Finite Group Databasebeta

← C3×#35321 #10653928

Subgroups — C3×#35321 #10653928

Order 768.

1558 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2863C212yesmaximal, complemented
2C3×#2855C212yesmaximal
3C3×#2647C211yesmaximal, characteristic, complemented
4C3×#2642C211yesmaximal, characteristic, complemented
5C3×#2629C211yesmaximal, characteristic
6C3×#2621C211yesmaximal, characteristic
7C3×#2585C214yesmaximal, complemented
8C3×#2573C214yesmaximal
9C3×#2524C214yesmaximal, complemented
10C3×#2515C214yesmaximal
11C3×#2323C212yesmaximal
12C3×#2320C212yesmaximal, complemented
13C3×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
14C3×C4×#125C211yesmaximal, characteristic, complemented
15C3×C4×#117C211yesmaximal, characteristic
16#35321C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#125C2211yescharacteristic
19C2×C3×C22⋊Q8C2212yes—
20C2×C3×C4×Q8C2212yes—
21C2×C3×#117C2211yescharacteristic
22C2×C3×#117C2211yescharacteristic
23C22×C3×C42C2211yescharacteristic
24C3×#400C2214yes—
25C3×#400C2214yes—
26C3×#400C2214yes—
27C3×#398C2214yes—
28C3×#398C2214yes—
29C3×#398C2214yes—
30C3×#397C2214yes—
31C3×#396C2214yes—
32C3×#391C2212yes—
33C3×#391C2212yes—
34C3×#391C2212yes—
35C3×#391C2212yes—
36C3×#389C2212yes—
37C3×#389C2212yes—
38C3×#389C2212yes—
39C3×#389C2212yes—
40C3×#389C2214yes—
41C3×#389C2214yes—
42C3×#388C2214yes—
43C3×#388C2214yes—
44C3×#387C2214yes—
45C3×#387C2214yes—
46C3×#387C2214yes—
47C3×#387C2214yes—
48C3×#387C2214yes—
49C3×#387C2214yes—
50C3×#387C2214yes—
51C3×#387C2214yes—
52C3×#387C2214yes—
53C3×#385C2214yes—
54C3×#384C2211yescharacteristic
55C3×#383C2214yes—
56C3×#383C2214yes—
57C3×#382C2211yescharacteristic
58C3×#382C2214yes—
59C3×#381C2214yes—
60C3×#380C2212yes—
61C3×C4×C4⋊C4C2214yes—
62C3×C4×C4⋊C4C2212yes—
63C3×C4×C4⋊C4C2212yes—
64C3×C4×C4⋊C4C2214yes—
65C3×C4×C4⋊C4C2212yes—
66C3×C4×C22⋊C4C2212yes—
67C3×C4×C22⋊C4C2212yes—
68C3×#376C2211yescharacteristic
69C3×#376C2211yescharacteristic
70C3×C43C2211yescharacteristic
71C3×C43C2211yescharacteristic
72#2863C2×C312yescomplemented
73#2855C2×C312yes—
74#2647C2×C311yescharacteristic, complemented
75#2642C2×C311yescharacteristic, complemented
76#2629C2×C311yescharacteristic
77#2621C2×C311yescharacteristic
78#2585C2×C314yescomplemented
79#2573C2×C314yes—
80#2524C2×C314yescomplemented
81#2515C2×C314yes—
82#2323C2×C312yes—
83#2320C2×C312yescomplemented
84C4×C4⋊Q8C2×C311yescharacteristic, complemented
85C4×#125C2×C311yescharacteristic, complemented
86C4×#117C2×C311yescharacteristic
87C22×C3×Q8C2312yes—
88C23×C3×C4C2312yes—
89C23×C3×C4C2311yescharacteristic
90C3×C4⋊Q8—24no—
91C3×#125—24no—
92C3×C22⋊Q8—28no—
93C3×C4×Q8—28no—
94C3×#117—24no—
95C3×#117—24no—
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2312yes—
99C2×C3×C4⋊C4C2312yes—
100C2×C3×C4⋊C4C2312yes—
101C2×C3×C4⋊C4C2314yes—
102C2×C3×C4⋊C4C2314yes—
103C2×C3×C4⋊C4C2314yes—
104C2×C3×C4⋊C4C2312yes—
105C2×C3×C4⋊C4C2314yes—
106C2×C3×C4⋊C4C2314yes—
107C2×C3×C4⋊C4C2314yes—
108C2×C3×C4⋊C4C2314yes—
109C2×C3×C4⋊C4C2314yes—
110C2×C3×C4⋊C4C2314yes—
111C2×C3×C22⋊C4C2311yescharacteristic
112C2×C3×C22⋊C4C2311yescharacteristic
113C2×C3×C22⋊C4C2312yes—
114C2×C3×C22⋊C4C2314yes—
115C2×C3×C22⋊C4C2314yes—
116C2×C3×C42—24no—
117C2×C3×C42C2311yescharacteristic
118C2×C3×C42C2311yescharacteristic
119C2×C3×C42C2311yescharacteristic
120C2×C3×C42C2312yes—
121C2×C3×C42C2312yes—
122C2×C3×C42C2311yescharacteristic
123C2×C3×C42C2314yes—
124C2×C3×C42C2312yes—
125C2×C3×C42C2312yes—
126C2×C3×C42C2312yes—
127C2×C3×C42C2314yes—
128C2×C3×C42C2311yescharacteristic
129C2×C3×C42C2311yescharacteristic
130C3×#95C2312yes—
131C3×#95C2312yes—
132C3×#95C2312yes—
133C3×#95C2312yes—
134C3×#95C2314yes—
135C3×#95C2314yes—
136C3×#95C2314yes—
137C3×#95C2314yes—
138C3×#95C2314yes—
139C3×#95C2314yes—
140C3×#95C2314yes—
141C3×#95C2314yes—
142C3×#95C2314yes—
143C3×#95C2314yes—
144C3×#95C2314yes—
145C3×#95C2314yes—
146C3×#95C2314yes—
147C3×#95C2314yes—
148C3×#95C2314yes—
149C2×C4⋊Q8C22×C311yescharacteristic
150C2×#125C22×C311yescharacteristic
151C2×C22⋊Q8C22×C312yes—
152C2×C4×Q8C22×C312yes—
153C2×#117C22×C311yescharacteristic
154C2×#117C22×C311yescharacteristic
155C22×C42C22×C311yescharacteristic
156#400C22×C314yes—
157#400C22×C314yes—
158#400C22×C314yes—
159#398C22×C314yes—
160#398C22×C314yes—
161#398C22×C314yes—
162#397C22×C314yes—
163#396C22×C314yes—
164#391C22×C312yes—
165#391C22×C312yes—
166#391C22×C312yes—
167#391C22×C312yes—
168#389C22×C312yes—
169#389C22×C314yes—
170#389C22×C312yes—
171#389C22×C312yes—
172#389C22×C314yes—
173#389C22×C312yes—
174#388C22×C314yes—
175#388C22×C314yes—
176#387C22×C314yes—
177#387C22×C314yes—
178#387C22×C314yes—
179#387C22×C314yes—
180#387C22×C314yes—
181#387C22×C314yes—
182#387C22×C314yes—
183#387C22×C314yes—
184#387C22×C314yes—
185#385C22×C314yes—
186#384C22×C311yescharacteristic
187#383C22×C314yes—
188#383C22×C314yes—
189#382C22×C311yescharacteristic
190#382C22×C314yes—
191#381C22×C314yes—
192#380C22×C312yes—
193C4×C4⋊C4C22×C314yes—
194C4×C4⋊C4C22×C312yes—
195C4×C4⋊C4C22×C314yes—
196C4×C4⋊C4C22×C312yes—
197C4×C4⋊C4C22×C312yes—
198C4×C22⋊C4C22×C312yes—
199C4×C22⋊C4C22×C312yes—
200#376C22×C311yescharacteristic
201#376C22×C311yescharacteristic
202C43C22×C311yescharacteristic
203C43C22×C311yescharacteristic
204C24×C3C2411yescharacteristic
205C2×C3×Q8—24no—
206C2×C3×Q8—24no—
207C2×C3×Q8—24no—
208C22×C3×C4—22no—
209C22×C3×C4—22no—
210C22×C3×C4—24no—
211C22×C3×C4—24no—
212C22×C3×C4—42no—
213C22×C3×C4#4114yes—
214C22×C3×C4C2411yescharacteristic
215C22×C3×C4C2412yes—
216C22×C3×C4C2412yes—
217C22×C3×C4C2414yes—
218C22×C3×C4C2414yes—
219C22×C3×C4C2414yes—
220C22×C3×C4C2411yescharacteristic
221C22×C3×C4C2414yes—
222C22×C3×C4C2411yescharacteristic
223C22×C3×C4C2412yes—
224C22×C3×C4C2411yescharacteristic
225C22×C3×C4C2411yescharacteristic
226C22×C3×C4C2412yes—
227C22×C3×C4C2411yescharacteristic
228C3×C4⋊C4—22no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4—24no—
231C3×C4⋊C4—24no—
232C3×C4⋊C4—28no—
233C3×C4⋊C4—24no—
234C3×C4⋊C4—28no—
235C3×C4⋊C4—28no—
236C3×C4⋊C4—24no—
237C3×C4⋊C4—28no—
238C3×C4⋊C4—28no—
239C3×C4⋊C4—44no—
240C3×C4⋊C4—44no—
241C3×C4⋊C4—44no—
242C3×C4⋊C4—44no—
243C3×C22⋊C4—28no—
244C3×C22⋊C4—22no—
245C3×C22⋊C4—22no—
246C3×C22⋊C4—24no—
247C3×C22⋊C4—44no—
248C3×C42—22no—
249C3×C42—24no—
250C3×C42—22no—
251C3×C42—22no—
252C3×C42—22no—
253C3×C42—28no—
254C3×C42—24no—
255C3×C42—24no—
256C3×C42—42no—
257C3×C42—44no—
258C3×C42—42no—
259C3×C42#4114yes—
260C3×C42#4114yes—
261C22×Q8C23×C312yes—
262C23×C4C23×C312yes—
263C23×C4C23×C311yescharacteristic
264C4⋊Q8—24no—
265#125—24no—
266C22⋊Q8—28no—
267C4×Q8—28no—
268#117—24no—
269#117—24no—
270C2×C4⋊C4C23×C311yescharacteristic
271C2×C4⋊C4C23×C311yescharacteristic
272C2×C4⋊C4C23×C312yes—
273C2×C4⋊C4C23×C312yes—
274C2×C4⋊C4C23×C314yes—
275C2×C4⋊C4C23×C312yes—
276C2×C4⋊C4C23×C314yes—
277C2×C4⋊C4C23×C312yes—
278C2×C4⋊C4C23×C314yes—
279C2×C4⋊C4C23×C314yes—
280C2×C4⋊C4C23×C314yes—
281C2×C4⋊C4C23×C314yes—
282C2×C4⋊C4C23×C314yes—
283C2×C4⋊C4C23×C314yes—
284C2×C4⋊C4C23×C314yes—
285C2×C22⋊C4C23×C311yescharacteristic
286C2×C22⋊C4C23×C311yescharacteristic
287C2×C22⋊C4C23×C312yes—
288C2×C22⋊C4C23×C314yes—
289C2×C22⋊C4C23×C314yes—
290C2×C42—24no—
291C2×C42C23×C311yescharacteristic
292C2×C42C23×C311yescharacteristic
293C2×C42C23×C314yes—
294C2×C42C23×C311yescharacteristic
295C2×C42C23×C312yes—
296C2×C42C23×C312yes—
297C2×C42C23×C311yescharacteristic
298C2×C42C23×C312yes—
299C2×C42C23×C312yes—
300C2×C42C23×C312yes—
301C2×C42C23×C314yes—
302C2×C42C23×C311yescharacteristic
303C2×C42C23×C311yescharacteristic
304#95C23×C312yes—
305#95C23×C312yes—
306#95C23×C312yes—
307#95C23×C312yes—
308#95C23×C314yes—
309#95C23×C314yes—
310#95C23×C314yes—
311#95C23×C314yes—
312#95C23×C314yes—
313#95C23×C314yes—
314#95C23×C314yes—
315#95C23×C314yes—
316#95C23×C314yes—
317#95C23×C314yes—
318#95C23×C314yes—
319#95C23×C314yes—
320#95C23×C314yes—
321#95C23×C314yes—
322#95C23×C314yes—
323C23×C3—21no—
324C23×C3—21no—
325C23×C3—22no—
326C23×C3—22no—
327C23×C3C2×#4112yes—
328C23×C3C2511yescharacteristic
329C3×Q8—48no—
330C2×C3×C4—22no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—28no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—24no—
344C2×C3×C4—24no—
345C2×C3×C4—24no—
346C2×C3×C4—24no—
347C2×C3×C4—24no—
348C2×C3×C4—24no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—24no—
352C2×C3×C4—24no—
353C2×C3×C4—22no—
354C2×C3×C4—24no—
355C2×C3×C4—22no—
356C2×C3×C4—24no—
357C2×C3×C4—22no—
358C2×C3×C4—24no—
359C2×C3×C4—24no—
360C2×C3×C4—22no—
361C2×C3×C4—48no—
362C2×C3×C4C2×#4112yes—
363C2×C3×C4C2×#4112yes—
364C2×C3×C4C2×#4114yes—
365C2×C3×C4C2×#4112yes—
366C2×C3×C4C2×#4112yes—
367C2×C3×C4C2×#4112yes—
368C2×C3×C4C2×#4112yes—
369C2×C3×C42−1+414yes—
370C24C24×C311yescharacteristic
371C2×Q8—24no—
372C2×Q8—24no—
373C2×Q8—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—42no—
379C22×C4C3×#4114yes—
380C22×C4C24×C311yescharacteristic
381C22×C4C24×C312yes—
382C22×C4C24×C312yes—
383C22×C4C24×C314yes—
384C22×C4C24×C314yes—
385C22×C4C24×C314yes—
386C22×C4C24×C311yescharacteristic
387C22×C4C24×C314yes—
388C22×C4C24×C311yescharacteristic
389C22×C4C24×C312yes—
390C22×C4C24×C311yescharacteristic
391C22×C4C24×C311yescharacteristic
392C22×C4C24×C312yes—
393C22×C4C24×C311yescharacteristic
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—28no—
399C4⋊C4—24no—
400C4⋊C4—28no—
401C4⋊C4—28no—
402C4⋊C4—24no—
403C4⋊C4—28no—
404C4⋊C4—28no—
405C4⋊C4—44no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C22⋊C4—28no—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C22⋊C4—24no—
413C22⋊C4—44no—
414C42—22no—
415C42—24no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—28no—
420C42—24no—
421C42—24no—
422C42—42no—
423C42—44no—
424C42—42no—
425C42C3×#4114yes—
426C42C3×#4114yes—
427C22×C3—22no—
428C22×C3—24no—
429C22×C3—42no—
430C22×C3—41no—
431C22×C3—41no—
432C22×C3C22×#4111yescharacteristic
433C22×C3C22×#4111yescharacteristic
434C22×C3C22×#4111yescharacteristic
435C22×C3C2×2−1+412yes—
436C22×C3#58512yes—
437C3×C4—22no—
438C3×C4—24no—
439C3×C4—24no—
440C3×C4—22no—
441C3×C4—22no—
442C3×C4—44no—
443C3×C4—42no—
444C3×C4—44no—
445C3×C4—44no—
446C3×C4—42no—
447C3×C4—41no—
448C3×C4—41no—
449C3×C4—44no—
450C3×C4#52914yes—
451C23—21no—
452C23—21no—
453C23—22no—
454C23—22no—
455C23C2×C3×#4112yes—
456C23C25×C311yescharacteristic
457Q8—48no—
458C2×C4—22no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—28no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—48no—
490C2×C4C2×C3×#4112yes—
491C2×C4C2×C3×#4112yes—
492C2×C4C2×C3×#4114yes—
493C2×C4C2×C3×#4112yes—
494C2×C4C2×C3×#4112yes—
495C2×C4C2×C3×#4112yes—
496C2×C4C2×C3×#4112yes—
497C2×C4C3×2−1+414yes—
498C2×C3—42no—
499C2×C3C2×#52911yescharacteristic
500C2×C3#346112yes—
501C2×C3#349512yes—
502C2×C3#350911yescharacteristic
503C2×C3#351211yescharacteristic
504C22—22no—
505C22—24no—
506C22—42no—
507C22—41no—
508C22—41no—
509C22C22×C3×#4111yescharacteristic
510C22C22×C3×#4111yescharacteristic
511C22C22×C3×#4111yescharacteristic
512C22C2×C3×2−1+412yes—
513C22C3×#58512yes—
514C4—22no—
515C4—24no—
516C4—24no—
517C4—22no—
518C4—22no—
519C4—44no—
520C4—42no—
521C4—44no—
522C4—42no—
523C4—44no—
524C4—41no—
525C4—41no—
526C4—44no—
527C4C3×#52914yes—
528C3#3532111yescharacteristic, direct product factor, Sylow
529C2—42no—
530C2C2×C3×#52911yescharacteristic
531C2C3×#346112yes—
532C2C3×#349512yes—
533C2C3×#350911yescharacteristic
534C2C3×#351211yescharacteristic
subgroupG/H
center, socleC23×C3C25
commutator, Frattini, cosocleC23C25×C3
prime radicalC24×C3C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#2863C2semidirect
3. C3×#2647C2semidirect
4. C3×#2642C2semidirect
7. C3×#2585C2semidirect
9. C3×#2524C2semidirect
12. C3×#2320C2semidirect
13. C3×C4×C4⋊Q8C2semidirect
14. C3×C4×#125C2semidirect
16. #35321C3direct (N × H)
72. #2863C2×C3semidirect
74. #2647C2×C3semidirect
75. #2642C2×C3semidirect
78. #2585C2×C3semidirect
80. #2524C2×C3semidirect
83. #2320C2×C3semidirect
84. C4×C4⋊Q8C2×C3semidirect
85. C4×#125C2×C3semidirect
528. C3#35321direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#28632
C3×#28552
C3×#26472
C3×#26422
C3×#26292
C3×#26212
C3×#25852
C3×#25732
C3×#25242
C3×#25152
C3×#23232
C3×#23202
C3×C4×C4⋊Q82
C3×C4×#1252
C3×C4×#1172
#353213