Finite Group Databasebeta

← C3×#29696 #10648303

Subgroups — C3×#29696 #10648303

Order 768.

1012 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#542C211yesmaximal, characteristic
2C2×C3×#477C211yesmaximal, characteristic, complemented
3C2×C3×#474C211yesmaximal, characteristic, complemented
4C3×#2628C211yesmaximal, characteristic, complemented
5C3×#2057C212yesmaximal, complemented
6C3×#2042C212yesmaximal, complemented
7C3×#2036C212yesmaximal, complemented
8C3×#2033C212yesmaximal
9C3×#1805C211yesmaximal, characteristic, complemented
10C3×#1804C211yesmaximal, characteristic, complemented
11C3×#1798C211yesmaximal, characteristic, complemented
12#29696C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#542—24no—
14C3×#542—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#126—21no—
18C2×C3×#124—21no—
19C2×C3×#124C2211yescharacteristic
20C2×C3×#123—21no—
21C2×C3×C22⋊Q8—21no—
22C2×C3×C22⋊Q8C2211yescharacteristic
23C2×C3×#121C2211yescharacteristic
24C2×C3×C4×Q8C2211yescharacteristic
25C2×C3×C4×D4C2211yescharacteristic
26C2×C3×#117C2211yescharacteristic
27C3×#477—24no—
28C3×#474—24no—
29C2×C3×#107C2212yes—
30C2×C3×#107C2212yes—
31C2×C3×C4⋊C8C2211yescharacteristic
32C2×C3×C4⋊C8C2211yescharacteristic
33C2×C3×#103C2212yes—
34C2×C3×#103C2212yes—
35C2×C3×#102C2212yes—
36C2×C3×#102C2212yes—
37C2×C3×C22⋊C8C2212yes—
38C3×#400—21no—
39C3×#399—21no—
40C3×#395—21no—
41C3×#393—21no—
42C3×#389C2212yes—
43C3×#382C2211yescharacteristic
44C3×#380C2212yes—
45C3×#340C2212yes—
46C3×#340C2212yes—
47C3×#336C2212yes—
48C2×#542C2×C311yescharacteristic
49C2×#477C2×C311yescharacteristic, complemented
50C2×#474C2×C311yescharacteristic, complemented
51#2628C2×C311yescharacteristic, complemented
52#2057C2×C312yescomplemented
53#2042C2×C312yescomplemented
54#2036C2×C312yescomplemented
55#2033C2×C312yes—
56#1805C2×C311yescharacteristic, complemented
57#1804C2×C311yescharacteristic, complemented
58#1798C2×C311yescharacteristic, complemented
59C22×C3×Q8—21no—
60C22×C3×Q8—21no—
61C22×C3×Q8C2311yescharacteristic
62C22×C3×D4—21no—
63C22×C3×D4C2311yescharacteristic
64C23×C3×C4—21no—
65C23×C3×C4C2311yescharacteristic
66C22×C3×C8C2312yes—
67C22×C3×C8C2312yes—
68C3×C4⋊Q8—22no—
69C3×C4⋊Q8—22no—
70C3×C4⋊Q8—22no—
71C3×C4⋊Q8—22no—
72C3×#126—28no—
73C3×#124—28no—
74C3×#124—22no—
75C3×#124—22no—
76C3×#123—28no—
77C3×C22⋊Q8—28no—
78C3×C22⋊Q8—22no—
79C3×C22⋊Q8—22no—
80C3×#121—22no—
81C3×#121—22no—
82C3×C4×Q8—22no—
83C3×C4×Q8D414yes—
84C3×C4×D4—22no—
85C3×C4×D4D414yes—
86C3×#117—22no—
87C3×#117—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4D411yescharacteristic
94C2×C3×C4⋊C4D411yescharacteristic
95C2×C3×C4⋊C4C2312yes—
96C2×C3×C4⋊C4C2312yes—
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—22no—
105C2×C3×C22⋊C4D411yescharacteristic
106C2×C3×C22⋊C4D411yescharacteristic
107C2×C3×C42—21no—
108C2×C3×C42C2311yescharacteristic
109C2×C3×C42C2311yescharacteristic
110C3×#107—24no—
111C3×#107—24no—
112C3×C4⋊C8—22no—
113C3×C4⋊C8—22no—
114C3×#103—24no—
115C3×#103—42no—
116C3×#102—24no—
117C3×#102—42no—
118C3×C22⋊C8—42no—
119C3×#95—22no—
120C3×#95—21no—
121C3×#95—21no—
122C3×#95—22no—
123#542—24no—
124#542—24no—
125C2×C4⋊Q8C22×C311yescharacteristic
126C2×C4⋊Q8C22×C311yescharacteristic
127C2×#126—21no—
128C2×#124—21no—
129C2×#124C22×C311yescharacteristic
130C2×#123—21no—
131C2×C22⋊Q8—21no—
132C2×C22⋊Q8C22×C311yescharacteristic
133C2×#121C22×C311yescharacteristic
134C2×C4×Q8C22×C311yescharacteristic
135C2×C4×D4C22×C311yescharacteristic
136C2×#117C22×C311yescharacteristic
137#477—24no—
138#474—24no—
139C2×#107C22×C312yes—
140C2×#107C22×C312yes—
141C2×C4⋊C8C22×C311yescharacteristic
142C2×C4⋊C8C22×C311yescharacteristic
143C2×#103C22×C312yes—
144C2×#103C22×C312yes—
145C2×#102C22×C312yes—
146C2×#102C22×C312yes—
147C2×C22⋊C8C22×C312yes—
148#400—21no—
149#399—21no—
150#395—21no—
151#393—21no—
152#389C22×C312yes—
153#382C22×C311yescharacteristic
154#380C22×C312yes—
155#340C22×C312yes—
156#340C22×C312yes—
157#336C22×C312yes—
158C24×C3—21no—
159C24×C3C2×D411yescharacteristic
160C2×C3×Q8—24no—
161C2×C3×Q8—21no—
162C2×C3×Q8—21no—
163C2×C3×Q8—21no—
164C2×C3×Q8—21no—
165C2×C3×Q8—21no—
166C2×C3×Q8—42no—
167C2×C3×Q8—82no—
168C2×C3×Q8—82no—
169C2×C3×Q8C2×D412yes—
170C2×C3×D4—24no—
171C2×C3×D4—21no—
172C2×C3×D4—21no—
173C2×C3×D4—21no—
174C2×C3×D4—21no—
175C2×C3×D4—21no—
176C2×C3×D4—82no—
177C2×C3×D4C2×D412yes—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—24no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—82no—
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2411yescharacteristic
197C2×C3×C8—22no—
198C2×C3×C8—22no—
199C2×C3×C8—22no—
200C2×C3×C8—22no—
201C2×C3×C8—22no—
202C2×C3×C8—22no—
203C3×C4⋊C4—24no—
204C3×C4⋊C4—24no—
205C3×C4⋊C4—24no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—42no—
215C3×C4⋊C4—82no—
216C3×C4⋊C4C2×D412yes—
217C3×C4⋊C4C2×D412yes—
218C3×C22⋊C4—24no—
219C3×C22⋊C4—24no—
220C3×C22⋊C4—24no—
221C3×C22⋊C4—24no—
222C3×C22⋊C4—24no—
223C3×C22⋊C4—22no—
224C3×C22⋊C4—22no—
225C3×C22⋊C4—82no—
226C3×C42—24no—
227C3×C42—21no—
228C3×C42—21no—
229C3×C42C2×D412yes—
230C3×C42C2×D412yes—
231C22×Q8—21no—
232C22×Q8—21no—
233C22×Q8C23×C311yescharacteristic
234C22×D4—21no—
235C22×D4C23×C311yescharacteristic
236C23×C4—21no—
237C23×C4C23×C311yescharacteristic
238C22×C8C23×C312yes—
239C22×C8C23×C312yes—
240C4⋊Q8—22no—
241C4⋊Q8—22no—
242C4⋊Q8—22no—
243C4⋊Q8—22no—
244#126—28no—
245#124—28no—
246#124—22no—
247#124—22no—
248#123—28no—
249C22⋊Q8—28no—
250C22⋊Q8—22no—
251C22⋊Q8—22no—
252#121—22no—
253#121—22no—
254C4×Q8—22no—
255C4×Q8C3×D414yes—
256C4×D4—22no—
257C4×D4C3×D414yes—
258#117—22no—
259#117—22no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4C3×D411yescharacteristic
266C2×C4⋊C4C3×D411yescharacteristic
267C2×C4⋊C4C23×C312yes—
268C2×C4⋊C4C23×C312yes—
269C2×C4⋊C4C23×C311yescharacteristic
270C2×C4⋊C4C23×C311yescharacteristic
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4C3×D411yescharacteristic
278C2×C22⋊C4C3×D411yescharacteristic
279C2×C42—21no—
280C2×C42C23×C311yescharacteristic
281C2×C42C23×C311yescharacteristic
282#107—24no—
283#107—24no—
284C4⋊C8—22no—
285C4⋊C8—22no—
286#103—24no—
287#103—42no—
288#102—24no—
289#102—42no—
290C22⋊C8—42no—
291#95—22no—
292#95—21no—
293#95—21no—
294#95—22no—
295C23×C3—21no—
296C23×C3—22no—
297C23×C3—21no—
298C23×C3—21no—
299C23×C3—22no—
300C23×C3—22no—
301C23×C3—41no—
302C23×C3—41no—
303C23×C3—42no—
304C23×C3—42no—
305C23×C3C22×D411yescharacteristic
306C3×Q8—22no—
307C3×Q8—22no—
308C3×Q8—41no—
309C3×Q8—41no—
310C3×Q8—84no—
311C3×Q8—84no—
312C3×D4—22no—
313C3×D4—22no—
314C3×D4—41no—
315C3×D4—41no—
316C3×D4—84no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—21no—
320C2×C3×C4—22no—
321C2×C3×C4—21no—
322C2×C3×C4—22no—
323C2×C3×C4—21no—
324C2×C3×C4—22no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—84no—
344C2×C3×C4#12012yes—
345C2×C3×C4#12012yes—
346C2×C3×C4C22×D411yescharacteristic
347C2×C3×C4C22×D411yescharacteristic
348C2×C3×C4H(5,𝔽2)11yescharacteristic
349C2×C3×C42−1+411yescharacteristic
350C2×C3×C42−1+411yescharacteristic
351C2×C3×C42−1+411yescharacteristic
352C3×C8—42no—
353C3×C8—42no—
354C24—21no—
355C24C2×C3×D411yescharacteristic
356C2×Q8—24no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8—21no—
361C2×Q8—21no—
362C2×Q8—42no—
363C2×Q8—82no—
364C2×Q8—82no—
365C2×Q8C2×C3×D412yes—
366C2×D4—24no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4—82no—
373C2×D4C2×C3×D412yes—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—82no—
387C22×C4C2×C3×D411yescharacteristic
388C22×C4C2×C3×D411yescharacteristic
389C22×C4C2×C3×D411yescharacteristic
390C22×C4C2×C3×D411yescharacteristic
391C22×C4C2×C3×D411yescharacteristic
392C22×C4C24×C311yescharacteristic
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—42no—
411C4⋊C4—82no—
412C4⋊C4C2×C3×D412yes—
413C4⋊C4C2×C3×D412yes—
414C22⋊C4—24no—
415C22⋊C4—24no—
416C22⋊C4—24no—
417C22⋊C4—24no—
418C22⋊C4—24no—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—82no—
422C42—24no—
423C42—21no—
424C42—21no—
425C42C2×C3×D412yes—
426C42C2×C3×D412yes—
427C22×C3—41no—
428C22×C3—42no—
429C22×C3—42no—
430C22×C3—42no—
431C22×C3—42no—
432C22×C3—41no—
433C22×C3—41no—
434C22×C3—42no—
435C22×C3—82no—
436C22×C3—82no—
437C22×C3C2×#12011yescharacteristic
438C22×C3#53611yescharacteristic
439C22×C3#53711yescharacteristic
440C22×C3#57612yes—
441C22×C3#57812yes—
442C3×C4—22no—
443C3×C4—22no—
444C3×C4—41no—
445C3×C4—42no—
446C3×C4—41no—
447C3×C4—42no—
448C3×C4—42no—
449C3×C4—41no—
450C3×C4—82no—
451C3×C4—82no—
452C3×C4#53611yescharacteristic
453C3×C4#53611yescharacteristic
454C3×C4#53711yescharacteristic
455C3×C4#53711yescharacteristic
456C23—22no—
457C23—22no—
458C23—21no—
459C23—22no—
460C23—21no—
461C23—21no—
462C23—42no—
463C23—42no—
464C23—41no—
465C23—41no—
466C23C22×C3×D411yescharacteristic
467Q8—22no—
468Q8—22no—
469Q8—41no—
470Q8—41no—
471Q8—84no—
472Q8—84no—
473D4—22no—
474D4—22no—
475D4—41no—
476D4—41no—
477D4—84no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—84no—
505C2×C4C3×#12012yes—
506C2×C4C3×#12012yes—
507C2×C4C22×C3×D411yescharacteristic
508C2×C4C22×C3×D411yescharacteristic
509C2×C4C3×H(5,𝔽2)11yescharacteristic
510C2×C4C3×2−1+411yescharacteristic
511C2×C4C3×2−1+411yescharacteristic
512C2×C4C3×2−1+411yescharacteristic
513C8—42no—
514C8—42no—
515C2×C3—81no—
516C2×C3—82no—
517C2×C3#261611yescharacteristic
518C2×C3#300811yescharacteristic
519C2×C3#301111yescharacteristic
520C2×C3#324312yes—
521C2×C3#326212yes—
522C22—41no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—41no—
528C22—41no—
529C22—42no—
530C22—82no—
531C22—82no—
532C22C2×C3×#12011yescharacteristic
533C22C3×#53611yescharacteristic
534C22C3×#53711yescharacteristic
535C22C3×#57612yes—
536C22C3×#57812yes—
537C4—22no—
538C4—22no—
539C4—41no—
540C4—42no—
541C4—41no—
542C4—42no—
543C4—42no—
544C4—41no—
545C4—82no—
546C4—82no—
547C4C3×#53611yescharacteristic
548C4C3×#53611yescharacteristic
549C4C3×#53711yescharacteristic
550C4C3×#53711yescharacteristic
551C3#2969611yescharacteristic, direct product factor, Sylow
552C2—81no—
553C2—82no—
554C2C3×#261611yescharacteristic
555C2C3×#300811yescharacteristic
556C2C3×#301111yescharacteristic
557C2C3×#324312yes—
558C2C3×#326212yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#477C2semidirect
3. C2×C3×#474C2semidirect
4. C3×#2628C2semidirect
5. C3×#2057C2semidirect
6. C3×#2042C2semidirect
7. C3×#2036C2semidirect
9. C3×#1805C2semidirect
10. C3×#1804C2semidirect
11. C3×#1798C2semidirect
12. #29696C3direct (N × H)
49. C2×#477C2×C3semidirect
50. C2×#474C2×C3semidirect
51. #2628C2×C3semidirect
52. #2057C2×C3semidirect
53. #2042C2×C3semidirect
54. #2036C2×C3semidirect
56. #1805C2×C3semidirect
57. #1804C2×C3semidirect
58. #1798C2×C3semidirect
551. C3#29696direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5422
C2×C3×#4772
C2×C3×#4742
C3×#26282
C3×#20572
C3×#20422
C3×#20362
C3×#20332
C3×#18052
C3×#18042
C3×#17982
#296963