Finite Group Databasebeta

← C3×#37254 #10655861

Subgroups — C3×#37254 #10655861

Order 768.

1752 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2864C212yesmaximal, complemented
2C3×#2792C212yesmaximal, complemented
3C3×#2761C214yesmaximal, complemented
4C3×#2727C212yesmaximal, complemented
5C3×#2724C211yesmaximal, characteristic, complemented
6C3×#2723C214yesmaximal, complemented
7C3×#2709C212yesmaximal
8C3×#2683C211yesmaximal, characteristic, complemented
9C3×#2638C212yesmaximal, complemented
10C3×#2466C214yesmaximal, complemented
11C3×#2449C214yesmaximal, complemented
12C3×#2322C212yesmaximal, complemented
13C3×C4×#127C211yesmaximal, characteristic
14#37254C311yesmaximal, characteristic, direct product factor, Sylow
15C2×C3×#127C2211yescharacteristic
16C2×C3×#126C2214yes—
17C2×C3×#123C2212yes—
18C2×C3×C22⋊Q8C2212yes—
19C2×C3×C22⋊Q8C2214yes—
20C2×C3×C4×D4C2211yescharacteristic
21C2×C3×C4×D4C2214yes—
22C2×C3×C4×D4C2211yescharacteristic
23C22×C3×C4⋊C4C2212yes—
24C3×#399C2214yes—
25C3×#398C2212yes—
26C3×#398C2214yes—
27C3×#397C2214yes—
28C3×#397C2212yes—
29C3×#397C2214yes—
30C3×#396C2214yes—
31C3×#396C2214yes—
32C3×#395C2212yes—
33C3×#395C2214yes—
34C3×#394C2214yes—
35C3×#394C2214yes—
36C3×#394C2214yes—
37C3×#393C2212yes—
38C3×#390C2211yescharacteristic
39C3×#390C2214yes—
40C3×#390C2211yescharacteristic
41C3×#389C2212yes—
42C3×#389C2212yes—
43C3×#389C2212yes—
44C3×#389C2214yes—
45C3×#389C2212yes—
46C3×#389C2214yes—
47C3×#388C2214yes—
48C3×#388C2214yes—
49C3×#387C2214yes—
50C3×#387C2214yes—
51C3×#387C2214yes—
52C3×#387C2214yes—
53C3×#387C2214yes—
54C3×#386C2212yes—
55C3×#386C2214yes—
56C3×#386C2214yes—
57C3×#385C2214yes—
58C3×#385C2214yes—
59C3×#385C2214yes—
60C3×#385C2212yes—
61C3×#384C2211yescharacteristic
62C3×#382C2212yes—
63C3×#382C2212yes—
64C3×#381C2212yes—
65C3×#381C2212yes—
66C3×C4×C22⋊C4C2212yes—
67C3×C43C2211yescharacteristic
68#2864C2×C312yescomplemented
69#2792C2×C312yescomplemented
70#2761C2×C314yescomplemented
71#2727C2×C312yescomplemented
72#2724C2×C311yescharacteristic, complemented
73#2723C2×C314yescomplemented
74#2709C2×C312yes—
75#2683C2×C311yescharacteristic, complemented
76#2638C2×C312yescomplemented
77#2466C2×C314yescomplemented
78#2449C2×C314yescomplemented
79#2322C2×C312yescomplemented
80C4×#127C2×C311yescharacteristic
81C22×C3×Q8C2312yes—
82C22×C3×D4C2314yes—
83C22×C3×D4C2311yescharacteristic
84C22×C3×D4C2311yescharacteristic
85C23×C3×C4C2312yes—
86C23×C3×C4C2312yes—
87C23×C3×C4C2312yes—
88C23×C3×C4C2312yes—
89C3×#127—24no—
90C3×#126—48no—
91C3×#123—28no—
92C3×C22⋊Q8—48no—
93C3×C22⋊Q8—44no—
94C3×C4×D4—48no—
95C3×C4×D4—42no—
96C3×C4×D4—42no—
97C2×C3×C4⋊C4—44no—
98C2×C3×C4⋊C4C2312yes—
99C2×C3×C4⋊C4C2312yes—
100C2×C3×C4⋊C4C2314yes—
101C2×C3×C4⋊C4C2314yes—
102C2×C3×C4⋊C4C2314yes—
103C2×C3×C4⋊C4C2314yes—
104C2×C3×C4⋊C4C2312yes—
105C2×C3×C4⋊C4C2312yes—
106C2×C3×C4⋊C4C2314yes—
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C4⋊C4C2311yescharacteristic
109C2×C3×C4⋊C4C2314yes—
110C2×C3×C4⋊C4C2312yes—
111C2×C3×C4⋊C4C2314yes—
112C2×C3×C4⋊C4C2314yes—
113C2×C3×C22⋊C4C2312yes—
114C2×C3×C22⋊C4C2314yes—
115C2×C3×C22⋊C4C2314yes—
116C2×C3×C22⋊C4C2314yes—
117C2×C3×C22⋊C4C2312yes—
118C2×C3×C22⋊C4C2312yes—
119C2×C3×C22⋊C4C2314yes—
120C2×C3×C22⋊C4C2314yes—
121C2×C3×C22⋊C4C2314yes—
122C2×C3×C22⋊C4C2314yes—
123C2×C3×C22⋊C4C2314yes—
124C2×C3×C22⋊C4C2312yes—
125C2×C3×C42C2312yes—
126C2×C3×C42C2311yescharacteristic
127C2×C3×C42C2311yescharacteristic
128C2×C3×C42C2312yes—
129C2×C3×C42C2311yescharacteristic
130C2×C3×C42C2314yes—
131C3×#95C2314yes—
132C3×#95C2314yes—
133C3×#95C2314yes—
134C3×#95C2312yes—
135C3×#95C2312yes—
136C3×#95C2314yes—
137C3×#95C2314yes—
138C3×#95C2314yes—
139C3×#95C2314yes—
140C3×#95C2314yes—
141C3×#95C2314yes—
142C3×#95C2314yes—
143C2×#127C22×C311yescharacteristic
144C2×#126C22×C314yes—
145C2×#123C22×C312yes—
146C2×C22⋊Q8C22×C312yes—
147C2×C22⋊Q8C22×C314yes—
148C2×C4×D4C22×C311yescharacteristic
149C2×C4×D4C22×C311yescharacteristic
150C2×C4×D4C22×C314yes—
151C22×C4⋊C4C22×C312yes—
152#399C22×C314yes—
153#398C22×C312yes—
154#398C22×C314yes—
155#397C22×C314yes—
156#397C22×C312yes—
157#397C22×C314yes—
158#396C22×C314yes—
159#396C22×C314yes—
160#395C22×C312yes—
161#395C22×C314yes—
162#394C22×C314yes—
163#394C22×C314yes—
164#394C22×C314yes—
165#393C22×C312yes—
166#390C22×C311yescharacteristic
167#390C22×C314yes—
168#390C22×C311yescharacteristic
169#389C22×C314yes—
170#389C22×C312yes—
171#389C22×C312yes—
172#389C22×C312yes—
173#389C22×C312yes—
174#389C22×C314yes—
175#388C22×C314yes—
176#388C22×C314yes—
177#387C22×C314yes—
178#387C22×C314yes—
179#387C22×C314yes—
180#387C22×C314yes—
181#387C22×C314yes—
182#386C22×C312yes—
183#386C22×C314yes—
184#386C22×C314yes—
185#385C22×C314yes—
186#385C22×C312yes—
187#385C22×C314yes—
188#385C22×C314yes—
189#384C22×C311yescharacteristic
190#382C22×C312yes—
191#382C22×C312yes—
192#381C22×C312yes—
193#381C22×C312yes—
194C4×C22⋊C4C22×C312yes—
195C43C22×C311yescharacteristic
196C24×C3C2412yes—
197C24×C3C2412yes—
198C2×C3×Q8—44no—
199C2×C3×Q8—42no—
200C2×C3×D4—28no—
201C2×C3×D4—22no—
202C2×C3×D4—28no—
203C2×C3×D4—41no—
204C2×C3×D4—44no—
205C2×C3×D4—41no—
206C2×C3×D4—41no—
207C2×C3×D4—41no—
208C2×C3×D4#4114yes—
209C22×C3×C4—24no—
210C22×C3×C4—24no—
211C22×C3×C4—24no—
212C22×C3×C4—42no—
213C22×C3×C4—42no—
214C22×C3×C4—42no—
215C22×C3×C4—44no—
216C22×C3×C4—44no—
217C22×C3×C4—44no—
218C22×C3×C4C2411yescharacteristic
219C22×C3×C4C2414yes—
220C22×C3×C4C2414yes—
221C22×C3×C4C2414yes—
222C22×C3×C4C2412yes—
223C22×C3×C4C2414yes—
224C22×C3×C4C2412yes—
225C22×C3×C4C2411yescharacteristic
226C22×C3×C4C2411yescharacteristic
227C22×C3×C4C2412yes—
228C22×C3×C4C2412yes—
229C3×C4⋊C4—28no—
230C3×C4⋊C4—28no—
231C3×C4⋊C4—24no—
232C3×C4⋊C4—28no—
233C3×C4⋊C4—24no—
234C3×C4⋊C4—28no—
235C3×C4⋊C4—44no—
236C3×C4⋊C4—42no—
237C3×C4⋊C4—42no—
238C3×C4⋊C4—44no—
239C3×C4⋊C4—41no—
240C3×C4⋊C4—41no—
241C3×C4⋊C4—44no—
242C3×C4⋊C4—42no—
243C3×C4⋊C4—44no—
244C3×C22⋊C4—24no—
245C3×C22⋊C4—28no—
246C3×C22⋊C4—28no—
247C3×C22⋊C4—28no—
248C3×C22⋊C4—42no—
249C3×C22⋊C4—44no—
250C3×C22⋊C4—42no—
251C3×C22⋊C4—42no—
252C3×C22⋊C4—44no—
253C3×C22⋊C4—44no—
254C3×C22⋊C4—44no—
255C3×C22⋊C4—44no—
256C3×C42—22no—
257C3×C42—24no—
258C3×C42—22no—
259C3×C42—24no—
260C3×C42—41no—
261C3×C42—44no—
262C22×Q8C23×C312yes—
263C22×D4C23×C314yes—
264C22×D4C23×C311yescharacteristic
265C22×D4C23×C311yescharacteristic
266C23×C4C23×C312yes—
267C23×C4C23×C312yes—
268C23×C4C23×C312yes—
269C23×C4C23×C312yes—
270#127—24no—
271#126—48no—
272#123—28no—
273C22⋊Q8—48no—
274C22⋊Q8—44no—
275C4×D4—48no—
276C4×D4—42no—
277C4×D4—42no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4C23×C314yes—
280C2×C4⋊C4C23×C312yes—
281C2×C4⋊C4C23×C314yes—
282C2×C4⋊C4C23×C314yes—
283C2×C4⋊C4C23×C312yes—
284C2×C4⋊C4C23×C314yes—
285C2×C4⋊C4C23×C314yes—
286C2×C4⋊C4C23×C314yes—
287C2×C4⋊C4C23×C311yescharacteristic
288C2×C4⋊C4C23×C311yescharacteristic
289C2×C4⋊C4C23×C312yes—
290C2×C4⋊C4C23×C314yes—
291C2×C4⋊C4C23×C314yes—
292C2×C4⋊C4C23×C312yes—
293C2×C4⋊C4C23×C312yes—
294C2×C22⋊C4C23×C314yes—
295C2×C22⋊C4C23×C314yes—
296C2×C22⋊C4C23×C312yes—
297C2×C22⋊C4C23×C314yes—
298C2×C22⋊C4C23×C312yes—
299C2×C22⋊C4C23×C312yes—
300C2×C22⋊C4C23×C314yes—
301C2×C22⋊C4C23×C314yes—
302C2×C22⋊C4C23×C314yes—
303C2×C22⋊C4C23×C314yes—
304C2×C22⋊C4C23×C312yes—
305C2×C22⋊C4C23×C314yes—
306C2×C42C23×C311yescharacteristic
307C2×C42C23×C312yes—
308C2×C42C23×C311yescharacteristic
309C2×C42C23×C312yes—
310C2×C42C23×C314yes—
311C2×C42C23×C311yescharacteristic
312#95C23×C314yes—
313#95C23×C314yes—
314#95C23×C314yes—
315#95C23×C312yes—
316#95C23×C312yes—
317#95C23×C314yes—
318#95C23×C314yes—
319#95C23×C314yes—
320#95C23×C314yes—
321#95C23×C314yes—
322#95C23×C314yes—
323#95C23×C314yes—
324C23×C3—24no—
325C23×C3—22no—
326C23×C3—24no—
327C23×C3—22no—
328C23×C3—22no—
329C23×C3—24no—
330C23×C3—22no—
331C23×C3—24no—
332C23×C3—22no—
333C23×C3C2×#4114yes—
334C23×C3C2511yescharacteristic
335C3×Q8—84no—
336C3×D4—44no—
337C3×D4—416no—
338C3×D4—44no—
339C2×C3×C4—22no—
340C2×C3×C4—21no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—21no—
346C2×C3×C4—24no—
347C2×C3×C4—24no—
348C2×C3×C4—21no—
349C2×C3×C4—24no—
350C2×C3×C4—24no—
351C2×C3×C4—24no—
352C2×C3×C4—24no—
353C2×C3×C4—24no—
354C2×C3×C4—24no—
355C2×C3×C4—24no—
356C2×C3×C4—24no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—24no—
361C2×C3×C4—24no—
362C2×C3×C4—24no—
363C2×C3×C4—24no—
364C2×C3×C4—48no—
365C2×C3×C4—84no—
366C2×C3×C4—84no—
367C2×C3×C4—84no—
368C2×C3×C4C2×#4112yes—
369C2×C3×C4H(5,𝔽2)14yes—
370C2×C3×C4H(5,𝔽2)12yes—
371C2×C3×C42−1+414yes—
372C24C24×C312yes—
373C24C24×C312yes—
374C2×Q8—44no—
375C2×Q8—42no—
376C2×D4—28no—
377C2×D4—22no—
378C2×D4—28no—
379C2×D4—41no—
380C2×D4—44no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4—41no—
384C2×D4C3×#4114yes—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4C24×C311yescharacteristic
395C22×C4C24×C314yes—
396C22×C4C24×C314yes—
397C22×C4C24×C314yes—
398C22×C4C24×C312yes—
399C22×C4C24×C314yes—
400C22×C4C24×C312yes—
401C22×C4C24×C311yescharacteristic
402C22×C4C24×C311yescharacteristic
403C22×C4C24×C312yes—
404C22×C4C24×C312yes—
405C4⋊C4—28no—
406C4⋊C4—28no—
407C4⋊C4—24no—
408C4⋊C4—28no—
409C4⋊C4—24no—
410C4⋊C4—28no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—44no—
416C4⋊C4—42no—
417C4⋊C4—44no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C22⋊C4—24no—
421C22⋊C4—28no—
422C22⋊C4—28no—
423C22⋊C4—28no—
424C22⋊C4—42no—
425C22⋊C4—42no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C22⋊C4—42no—
431C22⋊C4—44no—
432C42—22no—
433C42—24no—
434C42—22no—
435C42—24no—
436C42—44no—
437C42—41no—
438C22×C3—24no—
439C22×C3—28no—
440C22×C3—42no—
441C22×C3—42no—
442C22×C3—44no—
443C22×C3—42no—
444C22×C3—44no—
445C22×C3—42no—
446C22×C3—42no—
447C22×C3—44no—
448C22×C3C22×#4111yescharacteristic
449C22×C3C2×H(5,𝔽2)11yescharacteristic
450C22×C3C2×H(5,𝔽2)12yes—
451C22×C3C2×2−1+412yes—
452C22×C3#58511yescharacteristic
453C3×C4—22no—
454C3×C4—24no—
455C3×C4—22no—
456C3×C4—24no—
457C3×C4—44no—
458C3×C4—44no—
459C3×C4—42no—
460C3×C4—42no—
461C3×C4—44no—
462C3×C4—41no—
463C3×C4—44no—
464C23—24no—
465C23—22no—
466C23—24no—
467C23—22no—
468C23—22no—
469C23—24no—
470C23—22no—
471C23—24no—
472C23—22no—
473C23C2×C3×#4114yes—
474C23C25×C311yescharacteristic
475Q8—84no—
476D4—44no—
477D4—416no—
478D4—44no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—21no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—21no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—48no—
505C2×C4—84no—
506C2×C4—84no—
507C2×C4—84no—
508C2×C4C2×C3×#4112yes—
509C2×C4C3×H(5,𝔽2)14yes—
510C2×C4C3×H(5,𝔽2)12yes—
511C2×C4C3×2−1+414yes—
512C2×C3—44no—
513C2×C3—82no—
514C2×C3#345911yescharacteristic
515C2×C3#352012yes—
516C2×C3#354311yescharacteristic
517C2×C3#355011yescharacteristic
518C2×C3#355912yes—
519C22—24no—
520C22—28no—
521C22—42no—
522C22—42no—
523C22—44no—
524C22—42no—
525C22—44no—
526C22—42no—
527C22—42no—
528C22—44no—
529C22C22×C3×#4111yescharacteristic
530C22C2×C3×H(5,𝔽2)11yescharacteristic
531C22C2×C3×H(5,𝔽2)12yes—
532C22C2×C3×2−1+412yes—
533C22C3×#58511yescharacteristic
534C4—22no—
535C4—24no—
536C4—22no—
537C4—24no—
538C4—44no—
539C4—44no—
540C4—42no—
541C4—44no—
542C4—42no—
543C4—41no—
544C4—44no—
545C3#3725411yescharacteristic, direct product factor, Sylow
546C2—44no—
547C2—82no—
548C2C3×#345911yescharacteristic
549C2C3×#352012yes—
550C2C3×#354311yescharacteristic
551C2C3×#355011yescharacteristic
552C2C3×#355912yes—
subgroupG/H
center, socleC23×C3C25
commutator, Frattini, cosocleC23C25×C3
prime radicalC2×C3×#127C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#2864C2semidirect
2. C3×#2792C2semidirect
3. C3×#2761C2semidirect
4. C3×#2727C2semidirect
5. C3×#2724C2semidirect
6. C3×#2723C2semidirect
8. C3×#2683C2semidirect
9. C3×#2638C2semidirect
10. C3×#2466C2semidirect
11. C3×#2449C2semidirect
12. C3×#2322C2semidirect
14. #37254C3direct (N × H)
68. #2864C2×C3semidirect
69. #2792C2×C3semidirect
70. #2761C2×C3semidirect
71. #2727C2×C3semidirect
72. #2724C2×C3semidirect
73. #2723C2×C3semidirect
75. #2683C2×C3semidirect
76. #2638C2×C3semidirect
77. #2466C2×C3semidirect
78. #2449C2×C3semidirect
79. #2322C2×C3semidirect
545. C3#37254direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#28642
C3×#27922
C3×#27612
C3×#27272
C3×#27242
C3×#27232
C3×#27092
C3×#26832
C3×#26382
C3×#24662
C3×#24492
C3×#23222
C3×C4×#1272
#372543