Finite Group Databasebeta

← C3×#33070 #10651677

Subgroups — C3×#33070 #10651677

Order 768.

834 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2864C211yesmaximal, characteristic, complemented
2C3×C42×D4C211yesmaximal, characteristic
3C3×#1949C211yesmaximal, characteristic, complemented
4C3×#1949C211yesmaximal, characteristic, complemented
5C3×#1921C211yesmaximal, characteristic, complemented
6C3×#1920C211yesmaximal, characteristic, complemented
7C3×#1919C211yesmaximal, characteristic, complemented
8C3×#1918C211yesmaximal, characteristic, complemented
9C3×#1864C211yesmaximal, characteristic
10C3×#1864C211yesmaximal, characteristic
11C3×#1862C211yesmaximal, characteristic
12C3×#1862C211yesmaximal, characteristic
13C3×#1831C211yesmaximal, characteristic, complemented
14C3×#1804C211yesmaximal, characteristic
15C3×#1804C211yesmaximal, characteristic
16#33070C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C22×C3×C42—21no—
21C2×C3×#107C2211yescharacteristic
22C2×C3×#107C2211yescharacteristic
23C2×C3×#106C2211yescharacteristic
24C2×C3×#106C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C2×C3×#102C2211yescharacteristic
29C2×C3×#102C2211yescharacteristic
30C2×C3×#102C2211yescharacteristic
31C3×#389C2211yescharacteristic
32C3×#389C2211yescharacteristic
33C3×#389C2211yescharacteristic
34C3×#389C2211yescharacteristic
35C3×#387—21no—
36C3×#387—21no—
37C3×#387—21no—
38C3×#387—21no—
39C3×#382C2211yescharacteristic
40C3×#382C2211yescharacteristic
41C3×C4×C4⋊C4C2211yescharacteristic
42C3×C4×C4⋊C4C2211yescharacteristic
43C3×C4×C4⋊C4C2211yescharacteristic
44C3×C4×C22⋊C4—21no—
45C3×C4×C22⋊C4—21no—
46C3×C4×C22⋊C4—21no—
47C3×C43C2211yescharacteristic
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#336C2211yescharacteristic
57C3×#336C2211yescharacteristic
58C3×#336C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#2864C2×C311yescharacteristic, complemented
61C42×D4C2×C311yescharacteristic
62#1949C2×C311yescharacteristic, complemented
63#1949C2×C311yescharacteristic, complemented
64#1921C2×C311yescharacteristic, complemented
65#1920C2×C311yescharacteristic, complemented
66#1919C2×C311yescharacteristic, complemented
67#1918C2×C311yescharacteristic, complemented
68#1864C2×C311yescharacteristic
69#1864C2×C311yescharacteristic
70#1862C2×C311yescharacteristic
71#1862C2×C311yescharacteristic
72#1831C2×C311yescharacteristic, complemented
73#1804C2×C311yescharacteristic
74#1804C2×C311yescharacteristic
75C22×C3×D4C2311yescharacteristic
76C23×C3×C4—21no—
77C23×C3×C4—21no—
78C23×C3×C4—21no—
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C3×C4×D4—22no—
84C3×C4×D4—22no—
85C3×C4×D4—22no—
86C3×C4×D4—22no—
87C3×C4×D4—22no—
88C3×C4×D4Q814yes—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C42—21no—
104C2×C3×C42—21no—
105C2×C3×C42—21no—
106C2×C3×C42—44no—
107C2×C3×C42D411yescharacteristic
108C2×C3×C42D411yescharacteristic
109C2×C3×C42D411yescharacteristic
110C2×C3×C42D411yescharacteristic
111C2×C3×C42C2311yescharacteristic
112C2×C3×C42C2311yescharacteristic
113C2×C3×C42C2311yescharacteristic
114C3×#107—22no—
115C3×#107—22no—
116C3×#106—22no—
117C3×#106—22no—
118C3×C4⋊C8—22no—
119C3×C4⋊C8—22no—
120C3×#102—22no—
121C3×#102—22no—
122C3×#102—22no—
123C3×#102—22no—
124C3×#95—21no—
125C3×#95—21no—
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—
135C2×C4×D4C22×C311yescharacteristic
136C2×C4×D4C22×C311yescharacteristic
137C2×C4×D4C22×C311yescharacteristic
138C22×C42—21no—
139C2×#107C22×C311yescharacteristic
140C2×#107C22×C311yescharacteristic
141C2×#106C22×C311yescharacteristic
142C2×#106C22×C311yescharacteristic
143C2×C4⋊C8C22×C311yescharacteristic
144C2×C4⋊C8C22×C311yescharacteristic
145C2×#102C22×C311yescharacteristic
146C2×#102C22×C311yescharacteristic
147C2×#102C22×C311yescharacteristic
148C2×#102C22×C311yescharacteristic
149#389C22×C311yescharacteristic
150#389C22×C311yescharacteristic
151#389C22×C311yescharacteristic
152#389C22×C311yescharacteristic
153#387—21no—
154#387—21no—
155#387—21no—
156#387—21no—
157#382C22×C311yescharacteristic
158#382C22×C311yescharacteristic
159C4×C4⋊C4C22×C311yescharacteristic
160C4×C4⋊C4C22×C311yescharacteristic
161C4×C4⋊C4C22×C311yescharacteristic
162C4×C22⋊C4—21no—
163C4×C22⋊C4—21no—
164C4×C22⋊C4—21no—
165C43C22×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#336C22×C311yescharacteristic
175#336C22×C311yescharacteristic
176#336C22×C311yescharacteristic
177#336C22×C311yescharacteristic
178C24×C3—21no—
179C2×C3×D4—21no—
180C2×C3×D4—21no—
181C2×C3×D4—21no—
182C2×C3×D4C2×Q812yes—
183C2×C3×D4#4112yes—
184C2×C3×D4#4112yes—
185C22×C3×C4—24no—
186C22×C3×C4—24no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—24no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—42no—
196C22×C3×C4—42no—
197C22×C3×C4—42no—
198C22×C3×C4—42no—
199C22×C3×C4—42no—
200C22×C3×C4—42no—
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2411yescharacteristic
208C2×C3×C8—21no—
209C2×C3×C8—21no—
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—
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—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—42no—
232C3×C4⋊C4—42no—
233C3×C4⋊C4—42no—
234C3×C4⋊C4—42no—
235C3×C4⋊C4C2×Q812yes—
236C3×C4⋊C4#4112yes—
237C3×C4⋊C4#4112yes—
238C3×C22⋊C4—24no—
239C3×C22⋊C4—24no—
240C3×C22⋊C4—24no—
241C3×C42—22no—
242C3×C42—22no—
243C3×C42—21no—
244C3×C42—21no—
245C3×C42—21no—
246C3×C42—21no—
247C3×C42—41no—
248C3×C42—41no—
249C3×C42—42no—
250C3×C42—42no—
251C3×C42—42no—
252C3×C42C2×Q812yes—
253C3×C42#4112yes—
254C22×D4C23×C311yescharacteristic
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—21no—
258C22×C8C23×C311yescharacteristic
259C22×C8C23×C311yescharacteristic
260C22×C8C23×C311yescharacteristic
261C22×C8C23×C311yescharacteristic
262C4×D4—22no—
263C4×D4—22no—
264C4×D4—22no—
265C4×D4—22no—
266C4×D4—22no—
267C4×D4C3×Q814yes—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4C23×C311yescharacteristic
273C2×C4⋊C4C23×C311yescharacteristic
274C2×C4⋊C4C23×C311yescharacteristic
275C2×C4⋊C4C23×C311yescharacteristic
276C2×C4⋊C4C23×C311yescharacteristic
277C2×C4⋊C4C23×C311yescharacteristic
278C2×C4⋊C4C23×C311yescharacteristic
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—44no—
286C2×C42C3×D411yescharacteristic
287C2×C42C3×D411yescharacteristic
288C2×C42C3×D411yescharacteristic
289C2×C42C3×D411yescharacteristic
290C2×C42C23×C311yescharacteristic
291C2×C42C23×C311yescharacteristic
292C2×C42C23×C311yescharacteristic
293#107—22no—
294#107—22no—
295#106—22no—
296#106—22no—
297C4⋊C8—22no—
298C4⋊C8—22no—
299#102—22no—
300#102—22no—
301#102—22no—
302#102—22no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
312#95—21no—
313#95—21no—
314C23×C3—22no—
315C23×C3—22no—
316C23×C3—22no—
317C23×C3—41no—
318C23×C3—41no—
319C23×C3—41no—
320C23×C3—41no—
321C23×C3C22×D411yescharacteristic
322C3×D4—22no—
323C3×D4—22no—
324C3×D4—22no—
325C3×D4#12514yes—
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—21no—
333C2×C3×C4—21no—
334C2×C3×C4—22no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—48no—
344C2×C3×C4—48no—
345C2×C3×C4—48no—
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—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×C4C22⋊Q812yes—
365C2×C3×C4C22⋊Q812yes—
366C2×C3×C4#13512yes—
367C2×C3×C4AGL(1,ℤ/8)12yes—
368C2×C3×C4C22×Q811yescharacteristic
369C2×C3×C4C2×#4111yescharacteristic
370C2×C3×C4C2×#4111yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374C3×C8—41no—
375C3×C8—41no—
376C3×C8—41no—
377C3×C8—41no—
378C24—21no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4C2×C3×Q812yes—
383C2×D4C3×#4112yes—
384C2×D4C3×#4112yes—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—24no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4C2×C3×D411yescharacteristic
402C22×C4C2×C3×D411yescharacteristic
403C22×C4C2×C3×D411yescharacteristic
404C22×C4C2×C3×D411yescharacteristic
405C22×C4C2×C3×D411yescharacteristic
406C22×C4C2×C3×D411yescharacteristic
407C22×C4C24×C311yescharacteristic
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4C2×C3×Q812yes—
436C4⋊C4C3×#4112yes—
437C4⋊C4C3×#4112yes—
438C22⋊C4—24no—
439C22⋊C4—24no—
440C22⋊C4—24no—
441C42—22no—
442C42—22no—
443C42—21no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42—41no—
448C42—41no—
449C42—42no—
450C42—42no—
451C42—42no—
452C42C2×C3×Q812yes—
453C42C3×#4112yes—
454C22×C3—24no—
455C22×C3—42no—
456C22×C3—42no—
457C22×C3—42no—
458C22×C3—42no—
459C22×C3—42no—
460C22×C3—42no—
461C22×C3C2×C22⋊Q811yescharacteristic
462C22×C3#52511yescharacteristic
463C22×C3#52511yescharacteristic
464C22×C3C2×#13511yescharacteristic
465C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
466C22×C3#57511yescharacteristic
467C22×C3#57511yescharacteristic
468C3×C4—22no—
469C3×C4—22no—
470C3×C4—22no—
471C3×C4—22no—
472C3×C4—42no—
473C3×C4—41no—
474C3×C4—42no—
475C3×C4—41no—
476C3×C4—42no—
477C3×C4—81no—
478C3×C4—81no—
479C3×C4—81no—
480C3×C4—81no—
481C3×C4C2×#12511yescharacteristic
482C3×C4#52911yescharacteristic
483C3×C4#52911yescharacteristic
484C3×C4#53311yescharacteristic
485C23—22no—
486C23—22no—
487C23—22no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23C22×C3×D411yescharacteristic
493D4—22no—
494D4—22no—
495D4—22no—
496D4C3×#12514yes—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—48no—
515C2×C4—48no—
516C2×C4—48no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
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×C4C3×C22⋊Q812yes—
536C2×C4C3×C22⋊Q812yes—
537C2×C4C3×#13512yes—
538C2×C4C3×AGL(1,ℤ/8)12yes—
539C2×C4C22×C3×Q811yescharacteristic
540C2×C4C2×C3×#4111yescharacteristic
541C2×C4C2×C3×#4111yescharacteristic
542C2×C4C2×C3×#4111yescharacteristic
543C2×C4C2×C3×#4111yescharacteristic
544C2×C4C2×C3×#4111yescharacteristic
545C8—41no—
546C8—41no—
547C8—41no—
548C8—41no—
549C2×C3—44no—
550C2×C3#285111yescharacteristic
551C2×C3#307611yescharacteristic
552C2×C3#307711yescharacteristic
553C2×C3#310211yescharacteristic
554C2×C3#310511yescharacteristic
555C2×C3#311111yescharacteristic
556C2×C3#311411yescharacteristic
557C22—24no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22C2×C3×C22⋊Q811yescharacteristic
565C22C3×#52511yescharacteristic
566C22C3×#52511yescharacteristic
567C22C2×C3×#13511yescharacteristic
568C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
569C22C3×#57511yescharacteristic
570C22C3×#57511yescharacteristic
571C4—22no—
572C4—22no—
573C4—22no—
574C4—22no—
575C4—42no—
576C4—41no—
577C4—42no—
578C4—41no—
579C4—42no—
580C4—81no—
581C4—81no—
582C4—81no—
583C4—81no—
584C4C2×C3×#12511yescharacteristic
585C4C3×#52911yescharacteristic
586C4C3×#52911yescharacteristic
587C4C3×#53311yescharacteristic
588C3#3307011yescharacteristic, direct product factor, Sylow
589C2—44no—
590C2C3×#285111yescharacteristic
591C2C3×#307611yescharacteristic
592C2C3×#307711yescharacteristic
593C2C3×#310211yescharacteristic
594C2C3×#310511yescharacteristic
595C2C3×#311111yescharacteristic
596C2C3×#311411yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C43 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#2864C2semidirect
3. C3×#1949C2semidirect
4. C3×#1949C2semidirect
5. C3×#1921C2semidirect
6. C3×#1920C2semidirect
7. C3×#1919C2semidirect
8. C3×#1918C2semidirect
13. C3×#1831C2semidirect
16. #33070C3direct (N × H)
60. #2864C2×C3semidirect
62. #1949C2×C3semidirect
63. #1949C2×C3semidirect
64. #1921C2×C3semidirect
65. #1920C2×C3semidirect
66. #1919C2×C3semidirect
67. #1918C2×C3semidirect
72. #1831C2×C3semidirect
588. C3#33070direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#28642
C3×C42×D42
C3×#19492
C3×#19212
C3×#19202
C3×#19192
C3×#19182
C3×#18642
C3×#18622
C3×#18312
C3×#18042
#330703