Finite Group Databasebeta

← C3×#24351 #10642958

Subgroups — C3×#24351 #10642958

Order 768.

1122 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3501C211yesmaximal, characteristic
2C3×#2855C212yesmaximal, complemented
3C3×C4×#123C211yesmaximal, characteristic
4C3×#2062C212yesmaximal, complemented
5C3×#2030C212yesmaximal, complemented
6C3×#1996C212yesmaximal, complemented
7C3×#1890C212yesmaximal
8C3×#1867C212yesmaximal
9C3×#1799C211yesmaximal, characteristic
10#24351C311yesmaximal, characteristic, direct product factor, Sylow
11C3×#554—24no—
12C3×#554—22no—
13C3×D4×Q8—22no—
14C3×#544—21no—
15C3×#543—21no—
16C3×#543C2212yes—
17C3×#533—21no—
18C2×C3×#123C2211yescharacteristic
19C2×C3×C22⋊Q8—21no—
20C2×C3×C22⋊Q8C2212yes—
21C3×#518C2212yes—
22C2×C3×C4×D4C2211yescharacteristic
23C22×C3×C42C2211yescharacteristic
24C2×C3×#99C2212yes—
25C3×#391—22no—
26C3×#389—22no—
27C3×#388—21no—
28C3×#387—21no—
29C3×#387—24no—
30C3×#386C2211yescharacteristic
31C3×#385C2211yescharacteristic
32C3×#385C2214yes—
33C3×#381C2211yescharacteristic
34C3×#381C2214yes—
35C3×#380C2212yes—
36C3×C4×C4⋊C4—21no—
37C3×C4×C22⋊C4—21no—
38C3×C4×C22⋊C4C2211yescharacteristic
39C3×#342C2214yes—
40C3×#342C2214yes—
41C3×#342C2212yes—
42#3501C2×C311yescharacteristic
43#2855C2×C312yescomplemented
44C4×#123C2×C311yescharacteristic
45#2062C2×C312yescomplemented
46#2030C2×C312yescomplemented
47#1996C2×C312yescomplemented
48#1890C2×C312yes—
49#1867C2×C312yes—
50#1799C2×C311yescharacteristic
51C22×C3×Q8—21no—
52C22×C3×Q8D412yes—
53C22×C3×D4C2311yescharacteristic
54C23×C3×C4D411yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C3×C4⋊Q8—21no—
59C3×C4⋊Q8—21no—
60C3×C4⋊Q8—24no—
61C3×#125—22no—
62C3×#125—24no—
63C3×#123—22no—
64C3×#123Q814yescomplemented
65C3×C22⋊Q8—22no—
66C3×C22⋊Q8—22no—
67C3×C22⋊Q8—24no—
68C3×C22⋊Q8—24no—
69C3×C22⋊Q8—22no—
70C3×C22⋊Q8—22no—
71C3×C22⋊Q8—22no—
72C3×C22⋊Q8—22no—
73C3×C22⋊Q8—22no—
74C3×C22⋊Q8—22no—
75C3×C22⋊Q8—24no—
76C3×C22⋊Q8—24no—
77C3×C4×Q8—22no—
78C3×C4×Q8—22no—
79C3×C4×D4—22no—
80C3×C4×D4—22no—
81C3×C4×D4—24no—
82C3×C4×D4—24no—
83C3×#117—22no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—22no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—22no—
89C2×C3×C4⋊C4—24no—
90C2×C3×C4⋊C4D411yescharacteristic
91C2×C3×C4⋊C4D414yes—
92C2×C3×C4⋊C4D412yes—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—22no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4D411yescharacteristic
97C2×C3×C22⋊C4C2311yescharacteristic
98C2×C3×C22⋊C4C2314yes—
99C2×C3×C22⋊C4C2311yescharacteristic
100C2×C3×C22⋊C4C2314yes—
101C2×C3×C22⋊C4C2311yescharacteristic
102C2×C3×C42—21no—
103C2×C3×C42—21no—
104C2×C3×C42—22no—
105C2×C3×C42—21no—
106C2×C3×C42—22no—
107C2×C3×C42D411yescharacteristic
108C3×#99—24no—
109C3×#95—21no—
110C3×#95—21no—
111C3×#95—21no—
112C3×#95—21no—
113C3×#95—21no—
114C3×#95—22no—
115C3×#95—24no—
116C3×#95—24no—
117C3×#95—24no—
118#554—24no—
119#554—22no—
120D4×Q8—22no—
121#544—21no—
122#543—21no—
123#543C22×C312yes—
124#533—21no—
125C2×#123C22×C311yescharacteristic
126C2×C22⋊Q8—21no—
127C2×C22⋊Q8C22×C312yes—
128#518C22×C312yes—
129C2×C4×D4C22×C311yescharacteristic
130C22×C42C22×C311yescharacteristic
131C2×#99C22×C312yes—
132#391—22no—
133#389—22no—
134#388—21no—
135#387—24no—
136#387—21no—
137#386C22×C311yescharacteristic
138#385C22×C311yescharacteristic
139#385C22×C314yes—
140#381C22×C314yes—
141#381C22×C311yescharacteristic
142#380C22×C312yes—
143C4×C4⋊C4—21no—
144C4×C22⋊C4—21no—
145C4×C22⋊C4C22×C311yescharacteristic
146#342C22×C312yes—
147#342C22×C314yes—
148#342C22×C314yes—
149C24×C3C2×D411yescharacteristic
150C24×C3C2411yescharacteristic
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—44no—
156C2×C3×Q8—82no—
157C2×C3×D4—22no—
158C2×C3×D4—22no—
159C2×C3×D4C2×Q812yes—
160C2×C3×D4#4112yes—
161C22×C3×C4—22no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—22no—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4—21no—
173C22×C3×C4—24no—
174C22×C3×C4—42no—
175C22×C3×C4—42no—
176C22×C3×C4—42no—
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D414yes—
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D414yes—
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2×Q812yescomplemented
187C3×C4⋊C4—22no—
188C3×C4⋊C4—24no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—24no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—24no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—24no—
198C3×C4⋊C4—24no—
199C3×C4⋊C4—44no—
200C3×C4⋊C4—42no—
201C3×C4⋊C4—48no—
202C3×C22⋊C4—24no—
203C3×C22⋊C4—24no—
204C3×C22⋊C4—24no—
205C3×C22⋊C4—22no—
206C3×C22⋊C4—24no—
207C3×C22⋊C4—21no—
208C3×C22⋊C4—22no—
209C3×C22⋊C4—24no—
210C3×C22⋊C4—22no—
211C3×C22⋊C4—24no—
212C3×C22⋊C4—42no—
213C3×C22⋊C4C2×Q812yescomplemented
214C3×C42—24no—
215C3×C42—21no—
216C3×C42—21no—
217C3×C42—42no—
218C3×C42—41no—
219C3×C42—42no—
220C3×C42—41no—
221C22×Q8—21no—
222C22×Q8C3×D412yes—
223C22×D4C23×C311yescharacteristic
224C23×C4C3×D411yescharacteristic
225C23×C4C23×C311yescharacteristic
226C23×C4C23×C311yescharacteristic
227C23×C4C23×C311yescharacteristic
228C4⋊Q8—21no—
229C4⋊Q8—21no—
230C4⋊Q8—24no—
231#125—22no—
232#125—24no—
233#123—22no—
234#123C3×Q814yescomplemented
235C22⋊Q8—22no—
236C22⋊Q8—22no—
237C22⋊Q8—24no—
238C22⋊Q8—24no—
239C22⋊Q8—22no—
240C22⋊Q8—22no—
241C22⋊Q8—22no—
242C22⋊Q8—22no—
243C22⋊Q8—22no—
244C22⋊Q8—22no—
245C22⋊Q8—24no—
246C22⋊Q8—24no—
247C4×Q8—22no—
248C4×Q8—22no—
249C4×D4—22no—
250C4×D4—22no—
251C4×D4—24no—
252C4×D4—24no—
253#117—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4C3×D411yescharacteristic
261C2×C4⋊C4C3×D414yes—
262C2×C4⋊C4C3×D412yes—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4C3×D411yescharacteristic
267C2×C22⋊C4C23×C311yescharacteristic
268C2×C22⋊C4C23×C314yes—
269C2×C22⋊C4C23×C311yescharacteristic
270C2×C22⋊C4C23×C314yes—
271C2×C22⋊C4C23×C311yescharacteristic
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—22no—
275C2×C42—21no—
276C2×C42—22no—
277C2×C42C3×D411yescharacteristic
278#99—24no—
279#95—21no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—22no—
285#95—24no—
286#95—24no—
287#95—24no—
288C23×C3—21no—
289C23×C3—21no—
290C23×C3—41no—
291C23×C3—41no—
292C23×C3—41no—
293C23×C3—41no—
294C23×C3C22⋊Q812yes—
295C23×C3C22×D411yescharacteristic
296C23×C3C22×D412yes—
297C23×C3C22×Q811yescharacteristic
298C23×C3C2×#4111yescharacteristic
299C23×C32−1+412yes—
300C3×Q8—44no—
301C3×Q8—82no—
302C3×Q8—82no—
303C3×Q8—82no—
304C3×D4—24no—
305C3×D4—24no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—22no—
309C2×C3×C4—24no—
310C2×C3×C4—22no—
311C2×C3×C4—24no—
312C2×C3×C4—21no—
313C2×C3×C4—24no—
314C2×C3×C4—24no—
315C2×C3×C4—21no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—24no—
319C2×C3×C4—42no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—42no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—44no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—44no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—44no—
337C2×C3×C4—44no—
338C2×C3×C4—41no—
339C2×C3×C4—44no—
340C2×C3×C4—81no—
341C2×C3×C4—82no—
342C2×C3×C4—82no—
343C2×C3×C4#12011yescharacteristic
344C2×C3×C4#12012yes—
345C2×C3×C4#12011yescharacteristic
346C2×C3×C4C22⋊Q812yes—
347C24C2×C3×D411yescharacteristic
348C24C24×C311yescharacteristic
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—44no—
354C2×Q8—82no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4C2×C3×Q812yes—
358C2×D4C3×#4112yes—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—24no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4C2×C3×D411yescharacteristic
376C22×C4C2×C3×D412yes—
377C22×C4C2×C3×D411yescharacteristic
378C22×C4C2×C3×D414yes—
379C22×C4C2×C3×D411yescharacteristic
380C22×C4C2×C3×D411yescharacteristic
381C22×C4C2×C3×D414yes—
382C22×C4C2×C3×D411yescharacteristic
383C22×C4C2×C3×D412yes—
384C22×C4C2×C3×Q812yescomplemented
385C4⋊C4—22no—
386C4⋊C4—24no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—24no—
390C4⋊C4—22no—
391C4⋊C4—24no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—44no—
398C4⋊C4—42no—
399C4⋊C4—48no—
400C22⋊C4—24no—
401C22⋊C4—24no—
402C22⋊C4—24no—
403C22⋊C4—22no—
404C22⋊C4—24no—
405C22⋊C4—21no—
406C22⋊C4—22no—
407C22⋊C4—24no—
408C22⋊C4—22no—
409C22⋊C4—24no—
410C22⋊C4—42no—
411C22⋊C4C2×C3×Q812yescomplemented
412C42—24no—
413C42—21no—
414C42—21no—
415C42—42no—
416C42—41no—
417C42—42no—
418C42—41no—
419C22×C3—22no—
420C22×C3—21no—
421C22×C3—22no—
422C22×C3—21no—
423C22×C3—22no—
424C22×C3—41no—
425C22×C3—41no—
426C22×C3—42no—
427C22×C3—42no—
428C22×C3—41no—
429C22×C3—41no—
430C22×C3—81no—
431C22×C3C2×#12011yescharacteristic
432C22×C3C2×C22⋊Q811yescharacteristic
433C22×C3#53611yescharacteristic
434C22×C3#54812yes—
435C22×C3D4×Q812yes—
436C3×C4—22no—
437C3×C4—21no—
438C3×C4—22no—
439C3×C4—42no—
440C3×C4—42no—
441C3×C4—41no—
442C3×C4—44no—
443C3×C4—42no—
444C3×C4—44no—
445C3×C4—41no—
446C3×C4—84no—
447C3×C4—81no—
448C3×C4#45812yes—
449C23—21no—
450C23—21no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23C3×C22⋊Q812yes—
456C23C22×C3×D411yescharacteristic
457C23C22×C3×D412yes—
458C23C22×C3×Q811yescharacteristic
459C23C2×C3×#4111yescharacteristic
460C23C3×2−1+412yes—
461Q8—44no—
462Q8—82no—
463Q8—82no—
464Q8—82no—
465D4—24no—
466D4—24no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—21no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—44no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—41no—
500C2×C4—44no—
501C2×C4—81no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4C3×#12011yescharacteristic
505C2×C4C3×#12012yes—
506C2×C4C3×#12011yescharacteristic
507C2×C4C3×C22⋊Q812yes—
508C2×C3—21no—
509C2×C3—22no—
510C2×C3—21no—
511C2×C3—22no—
512C2×C3—42no—
513C2×C3#240911yescharacteristic
514C2×C3C2×#45811yescharacteristic
515C2×C3#302511yescharacteristic
516C22—22no—
517C22—21no—
518C22—22no—
519C22—21no—
520C22—22no—
521C22—41no—
522C22—41no—
523C22—42no—
524C22—42no—
525C22—41no—
526C22—41no—
527C22—81no—
528C22C2×C3×#12011yescharacteristic
529C22C2×C3×C22⋊Q811yescharacteristic
530C22C3×#53611yescharacteristic
531C22C3×#54812yes—
532C22C3×D4×Q812yes—
533C4—22no—
534C4—21no—
535C4—22no—
536C4—42no—
537C4—42no—
538C4—41no—
539C4—44no—
540C4—42no—
541C4—44no—
542C4—41no—
543C4—84no—
544C4—81no—
545C4C3×#45812yes—
546C3#2435111yescharacteristic, direct product factor, Sylow
547C2—21no—
548C2—22no—
549C2—21no—
550C2—22no—
551C2—42no—
552C2C3×#240911yescharacteristic
553C2C2×C3×#45811yescharacteristic
554C2C3×#302511yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC24C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2855C2semidirect
4. C3×#2062C2semidirect
5. C3×#2030C2semidirect
6. C3×#1996C2semidirect
10. #24351C3direct (N × H)
43. #2855C2×C3semidirect
45. #2062C2×C3semidirect
46. #2030C2×C3semidirect
47. #1996C2×C3semidirect
64. C3×#123Q8semidirect
186. C22×C3×C4C2×Q8semidirect
213. C3×C22⋊C4C2×Q8semidirect
234. #123C3×Q8semidirect
384. C22×C4C2×C3×Q8semidirect
411. C22⋊C4C2×C3×Q8semidirect
546. C3#24351direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35012
C3×#28552
C3×C4×#1232
C3×#20622
C3×#20302
C3×#19962
C3×#18902
C3×#18672
C3×#17992
#243513