Finite Group Databasebeta

← C3×#24341 #10642948

Subgroups — C3×#24341 #10642948

Order 768.

1068 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×Q82C211yesmaximal, characteristic, complemented
2C2×C3×#474C211yesmaximal, characteristic
3C2×C3×#440C211yesmaximal, characteristic
4C3×#2580C211yesmaximal, characteristic
5C3×#2057C212yesmaximal, complemented
6C3×#2023C212yesmaximal
7C3×#1993C211yesmaximal, characteristic, complemented
8C3×#1985C211yesmaximal, characteristic, complemented
9C3×#1880C212yesmaximal
10C3×#1863C212yesmaximal, complemented
11C3×#1805C211yesmaximal, characteristic, complemented
12#24341C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2211yescharacteristic
14C3×Q82—24no—
15C3×Q82—24no—
16C2×C3×C4⋊Q8—22no—
17C2×C3×C4⋊Q8—21no—
18C2×C3×C4⋊Q8C2212yes—
19C2×C3×C4⋊Q8C2211yescharacteristic
20C2×C3×C4×Q8—21no—
21C2×C3×C4×Q8C2212yes—
22C2×C3×C4×Q8C2211yescharacteristic
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×D4C2211yescharacteristic
25C3×#474—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×#399—21no—
36C3×#393—21no—
37C3×#390C2212yes—
38C3×#389C2212yes—
39C3×#385—22no—
40C3×#382C2211yescharacteristic
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#340C2212yes—
44C3×#336C2212yes—
45C3×#336C2212yes—
46C2×Q82C2×C311yescharacteristic, complemented
47C2×#474C2×C311yescharacteristic
48C2×#440C2×C311yescharacteristic
49#2580C2×C311yescharacteristic
50#2057C2×C312yescomplemented
51#2023C2×C312yes—
52#1993C2×C311yescharacteristic, complemented
53#1985C2×C311yescharacteristic, complemented
54#1880C2×C312yes—
55#1863C2×C312yescomplemented
56#1805C2×C311yescharacteristic, complemented
57C22×C3×Q8—22no—
58C22×C3×Q8—21no—
59C22×C3×Q8D411yescharacteristic
60C22×C3×Q8C2311yescharacteristic
61C22×C3×D4C2311yescharacteristic
62C23×C3×C4—21no—
63C2×C3×SD16—22no—
64C2×C3×SD16—22no—
65C2×C3×SD16Q814yescomplemented
66C22×C3×C8C2311yescharacteristic
67C22×C3×C8C2312yes—
68C22×C3×C8C2311yescharacteristic
69C3×C4⋊Q8—216no—
70C3×C4⋊Q8—28no—
71C3×C4⋊Q8—24no—
72C3×C4⋊Q8—24no—
73C3×C4⋊Q8—22no—
74C3×C4⋊Q8D414yes—
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⋊C4D412yes—
90C2×C3×C4⋊C4D411yescharacteristic
91C2×C3×C4⋊C4C2312yes—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2312yes—
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C22⋊C4—22no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—22no—
98C2×C3×C42—22no—
99C2×C3×C42—21no—
100C2×C3×C42C2312yes—
101C2×C3×C42C2311yescharacteristic
102C3×#107—22no—
103C3×#107—24no—
104C3×C4⋊C8—22no—
105C3×#103—22no—
106C3×#103—24no—
107C3×#102—22no—
108C3×#102—24no—
109C3×#97—22no—
110C3×#95—21no—
111C3×#95—21no—
112C3×#95—22no—
113C22×SD16C22×C311yescharacteristic
114Q82—24no—
115Q82—24no—
116C2×C4⋊Q8—22no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8C22×C312yes—
119C2×C4⋊Q8C22×C311yescharacteristic
120C2×C4×Q8—21no—
121C2×C4×Q8C22×C312yes—
122C2×C4×Q8C22×C311yescharacteristic
123C2×C4×Q8C22×C311yescharacteristic
124C2×C4×D4C22×C311yescharacteristic
125#474—24no—
126#440—24no—
127C2×#107C22×C311yescharacteristic
128C2×#107C22×C312yes—
129C2×C4⋊C8C22×C311yescharacteristic
130C2×#103C22×C312yes—
131C2×#103C22×C311yescharacteristic
132C2×#102C22×C312yes—
133C2×#102C22×C311yescharacteristic
134C2×#97C22×C311yescharacteristic
135#399—21no—
136#393—21no—
137#390C22×C312yes—
138#389C22×C312yes—
139#385—22no—
140#382C22×C311yescharacteristic
141#340C22×C312yes—
142#340C22×C312yes—
143#340C22×C312yes—
144#336C22×C312yes—
145#336C22×C312yes—
146C24×C3—21no—
147C2×C3×Q8—28no—
148C2×C3×Q8—22no—
149C2×C3×Q8—24no—
150C2×C3×Q8—24no—
151C2×C3×Q8—21no—
152C2×C3×Q8—21no—
153C2×C3×Q8—21no—
154C2×C3×Q8—82no—
155C2×C3×Q8—82no—
156C2×C3×Q8—82no—
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—22no—
168C22×C3×C4—21no—
169C22×C3×C4—22no—
170C22×C3×C4—21no—
171C22×C3×C4—42no—
172C22×C3×C4—82no—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2411yescharacteristic
178C3×SD16—28no—
179C2×C3×C8—21no—
180C2×C3×C8—21no—
181C2×C3×C8—21no—
182C2×C3×C8—22no—
183C2×C3×C8—22no—
184C2×C3×C8—21no—
185C2×C3×C8—21no—
186C2×C3×C8—22no—
187C2×C3×C8C2×Q812yes—
188C3×C4⋊C4—28no—
189C3×C4⋊C4—28no—
190C3×C4⋊C4—24no—
191C3×C4⋊C4—24no—
192C3×C4⋊C4—24no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—82no—
201C3×C4⋊C4C2×D414yes—
202C3×C4⋊C4C2×D412yes—
203C3×C22⋊C4—42no—
204C3×C22⋊C4—44no—
205C3×C22⋊C4—82no—
206C3×C42—28no—
207C3×C42—24no—
208C3×C42—22no—
209C3×C42—22no—
210C3×C42C2×D412yes—
211C3×C42C2×D412yes—
212C22×Q8—22no—
213C22×Q8—21no—
214C22×Q8C3×D411yescharacteristic
215C22×Q8C23×C311yescharacteristic
216C22×D4C23×C311yescharacteristic
217C23×C4—21no—
218C2×SD16—22no—
219C2×SD16—22no—
220C2×SD16C3×Q814yescomplemented
221C22×C8C23×C311yescharacteristic
222C22×C8C23×C312yes—
223C22×C8C23×C311yescharacteristic
224C4⋊Q8—216no—
225C4⋊Q8—28no—
226C4⋊Q8—24no—
227C4⋊Q8—24no—
228C4⋊Q8—22no—
229C4⋊Q8C3×D414yes—
230C4×Q8—28no—
231C4×Q8—24no—
232C4×Q8—24no—
233C4×Q8—22no—
234C4×Q8—22no—
235C4×Q8—22no—
236C4×Q8C3×D414yes—
237C4×D4—22no—
238C4×D4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4C3×D412yes—
245C2×C4⋊C4C3×D411yescharacteristic
246C2×C4⋊C4C23×C311yescharacteristic
247C2×C4⋊C4C23×C312yes—
248C2×C4⋊C4C23×C312yes—
249C2×C4⋊C4C23×C311yescharacteristic
250C2×C22⋊C4—21no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C42—22no—
254C2×C42—21no—
255C2×C42C23×C312yes—
256C2×C42C23×C311yescharacteristic
257#107—22no—
258#107—24no—
259C4⋊C8—22no—
260#103—22no—
261#103—24no—
262#102—22no—
263#102—24no—
264#97—22no—
265#95—21no—
266#95—21no—
267#95—22no—
268C23×C3—22no—
269C23×C3—41no—
270C23×C3—41no—
271C23×C3—42no—
272C23×C3—42no—
273C23×C3C22×D411yescharacteristic
274C3×Q8—28no—
275C3×Q8—22no—
276C3×Q8—22no—
277C3×Q8—22no—
278C3×Q8—84no—
279C3×Q8—88no—
280C3×Q8C22⋊Q814yes—
281C3×D4—22no—
282C3×D4—22no—
283C3×D4—41no—
284C3×D4—41no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
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—22no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—41no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—84no—
306C2×C3×C4#12012yes—
307C2×C3×C4#12012yes—
308C2×C3×C4#13714yes—
309C2×C3×C4C22×D411yescharacteristic
310C2×C3×C4C22×D411yescharacteristic
311C2×C3×C4C22×Q811yescharacteristic
312C2×C3×C4C2×#4111yescharacteristic
313C2×C3×C42−1+411yescharacteristic
314C2×C3×C42−1+411yescharacteristic
315C3×C8—22no—
316C3×C8—41no—
317C3×C8—42no—
318C24—21no—
319C2×Q8—28no—
320C2×Q8—22no—
321C2×Q8—24no—
322C2×Q8—24no—
323C2×Q8—21no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8—82no—
327C2×Q8—82no—
328C2×Q8—82no—
329C2×Q8C2×C3×D412yes—
330C2×Q8C2×C3×Q812yes—
331C2×Q8C3×#4112yes—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—21no—
336C2×D4—21no—
337C2×D4C2×C3×Q812yes—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—42no—
344C22×C4—82no—
345C22×C4C2×C3×D412yes—
346C22×C4C2×C3×D411yescharacteristic
347C22×C4C2×C3×D411yescharacteristic
348C22×C4C2×C3×D412yes—
349C22×C4C24×C311yescharacteristic
350SD16—28no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—22no—
359C2×C8C2×C3×Q812yes—
360C4⋊C4—28no—
361C4⋊C4—28no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—82no—
373C4⋊C4C2×C3×D414yes—
374C4⋊C4C2×C3×D412yes—
375C22⋊C4—42no—
376C22⋊C4—44no—
377C22⋊C4—82no—
378C42—28no—
379C42—24no—
380C42—22no—
381C42—22no—
382C42C2×C3×D412yes—
383C42C2×C3×D412yes—
384C22×C3—44no—
385C22×C3—42no—
386C22×C3—82no—
387C22×C3—81no—
388C22×C3—81no—
389C22×C3C2×#12011yescharacteristic
390C22×C3#54811yescharacteristic
391C22×C3D4×Q811yescharacteristic
392C22×C3C2×#13712yes—
393C22×C3#57612yes—
394C3×C4—24no—
395C3×C4—22no—
396C3×C4—24no—
397C3×C4—22no—
398C3×C4—42no—
399C3×C4—44no—
400C3×C4—42no—
401C3×C4—82no—
402C3×C4—81no—
403C3×C4C2×C22⋊Q811yescharacteristic
404C3×C4#53611yescharacteristic
405C3×C4#54811yescharacteristic
406C3×C4D4×Q811yescharacteristic
407C23—22no—
408C23—42no—
409C23—42no—
410C23—41no—
411C23—41no—
412C23C22×C3×D411yescharacteristic
413Q8—28no—
414Q8—22no—
415Q8—22no—
416Q8—22no—
417Q8—84no—
418Q8—88no—
419Q8C3×C22⋊Q814yes—
420D4—22no—
421D4—22no—
422D4—41no—
423D4—41no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—22no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—84no—
445C2×C4C3×#12012yes—
446C2×C4C3×#12012yes—
447C2×C4C3×#13714yes—
448C2×C4C22×C3×D411yescharacteristic
449C2×C4C22×C3×D411yescharacteristic
450C2×C4C22×C3×Q811yescharacteristic
451C2×C4C2×C3×#4111yescharacteristic
452C2×C4C3×2−1+411yescharacteristic
453C2×C4C3×2−1+411yescharacteristic
454C8—22no—
455C8—41no—
456C8—42no—
457C2×C3—82no—
458C2×C3#240911yescharacteristic
459C2×C3#300411yescharacteristic
460C2×C3#300811yescharacteristic
461C2×C3#337512yes—
462C2×C3#338512yes—
463C22—44no—
464C22—42no—
465C22—81no—
466C22—81no—
467C22—82no—
468C22C2×C3×#12011yescharacteristic
469C22C3×#54811yescharacteristic
470C22C3×D4×Q811yescharacteristic
471C22C2×C3×#13712yes—
472C22C3×#57612yes—
473C4—24no—
474C4—22no—
475C4—24no—
476C4—22no—
477C4—42no—
478C4—44no—
479C4—42no—
480C4—81no—
481C4—82no—
482C4C2×C3×C22⋊Q811yescharacteristic
483C4C3×#53611yescharacteristic
484C4C3×#54811yescharacteristic
485C4C3×D4×Q811yescharacteristic
486C3#2434111yescharacteristic, direct product factor, Sylow
487C2—82no—
488C2C3×#240911yescharacteristic
489C2C3×#300411yescharacteristic
490C2C3×#300811yescharacteristic
491C2C3×#337512yes—
492C2C3×#338512yes—
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×Q82C2semidirect
5. C3×#2057C2semidirect
7. C3×#1993C2semidirect
8. C3×#1985C2semidirect
10. C3×#1863C2semidirect
11. C3×#1805C2semidirect
12. #24341C3direct (N × H)
46. C2×Q82C2×C3semidirect
50. #2057C2×C3semidirect
52. #1993C2×C3semidirect
53. #1985C2×C3semidirect
55. #1863C2×C3semidirect
56. #1805C2×C3semidirect
65. C2×C3×SD16Q8semidirect
220. C2×SD16C3×Q8semidirect
486. C3#24341direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×Q822
C2×C3×#4742
C2×C3×#4402
C3×#25802
C3×#20572
C3×#20232
C3×#19932
C3×#19852
C3×#18802
C3×#18632
C3×#18052
#243413