Finite Group Databasebeta

← C3×#25609 #10644216

Subgroups — C3×#25609 #10644216

Order 768.

1052 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#541C211yesmaximal, characteristic
2C2×C3×#494C211yesmaximal, characteristic, complemented
3C2×C3×#491C211yesmaximal, characteristic, complemented
4C3×#2621C211yesmaximal, characteristic
5C3×#2001C214yesmaximal
6C3×#1991C211yesmaximal, characteristic, complemented
7C3×#1960C211yesmaximal, characteristic, complemented
8C3×#1878C214yesmaximal, complemented
9C3×#1750C211yesmaximal, characteristic
10#25609C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×Q16C2214yes—
12C3×#541—24no—
13C3×#541—24no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#126—22no—
17C2×C3×#125—22no—
18C2×C3×#125C2211yescharacteristic
19C2×C3×C22⋊Q8C2212yes—
20C2×C3×C22⋊Q8C2212yes—
21C2×C3×C4×Q8C2212yes—
22C2×C3×#117C2211yescharacteristic
23C3×#494—24no—
24C3×#491—24no—
25C2×C3×#103C2214yes—
26C2×C3×#103C2214yes—
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×#541C2×C311yescharacteristic
37C2×#494C2×C311yescharacteristic, complemented
38C2×#491C2×C311yescharacteristic, complemented
39#2621C2×C311yescharacteristic
40#2001C2×C314yes—
41#1991C2×C311yescharacteristic, complemented
42#1960C2×C311yescharacteristic, complemented
43#1878C2×C314yescomplemented
44#1750C2×C311yescharacteristic
45C22×C3×Q8C2312yes—
46C22×C3×Q8C2312yes—
47C23×C3×C4C2311yescharacteristic
48C2×C3×Q16—28no—
49C2×C3×Q16—28no—
50C2×C3×Q16—44no—
51C22×C3×C8C2312yes—
52C22×C3×C8C2312yes—
53C3×C4⋊Q8—22no—
54C3×C4⋊Q8—22no—
55C3×C4⋊Q8—22no—
56C3×C4⋊Q8D414yes—
57C3×#126—216no—
58C3×#125—216no—
59C3×#125—22no—
60C3×#125D414yes—
61C3×C22⋊Q8—24no—
62C3×C22⋊Q8—24no—
63C3×C22⋊Q8—44no—
64C3×C4×Q8—24no—
65C3×C4×Q8—24no—
66C3×#117—22no—
67C3×#117—22no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—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⋊C4D412yes—
80C2×C3×C42—22no—
81C2×C3×C42C2311yescharacteristic
82C2×C3×C42C2311yescharacteristic
83C3×#103—28no—
84C3×#103—44no—
85C3×C22⋊C8—24no—
86C3×#97—22no—
87C3×C4×C8—22no—
88C3×#95—22no—
89C3×#95—24no—
90C22×Q16C22×C314yes—
91#541—24no—
92#541—24no—
93C2×C4⋊Q8C22×C311yescharacteristic
94C2×C4⋊Q8C22×C311yescharacteristic
95C2×#126—22no—
96C2×#125—22no—
97C2×#125C22×C311yescharacteristic
98C2×C22⋊Q8C22×C312yes—
99C2×C22⋊Q8C22×C312yes—
100C2×C4×Q8C22×C312yes—
101C2×#117C22×C311yescharacteristic
102#494—24no—
103#491—24no—
104C2×#103C22×C314yes—
105C2×#103C22×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×C3C2×D411yescharacteristic
116C2×C3×Q8—22no—
117C2×C3×Q8—22no—
118C2×C3×Q8—22no—
119C2×C3×Q8—24no—
120C2×C3×Q8—24no—
121C2×C3×Q8—24no—
122C2×C3×Q8—24no—
123C2×C3×Q8C2×D414yes—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4C2×D412yes—
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D412yes—
135C22×C3×C4C2411yescharacteristic
136C3×Q16—416no—
137C2×C3×C8—22no—
138C2×C3×C8—22no—
139C2×C3×C8—22no—
140C2×C3×C8—22no—
141C2×C3×C8—22no—
142C2×C3×C8#4114yes—
143C3×C4⋊C4—28no—
144C3×C4⋊C4—28no—
145C3×C4⋊C4—28no—
146C3×C4⋊C4—28no—
147C3×C4⋊C4—24no—
148C3×C4⋊C4—22no—
149C3×C4⋊C4—24no—
150C3×C4⋊C4—44no—
151C3×C4⋊C4—44no—
152C3×C4⋊C4C2×D414yes—
153C3×C22⋊C4—28no—
154C3×C22⋊C4—24no—
155C3×C22⋊C4—44no—
156C3×C42—28no—
157C3×C42—21no—
158C3×C42—21no—
159C3×C42C2×D412yes—
160C3×C42C2×D412yes—
161C22×Q8C23×C312yes—
162C22×Q8C23×C312yes—
163C23×C4C23×C311yescharacteristic
164C2×Q16—28no—
165C2×Q16—28no—
166C2×Q16—44no—
167C22×C8C23×C312yes—
168C22×C8C23×C312yes—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172C4⋊Q8C3×D414yes—
173#126—216no—
174#125—216no—
175#125—22no—
176#125C3×D414yes—
177C22⋊Q8—24no—
178C22⋊Q8—24no—
179C22⋊Q8—44no—
180C4×Q8—24no—
181C4×Q8—24no—
182#117—22no—
183#117—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4C3×D412yes—
191C2×C4⋊C4C23×C312yes—
192C2×C4⋊C4C23×C312yes—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4C3×D412yes—
196C2×C42—22no—
197C2×C42C23×C311yescharacteristic
198C2×C42C23×C311yescharacteristic
199#103—28no—
200#103—44no—
201C22⋊C8—24no—
202#97—22no—
203C4×C8—22no—
204#95—22no—
205#95—24no—
206C23×C3—21no—
207C23×C3—21no—
208C23×C3—21no—
209C23×C3—24no—
210C23×C3C22×D411yescharacteristic
211C3×Q8—28no—
212C3×Q8—44no—
213C3×Q8—44no—
214C3×Q8—44no—
215C2×C3×C4—24no—
216C2×C3×C4—22no—
217C2×C3×C4—24no—
218C2×C3×C4—24no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—24no—
223C2×C3×C4—42no—
224C2×C3×C4—42no—
225C2×C3×C4—44no—
226C2×C3×C4—42no—
227C2×C3×C4—42no—
228C2×C3×C4—42no—
229C2×C3×C4—42no—
230C2×C3×C4—42no—
231C2×C3×C4—42no—
232C2×C3×C4—44no—
233C2×C3×C4#12014yes—
234C2×C3×C4C22×D411yescharacteristic
235C2×C3×C4C22×D411yescharacteristic
236C2×C3×C4C2×#4112yes—
237C2×C3×C4H(5,𝔽2)12yes—
238C3×C8—24no—
239C3×C8—42no—
240C24C2×C3×D411yescharacteristic
241C2×Q8—22no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—24no—
245C2×Q8—24no—
246C2×Q8—24no—
247C2×Q8—24no—
248C2×Q8C2×C3×D414yes—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4C2×C3×D412yes—
258C22×C4C2×C3×D411yescharacteristic
259C22×C4C2×C3×D412yes—
260C22×C4C24×C311yescharacteristic
261Q16—416no—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8C3×#4114yes—
268C4⋊C4—28no—
269C4⋊C4—28no—
270C4⋊C4—28no—
271C4⋊C4—28no—
272C4⋊C4—24no—
273C4⋊C4—22no—
274C4⋊C4—24no—
275C4⋊C4—44no—
276C4⋊C4—44no—
277C4⋊C4C2×C3×D414yes—
278C22⋊C4—28no—
279C22⋊C4—24no—
280C22⋊C4—44no—
281C42—28no—
282C42—21no—
283C42—21no—
284C42C2×C3×D412yes—
285C42C2×C3×D412yes—
286C22×C3—42no—
287C22×C3—41no—
288C22×C3—42no—
289C22×C3—42no—
290C22×C3C2×#12011yescharacteristic
291C22×C3#53211yescharacteristic
292C22×C3#53511yescharacteristic
293C22×C3#57814yes—
294C3×C4—24no—
295C3×C4—41no—
296C3×C4—44no—
297C3×C4—42no—
298C3×C4—44no—
299C3×C4—82no—
300C3×C4—82no—
301C3×C4#54612yes—
302C3×C4#55012yes—
303C23—24no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23C22×C3×D411yescharacteristic
308Q8—28no—
309Q8—44no—
310Q8—44no—
311Q8—44no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—22no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—44no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—44no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4C3×#12014yes—
331C2×C4C22×C3×D411yescharacteristic
332C2×C4C22×C3×D411yescharacteristic
333C2×C4C2×C3×#4112yes—
334C2×C4C3×H(5,𝔽2)12yes—
335C8—24no—
336C8—42no—
337C2×C3—81no—
338C2×C3#243111yescharacteristic
339C2×C3#301112yes—
340C2×C3#317712yes—
341C2×C3#322312yes—
342C22—42no—
343C22—41no—
344C22—42no—
345C22—42no—
346C22C2×C3×#12011yescharacteristic
347C22C3×#53211yescharacteristic
348C22C3×#53511yescharacteristic
349C22C3×#57814yes—
350C4—24no—
351C4—41no—
352C4—44no—
353C4—42no—
354C4—44no—
355C4—82no—
356C4—82no—
357C4C3×#54612yes—
358C4C3×#55012yes—
359C3#2560911yescharacteristic, direct product factor, Sylow
360C2—81no—
361C2C3×#243111yescharacteristic
362C2C3×#301112yes—
363C2C3×#317712yes—
364C2C3×#322312yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
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×#494C2semidirect
3. C2×C3×#491C2semidirect
6. C3×#1991C2semidirect
7. C3×#1960C2semidirect
8. C3×#1878C2semidirect
10. #25609C3direct (N × H)
37. C2×#494C2×C3semidirect
38. C2×#491C2×C3semidirect
41. #1991C2×C3semidirect
42. #1960C2×C3semidirect
43. #1878C2×C3semidirect
359. C3#25609direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5412
C2×C3×#4942
C2×C3×#4912
C3×#26212
C3×#20012
C3×#19912
C3×#19602
C3×#18782
C3×#17502
#256093