Finite Group Databasebeta

← C3×#35737 #10654344

Subgroups — C3×#35737 #10654344

Order 768.

2990 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×Q8C211yesmaximal, characteristic, complemented
2C2×C3×#548C212yesmaximal, complemented
3C2×C3×#520C211yesmaximal, characteristic, complemented
4C3×#2621C213yesmaximal, complemented
5C3×#2547C213yesmaximal, complemented
6C3×#2535C213yesmaximal, complemented
7C3×#2427C216yesmaximal, complemented
8C3×#2417C216yesmaximal, complemented
9C3×#2392C213yesmaximal, complemented
10C3×#2351C211yesmaximal, characteristic, complemented
11C3×Q8×C22⋊C4C212yesmaximal, complemented
12#35737C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×C3×Q8C2212yes—
15C3×D4×Q8—28no—
16C3×#548—216no—
17C2×C3×C4⋊Q8C2213yes—
18C2×C3×#124C2216yes—
19C2×C3×C22⋊Q8C2216yes—
20C2×C3×C22⋊Q8C2216yes—
21C2×C3×#121C2216yes—
22C3×#520—28no—
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×Q8C2212yes—
25C2×C3×C4×Q8C2211yescharacteristic
26C2×C3×C4×D4C2213yes—
27C2×C3×C4×D4C2213yes—
28C2×C3×C4×D4C2216yes—
29C2×C3×C4×D4C2213yes—
30C2×C3×#117C2213yes—
31C22×C3×C4⋊C4C2213yes—
32C3×#398C2213yes—
33C3×#398C2216yes—
34C3×#397C2216yes—
35C3×#397C2216yes—
36C3×#397C2213yes—
37C3×#396C2216yes—
38C3×#394C2211yescharacteristic
39C3×#394C2216yes—
40C3×#392C2213yes—
41C3×#391C2216yes—
42C3×#391C2216yes—
43C3×#389C2216yes—
44C3×#386C22112yes—
45C3×#382C2213yes—
46C3×#380C22112yes—
47C3×#380C2216yes—
48C3×C4×C4⋊C4C2213yes—
49C3×C4×C22⋊C4C2216yes—
50C2×D4×Q8C2×C311yescharacteristic, complemented
51C2×#548C2×C312yescomplemented
52C2×#520C2×C311yescharacteristic, complemented
53#2621C2×C313yescomplemented
54#2547C2×C313yescomplemented
55#2535C2×C313yescomplemented
56#2427C2×C316yescomplemented
57#2417C2×C316yescomplemented
58#2392C2×C313yescomplemented
59#2351C2×C311yescharacteristic, complemented
60Q8×C22⋊C4C2×C312yescomplemented
61C2×C3×#41—24no—
62C2×C3×#41—24no—
63C2×C3×#41D418yescomplemented
64C22×C3×Q8—28no—
65C22×C3×Q8—28no—
66C22×C3×Q8—28no—
67C22×C3×Q8C2316yes—
68C22×C3×Q8C2311yescharacteristic
69C22×C3×D4C2311yescharacteristic
70C22×C3×D4C2316yes—
71C22×C3×D4C2313yes—
72C23×C3×C4C2316yes—
73C23×C3×C4C2313yes—
74C23×C3×C4C2316yes—
75C3×C4⋊Q8—212no—
76C3×#124—224no—
77C3×C22⋊Q8—224no—
78C3×C22⋊Q8—224no—
79C3×#121—224no—
80C3×C4×Q8—24no—
81C3×C4×Q8—28no—
82C3×C4×Q8D418yescomplemented
83C3×C4×D4—224no—
84C3×C4×D4—212no—
85C3×C4×D4—212no—
86C3×C4×D4—212no—
87C3×#117—212no—
88C2×C3×C4⋊C4—212no—
89C2×C3×C4⋊C4C2313yes—
90C2×C3×C4⋊C4C2316yes—
91C2×C3×C4⋊C4C2316yes—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2313yes—
94C2×C3×C4⋊C4C2316yes—
95C2×C3×C4⋊C4C2313yes—
96C2×C3×C4⋊C4C2316yes—
97C2×C3×C4⋊C4C2313yes—
98C2×C3×C22⋊C4C2312yes—
99C2×C3×C22⋊C4C2316yes—
100C2×C3×C22⋊C4C23112yes—
101C2×C3×C22⋊C4C2316yes—
102C2×C3×C22⋊C4C2313yes—
103C2×C3×C22⋊C4C2313yes—
104C2×C3×C22⋊C4C2312yes—
105C2×C3×C22⋊C4C2316yes—
106C2×C3×C42C2313yes—
107C2×C3×C42C2316yes—
108C2×C3×C42C2313yes—
109C3×#95C2316yes—
110C3×#95C2316yes—
111C3×#95C2311yescharacteristic
112C3×#95C2313yes—
113C3×#95C23112yes—
114C3×#95C2316yes—
115C22×#41C22×C311yescharacteristic
116C23×Q8C22×C312yes—
117D4×Q8—28no—
118#548—216no—
119C2×C4⋊Q8C22×C313yes—
120C2×#124C22×C316yes—
121C2×C22⋊Q8C22×C316yes—
122C2×C22⋊Q8C22×C316yes—
123C2×#121C22×C316yes—
124#520—28no—
125C2×C4×Q8C22×C311yescharacteristic
126C2×C4×Q8C22×C312yes—
127C2×C4×Q8C22×C311yescharacteristic
128C2×C4×D4C22×C316yes—
129C2×C4×D4C22×C313yes—
130C2×C4×D4C22×C313yes—
131C2×C4×D4C22×C313yes—
132C2×#117C22×C313yes—
133C22×C4⋊C4C22×C313yes—
134#398C22×C313yes—
135#398C22×C316yes—
136#397C22×C316yes—
137#397C22×C316yes—
138#397C22×C313yes—
139#396C22×C316yes—
140#394C22×C311yescharacteristic
141#394C22×C316yes—
142#392C22×C313yes—
143#391C22×C316yes—
144#391C22×C316yes—
145#389C22×C316yes—
146#386C22×C3112yes—
147#382C22×C313yes—
148#380C22×C316yes—
149#380C22×C3112yes—
150C4×C4⋊C4C22×C313yes—
151C4×C22⋊C4C22×C316yes—
152C24×C3C2413yes—
153C24×C3C2412yes—
154C3×#41—232no—
155C2×C3×Q8—212no—
156C2×C3×Q8—212no—
157C2×C3×Q8—212no—
158C2×C3×Q8—432no—
159C2×C3×Q8C2×D414yes—
160C2×C3×Q8C2×D414yes—
161C2×C3×Q8#4114yes—
162C2×C3×D4—24no—
163C2×C3×D4—212no—
164C2×C3×D4—212no—
165C2×C3×D4—22no—
166C2×C3×D4—26no—
167C2×C3×D4—26no—
168C2×C3×D4—46no—
169C2×C3×D4C2×D4112yes—
170C22×C3×C4—26no—
171C22×C3×C4—26no—
172C22×C3×C4—212no—
173C22×C3×C4—212no—
174C22×C3×C4—212no—
175C22×C3×C4—212no—
176C22×C3×C4—412no—
177C22×C3×C4C2×D4112yes—
178C22×C3×C4C2411yescharacteristic
179C22×C3×C4C2416yes—
180C22×C3×C4C2413yes—
181C22×C3×C4C2412yes—
182C22×C3×C4C2416yes—
183C22×C3×C4C2411yescharacteristic
184C22×C3×C4C2413yes—
185C22×C3×C4C2411yescharacteristic
186C22×C3×C4C2413yes—
187C3×C4⋊C4—26no—
188C3×C4⋊C4—26no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—212no—
191C3×C4⋊C4—26no—
192C3×C4⋊C4—212no—
193C3×C4⋊C4—212no—
194C3×C4⋊C4—46no—
195C3×C4⋊C4C2×D4112yes—
196C3×C22⋊C4—212no—
197C3×C22⋊C4—224no—
198C3×C22⋊C4—24no—
199C3×C22⋊C4—26no—
200C3×C22⋊C4—26no—
201C3×C22⋊C4—24no—
202C3×C22⋊C4—212no—
203C3×C22⋊C4—46no—
204C3×C42—26no—
205C3×C42—212no—
206C3×C42C2×D4112yes—
207C2×#41—24no—
208C2×#41—24no—
209C2×#41C3×D418yescomplemented
210C22×Q8—28no—
211C22×Q8—28no—
212C22×Q8—28no—
213C22×Q8C23×C316yes—
214C22×Q8C23×C311yescharacteristic
215C22×D4C23×C311yescharacteristic
216C22×D4C23×C316yes—
217C22×D4C23×C313yes—
218C23×C4C23×C316yes—
219C23×C4C23×C313yes—
220C23×C4C23×C316yes—
221C4⋊Q8—212no—
222#124—224no—
223C22⋊Q8—224no—
224C22⋊Q8—224no—
225#121—224no—
226C4×Q8—24no—
227C4×Q8—28no—
228C4×Q8C3×D418yescomplemented
229C4×D4—224no—
230C4×D4—212no—
231C4×D4—212no—
232C4×D4—212no—
233#117—212no—
234C2×C4⋊C4—212no—
235C2×C4⋊C4C23×C313yes—
236C2×C4⋊C4C23×C313yes—
237C2×C4⋊C4C23×C311yescharacteristic
238C2×C4⋊C4C23×C316yes—
239C2×C4⋊C4C23×C313yes—
240C2×C4⋊C4C23×C316yes—
241C2×C4⋊C4C23×C316yes—
242C2×C4⋊C4C23×C316yes—
243C2×C4⋊C4C23×C313yes—
244C2×C22⋊C4C23×C313yes—
245C2×C22⋊C4C23×C313yes—
246C2×C22⋊C4C23×C312yes—
247C2×C22⋊C4C23×C316yes—
248C2×C22⋊C4C23×C316yes—
249C2×C22⋊C4C23×C3112yes—
250C2×C22⋊C4C23×C312yes—
251C2×C22⋊C4C23×C316yes—
252C2×C42C23×C313yes—
253C2×C42C23×C316yes—
254C2×C42C23×C313yes—
255#95C23×C316yes—
256#95C23×C316yes—
257#95C23×C311yescharacteristic
258#95C23×C313yes—
259#95C23×C3112yes—
260#95C23×C316yes—
261C23×C3—22no—
262C23×C3—23no—
263C23×C3—22no—
264C23×C3—22no—
265C23×C3—23no—
266C23×C3—22no—
267C23×C3—26no—
268C23×C3—22no—
269C23×C3—26no—
270C23×C3—22no—
271C23×C3C22×D416yes—
272C23×C32−1+414yes—
273C23×C3C2511yescharacteristic
274C3×Q8—424no—
275C3×Q8#121116yes—
276C3×D4—28no—
277C3×D4—224no—
278C3×D4—424no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—22no—
282C2×C3×C4—26no—
283C2×C3×C4—224no—
284C2×C3×C4—22no—
285C2×C3×C4—21no—
286C2×C3×C4—26no—
287C2×C3×C4—23no—
288C2×C3×C4—26no—
289C2×C3×C4—26no—
290C2×C3×C4—26no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—26no—
294C2×C3×C4—26no—
295C2×C3×C4—26no—
296C2×C3×C4—22no—
297C2×C3×C4—424no—
298C2×C3×C4—812no—
299C2×C3×C4C22×D412yes—
300C2×C3×C4C22×D416yes—
301C2×C3×C4C22×D412yes—
302C2×C3×C4C22×D416yes—
303C2×C3×C4C22×D416yes—
304C2×C3×C4C2×#4116yes—
305C24C24×C313yes—
306C24C24×C312yes—
307#41—232no—
308C2×Q8—212no—
309C2×Q8—212no—
310C2×Q8—212no—
311C2×Q8—432no—
312C2×Q8C2×C3×D414yes—
313C2×Q8C2×C3×D414yes—
314C2×Q8C3×#4114yes—
315C2×D4—24no—
316C2×D4—212no—
317C2×D4—212no—
318C2×D4—22no—
319C2×D4—26no—
320C2×D4—26no—
321C2×D4—46no—
322C2×D4C2×C3×D4112yes—
323C22×C4—26no—
324C22×C4—26no—
325C22×C4—212no—
326C22×C4—212no—
327C22×C4—212no—
328C22×C4—212no—
329C22×C4—412no—
330C22×C4C2×C3×D4112yes—
331C22×C4C24×C311yescharacteristic
332C22×C4C24×C316yes—
333C22×C4C24×C313yes—
334C22×C4C24×C312yes—
335C22×C4C24×C316yes—
336C22×C4C24×C311yescharacteristic
337C22×C4C24×C313yes—
338C22×C4C24×C311yescharacteristic
339C22×C4C24×C313yes—
340C4⋊C4—26no—
341C4⋊C4—26no—
342C4⋊C4—22no—
343C4⋊C4—212no—
344C4⋊C4—26no—
345C4⋊C4—212no—
346C4⋊C4—212no—
347C4⋊C4—46no—
348C4⋊C4C2×C3×D4112yes—
349C22⋊C4—212no—
350C22⋊C4—224no—
351C22⋊C4—24no—
352C22⋊C4—26no—
353C22⋊C4—26no—
354C22⋊C4—24no—
355C22⋊C4—212no—
356C22⋊C4—46no—
357C42—26no—
358C42—212no—
359C42C2×C3×D4112yes—
360C22×C3—26no—
361C22×C3—24no—
362C22×C3—24no—
363C22×C3—212no—
364C22×C3—24no—
365C22×C3—42no—
366C22×C3—43no—
367C22×C3—42no—
368C22×C3—43no—
369C22×C3—42no—
370C22×C3—42no—
371C22×C3—46no—
372C22×C3C23×D411yescharacteristic
373C22×C3C23×D411yescharacteristic
374C22×C3C22×#4111yescharacteristic
375C22×C3C2×2−1+412yes—
376C22×C3#58512yes—
377C3×C4—22no—
378C3×C4—22no—
379C3×C4—26no—
380C3×C4—46no—
381C3×C4—41no—
382C3×C4—42no—
383C3×C4—46no—
384C3×C4—43no—
385C3×C4C2×#121112yes—
386C23—22no—
387C23—23no—
388C23—22no—
389C23—22no—
390C23—23no—
391C23—22no—
392C23—26no—
393C23—22no—
394C23—26no—
395C23—22no—
396C23C22×C3×D416yes—
397C23C3×2−1+414yes—
398C23C25×C311yescharacteristic
399Q8—424no—
400Q8C3×#121116yes—
401D4—28no—
402D4—224no—
403D4—424no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—26no—
408C2×C4—224no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—26no—
412C2×C4—23no—
413C2×C4—26no—
414C2×C4—26no—
415C2×C4—26no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—26no—
419C2×C4—26no—
420C2×C4—26no—
421C2×C4—22no—
422C2×C4—424no—
423C2×C4—812no—
424C2×C4C22×C3×D412yes—
425C2×C4C22×C3×D416yes—
426C2×C4C22×C3×D412yes—
427C2×C4C22×C3×D416yes—
428C2×C4C22×C3×D416yes—
429C2×C4C2×C3×#4116yes—
430C2×C3—46no—
431C2×C3—44no—
432C2×C3C22×#12111yescharacteristic
433C2×C3#348611yescharacteristic
434C2×C3#348912yes—
435C2×C3#350011yescharacteristic
436C2×C3#351612yes—
437C22—26no—
438C22—24no—
439C22—24no—
440C22—212no—
441C22—24no—
442C22—42no—
443C22—43no—
444C22—42no—
445C22—43no—
446C22—42no—
447C22—42no—
448C22—46no—
449C22C23×C3×D411yescharacteristic
450C22C23×C3×D411yescharacteristic
451C22C22×C3×#4111yescharacteristic
452C22C2×C3×2−1+412yes—
453C22C3×#58512yes—
454C4—22no—
455C4—22no—
456C4—26no—
457C4—46no—
458C4—41no—
459C4—46no—
460C4—42no—
461C4—43no—
462C4C2×C3×#121112yes—
463C3#3573711yescharacteristic, direct product factor, Sylow
464C2—46no—
465C2—44no—
466C2C22×C3×#12111yescharacteristic
467C2C3×#348611yescharacteristic
468C2C3×#348912yes—
469C2C3×#350011yescharacteristic
470C2C3×#351612yes—
subgroupG/H
center, socleC23×C3C25
commutator, Frattini, cosocleC23C25×C3
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×D4×Q8C2semidirect
2. C2×C3×#548C2semidirect
3. C2×C3×#520C2semidirect
4. C3×#2621C2semidirect
5. C3×#2547C2semidirect
6. C3×#2535C2semidirect
7. C3×#2427C2semidirect
8. C3×#2417C2semidirect
9. C3×#2392C2semidirect
10. C3×#2351C2semidirect
11. C3×Q8×C22⋊C4C2semidirect
12. #35737C3direct (N × H)
50. C2×D4×Q8C2×C3semidirect
51. C2×#548C2×C3semidirect
52. C2×#520C2×C3semidirect
53. #2621C2×C3semidirect
54. #2547C2×C3semidirect
55. #2535C2×C3semidirect
56. #2427C2×C3semidirect
57. #2417C2×C3semidirect
58. #2392C2×C3semidirect
59. #2351C2×C3semidirect
60. Q8×C22⋊C4C2×C3semidirect
63. C2×C3×#41D4semidirect
82. C3×C4×Q8D4semidirect
209. C2×#41C3×D4semidirect
228. C4×Q8C3×D4semidirect
463. C3#35737direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D4×Q82
C2×C3×#5482
C2×C3×#5202
C3×#26212
C3×#25472
C3×#25352
C3×#24272
C3×#24172
C3×#23922
C3×#23512
C3×Q8×C22⋊C42
#357373