Finite Group Databasebeta

← C3×#24342 #10642949

Subgroups — C3×#24342 #10642949

Order 768.

1248 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×Q8C211yesmaximal, characteristic
2C2×C3×#477C211yesmaximal, characteristic, complemented
3C2×C3×#440C211yesmaximal, characteristic
4C3×#2581C211yesmaximal, characteristic, complemented
5C3×#2057C212yesmaximal, complemented
6C3×#2024C212yesmaximal, complemented
7C3×#1993C211yesmaximal, characteristic, complemented
8C3×#1985C211yesmaximal, characteristic, complemented
9C3×#1877C212yesmaximal
10C3×#1862C212yesmaximal
11C3×#1804C211yesmaximal, characteristic
12#24342C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8—21no—
14C22×C3×SD16C2211yescharacteristic
15C3×D4×Q8—24no—
16C3×D4×Q8—24no—
17C2×C3×C4⋊Q8C2212yes—
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×C22⋊Q8—22no—
20C2×C3×C22⋊Q8—21no—
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×Q8C2211yescharacteristic
23C2×C3×C4×D4C2212yes—
24C2×C3×C4×D4C2211yescharacteristic
25C3×#477—24no—
26C3×#440—24no—
27C2×C3×#107C2211yescharacteristic
28C2×C3×#107C2212yes—
29C2×C3×C4⋊C8C2211yescharacteristic
30C2×C3×#103C2211yescharacteristic
31C2×C3×#103C2212yes—
32C2×C3×#102C2211yescharacteristic
33C2×C3×#102C2212yes—
34C2×C3×#97C2211yescharacteristic
35C3×#400—21no—
36C3×#395—21no—
37C3×#391C2212yes—
38C3×#389C2212yes—
39C3×#387—22no—
40C3×#382C2211yescharacteristic
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#340C2212yes—
44C3×#336C2212yes—
45C3×#336C2212yes—
46C2×D4×Q8C2×C311yescharacteristic
47C2×#477C2×C311yescharacteristic, complemented
48C2×#440C2×C311yescharacteristic
49#2581C2×C311yescharacteristic, complemented
50#2057C2×C312yescomplemented
51#2024C2×C312yescomplemented
52#1993C2×C311yescharacteristic, complemented
53#1985C2×C311yescharacteristic, complemented
54#1877C2×C312yes—
55#1862C2×C312yes—
56#1804C2×C311yescharacteristic
57C22×C3×Q8—28no—
58C22×C3×Q8—21no—
59C22×C3×Q8—22no—
60C22×C3×Q8—48no—
61C22×C3×Q8D411yescharacteristic
62C22×C3×Q8C2311yescharacteristic
63C22×C3×D4C2311yescharacteristic
64C23×C3×C4—22no—
65C23×C3×C4—21no—
66C2×C3×SD16—22no—
67C2×C3×SD16—22no—
68C2×C3×SD16Q814yescomplemented
69C22×C3×C8C2311yescharacteristic
70C22×C3×C8C2312yes—
71C22×C3×C8C2311yescharacteristic
72C3×C4⋊Q8—24no—
73C3×C4⋊Q8—24no—
74C3×C4⋊Q8—22no—
75C3×C4⋊Q8D414yes—
76C3×C22⋊Q8—28no—
77C3×C22⋊Q8—216no—
78C3×C4×Q8—22no—
79C3×C4×Q8—22no—
80C3×C4×Q8—22no—
81C3×C4×Q8—22no—
82C3×C4×D4—24no—
83C3×C4×D4—24no—
84C3×C4×D4—22no—
85C3×C4×D4D414yes—
86C2×C3×C4⋊C4—22no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—22no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4D412yes—
91C2×C3×C4⋊C4D411yescharacteristic
92C2×C3×C4⋊C4C2312yes—
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2312yes—
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4—22no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C42—21no—
99C2×C3×C42C2312yes—
100C2×C3×C42C2311yescharacteristic
101C3×#107—22no—
102C3×#107—24no—
103C3×C4⋊C8—22no—
104C3×#103—22no—
105C3×#103—24no—
106C3×#102—22no—
107C3×#102—24no—
108C3×#97—22no—
109C3×#95—21no—
110C3×#95—21no—
111C3×#95—22no—
112C3×#95—22no—
113C3×#95—22no—
114C23×Q8—21no—
115C22×SD16C22×C311yescharacteristic
116D4×Q8—24no—
117D4×Q8—24no—
118C2×C4⋊Q8C22×C312yes—
119C2×C4⋊Q8C22×C311yescharacteristic
120C2×C22⋊Q8—22no—
121C2×C22⋊Q8—21no—
122C2×C4×Q8C22×C311yescharacteristic
123C2×C4×Q8C22×C311yescharacteristic
124C2×C4×D4C22×C312yes—
125C2×C4×D4C22×C311yescharacteristic
126#477—24no—
127#440—24no—
128C2×#107C22×C312yes—
129C2×#107C22×C311yescharacteristic
130C2×C4⋊C8C22×C311yescharacteristic
131C2×#103C22×C312yes—
132C2×#103C22×C311yescharacteristic
133C2×#102C22×C312yes—
134C2×#102C22×C311yescharacteristic
135C2×#97C22×C311yescharacteristic
136#400—21no—
137#395—21no—
138#391C22×C312yes—
139#389C22×C312yes—
140#387—22no—
141#382C22×C311yescharacteristic
142#340C22×C312yes—
143#340C22×C312yes—
144#340C22×C312yes—
145#336C22×C312yes—
146#336C22×C312yes—
147C24×C3—21no—
148C2×C3×Q8—28no—
149C2×C3×Q8—22no—
150C2×C3×Q8—24no—
151C2×C3×Q8—24no—
152C2×C3×Q8—21no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8—44no—
158C2×C3×Q8—44no—
159C2×C3×Q8—432no—
160C2×C3×Q8—44no—
161C2×C3×Q8C2×Q812yescomplemented
162C2×C3×D4—21no—
163C2×C3×D4—21no—
164C2×C3×D4—21no—
165C2×C3×D4C2×D412yes—
166C2×C3×D4C2×Q812yes—
167C2×C3×D4#4112yes—
168C22×C3×C4—28no—
169C22×C3×C4—21no—
170C22×C3×C4—22no—
171C22×C3×C4—21no—
172C22×C3×C4—24no—
173C22×C3×C4—22no—
174C22×C3×C4—21no—
175C22×C3×C4—44no—
176C22×C3×C4—44no—
177C22×C3×C4—44no—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2411yescharacteristic
183C3×SD16—28no—
184C2×C3×C8—21no—
185C2×C3×C8—21no—
186C2×C3×C8—21no—
187C2×C3×C8—22no—
188C2×C3×C8—22no—
189C2×C3×C8—21no—
190C2×C3×C8—21no—
191C2×C3×C8—22no—
192C2×C3×C8C2×Q812yescomplemented
193C3×C4⋊C4—28no—
194C3×C4⋊C4—24no—
195C3×C4⋊C4—24no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—42no—
204C3×C4⋊C4—44no—
205C3×C4⋊C4C2×D414yes—
206C3×C4⋊C4C2×D412yes—
207C3×C22⋊C4—28no—
208C3×C22⋊C4—24no—
209C3×C42—22no—
210C3×C42—22no—
211C3×C42—42no—
212C3×C42C2×D412yes—
213C3×C42C2×D412yes—
214C22×Q8—28no—
215C22×Q8—21no—
216C22×Q8—22no—
217C22×Q8—48no—
218C22×Q8C3×D411yescharacteristic
219C22×Q8C23×C311yescharacteristic
220C22×D4C23×C311yescharacteristic
221C23×C4—22no—
222C23×C4—21no—
223C2×SD16—22no—
224C2×SD16—22no—
225C2×SD16C3×Q814yescomplemented
226C22×C8C23×C311yescharacteristic
227C22×C8C23×C312yes—
228C22×C8C23×C311yescharacteristic
229C4⋊Q8—24no—
230C4⋊Q8—24no—
231C4⋊Q8—22no—
232C4⋊Q8C3×D414yes—
233C22⋊Q8—28no—
234C22⋊Q8—216no—
235C4×Q8—22no—
236C4×Q8—22no—
237C4×Q8—22no—
238C4×Q8—22no—
239C4×D4—24no—
240C4×D4—24no—
241C4×D4—22no—
242C4×D4C3×D414yes—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4C3×D412yes—
248C2×C4⋊C4C3×D411yescharacteristic
249C2×C4⋊C4C23×C311yescharacteristic
250C2×C4⋊C4C23×C312yes—
251C2×C4⋊C4C23×C312yes—
252C2×C4⋊C4C23×C311yescharacteristic
253C2×C22⋊C4—22no—
254C2×C22⋊C4—21no—
255C2×C42—21no—
256C2×C42C23×C312yes—
257C2×C42C23×C311yescharacteristic
258#107—22no—
259#107—24no—
260C4⋊C8—22no—
261#103—22no—
262#103—24no—
263#102—22no—
264#102—24no—
265#97—22no—
266#95—21no—
267#95—21no—
268#95—22no—
269#95—22no—
270#95—22no—
271C23×C3—22no—
272C23×C3—22no—
273C23×C3—22no—
274C23×C3—42no—
275C23×C3—42no—
276C23×C3C22×D411yescharacteristic
277C3×Q8—28no—
278C3×Q8—22no—
279C3×Q8—22no—
280C3×Q8—48no—
281C3×Q8—416no—
282C3×Q8—41no—
283C3×Q8—41no—
284C3×D4—22no—
285C3×D4—22no—
286C3×D4—22no—
287C3×D4C22⋊Q814yes—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—24no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—24no—
297C2×C3×C4—48no—
298C2×C3×C4—416no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—41no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4#12012yes—
310C2×C3×C4#12012yes—
311C2×C3×C4AGL(1,ℤ/8)14yes—
312C2×C3×C4C22×D411yescharacteristic
313C2×C3×C4C22×D411yescharacteristic
314C2×C3×C4C22×Q811yescharacteristic
315C2×C3×C4C2×#4111yescharacteristic
316C2×C3×C42−1+411yescharacteristic
317C2×C3×C42−1+411yescharacteristic
318C3×C8—22no—
319C3×C8—41no—
320C3×C8—42no—
321C24—21no—
322C2×Q8—28no—
323C2×Q8—22no—
324C2×Q8—24no—
325C2×Q8—24no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—21no—
331C2×Q8—44no—
332C2×Q8—44no—
333C2×Q8—432no—
334C2×Q8—44no—
335C2×Q8C2×C3×Q812yescomplemented
336C2×D4—21no—
337C2×D4—21no—
338C2×D4—21no—
339C2×D4C2×C3×D412yes—
340C2×D4C2×C3×Q812yes—
341C2×D4C3×#4112yes—
342C22×C4—28no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4—44no—
352C22×C4C2×C3×D412yes—
353C22×C4C2×C3×D411yescharacteristic
354C22×C4C2×C3×D411yescharacteristic
355C22×C4C2×C3×D412yes—
356C22×C4C24×C311yescharacteristic
357SD16—28no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—22no—
366C2×C8C2×C3×Q812yescomplemented
367C4⋊C4—28no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—42no—
378C4⋊C4—44no—
379C4⋊C4C2×C3×D414yes—
380C4⋊C4C2×C3×D412yes—
381C22⋊C4—28no—
382C22⋊C4—24no—
383C42—22no—
384C42—22no—
385C42—42no—
386C42C2×C3×D412yes—
387C42C2×C3×D412yes—
388C22×C3—24no—
389C22×C3—44no—
390C22×C3—44no—
391C22×C3—42no—
392C22×C3—42no—
393C22×C3C2×#12011yescharacteristic
394C22×C3#54811yescharacteristic
395C22×C3D4×Q811yescharacteristic
396C22×C3C2×AGL(1,ℤ/8)12yes—
397C22×C3#57812yes—
398C3×C4—22no—
399C3×C4—24no—
400C3×C4—22no—
401C3×C4—42no—
402C3×C4—44no—
403C3×C4—42no—
404C3×C4—42no—
405C3×C4—82no—
406C3×C4—81no—
407C3×C4C2×C22⋊Q811yescharacteristic
408C3×C4#53611yescharacteristic
409C3×C4#54811yescharacteristic
410C3×C4D4×Q811yescharacteristic
411C23—22no—
412C23—22no—
413C23—22no—
414C23—42no—
415C23—42no—
416C23C22×C3×D411yescharacteristic
417Q8—28no—
418Q8—22no—
419Q8—22no—
420Q8—48no—
421Q8—416no—
422Q8—41no—
423Q8—41no—
424D4—22no—
425D4—22no—
426D4—22no—
427D4C3×C22⋊Q814yes—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—48no—
438C2×C4—416no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4C3×#12012yes—
450C2×C4C3×#12012yes—
451C2×C4C3×AGL(1,ℤ/8)14yes—
452C2×C4C22×C3×D411yescharacteristic
453C2×C4C22×C3×D411yescharacteristic
454C2×C4C22×C3×Q811yescharacteristic
455C2×C4C2×C3×#4111yescharacteristic
456C2×C4C3×2−1+411yescharacteristic
457C2×C4C3×2−1+411yescharacteristic
458C8—22no—
459C8—41no—
460C8—42no—
461C2×C3—44no—
462C2×C3#240911yescharacteristic
463C2×C3#300311yescharacteristic
464C2×C3#301111yescharacteristic
465C2×C3#336712yes—
466C2×C3#338612yes—
467C22—24no—
468C22—44no—
469C22—42no—
470C22—42no—
471C22—44no—
472C22C2×C3×#12011yescharacteristic
473C22C3×#54811yescharacteristic
474C22C3×D4×Q811yescharacteristic
475C22C2×C3×AGL(1,ℤ/8)12yes—
476C22C3×#57812yes—
477C4—22no—
478C4—24no—
479C4—22no—
480C4—42no—
481C4—44no—
482C4—42no—
483C4—42no—
484C4—81no—
485C4—82no—
486C4C2×C3×C22⋊Q811yescharacteristic
487C4C3×#53611yescharacteristic
488C4C3×#54811yescharacteristic
489C4C3×D4×Q811yescharacteristic
490C3#2434211yescharacteristic, direct product factor, Sylow
491C2—44no—
492C2C3×#240911yescharacteristic
493C2C3×#300311yescharacteristic
494C2C3×#301111yescharacteristic
495C2C3×#336712yes—
496C2C3×#338612yes—
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
2. C2×C3×#477C2semidirect
4. C3×#2581C2semidirect
5. C3×#2057C2semidirect
6. C3×#2024C2semidirect
7. C3×#1993C2semidirect
8. C3×#1985C2semidirect
12. #24342C3direct (N × H)
47. C2×#477C2×C3semidirect
49. #2581C2×C3semidirect
50. #2057C2×C3semidirect
51. #2024C2×C3semidirect
52. #1993C2×C3semidirect
53. #1985C2×C3semidirect
68. C2×C3×SD16Q8semidirect
161. C2×C3×Q8C2×Q8semidirect
192. C2×C3×C8C2×Q8semidirect
225. C2×SD16C3×Q8semidirect
335. C2×Q8C2×C3×Q8semidirect
366. C2×C8C2×C3×Q8semidirect
490. C3#24342direct (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×#4772
C2×C3×#4402
C3×#25812
C3×#20572
C3×#20242
C3×#19932
C3×#19852
C3×#18772
C3×#18622
C3×#18042
#243423