Finite Group Databasebeta

← C3×#29605 #10648212

Subgroups — C3×#29605 #10648212

Order 768.

1268 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#536C211yesmaximal, characteristic
2C2×C3×#485C211yesmaximal, characteristic, complemented
3C2×C3×#451C211yesmaximal, characteristic, complemented
4C3×#2616C211yesmaximal, characteristic
5C3×#2036C212yesmaximal, complemented
6C3×#2035C212yesmaximal, complemented
7C3×#2001C212yesmaximal, complemented
8C3×#2001C212yesmaximal, complemented
9C3×#1825C211yesmaximal, characteristic, complemented
10C3×#1787C211yesmaximal, characteristic, complemented
11C3×#1780C211yesmaximal, characteristic
12#29605C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×C3×Q8C2211yescharacteristic
15C22×C3×Q16C2212yes—
16C3×#536—24no—
17C3×#536—24no—
18C2×C3×C4⋊Q8—21no—
19C2×C3×#124—21no—
20C2×C3×#123—22no—
21C2×C3×C22⋊Q8C2211yescharacteristic
22C2×C3×C22⋊Q8C2212yes—
23C2×C3×C22⋊Q8C2212yes—
24C2×C3×C22⋊Q8C2212yes—
25C2×C3×#117C2211yescharacteristic
26C3×#485—24no—
27C3×#451—24no—
28C2×C3×#107C2211yescharacteristic
29C2×C3×#106C2211yescharacteristic
30C2×C3×#103C2211yescharacteristic
31C2×C3×#103C2211yescharacteristic
32C2×C3×#103C2212yes—
33C2×C3×#103C2212yes—
34C2×C3×#103C2212yes—
35C2×C3×C22⋊C8C2211yescharacteristic
36C2×C3×C22⋊C8C2211yescharacteristic
37C2×C3×C22⋊C8C2212yes—
38C2×C3×C22⋊C8C2212yes—
39C3×#395—21no—
40C3×#394—22no—
41C3×#394—21no—
42C3×#380C2212yes—
43C3×#380C2211yescharacteristic
44C3×#340C2212yes—
45C3×#340C2212yes—
46C2×#536C2×C311yescharacteristic
47C2×#485C2×C311yescharacteristic, complemented
48C2×#451C2×C311yescharacteristic, complemented
49#2616C2×C311yescharacteristic
50#2036C2×C312yescomplemented
51#2035C2×C312yescomplemented
52#2001C2×C312yescomplemented
53#2001C2×C312yescomplemented
54#1825C2×C311yescharacteristic, complemented
55#1787C2×C311yescharacteristic, complemented
56#1780C2×C311yescharacteristic
57C2×C3×#41—22no—
58C2×C3×#41—22no—
59C2×C3×#41—42no—
60C2×C3×#41—42no—
61C22×C3×Q8—21no—
62C22×C3×Q8—22no—
63C22×C3×Q8—42no—
64C22×C3×Q8—42no—
65C22×C3×Q8D414yes—
66C22×C3×Q8D411yescharacteristic
67C22×C3×Q8C2311yescharacteristic
68C22×C3×Q8C2312yes—
69C22×C3×D4D412yes—
70C22×C3×D4D411yescharacteristic
71C23×C3×C4—21no—
72C23×C3×C4C2312yes—
73C23×C3×C4C2311yescharacteristic
74C2×C3×Q16—24no—
75C2×C3×Q16—42no—
76C2×C3×Q16—42no—
77C22×C3×C8C2312yes—
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C3×C4⋊Q8—28no—
81C3×#124—28no—
82C3×#123—216no—
83C3×C22⋊Q8—24no—
84C3×C22⋊Q8—24no—
85C3×C22⋊Q8—24no—
86C3×C22⋊Q8—24no—
87C3×C22⋊Q8—22no—
88C3×C22⋊Q8—22no—
89C3×C22⋊Q8—42no—
90C3×C22⋊Q8—42no—
91C3×#117—22no—
92C3×#117D414yes—
93C2×C3×C4⋊C4—22no—
94C2×C3×C4⋊C4—22no—
95C2×C3×C4⋊C4—22no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2312yes—
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—22no—
103C2×C3×C22⋊C4—22no—
104C2×C3×C22⋊C4—22no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—22no—
107C2×C3×C22⋊C4—21no—
108C2×C3×C42—21no—
109C3×#107—22no—
110C3×#106—22no—
111C3×#103—24no—
112C3×#103—22no—
113C3×#103—22no—
114C3×#103—42no—
115C3×#103—42no—
116C3×C22⋊C8—22no—
117C3×C22⋊C8—22no—
118C3×C22⋊C8—42no—
119C3×C22⋊C8—42no—
120C3×#95—22no—
121C3×#95—21no—
122C22×#41C22×C311yescharacteristic
123C23×Q8C22×C311yescharacteristic
124C22×Q16C22×C312yes—
125#536—24no—
126#536—24no—
127C2×C4⋊Q8—21no—
128C2×#124—21no—
129C2×#123—22no—
130C2×C22⋊Q8C22×C311yescharacteristic
131C2×C22⋊Q8C22×C312yes—
132C2×C22⋊Q8C22×C312yes—
133C2×C22⋊Q8C22×C312yes—
134C2×#117C22×C311yescharacteristic
135#485—24no—
136#451—24no—
137C2×#107C22×C311yescharacteristic
138C2×#106C22×C311yescharacteristic
139C2×#103C22×C311yescharacteristic
140C2×#103C22×C311yescharacteristic
141C2×#103C22×C312yes—
142C2×#103C22×C312yes—
143C2×#103C22×C312yes—
144C2×C22⋊C8C22×C311yescharacteristic
145C2×C22⋊C8C22×C311yescharacteristic
146C2×C22⋊C8C22×C312yes—
147C2×C22⋊C8C22×C312yes—
148#395—21no—
149#394—22no—
150#394—21no—
151#380C22×C311yescharacteristic
152#380C22×C312yes—
153#340C22×C312yes—
154#340C22×C312yes—
155C24×C3C2×D412yes—
156C24×C3C2×D411yescharacteristic
157C3×#41—48no—
158C3×#41—44no—
159C3×#41—44no—
160C2×C3×Q8—24no—
161C2×C3×Q8—22no—
162C2×C3×Q8—22no—
163C2×C3×Q8—21no—
164C2×C3×Q8—21no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8—22no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—42no—
172C2×C3×Q8—48no—
173C2×C3×Q8—44no—
174C2×C3×Q8—44no—
175C2×C3×Q8—44no—
176C2×C3×Q8C2×D414yes—
177C2×C3×D4—24no—
178C2×C3×D4—22no—
179C2×C3×D4—42no—
180C2×C3×D4—42no—
181C2×C3×D4—42no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—24no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—22no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4—21no—
195C22×C3×C4—22no—
196C22×C3×C4—21no—
197C22×C3×C4—44no—
198C22×C3×C4C2×D412yes—
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D412yes—
201C22×C3×C4C2×D412yes—
202C22×C3×C4C2411yescharacteristic
203C3×Q16—48no—
204C2×C3×C8—21no—
205C2×C3×C8—21no—
206C2×C3×C8—22no—
207C2×C3×C8—22no—
208C2×C3×C8—22no—
209C2×C3×C8—22no—
210C2×C3×C8—22no—
211C3×C4⋊C4—28no—
212C3×C4⋊C4—28no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—22no—
218C3×C4⋊C4—22no—
219C3×C4⋊C4—44no—
220C3×C4⋊C4—42no—
221C3×C4⋊C4C2×D412yes—
222C3×C4⋊C4C2×D412yes—
223C3×C22⋊C4—28no—
224C3×C22⋊C4—28no—
225C3×C22⋊C4—24no—
226C3×C22⋊C4—44no—
227C3×C22⋊C4—42no—
228C3×C22⋊C4—81no—
229C3×C22⋊C4—82no—
230C3×C42—24no—
231C2×#41—22no—
232C2×#41—22no—
233C2×#41—42no—
234C2×#41—42no—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×Q8—42no—
238C22×Q8—42no—
239C22×Q8C3×D414yes—
240C22×Q8C3×D411yescharacteristic
241C22×Q8C23×C311yescharacteristic
242C22×Q8C23×C312yes—
243C22×D4C3×D412yes—
244C22×D4C3×D411yescharacteristic
245C23×C4—21no—
246C23×C4C23×C312yes—
247C23×C4C23×C311yescharacteristic
248C2×Q16—24no—
249C2×Q16—42no—
250C2×Q16—42no—
251C22×C8C23×C312yes—
252C22×C8C23×C311yescharacteristic
253C22×C8C23×C311yescharacteristic
254C4⋊Q8—28no—
255#124—28no—
256#123—216no—
257C22⋊Q8—24no—
258C22⋊Q8—24no—
259C22⋊Q8—24no—
260C22⋊Q8—24no—
261C22⋊Q8—22no—
262C22⋊Q8—22no—
263C22⋊Q8—42no—
264C22⋊Q8—42no—
265#117—22no—
266#117C3×D414yes—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4C23×C312yes—
272C2×C4⋊C4C23×C311yescharacteristic
273C2×C4⋊C4C23×C311yescharacteristic
274C2×C4⋊C4C23×C311yescharacteristic
275C2×C22⋊C4—21no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—21no—
282C2×C42—21no—
283#107—22no—
284#106—22no—
285#103—24no—
286#103—22no—
287#103—22no—
288#103—42no—
289#103—42no—
290C22⋊C8—22no—
291C22⋊C8—22no—
292C22⋊C8—42no—
293C22⋊C8—42no—
294#95—22no—
295#95—21no—
296C23×C3—22no—
297C23×C3—21no—
298C23×C3—22no—
299C23×C3—21no—
300C23×C3—22no—
301C23×C3—22no—
302C23×C3—22no—
303C23×C3—22no—
304C23×C3—22no—
305C23×C3—22no—
306C23×C3—22no—
307C23×C3#12012yes—
308C23×C3C22×D411yescharacteristic
309C3×Q8—24no—
310C3×Q8—24no—
311C3×Q8—48no—
312C3×Q8—44no—
313C3×Q8—42no—
314C3×Q8—42no—
315C3×Q8—42no—
316C3×Q8—41no—
317C3×Q8—41no—
318C3×D4—44no—
319C3×D4—48no—
320C2×C3×C4—22no—
321C2×C3×C4—24no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—24no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—48no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—41no—
334C2×C3×C4—42no—
335C2×C3×C4—41no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—41no—
346C2×C3×C4—42no—
347C2×C3×C4—41no—
348C2×C3×C4#12012yes—
349C2×C3×C4C22×D411yescharacteristic
350C2×C3×C4C22×D411yescharacteristic
351C2×C3×C4H(5,𝔽2)12yes—
352C2×C3×C4H(5,𝔽2)12yes—
353C3×C8—42no—
354C3×C8—41no—
355C3×C8—41no—
356C24C2×C3×D412yes—
357C24C2×C3×D411yescharacteristic
358#41—48no—
359#41—44no—
360#41—44no—
361C2×Q8—24no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—42no—
373C2×Q8—48no—
374C2×Q8—44no—
375C2×Q8—44no—
376C2×Q8—44no—
377C2×Q8C2×C3×D414yes—
378C2×D4—24no—
379C2×D4—22no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—42no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—44no—
399C22×C4C2×C3×D412yes—
400C22×C4C2×C3×D411yescharacteristic
401C22×C4C2×C3×D412yes—
402C22×C4C2×C3×D412yes—
403C22×C4C24×C311yescharacteristic
404Q16—48no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C4⋊C4—28no—
413C4⋊C4—28no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—44no—
421C4⋊C4—42no—
422C4⋊C4C2×C3×D412yes—
423C4⋊C4C2×C3×D412yes—
424C22⋊C4—28no—
425C22⋊C4—28no—
426C22⋊C4—24no—
427C22⋊C4—44no—
428C22⋊C4—42no—
429C22⋊C4—81no—
430C22⋊C4—82no—
431C42—24no—
432C22×C3—22no—
433C22×C3—22no—
434C22×C3—22no—
435C22×C3—42no—
436C22×C3—42no—
437C22×C3—42no—
438C22×C3—42no—
439C22×C3—42no—
440C22×C3—42no—
441C22×C3—42no—
442C22×C3—42no—
443C22×C3—42no—
444C22×C3C2×#12011yescharacteristic
445C22×C3#53412yes—
446C22×C3#57711yescharacteristic
447C22×C3#57812yes—
448C22×C3#57811yescharacteristic
449C3×C4—22no—
450C3×C4—42no—
451C3×C4—44no—
452C3×C4—42no—
453C3×C4—42no—
454C3×C4—81no—
455C3×C4—82no—
456C3×C4—81no—
457C3×C4#53412yes—
458C3×C4#53511yescharacteristic
459C3×C4#53511yescharacteristic
460C23—21no—
461C23—22no—
462C23—21no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23C3×#12012yes—
472C23C22×C3×D411yescharacteristic
473Q8—24no—
474Q8—24no—
475Q8—48no—
476Q8—44no—
477Q8—42no—
478Q8—42no—
479Q8—42no—
480Q8—41no—
481Q8—41no—
482D4—44no—
483D4—48no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—48no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4C3×#12012yes—
513C2×C4C22×C3×D411yescharacteristic
514C2×C4C22×C3×D411yescharacteristic
515C2×C4C3×H(5,𝔽2)12yes—
516C2×C4C3×H(5,𝔽2)12yes—
517C8—42no—
518C8—41no—
519C8—41no—
520C2×C3—42no—
521C2×C3—82no—
522C2×C3#261311yescharacteristic
523C2×C3#301111yescharacteristic
524C2×C3#301311yescharacteristic
525C2×C3#320112yes—
526C2×C3#320212yes—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22C2×C3×#12011yescharacteristic
540C22C3×#53412yes—
541C22C3×#57711yescharacteristic
542C22C3×#57812yes—
543C22C3×#57811yescharacteristic
544C4—22no—
545C4—42no—
546C4—42no—
547C4—42no—
548C4—44no—
549C4—82no—
550C4—81no—
551C4—81no—
552C4C3×#53412yes—
553C4C3×#53511yescharacteristic
554C4C3×#53511yescharacteristic
555C3#2960511yescharacteristic, direct product factor, Sylow
556C2—42no—
557C2—82no—
558C2C3×#261311yescharacteristic
559C2C3×#301111yescharacteristic
560C2C3×#301311yescharacteristic
561C2C3×#320112yes—
562C2C3×#320212yes—
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×#41 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#485C2semidirect
3. C2×C3×#451C2semidirect
5. C3×#2036C2semidirect
6. C3×#2035C2semidirect
7. C3×#2001C2semidirect
8. C3×#2001C2semidirect
9. C3×#1825C2semidirect
10. C3×#1787C2semidirect
12. #29605C3direct (N × H)
47. C2×#485C2×C3semidirect
48. C2×#451C2×C3semidirect
50. #2036C2×C3semidirect
51. #2035C2×C3semidirect
52. #2001C2×C3semidirect
53. #2001C2×C3semidirect
54. #1825C2×C3semidirect
55. #1787C2×C3semidirect
555. C3#29605direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5362
C2×C3×#4852
C2×C3×#4512
C3×#26162
C3×#20362
C3×#20352
C3×#20012
C3×#18252
C3×#17872
C3×#17802
#296053