Finite Group Databasebeta

← C3×#32992 #10651599

Subgroups — C3×#32992 #10651599

Order 768.

1304 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3428C211yesmaximal, characteristic, complemented
2C3×#3094C212yesmaximal, complemented
3C3×#2431C211yesmaximal, characteristic
4C3×#1888C212yesmaximal, complemented
5C3×#1887C212yesmaximal, complemented
6C3×#1857C212yesmaximal, complemented
7C3×#1856C212yesmaximal, complemented
8C3×#1826C212yesmaximal, complemented
9C3×#1785C211yesmaximal, characteristic, complemented
10#32992C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×#41C2211yescharacteristic
12C2×C3×#127C2211yescharacteristic
13C2×C3×#124C2211yescharacteristic
14C2×C3×#121C2211yescharacteristic
15C3×#520—22no—
16C3×#520—22no—
17C3×#519C2212yes—
18C3×#518—22no—
19C3×#518C2212yes—
20C3×C4×#41—24no—
21C2×C3×#117—22no—
22C2×C3×#117C2212yes—
23C3×#482—24no—
24C3×#481—22no—
25C3×#480—22no—
26C3×#428C2214yes—
27C3×#418C2214yes—
28C2×C3×#101C2212yes—
29C2×C3×#100C2212yes—
30C3×#407C2212yes—
31C3×#392—22no—
32C3×#391C2211yescharacteristic
33C3×#390C2211yescharacteristic
34C3×#386—22no—
35C3×C4×C22⋊C4C2211yescharacteristic
36C3×#339C2214yes—
37C3×#339C2214yes—
38#3428C2×C311yescharacteristic, complemented
39#3094C2×C312yescomplemented
40#2431C2×C311yescharacteristic
41#1888C2×C312yescomplemented
42#1887C2×C312yescomplemented
43#1857C2×C312yescomplemented
44#1856C2×C312yescomplemented
45#1826C2×C312yescomplemented
46#1785C2×C311yescharacteristic, complemented
47C2×C3×#41—24no—
48C2×C3×#41—24no—
49C2×C3×#41—22no—
50C2×C3×#41—22no—
51C22×C3×Q8C2311yescharacteristic
52C22×C3×D4—22no—
53C22×C3×D4—21no—
54C22×C3×D4D412yes—
55C22×C3×D4C2311yescharacteristic
56C22×C3×D4C2311yescharacteristic
57C23×C3×C4D412yes—
58C23×C3×C4C2311yescharacteristic
59C2×C3×M16C2314yes—
60C3×#127—24no—
61C3×#124—24no—
62C3×#121—22no—
63C3×#121—22no—
64C3×C4×Q8—24no—
65C3×C4×Q8—24no—
66C3×C4×D4—28no—
67C3×C4×D4—28no—
68C3×C4×D4—24no—
69C3×C4×D4—24no—
70C3×#117—24no—
71C3×#117—24no—
72C3×#117—22no—
73C3×#117—24no—
74C3×#117—22no—
75C3×#117D414yescomplemented
76C3×#117C2314yes—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4D412yescomplemented
80C2×C3×C4⋊C4D412yescomplemented
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C22⋊C4—24no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—21no—
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4—22no—
87C2×C3×C22⋊C4—22no—
88C2×C3×C42—24no—
89C2×C3×C42—22no—
90C2×C3×C42C2311yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C3×#107—24no—
93C3×#106—24no—
94C3×#102—24no—
95C3×#102—24no—
96C3×#101—24no—
97C3×#100—24no—
98C3×C22⋊C8—22no—
99C3×C22⋊C8—22no—
100C3×#95—21no—
101C3×#95—22no—
102C22×#41C22×C311yescharacteristic
103C2×#127C22×C311yescharacteristic
104C2×#124C22×C311yescharacteristic
105C2×#121C22×C311yescharacteristic
106#520—22no—
107#520—22no—
108#519C22×C312yes—
109#518—22no—
110#518C22×C312yes—
111C4×#41—24no—
112C2×#117—22no—
113C2×#117C22×C312yes—
114#482—24no—
115#481—22no—
116#480—22no—
117#428C22×C314yes—
118#418C22×C314yes—
119C2×#101C22×C312yes—
120C2×#100C22×C312yes—
121#407C22×C312yes—
122#392—22no—
123#391C22×C311yescharacteristic
124#390C22×C311yescharacteristic
125#386—22no—
126C4×C22⋊C4C22×C311yescharacteristic
127#339C22×C314yes—
128#339C22×C314yes—
129C24×C3—21no—
130C24×C3C2×D411yescharacteristic
131C24×C3C2×D412yes—
132C3×#41—216no—
133C3×#41—216no—
134C2×C3×Q8—22no—
135C2×C3×Q8—22no—
136C2×C3×Q8#4114yes—
137C2×C3×D4—24no—
138C2×C3×D4—24no—
139C2×C3×D4—24no—
140C2×C3×D4—22no—
141C2×C3×D4—22no—
142C2×C3×D4—22no—
143C2×C3×D4—22no—
144C2×C3×D4—22no—
145C2×C3×D4—44no—
146C2×C3×D4—42no—
147C2×C3×D4—44no—
148C2×C3×D4—82no—
149C2×C3×D4—82no—
150C2×C3×D4#4114yes—
151C22×C3×C4—24no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—24no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—21no—
158C22×C3×C4—24no—
159C22×C3×C4—24no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D412yes—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2411yescharacteristic
169C3×M16—24no—
170C3×M16—24no—
171C2×C3×C8—24no—
172C3×C4⋊C4—24no—
173C3×C4⋊C4—24no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4C2×D414yes—
178C3×C4⋊C4C2×D414yes—
179C3×C22⋊C4—28no—
180C3×C22⋊C4—24no—
181C3×C22⋊C4—24no—
182C3×C22⋊C4—42no—
183C3×C22⋊C4—41no—
184C3×C22⋊C4—41no—
185C3×C22⋊C4—44no—
186C3×C22⋊C4—82no—
187C3×C42—28no—
188C3×C42—24no—
189C3×C42—24no—
190C3×C42—22no—
191C3×C42—22no—
192C2×#41—24no—
193C2×#41—24no—
194C2×#41—22no—
195C2×#41—22no—
196C22×Q8C23×C311yescharacteristic
197C22×D4—22no—
198C22×D4—21no—
199C22×D4C3×D412yes—
200C22×D4C23×C311yescharacteristic
201C22×D4C23×C311yescharacteristic
202C23×C4C3×D412yes—
203C23×C4C23×C311yescharacteristic
204C2×M16C23×C314yes—
205#127—24no—
206#124—24no—
207#121—22no—
208#121—22no—
209C4×Q8—24no—
210C4×Q8—24no—
211C4×D4—28no—
212C4×D4—28no—
213C4×D4—24no—
214C4×D4—24no—
215#117—24no—
216#117—24no—
217#117—22no—
218#117—24no—
219#117—22no—
220#117C3×D414yescomplemented
221#117C23×C314yes—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4C3×D412yescomplemented
225C2×C4⋊C4C3×D412yescomplemented
226C2×C4⋊C4C23×C311yescharacteristic
227C2×C22⋊C4—24no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C42—24no—
234C2×C42—22no—
235C2×C42C23×C311yescharacteristic
236C2×C42C23×C311yescharacteristic
237#107—24no—
238#106—24no—
239#102—24no—
240#102—24no—
241#101—24no—
242#100—24no—
243C22⋊C8—22no—
244C22⋊C8—22no—
245#95—21no—
246#95—22no—
247C23×C3—22no—
248C23×C3—21no—
249C23×C3—21no—
250C23×C3—22no—
251C23×C3—41no—
252C23×C3—41no—
253C23×C3—42no—
254C23×C3—42no—
255C23×C3—42no—
256C23×C3—42no—
257C23×C3—41no—
258C23×C3—41no—
259C23×C3—41no—
260C23×C3#12012yes—
261C23×C3#12314yes—
262C23×C3C22×D411yescharacteristic
263C3×Q8—28no—
264C3×D4—216no—
265C3×D4—28no—
266C3×D4—44no—
267C3×D4—88no—
268C3×D4—84no—
269C2×C3×C4—28no—
270C2×C3×C4—28no—
271C2×C3×C4—22no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—22no—
275C2×C3×C4—24no—
276C2×C3×C4—24no—
277C2×C3×C4—22no—
278C2×C3×C4—24no—
279C2×C3×C4—24no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4—42no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—82no—
288C2×C3×C4—82no—
289C2×C3×C4—84no—
290C2×C3×C4—84no—
291C2×C3×C4#12011yescharacteristic
292C2×C3×C4#12011yescharacteristic
293C2×C3×C4#12314yes—
294C2×C3×C4C22×D412yes—
295C2×C3×C4C2×#4112yes—
296C2×C3×C4C2×#4112yes—
297C3×C8—44no—
298C24—21no—
299C24C2×C3×D411yescharacteristic
300C24C2×C3×D412yes—
301#41—216no—
302#41—216no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8C3×#4114yes—
306C2×D4—24no—
307C2×D4—24no—
308C2×D4—24no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—44no—
315C2×D4—42no—
316C2×D4—44no—
317C2×D4—82no—
318C2×D4—82no—
319C2×D4C3×#4114yes—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4C2×C3×D411yescharacteristic
334C22×C4C2×C3×D412yes—
335C22×C4C2×C3×D412yes—
336C22×C4C2×C3×D412yes—
337C22×C4C24×C311yescharacteristic
338M16—24no—
339M16—24no—
340C2×C8—24no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4C2×C3×D414yes—
347C4⋊C4C2×C3×D414yes—
348C22⋊C4—28no—
349C22⋊C4—24no—
350C22⋊C4—24no—
351C22⋊C4—42no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—44no—
355C22⋊C4—82no—
356C42—28no—
357C42—24no—
358C42—24no—
359C42—22no—
360C42—22no—
361C22×C3—21no—
362C22×C3—21no—
363C22×C3—22no—
364C22×C3—24no—
365C22×C3—22no—
366C22×C3—22no—
367C22×C3—44no—
368C22×C3—42no—
369C22×C3—42no—
370C22×C3—81no—
371C22×C3—81no—
372C22×C3—82no—
373C22×C3—82no—
374C22×C3—82no—
375C22×C3—81no—
376C22×C3—81no—
377C22×C3—81no—
378C22×C3C2×#12011yescharacteristic
379C22×C3C2×#12311yescharacteristic
380C22×C3C2×#12311yescharacteristic
381C3×C4—21no—
382C3×C4—21no—
383C3×C4—24no—
384C3×C4—42no—
385C3×C4—44no—
386C3×C4—44no—
387C3×C4—81no—
388C3×C4—81no—
389C3×C4#52514yes—
390C23—22no—
391C23—21no—
392C23—21no—
393C23—22no—
394C23—41no—
395C23—41no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23C3×#12012yes—
404C23C3×#12314yes—
405C23C22×C3×D411yescharacteristic
406Q8—28no—
407D4—216no—
408D4—28no—
409D4—44no—
410D4—88no—
411D4—84no—
412C2×C4—28no—
413C2×C4—28no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—84no—
433C2×C4—84no—
434C2×C4C3×#12011yescharacteristic
435C2×C4C3×#12011yescharacteristic
436C2×C4C3×#12314yes—
437C2×C4C22×C3×D412yes—
438C2×C4C2×C3×#4112yes—
439C2×C4C2×C3×#4112yes—
440C8—44no—
441C2×C3—21no—
442C2×C3—21no—
443C2×C3—42no—
444C2×C3—44no—
445C2×C3—161no—
446C2×C3#284911yescharacteristic
447C2×C3#301811yescharacteristic
448C2×C3#302111yescharacteristic
449C22—21no—
450C22—21no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—24no—
455C22—44no—
456C22—42no—
457C22—42no—
458C22—81no—
459C22—81no—
460C22—82no—
461C22—82no—
462C22—82no—
463C22—81no—
464C22—81no—
465C22—81no—
466C22C2×C3×#12011yescharacteristic
467C22C2×C3×#12311yescharacteristic
468C22C2×C3×#12311yescharacteristic
469C4—21no—
470C4—21no—
471C4—24no—
472C4—42no—
473C4—44no—
474C4—44no—
475C4—81no—
476C4—81no—
477C4C3×#52514yes—
478C3#3299211yescharacteristic, direct product factor, Sylow
479C2—21no—
480C2—21no—
481C2—42no—
482C2—44no—
483C2—161no—
484C2C3×#284911yescharacteristic
485C2C3×#301811yescharacteristic
486C2C3×#302111yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#2431C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3428C2semidirect
2. C3×#3094C2semidirect
4. C3×#1888C2semidirect
5. C3×#1887C2semidirect
6. C3×#1857C2semidirect
7. C3×#1856C2semidirect
8. C3×#1826C2semidirect
9. C3×#1785C2semidirect
10. #32992C3direct (N × H)
38. #3428C2×C3semidirect
39. #3094C2×C3semidirect
41. #1888C2×C3semidirect
42. #1887C2×C3semidirect
43. #1857C2×C3semidirect
44. #1856C2×C3semidirect
45. #1826C2×C3semidirect
46. #1785C2×C3semidirect
75. C3×#117D4semidirect
79. C2×C3×C4⋊C4D4semidirect
80. C2×C3×C4⋊C4D4semidirect
220. #117C3×D4semidirect
224. C2×C4⋊C4C3×D4semidirect
225. C2×C4⋊C4C3×D4semidirect
478. C3#32992direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34282
C3×#30942
C3×#24312
C3×#18882
C3×#18872
C3×#18572
C3×#18562
C3×#18262
C3×#17852
#329923