Finite Group Databasebeta

← C3×#29717 #10648324

Subgroups — C3×#29717 #10648324

Order 768.

1016 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×#452C212yesmaximal
2C3×#2857C211yesmaximal, characteristic, complemented
3C3×#2297C211yesmaximal, characteristic
4C3×#2072C211yesmaximal, characteristic, complemented
5C3×#2071C211yesmaximal, characteristic, complemented
6C3×#2056C212yesmaximal, complemented
7C3×#2055C212yesmaximal, complemented
8C3×#2016C211yesmaximal, characteristic, complemented
9C3×#2013C211yesmaximal, characteristic, complemented
10C3×#1873C212yesmaximal, complemented
11C3×#1797C211yesmaximal, characteristic, complemented
12#29717C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C22×C3×Q16C2212yes—
15C22×C3×SD16C2212yes—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×C22⋊Q8C2212yes—
18C2×C3×#117C2211yescharacteristic
19C22×C3×C42C2211yescharacteristic
20C3×#452—28no—
21C2×C3×C4⋊C8C2212yes—
22C2×C3×#103C2212yes—
23C2×C3×#103C2212yes—
24C2×C3×#103C2212yes—
25C2×C3×#102C2212yes—
26C2×C3×C22⋊C8C2212yes—
27C3×#391C2211yescharacteristic
28C3×#391C2212yes—
29C3×#391C2211yescharacteristic
30C3×#390C2211yescharacteristic
31C3×#390C2211yescharacteristic
32C3×#389—21no—
33C3×#389—21no—
34C3×#386—21no—
35C3×#386—21no—
36C3×#384C2211yescharacteristic
37C3×#380—21no—
38C3×#380—21no—
39C3×C4×C22⋊C4—21no—
40C3×C4×C22⋊C4—21no—
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#336C2211yescharacteristic
44C3×#336C2211yescharacteristic
45C2×#452C2×C312yes—
46#2857C2×C311yescharacteristic, complemented
47#2297C2×C311yescharacteristic
48#2072C2×C311yescharacteristic, complemented
49#2071C2×C311yescharacteristic, complemented
50#2056C2×C312yescomplemented
51#2055C2×C312yescomplemented
52#2016C2×C311yescharacteristic, complemented
53#2013C2×C311yescharacteristic, complemented
54#1873C2×C312yescomplemented
55#1797C2×C311yescharacteristic, complemented
56C2×C3×#41—22no—
57C2×C3×#41—42no—
58C2×C3×#41—42no—
59C2×C3×#41D414yes—
60C22×C3×Q8C2312yes—
61C22×C3×Q8C2311yescharacteristic
62C22×C3×D4—21no—
63C22×C3×D4C2311yescharacteristic
64C23×C3×C4—21no—
65C23×C3×C4D411yescharacteristic
66C23×C3×C4D411yescharacteristic
67C23×C3×C4C2311yescharacteristic
68C2×C3×Q16—24no—
69C2×C3×Q16—42no—
70C2×C3×Q16—42no—
71C2×C3×SD16—24no—
72C2×C3×SD16—42no—
73C2×C3×SD16—42no—
74C22×C3×C8C2312yes—
75C22×C3×C8C2312yes—
76C3×C4⋊Q8—42no—
77C3×C22⋊Q8—24no—
78C3×C22⋊Q8—24no—
79C3×#117—42no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—22no—
85C2×C3×C4⋊C4C2312yes—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C22⋊C4—22no—
89C2×C3×C22⋊C4—21no—
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C42—21no—
95C2×C3×C42—42no—
96C2×C3×C42D411yescharacteristic
97C2×C3×C42D411yescharacteristic
98C2×C3×C42C2311yescharacteristic
99C2×C3×C42C2311yescharacteristic
100C3×C4⋊C8—42no—
101C3×#103—24no—
102C3×#103—42no—
103C3×#103—42no—
104C3×#102—24no—
105C3×C22⋊C8—24no—
106C3×#95—21no—
107C3×#95—21no—
108C3×#95—21no—
109C3×#95—21no—
110C3×#95—22no—
111C3×#95—22no—
112C22×#41C22×C311yescharacteristic
113C22×Q16C22×C312yes—
114C22×SD16C22×C312yes—
115C2×C4⋊Q8C22×C311yescharacteristic
116C2×C22⋊Q8C22×C312yes—
117C2×#117C22×C311yescharacteristic
118C22×C42C22×C311yescharacteristic
119#452—28no—
120C2×C4⋊C8C22×C312yes—
121C2×#103C22×C312yes—
122C2×#103C22×C312yes—
123C2×#103C22×C312yes—
124C2×#102C22×C312yes—
125C2×C22⋊C8C22×C312yes—
126#391C22×C312yes—
127#391C22×C311yescharacteristic
128#391C22×C311yescharacteristic
129#390C22×C311yescharacteristic
130#390C22×C311yescharacteristic
131#389—21no—
132#389—21no—
133#386—21no—
134#386—21no—
135#384C22×C311yescharacteristic
136#380—21no—
137#380—21no—
138C4×C22⋊C4—21no—
139C4×C22⋊C4—21no—
140#340C22×C312yes—
141#340C22×C312yes—
142#336C22×C311yescharacteristic
143#336C22×C311yescharacteristic
144C24×C3—21no—
145C24×C3C2×D411yescharacteristic
146C3×#41—48no—
147C3×#41—48no—
148C2×C3×Q8—21no—
149C2×C3×Q8—22no—
150C2×C3×Q8—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8—22no—
157C2×C3×Q8C2×D412yes—
158C2×C3×D4—24no—
159C2×C3×D4—21no—
160C2×C3×D4—22no—
161C2×C3×D4—22no—
162C2×C3×D4—42no—
163C2×C3×D4—42no—
164C2×C3×D4C2×D412yes—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—21no—
168C22×C3×C4—24no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—41no—
177C22×C3×C4—41no—
178C22×C3×C4—41no—
179C22×C3×C4—42no—
180C22×C3×C4—42no—
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4#4112yes—
188C22×C3×C4C2411yescharacteristic
189C3×Q16—48no—
190C3×SD16—48no—
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8—22no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—42no—
202C3×C4⋊C4—42no—
203C3×C4⋊C4—42no—
204C3×C4⋊C4—44no—
205C3×C4⋊C4—42no—
206C3×C22⋊C4—42no—
207C3×C22⋊C4—42no—
208C3×C22⋊C4—44no—
209C3×C22⋊C4—42no—
210C3×C22⋊C4—42no—
211C3×C22⋊C4—42no—
212C3×C42—22no—
213C3×C42—22no—
214C3×C42—41no—
215C3×C42—41no—
216C3×C42—42no—
217C2×#41—22no—
218C2×#41—42no—
219C2×#41—42no—
220C2×#41C3×D414yes—
221C22×Q8C23×C312yes—
222C22×Q8C23×C311yescharacteristic
223C22×D4—21no—
224C22×D4C23×C311yescharacteristic
225C23×C4—21no—
226C23×C4C3×D411yescharacteristic
227C23×C4C3×D411yescharacteristic
228C23×C4C23×C311yescharacteristic
229C2×Q16—24no—
230C2×Q16—42no—
231C2×Q16—42no—
232C2×SD16—24no—
233C2×SD16—42no—
234C2×SD16—42no—
235C22×C8C23×C312yes—
236C22×C8C23×C312yes—
237C4⋊Q8—42no—
238C22⋊Q8—24no—
239C22⋊Q8—24no—
240#117—42no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4C23×C311yescharacteristic
247C2×C4⋊C4C23×C311yescharacteristic
248C2×C4⋊C4C23×C312yes—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C42—21no—
256C2×C42—42no—
257C2×C42C3×D411yescharacteristic
258C2×C42C3×D411yescharacteristic
259C2×C42C23×C311yescharacteristic
260C2×C42C23×C311yescharacteristic
261C4⋊C8—42no—
262#103—24no—
263#103—42no—
264#103—42no—
265#102—24no—
266C22⋊C8—24no—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—22no—
272#95—22no—
273C23×C3—21no—
274C23×C3—21no—
275C23×C3—21no—
276C23×C3—21no—
277C23×C3—21no—
278C23×C3—21no—
279C23×C3—22no—
280C23×C3—41no—
281C23×C3—41no—
282C23×C3—41no—
283C23×C3—41no—
284C23×C3—41no—
285C23×C3—41no—
286C23×C3#12112yes—
287C23×C3C22×D411yescharacteristic
288C3×Q8—24no—
289C3×Q8—42no—
290C3×Q8—42no—
291C3×Q8—42no—
292C3×Q8—42no—
293C3×Q8—42no—
294C3×D4—24no—
295C3×D4—48no—
296C3×D4—42no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—24no—
306C2×C3×C4—24no—
307C2×C3×C4—22no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—44no—
312C2×C3×C4—44no—
313C2×C3×C4—48no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—41no—
320C2×C3×C4—41no—
321C2×C3×C4—42no—
322C2×C3×C4—41no—
323C2×C3×C4—42no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4—42no—
328C2×C3×C4#12112yes—
329C2×C3×C4#13512yes—
330C2×C3×C4#13712yes—
331C2×C3×C4#13712yes—
332C2×C3×C4C22×D411yescharacteristic
333C2×C3×C4C2×#4111yescharacteristic
334C2×C3×C4H(5,𝔽2)12yes—
335C2×C3×C4H(5,𝔽2)12yes—
336C3×C8—42no—
337C3×C8—42no—
338C24—21no—
339C24C2×C3×D411yescharacteristic
340#41—48no—
341#41—48no—
342C2×Q8—21no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8C2×C3×D412yes—
352C2×D4—24no—
353C2×D4—21no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4C2×C3×D412yes—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4C2×C3×D411yescharacteristic
376C22×C4C2×C3×D411yescharacteristic
377C22×C4C2×C3×D412yes—
378C22×C4C2×C3×D411yescharacteristic
379C22×C4C2×C3×D411yescharacteristic
380C22×C4C2×C3×D411yescharacteristic
381C22×C4C3×#4112yes—
382C22×C4C24×C311yescharacteristic
383Q16—48no—
384SD16—48no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—44no—
399C4⋊C4—42no—
400C22⋊C4—42no—
401C22⋊C4—42no—
402C22⋊C4—44no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C42—22no—
407C42—22no—
408C42—41no—
409C42—41no—
410C42—42no—
411C22×C3—22no—
412C22×C3—22no—
413C22×C3—22no—
414C22×C3—41no—
415C22×C3—41no—
416C22×C3—41no—
417C22×C3—41no—
418C22×C3—42no—
419C22×C3—42no—
420C22×C3—42no—
421C22×C3—81no—
422C22×C3—81no—
423C22×C3—81no—
424C22×C3—81no—
425C22×C3C2×#12111yescharacteristic
426C22×C3#53411yescharacteristic
427C22×C3#53511yescharacteristic
428C22×C3C2×#13511yescharacteristic
429C22×C3C2×#13711yescharacteristic
430C22×C3C2×#13711yescharacteristic
431C22×C3#57511yescharacteristic
432C3×C4—22no—
433C3×C4—22no—
434C3×C4—22no—
435C3×C4—22no—
436C3×C4—42no—
437C3×C4—41no—
438C3×C4—82no—
439C3×C4—81no—
440C3×C4—81no—
441C3×C4—82no—
442C3×C4#53412yes—
443C3×C4#53812yes—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—22no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23C3×#12112yes—
458C23C22×C3×D411yescharacteristic
459Q8—24no—
460Q8—42no—
461Q8—42no—
462Q8—42no—
463Q8—42no—
464Q8—42no—
465D4—24no—
466D4—48no—
467D4—42no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—21no—
482C2×C4—44no—
483C2×C4—48no—
484C2×C4—44no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4C3×#12112yes—
500C2×C4C3×#13512yes—
501C2×C4C3×#13712yes—
502C2×C4C3×#13712yes—
503C2×C4C22×C3×D411yescharacteristic
504C2×C4C2×C3×#4111yescharacteristic
505C2×C4C3×H(5,𝔽2)12yes—
506C2×C4C3×H(5,𝔽2)12yes—
507C8—42no—
508C8—42no—
509C2×C3—42no—
510C2×C3—82no—
511C2×C3#261711yescharacteristic
512C2×C3#303511yescharacteristic
513C2×C3#303911yescharacteristic
514C2×C3#319011yescharacteristic
515C2×C3#319611yescharacteristic
516C2×C3#321011yescharacteristic
517C2×C3#321611yescharacteristic
518C22—22no—
519C22—22no—
520C22—22no—
521C22—41no—
522C22—41no—
523C22—41no—
524C22—41no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—81no—
529C22—81no—
530C22—81no—
531C22—81no—
532C22C2×C3×#12111yescharacteristic
533C22C3×#53411yescharacteristic
534C22C3×#53511yescharacteristic
535C22C2×C3×#13511yescharacteristic
536C22C2×C3×#13711yescharacteristic
537C22C2×C3×#13711yescharacteristic
538C22C3×#57511yescharacteristic
539C4—22no—
540C4—22no—
541C4—22no—
542C4—22no—
543C4—42no—
544C4—41no—
545C4—82no—
546C4—81no—
547C4—81no—
548C4—82no—
549C4C3×#53412yes—
550C4C3×#53812yes—
551C3#2971711yescharacteristic, direct product factor, Sylow
552C2—42no—
553C2—82no—
554C2C3×#261711yescharacteristic
555C2C3×#303511yescharacteristic
556C2C3×#303911yescharacteristic
557C2C3×#319011yescharacteristic
558C2C3×#319611yescharacteristic
559C2C3×#321011yescharacteristic
560C2C3×#321611yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×#41C22
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. C3×#2857C2semidirect
4. C3×#2072C2semidirect
5. C3×#2071C2semidirect
6. C3×#2056C2semidirect
7. C3×#2055C2semidirect
8. C3×#2016C2semidirect
9. C3×#2013C2semidirect
10. C3×#1873C2semidirect
11. C3×#1797C2semidirect
12. #29717C3direct (N × H)
46. #2857C2×C3semidirect
48. #2072C2×C3semidirect
49. #2071C2×C3semidirect
50. #2056C2×C3semidirect
51. #2055C2×C3semidirect
52. #2016C2×C3semidirect
53. #2013C2×C3semidirect
54. #1873C2×C3semidirect
55. #1797C2×C3semidirect
551. C3#29717direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4522
C3×#28572
C3×#22972
C3×#20722
C3×#20712
C3×#20562
C3×#20552
C3×#20162
C3×#20132
C3×#18732
C3×#17972
#297173