaximal, characteristic
14C3×#1771C211yesmaximal, characteristic, complemented
15C3×#1762C211yesmaximal, characteristic
16#27739C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×#125—21no—
18C2×C3×#125C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C2×C3×C4⋊C8C2211yescharacteristic
21C2×C3×#102C2211yescharacteristic
22C2×C3×#102C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×#97C2211yescharacteristic
26C2×C3×#97C2211yescharacteristic
27C2×C3×C4×C8C2211yescharacteristic
28C3×#398—21no—
29C3×#398—21no—
30C3×#396—21no—
31C3×#396—21no—
32C3×#390C2211yescharacteristic
33C3×#390C2211yescharacteristic
34C3×#389—21no—
35C3×#389C2211yescharacteristic
36C3×#389C2211yescharacteristic
37C3×#385—21no—
38C3×#385—21no—
39C3×#384C2211yescharacteristic
40C3×#384C2211yescharacteristic
41C3×#384C2211yescharacteristic
42C3×#384C2211yescharacteristic
43C3×#382C2211yescharacteristic
44C3×C4×C4⋊C4C2211yescharacteristic
45C3×C4×C4⋊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#2566C2×C311yescharacteristic
61#2552C2×C311yescharacteristic, complemented
62#2101C2×C311yescharacteristic, complemented
63#2100C2×C311yescharacteristic, complemented
64#2071C2×C311yescharacteristic
65#2071C2×C311yescharacteristic
66#1992C2×C311yescharacteristic, complemented
67#1989C2×C311yescharacteristic, complemented
68#1936C2×C311yescharacteristic
69#1934C2×C311yescharacteristic
70#1836C2×C311yescharacteristic, complemented
71#1834C2×C311yescharacteristic, complemented
72#1802C2×C311yescharacteristic
73#1771C2×C311yescharacteristic, complemented
74#1762C2×C311yescharacteristic
75C22×C3×D4C2311yescharacteristic
76C23×C3×C4—21no—
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C3×#125—22no—
82C3×#125—22no—
83C3×#125—44no—
84C3×C4×D4—22no—
85C3×C4×D4—22no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
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⋊C4—21no—
98C2×C3×C4⋊C4D411yescharacteristic
99C2×C3×C4⋊C4D411yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C4⋊C4C2311yescharacteristic
104C2×C3×C4⋊C4C2311yescharacteristic
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C22⋊C4—21no—
108C2×C3×C22⋊C4—21no—
109C2×C3×C22⋊C4—21no—
110C2×C3×C22⋊C4—21no—
111C2×C3×C22⋊C4—21no—
112C2×C3×C42—21no—
113C2×C3×C42C2311yescharacteristic
114C2×C3×C42C2311yescharacteristic
115C2×C3×C42C2311yescharacteristic
116C3×C4⋊C8—22no—
117C3×#102—22no—
118C3×#102—22no—
119C3×#102—22no—
120C3×#102—22no—
121C3×#97—22no—
122C3×#97—22no—
123C3×C4×C8—22no—
124C3×#95—21no—
125C3×#95—21no—
126C3×#95—21no—
127C3×#95—21no—
128C3×#95—21no—
129C3×#95D411yescharacteristic
130C3×#95D411yescharacteristic
131C2×#125—21no—
132C2×#125C22×C311yescharacteristic
133C2×C4×D4C22×C311yescharacteristic
134C2×C4⋊C8C22×C311yescharacteristic
135C2×#102C22×C311yescharacteristic
136C2×#102C22×C311yescharacteristic
137C2×#102C22×C311yescharacteristic
138C2×#102C22×C311yescharacteristic
139C2×#97C22×C311yescharacteristic
140C2×#97C22×C311yescharacteristic
141C2×C4×C8C22×C311yescharacteristic
142#398—21no—
143#398—21no—
144#396—21no—
145#396—21no—
146#390C22×C311yescharacteristic
147#390C22×C311yescharacteristic
148#389—21no—
149#389C22×C311yescharacteristic
150#389C22×C311yescharacteristic
151#385—21no—
152#385—21no—
153#384C22×C311yescharacteristic
154#384C22×C311yescharacteristic
155#384C22×C311yescharacteristic
156#384C22×C311yescharacteristic
157#382C22×C311yescharacteristic
158C4×C4⋊C4C22×C311yescharacteristic
159C4×C4⋊C4C22×C311yescharacteristic
160#340C22×C311yescharacteristic
161#340C22×C311yescharacteristic
162#340C22×C311yescharacteristic
163#340C22×C311yescharacteristic
164#340C22×C311yescharacteristic
165#340C22×C311yescharacteristic
166#340C22×C311yescharacteristic
167#340C22×C311yescharacteristic
168#340C22×C311yescharacteristic
169#340C22×C311yescharacteristic
170#340C22×C311yescharacteristic
171#340C22×C311yescharacteristic
172#336C22×C311yescharacteristic
173#336C22×C311yescharacteristic
174C24×C3—21no—
175C2×C3×D4—21no—
176C2×C3×D4—21no—
177C2×C3×D4—21no—
178C2×C3×D4—21no—
179C2×C3×D4—21no—
180C2×C3×D4#4112yes—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—42no—
189C22×C3×C4—81no—
190C22×C3×C4—81no—
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2411yescharacteristic
198C2×C3×C8—21no—
199C2×C3×C8—21no—
200C2×C3×C8—21no—
201C2×C3×C8—21no—
202C2×C3×C8—21no—
203C2×C3×C8—21no—
204C2×C3×C8—21no—
205C2×C3×C8—21no—
206C2×C3×C8—21no—
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C2×C3×C8#4112yes—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—42no—
227C3×C4⋊C4—42no—
228C3×C4⋊C4—42no—
229C3×C4⋊C4—42no—
230C3×C4⋊C4—42no—
231C3×C4⋊C4—42no—
232C3×C4⋊C4—42no—
233C3×C4⋊C4—81no—
234C3×C4⋊C4—81no—
235C3×C4⋊C4—81no—
236C3×C4⋊C4—81no—
237C3×C4⋊C4—81no—
238C3×C22⋊C4—42no—
239C3×C22⋊C4—42no—
240C3×C22⋊C4—42no—
241C3×C22⋊C4—81no—
242C3×C22⋊C4—81no—
243C3×C42—21no—
244C3×C42—21no—
245C3×C42—21no—
246C3×C42—21no—
247C3×C42—42no—
248C3×C42#4112yes—
249C3×C42#4112yes—
250C22×D4C23×C311yescharacteristic
251C23×C4—21no—
252C22×C8C23×C311yescharacteristic
253C22×C8C23×C311yescharacteristic
254C22×C8C23×C311yescharacteristic
255C22×C8C23×C311yescharacteristic
256#125—22no—
257#125—22no—
258#125—44no—
259C4×D4—22no—
260C4×D4—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4C3×D411yescharacteristic
274C2×C4⋊C4C3×D411yescharacteristic
275C2×C4⋊C4C23×C311yescharacteristic
276C2×C4⋊C4C23×C311yescharacteristic
277C2×C4⋊C4C23×C311yescharacteristic
278C2×C4⋊C4C23×C311yescharacteristic
279C2×C4⋊C4C23×C311yescharacteristic
280C2×C4⋊C4C23×C311yescharacteristic
281C2×C4⋊C4C23×C311yescharacteristic
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C42—21no—
288C2×C42C23×C311yescharacteristic
289C2×C42C23×C311yescharacteristic
290C2×C42C23×C311yescharacteristic
291C4⋊C8—22no—
292#102—22no—
293#102—22no—
294#102—22no—
295#102—22no—
296#97—22no—
297#97—22no—
298C4×C8—22no—
299#95—21no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95C3×D411yescharacteristic
305#95C3×D411yescharacteristic
306C23×C3—22no—
307C23×C3—41no—
308C23×C3—41no—
309C23×C3—41no—
310C23×C3—41no—
311C23×C3—41no—
312C23×C3—41no—
313C23×C3C22×D411yescharacteristic
314C3×D4—22no—
315C3×D4—22no—
316C3×D4—41no—
317C3×D4—41no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—84no—
354C2×C3×C4#12412yes—
355C2×C3×C4#12412yes—
356C2×C3×C4AGL(1,ℤ/8)12yes—
357C2×C3×C4#13712yes—
358C2×C3×C4C2×#4111yescharacteristic
359C2×C3×C4C2×#4111yescharacteristic
360C2×C3×C4C2×#4111yescharacteristic
361C2×C3×C4C2×#4111yescharacteristic
362C2×C3×C42−1+411yescharacteristic
363C2×C3×C42−1+411yescharacteristic
364C3×C8—22no—
365C3×C8—41no—
366C3×C8—41no—
367C3×C8—41no—
368C24—21no—
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—21no—
374C2×D4C3×#4112yes—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—42no—
383C22×C4—81no—
384C22×C4—81no—
385C22×C4C2×C3×D411yescharacteristic
386C22×C4C2×C3×D411yescharacteristic
387C22×C4C2×C3×D411yescharacteristic
388C22×C4C2×C3×D411yescharacteristic
389C22×C4C2×C3×D411yescharacteristic
390C22×C4C2×C3×D411yescharacteristic
391C22×C4C24×C311yescharacteristic
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8C3×#4112yes—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—81no—
428C4⋊C4—81no—
429C4⋊C4—81no—
430C4⋊C4—81no—
431C4⋊C4—81no—
432C22⋊C4—42no—
433C22⋊C4—42no—
434C22⋊C4—42no—
435C22⋊C4—81no—
436C22⋊C4—81no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42—21no—
441C42—42no—
442C42C3×#4112yes—
443C42C3×#4112yes—
444C22×C3—42no—
445C22×C3—42no—
446C22×C3—42no—
447C22×C3—81no—
448C22×C3—81no—
449C22×C3—81no—
450C22×C3—81no—
451C22×C3C2×#12411yescharacteristic
452C22×C3#54711yescharacteristic
453C22×C3#54811yescharacteristic
454C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
455C22×C3C2×#13711yescharacteristic
456C22×C3#57611yescharacteristic
457C22×C3#57711yescharacteristic
458C3×C4—22no—
459C3×C4—22no—
460C3×C4—22no—
461C3×C4—22no—
462C3×C4—41no—
463C3×C4—41no—
464C3×C4—81no—
465C3×C4—81no—
466C3×C4—81no—
467C3×C4—81no—
468C3×C4—81no—
469C3×C4—81no—
470C3×C4—81no—
471C3×C4#52911yescharacteristic
472C3×C4#54111yescharacteristic
473C3×C4#55211yescharacteristic
474C3×C4#55211yescharacteristic
475C23—22no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23C22×C3×D411yescharacteristic
483D4—22no—
484D4—22no—
485D4—41no—
486D4—41no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—84no—
523C2×C4C3×#12412yes—
524C2×C4C3×#12412yes—
525C2×C4C3×AGL(1,ℤ/8)12yes—
526C2×C4C3×#13712yes—
527C2×C4C2×C3×#4111yescharacteristic
528C2×C4C2×C3×#4111yescharacteristic
529C2×C4C2×C3×#4111yescharacteristic
530C2×C4C2×C3×#4111yescharacteristic
531C2×C4C3×2−1+411yescharacteristic
532C2×C4C3×2−1+411yescharacteristic
533C8—22no—
534C8—41no—
535C8—41no—
536C8—41no—
537C2×C3—82no—
538C2×C3#248911yescharacteristic
539C2×C3#313611yescharacteristic
540C2×C3#314111yescharacteristic
541C2×C3#334511yescharacteristic
542C2×C3#334611yescharacteristic
543C2×C3#336611yescharacteristic
544C2×C3#337211yescharacteristic
545C22—42no—
546C22—42no—
547C22—42no—
548C22—81no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22C2×C3×#12411yescharacteristic
553C22C3×#54711yescharacteristic
554C22C3×#54811yescharacteristic
555C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
556C22C2×C3×#13711yescharacteristic
557C22C3×#57611yescharacteristic
558C22C3×#57711yescharacteristic
559C4—22no—
560C4—22no—
561C4—22no—
562C4—22no—
563C4—41no—
564C4—41no—
565C4—81no—
566C4—81no—
567C4—81no—
568C4—81no—
569C4—81no—
570C4—81no—
571C4—81no—
572C4C3×#52911yescharacteristic
573C4C3×#54111yescharacteristic
574C4C3×#55211yescharacteristic
575C4C3×#55211yescharacteristic
576C3#2773911yescharacteristic, direct product factor, Sylow
577C2—82no—
578C2C3×#248911yescharacteristic
579C2C3×#313611yescharacteristic
580C2C3×#314111yescharacteristic
581C2C3×#334511yescharacteristic
582C2C3×#334611yescharacteristic
583C2C3×#336611yescharacteristic
584C2C3×#337211yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C4×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2552C2semidirect
3. C3×#2101C2semidirect
4. C3×#2100C2semidirect
7. C3×#1992C2semidirect
8. C3×#1989C2semidirect
11. C3×#1836C2semidirect
12. C3×#1834C2semidirect
14. C3×#1771C2semidirect
16. #27739C3direct (N × H)
61. #2552C2×C3semidirect
62. #2101C2×C3semidirect
63. #2100C2×C3semidirect
66. #1992C2×C3semidirect
67. #1989C2×C3semidirect
70. #1836C2×C3semidirect
71. #1834C2×C3semidirect
73. #1771C2×C3semidirect
576. C3#27739direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#25662
C3×#25522
C3×#21012
C3×#21002
C3×#20712
C3×#19922
C3×#19892
C3×#19362
C3×#19342
C3×#18362
C3×#18342
C3×#18022
C3×#17712
C3×#17622
#277393