Finite Group Databasebeta

← C3×#25606 #10644213

Subgroups — C3×#25606 #10644213

Order 768.

1180 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#540C211yesmaximal, characteristic
2C2×C3×#492C211yesmaximal, characteristic, complemented
3C2×C3×#491C211yesmaximal, characteristic, complemented
4C3×#2621C211yesmaximal, characteristic, complemented
5C3×#2000C214yesmaximal, complemented
6C3×#1991C211yesmaximal, characteristic, complemented
7C3×#1961C211yesmaximal, characteristic, complemented
8C3×#1877C214yesmaximal, complemented
9C3×#1750C211yesmaximal, characteristic, complemented
10#25606C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×SD16C2214yes—
12C3×#540—24no—
13C3×#540—24no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×#127C2211yescharacteristic
16C2×C3×#125C2211yescharacteristic
17C2×C3×#123—22no—
18C2×C3×C22⋊Q8C2212yes—
19C2×C3×#121—22no—
20C2×C3×#121C2212yes—
21C2×C3×C4×D4C2212yes—
22C2×C3×#117C2211yescharacteristic
23C3×#492—24no—
24C3×#491—24no—
25C2×C3×#103C2214yes—
26C2×C3×#102C2214yes—
27C2×C3×C22⋊C8C2212yes—
28C2×C3×#97C2211yescharacteristic
29C2×C3×C4×C8C2211yescharacteristic
30C3×#398—22no—
31C3×#397—22no—
32C3×#391C2212yes—
33C3×#382C2211yescharacteristic
34C3×#340C2214yes—
35C3×#336C2212yes—
36C2×#540C2×C311yescharacteristic
37C2×#492C2×C311yescharacteristic, complemented
38C2×#491C2×C311yescharacteristic, complemented
39#2621C2×C311yescharacteristic, complemented
40#2000C2×C314yescomplemented
41#1991C2×C311yescharacteristic, complemented
42#1961C2×C311yescharacteristic, complemented
43#1877C2×C314yescomplemented
44#1750C2×C311yescharacteristic, complemented
45C22×C3×Q8C2312yes—
46C22×C3×D4—22no—
47C22×C3×D4—22no—
48C22×C3×D4C2312yes—
49C23×C3×C4—22no—
50C23×C3×C4C2311yescharacteristic
51C2×C3×SD16—28no—
52C2×C3×SD16—28no—
53C2×C3×SD16—44no—
54C22×C3×C8C2312yes—
55C22×C3×C8C2312yes—
56C3×C4⋊Q8—22no—
57C3×C4⋊Q8—22no—
58C3×#127—22no—
59C3×#127D414yes—
60C3×#125—22no—
61C3×#125D414yes—
62C3×#123—216no—
63C3×C22⋊Q8—44no—
64C3×#121—216no—
65C3×#121—24no—
66C3×#121—24no—
67C3×C4×D4—24no—
68C3×C4×D4—24no—
69C3×#117—22no—
70C3×#117—22no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4D412yes—
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4D412yes—
81C2×C3×C42C2311yescharacteristic
82C2×C3×C42C2311yescharacteristic
83C3×#103—28no—
84C3×#102—44no—
85C3×C22⋊C8—24no—
86C3×#97—22no—
87C3×C4×C8—22no—
88C3×#95—22no—
89C3×#95—24no—
90C22×SD16C22×C314yes—
91#540—24no—
92#540—24no—
93C2×C4⋊Q8C22×C311yescharacteristic
94C2×#127C22×C311yescharacteristic
95C2×#125C22×C311yescharacteristic
96C2×#123—22no—
97C2×C22⋊Q8C22×C312yes—
98C2×#121—22no—
99C2×#121C22×C312yes—
100C2×C4×D4C22×C312yes—
101C2×#117C22×C311yescharacteristic
102#492—24no—
103#491—24no—
104C2×#103C22×C314yes—
105C2×#102C22×C314yes—
106C2×C22⋊C8C22×C312yes—
107C2×#97C22×C311yescharacteristic
108C2×C4×C8C22×C311yescharacteristic
109#398—22no—
110#397—22no—
111#391C22×C312yes—
112#382C22×C311yescharacteristic
113#340C22×C314yes—
114#336C22×C312yes—
115C24×C3—22no—
116C24×C3C2×D411yescharacteristic
117C2×C3×Q8—22no—
118C2×C3×Q8—22no—
119C2×C3×Q8—24no—
120C2×C3×Q8—24no—
121C2×C3×D4—28no—
122C2×C3×D4—28no—
123C2×C3×D4—22no—
124C2×C3×D4—24no—
125C2×C3×D4—24no—
126C2×C3×D4—84no—
127C2×C3×D4—84no—
128C2×C3×D4C2×D414yes—
129C22×C3×C4—22no—
130C22×C3×C4—28no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—84no—
138C22×C3×C4C2×D412yes—
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D412yes—
141C22×C3×C4C2411yescharacteristic
142C3×SD16—416no—
143C2×C3×C8—22no—
144C2×C3×C8—22no—
145C2×C3×C8—22no—
146C2×C3×C8—22no—
147C2×C3×C8—22no—
148C2×C3×C8#4114yes—
149C3×C4⋊C4—28no—
150C3×C4⋊C4—24no—
151C3×C4⋊C4—22no—
152C3×C4⋊C4—24no—
153C3×C4⋊C4—44no—
154C3×C4⋊C4—44no—
155C3×C4⋊C4C2×D414yes—
156C3×C22⋊C4—28no—
157C3×C22⋊C4—28no—
158C3×C22⋊C4—24no—
159C3×C22⋊C4—44no—
160C3×C42—21no—
161C3×C42—21no—
162C3×C42C2×D412yes—
163C3×C42C2×D412yes—
164C22×Q8C23×C312yes—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4C23×C312yes—
168C23×C4—22no—
169C23×C4C23×C311yescharacteristic
170C2×SD16—28no—
171C2×SD16—28no—
172C2×SD16—44no—
173C22×C8C23×C312yes—
174C22×C8C23×C312yes—
175C4⋊Q8—22no—
176C4⋊Q8—22no—
177#127—22no—
178#127C3×D414yes—
179#125—22no—
180#125C3×D414yes—
181#123—216no—
182C22⋊Q8—44no—
183#121—216no—
184#121—24no—
185#121—24no—
186C4×D4—24no—
187C4×D4—24no—
188#117—22no—
189#117—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4C3×D412yes—
194C2×C4⋊C4C23×C312yes—
195C2×C4⋊C4C23×C312yes—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4C3×D412yes—
200C2×C42C23×C311yescharacteristic
201C2×C42C23×C311yescharacteristic
202#103—28no—
203#102—44no—
204C22⋊C8—24no—
205#97—22no—
206C4×C8—22no—
207#95—22no—
208#95—24no—
209C23×C3—21no—
210C23×C3—24no—
211C23×C3—21no—
212C23×C3—21no—
213C23×C3—24no—
214C23×C3—42no—
215C23×C3—42no—
216C23×C3—44no—
217C23×C3—44no—
218C23×C3C22×D411yescharacteristic
219C3×Q8—44no—
220C3×Q8—44no—
221C3×D4—28no—
222C3×D4—44no—
223C3×D4—88no—
224C3×D4—88no—
225C2×C3×C4—24no—
226C2×C3×C4—22no—
227C2×C3×C4—21no—
228C2×C3×C4—24no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—24no—
232C2×C3×C4—44no—
233C2×C3×C4—42no—
234C2×C3×C4—42no—
235C2×C3×C4—42no—
236C2×C3×C4—42no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4—44no—
240C2×C3×C4—88no—
241C2×C3×C4#12014yes—
242C2×C3×C4C22×D411yescharacteristic
243C2×C3×C4C22×D411yescharacteristic
244C2×C3×C4C2×#4112yes—
245C2×C3×C4H(5,𝔽2)12yes—
246C3×C8—24no—
247C3×C8—42no—
248C24—22no—
249C24C2×C3×D411yescharacteristic
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—24no—
253C2×Q8—24no—
254C2×D4—28no—
255C2×D4—28no—
256C2×D4—22no—
257C2×D4—24no—
258C2×D4—24no—
259C2×D4—84no—
260C2×D4—84no—
261C2×D4C2×C3×D414yes—
262C22×C4—22no—
263C22×C4—28no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—84no—
271C22×C4C2×C3×D412yes—
272C22×C4C2×C3×D411yescharacteristic
273C22×C4C2×C3×D412yes—
274C22×C4C24×C311yescharacteristic
275SD16—416no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8C3×#4114yes—
282C4⋊C4—28no—
283C4⋊C4—24no—
284C4⋊C4—22no—
285C4⋊C4—24no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4C2×C3×D414yes—
289C22⋊C4—28no—
290C22⋊C4—28no—
291C22⋊C4—24no—
292C22⋊C4—44no—
293C42—21no—
294C42—21no—
295C42C2×C3×D412yes—
296C42C2×C3×D412yes—
297C22×C3—42no—
298C22×C3—41no—
299C22×C3—44no—
300C22×C3—42no—
301C22×C3—44no—
302C22×C3—42no—
303C22×C3—44no—
304C22×C3—84no—
305C22×C3—84no—
306C22×C3C2×#12011yescharacteristic
307C22×C3#53211yescharacteristic
308C22×C3#53511yescharacteristic
309C22×C3#57714yes—
310C3×C4—24no—
311C3×C4—41no—
312C3×C4—42no—
313C3×C4—44no—
314C3×C4—82no—
315C3×C4—82no—
316C3×C4#54612yes—
317C3×C4#55012yes—
318C23—24no—
319C23—21no—
320C23—24no—
321C23—21no—
322C23—21no—
323C23—44no—
324C23—44no—
325C23—42no—
326C23—42no—
327C23C22×C3×D411yescharacteristic
328Q8—44no—
329Q8—44no—
330D4—28no—
331D4—44no—
332D4—88no—
333D4—88no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—24no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—44no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—88no—
350C2×C4C3×#12014yes—
351C2×C4C22×C3×D411yescharacteristic
352C2×C4C22×C3×D411yescharacteristic
353C2×C4C2×C3×#4112yes—
354C2×C4C3×H(5,𝔽2)12yes—
355C8—24no—
356C8—42no—
357C2×C3—81no—
358C2×C3—84no—
359C2×C3#243111yescharacteristic
360C2×C3#300912yes—
361C2×C3#317412yes—
362C2×C3#322112yes—
363C22—42no—
364C22—41no—
365C22—44no—
366C22—42no—
367C22—44no—
368C22—42no—
369C22—44no—
370C22—84no—
371C22—84no—
372C22C2×C3×#12011yescharacteristic
373C22C3×#53211yescharacteristic
374C22C3×#53511yescharacteristic
375C22C3×#57714yes—
376C4—24no—
377C4—41no—
378C4—42no—
379C4—44no—
380C4—82no—
381C4—82no—
382C4C3×#54612yes—
383C4C3×#55012yes—
384C3#2560611yescharacteristic, direct product factor, Sylow
385C2—81no—
386C2—84no—
387C2C3×#243111yescharacteristic
388C2C3×#300912yes—
389C2C3×#317412yes—
390C2C3×#322112yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#540C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#492C2semidirect
3. C2×C3×#491C2semidirect
4. C3×#2621C2semidirect
5. C3×#2000C2semidirect
6. C3×#1991C2semidirect
7. C3×#1961C2semidirect
8. C3×#1877C2semidirect
9. C3×#1750C2semidirect
10. #25606C3direct (N × H)
37. C2×#492C2×C3semidirect
38. C2×#491C2×C3semidirect
39. #2621C2×C3semidirect
40. #2000C2×C3semidirect
41. #1991C2×C3semidirect
42. #1961C2×C3semidirect
43. #1877C2×C3semidirect
44. #1750C2×C3semidirect
384. C3#25606direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5402
C2×C3×#4922
C2×C3×#4912
C3×#26212
C3×#20002
C3×#19912
C3×#19612
C3×#18772
C3×#17502
#256063