Finite Group Databasebeta

← C3×#27707 #10646314

Subgroups — C3×#27707 #10646314

Order 768.

714 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2502C211yesmaximal, characteristic
2C3×#2441C211yesmaximal, characteristic
3C3×#2100C211yesmaximal, characteristic, complemented
4C3×#2099C211yesmaximal, characteristic, complemented
5C3×#2072C211yesmaximal, characteristic, complemented
6C3×#2026C211yesmaximal, characteristic, complemented
7C3×#1992C211yesmaximal, characteristic, complemented
8C3×#1935C211yesmaximal, characteristic, complemented
9C3×#1933C211yesmaximal, characteristic, complemented
10C3×#1894C211yesmaximal, characteristic
11C3×#1873C211yesmaximal, characteristic
12C3×#1834C211yesmaximal, characteristic, complemented
13C3×#1823C211yesmaximal, characteristic
14C3×#1753C211yesmaximal, characteristic
15C3×#1736C211yesmaximal, characteristic
16#27707C311yesmaximal, characteristic, direct product factor, Sylow
17C22×C3×M16C2211yescharacteristic
18C2×C3×#125C2211yescharacteristic
19C2×C3×C22⋊Q8C2211yescharacteristic
20C2×C3×#117C2211yescharacteristic
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×C4×C8C2211yescharacteristic
29C3×#399—21no—
30C3×#398—21no—
31C3×#398—21no—
32C3×#396—21no—
33C3×#391C2211yescharacteristic
34C3×#391C2211yescharacteristic
35C3×#389—21no—
36C3×#389C2211yescharacteristic
37C3×#389C2211yescharacteristic
38C3×#387—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#2441C2×C311yescharacteristic
62#2100C2×C311yescharacteristic, complemented
63#2099C2×C311yescharacteristic, complemented
64#2072C2×C311yescharacteristic, complemented
65#2026C2×C311yescharacteristic, complemented
66#1992C2×C311yescharacteristic, complemented
67#1935C2×C311yescharacteristic, complemented
68#1933C2×C311yescharacteristic, complemented
69#1894C2×C311yescharacteristic
70#1873C2×C311yescharacteristic
71#1834C2×C311yescharacteristic, complemented
72#1823C2×C311yescharacteristic
73#1753C2×C311yescharacteristic
74#1736C2×C311yescharacteristic
75C22×C3×Q8C2311yescharacteristic
76C23×C3×C4—21no—
77C23×C3×C4C2311yescharacteristic
78C2×C3×M16—22no—
79C2×C3×M16—22no—
80C2×C3×M16—41no—
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C3×#125—22no—
86C3×#125—41no—
87C3×C22⋊Q8—22no—
88C3×C22⋊Q8—22no—
89C3×#117—22no—
90C3×#117—41no—
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—21no—
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⋊C4—21no—
115C2×C3×C22⋊C4D411yescharacteristic
116C2×C3×C22⋊C4D411yescharacteristic
117C2×C3×C42—21no—
118C2×C3×C42C2311yescharacteristic
119C2×C3×C42C2311yescharacteristic
120C3×#103—22no—
121C3×#103—22no—
122C3×#103—41no—
123C3×#103—41no—
124C3×C22⋊C8—22no—
125C3×#97—41no—
126C3×C4×C8—22no—
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
136C22×M16C22×C311yescharacteristic
137C2×#125C22×C311yescharacteristic
138C2×C22⋊Q8C22×C311yescharacteristic
139C2×#117C22×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×C4×C8C22×C311yescharacteristic
148#399—21no—
149#398—21no—
150#398—21no—
151#396—21no—
152#391C22×C311yescharacteristic
153#391C22×C311yescharacteristic
154#389—21no—
155#389C22×C311yescharacteristic
156#389C22×C311yescharacteristic
157#387—21no—
158#385—21no—
159#385—21no—
160#384C22×C311yescharacteristic
161#384C22×C311yescharacteristic
162#384C22×C311yescharacteristic
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—81no—
200C22×C3×C4—81no—
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4#4112yes—
207C22×C3×C4#4112yes—
208C22×C3×C4C2411yescharacteristic
209C3×M16—44no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
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#4112yes—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
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—42no—
236C3×C4⋊C4—42no—
237C3×C4⋊C4—42no—
238C3×C4⋊C4—42no—
239C3×C4⋊C4—42no—
240C3×C4⋊C4—42no—
241C3×C4⋊C4—81no—
242C3×C4⋊C4—81no—
243C3×C4⋊C4—81no—
244C3×C4⋊C4—81no—
245C3×C4⋊C4—81no—
246C3×C4⋊C4—81no—
247C3×C4⋊C4#4112yes—
248C3×C22⋊C4—22no—
249C3×C22⋊C4—22no—
250C3×C22⋊C4—42no—
251C3×C22⋊C4—42no—
252C3×C22⋊C4—81no—
253C3×C42—21no—
254C3×C42—21no—
255C3×C42—21no—
256C3×C42—21no—
257C3×C42—42no—
258C22×Q8C23×C311yescharacteristic
259C23×C4—21no—
260C23×C4C23×C311yescharacteristic
261C2×M16—22no—
262C2×M16—22no—
263C2×M16—41no—
264C22×C8C23×C311yescharacteristic
265C22×C8C23×C311yescharacteristic
266C22×C8C23×C311yescharacteristic
267C22×C8C23×C311yescharacteristic
268#125—22no—
269#125—41no—
270C22⋊Q8—22no—
271C22⋊Q8—22no—
272#117—22no—
273#117—41no—
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—21no—
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#95C3×D411yescharacteristic
318#95C3×D411yescharacteristic
319C23×C3—21no—
320C23×C3—21no—
321C23×C3—21no—
322C23×C3—21no—
323C23×C3—21no—
324C23×C3—21no—
325C23×C3#12412yes—
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—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—22no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
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—84no—
372C2×C3×C4#12412yes—
373C2×C3×C4#13712yes—
374C2×C3×C4#13712yes—
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C2×C3×C4C2×#4111yescharacteristic
379C2×C3×C4H(5,𝔽2)11yescharacteristic
380C2×C3×C42−1+411yescharacteristic
381C3×C8—22no—
382C3×C8—41no—
383C3×C8—41no—
384C3×C8—41no—
385C24C2×C3×D411yescharacteristic
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—21no—
392C22×C4—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—42no—
405C22×C4—81no—
406C22×C4—81no—
407C22×C4C2×C3×D411yescharacteristic
408C22×C4C2×C3×D411yescharacteristic
409C22×C4C2×C3×D411yescharacteristic
410C22×C4C2×C3×D411yescharacteristic
411C22×C4C2×C3×D411yescharacteristic
412C22×C4C3×#4112yes—
413C22×C4C3×#4112yes—
414C22×C4C24×C311yescharacteristic
415M16—44no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
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×C8C3×#4112yes—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
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—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—81no—
448C4⋊C4—81no—
449C4⋊C4—81no—
450C4⋊C4—81no—
451C4⋊C4—81no—
452C4⋊C4—81no—
453C4⋊C4C3×#4112yes—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—42no—
457C22⋊C4—42no—
458C22⋊C4—81no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—42no—
464C22×C3—22no—
465C22×C3—22no—
466C22×C3—22no—
467C22×C3—41no—
468C22×C3—41no—
469C22×C3—41no—
470C22×C3—41no—
471C22×C3C2×#12411yescharacteristic
472C22×C3#54611yescharacteristic
473C22×C3#54711yescharacteristic
474C22×C3C2×#13711yescharacteristic
475C22×C3C2×#13711yescharacteristic
476C22×C3#57611yescharacteristic
477C22×C3#57711yescharacteristic
478C3×C4—22no—
479C3×C4—22no—
480C3×C4—22no—
481C3×C4—41no—
482C3×C4—42no—
483C3×C4—41no—
484C3×C4—81no—
485C3×C4—81no—
486C3×C4—81no—
487C3×C4—81no—
488C3×C4—81no—
489C3×C4—81no—
490C3×C4—81no—
491C3×C4#52911yescharacteristic
492C3×C4#53911yescharacteristic
493C3×C4#55111yescharacteristic
494C3×C4#55211yescharacteristic
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23C3×#12412yes—
502C23C22×C3×D411yescharacteristic
503Q8—41no—
504Q8—41no—
505Q8—41no—
506Q8—41no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
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—84no—
548C2×C4C3×#12412yes—
549C2×C4C3×#13712yes—
550C2×C4C3×#13712yes—
551C2×C4C2×C3×#4111yescharacteristic
552C2×C4C2×C3×#4111yescharacteristic
553C2×C4C2×C3×#4111yescharacteristic
554C2×C4C2×C3×#4111yescharacteristic
555C2×C4C3×H(5,𝔽2)11yescharacteristic
556C2×C4C3×2−1+411yescharacteristic
557C8—22no—
558C8—41no—
559C8—41no—
560C8—41no—
561C2×C3—42no—
562C2×C3#248811yescharacteristic
563C2×C3#313611yescharacteristic
564C2×C3#313911yescharacteristic
565C2×C3#333111yescharacteristic
566C2×C3#333311yescharacteristic
567C2×C3#334411yescharacteristic
568C2×C3#334611yescharacteristic
569C22—22no—
570C22—22no—
571C22—22no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22C2×C3×#12411yescharacteristic
577C22C3×#54611yescharacteristic
578C22C3×#54711yescharacteristic
579C22C2×C3×#13711yescharacteristic
580C22C2×C3×#13711yescharacteristic
581C22C3×#57611yescharacteristic
582C22C3×#57711yescharacteristic
583C4—22no—
584C4—22no—
585C4—22no—
586C4—41no—
587C4—42no—
588C4—41no—
589C4—81no—
590C4—81no—
591C4—81no—
592C4—81no—
593C4—81no—
594C4—81no—
595C4—81no—
596C4C3×#52911yescharacteristic
597C4C3×#53911yescharacteristic
598C4C3×#55111yescharacteristic
599C4C3×#55211yescharacteristic
600C3#2770711yescharacteristic, direct product factor, Sylow
601C2—42no—
602C2C3×#248811yescharacteristic
603C2C3×#313611yescharacteristic
604C2C3×#313911yescharacteristic
605C2C3×#333111yescharacteristic
606C2C3×#333311yescharacteristic
607C2C3×#334411yescharacteristic
608C2C3×#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×#2100C2semidirect
4. C3×#2099C2semidirect
5. C3×#2072C2semidirect
6. C3×#2026C2semidirect
7. C3×#1992C2semidirect
8. C3×#1935C2semidirect
9. C3×#1933C2semidirect
12. C3×#1834C2semidirect
16. #27707C3direct (N × H)
62. #2100C2×C3semidirect
63. #2099C2×C3semidirect
64. #2072C2×C3semidirect
65. #2026C2×C3semidirect
66. #1992C2×C3semidirect
67. #1935C2×C3semidirect
68. #1933C2×C3semidirect
71. #1834C2×C3semidirect
600. C3#27707direct (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×#24412
C3×#21002
C3×#20992
C3×#20722
C3×#20262
C3×#19922
C3×#19352
C3×#19332
C3×#18942
C3×#18732
C3×#18342
C3×#18232
C3×#17532
C3×#17362
#277073