Finite Group Databasebeta

← C3×#29694 #10648301

Subgroups — C3×#29694 #10648301

Order 768.

1304 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#532C211yesmaximal, characteristic
2C2×C3×#478C212yesmaximal, complemented
3C3×#2851C211yesmaximal, characteristic, complemented
4C3×#2058C211yesmaximal, characteristic, complemented
5C3×#2057C211yesmaximal, characteristic, complemented
6C3×#2041C211yesmaximal, characteristic, complemented
7C3×#2041C211yesmaximal, characteristic, complemented
8C3×#2034C212yesmaximal, complemented
9C3×#2033C212yesmaximal, complemented
10C3×#1804C212yesmaximal, complemented
11C3×#1797C211yesmaximal, characteristic, complemented
12#29694C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41—21no—
14C3×#532—28no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#127C2211yescharacteristic
17C2×C3×#125C2211yescharacteristic
18C2×C3×#124—21no—
19C2×C3×#121—21no—
20C2×C3×#121C2212yes—
21C2×C3×C4×D4—21no—
22C2×C3×C4×D4C2212yes—
23C22×C3×C42C2211yescharacteristic
24C3×#478—28no—
25C2×C3×#107C2212yes—
26C2×C3×#106C2212yes—
27C2×C3×C4⋊C8C2212yes—
28C2×C3×#102C2212yes—
29C2×C3×#102C2212yes—
30C2×C3×#102C2212yes—
31C2×C3×#102C2212yes—
32C2×C3×C22⋊C8C2212yes—
33C3×#389C2211yescharacteristic
34C3×#389C2211yescharacteristic
35C3×#387—21no—
36C3×#387—21no—
37C3×#385—21no—
38C3×#385—21no—
39C3×#382C2211yescharacteristic
40C3×#380C2211yescharacteristic
41C3×#380C2211yescharacteristic
42C3×#340C2212yes—
43C3×#340C2212yes—
44C3×#336C2211yescharacteristic
45C3×#336C2211yescharacteristic
46C2×#532C2×C311yescharacteristic
47C2×#478C2×C312yescomplemented
48#2851C2×C311yescharacteristic, complemented
49#2058C2×C311yescharacteristic, complemented
50#2057C2×C311yescharacteristic, complemented
51#2041C2×C311yescharacteristic, complemented
52#2041C2×C311yescharacteristic, complemented
53#2034C2×C312yescomplemented
54#2033C2×C312yescomplemented
55#1804C2×C312yescomplemented
56#1797C2×C311yescharacteristic, complemented
57C2×C3×#41—28no—
58C2×C3×#41—44no—
59C2×C3×#41—44no—
60C22×C3×Q8—21no—
61C22×C3×D4—22no—
62C22×C3×D4—21no—
63C22×C3×D4—21no—
64C22×C3×D4C2312yes—
65C23×C3×C4—22no—
66C23×C3×C4D411yescharacteristic
67C23×C3×C4D411yescharacteristic
68C23×C3×C4C2311yescharacteristic
69C22×C3×C8C2312yes—
70C22×C3×C8C2312yes—
71C3×C4⋊Q8—24no—
72C3×#127—24no—
73C3×#125—24no—
74C3×#124—28no—
75C3×#121—28no—
76C3×#121—24no—
77C3×#121—24no—
78C3×C4×D4—28no—
79C3×C4×D4—24no—
80C3×C4×D4D418yes—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2312yes—
91C2×C3×C22⋊C4—22no—
92C2×C3×C22⋊C4—22no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C42—24no—
96C2×C3×C42D411yescharacteristic
97C2×C3×C42D411yescharacteristic
98C2×C3×C42C2311yescharacteristic
99C2×C3×C42C2311yescharacteristic
100C3×#107—24no—
101C3×#106—24no—
102C3×C4⋊C8—24no—
103C3×#102—24no—
104C3×#102—24no—
105C3×#102—42no—
106C3×#102—42no—
107C3×C22⋊C8—42no—
108C3×#95—21no—
109C3×#95—21no—
110C3×#95—21no—
111C3×#95—21no—
112C3×#95—21no—
113C3×#95—21no—
114C22×#41—21no—
115#532—28no—
116C2×C4⋊Q8C22×C311yescharacteristic
117C2×#127C22×C311yescharacteristic
118C2×#125C22×C311yescharacteristic
119C2×#124—21no—
120C2×#121—21no—
121C2×#121C22×C312yes—
122C2×C4×D4—21no—
123C2×C4×D4C22×C312yes—
124C22×C42C22×C311yescharacteristic
125#478—28no—
126C2×#107C22×C312yes—
127C2×#106C22×C312yes—
128C2×C4⋊C8C22×C312yes—
129C2×#102C22×C312yes—
130C2×#102C22×C312yes—
131C2×#102C22×C312yes—
132C2×#102C22×C312yes—
133C2×C22⋊C8C22×C312yes—
134#389C22×C311yescharacteristic
135#389C22×C311yescharacteristic
136#387—21no—
137#387—21no—
138#385—21no—
139#385—21no—
140#382C22×C311yescharacteristic
141#380C22×C311yescharacteristic
142#380C22×C311yescharacteristic
143#340C22×C312yes—
144#340C22×C312yes—
145#336C22×C311yescharacteristic
146#336C22×C311yescharacteristic
147C24×C3—22no—
148C24×C3C2×D411yescharacteristic
149C3×#41—432no—
150C2×C3×Q8—24no—
151C2×C3×Q8—42no—
152C2×C3×Q8—42no—
153C2×C3×D4—24no—
154C2×C3×D4—28no—
155C2×C3×D4—24no—
156C2×C3×D4—22no—
157C2×C3×D4—22no—
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4—22no—
161C2×C3×D4—42no—
162C2×C3×D4—42no—
163C2×C3×D4—44no—
164C2×C3×D4—44no—
165C2×C3×D4—42no—
166C2×C3×D4—42no—
167C2×C3×D4C2×D414yes—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—28no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—21no—
177C22×C3×C4—22no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—41no—
184C22×C3×C4—44no—
185C22×C3×C4—44no—
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2411yescharacteristic
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8—22no—
197C2×C3×C8—22no—
198C3×C4⋊C4—24no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—42no—
207C3×C4⋊C4—42no—
208C3×C4⋊C4—42no—
209C3×C4⋊C4—42no—
210C3×C4⋊C4C2×D414yes—
211C3×C22⋊C4—28no—
212C3×C22⋊C4—28no—
213C3×C22⋊C4—42no—
214C3×C22⋊C4—42no—
215C3×C42—22no—
216C3×C42—22no—
217C3×C42—22no—
218C3×C42C2×D414yes—
219C2×#41—28no—
220C2×#41—44no—
221C2×#41—44no—
222C22×Q8—21no—
223C22×D4—22no—
224C22×D4—21no—
225C22×D4—21no—
226C22×D4C23×C312yes—
227C23×C4—22no—
228C23×C4C3×D411yescharacteristic
229C23×C4C3×D411yescharacteristic
230C23×C4C23×C311yescharacteristic
231C22×C8C23×C312yes—
232C22×C8C23×C312yes—
233C4⋊Q8—24no—
234#127—24no—
235#125—24no—
236#124—28no—
237#121—28no—
238#121—24no—
239#121—24no—
240C4×D4—28no—
241C4×D4—24no—
242C4×D4C3×D418yes—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4C23×C311yescharacteristic
249C2×C4⋊C4C23×C311yescharacteristic
250C2×C4⋊C4C23×C311yescharacteristic
251C2×C4⋊C4C23×C311yescharacteristic
252C2×C4⋊C4C23×C312yes—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C42—24no—
258C2×C42C3×D411yescharacteristic
259C2×C42C3×D411yescharacteristic
260C2×C42C23×C311yescharacteristic
261C2×C42C23×C311yescharacteristic
262#107—24no—
263#106—24no—
264C4⋊C8—24no—
265#102—24no—
266#102—24no—
267#102—42no—
268#102—42no—
269C22⋊C8—42no—
270#95—21no—
271#95—21no—
272#95—21no—
273#95—21no—
274#95—21no—
275#95—21no—
276C23×C3—21no—
277C23×C3—21no—
278C23×C3—21no—
279C23×C3—21no—
280C23×C3—21no—
281C23×C3—21no—
282C23×C3—24no—
283C23×C3—42no—
284C23×C3—42no—
285C23×C3—42no—
286C23×C3—42no—
287C23×C3—42no—
288C23×C3—42no—
289C23×C3AGL(1,ℤ/8)12yes—
290C23×C3C22×D411yescharacteristic
291C3×Q8—48no—
292C3×D4—24no—
293C3×D4—24no—
294C3×D4—48no—
295C3×D4—48no—
296C3×D4—416no—
297C3×D4—42no—
298C3×D4—42no—
299C2×C3×C4—28no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—24no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—44no—
313C2×C3×C4—416no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4#12012yes—
331C2×C3×C4#12012yes—
332C2×C3×C4#13512yes—
333C2×C3×C4AGL(1,ℤ/8)12yes—
334C2×C3×C4C22×D412yes—
335C2×C3×C4H(5,𝔽2)11yescharacteristic
336C2×C3×C42−1+411yescharacteristic
337C2×C3×C42−1+412yes—
338C3×C8—42no—
339C3×C8—42no—
340C24—22no—
341C24C2×C3×D411yescharacteristic
342#41—432no—
343C2×Q8—24no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×D4—24no—
347C2×D4—28no—
348C2×D4—24no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4—44no—
357C2×D4—44no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4C2×C3×D414yes—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—28no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—41no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4C2×C3×D411yescharacteristic
380C22×C4C2×C3×D411yescharacteristic
381C22×C4C2×C3×D411yescharacteristic
382C22×C4C2×C3×D411yescharacteristic
383C22×C4C2×C3×D411yescharacteristic
384C22×C4C24×C311yescharacteristic
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C4⋊C4—24no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4C2×C3×D414yes—
404C22⋊C4—28no—
405C22⋊C4—28no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C42—22no—
409C42—22no—
410C42—22no—
411C42C2×C3×D414yes—
412C22×C3—22no—
413C22×C3—22no—
414C22×C3—22no—
415C22×C3—41no—
416C22×C3—41no—
417C22×C3—41no—
418C22×C3—41no—
419C22×C3—44no—
420C22×C3—44no—
421C22×C3—44no—
422C22×C3—82no—
423C22×C3—82no—
424C22×C3—82no—
425C22×C3—82no—
426C22×C3C2×#12011yescharacteristic
427C22×C3#53611yescharacteristic
428C22×C3#53711yescharacteristic
429C22×C3C2×#13511yescharacteristic
430C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
431C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
432C22×C3#57511yescharacteristic
433C3×C4—22no—
434C3×C4—22no—
435C3×C4—22no—
436C3×C4—22no—
437C3×C4—41no—
438C3×C4—44no—
439C3×C4—81no—
440C3×C4—81no—
441C3×C4—81no—
442C3×C4—81no—
443C3×C4#53612yes—
444C3×C4#53712yes—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—24no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23C3×AGL(1,ℤ/8)12yes—
459C23C22×C3×D411yescharacteristic
460Q8—48no—
461D4—24no—
462D4—24no—
463D4—48no—
464D4—48no—
465D4—416no—
466D4—42no—
467D4—42no—
468C2×C4—28no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—24no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—44no—
482C2×C4—416no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4C3×#12012yes—
500C2×C4C3×#12012yes—
501C2×C4C3×#13512yes—
502C2×C4C3×AGL(1,ℤ/8)12yes—
503C2×C4C22×C3×D412yes—
504C2×C4C3×H(5,𝔽2)11yescharacteristic
505C2×C4C3×2−1+411yescharacteristic
506C2×C4C3×2−1+412yes—
507C8—42no—
508C8—42no—
509C2×C3—42no—
510C2×C3—84no—
511C2×C3#261611yescharacteristic
512C2×C3#300211yescharacteristic
513C2×C3#300311yescharacteristic
514C2×C3#322811yescharacteristic
515C2×C3#323411yescharacteristic
516C2×C3#324711yescharacteristic
517C2×C3#325111yescharacteristic
518C22—22no—
519C22—22no—
520C22—22no—
521C22—41no—
522C22—41no—
523C22—41no—
524C22—41no—
525C22—44no—
526C22—44no—
527C22—44no—
528C22—82no—
529C22—82no—
530C22—82no—
531C22—82no—
532C22C2×C3×#12011yescharacteristic
533C22C3×#53611yescharacteristic
534C22C3×#53711yescharacteristic
535C22C2×C3×#13511yescharacteristic
536C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
537C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
538C22C3×#57511yescharacteristic
539C4—22no—
540C4—22no—
541C4—22no—
542C4—22no—
543C4—41no—
544C4—44no—
545C4—81no—
546C4—81no—
547C4—81no—
548C4—81no—
549C4C3×#53612yes—
550C4C3×#53712yes—
551C3#2969411yescharacteristic, direct product factor, Sylow
552C2—42no—
553C2—84no—
554C2C3×#261611yescharacteristic
555C2C3×#300211yescharacteristic
556C2C3×#300311yescharacteristic
557C2C3×#322811yescharacteristic
558C2C3×#323411yescharacteristic
559C2C3×#324711yescharacteristic
560C2C3×#325111yescharacteristic
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×#478C2semidirect
3. C3×#2851C2semidirect
4. C3×#2058C2semidirect
5. C3×#2057C2semidirect
6. C3×#2041C2semidirect
7. C3×#2041C2semidirect
8. C3×#2034C2semidirect
9. C3×#2033C2semidirect
10. C3×#1804C2semidirect
11. C3×#1797C2semidirect
12. #29694C3direct (N × H)
47. C2×#478C2×C3semidirect
48. #2851C2×C3semidirect
49. #2058C2×C3semidirect
50. #2057C2×C3semidirect
51. #2041C2×C3semidirect
52. #2041C2×C3semidirect
53. #2034C2×C3semidirect
54. #2033C2×C3semidirect
55. #1804C2×C3semidirect
56. #1797C2×C3semidirect
551. C3#29694direct (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×#4782
C3×#28512
C3×#20582
C3×#20572
C3×#20412
C3×#20342
C3×#20332
C3×#18042
C3×#17972
#296943