Finite Group Databasebeta

← C3×#29333 #10647940

Subgroups — C3×#29333 #10647940

Order 768.

808 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2850C211yesmaximal, characteristic
2C3×#2296C211yesmaximal, characteristic, complemented
3C3×#2101C212yesmaximal, complemented
4C3×#2042C212yesmaximal, complemented
5C3×#1934C212yesmaximal, complemented
6C3×#1933C212yesmaximal, complemented
7C3×#1875C211yesmaximal, characteristic
8C3×#1875C211yesmaximal, characteristic
9C3×#1873C211yesmaximal, characteristic, complemented
10C3×#1873C211yesmaximal, characteristic, complemented
11C3×C4×C22⋊C8C211yesmaximal, characteristic, complemented
12#29333C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#124C2211yescharacteristic
14C2×C3×C22⋊Q8C2211yescharacteristic
15C2×C3×#121C2211yescharacteristic
16C2×C3×#117C2211yescharacteristic
17C2×C3×#117C2211yescharacteristic
18C22×C3×C42C2211yescharacteristic
19C2×C3×#103C2212yes—
20C2×C3×#103C2212yes—
21C2×C3×#102C2212yes—
22C2×C3×#102C2212yes—
23C2×C3×C22⋊C8C2211yescharacteristic
24C2×C3×C22⋊C8C2211yescharacteristic
25C2×C3×C4×C8C2212yes—
26C3×#391C2211yescharacteristic
27C3×#390C2211yescharacteristic
28C3×#389C2212yes—
29C3×#389C2212yes—
30C3×#388—22no—
31C3×#386—22no—
32C3×#385—22no—
33C3×#382C2211yescharacteristic
34C3×C4×C4⋊C4—22no—
35C3×#340C2212yes—
36C3×#340C2212yes—
37C3×#340C2212yes—
38C3×#340C2212yes—
39C3×#336C2212yes—
40#2850C2×C311yescharacteristic
41#2296C2×C311yescharacteristic, complemented
42#2101C2×C312yescomplemented
43#2042C2×C312yescomplemented
44#1934C2×C312yescomplemented
45#1933C2×C312yescomplemented
46#1875C2×C311yescharacteristic
47#1875C2×C311yescharacteristic
48#1873C2×C311yescharacteristic, complemented
49#1873C2×C311yescharacteristic, complemented
50C4×C22⋊C8C2×C311yescharacteristic, complemented
51C22×C3×Q8C2311yescharacteristic
52C22×C3×D4—21no—
53C22×C3×D4C2311yescharacteristic
54C23×C3×C4D412yes—
55C23×C3×C4C2311yescharacteristic
56C22×C3×C8C2312yes—
57C22×C3×C8C2312yes—
58C3×#124—42no—
59C3×C22⋊Q8—22no—
60C3×C22⋊Q8—22no—
61C3×#121—22no—
62C3×#121—22no—
63C3×#117—24no—
64C3×#117—42no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C4⋊C4C2312yes—
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C22⋊C4—21no—
73C2×C3×C22⋊C4—21no—
74C2×C3×C22⋊C4—21no—
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4—22no—
77C2×C3×C22⋊C4—21no—
78C2×C3×C42—21no—
79C2×C3×C42—21no—
80C2×C3×C42—42no—
81C2×C3×C42D412yes—
82C2×C3×C42C2311yescharacteristic
83C2×C3×C42C2311yescharacteristic
84C3×#103—24no—
85C3×#103—42no—
86C3×#102—24no—
87C3×#102—42no—
88C3×C22⋊C8—22no—
89C3×C22⋊C8—22no—
90C3×C4×C8—42no—
91C3×#95—22no—
92C3×#95—22no—
93C3×#95—22no—
94C3×#95—22no—
95C2×#124C22×C311yescharacteristic
96C2×C22⋊Q8C22×C311yescharacteristic
97C2×#121C22×C311yescharacteristic
98C2×#117C22×C311yescharacteristic
99C2×#117C22×C311yescharacteristic
100C22×C42C22×C311yescharacteristic
101C2×#103C22×C312yes—
102C2×#103C22×C312yes—
103C2×#102C22×C312yes—
104C2×#102C22×C312yes—
105C2×C22⋊C8C22×C311yescharacteristic
106C2×C22⋊C8C22×C311yescharacteristic
107C2×C4×C8C22×C312yes—
108#391C22×C311yescharacteristic
109#390C22×C311yescharacteristic
110#389C22×C312yes—
111#389C22×C312yes—
112#388—22no—
113#386—22no—
114#385—22no—
115#382C22×C311yescharacteristic
116C4×C4⋊C4—22no—
117#340C22×C312yes—
118#340C22×C312yes—
119#340C22×C312yes—
120#340C22×C312yes—
121#336C22×C312yes—
122C24×C3—21no—
123C24×C3C2×D411yescharacteristic
124C2×C3×Q8—22no—
125C2×C3×Q8—21no—
126C2×C3×Q8—21no—
127C2×C3×Q8—22no—
128C2×C3×D4—22no—
129C2×C3×D4—21no—
130C2×C3×D4—21no—
131C2×C3×D4—22no—
132C2×C3×D4—44no—
133C2×C3×D4—42no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—24no—
138C22×C3×C4—21no—
139C22×C3×C4—22no—
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4—42no—
143C22×C3×C4—42no—
144C22×C3×C4C2×D412yes—
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D412yes—
147C22×C3×C4#4112yes—
148C22×C3×C4#4112yes—
149C22×C3×C4C2411yescharacteristic
150C2×C3×C8—22no—
151C2×C3×C8—22no—
152C2×C3×C8—22no—
153C2×C3×C8—22no—
154C2×C3×C8—22no—
155C2×C3×C8—22no—
156C3×C4⋊C4—22no—
157C3×C4⋊C4—21no—
158C3×C4⋊C4—21no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4—21no—
161C3×C4⋊C4—22no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—44no—
164C3×C4⋊C4—42no—
165C3×C4⋊C4—44no—
166C3×C4⋊C4#4114yes—
167C3×C22⋊C4—24no—
168C3×C22⋊C4—44no—
169C3×C22⋊C4—42no—
170C3×C22⋊C4—42no—
171C3×C22⋊C4—42no—
172C3×C22⋊C4—44no—
173C3×C42—24no—
174C3×C42—22no—
175C3×C42—22no—
176C3×C42—42no—
177C3×C42—42no—
178C22×Q8C23×C311yescharacteristic
179C22×D4—21no—
180C22×D4C23×C311yescharacteristic
181C23×C4C3×D412yes—
182C23×C4C23×C311yescharacteristic
183C22×C8C23×C312yes—
184C22×C8C23×C312yes—
185#124—42no—
186C22⋊Q8—22no—
187C22⋊Q8—22no—
188#121—22no—
189#121—22no—
190#117—24no—
191#117—42no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4C23×C312yes—
196C2×C4⋊C4C23×C311yescharacteristic
197C2×C4⋊C4C23×C311yescharacteristic
198C2×C4⋊C4C23×C312yes—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—42no—
208C2×C42C3×D412yes—
209C2×C42C23×C311yescharacteristic
210C2×C42C23×C311yescharacteristic
211#103—24no—
212#103—42no—
213#102—24no—
214#102—42no—
215C22⋊C8—22no—
216C22⋊C8—22no—
217C4×C8—42no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222C23×C3—21no—
223C23×C3—21no—
224C23×C3—22no—
225C23×C3—22no—
226C23×C3—41no—
227C23×C3—41no—
228C23×C3—41no—
229C23×C3—42no—
230C23×C3—42no—
231C23×C3#12412yes—
232C23×C3C22×D411yescharacteristic
233C3×Q8—42no—
234C3×Q8—42no—
235C3×D4—42no—
236C3×D4—42no—
237C3×D4—84no—
238C2×C3×C4—21no—
239C2×C3×C4—22no—
240C2×C3×C4—21no—
241C2×C3×C4—22no—
242C2×C3×C4—24no—
243C2×C3×C4—24no—
244C2×C3×C4—24no—
245C2×C3×C4—22no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—48no—
249C2×C3×C4—42no—
250C2×C3×C4—42no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—42no—
256C2×C3×C4—41no—
257C2×C3×C4—42no—
258C2×C3×C4—42no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4#12412yes—
262C2×C3×C4#13514yes—
263C2×C3×C4C2×#4112yes—
264C2×C3×C4C2×#4111yescharacteristic
265C2×C3×C4C2×#4111yescharacteristic
266C2×C3×C4H(5,𝔽2)12yes—
267C3×C8—42no—
268C3×C8—42no—
269C24—21no—
270C24C2×C3×D411yescharacteristic
271C2×Q8—22no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—22no—
275C2×D4—22no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—22no—
279C2×D4—44no—
280C2×D4—42no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4C2×C3×D412yes—
292C22×C4C2×C3×D411yescharacteristic
293C22×C4C2×C3×D412yes—
294C22×C4C3×#4112yes—
295C22×C4C3×#4112yes—
296C22×C4C24×C311yescharacteristic
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C4⋊C4—22no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—44no—
311C4⋊C4—42no—
312C4⋊C4—44no—
313C4⋊C4C3×#4114yes—
314C22⋊C4—24no—
315C22⋊C4—44no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—44no—
320C42—24no—
321C42—22no—
322C42—22no—
323C42—42no—
324C42—42no—
325C22×C3—22no—
326C22×C3—22no—
327C22×C3—22no—
328C22×C3—42no—
329C22×C3—42no—
330C22×C3—82no—
331C22×C3—81no—
332C22×C3—81no—
333C22×C3—81no—
334C22×C3—82no—
335C22×C3#52511yescharacteristic
336C22×C3C2×#12411yescharacteristic
337C22×C3#53411yescharacteristic
338C22×C3C2×#13512yes—
339C22×C3#57512yes—
340C3×C4—22no—
341C3×C4—24no—
342C3×C4—44no—
343C3×C4—42no—
344C3×C4—81no—
345C3×C4—81no—
346C3×C4—81no—
347C3×C4—82no—
348C3×C4#55312yes—
349C3×C4#55312yes—
350C23—22no—
351C23—22no—
352C23—21no—
353C23—21no—
354C23—42no—
355C23—42no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23C3×#12412yes—
360C23C22×C3×D411yescharacteristic
361Q8—42no—
362Q8—42no—
363D4—42no—
364D4—42no—
365D4—84no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—22no—
376C2×C4—48no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4C3×#12412yes—
390C2×C4C3×#13514yes—
391C2×C4C2×C3×#4112yes—
392C2×C4C2×C3×#4111yescharacteristic
393C2×C4C2×C3×#4111yescharacteristic
394C2×C4C3×H(5,𝔽2)12yes—
395C8—42no—
396C8—42no—
397C2×C3—42no—
398C2×C3—161no—
399C2×C3#259111yescharacteristic
400C2×C3#310212yes—
401C2×C3#313111yescharacteristic
402C2×C3#314011yescharacteristic
403C2×C3#319112yes—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—42no—
408C22—42no—
409C22—81no—
410C22—81no—
411C22—82no—
412C22—82no—
413C22—81no—
414C22C3×#52511yescharacteristic
415C22C2×C3×#12411yescharacteristic
416C22C3×#53411yescharacteristic
417C22C2×C3×#13512yes—
418C22C3×#57512yes—
419C4—24no—
420C4—22no—
421C4—42no—
422C4—44no—
423C4—81no—
424C4—81no—
425C4—82no—
426C4—81no—
427C4C3×#55312yes—
428C4C3×#55312yes—
429C3#2933311yescharacteristic, direct product factor, Sylow
430C2—42no—
431C2—161no—
432C2C3×#259111yescharacteristic
433C2C3×#310212yes—
434C2C3×#313111yescharacteristic
435C2C3×#314011yescharacteristic
436C2C3×#319112yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2296C2semidirect
3. C3×#2101C2semidirect
4. C3×#2042C2semidirect
5. C3×#1934C2semidirect
6. C3×#1933C2semidirect
9. C3×#1873C2semidirect
10. C3×#1873C2semidirect
11. C3×C4×C22⋊C8C2semidirect
12. #29333C3direct (N × H)
41. #2296C2×C3semidirect
42. #2101C2×C3semidirect
43. #2042C2×C3semidirect
44. #1934C2×C3semidirect
45. #1933C2×C3semidirect
48. #1873C2×C3semidirect
49. #1873C2×C3semidirect
50. C4×C22⋊C8C2×C3semidirect
429. C3#29333direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#28502
C3×#22962
C3×#21012
C3×#20422
C3×#19342
C3×#19332
C3×#18752
C3×#18732
C3×C4×C22⋊C82
#293333