Finite Group Databasebeta

← C3×#32938 #10651545

Subgroups — C3×#32938 #10651545

Order 768.

1436 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
1C22×C3×#131C211yesmaximal, characteristic, complemented
2C2×C3×#468C212yesmaximal, complemented
3C2×C3×#465C211yesmaximal, characteristic, complemented
4C3×#2616C211yesmaximal, characteristic, complemented
5C3×#2615C211yesmaximal, characteristic
6C3×#1895C211yesmaximal, characteristic, complemented
7C3×#1894C211yesmaximal, characteristic, complemented
8C3×#1893C212yesmaximal, complemented
9C3×#1892C212yesmaximal, complemented
10C3×#1809C212yesmaximal, complemented
11C3×#1808C211yesmaximal, characteristic, complemented
12#32938C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C22×C3×SD16C2211yescharacteristic
15C2×C3×#131—28no—
16C2×C3×#131—24no—
17C22×C3×M16C2211yescharacteristic
18C22×C3×M16C2212yes—
19C23×C3×C8C2212yes—
20C23×C3×C8C2211yescharacteristic
21C2×C3×C22⋊Q8C2212yes—
22C2×C3×C22⋊Q8C2211yescharacteristic
23C2×C3×#121C2212yes—
24C2×C3×#121C2211yescharacteristic
25C3×#468—28no—
26C3×#465—24no—
27C2×C3×#107C2212yes—
28C2×C3×#106C2211yescharacteristic
29C2×C3×#103C2211yescharacteristic
30C2×C3×#103C2212yes—
31C2×C3×#102C2211yescharacteristic
32C2×C3×#102C2212yes—
33C3×#395—21no—
34C3×#394—21no—
35C3×#394—22no—
36C3×#393—21no—
37C3×#392—21no—
38C3×#392—22no—
39C3×#380C2211yescharacteristic
40C3×#380C2212yes—
41C3×#380C2211yescharacteristic
42C3×#380C2212yes—
43C3×#340C2212yes—
44C3×#340C2212yes—
45C3×#340C2212yes—
46C22×#131C2×C311yescharacteristic, complemented
47C2×#468C2×C312yescomplemented
48C2×#465C2×C311yescharacteristic, complemented
49#2616C2×C311yescharacteristic, complemented
50#2615C2×C311yescharacteristic
51#1895C2×C311yescharacteristic, complemented
52#1894C2×C311yescharacteristic, complemented
53#1893C2×C312yescomplemented
54#1892C2×C312yescomplemented
55#1809C2×C312yescomplemented
56#1808C2×C311yescharacteristic, complemented
57C2×C3×#41—24no—
58C2×C3×#41—22no—
59C2×C3×#41—22no—
60C2×C3×#41—24no—
61C22×C3×Q8D411yescharacteristic
62C22×C3×Q8C2311yescharacteristic
63C22×C3×D4—22no—
64C22×C3×D4—21no—
65C22×C3×D4D412yes—
66C22×C3×D4D411yescharacteristic
67C22×C3×D4C2311yescharacteristic
68C23×C3×C4C2312yes—
69C23×C3×C4C2311yescharacteristic
70C2×C3×SD16—22no—
71C2×C3×SD16—24no—
72C3×#131—232no—
73C2×C3×M16—22no—
74C2×C3×M16—24no—
75C2×C3×M16—24no—
76C2×C3×M16—24no—
77C2×C3×M16D418yescomplemented
78C22×C3×C8—24no—
79C22×C3×C8—24no—
80C22×C3×C8—24no—
81C22×C3×C8—24no—
82C22×C3×C8D414yescomplemented
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2312yes—
85C22×C3×C8C2311yescharacteristic
86C3×C22⋊Q8—22no—
87C3×C22⋊Q8—24no—
88C3×C22⋊Q8—24no—
89C3×C22⋊Q8—22no—
90C3×#121—22no—
91C3×#121—24no—
92C3×#121—24no—
93C3×#121—22no—
94C2×C3×C4⋊C4—22no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C4⋊C4C2312yes—
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2312yes—
100C2×C3×C22⋊C4—22no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—22no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—22no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C22⋊C4—22no—
108C3×#107—24no—
109C3×#106—22no—
110C3×#103—22no—
111C3×#103—24no—
112C3×#102—22no—
113C3×#102—24no—
114C3×#95—21no—
115C3×#95—22no—
116C3×#95—21no—
117C3×#95—22no—
118C22×#41C22×C311yescharacteristic
119C22×SD16C22×C311yescharacteristic
120C2×#131—28no—
121C2×#131—24no—
122C22×M16C22×C311yescharacteristic
123C22×M16C22×C312yes—
124C23×C8C22×C312yes—
125C23×C8C22×C311yescharacteristic
126C2×C22⋊Q8C22×C312yes—
127C2×C22⋊Q8C22×C311yescharacteristic
128C2×#121C22×C312yes—
129C2×#121C22×C311yescharacteristic
130#468—28no—
131#465—24no—
132C2×#107C22×C312yes—
133C2×#106C22×C311yescharacteristic
134C2×#103C22×C311yescharacteristic
135C2×#103C22×C312yes—
136C2×#102C22×C311yescharacteristic
137C2×#102C22×C312yes—
138#395—21no—
139#394—21no—
140#394—22no—
141#393—21no—
142#392—21no—
143#392—22no—
144#380C22×C311yescharacteristic
145#380C22×C311yescharacteristic
146#380C22×C312yes—
147#380C22×C312yes—
148#340C22×C312yes—
149#340C22×C312yes—
150#340C22×C312yes—
151C24×C3—21no—
152C24×C3C2×D411yescharacteristic
153C24×C3C2×D412yes—
154C3×#41—28no—
155C3×#41—216no—
156C3×#41—28no—
157C2×C3×Q8—22no—
158C2×C3×Q8—24no—
159C2×C3×Q8—21no—
160C2×C3×Q8—22no—
161C2×C3×Q8—21no—
162C2×C3×Q8—22no—
163C2×C3×D4—24no—
164C2×C3×D4—24no—
165C2×C3×D4—24no—
166C2×C3×D4—22no—
167C2×C3×D4—24no—
168C2×C3×D4—21no—
169C2×C3×D4—22no—
170C2×C3×D4—21no—
171C2×C3×D4—22no—
172C2×C3×D4—42no—
173C2×C3×D4—44no—
174C2×C3×D4—82no—
175C2×C3×D4—82no—
176C2×C3×D4—82no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—21no—
185C22×C3×C4—22no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4C2×D412yes—
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D414yes—
191C22×C3×C4C2×D412yes—
192C22×C3×C4#4112yes—
193C22×C3×C4#4114yes—
194C22×C3×C4C2411yescharacteristic
195C3×SD16—44no—
196C3×M16—28no—
197C3×M16—216no—
198C2×C3×C8—216no—
199C2×C3×C8—22no—
200C2×C3×C8—22no—
201C2×C3×C8—22no—
202C2×C3×C8—28no—
203C2×C3×C8C2×D412yes—
204C2×C3×C8C2×D414yes—
205C2×C3×C8C2×D412yes—
206C2×C3×C8C2×D414yes—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—42no—
216C3×C4⋊C4—44no—
217C3×C22⋊C4—42no—
218C3×C22⋊C4—44no—
219C3×C22⋊C4—42no—
220C3×C22⋊C4—44no—
221C3×C22⋊C4—81no—
222C3×C22⋊C4—82no—
223C3×C22⋊C4—81no—
224C3×C22⋊C4—82no—
225C2×#41—24no—
226C2×#41—22no—
227C2×#41—22no—
228C2×#41—24no—
229C22×Q8C3×D411yescharacteristic
230C22×Q8C23×C311yescharacteristic
231C22×D4—22no—
232C22×D4—21no—
233C22×D4C3×D412yes—
234C22×D4C3×D411yescharacteristic
235C22×D4C23×C311yescharacteristic
236C23×C4C23×C312yes—
237C23×C4C23×C311yescharacteristic
238C2×SD16—22no—
239C2×SD16—24no—
240#131—232no—
241C2×M16—22no—
242C2×M16—24no—
243C2×M16—24no—
244C2×M16—24no—
245C2×M16C3×D418yescomplemented
246C22×C8—24no—
247C22×C8—24no—
248C22×C8—24no—
249C22×C8—24no—
250C22×C8C3×D414yescomplemented
251C22×C8C23×C311yescharacteristic
252C22×C8C23×C312yes—
253C22×C8C23×C311yescharacteristic
254C22⋊Q8—22no—
255C22⋊Q8—24no—
256C22⋊Q8—24no—
257C22⋊Q8—22no—
258#121—22no—
259#121—24no—
260#121—24no—
261#121—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4C23×C311yescharacteristic
265C2×C4⋊C4C23×C312yes—
266C2×C4⋊C4C23×C311yescharacteristic
267C2×C4⋊C4C23×C312yes—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—22no—
276#107—24no—
277#106—22no—
278#103—22no—
279#103—24no—
280#102—22no—
281#102—24no—
282#95—21no—
283#95—22no—
284#95—21no—
285#95—22no—
286C23×C3—22no—
287C23×C3—22no—
288C23×C3—22no—
289C23×C3—21no—
290C23×C3—22no—
291C23×C3—22no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3—21no—
295C23×C3—22no—
296C23×C3—41no—
297C23×C3—42no—
298C23×C3—41no—
299C23×C3—42no—
300C23×C3—41no—
301C23×C3#12112yes—
302C23×C3#12114yes—
303C23×C3C22×D411yescharacteristic
304C3×Q8—28no—
305C3×Q8—41no—
306C3×Q8—41no—
307C3×Q8—42no—
308C3×D4—216no—
309C3×D4—28no—
310C3×D4—41no—
311C3×D4—41no—
312C3×D4—42no—
313C3×D4—84no—
314C3×D4—88no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—24no—
319C2×C3×C4—22no—
320C2×C3×C4—24no—
321C2×C3×C4—22no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—41no—
326C2×C3×C4—42no—
327C2×C3×C4—41no—
328C2×C3×C4—42no—
329C2×C3×C4—41no—
330C2×C3×C4—42no—
331C2×C3×C4—41no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—41no—
338C2×C3×C4#12112yes—
339C2×C3×C4#12114yes—
340C2×C3×C4C22×D411yescharacteristic
341C2×C3×C4C22×D412yes—
342C2×C3×C4C2×#4111yescharacteristic
343C2×C3×C4C2×#4112yes—
344C3×C8—22no—
345C3×C8—24no—
346C3×C8#12014yes—
347C24—21no—
348C24C2×C3×D411yescharacteristic
349C24C2×C3×D412yes—
350#41—28no—
351#41—216no—
352#41—28no—
353C2×Q8—22no—
354C2×Q8—24no—
355C2×Q8—21no—
356C2×Q8—22no—
357C2×Q8—21no—
358C2×Q8—22no—
359C2×D4—24no—
360C2×D4—24no—
361C2×D4—24no—
362C2×D4—22no—
363C2×D4—24no—
364C2×D4—21no—
365C2×D4—22no—
366C2×D4—21no—
367C2×D4—22no—
368C2×D4—42no—
369C2×D4—44no—
370C2×D4—82no—
371C2×D4—82no—
372C2×D4—82no—
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×C4C2×C3×D412yes—
385C22×C4C2×C3×D411yescharacteristic
386C22×C4C2×C3×D414yes—
387C22×C4C2×C3×D412yes—
388C22×C4C3×#4112yes—
389C22×C4C3×#4114yes—
390C22×C4C24×C311yescharacteristic
391SD16—44no—
392M16—28no—
393M16—216no—
394C2×C8—216no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—28no—
399C2×C8C2×C3×D412yes—
400C2×C8C2×C3×D414yes—
401C2×C8C2×C3×D412yes—
402C2×C8C2×C3×D414yes—
403C4⋊C4—21no—
404C4⋊C4—22no—
405C4⋊C4—21no—
406C4⋊C4—22no—
407C4⋊C4—21no—
408C4⋊C4—22no—
409C4⋊C4—21no—
410C4⋊C4—22no—
411C4⋊C4—42no—
412C4⋊C4—44no—
413C22⋊C4—42no—
414C22⋊C4—44no—
415C22⋊C4—42no—
416C22⋊C4—44no—
417C22⋊C4—81no—
418C22⋊C4—82no—
419C22⋊C4—81no—
420C22⋊C4—82no—
421C22×C3—22no—
422C22×C3—24no—
423C22×C3—22no—
424C22×C3—22no—
425C22×C3—24no—
426C22×C3—24no—
427C22×C3—42no—
428C22×C3—42no—
429C22×C3—42no—
430C22×C3—42no—
431C22×C3—42no—
432C22×C3—42no—
433C22×C3—81no—
434C22×C3—81no—
435C22×C3—81no—
436C22×C3—82no—
437C22×C3—82no—
438C22×C3C2×#12111yescharacteristic
439C22×C3C2×#12112yes—
440C22×C3#57611yescharacteristic
441C22×C3#57712yes—
442C22×C3#57811yescharacteristic
443C3×C4—24no—
444C3×C4—22no—
445C3×C4—81no—
446C3×C4—81no—
447C3×C4—81no—
448C3×C4—82no—
449C3×C4—82no—
450C3×C4C2×#12011yescharacteristic
451C3×C4#52512yes—
452C3×C4#52511yescharacteristic
453C23—21no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—21no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—41no—
464C23—42no—
465C23—41no—
466C23—41no—
467C23—42no—
468C23C3×#12112yes—
469C23C3×#12114yes—
470C23C22×C3×D411yescharacteristic
471Q8—28no—
472Q8—41no—
473Q8—41no—
474Q8—42no—
475D4—216no—
476D4—28no—
477D4—41no—
478D4—41no—
479D4—42no—
480D4—84no—
481D4—88no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4C3×#12112yes—
506C2×C4C3×#12114yes—
507C2×C4C22×C3×D411yescharacteristic
508C2×C4C22×C3×D412yes—
509C2×C4C2×C3×#4111yescharacteristic
510C2×C4C2×C3×#4112yes—
511C8—22no—
512C8—24no—
513C8C3×#12014yes—
514C2×C3—42no—
515C2×C3—44no—
516C2×C3—161no—
517C2×C3#284711yescharacteristic
518C2×C3#305711yescharacteristic
519C2×C3#306111yescharacteristic
520C2×C3#306212yes—
521C2×C3#306312yes—
522C22—22no—
523C22—22no—
524C22—24no—
525C22—24no—
526C22—22no—
527C22—24no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—81no—
535C22—81no—
536C22—82no—
537C22—82no—
538C22—81no—
539C22C2×C3×#12111yescharacteristic
540C22C2×C3×#12112yes—
541C22C3×#57611yescharacteristic
542C22C3×#57712yes—
543C22C3×#57811yescharacteristic
544C4—24no—
545C4—22no—
546C4—81no—
547C4—81no—
548C4—82no—
549C4—82no—
550C4—81no—
551C4C2×C3×#12011yescharacteristic
552C4C3×#52512yes—
553C4C3×#52511yescharacteristic
554C3#3293811yescharacteristic, direct product factor, Sylow
555C2—42no—
556C2—44no—
557C2—161no—
558C2C3×#284711yescharacteristic
559C2C3×#305711yescharacteristic
560C2C3×#306111yescharacteristic
561C2C3×#306212yes—
562C2C3×#306312yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#2615C2
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
1. C22×C3×#131C2semidirect
2. C2×C3×#468C2semidirect
3. C2×C3×#465C2semidirect
4. C3×#2616C2semidirect
6. C3×#1895C2semidirect
7. C3×#1894C2semidirect
8. C3×#1893C2semidirect
9. C3×#1892C2semidirect
10. C3×#1809C2semidirect
11. C3×#1808C2semidirect
12. #32938C3direct (N × H)
46. C22×#131C2×C3semidirect
47. C2×#468C2×C3semidirect
48. C2×#465C2×C3semidirect
49. #2616C2×C3semidirect
51. #1895C2×C3semidirect
52. #1894C2×C3semidirect
53. #1893C2×C3semidirect
54. #1892C2×C3semidirect
55. #1809C2×C3semidirect
56. #1808C2×C3semidirect
77. C2×C3×M16D4semidirect
82. C22×C3×C8D4semidirect
245. C2×M16C3×D4semidirect
250. C22×C8C3×D4semidirect
554. C3#32938direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1312
C2×C3×#4682
C2×C3×#4652
C3×#26162
C3×#26152
C3×#18952
C3×#18942
C3×#18932
C3×#18922
C3×#18092
C3×#18082
#329383