Finite Group Databasebeta

← C3×#9524 #10628131

Subgroups — C3×#9524 #10628131

Order 768.

686 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3007C211yesmaximal, characteristic
2C3×#2332C211yesmaximal, characteristic, complemented
3C3×#1789C211yesmaximal, characteristic
4C3×#1782C211yesmaximal, characteristic
5C3×#1511C211yesmaximal, characteristic, complemented
6C3×#1306C211yesmaximal, characteristic, complemented
7C3×#1284C211yesmaximal, characteristic, complemented
8#9524C311yesmaximal, characteristic, direct product factor, Sylow
9C22×C3×#41C2211yescharacteristic
10C2×C3×#137—21no—
11C2×C3×C22⋊Q8C2211yescharacteristic
12C3×#452—22no—
13C3×#451C412yescomplemented
14C3×#450C412yescomplemented
15C2×C3×#104—21no—
16C2×C3×#104—21no—
17C3×#417—21no—
18C2×C3×#99—21no—
19C2×C3×#99—21no—
20C3×#407C2211yescharacteristic
21C3×#407C2211yescharacteristic
22C3×#391—21no—
23C3×#388—21no—
24C3×#388—21no—
25C3×#387—21no—
26C3×#387—21no—
27C3×#381C2211yescharacteristic
28C3×#381C2211yescharacteristic
29C3×C4×C4⋊C4—21no—
30C3×C4×C22⋊C4C2211yescharacteristic
31C3×#328—22no—
32#3007C2×C311yescharacteristic
33#2332C2×C311yescharacteristic, complemented
34#1789C2×C311yescharacteristic
35#1782C2×C311yescharacteristic
36#1511C2×C311yescharacteristic, complemented
37#1306C2×C311yescharacteristic, complemented
38#1284C2×C311yescharacteristic, complemented
39C2×C3×#41—21no—
40C2×C3×#41—21no—
41C2×C3×#41—21no—
42C2×C3×#41—21no—
43C2×C3×#41—42no—
44C2×C3×#41—42no—
45C22×C3×Q8C2×C411yescharacteristic
46C22×C3×Q8D411yescharacteristic
47C22×C3×D4—21no—
48C22×C3×D4C2×C411yescharacteristic
49C23×C3×C4—21no—
50C23×C3×C4C2311yescharacteristic
51C3×#137—42no—
52C3×#137—42no—
53C2×C3×Q16—22no—
54C2×C3×SD16—22no—
55C2×C3×M16—21no—
56C2×C3×M16—21no—
57C3×C22⋊Q8—21no—
58C3×C22⋊Q8—21no—
59C3×C22⋊Q8C2×C412yes—
60C3×C22⋊Q8D412yes—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4D411yescharacteristic
65C2×C3×C22⋊C4—21no—
66C2×C3×C22⋊C4—21no—
67C2×C3×C22⋊C4—21no—
68C2×C3×C22⋊C4D411yescharacteristic
69C2×C3×C22⋊C4D411yescharacteristic
70C2×C3×C42—21no—
71C2×C3×C42—21no—
72C2×C3×C42D411yescharacteristic
73C2×C3×C42D411yescharacteristic
74C3×#104—42no—
75C3×#104—42no—
76C3×#103—22no—
77C3×#102—22no—
78C3×#99—42no—
79C3×#99—42no—
80C3×C22⋊C8—21no—
81C3×C22⋊C8—21no—
82C3×C22⋊C8—21no—
83C3×C22⋊C8C2×C412yes—
84C3×#95—21no—
85C3×#95—21no—
86C3×#95—21no—
87C3×#95—21no—
88C3×#95—21no—
89C3×#95—21no—
90C3×#95—21no—
91C3×#95—21no—
92C3×#95—21no—
93C22×#41C22×C311yescharacteristic
94C2×#137—21no—
95C2×C22⋊Q8C22×C311yescharacteristic
96#452—22no—
97#451C3×C412yescomplemented
98#450C3×C412yescomplemented
99C2×#104—21no—
100C2×#104—21no—
101#417—21no—
102C2×#99—21no—
103C2×#99—21no—
104#407C22×C311yescharacteristic
105#407C22×C311yescharacteristic
106#391—21no—
107#388—21no—
108#388—21no—
109#387—21no—
110#387—21no—
111#381C22×C311yescharacteristic
112#381C22×C311yescharacteristic
113C4×C4⋊C4—21no—
114C4×C22⋊C4C22×C311yescharacteristic
115#328—22no—
116C24×C3—21no—
117C24×C3C2×D411yescharacteristic
118C3×#41—22no—
119C3×#41—22no—
120C3×#41—22no—
121C3×#41—22no—
122C3×#41—41no—
123C3×#41—41no—
124C3×#41—41no—
125C3×#41—41no—
126C3×#41—42no—
127C3×#41—42no—
128C3×#41—42no—
129C3×#41—42no—
130C2×C3×Q8—21no—
131C2×C3×Q8—21no—
132C2×C3×Q8—21no—
133C2×C3×Q8—42no—
134C2×C3×Q8—41no—
135C2×C3×Q8—41no—
136C2×C3×Q8C22⋊C412yes—
137C2×C3×D4—22no—
138C2×C3×D4—22no—
139C2×C3×D4—21no—
140C2×C3×D4—21no—
141C2×C3×D4—42no—
142C2×C3×D4—41no—
143C2×C3×D4—41no—
144C2×C3×D4—41no—
145C2×C3×D4—41no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—41no—
154C22×C3×C4—41no—
155C22×C3×C4—42no—
156C22×C3×C4—42no—
157C22×C3×C4—41no—
158C22×C3×C4—41no—
159C22×C3×C4—41no—
160C22×C3×C4C22×C411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4#4111yescharacteristic
165C22×C3×C4#4111yescharacteristic
166C3×Q16—44no—
167C3×SD16—44no—
168C3×M16—41no—
169C3×M16—41no—
170C3×M16—41no—
171C3×M16—41no—
172C2×C3×C8—22no—
173C2×C3×C8—41no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—42no—
178C3×C4⋊C4—81no—
179C3×C4⋊C4C22⋊C412yes—
180C3×C22⋊C4—22no—
181C3×C22⋊C4—22no—
182C3×C22⋊C4—42no—
183C3×C22⋊C4—41no—
184C3×C22⋊C4—41no—
185C3×C22⋊C4—41no—
186C3×C22⋊C4—41no—
187C3×C42—41no—
188C3×C42—41no—
189C3×C42—42no—
190C3×C42—81no—
191C2×#41—21no—
192C2×#41—21no—
193C2×#41—21no—
194C2×#41—21no—
195C2×#41—42no—
196C2×#41—42no—
197C22×Q8C2×C3×C411yescharacteristic
198C22×Q8C3×D411yescharacteristic
199C22×D4—21no—
200C22×D4C2×C3×C411yescharacteristic
201C23×C4—21no—
202C23×C4C23×C311yescharacteristic
203#137—42no—
204#137—42no—
205C2×Q16—22no—
206C2×SD16—22no—
207C2×M16—21no—
208C2×M16—21no—
209C22⋊Q8—21no—
210C22⋊Q8—21no—
211C22⋊Q8C2×C3×C412yes—
212C22⋊Q8C3×D412yes—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4C3×D411yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4C3×D411yescharacteristic
221C2×C22⋊C4C3×D411yescharacteristic
222C2×C42—21no—
223C2×C42—21no—
224C2×C42C3×D411yescharacteristic
225C2×C42C3×D411yescharacteristic
226#104—42no—
227#104—42no—
228#103—22no—
229#102—22no—
230#99—42no—
231#99—42no—
232C22⋊C8—21no—
233C22⋊C8—21no—
234C22⋊C8—21no—
235C22⋊C8C2×C3×C412yes—
236#95—21no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—21no—
242#95—21no—
243#95—21no—
244#95—21no—
245C23×C3—21no—
246C23×C3—21no—
247C23×C3—22no—
248C23×C3—41no—
249C23×C3—41no—
250C23×C3—41no—
251C23×C3—41no—
252C23×C3—81no—
253C23×C3—81no—
254C23×C3C4×D411yescharacteristic
255C23×C3#12011yescharacteristic
256C23×C3#12111yescharacteristic
257C3×Q8—22no—
258C3×Q8—22no—
259C3×Q8—42no—
260C3×Q8—41no—
261C3×Q8—41no—
262C3×Q8—81no—
263C3×D4—22no—
264C3×D4—22no—
265C3×D4—41no—
266C3×D4—41no—
267C3×D4—42no—
268C3×D4—42no—
269C3×D4—42no—
270C3×D4—42no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—41no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—81no—
305C2×C3×C4—81no—
306C2×C3×C4—81no—
307C2×C3×C4C2×C22⋊C411yescharacteristic
308C2×C3×C4C4×D411yescharacteristic
309C2×C3×C4#12111yescharacteristic
310C2×C3×C4#12311yescharacteristic
311C3×C8—42no—
312C3×C8—81no—
313C24—21no—
314C24C2×C3×D411yescharacteristic
315#41—22no—
316#41—22no—
317#41—22no—
318#41—22no—
319#41—41no—
320#41—41no—
321#41—41no—
322#41—41no—
323#41—42no—
324#41—42no—
325#41—42no—
326#41—42no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—42no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8C3×C22⋊C412yes—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—21no—
337C2×D4—21no—
338C2×D4—42no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—41no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4C22×C3×C411yescharacteristic
358C22×C4C2×C3×D411yescharacteristic
359C22×C4C2×C3×D411yescharacteristic
360C22×C4C2×C3×D411yescharacteristic
361C22×C4C3×#4111yescharacteristic
362C22×C4C3×#4111yescharacteristic
363Q16—44no—
364SD16—44no—
365M16—41no—
366M16—41no—
367M16—41no—
368M16—41no—
369C2×C8—22no—
370C2×C8—41no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—21no—
374C4⋊C4—42no—
375C4⋊C4—81no—
376C4⋊C4C3×C22⋊C412yes—
377C22⋊C4—22no—
378C22⋊C4—22no—
379C22⋊C4—42no—
380C22⋊C4—41no—
381C22⋊C4—41no—
382C22⋊C4—41no—
383C22⋊C4—41no—
384C42—41no—
385C42—41no—
386C42—42no—
387C42—81no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—42no—
397C22×C3—81no—
398C22×C3—81no—
399C22×C3—82no—
400C22×C3—82no—
401C22×C3—81no—
402C22×C3—81no—
403C22×C3#38611yescharacteristic
404C22×C3B(4,𝔽2)11yescharacteristic
405C22×C3B(4,𝔽2)11yescharacteristic
406C3×C4—21no—
407C3×C4—21no—
408C3×C4—21no—
409C3×C4—21no—
410C3×C4—42no—
411C3×C4—42no—
412C3×C4—42no—
413C3×C4—42no—
414C3×C4—81no—
415C3×C4—81no—
416C3×C4—81no—
417C3×C4—161no—
418C23—21no—
419C23—21no—
420C23—22no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—81no—
426C23—81no—
427C23C3×C4×D411yescharacteristic
428C23C3×#12011yescharacteristic
429C23C3×#12111yescharacteristic
430Q8—22no—
431Q8—22no—
432Q8—42no—
433Q8—41no—
434Q8—41no—
435Q8—81no—
436D4—22no—
437D4—22no—
438D4—41no—
439D4—41no—
440D4—42no—
441D4—42no—
442D4—42no—
443D4—42no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4C2×C3×C22⋊C411yescharacteristic
481C2×C4C3×C4×D411yescharacteristic
482C2×C4C3×#12111yescharacteristic
483C2×C4C3×#12311yescharacteristic
484C8—42no—
485C8—81no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—41no—
489C2×C3—41no—
490C2×C3—82no—
491C2×C3#188911yescharacteristic
492C2×C3#190511yescharacteristic
493C2×C3#190611yescharacteristic
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—42no—
503C22—81no—
504C22—81no—
505C22—82no—
506C22—82no—
507C22—81no—
508C22—81no—
509C22C3×#38611yescharacteristic
510C22C3×B(4,𝔽2)11yescharacteristic
511C22C3×B(4,𝔽2)11yescharacteristic
512C4—21no—
513C4—21no—
514C4—21no—
515C4—21no—
516C4—42no—
517C4—42no—
518C4—42no—
519C4—42no—
520C4—81no—
521C4—81no—
522C4—81no—
523C4—161no—
524C3#952411yescharacteristic, direct product factor, Sylow
525C2—21no—
526C2—21no—
527C2—41no—
528C2—41no—
529C2—82no—
530C2C3×#188911yescharacteristic
531C2C3×#190511yescharacteristic
532C2C3×#190611yescharacteristic
subgroupG/H
center, socleC22×C3#386
commutatorC22×C4C22×C3×C4
Frattini, cosocleC23×C4C23×C3
prime radicalC22×C3×#41C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2332C2semidirect
5. C3×#1511C2semidirect
6. C3×#1306C2semidirect
7. C3×#1284C2semidirect
8. #9524C3direct (N × H)
13. C3×#451C4semidirect
14. C3×#450C4semidirect
33. #2332C2×C3semidirect
36. #1511C2×C3semidirect
37. #1306C2×C3semidirect
38. #1284C2×C3semidirect
97. #451C3×C4semidirect
98. #450C3×C4semidirect
524. C3#9524direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#30072
C3×#23322
C3×#17892
C3×#17822
C3×#15112
C3×#13062
C3×#12842
#95243