Finite Group Databasebeta

← C3×#29668 #10648275

Subgroups — C3×#29668 #10648275

Order 768.

1158 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#533C211yesmaximal, characteristic, complemented
2C2×C3×#464C212yesmaximal, complemented
3C3×#2850C211yesmaximal, characteristic
4C3×#2024C211yesmaximal, characteristic, complemented
5C3×#2023C211yesmaximal, characteristic, complemented
6C3×#2016C211yesmaximal, characteristic, complemented
7C3×#2013C211yesmaximal, characteristic, complemented
8C3×#2001C212yesmaximal, complemented
9C3×#2000C212yesmaximal
10C3×#1805C212yesmaximal, complemented
11C3×#1797C211yesmaximal, characteristic, complemented
12#29668C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×Q16C2212yes—
14C22×C3×SD16C2212yes—
15C3×#533—28no—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#125—21no—
19C2×C3×#124C2211yescharacteristic
20C2×C3×C22⋊Q8—21no—
21C2×C3×C22⋊Q8C2212yes—
22C2×C3×C22⋊Q8C2211yescharacteristic
23C2×C3×#121C2211yescharacteristic
24C2×C3×C4×Q8—21no—
25C2×C3×C4×Q8C2212yes—
26C2×C3×#117—21no—
27C22×C3×C42C2211yescharacteristic
28C3×#464—28no—
29C2×C3×C4⋊C8C2212yes—
30C2×C3×#103C2212yes—
31C2×C3×#103C2212yes—
32C2×C3×#103C2212yes—
33C2×C3×#102C2212yes—
34C2×C3×C22⋊C8C2212yes—
35C3×#391C2211yescharacteristic
36C3×#390C2211yescharacteristic
37C3×#388—21no—
38C3×#388—21no—
39C3×#386—21no—
40C3×#386—21no—
41C3×#382C2211yescharacteristic
42C3×#340C2212yes—
43C3×#340C2212yes—
44C3×#336C2211yescharacteristic
45C3×#336C2211yescharacteristic
46C2×#533C2×C311yescharacteristic, complemented
47C2×#464C2×C312yescomplemented
48#2850C2×C311yescharacteristic
49#2024C2×C311yescharacteristic, complemented
50#2023C2×C311yescharacteristic, complemented
51#2016C2×C311yescharacteristic, complemented
52#2013C2×C311yescharacteristic, complemented
53#2001C2×C312yescomplemented
54#2000C2×C312yes—
55#1805C2×C312yescomplemented
56#1797C2×C311yescharacteristic, complemented
57C22×C3×Q8—21no—
58C22×C3×Q8C2311yescharacteristic
59C22×C3×Q8C2312yes—
60C22×C3×D4—21no—
61C22×C3×D4C2311yescharacteristic
62C23×C3×C4D411yescharacteristic
63C23×C3×C4D411yescharacteristic
64C23×C3×C4C2311yescharacteristic
65C2×C3×Q16—24no—
66C2×C3×Q16—42no—
67C2×C3×Q16—42no—
68C2×C3×SD16—24no—
69C2×C3×SD16—42no—
70C2×C3×SD16—42no—
71C22×C3×C8C2312yes—
72C22×C3×C8C2312yes—
73C3×C4⋊Q8—24no—
74C3×C4⋊Q8—24no—
75C3×#125—28no—
76C3×#124—24no—
77C3×C22⋊Q8—28no—
78C3×C22⋊Q8—24no—
79C3×C22⋊Q8—24no—
80C3×C22⋊Q8—42no—
81C3×#121—42no—
82C3×C4×Q8—28no—
83C3×C4×Q8—24no—
84C3×C4×Q8D418yescomplemented
85C3×#117—28no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—22no—
90C2×C3×C4⋊C4—22no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2312yes—
95C2×C3×C22⋊C4—22no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C42—22no—
103C2×C3×C42—24no—
104C2×C3×C42D411yescharacteristic
105C2×C3×C42D411yescharacteristic
106C2×C3×C42C2311yescharacteristic
107C2×C3×C42C2311yescharacteristic
108C3×C4⋊C8—24no—
109C3×#103—24no—
110C3×#103—42no—
111C3×#103—42no—
112C3×#102—24no—
113C3×C22⋊C8—42no—
114C3×#95—21no—
115C3×#95—21no—
116C3×#95—21no—
117C3×#95—21no—
118C22×Q16C22×C312yes—
119C22×SD16C22×C312yes—
120#533—28no—
121C2×C4⋊Q8C22×C311yescharacteristic
122C2×C4⋊Q8C22×C311yescharacteristic
123C2×#125—21no—
124C2×#124C22×C311yescharacteristic
125C2×C22⋊Q8—21no—
126C2×C22⋊Q8C22×C312yes—
127C2×C22⋊Q8C22×C311yescharacteristic
128C2×#121C22×C311yescharacteristic
129C2×C4×Q8—21no—
130C2×C4×Q8C22×C312yes—
131C2×#117—21no—
132C22×C42C22×C311yescharacteristic
133#464—28no—
134C2×C4⋊C8C22×C312yes—
135C2×#103C22×C312yes—
136C2×#103C22×C312yes—
137C2×#103C22×C312yes—
138C2×#102C22×C312yes—
139C2×C22⋊C8C22×C312yes—
140#391C22×C311yescharacteristic
141#390C22×C311yescharacteristic
142#388—21no—
143#388—21no—
144#386—21no—
145#386—21no—
146#382C22×C311yescharacteristic
147#340C22×C312yes—
148#340C22×C312yes—
149#336C22×C311yescharacteristic
150#336C22×C311yescharacteristic
151C24×C3—21no—
152C24×C3C2×D411yescharacteristic
153C2×C3×Q8—24no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8—22no—
157C2×C3×Q8—22no—
158C2×C3×Q8—22no—
159C2×C3×Q8—22no—
160C2×C3×Q8—21no—
161C2×C3×Q8—22no—
162C2×C3×Q8—21no—
163C2×C3×Q8—42no—
164C2×C3×Q8—42no—
165C2×C3×Q8C2×D414yes—
166C2×C3×D4—22no—
167C2×C3×D4—22no—
168C2×C3×D4—21no—
169C2×C3×D4—21no—
170C2×C3×D4—42no—
171C2×C3×D4—42no—
172C2×C3×D4—42no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—21no—
182C22×C3×C4—22no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—41no—
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2411yescharacteristic
194C3×Q16—48no—
195C3×SD16—48no—
196C2×C3×C8—22no—
197C2×C3×C8—22no—
198C2×C3×C8—22no—
199C2×C3×C8—22no—
200C2×C3×C8—22no—
201C2×C3×C8—22no—
202C3×C4⋊C4—24no—
203C3×C4⋊C4—24no—
204C3×C4⋊C4—28no—
205C3×C4⋊C4—24no—
206C3×C4⋊C4—28no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—42no—
211C3×C4⋊C4C2×D414yes—
212C3×C22⋊C4—28no—
213C3×C22⋊C4—42no—
214C3×C22⋊C4—42no—
215C3×C22⋊C4—42no—
216C3×C22⋊C4—42no—
217C3×C22⋊C4—42no—
218C3×C22⋊C4—42no—
219C3×C42—22no—
220C3×C42—22no—
221C3×C42—28no—
222C3×C42—22no—
223C3×C42C2×D414yes—
224C22×Q8—21no—
225C22×Q8C23×C311yescharacteristic
226C22×Q8C23×C312yes—
227C22×D4—21no—
228C22×D4C23×C311yescharacteristic
229C23×C4C3×D411yescharacteristic
230C23×C4C3×D411yescharacteristic
231C23×C4C23×C311yescharacteristic
232C2×Q16—24no—
233C2×Q16—42no—
234C2×Q16—42no—
235C2×SD16—24no—
236C2×SD16—42no—
237C2×SD16—42no—
238C22×C8C23×C312yes—
239C22×C8C23×C312yes—
240C4⋊Q8—24no—
241C4⋊Q8—24no—
242#125—28no—
243#124—24no—
244C22⋊Q8—28no—
245C22⋊Q8—24no—
246C22⋊Q8—24no—
247C22⋊Q8—42no—
248#121—42no—
249C4×Q8—28no—
250C4×Q8—24no—
251C4×Q8C3×D418yescomplemented
252#117—28no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4C23×C311yescharacteristic
260C2×C4⋊C4C23×C311yescharacteristic
261C2×C4⋊C4C23×C312yes—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C42—22no—
270C2×C42—24no—
271C2×C42C3×D411yescharacteristic
272C2×C42C3×D411yescharacteristic
273C2×C42C23×C311yescharacteristic
274C2×C42C23×C311yescharacteristic
275C4⋊C8—24no—
276#103—24no—
277#103—42no—
278#103—42no—
279#102—24no—
280C22⋊C8—42no—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—21no—
285C23×C3—21no—
286C23×C3—21no—
287C23×C3—21no—
288C23×C3—21no—
289C23×C3—21no—
290C23×C3—21no—
291C23×C3—41no—
292C23×C3—41no—
293C23×C3—41no—
294C23×C3—41no—
295C23×C3—41no—
296C23×C3—41no—
297C23×C3—41no—
298C23×C3#13712yes—
299C23×C3C22×D411yescharacteristic
300C3×Q8—24no—
301C3×Q8—24no—
302C3×Q8—48no—
303C3×Q8—42no—
304C3×Q8—42no—
305C3×Q8—42no—
306C3×Q8—42no—
307C3×D4—42no—
308C3×D4—42no—
309C3×D4—84no—
310C2×C3×C4—28no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—24no—
321C2×C3×C4—24no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—44no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4#12012yes—
341C2×C3×C4#12012yes—
342C2×C3×C4#13512yes—
343C2×C3×C4#13712yes—
344C2×C3×C4C22×D412yes—
345C2×C3×C4H(5,𝔽2)11yescharacteristic
346C2×C3×C4H(5,𝔽2)12yes—
347C2×C3×C42−1+411yescharacteristic
348C3×C8—42no—
349C3×C8—42no—
350C24—21no—
351C24C2×C3×D411yescharacteristic
352C2×Q8—24no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—21no—
360C2×Q8—22no—
361C2×Q8—21no—
362C2×Q8—42no—
363C2×Q8—42no—
364C2×Q8C2×C3×D414yes—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—42no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—41no—
387C22×C4C2×C3×D411yescharacteristic
388C22×C4C2×C3×D411yescharacteristic
389C22×C4C2×C3×D411yescharacteristic
390C22×C4C2×C3×D411yescharacteristic
391C22×C4C2×C3×D411yescharacteristic
392C22×C4C24×C311yescharacteristic
393Q16—48no—
394SD16—48no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—28no—
404C4⋊C4—24no—
405C4⋊C4—28no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—42no—
410C4⋊C4C2×C3×D414yes—
411C22⋊C4—28no—
412C22⋊C4—42no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C22⋊C4—42no—
417C22⋊C4—42no—
418C42—22no—
419C42—22no—
420C42—28no—
421C42—22no—
422C42C2×C3×D414yes—
423C22×C3—22no—
424C22×C3—22no—
425C22×C3—22no—
426C22×C3—41no—
427C22×C3—41no—
428C22×C3—41no—
429C22×C3—41no—
430C22×C3—81no—
431C22×C3—81no—
432C22×C3—81no—
433C22×C3—81no—
434C22×C3—81no—
435C22×C3—81no—
436C22×C3—81no—
437C22×C3C2×#12011yescharacteristic
438C22×C3#53511yescharacteristic
439C22×C3#53711yescharacteristic
440C22×C3C2×#13511yescharacteristic
441C22×C3C2×#13711yescharacteristic
442C22×C3C2×#13711yescharacteristic
443C22×C3#57511yescharacteristic
444C3×C4—22no—
445C3×C4—22no—
446C3×C4—22no—
447C3×C4—22no—
448C3×C4—41no—
449C3×C4—44no—
450C3×C4—44no—
451C3×C4—81no—
452C3×C4—81no—
453C3×C4—81no—
454C3×C4#53512yes—
455C3×C4#53712yes—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23C3×#13712yes—
470C23C22×C3×D411yescharacteristic
471Q8—24no—
472Q8—24no—
473Q8—48no—
474Q8—42no—
475Q8—42no—
476Q8—42no—
477Q8—42no—
478D4—42no—
479D4—42no—
480D4—84no—
481C2×C4—28no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4C3×#12012yes—
512C2×C4C3×#12012yes—
513C2×C4C3×#13512yes—
514C2×C4C3×#13712yes—
515C2×C4C22×C3×D412yes—
516C2×C4C3×H(5,𝔽2)11yescharacteristic
517C2×C4C3×H(5,𝔽2)12yes—
518C2×C4C3×2−1+411yescharacteristic
519C8—42no—
520C8—42no—
521C2×C3—42no—
522C2×C3—161no—
523C2×C3#261511yescharacteristic
524C2×C3#300211yescharacteristic
525C2×C3#300411yescharacteristic
526C2×C3#321011yescharacteristic
527C2×C3#321511yescharacteristic
528C2×C3#324811yescharacteristic
529C2×C3#325211yescharacteristic
530C22—22no—
531C22—22no—
532C22—22no—
533C22—41no—
534C22—41no—
535C22—41no—
536C22—41no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22C2×C3×#12011yescharacteristic
545C22C3×#53511yescharacteristic
546C22C3×#53711yescharacteristic
547C22C2×C3×#13511yescharacteristic
548C22C2×C3×#13711yescharacteristic
549C22C2×C3×#13711yescharacteristic
550C22C3×#57511yescharacteristic
551C4—22no—
552C4—22no—
553C4—22no—
554C4—22no—
555C4—44no—
556C4—41no—
557C4—44no—
558C4—81no—
559C4—81no—
560C4—81no—
561C4C3×#53512yes—
562C4C3×#53712yes—
563C3#2966811yescharacteristic, direct product factor, Sylow
564C2—42no—
565C2—161no—
566C2C3×#261511yescharacteristic
567C2C3×#300211yescharacteristic
568C2C3×#300411yescharacteristic
569C2C3×#321011yescharacteristic
570C2C3×#321511yescharacteristic
571C2C3×#324811yescharacteristic
572C2C3×#325211yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
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
1. C2×C3×#533C2semidirect
2. C2×C3×#464C2semidirect
4. C3×#2024C2semidirect
5. C3×#2023C2semidirect
6. C3×#2016C2semidirect
7. C3×#2013C2semidirect
8. C3×#2001C2semidirect
10. C3×#1805C2semidirect
11. C3×#1797C2semidirect
12. #29668C3direct (N × H)
46. C2×#533C2×C3semidirect
47. C2×#464C2×C3semidirect
49. #2024C2×C3semidirect
50. #2023C2×C3semidirect
51. #2016C2×C3semidirect
52. #2013C2×C3semidirect
53. #2001C2×C3semidirect
55. #1805C2×C3semidirect
56. #1797C2×C3semidirect
84. C3×C4×Q8D4semidirect
251. C4×Q8C3×D4semidirect
563. C3#29668direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5332
C2×C3×#4642
C3×#28502
C3×#20242
C3×#20232
C3×#20162
C3×#20132
C3×#20012
C3×#20002
C3×#18052
C3×#17972
#296683