Finite Group Databasebeta

← C3×#18604 #10637211

Subgroups — C3×#18604 #10637211

Order 768.

900 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3507C211yesmaximal, characteristic, complemented
2C3×#3214C211yesmaximal, characteristic
3C3×#3182C211yesmaximal, characteristic, complemented
4C3×#3103C211yesmaximal, characteristic, complemented
5C3×#1650C212yesmaximal, complemented
6C3×#1643C212yesmaximal, complemented
7C3×#1628C212yesmaximal, complemented
8C3×#1622C212yesmaximal, complemented
9C3×#1554C211yesmaximal, characteristic, complemented
10C3×#1491C211yesmaximal, characteristic, complemented
11C3×#1484C211yesmaximal, characteristic, complemented
12#18604C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#559—21no—
14C3×#557—21no—
15C3×#556—22no—
16C3×#556C2212yes—
17C3×#554C2212yes—
18C3×#552—22no—
19C3×#539—22no—
20C3×#536—21no—
21C3×#533C2211yescharacteristic
22C3×#532C2211yescharacteristic
23C3×#532C2211yescharacteristic
24C3×#529—22no—
25C3×#520—21no—
26C2×C3×C4×Q8C2211yescharacteristic
27C3×#500C2211yescharacteristic
28C3×#496—21no—
29C3×#492C2211yescharacteristic
30C3×#489—21no—
31C3×#487C2211yescharacteristic
32C3×#486C2211yescharacteristic
33C3×#481—21no—
34C3×#475C2212yes—
35C3×#468—22no—
36C3×#461C2212yes—
37C3×#460C2212yes—
38C3×#459C2212yes—
39C3×#449—22no—
40C3×#448—21no—
41C3×#440—22no—
42C3×C4×SD16—22no—
43C3×#432C2211yescharacteristic
44C3×#431C2211yescharacteristic
45C3×C4×M16C2211yescharacteristic
46C3×#333C2212yes—
47C3×#332C2212yes—
48C3×Q8⋊C8C2212yes—
49C3×Q8⋊C8C2212yes—
50C3×D4⋊C8C2212yes—
51C3×D4⋊C8C2212yes—
52#3507C2×C311yescharacteristic, complemented
53#3214C2×C311yescharacteristic
54#3182C2×C311yescharacteristic, complemented
55#3103C2×C311yescharacteristic, complemented
56#1650C2×C312yescomplemented
57#1643C2×C312yescomplemented
58#1628C2×C312yescomplemented
59#1622C2×C312yescomplemented
60#1554C2×C311yescharacteristic, complemented
61#1491C2×C311yescharacteristic, complemented
62#1484C2×C311yescharacteristic, complemented
63C2×C3×#41—21no—
64C2×C3×#41—21no—
65C22×C3×Q8D411yescharacteristic
66C2×C3×SD16—22no—
67C2×C3×SD16—22no—
68C2×C3×SD16—22no—
69C2×C3×D8—22no—
70C2×C3×M16—21no—
71C3×C4⋊Q8—22no—
72C3×C4⋊Q8D411yescharacteristic
73C3×C4⋊Q8C2311yescharacteristic
74C3×C4⋊Q8C2311yescharacteristic
75C3×C4⋊Q8C2311yescharacteristic
76C3×#127D411yescharacteristic
77C3×#127C2311yescharacteristic
78C3×#126—24no—
79C3×#126—24no—
80C3×#125—24no—
81C3×#125—22no—
82C3×#125—21no—
83C3×#124—22no—
84C3×#124—22no—
85C3×#124—21no—
86C3×#124—21no—
87C3×#123—24no—
88C3×#123—24no—
89C3×C22⋊Q8—22no—
90C3×C22⋊Q8—22no—
91C3×C22⋊Q8—24no—
92C3×C22⋊Q8—21no—
93C3×C22⋊Q8D412yes—
94C3×#121—21no—
95C3×#121—21no—
96C3×#121—21no—
97C3×#121D412yes—
98C3×C4×Q8—22no—
99C3×C4×Q8—22no—
100C3×C4×Q8—21no—
101C3×C4×Q8—21no—
102C3×C4×Q8—21no—
103C3×C4×Q8D412yes—
104C3×C4×Q8D412yes—
105C3×C4×Q8C2312yes—
106C3×C4×D4—22no—
107C3×C4×D4—24no—
108C3×C4×D4—22no—
109C3×C4×D4—21no—
110C3×C4×D4—21no—
111C3×C4×D4—21no—
112C3×C4×D4C2312yes—
113C3×#117—24no—
114C3×#117—21no—
115C3×#117—22no—
116C2×C3×C4⋊C4—21no—
117C2×C3×C4⋊C4—22no—
118C2×C3×C4⋊C4D411yescharacteristic
119C2×C3×C42—21no—
120C2×C3×C42C2311yescharacteristic
121C3×#107—22no—
122C3×#106—22no—
123C3×C4⋊C8C2312yes—
124C3×C4⋊C8C2312yes—
125C3×#103—22no—
126C3×#103—22no—
127C3×#103—22no—
128C3×#102—22no—
129C3×#102—22no—
130C3×#102—22no—
131C3×C22⋊C8—21no—
132C3×C22⋊C8—21no—
133C3×#97—21no—
134C3×#97—21no—
135C3×C4×C8—21no—
136C3×C4×C8C2311yescharacteristic
137C3×C4×C8C2311yescharacteristic
138#559—21no—
139#557—21no—
140#556—22no—
141#556C22×C312yes—
142#554C22×C312yes—
143#552—22no—
144#539—22no—
145#536—21no—
146#533C22×C311yescharacteristic
147#532C22×C311yescharacteristic
148#532C22×C311yescharacteristic
149#529—22no—
150#520—21no—
151C2×C4×Q8C22×C311yescharacteristic
152#500C22×C311yescharacteristic
153#496—21no—
154#492C22×C311yescharacteristic
155#489—21no—
156#487C22×C311yescharacteristic
157#486C22×C311yescharacteristic
158#481—21no—
159#475C22×C312yes—
160#468—22no—
161#461C22×C312yes—
162#460C22×C312yes—
163#459C22×C312yes—
164#449—22no—
165#448—21no—
166#440—22no—
167C4×SD16—22no—
168#432C22×C311yescharacteristic
169#431C22×C311yescharacteristic
170C4×M16C22×C311yescharacteristic
171#333C22×C312yes—
172#332C22×C312yes—
173Q8⋊C8C22×C312yes—
174Q8⋊C8C22×C312yes—
175D4⋊C8C22×C312yes—
176D4⋊C8C22×C312yes—
177C3×#41—44no—
178C3×#41—82no—
179C2×C3×Q8—21no—
180C2×C3×Q8—21no—
181C2×C3×Q8—21no—
182C2×C3×Q8—21no—
183C2×C3×Q8—42no—
184C2×C3×Q8C2×D412yes—
185C2×C3×Q8C2×D412yes—
186C2×C3×D4—22no—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4—21no—
190C2×C3×D4—21no—
191C2×C3×D4—21no—
192C2×C3×D4—41no—
193C2×C3×D4C2×D412yes—
194C22×C3×C4—21no—
195C22×C3×C4—24no—
196C22×C3×C4—21no—
197C22×C3×C4—22no—
198C22×C3×C4—41no—
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C3×SD16—44no—
203C3×SD16—44no—
204C3×SD16—82no—
205C3×D8—82no—
206C3×M16—42no—
207C2×C3×C8—21no—
208C2×C3×C8—22no—
209C2×C3×C8—21no—
210C2×C3×C8—22no—
211C3×C4⋊C4—24no—
212C3×C4⋊C4—24no—
213C3×C4⋊C4—24no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—22no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4C2×D412yes—
226C3×C4⋊C4C2×D412yes—
227C3×C4⋊C4C2×D412yes—
228C3×C22⋊C4—24no—
229C3×C22⋊C4—24no—
230C3×C22⋊C4—24no—
231C3×C22⋊C4—22no—
232C3×C22⋊C4—22no—
233C3×C22⋊C4—22no—
234C3×C22⋊C4—41no—
235C3×C22⋊C4—41no—
236C3×C42—24no—
237C3×C42—22no—
238C3×C42—22no—
239C3×C42—22no—
240C3×C42C2×D411yescharacteristic
241C3×C42C2×D411yescharacteristic
242C3×C42C2×D411yescharacteristic
243C3×C42C2411yescharacteristic
244C2×#41—21no—
245C2×#41—21no—
246C22×Q8C3×D411yescharacteristic
247C2×SD16—22no—
248C2×SD16—22no—
249C2×SD16—22no—
250C2×D8—22no—
251C2×M16—21no—
252C4⋊Q8—22no—
253C4⋊Q8C3×D411yescharacteristic
254C4⋊Q8C23×C311yescharacteristic
255C4⋊Q8C23×C311yescharacteristic
256C4⋊Q8C23×C311yescharacteristic
257#127C3×D411yescharacteristic
258#127C23×C311yescharacteristic
259#126—24no—
260#126—24no—
261#125—24no—
262#125—22no—
263#125—21no—
264#124—22no—
265#124—22no—
266#124—21no—
267#124—21no—
268#123—24no—
269#123—24no—
270C22⋊Q8—22no—
271C22⋊Q8—22no—
272C22⋊Q8—24no—
273C22⋊Q8—21no—
274C22⋊Q8C3×D412yes—
275#121—21no—
276#121—21no—
277#121—21no—
278#121C3×D412yes—
279C4×Q8—22no—
280C4×Q8—22no—
281C4×Q8—21no—
282C4×Q8—21no—
283C4×Q8—21no—
284C4×Q8C3×D412yes—
285C4×Q8C3×D412yes—
286C4×Q8C23×C312yes—
287C4×D4—24no—
288C4×D4—22no—
289C4×D4—22no—
290C4×D4—21no—
291C4×D4—21no—
292C4×D4—21no—
293C4×D4C23×C312yes—
294#117—24no—
295#117—21no—
296#117—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4C3×D411yescharacteristic
300C2×C42—21no—
301C2×C42C23×C311yescharacteristic
302#107—22no—
303#106—22no—
304C4⋊C8C23×C312yes—
305C4⋊C8C23×C312yes—
306#103—22no—
307#103—22no—
308#103—22no—
309#102—22no—
310#102—22no—
311#102—22no—
312C22⋊C8—21no—
313C22⋊C8—21no—
314#97—21no—
315#97—21no—
316C4×C8—21no—
317C4×C8C23×C311yescharacteristic
318C4×C8C23×C311yescharacteristic
319C23×C3—22no—
320C23×C3—41no—
321C23×C3#12011yescharacteristic
322C3×Q8—22no—
323C3×Q8—22no—
324C3×Q8—42no—
325C3×Q8—42no—
326C3×Q8—42no—
327C3×Q8—42no—
328C3×Q8—42no—
329C3×D4—22no—
330C3×D4—22no—
331C3×D4—42no—
332C3×D4—42no—
333C3×D4—44no—
334C3×D4—42no—
335C3×D4—42no—
336C3×D4—81no—
337C3×D4—82no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—41no—
346C2×C3×C4—42no—
347C2×C3×C4—44no—
348C2×C3×C4—41no—
349C2×C3×C4—82no—
350C2×C3×C4—81no—
351C2×C3×C4—84no—
352C2×C3×C4#12011yescharacteristic
353C2×C3×C4#12011yescharacteristic
354C2×C3×C4#12011yescharacteristic
355C2×C3×C4C22×D411yescharacteristic
356C2×C3×C4C22×D411yescharacteristic
357C2×C3×C4C22×D411yescharacteristic
358C3×C8—22no—
359C3×C8—41no—
360C3×C8—42no—
361C3×C8—42no—
362#41—44no—
363#41—82no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8—42no—
369C2×Q8C2×C3×D412yes—
370C2×Q8C2×C3×D412yes—
371C2×D4—22no—
372C2×D4—21no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—41no—
378C2×D4C2×C3×D412yes—
379C22×C4—21no—
380C22×C4—24no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—41no—
384C22×C4C2×C3×D411yescharacteristic
385C22×C4C2×C3×D411yescharacteristic
386C22×C4C2×C3×D411yescharacteristic
387SD16—44no—
388SD16—44no—
389SD16—82no—
390D8—82no—
391M16—42no—
392C2×C8—21no—
393C2×C8—22no—
394C2×C8—21no—
395C2×C8—22no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—22no—
400C4⋊C4—21no—
401C4⋊C4—22no—
402C4⋊C4—21no—
403C4⋊C4—22no—
404C4⋊C4—21no—
405C4⋊C4—22no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4C2×C3×D412yes—
411C4⋊C4C2×C3×D412yes—
412C4⋊C4C2×C3×D412yes—
413C22⋊C4—24no—
414C22⋊C4—24no—
415C22⋊C4—24no—
416C22⋊C4—22no—
417C22⋊C4—22no—
418C22⋊C4—22no—
419C22⋊C4—41no—
420C22⋊C4—41no—
421C42—24no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42C2×C3×D411yescharacteristic
426C42C2×C3×D411yescharacteristic
427C42C2×C3×D411yescharacteristic
428C42C24×C311yescharacteristic
429C22×C3—21no—
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—42no—
433C22×C3—42no—
434C22×C3—42no—
435C22×C3—81no—
436C22×C3—81no—
437C22×C3—81no—
438C22×C3C2×#12011yescharacteristic
439C3×C4—21no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—42no—
444C3×C4—42no—
445C3×C4—42no—
446C3×C4—42no—
447C3×C4—42no—
448C3×C4—81no—
449C3×C4#57512yes—
450C3×C4#57712yes—
451C23—22no—
452C23—41no—
453C23C3×#12011yescharacteristic
454Q8—22no—
455Q8—22no—
456Q8—42no—
457Q8—42no—
458Q8—42no—
459Q8—42no—
460Q8—42no—
461D4—22no—
462D4—22no—
463D4—42no—
464D4—42no—
465D4—44no—
466D4—42no—
467D4—42no—
468D4—81no—
469D4—82no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—41no—
481C2×C4—82no—
482C2×C4—81no—
483C2×C4—84no—
484C2×C4C3×#12011yescharacteristic
485C2×C4C3×#12011yescharacteristic
486C2×C4C3×#12011yescharacteristic
487C2×C4C22×C3×D411yescharacteristic
488C2×C4C22×C3×D411yescharacteristic
489C2×C4C22×C3×D411yescharacteristic
490C8—22no—
491C8—41no—
492C8—42no—
493C8—42no—
494C2×C3—41no—
495C2×C3—82no—
496C2×C3—161no—
497C2×C3#300511yescharacteristic
498C2×C3#301011yescharacteristic
499C2×C3#301811yescharacteristic
500C22—21no—
501C22—21no—
502C22—21no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22C2×C3×#12011yescharacteristic
510C4—21no—
511C4—21no—
512C4—21no—
513C4—21no—
514C4—42no—
515C4—42no—
516C4—42no—
517C4—42no—
518C4—42no—
519C4—81no—
520C4C3×#57512yes—
521C4C3×#57712yes—
522C3#1860411yescharacteristic, direct product factor, Sylow
523C2—41no—
524C2—82no—
525C2—161no—
526C2C3×#300511yescharacteristic
527C2C3×#301011yescharacteristic
528C2C3×#301811yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC42C24×C3
prime radicalC3×#3214C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3507C2semidirect
3. C3×#3182C2semidirect
4. C3×#3103C2semidirect
5. C3×#1650C2semidirect
6. C3×#1643C2semidirect
7. C3×#1628C2semidirect
8. C3×#1622C2semidirect
9. C3×#1554C2semidirect
10. C3×#1491C2semidirect
11. C3×#1484C2semidirect
12. #18604C3direct (N × H)
52. #3507C2×C3semidirect
54. #3182C2×C3semidirect
55. #3103C2×C3semidirect
56. #1650C2×C3semidirect
57. #1643C2×C3semidirect
58. #1628C2×C3semidirect
59. #1622C2×C3semidirect
60. #1554C2×C3semidirect
61. #1491C2×C3semidirect
62. #1484C2×C3semidirect
522. C3#18604direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35072
C3×#32142
C3×#31822
C3×#31032
C3×#16502
C3×#16432
C3×#16282
C3×#16222
C3×#15542
C3×#14912
C3×#14842
#186043