Finite Group Databasebeta

← C3×#25686 #10644293

Subgroups — C3×#25686 #10644293

Order 768.

968 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#541C211yesmaximal, characteristic
2C2×C3×#501C211yesmaximal, characteristic, complemented
3C2×C3×#487C211yesmaximal, characteristic, complemented
4C3×#2628C211yesmaximal, characteristic
5C3×#2036C212yesmaximal
6C3×#2035C212yesmaximal
7C3×#1991C211yesmaximal, characteristic, complemented
8C3×#1960C211yesmaximal, characteristic, complemented
9C3×#1865C212yesmaximal, complemented
10C3×#1863C212yesmaximal, complemented
11C3×#1750C211yesmaximal, characteristic
12#25686C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#541—24no—
14C3×#541—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#126—21no—
18C2×C3×#126—21no—
19C2×C3×#125—21no—
20C2×C3×#125—21no—
21C2×C3×#125C2211yescharacteristic
22C2×C3×C22⋊Q8C2211yescharacteristic
23C2×C3×C22⋊Q8C2211yescharacteristic
24C2×C3×C4×Q8C2211yescharacteristic
25C2×C3×C4×Q8C2211yescharacteristic
26C2×C3×#117C2211yescharacteristic
27C3×#501—24no—
28C3×#487—24no—
29C2×C3×#107C2212yes—
30C2×C3×#106C2212yes—
31C2×C3×#103C2212yes—
32C2×C3×#103C2212yes—
33C2×C3×#103C2212yes—
34C2×C3×#103C2212yes—
35C2×C3×C22⋊C8C2212yes—
36C2×C3×#97C2211yescharacteristic
37C2×C3×C4×C8C2211yescharacteristic
38C3×#400—21no—
39C3×#399—21no—
40C3×#395—21no—
41C3×#393—21no—
42C3×#389C2212yes—
43C3×#382C2211yescharacteristic
44C3×#380C2212yes—
45C3×#340C2212yes—
46C3×#340C2212yes—
47C3×#336C2212yes—
48C2×#541C2×C311yescharacteristic
49C2×#501C2×C311yescharacteristic, complemented
50C2×#487C2×C311yescharacteristic, complemented
51#2628C2×C311yescharacteristic
52#2036C2×C312yes—
53#2035C2×C312yes—
54#1991C2×C311yescharacteristic, complemented
55#1960C2×C311yescharacteristic, complemented
56#1865C2×C312yescomplemented
57#1863C2×C312yescomplemented
58#1750C2×C311yescharacteristic
59C22×C3×Q8—21no—
60C22×C3×Q8C2311yescharacteristic
61C22×C3×Q8C2311yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C22×C3×C8C2312yes—
64C22×C3×C8C2312yes—
65C3×C4⋊Q8—22no—
66C3×C4⋊Q8—22no—
67C3×C4⋊Q8—22no—
68C3×C4⋊Q8D414yes—
69C3×#126—28no—
70C3×#126—28no—
71C3×#125—28no—
72C3×#125—28no—
73C3×#125—22no—
74C3×#125D414yes—
75C3×C22⋊Q8—22no—
76C3×C22⋊Q8—22no—
77C3×C22⋊Q8—22no—
78C3×C22⋊Q8—22no—
79C3×C4×Q8—22no—
80C3×C4×Q8—22no—
81C3×C4×Q8—22no—
82C3×C4×Q8—22no—
83C3×#117—22no—
84C3×#117—22no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
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⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—22no—
95C2×C3×C4⋊C4D411yescharacteristic
96C2×C3×C4⋊C4D411yescharacteristic
97C2×C3×C4⋊C4C2312yes—
98C2×C3×C4⋊C4C2312yes—
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—22no—
104C2×C3×C22⋊C4D411yescharacteristic
105C2×C3×C22⋊C4D411yescharacteristic
106C2×C3×C42—21no—
107C2×C3×C42—21no—
108C2×C3×C42C2311yescharacteristic
109C2×C3×C42C2311yescharacteristic
110C3×#107—24no—
111C3×#106—24no—
112C3×#103—24no—
113C3×#103—24no—
114C3×#103—24no—
115C3×#103—24no—
116C3×C22⋊C8—24no—
117C3×#97—22no—
118C3×C4×C8—22no—
119C3×#95—22no—
120C3×#95—21no—
121C3×#95—21no—
122C3×#95—22no—
123#541—24no—
124#541—24no—
125C2×C4⋊Q8C22×C311yescharacteristic
126C2×C4⋊Q8C22×C311yescharacteristic
127C2×#126—21no—
128C2×#126—21no—
129C2×#125—21no—
130C2×#125—21no—
131C2×#125C22×C311yescharacteristic
132C2×C22⋊Q8C22×C311yescharacteristic
133C2×C22⋊Q8C22×C311yescharacteristic
134C2×C4×Q8C22×C311yescharacteristic
135C2×C4×Q8C22×C311yescharacteristic
136C2×#117C22×C311yescharacteristic
137#501—24no—
138#487—24no—
139C2×#107C22×C312yes—
140C2×#106C22×C312yes—
141C2×#103C22×C312yes—
142C2×#103C22×C312yes—
143C2×#103C22×C312yes—
144C2×#103C22×C312yes—
145C2×C22⋊C8C22×C312yes—
146C2×#97C22×C311yescharacteristic
147C2×C4×C8C22×C311yescharacteristic
148#400—21no—
149#399—21no—
150#395—21no—
151#393—21no—
152#389C22×C312yes—
153#382C22×C311yescharacteristic
154#380C22×C312yes—
155#340C22×C312yes—
156#340C22×C312yes—
157#336C22×C312yes—
158C24×C3C2×D411yescharacteristic
159C2×C3×Q8—21no—
160C2×C3×Q8—21no—
161C2×C3×Q8—22no—
162C2×C3×Q8—22no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8—42no—
166C2×C3×Q8—82no—
167C2×C3×Q8C2×D412yes—
168C2×C3×Q8C2×D412yes—
169C22×C3×C4—21no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2411yescharacteristic
185C2×C3×C8—22no—
186C2×C3×C8—22no—
187C2×C3×C8—22no—
188C2×C3×C8—22no—
189C2×C3×C8—22no—
190C2×C3×C8#4114yes—
191C3×C4⋊C4—24no—
192C3×C4⋊C4—24no—
193C3×C4⋊C4—24no—
194C3×C4⋊C4—24no—
195C3×C4⋊C4—24no—
196C3×C4⋊C4—24no—
197C3×C4⋊C4—24no—
198C3×C4⋊C4—24no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—42no—
208C3×C4⋊C4—82no—
209C3×C4⋊C4C2×D412yes—
210C3×C4⋊C4C2×D412yes—
211C3×C22⋊C4—24no—
212C3×C22⋊C4—24no—
213C3×C22⋊C4—22no—
214C3×C22⋊C4—22no—
215C3×C22⋊C4—82no—
216C3×C42—24no—
217C3×C42—24no—
218C3×C42—21no—
219C3×C42—21no—
220C3×C42C2×D412yes—
221C3×C42C2×D412yes—
222C22×Q8—21no—
223C22×Q8C23×C311yescharacteristic
224C22×Q8C23×C311yescharacteristic
225C23×C4C23×C311yescharacteristic
226C22×C8C23×C312yes—
227C22×C8C23×C312yes—
228C4⋊Q8—22no—
229C4⋊Q8—22no—
230C4⋊Q8—22no—
231C4⋊Q8C3×D414yes—
232#126—28no—
233#126—28no—
234#125—28no—
235#125—28no—
236#125—22no—
237#125C3×D414yes—
238C22⋊Q8—22no—
239C22⋊Q8—22no—
240C22⋊Q8—22no—
241C22⋊Q8—22no—
242C4×Q8—22no—
243C4×Q8—22no—
244C4×Q8—22no—
245C4×Q8—22no—
246#117—22no—
247#117—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4C3×D411yescharacteristic
259C2×C4⋊C4C3×D411yescharacteristic
260C2×C4⋊C4C23×C312yes—
261C2×C4⋊C4C23×C312yes—
262C2×C4⋊C4C23×C311yescharacteristic
263C2×C4⋊C4C23×C311yescharacteristic
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4C3×D411yescharacteristic
268C2×C22⋊C4C3×D411yescharacteristic
269C2×C42—21no—
270C2×C42—21no—
271C2×C42C23×C311yescharacteristic
272C2×C42C23×C311yescharacteristic
273#107—24no—
274#106—24no—
275#103—24no—
276#103—24no—
277#103—24no—
278#103—24no—
279C22⋊C8—24no—
280#97—22no—
281C4×C8—22no—
282#95—22no—
283#95—21no—
284#95—21no—
285#95—22no—
286C23×C3—21no—
287C23×C3—21no—
288C23×C3—21no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3C22×D411yescharacteristic
292C3×Q8—24no—
293C3×Q8—24no—
294C3×Q8—42no—
295C3×Q8—42no—
296C3×Q8—84no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4#12012yes—
325C2×C3×C4#12012yes—
326C2×C3×C4C22×D411yescharacteristic
327C2×C3×C4C22×D411yescharacteristic
328C2×C3×C4C2×#4112yes—
329C2×C3×C42−1+412yes—
330C3×C8—24no—
331C3×C8—42no—
332C24C2×C3×D411yescharacteristic
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—42no—
340C2×Q8—82no—
341C2×Q8C2×C3×D412yes—
342C2×Q8C2×C3×D412yes—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4C2×C3×D411yescharacteristic
354C22×C4C2×C3×D411yescharacteristic
355C22×C4C2×C3×D411yescharacteristic
356C22×C4C2×C3×D411yescharacteristic
357C22×C4C2×C3×D411yescharacteristic
358C22×C4C24×C311yescharacteristic
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8C3×#4114yes—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—42no—
382C4⋊C4—82no—
383C4⋊C4C2×C3×D412yes—
384C4⋊C4C2×C3×D412yes—
385C22⋊C4—24no—
386C22⋊C4—24no—
387C22⋊C4—22no—
388C22⋊C4—22no—
389C22⋊C4—82no—
390C42—24no—
391C42—24no—
392C42—21no—
393C42—21no—
394C42C2×C3×D412yes—
395C42C2×C3×D412yes—
396C22×C3—42no—
397C22×C3—41no—
398C22×C3—42no—
399C22×C3—41no—
400C22×C3—41no—
401C22×C3C2×#12011yescharacteristic
402C22×C3#53211yescharacteristic
403C22×C3#53611yescharacteristic
404C22×C3#57712yes—
405C22×C3#57812yes—
406C3×C4—22no—
407C3×C4—22no—
408C3×C4—41no—
409C3×C4—42no—
410C3×C4—42no—
411C3×C4—41no—
412C3×C4—42no—
413C3×C4—41no—
414C3×C4—42no—
415C3×C4—82no—
416C3×C4—82no—
417C3×C4#54712yes—
418C3×C4#54812yes—
419C23—22no—
420C23—22no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23C22×C3×D411yescharacteristic
425Q8—24no—
426Q8—24no—
427Q8—42no—
428Q8—42no—
429Q8—84no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4C3×#12012yes—
458C2×C4C3×#12012yes—
459C2×C4C22×C3×D411yescharacteristic
460C2×C4C22×C3×D411yescharacteristic
461C2×C4C2×C3×#4112yes—
462C2×C4C3×2−1+412yes—
463C8—24no—
464C8—42no—
465C2×C3—81no—
466C2×C3#243211yescharacteristic
467C2×C3#301011yescharacteristic
468C2×C3#301111yescharacteristic
469C2×C3#317812yes—
470C2×C3#324212yes—
471C22—42no—
472C22—41no—
473C22—42no—
474C22—41no—
475C22—41no—
476C22C2×C3×#12011yescharacteristic
477C22C3×#53211yescharacteristic
478C22C3×#53611yescharacteristic
479C22C3×#57712yes—
480C22C3×#57812yes—
481C4—22no—
482C4—22no—
483C4—41no—
484C4—42no—
485C4—42no—
486C4—41no—
487C4—42no—
488C4—41no—
489C4—42no—
490C4—82no—
491C4—82no—
492C4C3×#54712yes—
493C4C3×#54812yes—
494C3#2568611yescharacteristic, direct product factor, Sylow
495C2—81no—
496C2C3×#243211yescharacteristic
497C2C3×#301011yescharacteristic
498C2C3×#301111yescharacteristic
499C2C3×#317812yes—
500C2C3×#324212yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#501C2semidirect
3. C2×C3×#487C2semidirect
7. C3×#1991C2semidirect
8. C3×#1960C2semidirect
9. C3×#1865C2semidirect
10. C3×#1863C2semidirect
12. #25686C3direct (N × H)
49. C2×#501C2×C3semidirect
50. C2×#487C2×C3semidirect
54. #1991C2×C3semidirect
55. #1960C2×C3semidirect
56. #1865C2×C3semidirect
57. #1863C2×C3semidirect
494. C3#25686direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5412
C2×C3×#5012
C2×C3×#4872
C3×#26282
C3×#20362
C3×#20352
C3×#19912
C3×#19602
C3×#18652
C3×#18632
C3×#17502
#256863