Finite Group Databasebeta

← C3×#32988 #10651595

Subgroups — C3×#32988 #10651595

Order 768.

1300 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#520C211yesmaximal, characteristic
2C2×C3×#485C211yesmaximal, characteristic, complemented
3C2×C3×#483C211yesmaximal, characteristic, complemented
4C3×#2616C211yesmaximal, characteristic
5C3×#1873C212yesmaximal, complemented
6C3×#1872C212yesmaximal, complemented
7C3×#1854C212yesmaximal, complemented
8C3×#1853C212yesmaximal, complemented
9C3×#1825C211yesmaximal, characteristic, complemented
10C3×#1824C211yesmaximal, characteristic, complemented
11C3×#1780C211yesmaximal, characteristic
12#32988C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C2×C3×C22⋊Q8C2212yes—
15C2×C3×C22⋊Q8C2211yescharacteristic
16C3×#520—24no—
17C3×#520—24no—
18C2×C3×C4×Q8—21no—
19C2×C3×C4×D4—22no—
20C2×C3×C4×D4—21no—
21C2×C3×#117C2212yes—
22C2×C3×#117C2211yescharacteristic
23C22×C3×C4⋊C4C2212yes—
24C22×C3×C4⋊C4C2211yescharacteristic
25C3×#485—24no—
26C3×#483—24no—
27C2×C3×#107C2211yescharacteristic
28C2×C3×#106C2211yescharacteristic
29C2×C3×#106C2212yes—
30C2×C3×#103C2211yescharacteristic
31C2×C3×#103C2212yes—
32C2×C3×#103C2211yescharacteristic
33C2×C3×C22⋊C8C2211yescharacteristic
34C2×C3×C22⋊C8C2211yescharacteristic
35C2×C3×C22⋊C8C2212yes—
36C2×C3×C22⋊C8C2212yes—
37C3×#395—21no—
38C3×#394—21no—
39C3×#394—22no—
40C3×#380C2211yescharacteristic
41C3×#380C2212yes—
42C3×#340C2212yes—
43C3×#340C2212yes—
44C3×#340C2212yes—
45C3×#340C2212yes—
46C2×#520C2×C311yescharacteristic
47C2×#485C2×C311yescharacteristic, complemented
48C2×#483C2×C311yescharacteristic, complemented
49#2616C2×C311yescharacteristic
50#1873C2×C312yescomplemented
51#1872C2×C312yescomplemented
52#1854C2×C312yescomplemented
53#1853C2×C312yescomplemented
54#1825C2×C311yescharacteristic, complemented
55#1824C2×C311yescharacteristic, complemented
56#1780C2×C311yescharacteristic
57C2×C3×#41—22no—
58C2×C3×#41—22no—
59C2×C3×#41—24no—
60C2×C3×#41—24no—
61C22×C3×Q8D411yescharacteristic
62C22×C3×Q8C2311yescharacteristic
63C22×C3×D4D411yescharacteristic
64C22×C3×D4D412yes—
65C23×C3×C4—22no—
66C23×C3×C4—21no—
67C23×C3×C4C2311yescharacteristic
68C23×C3×C4C2312yes—
69C22×C3×C8C2311yescharacteristic
70C22×C3×C8C2312yes—
71C22×C3×C8C2311yescharacteristic
72C3×C22⋊Q8—22no—
73C3×C22⋊Q8—22no—
74C3×C22⋊Q8—24no—
75C3×C22⋊Q8—24no—
76C3×C4×Q8—28no—
77C3×C4×D4—216no—
78C3×C4×D4—28no—
79C3×#117—22no—
80C3×#117—24no—
81C3×#117—24no—
82C3×#117D414yes—
83C2×C3×C4⋊C4—22no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—22no—
86C2×C3×C4⋊C4—24no—
87C2×C3×C4⋊C4—24no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—22no—
90C2×C3×C4⋊C4D414yes—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2312yes—
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2312yes—
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4—22no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4—22no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—22no—
102C2×C3×C42—22no—
103C2×C3×C42—21no—
104C3×#107—22no—
105C3×#106—24no—
106C3×#106—22no—
107C3×#103—24no—
108C3×#103—22no—
109C3×#103—22no—
110C3×C22⋊C8—22no—
111C3×C22⋊C8—22no—
112C3×C22⋊C8—24no—
113C3×C22⋊C8—24no—
114C3×#95—21no—
115C3×#95—22no—
116C22×#41C22×C311yescharacteristic
117C2×C22⋊Q8C22×C311yescharacteristic
118C2×C22⋊Q8C22×C312yes—
119#520—24no—
120#520—24no—
121C2×C4×Q8—21no—
122C2×C4×D4—22no—
123C2×C4×D4—21no—
124C2×#117C22×C312yes—
125C2×#117C22×C311yescharacteristic
126C22×C4⋊C4C22×C312yes—
127C22×C4⋊C4C22×C311yescharacteristic
128#485—24no—
129#483—24no—
130C2×#107C22×C311yescharacteristic
131C2×#106C22×C312yes—
132C2×#106C22×C311yescharacteristic
133C2×#103C22×C312yes—
134C2×#103C22×C311yescharacteristic
135C2×#103C22×C311yescharacteristic
136C2×C22⋊C8C22×C311yescharacteristic
137C2×C22⋊C8C22×C311yescharacteristic
138C2×C22⋊C8C22×C312yes—
139C2×C22⋊C8C22×C312yes—
140#395—21no—
141#394—21no—
142#394—22no—
143#380C22×C311yescharacteristic
144#380C22×C312yes—
145#340C22×C312yes—
146#340C22×C312yes—
147#340C22×C312yes—
148#340C22×C312yes—
149C24×C3C2×D411yescharacteristic
150C24×C3C2×D412yes—
151C3×#41—28no—
152C3×#41—28no—
153C3×#41—216no—
154C2×C3×Q8—22no—
155C2×C3×Q8—24no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—22no—
159C2×C3×Q8—22no—
160C2×C3×D4—24no—
161C2×C3×D4—24no—
162C2×C3×D4—24no—
163C2×C3×D4—22no—
164C2×C3×D4—24no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—21no—
172C22×C3×C4—22no—
173C22×C3×C4—21no—
174C22×C3×C4—28no—
175C22×C3×C4—24no—
176C22×C3×C4—21no—
177C22×C3×C4—22no—
178C22×C3×C4—21no—
179C22×C3×C4—82no—
180C22×C3×C4—82no—
181C22×C3×C4—82no—
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2×D412yes—
186C22×C3×C4#4114yes—
187C22×C3×C4#4114yes—
188C22×C3×C4C2411yescharacteristic
189C2×C3×C8—22no—
190C2×C3×C8—21no—
191C2×C3×C8—21no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C3×C4⋊C4—28no—
197C3×C4⋊C4—24no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—42no—
206C3×C4⋊C4—44no—
207C3×C4⋊C4C2×D412yes—
208C3×C4⋊C4C2×D414yes—
209C3×C4⋊C4C2×D412yes—
210C3×C22⋊C4—28no—
211C3×C22⋊C4—24no—
212C3×C22⋊C4—42no—
213C3×C22⋊C4—44no—
214C3×C22⋊C4—81no—
215C3×C22⋊C4—82no—
216C3×C42—28no—
217C3×C42—24no—
218C2×#41—22no—
219C2×#41—22no—
220C2×#41—24no—
221C2×#41—24no—
222C22×Q8C3×D411yescharacteristic
223C22×Q8C23×C311yescharacteristic
224C22×D4C3×D411yescharacteristic
225C22×D4C3×D412yes—
226C23×C4—22no—
227C23×C4—21no—
228C23×C4C23×C311yescharacteristic
229C23×C4C23×C312yes—
230C22×C8C23×C311yescharacteristic
231C22×C8C23×C312yes—
232C22×C8C23×C311yescharacteristic
233C22⋊Q8—22no—
234C22⋊Q8—22no—
235C22⋊Q8—24no—
236C22⋊Q8—24no—
237C4×Q8—28no—
238C4×D4—216no—
239C4×D4—28no—
240#117—22no—
241#117—24no—
242#117—24no—
243#117C3×D414yes—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4C3×D414yes—
252C2×C4⋊C4C23×C312yes—
253C2×C4⋊C4C23×C311yescharacteristic
254C2×C4⋊C4C23×C311yescharacteristic
255C2×C4⋊C4C23×C312yes—
256C2×C4⋊C4C23×C311yescharacteristic
257C2×C22⋊C4—22no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—22no—
263C2×C42—22no—
264C2×C42—21no—
265#107—22no—
266#106—24no—
267#106—22no—
268#103—24no—
269#103—22no—
270#103—22no—
271C22⋊C8—22no—
272C22⋊C8—22no—
273C22⋊C8—24no—
274C22⋊C8—24no—
275#95—21no—
276#95—22no—
277C23×C3—22no—
278C23×C3—22no—
279C23×C3—22no—
280C23×C3—22no—
281C23×C3—21no—
282C23×C3—22no—
283C23×C3—22no—
284C23×C3—22no—
285C23×C3—22no—
286C23×C3—21no—
287C23×C3#12012yes—
288C23×C3#12314yes—
289C23×C3C22×D411yescharacteristic
290C3×Q8—28no—
291C3×Q8—42no—
292C3×Q8—41no—
293C3×Q8—41no—
294C3×D4—28no—
295C3×D4—216no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—24no—
302C2×C3×C4—24no—
303C2×C3×C4—24no—
304C2×C3×C4—24no—
305C2×C3×C4—24no—
306C2×C3×C4—22no—
307C2×C3×C4—24no—
308C2×C3×C4—22no—
309C2×C3×C4—41no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—41no—
320C2×C3×C4—42no—
321C2×C3×C4—41no—
322C2×C3×C4—84no—
323C2×C3×C4—88no—
324C2×C3×C4#12012yes—
325C2×C3×C4#12314yes—
326C2×C3×C4C22×D411yescharacteristic
327C2×C3×C4C22×D411yescharacteristic
328C2×C3×C4C2×#4112yes—
329C2×C3×C4C2×#4112yes—
330C3×C8—42no—
331C3×C8—41no—
332C3×C8—41no—
333C24C2×C3×D411yescharacteristic
334C24C2×C3×D412yes—
335#41—28no—
336#41—28no—
337#41—216no—
338C2×Q8—22no—
339C2×Q8—24no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×D4—24no—
345C2×D4—24no—
346C2×D4—24no—
347C2×D4—22no—
348C2×D4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—28no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—82no—
364C22×C4—82no—
365C22×C4—82no—
366C22×C4C2×C3×D411yescharacteristic
367C22×C4C2×C3×D412yes—
368C22×C4C2×C3×D412yes—
369C22×C4C2×C3×D412yes—
370C22×C4C3×#4114yes—
371C22×C4C3×#4114yes—
372C22×C4C24×C311yescharacteristic
373C2×C8—22no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C4⋊C4—28no—
381C4⋊C4—24no—
382C4⋊C4—22no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—42no—
390C4⋊C4—44no—
391C4⋊C4C2×C3×D412yes—
392C4⋊C4C2×C3×D414yes—
393C4⋊C4C2×C3×D412yes—
394C22⋊C4—28no—
395C22⋊C4—24no—
396C22⋊C4—42no—
397C22⋊C4—44no—
398C22⋊C4—82no—
399C22⋊C4—81no—
400C42—28no—
401C42—24no—
402C22×C3—24no—
403C22×C3—24no—
404C22×C3—22no—
405C22×C3—24no—
406C22×C3—22no—
407C22×C3—22no—
408C22×C3—42no—
409C22×C3—42no—
410C22×C3—42no—
411C22×C3—42no—
412C22×C3—42no—
413C22×C3—42no—
414C22×C3C2×#12011yescharacteristic
415C22×C3C2×#12312yes—
416C22×C3#57712yes—
417C22×C3#57711yescharacteristic
418C22×C3#57811yescharacteristic
419C3×C4—22no—
420C3×C4—24no—
421C3×C4—44no—
422C3×C4—42no—
423C3×C4—42no—
424C3×C4—82no—
425C3×C4—81no—
426C3×C4—81no—
427C3×C4#52511yescharacteristic
428C3×C4#52512yes—
429C3×C4#52511yescharacteristic
430C23—21no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—21no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23C3×#12012yes—
441C23C3×#12314yes—
442C23C22×C3×D411yescharacteristic
443Q8—28no—
444Q8—42no—
445Q8—41no—
446Q8—41no—
447D4—28no—
448D4—216no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—84no—
476C2×C4—88no—
477C2×C4C3×#12012yes—
478C2×C4C3×#12314yes—
479C2×C4C22×C3×D411yescharacteristic
480C2×C4C22×C3×D411yescharacteristic
481C2×C4C2×C3×#4112yes—
482C2×C4C2×C3×#4112yes—
483C8—42no—
484C8—41no—
485C8—41no—
486C2×C3—44no—
487C2×C3—42no—
488C2×C3#284911yescharacteristic
489C2×C3#300911yescharacteristic
490C2×C3#301311yescharacteristic
491C2×C3#309712yes—
492C2×C3#309912yes—
493C22—22no—
494C22—22no—
495C22—24no—
496C22—24no—
497C22—22no—
498C22—24no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22C2×C3×#12011yescharacteristic
506C22C2×C3×#12312yes—
507C22C3×#57712yes—
508C22C3×#57711yescharacteristic
509C22C3×#57811yescharacteristic
510C4—22no—
511C4—24no—
512C4—44no—
513C4—42no—
514C4—42no—
515C4—81no—
516C4—82no—
517C4—81no—
518C4C3×#52511yescharacteristic
519C4C3×#52512yes—
520C4C3×#52511yescharacteristic
521C3#3298811yescharacteristic, direct product factor, Sylow
522C2—42no—
523C2—44no—
524C2C3×#284911yescharacteristic
525C2C3×#300911yescharacteristic
526C2C3×#301311yescharacteristic
527C2C3×#309712yes—
528C2C3×#309912yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×#41 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#485C2semidirect
3. C2×C3×#483C2semidirect
5. C3×#1873C2semidirect
6. C3×#1872C2semidirect
7. C3×#1854C2semidirect
8. C3×#1853C2semidirect
9. C3×#1825C2semidirect
10. C3×#1824C2semidirect
12. #32988C3direct (N × H)
47. C2×#485C2×C3semidirect
48. C2×#483C2×C3semidirect
50. #1873C2×C3semidirect
51. #1872C2×C3semidirect
52. #1854C2×C3semidirect
53. #1853C2×C3semidirect
54. #1825C2×C3semidirect
55. #1824C2×C3semidirect
521. C3#32988direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5202
C2×C3×#4852
C2×C3×#4832
C3×#26162
C3×#18732
C3×#18722
C3×#18542
C3×#18532
C3×#18252
C3×#18242
C3×#17802
#329883