Finite Group Databasebeta

← C3×#33021 #10651628

Subgroups — C3×#33021 #10651628

Order 768.

1442 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3428C211yesmaximal, characteristic, complemented
2C3×#2482C212yesmaximal, complemented
3C3×#2404C211yesmaximal, characteristic
4C3×#1889C214yesmaximal, complemented
5C3×#1867C214yesmaximal, complemented
6C3×#1820C212yesmaximal, complemented
7C3×#1782C211yesmaximal, characteristic, complemented
8#33021C311yesmaximal, characteristic, direct product factor, Sylow
9C22×C3×#41C2211yescharacteristic
10C2×C3×#120C2212yes—
11C3×#520—22no—
12C3×#520—22no—
13C3×#519—21no—
14C3×#518—21no—
15C3×#518C2214yes—
16C3×C4×#41—24no—
17C2×C3×#117—22no—
18C2×C3×#117C2212yes—
19C22×C3×C22⋊C4C2211yescharacteristic
20C2×C3×#99C2214yes—
21C3×#399—22no—
22C3×#397—22no—
23C3×#393—21no—
24C3×#392—21no—
25C3×#388—24no—
26C3×#386—22no—
27C3×#386C2212yes—
28C3×#385C2214yes—
29C3×#381C2211yescharacteristic
30C3×#381C2212yes—
31C3×#379C2214yes—
32C3×#342C2218yes—
33#3428C2×C311yescharacteristic, complemented
34#2482C2×C312yescomplemented
35#2404C2×C311yescharacteristic
36#1889C2×C314yescomplemented
37#1867C2×C314yescomplemented
38#1820C2×C312yescomplemented
39#1782C2×C311yescharacteristic, complemented
40C25×C3C2311yescharacteristic
41C2×C3×#41—28no—
42C2×C3×#41—22no—
43C2×C3×#41—22no—
44C22×C3×Q8D411yescharacteristic
45C22×C3×D4—22no—
46C22×C3×D4D411yescharacteristic
47C22×C3×D4C2312yes—
48C23×C3×C4D412yes—
49C23×C3×C4C2311yescharacteristic
50C23×C3×C4C2311yescharacteristic
51C3×#120—28no—
52C3×C4×Q8—24no—
53C3×C4×Q8—24no—
54C3×C4×D4—24no—
55C3×C4×D4—24no—
56C3×C4×D4—28no—
57C3×C4×D4—28no—
58C3×#117—24no—
59C3×#117—24no—
60C3×#117—22no—
61C3×#117—24no—
62C3×#117—22no—
63C3×#117—22no—
64C3×#117—22no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—24no—
68C2×C3×C4⋊C4D412yescomplemented
69C2×C3×C4⋊C4D412yescomplemented
70C2×C3×C22⋊C4—24no—
71C2×C3×C22⋊C4—24no—
72C2×C3×C22⋊C4—21no—
73C2×C3×C22⋊C4—22no—
74C2×C3×C22⋊C4—22no—
75C2×C3×C22⋊C4—21no—
76C2×C3×C22⋊C4—22no—
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4—24no—
79C2×C3×C22⋊C4C2314yes—
80C2×C3×C22⋊C4C2312yes—
81C2×C3×C22⋊C4C2314yes—
82C2×C3×C42—24no—
83C2×C3×C42D414yescomplemented
84C3×#99—28no—
85C3×#95—21no—
86C3×#95—22no—
87C3×#95—21no—
88C3×#95—22no—
89C3×#95—24no—
90C3×#95—22no—
91C22×#41C22×C311yescharacteristic
92C2×#120C22×C312yes—
93#520—22no—
94#520—22no—
95#519—21no—
96#518—21no—
97#518C22×C314yes—
98C4×#41—24no—
99C2×#117—22no—
100C2×#117C22×C312yes—
101C22×C22⋊C4C22×C311yescharacteristic
102C2×#99C22×C314yes—
103#399—22no—
104#397—22no—
105#393—21no—
106#392—21no—
107#388—24no—
108#386—22no—
109#386C22×C312yes—
110#385C22×C314yes—
111#381C22×C311yescharacteristic
112#381C22×C312yes—
113#379C22×C314yes—
114#342C22×C318yes—
115C24×C3—22no—
116C24×C3—21no—
117C24×C3—24no—
118C24×C3—42no—
119C24×C3—81no—
120C24×C3C2×D412yes—
121C24×C3C2411yescharacteristic
122C3×#41—216no—
123C3×#41—216no—
124C2×C3×Q8—22no—
125C2×C3×Q8—24no—
126C2×C3×D4—24no—
127C2×C3×D4—24no—
128C2×C3×D4—24no—
129C2×C3×D4—22no—
130C2×C3×D4—44no—
131C2×C3×D4—44no—
132C2×C3×D4—82no—
133C2×C3×D4#4118yes—
134C22×C3×C4—24no—
135C22×C3×C4—24no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—24no—
139C22×C3×C4—24no—
140C22×C3×C4—24no—
141C22×C3×C4—24no—
142C22×C3×C4—22no—
143C22×C3×C4—21no—
144C22×C3×C4—22no—
145C22×C3×C4—81no—
146C22×C3×C4C2×D412yes—
147C22×C3×C4C2×D412yes—
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×D412yes—
151C22×C3×C4C2×D414yes—
152C22×C3×C4C2×D414yes—
153C3×C4⋊C4—24no—
154C3×C4⋊C4—24no—
155C3×C4⋊C4—22no—
156C3×C4⋊C4—22no—
157C3×C4⋊C4—22no—
158C3×C4⋊C4—22no—
159C3×C4⋊C4—84no—
160C3×C22⋊C4—28no—
161C3×C22⋊C4—24no—
162C3×C22⋊C4—24no—
163C3×C22⋊C4—28no—
164C3×C22⋊C4—24no—
165C3×C22⋊C4—48no—
166C3×C22⋊C4—41no—
167C3×C22⋊C4—42no—
168C3×C22⋊C4—41no—
169C3×C22⋊C4—44no—
170C3×C22⋊C4—84no—
171C3×C22⋊C4—82no—
172C3×C42—28no—
173C3×C42—24no—
174C3×C42—24no—
175C25C23×C311yescharacteristic
176C2×#41—28no—
177C2×#41—22no—
178C2×#41—22no—
179C22×Q8C3×D411yescharacteristic
180C22×D4—22no—
181C22×D4C3×D411yescharacteristic
182C22×D4C23×C312yes—
183C23×C4C3×D412yes—
184C23×C4C23×C311yescharacteristic
185C23×C4C23×C311yescharacteristic
186#120—28no—
187C4×Q8—24no—
188C4×Q8—24no—
189C4×D4—24no—
190C4×D4—24no—
191C4×D4—28no—
192C4×D4—28no—
193#117—24no—
194#117—24no—
195#117—22no—
196#117—24no—
197#117—22no—
198#117—22no—
199#117—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4C3×D412yescomplemented
204C2×C4⋊C4C3×D412yescomplemented
205C2×C22⋊C4—24no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4C23×C312yes—
215C2×C22⋊C4C23×C314yes—
216C2×C22⋊C4C23×C314yes—
217C2×C42—24no—
218C2×C42C3×D414yescomplemented
219#99—28no—
220#95—21no—
221#95—22no—
222#95—21no—
223#95—22no—
224#95—24no—
225#95—22no—
226C23×C3—22no—
227C23×C3—22no—
228C23×C3—24no—
229C23×C3—22no—
230C23×C3—41no—
231C23×C3—41no—
232C23×C3—42no—
233C23×C3—42no—
234C23×C3—44no—
235C23×C3—42no—
236C23×C3—42no—
237C23×C3—41no—
238C23×C3—41no—
239C23×C3—42no—
240C23×C3—44no—
241C23×C3—88no—
242C23×C3#12314yes—
243C23×C3C22×D411yescharacteristic
244C23×C3C22×D412yes—
245C23×C3C2×#4114yes—
246C3×Q8—28no—
247C3×D4—28no—
248C3×D4—216no—
249C3×D4—88no—
250C2×C3×C4—28no—
251C2×C3×C4—28no—
252C2×C3×C4—22no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—24no—
257C2×C3×C4—24no—
258C2×C3×C4—24no—
259C2×C3×C4—24no—
260C2×C3×C4—44no—
261C2×C3×C4—44no—
262C2×C3×C4—44no—
263C2×C3×C4—44no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—44no—
267C2×C3×C4—42no—
268C2×C3×C4—41no—
269C2×C3×C4—42no—
270C2×C3×C4—44no—
271C2×C3×C4—84no—
272C2×C3×C4—84no—
273C2×C3×C4#12011yescharacteristic
274C2×C3×C4#12012yes—
275C2×C3×C4#12011yescharacteristic
276C2×C3×C4#12314yes—
277C24—22no—
278C24—21no—
279C24—24no—
280C24—42no—
281C24—81no—
282C24C2×C3×D412yes—
283C24C24×C311yescharacteristic
284#41—216no—
285#41—216no—
286C2×Q8—22no—
287C2×Q8—24no—
288C2×D4—24no—
289C2×D4—24no—
290C2×D4—24no—
291C2×D4—22no—
292C2×D4—44no—
293C2×D4—44no—
294C2×D4—82no—
295C2×D4C3×#4118yes—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—81no—
308C22×C4C2×C3×D412yes—
309C22×C4C2×C3×D412yes—
310C22×C4C2×C3×D411yescharacteristic
311C22×C4C2×C3×D411yescharacteristic
312C22×C4C2×C3×D412yes—
313C22×C4C2×C3×D414yes—
314C22×C4C2×C3×D414yes—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—84no—
322C22⋊C4—28no—
323C22⋊C4—24no—
324C22⋊C4—24no—
325C22⋊C4—28no—
326C22⋊C4—24no—
327C22⋊C4—48no—
328C22⋊C4—41no—
329C22⋊C4—42no—
330C22⋊C4—41no—
331C22⋊C4—44no—
332C22⋊C4—84no—
333C22⋊C4—82no—
334C42—28no—
335C42—24no—
336C42—24no—
337C22×C3—22no—
338C22×C3—21no—
339C22×C3—22no—
340C22×C3—21no—
341C22×C3—24no—
342C22×C3—44no—
343C22×C3—44no—
344C22×C3—44no—
345C22×C3—48no—
346C22×C3—42no—
347C22×C3—42no—
348C22×C3—42no—
349C22×C3—82no—
350C22×C3—82no—
351C22×C3—84no—
352C22×C3—81no—
353C22×C3—81no—
354C22×C3C2×#12011yescharacteristic
355C22×C3C2×#12312yes—
356C22×C3#52514yes—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—22no—
360C3×C4—24no—
361C3×C4—44no—
362C3×C4—44no—
363C3×C4—84no—
364C3×C4—82no—
365C3×C4—81no—
366C23—22no—
367C23—22no—
368C23—24no—
369C23—22no—
370C23—41no—
371C23—41no—
372C23—42no—
373C23—42no—
374C23—44no—
375C23—42no—
376C23—42no—
377C23—41no—
378C23—41no—
379C23—42no—
380C23—44no—
381C23—88no—
382C23C3×#12314yes—
383C23C22×C3×D411yescharacteristic
384C23C22×C3×D412yes—
385C23C2×C3×#4114yes—
386Q8—28no—
387D4—28no—
388D4—216no—
389D4—88no—
390C2×C4—28no—
391C2×C4—28no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—44no—
411C2×C4—84no—
412C2×C4—84no—
413C2×C4C3×#12011yescharacteristic
414C2×C4C3×#12012yes—
415C2×C4C3×#12011yescharacteristic
416C2×C4C3×#12314yes—
417C2×C3—22no—
418C2×C3—24no—
419C2×C3—44no—
420C2×C3—82no—
421C2×C3#284911yescharacteristic
422C2×C3#302511yescharacteristic
423C2×C3#302711yescharacteristic
424C22—22no—
425C22—21no—
426C22—22no—
427C22—21no—
428C22—24no—
429C22—44no—
430C22—44no—
431C22—44no—
432C22—48no—
433C22—42no—
434C22—42no—
435C22—42no—
436C22—82no—
437C22—82no—
438C22—84no—
439C22—81no—
440C22—81no—
441C22C2×C3×#12011yescharacteristic
442C22C2×C3×#12312yes—
443C22C3×#52514yes—
444C4—21no—
445C4—21no—
446C4—22no—
447C4—24no—
448C4—44no—
449C4—44no—
450C4—82no—
451C4—84no—
452C4—81no—
453C3#3302111yescharacteristic, direct product factor, Sylow
454C2—22no—
455C2—24no—
456C2—44no—
457C2—82no—
458C2C3×#284911yescharacteristic
459C2C3×#302511yescharacteristic
460C2C3×#302711yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC24C24×C3
prime radicalC3×#2404C2
LCSG ▷ C24 ▷ 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×#2482C2semidirect
4. C3×#1889C2semidirect
5. C3×#1867C2semidirect
6. C3×#1820C2semidirect
7. C3×#1782C2semidirect
8. #33021C3direct (N × H)
33. #3428C2×C3semidirect
34. #2482C2×C3semidirect
36. #1889C2×C3semidirect
37. #1867C2×C3semidirect
38. #1820C2×C3semidirect
39. #1782C2×C3semidirect
68. C2×C3×C4⋊C4D4semidirect
69. C2×C3×C4⋊C4D4semidirect
83. C2×C3×C42D4semidirect
203. C2×C4⋊C4C3×D4semidirect
204. C2×C4⋊C4C3×D4semidirect
218. C2×C42C3×D4semidirect
453. C3#33021direct (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×#24822
C3×#24042
C3×#18892
C3×#18672
C3×#18202
C3×#17822
#330213