Finite Group Databasebeta

← C3×#29645 #10648252

Subgroups — C3×#29645 #10648252

Order 768.

1002 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#533C211yesmaximal, characteristic
2C2×C3×#485C212yesmaximal
3C3×#2855C211yesmaximal, characteristic
4C3×#2059C212yesmaximal, complemented
5C3×#2058C211yesmaximal, characteristic, complemented
6C3×#2057C211yesmaximal, characteristic, complemented
7C3×#2026C212yesmaximal, complemented
8C3×#2024C211yesmaximal, characteristic, complemented
9C3×#2024C211yesmaximal, characteristic, complemented
10C3×#1825C212yesmaximal
11C3×#1817C211yesmaximal, characteristic
12#29645C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#533—28no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#125C2212yes—
17C2×C3×C22⋊Q8—21no—
18C2×C3×C22⋊Q8—21no—
19C2×C3×C22⋊Q8C2211yescharacteristic
20C2×C3×C4×Q8—21no—
21C2×C3×C4×Q8—21no—
22C2×C3×#117C2212yes—
23C22×C3×C42C2211yescharacteristic
24C3×#485—28no—
25C2×C3×#107C2212yes—
26C2×C3×#106C2212yes—
27C2×C3×#103C2212yes—
28C2×C3×#103C2212yes—
29C2×C3×C22⋊C8C2212yes—
30C3×#391C2211yescharacteristic
31C3×#391C2211yescharacteristic
32C3×#389C2211yescharacteristic
33C3×#389C2211yescharacteristic
34C3×#387—21no—
35C3×#387—21no—
36C3×#385—21no—
37C3×#381—21no—
38C3×#380C2211yescharacteristic
39C3×#340C2212yes—
40C3×#340C2212yes—
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#336C2211yescharacteristic
44C3×#336C2211yescharacteristic
45C3×#336C2212yes—
46C2×#533C2×C311yescharacteristic
47C2×#485C2×C312yes—
48#2855C2×C311yescharacteristic
49#2059C2×C312yescomplemented
50#2058C2×C311yescharacteristic, complemented
51#2057C2×C311yescharacteristic, complemented
52#2026C2×C312yescomplemented
53#2024C2×C311yescharacteristic, complemented
54#2024C2×C311yescharacteristic, complemented
55#1825C2×C312yes—
56#1817C2×C311yescharacteristic
57C22×C3×Q8—21no—
58C22×C3×Q8—21no—
59C22×C3×Q8C2311yescharacteristic
60C23×C3×C4D411yescharacteristic
61C23×C3×C4D411yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C22×C3×C8C2312yes—
64C22×C3×C8C2312yes—
65C3×C4⋊Q8—24no—
66C3×C4⋊Q8—24no—
67C3×#125—24no—
68C3×#125—24no—
69C3×C22⋊Q8—28no—
70C3×C22⋊Q8—28no—
71C3×C22⋊Q8—24no—
72C3×C4×Q8—28no—
73C3×C4×Q8—28no—
74C3×#117—24no—
75C3×#117D418yes—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2312yes—
87C2×C3×C4⋊C4C2312yes—
88C2×C3×C22⋊C4—22no—
89C2×C3×C22⋊C4—21no—
90C2×C3×C22⋊C4—21no—
91C2×C3×C42—24no—
92C2×C3×C42—22no—
93C2×C3×C42D411yescharacteristic
94C2×C3×C42D411yescharacteristic
95C2×C3×C42C2311yescharacteristic
96C2×C3×C42C2311yescharacteristic
97C3×#107—24no—
98C3×#106—24no—
99C3×#103—24no—
100C3×#103—24no—
101C3×C22⋊C8—24no—
102C3×#95—21no—
103C3×#95—21no—
104C3×#95—21no—
105C3×#95—21no—
106C3×#95—21no—
107C3×#95—21no—
108C3×#95—21no—
109#533—28no—
110C2×C4⋊Q8C22×C311yescharacteristic
111C2×C4⋊Q8C22×C311yescharacteristic
112C2×#125C22×C312yes—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8C22×C311yescharacteristic
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×#117C22×C312yes—
119C22×C42C22×C311yescharacteristic
120#485—28no—
121C2×#107C22×C312yes—
122C2×#106C22×C312yes—
123C2×#103C22×C312yes—
124C2×#103C22×C312yes—
125C2×C22⋊C8C22×C312yes—
126#391C22×C311yescharacteristic
127#391C22×C311yescharacteristic
128#389C22×C311yescharacteristic
129#389C22×C311yescharacteristic
130#387—21no—
131#387—21no—
132#385—21no—
133#381—21no—
134#380C22×C311yescharacteristic
135#340C22×C312yes—
136#340C22×C312yes—
137#340C22×C312yes—
138#340C22×C312yes—
139#336C22×C311yescharacteristic
140#336C22×C311yescharacteristic
141#336C22×C312yes—
142C24×C3C2×D411yescharacteristic
143C2×C3×Q8—24no—
144C2×C3×Q8—24no—
145C2×C3×Q8—22no—
146C2×C3×Q8—22no—
147C2×C3×Q8—21no—
148C2×C3×Q8—21no—
149C2×C3×Q8—42no—
150C2×C3×Q8—42no—
151C2×C3×Q8—42no—
152C2×C3×Q8—42no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—41no—
167C22×C3×C4—41no—
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D414yes—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2411yescharacteristic
175C2×C3×C8—22no—
176C2×C3×C8—22no—
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C2×C3×C8—22no—
180C2×C3×C8—22no—
181C3×C4⋊C4—28no—
182C3×C4⋊C4—28no—
183C3×C4⋊C4—24no—
184C3×C4⋊C4—24no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—21no—
189C3×C4⋊C4—21no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—42no—
192C3×C4⋊C4—42no—
193C3×C4⋊C4—42no—
194C3×C4⋊C4C2×D414yes—
195C3×C4⋊C4C2×D414yes—
196C3×C22⋊C4—28no—
197C3×C22⋊C4—42no—
198C3×C22⋊C4—42no—
199C3×C42—22no—
200C3×C42—22no—
201C3×C42—28no—
202C3×C42—22no—
203C3×C42—22no—
204C22×Q8—21no—
205C22×Q8—21no—
206C22×Q8C23×C311yescharacteristic
207C23×C4C3×D411yescharacteristic
208C23×C4C3×D411yescharacteristic
209C23×C4C23×C311yescharacteristic
210C22×C8C23×C312yes—
211C22×C8C23×C312yes—
212C4⋊Q8—24no—
213C4⋊Q8—24no—
214#125—24no—
215#125—24no—
216C22⋊Q8—28no—
217C22⋊Q8—28no—
218C22⋊Q8—24no—
219C4×Q8—28no—
220C4×Q8—28no—
221#117—24no—
222#117C3×D418yes—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4C23×C311yescharacteristic
231C2×C4⋊C4C23×C311yescharacteristic
232C2×C4⋊C4C23×C311yescharacteristic
233C2×C4⋊C4C23×C312yes—
234C2×C4⋊C4C23×C312yes—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C42—22no—
239C2×C42—24no—
240C2×C42C3×D411yescharacteristic
241C2×C42C3×D411yescharacteristic
242C2×C42C23×C311yescharacteristic
243C2×C42C23×C311yescharacteristic
244#107—24no—
245#106—24no—
246#103—24no—
247#103—24no—
248C22⋊C8—24no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—21no—
256C23×C3—21no—
257C23×C3—21no—
258C23×C3—21no—
259C23×C3—21no—
260C23×C3—21no—
261C23×C3—21no—
262C23×C3#12012yes—
263C23×C3C22×D411yescharacteristic
264C3×Q8—48no—
265C3×Q8—48no—
266C3×Q8—42no—
267C3×Q8—42no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—24no—
275C2×C3×C4—24no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—24no—
279C2×C3×C4—24no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—44no—
285C2×C3×C4—44no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
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#12012yes—
303C2×C3×C4#13512yes—
304C2×C3×C4#13712yes—
305C2×C3×C4C22×D412yes—
306C2×C3×C42−1+412yes—
307C2×C3×C42−1+411yescharacteristic
308C2×C3×C42−1+411yescharacteristic
309C3×C8—42no—
310C3×C8—42no—
311C24C2×C3×D411yescharacteristic
312C2×Q8—24no—
313C2×Q8—24no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—21no—
317C2×Q8—21no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4C2×C3×D411yescharacteristic
338C22×C4C2×C3×D411yescharacteristic
339C22×C4C2×C3×D411yescharacteristic
340C22×C4C2×C3×D414yes—
341C22×C4C2×C3×D411yescharacteristic
342C22×C4C2×C3×D411yescharacteristic
343C22×C4C24×C311yescharacteristic
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—22no—
350C4⋊C4—28no—
351C4⋊C4—28no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—22no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4C2×C3×D414yes—
364C4⋊C4C2×C3×D414yes—
365C22⋊C4—28no—
366C22⋊C4—42no—
367C22⋊C4—42no—
368C42—22no—
369C42—22no—
370C42—28no—
371C42—22no—
372C42—22no—
373C22×C3—22no—
374C22×C3—22no—
375C22×C3—22no—
376C22×C3—41no—
377C22×C3—41no—
378C22×C3—41no—
379C22×C3—41no—
380C22×C3C2×#12011yescharacteristic
381C22×C3#53611yescharacteristic
382C22×C3#53611yescharacteristic
383C22×C3C2×#13511yescharacteristic
384C22×C3C2×#13711yescharacteristic
385C22×C3#57511yescharacteristic
386C22×C3#57511yescharacteristic
387C3×C4—22no—
388C3×C4—22no—
389C3×C4—22no—
390C3×C4—41no—
391C3×C4—44no—
392C3×C4—41no—
393C3×C4—44no—
394C3×C4—81no—
395C3×C4—81no—
396C3×C4—81no—
397C3×C4—81no—
398C3×C4#53612yes—
399C3×C4#53612yes—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23C3×#12012yes—
407C23C22×C3×D411yescharacteristic
408Q8—48no—
409Q8—48no—
410Q8—42no—
411Q8—42no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4C3×#12012yes—
447C2×C4C3×#13512yes—
448C2×C4C3×#13712yes—
449C2×C4C22×C3×D412yes—
450C2×C4C3×2−1+412yes—
451C2×C4C3×2−1+411yescharacteristic
452C2×C4C3×2−1+411yescharacteristic
453C8—42no—
454C8—42no—
455C2×C3—42no—
456C2×C3#261411yescharacteristic
457C2×C3#300211yescharacteristic
458C2×C3#300711yescharacteristic
459C2×C3#323111yescharacteristic
460C2×C3#323311yescharacteristic
461C2×C3#323711yescharacteristic
462C2×C3#324011yescharacteristic
463C22—22no—
464C22—22no—
465C22—22no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22C2×C3×#12011yescharacteristic
471C22C3×#53611yescharacteristic
472C22C3×#53611yescharacteristic
473C22C2×C3×#13511yescharacteristic
474C22C2×C3×#13711yescharacteristic
475C22C3×#57511yescharacteristic
476C22C3×#57511yescharacteristic
477C4—22no—
478C4—22no—
479C4—22no—
480C4—41no—
481C4—44no—
482C4—41no—
483C4—44no—
484C4—81no—
485C4—81no—
486C4—81no—
487C4—81no—
488C4C3×#53612yes—
489C4C3×#53612yes—
490C3#2964511yescharacteristic, direct product factor, Sylow
491C2—42no—
492C2C3×#261411yescharacteristic
493C2C3×#300211yescharacteristic
494C2C3×#300711yescharacteristic
495C2C3×#323111yescharacteristic
496C2C3×#323311yescharacteristic
497C2C3×#323711yescharacteristic
498C2C3×#324011yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
4. C3×#2059C2semidirect
5. C3×#2058C2semidirect
6. C3×#2057C2semidirect
7. C3×#2026C2semidirect
8. C3×#2024C2semidirect
9. C3×#2024C2semidirect
12. #29645C3direct (N × H)
49. #2059C2×C3semidirect
50. #2058C2×C3semidirect
51. #2057C2×C3semidirect
52. #2026C2×C3semidirect
53. #2024C2×C3semidirect
54. #2024C2×C3semidirect
490. C3#29645direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5332
C2×C3×#4852
C3×#28552
C3×#20592
C3×#20582
C3×#20572
C3×#20262
C3×#20242
C3×#18252
C3×#18172
#296453