Finite Group Databasebeta

← C3×#23576 #10642183

Subgroups — C3×#23576 #10642183

Order 768.

1424 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#532C211yesmaximal, characteristic
2C2×C3×#440C212yesmaximal, complemented
3C3×C4×C22⋊Q8C211yesmaximal, characteristic, complemented
4C3×#2057C211yesmaximal, characteristic, complemented
5C3×#2041C211yesmaximal, characteristic, complemented
6C3×#2033C212yesmaximal, complemented
7C3×#2024C211yesmaximal, characteristic, complemented
8C3×#2014C211yesmaximal, characteristic, complemented
9C3×#2000C212yesmaximal, complemented
10C3×#1762C212yesmaximal, complemented
11C3×#1749C211yesmaximal, characteristic, complemented
12#23576C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41—21no—
14C22×C3×SD16C2212yes—
15C3×#532—28no—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#127C2211yescharacteristic
18C2×C3×#124—21no—
19C2×C3×C22⋊Q8C2211yescharacteristic
20C2×C3×#121—21no—
21C2×C3×#121C2212yes—
22C2×C3×C4×Q8C2211yescharacteristic
23C2×C3×C4×D4—21no—
24C2×C3×C4×D4C2212yes—
25C22×C3×C42C2211yescharacteristic
26C3×#440—28no—
27C2×C3×#107C2212yes—
28C2×C3×#103C2212yes—
29C2×C3×#102C2212yes—
30C2×C3×#102C2212yes—
31C2×C3×#102C2212yes—
32C2×C3×C22⋊C8C2212yes—
33C2×C3×#97C2212yes—
34C3×#391C2211yescharacteristic
35C3×#389C2211yescharacteristic
36C3×#387—21no—
37C3×#385—21no—
38C3×#380C2211yescharacteristic
39C3×C4×C4⋊C4—21no—
40C3×C4×C4⋊C4C2211yescharacteristic
41C3×C4×C22⋊C4—21no—
42C3×#340C2212yes—
43C3×#340C2212yes—
44C3×#336C2211yescharacteristic
45C3×#336C2211yescharacteristic
46C2×#532C2×C311yescharacteristic
47C2×#440C2×C312yescomplemented
48C4×C22⋊Q8C2×C311yescharacteristic, complemented
49#2057C2×C311yescharacteristic, complemented
50#2041C2×C311yescharacteristic, complemented
51#2033C2×C312yescomplemented
52#2024C2×C311yescharacteristic, complemented
53#2014C2×C311yescharacteristic, complemented
54#2000C2×C312yescomplemented
55#1762C2×C312yescomplemented
56#1749C2×C311yescharacteristic, complemented
57C2×C3×#41—28no—
58C2×C3×#41—44no—
59C2×C3×#41—44no—
60C22×C3×Q8—21no—
61C22×C3×Q8C2311yescharacteristic
62C22×C3×D4—22no—
63C22×C3×D4—21no—
64C22×C3×D4—21no—
65C22×C3×D4C2312yes—
66C23×C3×C4—22no—
67C23×C3×C4D411yescharacteristic
68C23×C3×C4D411yescharacteristic
69C23×C3×C4C2311yescharacteristic
70C2×C3×SD16—24no—
71C2×C3×SD16—24no—
72C2×C3×SD16—24no—
73C22×C3×C8C2312yes—
74C22×C3×C8C2312yes—
75C3×C4⋊Q8—24no—
76C3×#127—24no—
77C3×#124—28no—
78C3×C22⋊Q8—22no—
79C3×C22⋊Q8—22no—
80C3×#121—28no—
81C3×#121—24no—
82C3×#121—24no—
83C3×C4×Q8—24no—
84C3×C4×D4—28no—
85C3×C4×D4—24no—
86C3×C4×D4D418yes—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2312yes—
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—22no—
97C2×C3×C22⋊C4—22no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C42—24no—
100C2×C3×C42—21no—
101C2×C3×C42—21no—
102C2×C3×C42D411yescharacteristic
103C2×C3×C42D411yescharacteristic
104C2×C3×C42C2311yescharacteristic
105C2×C3×C42C2311yescharacteristic
106C3×#107—24no—
107C3×#103—24no—
108C3×#102—24no—
109C3×#102—24no—
110C3×#102—24no—
111C3×C22⋊C8—24no—
112C3×#97—24no—
113C3×#95—21no—
114C3×#95—21no—
115C3×#95—21no—
116C3×#95—21no—
117C3×#95—21no—
118C22×#41—21no—
119C22×SD16C22×C312yes—
120#532—28no—
121C2×C4⋊Q8C22×C311yescharacteristic
122C2×#127C22×C311yescharacteristic
123C2×#124—21no—
124C2×C22⋊Q8C22×C311yescharacteristic
125C2×#121—21no—
126C2×#121C22×C312yes—
127C2×C4×Q8C22×C311yescharacteristic
128C2×C4×D4—21no—
129C2×C4×D4C22×C312yes—
130C22×C42C22×C311yescharacteristic
131#440—28no—
132C2×#107C22×C312yes—
133C2×#103C22×C312yes—
134C2×#102C22×C312yes—
135C2×#102C22×C312yes—
136C2×#102C22×C312yes—
137C2×C22⋊C8C22×C312yes—
138C2×#97C22×C312yes—
139#391C22×C311yescharacteristic
140#389C22×C311yescharacteristic
141#387—21no—
142#385—21no—
143#380C22×C311yescharacteristic
144C4×C4⋊C4—21no—
145C4×C4⋊C4C22×C311yescharacteristic
146C4×C22⋊C4—21no—
147#340C22×C312yes—
148#340C22×C312yes—
149#336C22×C311yescharacteristic
150#336C22×C311yescharacteristic
151C24×C3—22no—
152C24×C3C2×D411yescharacteristic
153C3×#41—432no—
154C2×C3×Q8—24no—
155C2×C3×Q8—21no—
156C2×C3×Q8—22no—
157C2×C3×Q8—22no—
158C2×C3×Q8—42no—
159C2×C3×Q8—42no—
160C2×C3×Q8#4112yescomplemented
161C2×C3×D4—24no—
162C2×C3×D4—28no—
163C2×C3×D4—24no—
164C2×C3×D4—22no—
165C2×C3×D4—22no—
166C2×C3×D4—22no—
167C2×C3×D4—22no—
168C2×C3×D4—22no—
169C2×C3×D4—42no—
170C2×C3×D4—42no—
171C2×C3×D4—44no—
172C2×C3×D4—44no—
173C2×C3×D4—42no—
174C2×C3×D4—42no—
175C2×C3×D4C2×D414yes—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—28no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—41no—
192C22×C3×C4—44no—
193C22×C3×C4—44no—
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2411yescharacteristic
200C3×SD16—216no—
201C2×C3×C8—22no—
202C2×C3×C8—22no—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C2×C3×C8#4114yescomplemented
207C3×C4⋊C4—24no—
208C3×C4⋊C4—24no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—42no—
214C3×C4⋊C4—42no—
215C3×C4⋊C4C2×D414yes—
216C3×C4⋊C4#4112yescomplemented
217C3×C22⋊C4—28no—
218C3×C22⋊C4—24no—
219C3×C22⋊C4—28no—
220C3×C22⋊C4—42no—
221C3×C42—22no—
222C3×C42—22no—
223C3×C42—22no—
224C3×C42—42no—
225C3×C42—42no—
226C3×C42C2×D414yes—
227C2×#41—28no—
228C2×#41—44no—
229C2×#41—44no—
230C22×Q8—21no—
231C22×Q8C23×C311yescharacteristic
232C22×D4—22no—
233C22×D4—21no—
234C22×D4—21no—
235C22×D4C23×C312yes—
236C23×C4—22no—
237C23×C4C3×D411yescharacteristic
238C23×C4C3×D411yescharacteristic
239C23×C4C23×C311yescharacteristic
240C2×SD16—24no—
241C2×SD16—24no—
242C2×SD16—24no—
243C22×C8C23×C312yes—
244C22×C8C23×C312yes—
245C4⋊Q8—24no—
246#127—24no—
247#124—28no—
248C22⋊Q8—22no—
249C22⋊Q8—22no—
250#121—28no—
251#121—24no—
252#121—24no—
253C4×Q8—24no—
254C4×D4—28no—
255C4×D4—24no—
256C4×D4C3×D418yes—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4C23×C311yescharacteristic
262C2×C4⋊C4C23×C311yescharacteristic
263C2×C4⋊C4C23×C312yes—
264C2×C4⋊C4C23×C311yescharacteristic
265C2×C22⋊C4—21no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—21no—
269C2×C42—24no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42C3×D411yescharacteristic
273C2×C42C3×D411yescharacteristic
274C2×C42C23×C311yescharacteristic
275C2×C42C23×C311yescharacteristic
276#107—24no—
277#103—24no—
278#102—24no—
279#102—24no—
280#102—24no—
281C22⋊C8—24no—
282#97—24no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—21no—
287#95—21no—
288C23×C3—21no—
289C23×C3—21no—
290C23×C3—21no—
291C23×C3—21no—
292C23×C3—21no—
293C23×C3—21no—
294C23×C3—24no—
295C23×C3—42no—
296C23×C3—42no—
297C23×C3—42no—
298C23×C3—42no—
299C23×C3—42no—
300C23×C3—42no—
301C23×C3AGL(1,ℤ/8)12yes—
302C23×C3C22×D411yescharacteristic
303C3×Q8—24no—
304C3×Q8—48no—
305C3×Q8—42no—
306C3×D4—24no—
307C3×D4—24no—
308C3×D4—48no—
309C3×D4—48no—
310C3×D4—416no—
311C3×D4—42no—
312C3×D4—42no—
313C2×C3×C4—28no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—24no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—44no—
329C2×C3×C4—416no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—41no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4#12012yes—
345C2×C3×C4#12012yes—
346C2×C3×C4AGL(1,ℤ/8)12yes—
347C2×C3×C4#13712yes—
348C2×C3×C4C22×D412yes—
349C2×C3×C4C2×#4111yescharacteristic
350C2×C3×C4C2×#4112yes—
351C2×C3×C4C2×#4111yescharacteristic
352C3×C8—24no—
353C3×C8—42no—
354C24—22no—
355C24C2×C3×D411yescharacteristic
356#41—432no—
357C2×Q8—24no—
358C2×Q8—21no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—42no—
362C2×Q8—42no—
363C2×Q8C3×#4112yescomplemented
364C2×D4—24no—
365C2×D4—28no—
366C2×D4—24no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4C2×C3×D414yes—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—28no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—41no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4C2×C3×D411yescharacteristic
398C22×C4C2×C3×D411yescharacteristic
399C22×C4C2×C3×D411yescharacteristic
400C22×C4C2×C3×D411yescharacteristic
401C22×C4C2×C3×D411yescharacteristic
402C22×C4C24×C311yescharacteristic
403SD16—216no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8C3×#4114yescomplemented
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—21no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4C2×C3×D414yes—
419C4⋊C4C3×#4112yescomplemented
420C22⋊C4—28no—
421C22⋊C4—24no—
422C22⋊C4—28no—
423C22⋊C4—42no—
424C42—22no—
425C42—22no—
426C42—22no—
427C42—42no—
428C42—42no—
429C42C2×C3×D414yes—
430C22×C3—22no—
431C22×C3—22no—
432C22×C3—22no—
433C22×C3—41no—
434C22×C3—41no—
435C22×C3—41no—
436C22×C3—41no—
437C22×C3—44no—
438C22×C3—44no—
439C22×C3—44no—
440C22×C3—82no—
441C22×C3—82no—
442C22×C3—82no—
443C22×C3—82no—
444C22×C3C2×#12011yescharacteristic
445C22×C3#52511yescharacteristic
446C22×C3#53211yescharacteristic
447C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
448C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
449C22×C3C2×#13711yescharacteristic
450C22×C3#57511yescharacteristic
451C3×C4—22no—
452C3×C4—22no—
453C3×C4—22no—
454C3×C4—22no—
455C3×C4—42no—
456C3×C4—41no—
457C3×C4—42no—
458C3×C4—44no—
459C3×C4—81no—
460C3×C4—81no—
461C3×C4#52512yes—
462C3×C4#53212yes—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—24no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23C3×AGL(1,ℤ/8)12yes—
477C23C22×C3×D411yescharacteristic
478Q8—24no—
479Q8—48no—
480Q8—42no—
481D4—24no—
482D4—24no—
483D4—48no—
484D4—48no—
485D4—416no—
486D4—42no—
487D4—42no—
488C2×C4—28no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—44no—
504C2×C4—416no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4C3×#12012yes—
520C2×C4C3×#12012yes—
521C2×C4C3×AGL(1,ℤ/8)12yes—
522C2×C4C3×#13712yes—
523C2×C4C22×C3×D412yes—
524C2×C4C2×C3×#4111yescharacteristic
525C2×C4C2×C3×#4112yes—
526C2×C4C2×C3×#4111yescharacteristic
527C8—24no—
528C8—42no—
529C2×C3—42no—
530C2×C3—84no—
531C2×C3#238811yescharacteristic
532C2×C3#300311yescharacteristic
533C2×C3#300711yescharacteristic
534C2×C3#311211yescharacteristic
535C2×C3#311511yescharacteristic
536C2×C3#318011yescharacteristic
537C2×C3#318211yescharacteristic
538C22—22no—
539C22—22no—
540C22—22no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—44no—
546C22—44no—
547C22—44no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22C2×C3×#12011yescharacteristic
553C22C3×#52511yescharacteristic
554C22C3×#53211yescharacteristic
555C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
556C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
557C22C2×C3×#13711yescharacteristic
558C22C3×#57511yescharacteristic
559C4—22no—
560C4—22no—
561C4—22no—
562C4—22no—
563C4—41no—
564C4—42no—
565C4—44no—
566C4—42no—
567C4—81no—
568C4—81no—
569C4C3×#52512yes—
570C4C3×#53212yes—
571C3#2357611yescharacteristic, direct product factor, Sylow
572C2—42no—
573C2—84no—
574C2C3×#238811yescharacteristic
575C2C3×#300311yescharacteristic
576C2C3×#300711yescharacteristic
577C2C3×#311211yescharacteristic
578C2C3×#311511yescharacteristic
579C2C3×#318011yescharacteristic
580C2C3×#318211yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#532C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#440C2semidirect
3. C3×C4×C22⋊Q8C2semidirect
4. C3×#2057C2semidirect
5. C3×#2041C2semidirect
6. C3×#2033C2semidirect
7. C3×#2024C2semidirect
8. C3×#2014C2semidirect
9. C3×#2000C2semidirect
10. C3×#1762C2semidirect
11. C3×#1749C2semidirect
12. #23576C3direct (N × H)
47. C2×#440C2×C3semidirect
48. C4×C22⋊Q8C2×C3semidirect
49. #2057C2×C3semidirect
50. #2041C2×C3semidirect
51. #2033C2×C3semidirect
52. #2024C2×C3semidirect
53. #2014C2×C3semidirect
54. #2000C2×C3semidirect
55. #1762C2×C3semidirect
56. #1749C2×C3semidirect
160. C2×C3×Q8#41semidirect
206. C2×C3×C8#41semidirect
216. C3×C4⋊C4#41semidirect
363. C2×Q8C3×#41semidirect
409. C2×C8C3×#41semidirect
419. C4⋊C4C3×#41semidirect
571. C3#23576direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5322
C2×C3×#4402
C3×C4×C22⋊Q82
C3×#20572
C3×#20412
C3×#20332
C3×#20242
C3×#20142
C3×#20002
C3×#17622
C3×#17492
#235763