Finite Group Databasebeta

← C3×#29453 #10648060

Subgroups — C3×#29453 #10648060

Order 768.

708 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2644C211yesmaximal, characteristic, complemented
2C3×#2319C211yesmaximal, characteristic, complemented
3C3×#2088C211yesmaximal, characteristic, complemented
4C3×#2067C211yesmaximal, characteristic, complemented
5C3×#2065C211yesmaximal, characteristic, complemented
6C3×#2054C211yesmaximal, characteristic
7C3×#1972C211yesmaximal, characteristic
8C3×#1944C211yesmaximal, characteristic, complemented
9C3×#1934C211yesmaximal, characteristic, complemented
10C3×#1934C211yesmaximal, characteristic, complemented
11C3×#1925C211yesmaximal, characteristic, complemented
12C3×#1919C211yesmaximal, characteristic, complemented
13C3×#1879C211yesmaximal, characteristic
14C3×#1864C211yesmaximal, characteristic, complemented
15C3×#1768C211yesmaximal, characteristic, complemented
16#29453C311yesmaximal, characteristic, direct product factor, Sylow
17C22×C3×D8C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×C3×#107C2211yescharacteristic
20C2×C3×#106C2211yescharacteristic
21C2×C3×#106C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C2×C3×#102C2211yescharacteristic
29C2×C3×#102C2211yescharacteristic
30C2×C3×#97C2211yescharacteristic
31C2×C3×C4×C8C2211yescharacteristic
32C3×#397—21no—
33C3×#397—21no—
34C3×#397—21no—
35C3×#396—21no—
36C3×#390C2211yescharacteristic
37C3×#390C2211yescharacteristic
38C3×#390C2211yescharacteristic
39C3×#390C2211yescharacteristic
40C3×#390C2211yescharacteristic
41C3×#389C2211yescharacteristic
42C3×#389C2211yescharacteristic
43C3×#389C2211yescharacteristic
44C3×#389C2211yescharacteristic
45C3×#389C2211yescharacteristic
46C3×#388—21no—
47C3×#388—21no—
48C3×#381—21no—
49C3×#380—21no—
50C3×C4×C4⋊C4C2211yescharacteristic
51C3×#376C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58C3×#336C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#2644C2×C311yescharacteristic, complemented
61#2319C2×C311yescharacteristic, complemented
62#2088C2×C311yescharacteristic, complemented
63#2067C2×C311yescharacteristic, complemented
64#2065C2×C311yescharacteristic, complemented
65#2054C2×C311yescharacteristic
66#1972C2×C311yescharacteristic
67#1944C2×C311yescharacteristic, complemented
68#1934C2×C311yescharacteristic, complemented
69#1934C2×C311yescharacteristic, complemented
70#1925C2×C311yescharacteristic, complemented
71#1919C2×C311yescharacteristic, complemented
72#1879C2×C311yescharacteristic
73#1864C2×C311yescharacteristic, complemented
74#1768C2×C311yescharacteristic, complemented
75C22×C3×D4C2311yescharacteristic
76C22×C3×D4C2311yescharacteristic
77C23×C3×C4—21no—
78C2×C3×D8—22no—
79C2×C3×D8—22no—
80C2×C3×D8—41no—
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C3×C4×D4—22no—
86C3×C4×D4—22no—
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⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C22⋊C4—21no—
108C2×C3×C22⋊C4—21no—
109C2×C3×C42—21no—
110C2×C3×C42C2311yescharacteristic
111C2×C3×C42C2311yescharacteristic
112C2×C3×C42C2311yescharacteristic
113C3×#107—41no—
114C3×#106—22no—
115C3×#106—22no—
116C3×C4⋊C8—22no—
117C3×C4⋊C8—22no—
118C3×#102—22no—
119C3×#102—22no—
120C3×#102—22no—
121C3×#102—22no—
122C3×#102—41no—
123C3×#102—41no—
124C3×#97—41no—
125C3×C4×C8—41no—
126C3×#95—21no—
127C3×#95—21no—
128C3×#95—21no—
129C3×#95—21no—
130C3×#95—21no—
131C3×#95—21no—
132C3×#95D411yescharacteristic
133C3×#95D411yescharacteristic
134C3×#95D411yescharacteristic
135C3×#95D411yescharacteristic
136C22×D8C22×C311yescharacteristic
137C2×C4×D4C22×C311yescharacteristic
138C2×#107C22×C311yescharacteristic
139C2×#106C22×C311yescharacteristic
140C2×#106C22×C311yescharacteristic
141C2×C4⋊C8C22×C311yescharacteristic
142C2×C4⋊C8C22×C311yescharacteristic
143C2×#102C22×C311yescharacteristic
144C2×#102C22×C311yescharacteristic
145C2×#102C22×C311yescharacteristic
146C2×#102C22×C311yescharacteristic
147C2×#102C22×C311yescharacteristic
148C2×#102C22×C311yescharacteristic
149C2×#97C22×C311yescharacteristic
150C2×C4×C8C22×C311yescharacteristic
151#397—21no—
152#397—21no—
153#397—21no—
154#396—21no—
155#390C22×C311yescharacteristic
156#390C22×C311yescharacteristic
157#390C22×C311yescharacteristic
158#390C22×C311yescharacteristic
159#390C22×C311yescharacteristic
160#389C22×C311yescharacteristic
161#389C22×C311yescharacteristic
162#389C22×C311yescharacteristic
163#389C22×C311yescharacteristic
164#389C22×C311yescharacteristic
165#388—21no—
166#388—21no—
167#381—21no—
168#380—21no—
169C4×C4⋊C4C22×C311yescharacteristic
170#376C22×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×C3—21no—
180C24×C3—21no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4—21no—
190C2×C3×D4—21no—
191C2×C3×D4—21no—
192C2×C3×D4#4112yes—
193C22×C3×C4—21no—
194C22×C3×C4—24no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—81no—
201C22×C3×C4—81no—
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2411yescharacteristic
209C3×D8—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#4112yes—
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—81no—
234C3×C4⋊C4—81no—
235C3×C4⋊C4—81no—
236C3×C4⋊C4—81no—
237C3×C4⋊C4—81no—
238C3×C4⋊C4#4112yes—
239C3×C22⋊C4—24no—
240C3×C22⋊C4—81no—
241C3×C22⋊C4—81no—
242C3×C22⋊C4—81no—
243C3×C22⋊C4—81no—
244C3×C22⋊C4—81no—
245C3×C22⋊C4—81no—
246C3×C22⋊C4—81no—
247C3×C22⋊C4—81no—
248C3×C22⋊C4—81no—
249C3×C22⋊C4—81no—
250C3×C42—21no—
251C3×C42—21no—
252C3×C42—21no—
253C3×C42—21no—
254C3×C42—21no—
255C3×C42—21no—
256C3×C42—81no—
257C22×D4C23×C311yescharacteristic
258C22×D4C23×C311yescharacteristic
259C23×C4—21no—
260C2×D8—22no—
261C2×D8—22no—
262C2×D8—41no—
263C22×C8C23×C311yescharacteristic
264C22×C8C23×C311yescharacteristic
265C22×C8C23×C311yescharacteristic
266C22×C8C23×C311yescharacteristic
267C4×D4—22no—
268C4×D4—22no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4C23×C311yescharacteristic
275C2×C4⋊C4C23×C311yescharacteristic
276C2×C4⋊C4C23×C311yescharacteristic
277C2×C4⋊C4C23×C311yescharacteristic
278C2×C4⋊C4C23×C311yescharacteristic
279C2×C4⋊C4C23×C311yescharacteristic
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C42—21no—
292C2×C42C23×C311yescharacteristic
293C2×C42C23×C311yescharacteristic
294C2×C42C23×C311yescharacteristic
295#107—41no—
296#106—22no—
297#106—22no—
298C4⋊C8—22no—
299C4⋊C8—22no—
300#102—22no—
301#102—22no—
302#102—22no—
303#102—22no—
304#102—41no—
305#102—41no—
306#97—41no—
307C4×C8—41no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95C3×D411yescharacteristic
315#95C3×D411yescharacteristic
316#95C3×D411yescharacteristic
317#95C3×D411yescharacteristic
318C23×C3—22no—
319C23×C3—41no—
320C23×C3—41no—
321C23×C3—41no—
322C23×C3—41no—
323C23×C3—41no—
324C23×C3—41no—
325C23×C3—41no—
326C23×C3—41no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3—41no—
330C23×C3—41no—
331C23×C3—41no—
332C23×C3C22×D411yescharacteristic
333C3×D4—22no—
334C3×D4—22no—
335C3×D4—41no—
336C3×D4—41no—
337C3×D4—41no—
338C3×D4—41no—
339C3×D4—41no—
340C3×D4—41no—
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—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
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—41no—
377C2×C3×C4—84no—
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—22no—
386C3×C8—41no—
387C3×C8—41no—
388C24—21no—
389C24—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4C3×#4112yes—
402C22×C4—21no—
403C22×C4—24no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
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×C4C2×C3×D411yescharacteristic
417C22×C4C24×C311yescharacteristic
418D8—44no—
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×C8C3×#4112yes—
430C2×C8C3×#4112yes—
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—21no—
442C4⋊C4—81no—
443C4⋊C4—81no—
444C4⋊C4—81no—
445C4⋊C4—81no—
446C4⋊C4—81no—
447C4⋊C4C3×#4112yes—
448C22⋊C4—24no—
449C22⋊C4—81no—
450C22⋊C4—81no—
451C22⋊C4—81no—
452C22⋊C4—81no—
453C22⋊C4—81no—
454C22⋊C4—81no—
455C22⋊C4—81no—
456C22⋊C4—81no—
457C22⋊C4—81no—
458C22⋊C4—81no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—81no—
466C22×C3—42no—
467C22×C3—42no—
468C22×C3—42no—
469C22×C3—81no—
470C22×C3—81no—
471C22×C3—81no—
472C22×C3—81no—
473C22×C3—81no—
474C22×C3—81no—
475C22×C3—81no—
476C22×C3—81no—
477C22×C3—81no—
478C22×C3—81no—
479C22×C3—81no—
480C22×C3#52511yescharacteristic
481C22×C3#53211yescharacteristic
482C22×C3#53711yescharacteristic
483C22×C3#57611yescharacteristic
484C22×C3#57611yescharacteristic
485C22×C3#57711yescharacteristic
486C22×C3#57711yescharacteristic
487C3×C4—41no—
488C3×C4—42no—
489C3×C4—41no—
490C3×C4—41no—
491C3×C4—41no—
492C3×C4—41no—
493C3×C4—41no—
494C3×C4—81no—
495C3×C4—81no—
496C3×C4—81no—
497C3×C4—81no—
498C3×C4—81no—
499C3×C4#55111yescharacteristic
500C3×C4#55211yescharacteristic
501C3×C4#55311yescharacteristic
502C3×C4#55611yescharacteristic
503C23—22no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23C22×C3×D411yescharacteristic
518D4—22no—
519D4—22no—
520D4—41no—
521D4—41no—
522D4—41no—
523D4—41no—
524D4—41no—
525D4—41no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
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—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—84no—
563C2×C4C2×C3×#4111yescharacteristic
564C2×C4C2×C3×#4111yescharacteristic
565C2×C4C2×C3×#4111yescharacteristic
566C2×C4C2×C3×#4111yescharacteristic
567C2×C4C3×H(5,𝔽2)11yescharacteristic
568C2×C4C3×2−1+411yescharacteristic
569C8—22no—
570C8—22no—
571C8—41no—
572C8—41no—
573C2×C3—82no—
574C2×C3—161no—
575C2×C3#260111yescharacteristic
576C2×C3#311811yescharacteristic
577C2×C3#311911yescharacteristic
578C2×C3#317911yescharacteristic
579C2×C3#317911yescharacteristic
580C2×C3#325911yescharacteristic
581C2×C3#326311yescharacteristic
582C22—42no—
583C22—42no—
584C22—42no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22—81no—
596C22C3×#52511yescharacteristic
597C22C3×#53211yescharacteristic
598C22C3×#53711yescharacteristic
599C22C3×#57611yescharacteristic
600C22C3×#57611yescharacteristic
601C22C3×#57711yescharacteristic
602C22C3×#57711yescharacteristic
603C4—41no—
604C4—41no—
605C4—41no—
606C4—41no—
607C4—41no—
608C4—42no—
609C4—41no—
610C4—81no—
611C4—81no—
612C4—81no—
613C4—81no—
614C4—81no—
615C4C3×#55111yescharacteristic
616C4C3×#55211yescharacteristic
617C4C3×#55311yescharacteristic
618C4C3×#55611yescharacteristic
619C3#2945311yescharacteristic, direct product factor, Sylow
620C2—82no—
621C2—161no—
622C2C3×#260111yescharacteristic
623C2C3×#311811yescharacteristic
624C2C3×#311911yescharacteristic
625C2C3×#317911yescharacteristic
626C2C3×#317911yescharacteristic
627C2C3×#325911yescharacteristic
628C2C3×#326311yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×#376 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#2644C2semidirect
2. C3×#2319C2semidirect
3. C3×#2088C2semidirect
4. C3×#2067C2semidirect
5. C3×#2065C2semidirect
8. C3×#1944C2semidirect
9. C3×#1934C2semidirect
10. C3×#1934C2semidirect
11. C3×#1925C2semidirect
12. C3×#1919C2semidirect
14. C3×#1864C2semidirect
15. C3×#1768C2semidirect
16. #29453C3direct (N × H)
60. #2644C2×C3semidirect
61. #2319C2×C3semidirect
62. #2088C2×C3semidirect
63. #2067C2×C3semidirect
64. #2065C2×C3semidirect
67. #1944C2×C3semidirect
68. #1934C2×C3semidirect
69. #1934C2×C3semidirect
70. #1925C2×C3semidirect
71. #1919C2×C3semidirect
73. #1864C2×C3semidirect
74. #1768C2×C3semidirect
619. C3#29453direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#26442
C3×#23192
C3×#20882
C3×#20672
C3×#20652
C3×#20542
C3×#19722
C3×#19442
C3×#19342
C3×#19252
C3×#19192
C3×#18792
C3×#18642
C3×#17682
#294533