Finite Group Databasebeta

← C3×#30384 #10648991

Subgroups — C3×#30384 #10648991

Order 768.

666 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2860C211yesmaximal, characteristic
2C3×#2859C211yesmaximal, characteristic, complemented
3C3×#2099C211yesmaximal, characteristic, complemented
4C3×#2093C211yesmaximal, characteristic
5C3×#2087C212yesmaximal, complemented
6C3×#2084C211yesmaximal, characteristic, complemented
7C3×#2071C211yesmaximal, characteristic
8C3×#2065C212yesmaximal
9C3×#1943C212yesmaximal, complemented
10C3×#1936C212yesmaximal
11C3×#1767C211yesmaximal, characteristic, complemented
12#30384C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#126—21no—
14C2×C3×#125—21no—
15C2×C3×#107C2212yes—
16C2×C3×#106C2212yes—
17C2×C3×C4⋊C8C2212yes—
18C2×C3×#102C2212yes—
19C2×C3×#102C2212yes—
20C2×C3×#97C2212yes—
21C3×#390C2211yescharacteristic
22C3×#390C2211yescharacteristic
23C3×#390C2211yescharacteristic
24C3×#389—21no—
25C3×#389—21no—
26C3×#389C2212yes—
27C3×#389C2211yescharacteristic
28C3×#389C2211yescharacteristic
29C3×#389C2212yes—
30C3×#389C2211yescharacteristic
31C3×#388—21no—
32C3×#388—21no—
33C3×#388—21no—
34C3×#384C2211yescharacteristic
35C3×#384C2211yescharacteristic
36C3×#382—21no—
37C3×C43C2211yescharacteristic
38C3×#340C2212yes—
39C3×#340C2212yes—
40C3×#340C2212yes—
41C3×#340C2212yes—
42C3×#336C2211yescharacteristic
43C3×#336C2211yescharacteristic
44#2860C2×C311yescharacteristic
45#2859C2×C311yescharacteristic, complemented
46#2099C2×C311yescharacteristic, complemented
47#2093C2×C311yescharacteristic
48#2087C2×C312yescomplemented
49#2084C2×C311yescharacteristic, complemented
50#2071C2×C311yescharacteristic
51#2065C2×C312yes—
52#1943C2×C312yescomplemented
53#1936C2×C312yes—
54#1767C2×C311yescharacteristic, complemented
55C22×C3×D4C2311yescharacteristic
56C22×C3×C8C2312yes—
57C22×C3×C8C2312yes—
58C3×#126—44no—
59C3×#125—44no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4C2312yes—
70C2×C3×C4⋊C4C2312yes—
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C22⋊C4—21no—
75C2×C3×C22⋊C4—21no—
76C2×C3×C22⋊C4—21no—
77C2×C3×C22⋊C4—21no—
78C2×C3×C22⋊C4—21no—
79C2×C3×C22⋊C4—21no—
80C2×C3×C42D411yescharacteristic
81C2×C3×C42D411yescharacteristic
82C2×C3×C42D411yescharacteristic
83C2×C3×C42D411yescharacteristic
84C2×C3×C42C2311yescharacteristic
85C2×C3×C42C2311yescharacteristic
86C2×C3×C42C2311yescharacteristic
87C3×#107—24no—
88C3×#106—42no—
89C3×C4⋊C8—24no—
90C3×#102—24no—
91C3×#102—42no—
92C3×#97—42no—
93C3×#95—22no—
94C3×#95—22no—
95C3×#95—21no—
96C3×#95—21no—
97C3×#95—21no—
98C2×#126—21no—
99C2×#125—21no—
100C2×#107C22×C312yes—
101C2×#106C22×C312yes—
102C2×C4⋊C8C22×C312yes—
103C2×#102C22×C312yes—
104C2×#102C22×C312yes—
105C2×#97C22×C312yes—
106#390C22×C311yescharacteristic
107#390C22×C311yescharacteristic
108#390C22×C311yescharacteristic
109#389—21no—
110#389—21no—
111#389C22×C312yes—
112#389C22×C311yescharacteristic
113#389C22×C311yescharacteristic
114#389C22×C312yes—
115#389C22×C311yescharacteristic
116#388—21no—
117#388—21no—
118#388—21no—
119#384C22×C311yescharacteristic
120#384C22×C311yescharacteristic
121#382—21no—
122C43C22×C311yescharacteristic
123#340C22×C312yes—
124#340C22×C312yes—
125#340C22×C312yes—
126#340C22×C312yes—
127#336C22×C311yescharacteristic
128#336C22×C311yescharacteristic
129C24×C3—21no—
130C2×C3×D4—22no—
131C2×C3×D4—21no—
132C2×C3×D4—22no—
133C2×C3×D4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—21no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2411yescharacteristic
146C2×C3×C8—22no—
147C2×C3×C8—22no—
148C2×C3×C8—22no—
149C2×C3×C8—22no—
150C2×C3×C8—22no—
151C2×C3×C8#4114yes—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—22no—
154C3×C4⋊C4—22no—
155C3×C4⋊C4—22no—
156C3×C4⋊C4—21no—
157C3×C4⋊C4—22no—
158C3×C4⋊C4—22no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4—42no—
161C3×C4⋊C4—44no—
162C3×C4⋊C4—44no—
163C3×C4⋊C4—42no—
164C3×C4⋊C4—42no—
165C3×C4⋊C4—42no—
166C3×C4⋊C4—42no—
167C3×C4⋊C4—42no—
168C3×C4⋊C4—42no—
169C3×C22⋊C4—42no—
170C3×C22⋊C4—42no—
171C3×C22⋊C4—42no—
172C3×C22⋊C4—42no—
173C3×C22⋊C4—42no—
174C3×C22⋊C4—42no—
175C3×C42—22no—
176C3×C42—22no—
177C3×C42—22no—
178C3×C42—22no—
179C3×C42—21no—
180C3×C42—22no—
181C3×C42—21no—
182C3×C42—41no—
183C22×D4C23×C311yescharacteristic
184C22×C8C23×C312yes—
185C22×C8C23×C312yes—
186#126—44no—
187#125—44no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4C23×C311yescharacteristic
198C2×C4⋊C4C23×C312yes—
199C2×C4⋊C4C23×C312yes—
200C2×C4⋊C4C23×C311yescharacteristic
201C2×C4⋊C4C23×C311yescharacteristic
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C42C3×D411yescharacteristic
209C2×C42C3×D411yescharacteristic
210C2×C42C3×D411yescharacteristic
211C2×C42C3×D411yescharacteristic
212C2×C42C23×C311yescharacteristic
213C2×C42C23×C311yescharacteristic
214C2×C42C23×C311yescharacteristic
215#107—24no—
216#106—42no—
217C4⋊C8—24no—
218#102—24no—
219#102—42no—
220#97—42no—
221#95—22no—
222#95—22no—
223#95—21no—
224#95—21no—
225#95—21no—
226C23×C3—41no—
227C23×C3—41no—
228C23×C3—41no—
229C23×C3—41no—
230C23×C3—41no—
231C23×C3—41no—
232C23×C3—41no—
233C23×C3C22×D411yescharacteristic
234C3×D4—42no—
235C3×D4—42no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—42no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—42no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—42no—
266C2×C3×C4AGL(1,ℤ/8)12yes—
267C2×C3×C4AGL(1,ℤ/8)12yes—
268C2×C3×C4#13712yes—
269C2×C3×C4C2×#4112yes—
270C2×C3×C4H(5,𝔽2)11yescharacteristic
271C2×C3×C42−1+412yes—
272C2×C3×C42−1+411yescharacteristic
273C3×C8—24no—
274C3×C8—42no—
275C24—21no—
276C2×D4—22no—
277C2×D4—21no—
278C2×D4—22no—
279C2×D4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4C2×C3×D411yescharacteristic
286C22×C4C2×C3×D411yescharacteristic
287C22×C4C2×C3×D411yescharacteristic
288C22×C4C2×C3×D411yescharacteristic
289C22×C4C2×C3×D411yescharacteristic
290C22×C4C2×C3×D411yescharacteristic
291C22×C4C24×C311yescharacteristic
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8C3×#4114yes—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—21no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—21no—
306C4⋊C4—44no—
307C4⋊C4—44no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—42no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—21no—
326C42—22no—
327C42—21no—
328C42—41no—
329C22×C3—81no—
330C22×C3—81no—
331C22×C3—81no—
332C22×C3—81no—
333C22×C3—81no—
334C22×C3—81no—
335C22×C3—81no—
336C22×C3#53211yescharacteristic
337C22×C3#53611yescharacteristic
338C22×C3#53711yescharacteristic
339C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
340C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
341C22×C3C2×#13711yescharacteristic
342C22×C3#57511yescharacteristic
343C3×C4—22no—
344C3×C4—22no—
345C3×C4—22no—
346C3×C4—41no—
347C3×C4—41no—
348C3×C4—41no—
349C3×C4—81no—
350C3×C4—82no—
351C3×C4—81no—
352C3×C4—82no—
353C3×C4—81no—
354C3×C4#53912yes—
355C3×C4#54112yes—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23C22×C3×D411yescharacteristic
364D4—42no—
365D4—42no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4C3×AGL(1,ℤ/8)12yes—
397C2×C4C3×AGL(1,ℤ/8)12yes—
398C2×C4C3×#13712yes—
399C2×C4C2×C3×#4112yes—
400C2×C4C3×H(5,𝔽2)11yescharacteristic
401C2×C4C3×2−1+412yes—
402C2×C4C3×2−1+411yescharacteristic
403C8—24no—
404C8—42no—
405C2×C3—161no—
406C2×C3#266411yescharacteristic
407C2×C3#318111yescharacteristic
408C2×C3#318411yescharacteristic
409C2×C3#323411yescharacteristic
410C2×C3#323811yescharacteristic
411C2×C3#325111yescharacteristic
412C2×C3#325411yescharacteristic
413C22—81no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22C3×#53211yescharacteristic
421C22C3×#53611yescharacteristic
422C22C3×#53711yescharacteristic
423C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
424C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
425C22C2×C3×#13711yescharacteristic
426C22C3×#57511yescharacteristic
427C4—22no—
428C4—22no—
429C4—22no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—81no—
434C4—82no—
435C4—81no—
436C4—82no—
437C4—81no—
438C4C3×#53912yes—
439C4C3×#54112yes—
440C3#3038411yescharacteristic, direct product factor, Sylow
441C2—161no—
442C2C3×#266411yescharacteristic
443C2C3×#318111yescharacteristic
444C2C3×#318411yescharacteristic
445C2C3×#323411yescharacteristic
446C2C3×#323811yescharacteristic
447C2C3×#325111yescharacteristic
448C2C3×#325411yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C43 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2859C2semidirect
3. C3×#2099C2semidirect
5. C3×#2087C2semidirect
6. C3×#2084C2semidirect
9. C3×#1943C2semidirect
11. C3×#1767C2semidirect
12. #30384C3direct (N × H)
45. #2859C2×C3semidirect
46. #2099C2×C3semidirect
48. #2087C2×C3semidirect
49. #2084C2×C3semidirect
52. #1943C2×C3semidirect
54. #1767C2×C3semidirect
440. C3#30384direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#28602
C3×#28592
C3×#20992
C3×#20932
C3×#20872
C3×#20842
C3×#20712
C3×#20652
C3×#19432
C3×#19362
C3×#17672
#303843