Finite Group Databasebeta

← C3×#24326 #10642933

Subgroups — C3×#24326 #10642933

Order 768.

1330 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×Q8C211yesmaximal, characteristic, complemented
2C2×C3×#474C211yesmaximal, characteristic, complemented
3C2×C3×C4×D8C211yesmaximal, characteristic
4C3×#2538C211yesmaximal, characteristic, complemented
5C3×#2057C212yesmaximal, complemented
6C3×#2023C212yesmaximal, complemented
7C3×#1990C211yesmaximal, characteristic, complemented
8C3×#1984C211yesmaximal, characteristic, complemented
9C3×#1879C212yesmaximal
10C3×#1862C212yesmaximal, complemented
11C3×#1802C211yesmaximal, characteristic, complemented
12#24326C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8—21no—
14C22×C3×D8C2211yescharacteristic
15C3×D4×Q8—24no—
16C3×D4×Q8—24no—
17C2×C3×C4⋊Q8C2212yes—
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×C22⋊Q8—21no—
20C2×C3×C22⋊Q8—21no—
21C2×C3×C22⋊Q8—22no—
22C2×C3×C4×Q8C2211yescharacteristic
23C2×C3×C4×D4C2212yes—
24C2×C3×C4×D4C2211yescharacteristic
25C2×C3×C4×D4C2211yescharacteristic
26C3×#474—24no—
27C3×C4×D8—24no—
28C2×C3×#107C2211yescharacteristic
29C2×C3×#107C2212yes—
30C2×C3×C4⋊C8C2211yescharacteristic
31C2×C3×#102C2211yescharacteristic
32C2×C3×#102C2212yes—
33C2×C3×#102C2211yescharacteristic
34C2×C3×#102C2212yes—
35C2×C3×C4×C8C2211yescharacteristic
36C3×#396—22no—
37C3×#390C2212yes—
38C3×#389C2212yes—
39C3×#384C2211yescharacteristic
40C3×#380—21no—
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#340C2212yes—
44C3×#336C2212yes—
45C3×#336C2212yes—
46C2×D4×Q8C2×C311yescharacteristic, complemented
47C2×#474C2×C311yescharacteristic, complemented
48C2×C4×D8C2×C311yescharacteristic
49#2538C2×C311yescharacteristic, complemented
50#2057C2×C312yescomplemented
51#2023C2×C312yescomplemented
52#1990C2×C311yescharacteristic, complemented
53#1984C2×C311yescharacteristic, complemented
54#1879C2×C312yes—
55#1862C2×C312yescomplemented
56#1802C2×C311yescharacteristic, complemented
57C22×C3×Q8—28no—
58C22×C3×Q8—22no—
59C22×C3×Q8—21no—
60C22×C3×Q8—44no—
61C22×C3×Q8—44no—
62C22×C3×Q8D411yescharacteristic
63C22×C3×D4C2311yescharacteristic
64C22×C3×D4C2311yescharacteristic
65C23×C3×C4—22no—
66C23×C3×C4—21no—
67C23×C3×C4—21no—
68C2×C3×D8—22no—
69C2×C3×D8—22no—
70C2×C3×D8Q814yescomplemented
71C22×C3×C8C2311yescharacteristic
72C22×C3×C8C2312yes—
73C22×C3×C8C2311yescharacteristic
74C3×C4⋊Q8—24no—
75C3×C4⋊Q8—24no—
76C3×C4⋊Q8—22no—
77C3×C4⋊Q8D414yes—
78C3×C22⋊Q8—28no—
79C3×C22⋊Q8—216no—
80C3×C22⋊Q8—44no—
81C3×C4×Q8—22no—
82C3×C4×Q8—22no—
83C3×C4×D4—24no—
84C3×C4×D4—24no—
85C3×C4×D4—22no—
86C3×C4×D4—22no—
87C3×C4×D4—22no—
88C3×C4×D4D414yes—
89C2×C3×C4⋊C4—22no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4D411yescharacteristic
95C2×C3×C4⋊C4D412yes—
96C2×C3×C4⋊C4C2312yes—
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2312yes—
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C22⋊C4—22no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—22no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—22no—
105C2×C3×C42C2312yes—
106C2×C3×C42C2311yescharacteristic
107C3×#107—22no—
108C3×#107—24no—
109C3×C4⋊C8—22no—
110C3×#102—22no—
111C3×#102—24no—
112C3×#102—22no—
113C3×#102—24no—
114C3×C4×C8—22no—
115C3×#95—22no—
116C23×Q8—21no—
117C22×D8C22×C311yescharacteristic
118D4×Q8—24no—
119D4×Q8—24no—
120C2×C4⋊Q8C22×C312yes—
121C2×C4⋊Q8C22×C311yescharacteristic
122C2×C22⋊Q8—21no—
123C2×C22⋊Q8—21no—
124C2×C22⋊Q8—22no—
125C2×C4×Q8C22×C311yescharacteristic
126C2×C4×D4C22×C312yes—
127C2×C4×D4C22×C311yescharacteristic
128C2×C4×D4C22×C311yescharacteristic
129#474—24no—
130C4×D8—24no—
131C2×#107C22×C311yescharacteristic
132C2×#107C22×C312yes—
133C2×C4⋊C8C22×C311yescharacteristic
134C2×#102C22×C312yes—
135C2×#102C22×C311yescharacteristic
136C2×#102C22×C312yes—
137C2×#102C22×C311yescharacteristic
138C2×C4×C8C22×C311yescharacteristic
139#396—22no—
140#390C22×C312yes—
141#389C22×C312yes—
142#384C22×C311yescharacteristic
143#380—21no—
144#340C22×C312yes—
145#340C22×C312yes—
146#340C22×C312yes—
147#336C22×C312yes—
148#336C22×C312yes—
149C24×C3—21no—
150C24×C3—21no—
151C2×C3×Q8—28no—
152C2×C3×Q8—22no—
153C2×C3×Q8—24no—
154C2×C3×Q8—22no—
155C2×C3×Q8—44no—
156C2×C3×Q8—432no—
157C2×C3×Q8—44no—
158C2×C3×Q8—42no—
159C2×C3×Q8—42no—
160C2×C3×Q8D814yescomplemented
161C2×C3×D4—21no—
162C2×C3×D4—21no—
163C2×C3×D4—21no—
164C2×C3×D4—21no—
165C2×C3×D4—21no—
166C2×C3×D4—21no—
167C2×C3×D4—21no—
168C2×C3×D4—21no—
169C2×C3×D4C2×D412yes—
170C2×C3×D4C2×Q812yes—
171C2×C3×D4C2×Q812yescomplemented
172C2×C3×D4#4112yes—
173C22×C3×C4—28no—
174C22×C3×C4—22no—
175C22×C3×C4—21no—
176C22×C3×C4—24no—
177C22×C3×C4—21no—
178C22×C3×C4—22no—
179C22×C3×C4—44no—
180C22×C3×C4—44no—
181C22×C3×C4—42no—
182C22×C3×C4—42no—
183C22×C3×C4—42no—
184C22×C3×C4—42no—
185C22×C3×C4—42no—
186C22×C3×C4C2×D412yes—
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D412yes—
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2411yescharacteristic
191C3×D8—28no—
192C2×C3×C8—21no—
193C2×C3×C8—21no—
194C2×C3×C8—21no—
195C2×C3×C8—22no—
196C2×C3×C8—22no—
197C2×C3×C8—21no—
198C2×C3×C8—21no—
199C2×C3×C8—22no—
200C2×C3×C8C2×Q812yescomplemented
201C3×C4⋊C4—28no—
202C3×C4⋊C4—24no—
203C3×C4⋊C4—24no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—42no—
212C3×C4⋊C4—44no—
213C3×C4⋊C4—81no—
214C3×C4⋊C4C2×D414yes—
215C3×C4⋊C4C2×D412yes—
216C3×C22⋊C4—28no—
217C3×C22⋊C4—24no—
218C3×C22⋊C4—42no—
219C3×C22⋊C4—44no—
220C3×C22⋊C4—82no—
221C3×C42—22no—
222C3×C42—22no—
223C3×C42C2×D412yes—
224C3×C42C2×D412yes—
225C22×Q8—28no—
226C22×Q8—22no—
227C22×Q8—21no—
228C22×Q8—44no—
229C22×Q8—44no—
230C22×Q8C3×D411yescharacteristic
231C22×D4C23×C311yescharacteristic
232C22×D4C23×C311yescharacteristic
233C23×C4—22no—
234C23×C4—21no—
235C23×C4—21no—
236C2×D8—22no—
237C2×D8—22no—
238C2×D8C3×Q814yescomplemented
239C22×C8C23×C311yescharacteristic
240C22×C8C23×C312yes—
241C22×C8C23×C311yescharacteristic
242C4⋊Q8—24no—
243C4⋊Q8—24no—
244C4⋊Q8—22no—
245C4⋊Q8C3×D414yes—
246C22⋊Q8—28no—
247C22⋊Q8—216no—
248C22⋊Q8—44no—
249C4×Q8—22no—
250C4×Q8—22no—
251C4×D4—24no—
252C4×D4—24no—
253C4×D4—22no—
254C4×D4—22no—
255C4×D4—22no—
256C4×D4C3×D414yes—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4C3×D411yescharacteristic
263C2×C4⋊C4C3×D412yes—
264C2×C4⋊C4C23×C311yescharacteristic
265C2×C4⋊C4C23×C312yes—
266C2×C4⋊C4C23×C312yes—
267C2×C4⋊C4C23×C311yescharacteristic
268C2×C22⋊C4—22no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C42C23×C312yes—
274C2×C42C23×C311yescharacteristic
275#107—22no—
276#107—24no—
277C4⋊C8—22no—
278#102—22no—
279#102—24no—
280#102—22no—
281#102—24no—
282C4×C8—22no—
283#95—22no—
284C23×C3—22no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3—22no—
288C23×C3—41no—
289C23×C3—41no—
290C23×C3—41no—
291C23×C3—41no—
292C23×C3—41no—
293C23×C3—41no—
294C23×C3—41no—
295C23×C3—41no—
296C23×C3—41no—
297C23×C3—41no—
298C23×C3C22×D411yescharacteristic
299C3×Q8—28no—
300C3×Q8—416no—
301C3×Q8—48no—
302C3×D4—22no—
303C3×D4—22no—
304C3×D4—22no—
305C3×D4—22no—
306C3×D4—22no—
307C3×D4—41no—
308C3×D4—41no—
309C3×D4C22⋊Q814yescomplemented
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—22no—
315C2×C3×C4—24no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—416no—
320C2×C3×C4—42no—
321C2×C3×C4—41no—
322C2×C3×C4—42no—
323C2×C3×C4—41no—
324C2×C3×C4—48no—
325C2×C3×C4—41no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—42no—
331C2×C3×C4—84no—
332C2×C3×C4#12012yes—
333C2×C3×C4#12012yes—
334C2×C3×C4C2×D814yes—
335C2×C3×C4C2×D812yes—
336C2×C3×C4AGL(1,ℤ/8)12yes—
337C2×C3×C4C22×D411yescharacteristic
338C2×C3×C4C22×D411yescharacteristic
339C2×C3×C4C22×Q811yescharacteristic
340C2×C3×C4C2×#4111yescharacteristic
341C2×C3×C42−1+411yescharacteristic
342C2×C3×C42−1+411yescharacteristic
343C3×C8—22no—
344C3×C8—41no—
345C3×C8—42no—
346C24—21no—
347C24—21no—
348C2×Q8—28no—
349C2×Q8—22no—
350C2×Q8—24no—
351C2×Q8—22no—
352C2×Q8—44no—
353C2×Q8—432no—
354C2×Q8—44no—
355C2×Q8—42no—
356C2×Q8—42no—
357C2×Q8C3×D814yescomplemented
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4C2×C3×D412yes—
367C2×D4C2×C3×Q812yes—
368C2×D4C2×C3×Q812yescomplemented
369C2×D4C3×#4112yes—
370C22×C4—28no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—24no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4C2×C3×D412yes—
384C22×C4C2×C3×D411yescharacteristic
385C22×C4C2×C3×D412yes—
386C22×C4C2×C3×D411yescharacteristic
387C22×C4C24×C311yescharacteristic
388D8—28no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—22no—
397C2×C8C2×C3×Q812yescomplemented
398C4⋊C4—28no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—42no—
409C4⋊C4—44no—
410C4⋊C4—81no—
411C4⋊C4C2×C3×D414yes—
412C4⋊C4C2×C3×D412yes—
413C22⋊C4—28no—
414C22⋊C4—24no—
415C22⋊C4—42no—
416C22⋊C4—44no—
417C22⋊C4—82no—
418C42—22no—
419C42—22no—
420C42C2×C3×D412yes—
421C42C2×C3×D412yes—
422C22×C3—24no—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—42no—
426C22×C3—42no—
427C22×C3—42no—
428C22×C3—42no—
429C22×C3—42no—
430C22×C3—42no—
431C22×C3—42no—
432C22×C3—81no—
433C22×C3—81no—
434C22×C3—81no—
435C22×C3—81no—
436C22×C3C2×#12011yescharacteristic
437C22×C3#54811yescharacteristic
438C22×C3D4×Q811yescharacteristic
439C22×C3C22×D811yescharacteristic
440C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
441C22×C3#57711yescharacteristic
442C22×C3#57811yescharacteristic
443C3×C4—24no—
444C3×C4—22no—
445C3×C4—44no—
446C3×C4—42no—
447C3×C4—42no—
448C3×C4—82no—
449C3×C4—81no—
450C3×C4#44714yes—
451C3×C4C2×C22⋊Q811yescharacteristic
452C3×C4#53611yescharacteristic
453C3×C4#54811yescharacteristic
454C3×C4D4×Q811yescharacteristic
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23C22×C3×D411yescharacteristic
470Q8—28no—
471Q8—416no—
472Q8—48no—
473D4—22no—
474D4—22no—
475D4—22no—
476D4—22no—
477D4—22no—
478D4—41no—
479D4—41no—
480D4C3×C22⋊Q814yescomplemented
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—416no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—48no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—84no—
503C2×C4C3×#12012yes—
504C2×C4C3×#12012yes—
505C2×C4C2×C3×D814yes—
506C2×C4C2×C3×D812yes—
507C2×C4C3×AGL(1,ℤ/8)12yes—
508C2×C4C22×C3×D411yescharacteristic
509C2×C4C22×C3×D411yescharacteristic
510C2×C4C22×C3×Q811yescharacteristic
511C2×C4C2×C3×#4111yescharacteristic
512C2×C4C3×2−1+411yescharacteristic
513C2×C4C3×2−1+411yescharacteristic
514C8—22no—
515C8—41no—
516C8—42no—
517C2×C3—44no—
518C2×C3—82no—
519C2×C3#240911yescharacteristic
520C2×C3C2×#44711yescharacteristic
521C2×C3#301311yescharacteristic
522C2×C3#335811yescharacteristic
523C2×C3#336711yescharacteristic
524C2×C3Q8×D811yescharacteristic
525C2×C3#338411yescharacteristic
526C22—24no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22C2×C3×#12011yescharacteristic
541C22C3×#54811yescharacteristic
542C22C3×D4×Q811yescharacteristic
543C22C22×C3×D811yescharacteristic
544C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
545C22C3×#57711yescharacteristic
546C22C3×#57811yescharacteristic
547C4—24no—
548C4—22no—
549C4—44no—
550C4—42no—
551C4—42no—
552C4—81no—
553C4—82no—
554C4C3×#44714yes—
555C4C2×C3×C22⋊Q811yescharacteristic
556C4C3×#53611yescharacteristic
557C4C3×#54811yescharacteristic
558C4C3×D4×Q811yescharacteristic
559C3#2432611yescharacteristic, direct product factor, Sylow
560C2—44no—
561C2—82no—
562C2C3×#240911yescharacteristic
563C2C2×C3×#44711yescharacteristic
564C2C3×#301311yescharacteristic
565C2C3×#335811yescharacteristic
566C2C3×#336711yescharacteristic
567C2C3×Q8×D811yescharacteristic
568C2C3×#338411yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×D4×Q8C2semidirect
2. C2×C3×#474C2semidirect
4. C3×#2538C2semidirect
5. C3×#2057C2semidirect
6. C3×#2023C2semidirect
7. C3×#1990C2semidirect
8. C3×#1984C2semidirect
10. C3×#1862C2semidirect
11. C3×#1802C2semidirect
12. #24326C3direct (N × H)
46. C2×D4×Q8C2×C3semidirect
47. C2×#474C2×C3semidirect
49. #2538C2×C3semidirect
50. #2057C2×C3semidirect
51. #2023C2×C3semidirect
52. #1990C2×C3semidirect
53. #1984C2×C3semidirect
55. #1862C2×C3semidirect
56. #1802C2×C3semidirect
70. C2×C3×D8Q8semidirect
160. C2×C3×Q8D8semidirect
171. C2×C3×D4C2×Q8semidirect
200. C2×C3×C8C2×Q8semidirect
238. C2×D8C3×Q8semidirect
309. C3×D4C22⋊Q8semidirect
357. C2×Q8C3×D8semidirect
368. C2×D4C2×C3×Q8semidirect
397. C2×C8C2×C3×Q8semidirect
480. D4C3×C22⋊Q8semidirect
559. C3#24326direct (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×#4742
C2×C3×C4×D82
C3×#25382
C3×#20572
C3×#20232
C3×#19902
C3×#19842
C3×#18792
C3×#18622
C3×#18022
#243263