Finite Group Databasebeta

← C3×#24328 #10642935

Subgroups — C3×#24328 #10642935

Order 768.

1070 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#557C211yesmaximal, characteristic, complemented
2C2×C3×#474C211yesmaximal, characteristic
3C2×C3×C4×SD16C211yesmaximal, characteristic
4C3×#2538C211yesmaximal, characteristic
5C3×#2059C212yesmaximal, complemented
6C3×#2025C212yesmaximal
7C3×#1990C211yesmaximal, characteristic, complemented
8C3×#1984C211yesmaximal, characteristic, complemented
9C3×#1880C212yesmaximal
10C3×#1863C212yesmaximal, complemented
11C3×#1803C211yesmaximal, characteristic, complemented
12#24328C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2211yescharacteristic
14C3×#557—24no—
15C3×#557—24no—
16C2×C3×C4⋊Q8—21no—
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#125—22no—
19C2×C3×#125C2212yes—
20C2×C3×C22⋊Q8—21no—
21C2×C3×C4×Q8—21no—
22C2×C3×C4×Q8C2212yes—
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×Q8C2211yescharacteristic
25C2×C3×C4×D4C2211yescharacteristic
26C3×#474—24no—
27C3×C4×SD16—24no—
28C2×C3×#107C2212yes—
29C2×C3×#106C2211yescharacteristic
30C2×C3×C4⋊C8C2211yescharacteristic
31C2×C3×#103C2211yescharacteristic
32C2×C3×#103C2212yes—
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×#557C2×C311yescharacteristic, complemented
47C2×#474C2×C311yescharacteristic
48C2×C4×SD16C2×C311yescharacteristic
49#2538C2×C311yescharacteristic
50#2059C2×C312yescomplemented
51#2025C2×C312yes—
52#1990C2×C311yescharacteristic, complemented
53#1984C2×C311yescharacteristic, complemented
54#1880C2×C312yes—
55#1863C2×C312yescomplemented
56#1803C2×C311yescharacteristic, complemented
57C22×C3×Q8—21no—
58C22×C3×Q8D411yescharacteristic
59C22×C3×Q8C2311yescharacteristic
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4—21no—
62C2×C3×SD16—22no—
63C2×C3×SD16—22no—
64C2×C3×SD16Q814yescomplemented
65C22×C3×C8C2311yescharacteristic
66C22×C3×C8C2312yes—
67C22×C3×C8C2311yescharacteristic
68C3×C4⋊Q8—28no—
69C3×C4⋊Q8—22no—
70C3×C4⋊Q8D414yes—
71C3×#125—216no—
72C3×#125—24no—
73C3×#125—24no—
74C3×C22⋊Q8—44no—
75C3×C4×Q8—28no—
76C3×C4×Q8—24no—
77C3×C4×Q8—24no—
78C3×C4×Q8—22no—
79C3×C4×Q8—22no—
80C3×C4×Q8—22no—
81C3×C4×Q8D414yes—
82C3×C4×D4—22no—
83C3×C4×D4—22no—
84C2×C3×C4⋊C4—22no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—22no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—22no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—22no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4D411yescharacteristic
93C2×C3×C4⋊C4D412yes—
94C2×C3×C4⋊C4C2312yes—
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C4⋊C4C2312yes—
98C2×C3×C22⋊C4—22no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—22no—
101C2×C3×C42—22no—
102C2×C3×C42—21no—
103C2×C3×C42C2312yes—
104C2×C3×C42C2311yescharacteristic
105C3×#107—24no—
106C3×#106—22no—
107C3×C4⋊C8—22no—
108C3×#103—22no—
109C3×#103—24no—
110C3×#102—22no—
111C3×#102—24no—
112C3×C4×C8—22no—
113C3×#95—22no—
114C22×SD16C22×C311yescharacteristic
115#557—24no—
116#557—24no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8C22×C311yescharacteristic
119C2×#125—22no—
120C2×#125C22×C312yes—
121C2×C22⋊Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8C22×C312yes—
124C2×C4×Q8C22×C311yescharacteristic
125C2×C4×Q8C22×C311yescharacteristic
126C2×C4×D4C22×C311yescharacteristic
127#474—24no—
128C4×SD16—24no—
129C2×#107C22×C312yes—
130C2×#106C22×C311yescharacteristic
131C2×C4⋊C8C22×C311yescharacteristic
132C2×#103C22×C312yes—
133C2×#103C22×C311yescharacteristic
134C2×#102C22×C312yes—
135C2×#102C22×C311yescharacteristic
136C2×C4×C8C22×C311yescharacteristic
137#396—22no—
138#390C22×C312yes—
139#389C22×C312yes—
140#384C22×C311yescharacteristic
141#380—21no—
142#340C22×C312yes—
143#340C22×C312yes—
144#340C22×C312yes—
145#336C22×C312yes—
146#336C22×C312yes—
147C24×C3—21no—
148C2×C3×Q8—22no—
149C2×C3×Q8—24no—
150C2×C3×Q8—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8—21no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—42no—
156C2×C3×Q8—42no—
157C2×C3×Q8C2×D412yes—
158C2×C3×Q8C2×Q812yes—
159C2×C3×Q8#4112yes—
160C2×C3×D4—21no—
161C2×C3×D4—21no—
162C2×C3×D4—21no—
163C2×C3×D4—21no—
164C2×C3×D4—21no—
165C2×C3×D4C2×Q812yes—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—22no—
169C22×C3×C4—21no—
170C22×C3×C4—22no—
171C22×C3×C4—42no—
172C22×C3×C4—42no—
173C22×C3×C4—42no—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2411yescharacteristic
179C3×SD16—28no—
180C2×C3×C8—21no—
181C2×C3×C8—21no—
182C2×C3×C8—21no—
183C2×C3×C8—22no—
184C2×C3×C8—22no—
185C2×C3×C8—21no—
186C2×C3×C8—21no—
187C2×C3×C8—22no—
188C2×C3×C8C2×Q812yes—
189C3×C4⋊C4—28no—
190C3×C4⋊C4—28no—
191C3×C4⋊C4—28no—
192C3×C4⋊C4—24no—
193C3×C4⋊C4—24no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—24no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—44no—
203C3×C4⋊C4—42no—
204C3×C4⋊C4—81no—
205C3×C4⋊C4C2×D412yes—
206C3×C4⋊C4C2×D414yes—
207C3×C22⋊C4—44no—
208C3×C22⋊C4—42no—
209C3×C22⋊C4—82no—
210C3×C42—28no—
211C3×C42—24no—
212C3×C42—22no—
213C3×C42—22no—
214C3×C42C2×D412yes—
215C3×C42C2×D412yes—
216C22×Q8—21no—
217C22×Q8C3×D411yescharacteristic
218C22×Q8C23×C311yescharacteristic
219C22×D4C23×C311yescharacteristic
220C23×C4—21no—
221C2×SD16—22no—
222C2×SD16—22no—
223C2×SD16C3×Q814yescomplemented
224C22×C8C23×C311yescharacteristic
225C22×C8C23×C312yes—
226C22×C8C23×C311yescharacteristic
227C4⋊Q8—28no—
228C4⋊Q8—22no—
229C4⋊Q8C3×D414yes—
230#125—216no—
231#125—24no—
232#125—24no—
233C22⋊Q8—44no—
234C4×Q8—28no—
235C4×Q8—24no—
236C4×Q8—24no—
237C4×Q8—22no—
238C4×Q8—22no—
239C4×Q8—22no—
240C4×Q8C3×D414yes—
241C4×D4—22no—
242C4×D4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4C3×D411yescharacteristic
252C2×C4⋊C4C3×D412yes—
253C2×C4⋊C4C23×C311yescharacteristic
254C2×C4⋊C4C23×C312yes—
255C2×C4⋊C4C23×C311yescharacteristic
256C2×C4⋊C4C23×C312yes—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C42—22no—
261C2×C42—21no—
262C2×C42C23×C312yes—
263C2×C42C23×C311yescharacteristic
264#107—24no—
265#106—22no—
266C4⋊C8—22no—
267#103—22no—
268#103—24no—
269#102—22no—
270#102—24no—
271C4×C8—22no—
272#95—22no—
273C23×C3—22no—
274C23×C3—41no—
275C23×C3—41no—
276C23×C3—41no—
277C23×C3—41no—
278C23×C3—41no—
279C23×C3—41no—
280C23×C3C22×D411yescharacteristic
281C3×Q8—22no—
282C3×Q8—22no—
283C3×Q8—22no—
284C3×Q8—44no—
285C3×Q8—48no—
286C3×Q8C22⋊Q814yescomplemented
287C3×D4—22no—
288C3×D4—22no—
289C3×D4—41no—
290C3×D4—41no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—24no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—42no—
310C2×C3×C4—41no—
311C2×C3×C4—42no—
312C2×C3×C4—41no—
313C2×C3×C4—42no—
314C2×C3×C4—84no—
315C2×C3×C4#12012yes—
316C2×C3×C4#12012yes—
317C2×C3×C4#13512yes—
318C2×C3×C4AGL(1,ℤ/8)12yes—
319C2×C3×C4C22×D411yescharacteristic
320C2×C3×C4C22×D411yescharacteristic
321C2×C3×C4C22×Q811yescharacteristic
322C2×C3×C4C2×#4111yescharacteristic
323C2×C3×C42−1+411yescharacteristic
324C2×C3×C42−1+411yescharacteristic
325C3×C8—22no—
326C3×C8—41no—
327C3×C8—42no—
328C24—21no—
329C2×Q8—22no—
330C2×Q8—24no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—42no—
337C2×Q8—42no—
338C2×Q8C2×C3×D412yes—
339C2×Q8C2×C3×Q812yes—
340C2×Q8C3×#4112yes—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4C2×C3×Q812yes—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4C2×C3×D412yes—
356C22×C4C2×C3×D412yes—
357C22×C4C2×C3×D411yescharacteristic
358C22×C4C2×C3×D411yescharacteristic
359C22×C4C24×C311yescharacteristic
360SD16—28no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—22no—
369C2×C8C2×C3×Q812yes—
370C4⋊C4—28no—
371C4⋊C4—28no—
372C4⋊C4—28no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—22no—
376C4⋊C4—24no—
377C4⋊C4—21no—
378C4⋊C4—22no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—44no—
384C4⋊C4—42no—
385C4⋊C4—81no—
386C4⋊C4C2×C3×D412yes—
387C4⋊C4C2×C3×D414yes—
388C22⋊C4—44no—
389C22⋊C4—42no—
390C22⋊C4—82no—
391C42—28no—
392C42—24no—
393C42—22no—
394C42—22no—
395C42C2×C3×D412yes—
396C42C2×C3×D412yes—
397C22×C3—42no—
398C22×C3—42no—
399C22×C3—42no—
400C22×C3—81no—
401C22×C3—81no—
402C22×C3—81no—
403C22×C3—81no—
404C22×C3C2×#12011yescharacteristic
405C22×C3#54811yescharacteristic
406C22×C3D4×Q811yescharacteristic
407C22×C3C2×#13511yescharacteristic
408C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
409C22×C3#57711yescharacteristic
410C22×C3#57811yescharacteristic
411C3×C4—24no—
412C3×C4—22no—
413C3×C4—42no—
414C3×C4—44no—
415C3×C4—42no—
416C3×C4—42no—
417C3×C4—82no—
418C3×C4—81no—
419C3×C4#44914yes—
420C3×C4C2×C22⋊Q811yescharacteristic
421C3×C4#53611yescharacteristic
422C3×C4#54811yescharacteristic
423C3×C4D4×Q811yescharacteristic
424C23—22no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23C22×C3×D411yescharacteristic
432Q8—22no—
433Q8—22no—
434Q8—22no—
435Q8—44no—
436Q8—48no—
437Q8C3×C22⋊Q814yescomplemented
438D4—22no—
439D4—22no—
440D4—41no—
441D4—41no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—84no—
466C2×C4C3×#12012yes—
467C2×C4C3×#12012yes—
468C2×C4C3×#13512yes—
469C2×C4C3×AGL(1,ℤ/8)12yes—
470C2×C4C22×C3×D411yescharacteristic
471C2×C4C22×C3×D411yescharacteristic
472C2×C4C22×C3×Q811yescharacteristic
473C2×C4C2×C3×#4111yescharacteristic
474C2×C4C3×2−1+411yescharacteristic
475C2×C4C3×2−1+411yescharacteristic
476C8—22no—
477C8—41no—
478C8—42no—
479C2×C3—82no—
480C2×C3#240911yescharacteristic
481C2×C3C2×#44911yescharacteristic
482C2×C3#301311yescharacteristic
483C2×C3#336411yescharacteristic
484C2×C3#336811yescharacteristic
485C2×C3#338111yescharacteristic
486C2×C3#338611yescharacteristic
487C22—42no—
488C22—42no—
489C22—42no—
490C22—81no—
491C22—81no—
492C22—81no—
493C22—81no—
494C22C2×C3×#12011yescharacteristic
495C22C3×#54811yescharacteristic
496C22C3×D4×Q811yescharacteristic
497C22C2×C3×#13511yescharacteristic
498C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
499C22C3×#57711yescharacteristic
500C22C3×#57811yescharacteristic
501C4—24no—
502C4—22no—
503C4—42no—
504C4—44no—
505C4—42no—
506C4—42no—
507C4—81no—
508C4—82no—
509C4C3×#44914yes—
510C4C2×C3×C22⋊Q811yescharacteristic
511C4C3×#53611yescharacteristic
512C4C3×#54811yescharacteristic
513C4C3×D4×Q811yescharacteristic
514C3#2432811yescharacteristic, direct product factor, Sylow
515C2—82no—
516C2C3×#240911yescharacteristic
517C2C2×C3×#44911yescharacteristic
518C2C3×#301311yescharacteristic
519C2C3×#336411yescharacteristic
520C2C3×#336811yescharacteristic
521C2C3×#338111yescharacteristic
522C2C3×#338611yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
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×#557C2semidirect
5. C3×#2059C2semidirect
7. C3×#1990C2semidirect
8. C3×#1984C2semidirect
10. C3×#1863C2semidirect
11. C3×#1803C2semidirect
12. #24328C3direct (N × H)
46. C2×#557C2×C3semidirect
50. #2059C2×C3semidirect
52. #1990C2×C3semidirect
53. #1984C2×C3semidirect
55. #1863C2×C3semidirect
56. #1803C2×C3semidirect
64. C2×C3×SD16Q8semidirect
223. C2×SD16C3×Q8semidirect
286. C3×Q8C22⋊Q8semidirect
437. Q8C3×C22⋊Q8semidirect
514. C3#24328direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5572
C2×C3×#4742
C2×C3×C4×SD162
C3×#25382
C3×#20592
C3×#20252
C3×#19902
C3×#19842
C3×#18802
C3×#18632
C3×#18032
#243283