Finite Group Databasebeta

← C3×#29673 #10648280

Subgroups — C3×#29673 #10648280

Order 768.

1160 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#540C211yesmaximal, characteristic
2C2×C3×#461C212yesmaximal, complemented
3C3×#2621C211yesmaximal, characteristic, complemented
4C3×#2026C212yesmaximal, complemented
5C3×#2014C212yesmaximal, complemented
6C3×#2000C214yesmaximal, complemented
7C3×#1804C212yesmaximal, complemented
8C3×#1798C211yesmaximal, characteristic, complemented
9#29673C311yesmaximal, characteristic, direct product factor, Sylow
10C22×C3×SD16C2214yes—
11C3×#540—28no—
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×C3×#127C2211yescharacteristic
14C2×C3×#125C2211yescharacteristic
15C2×C3×#123—22no—
16C2×C3×C22⋊Q8C2212yes—
17C2×C3×#121—22no—
18C2×C3×#121C2212yes—
19C2×C3×C4×D4C2212yes—
20C2×C3×#117C2211yescharacteristic
21C3×#461—28no—
22C2×C3×C4⋊C8C2212yes—
23C2×C3×#103C2214yes—
24C2×C3×#102C2214yes—
25C2×C3×C22⋊C8C2212yes—
26C3×#398—22no—
27C3×#397—22no—
28C3×#391C2212yes—
29C3×#382C2211yescharacteristic
30C3×#340C2214yes—
31C3×#336C2212yes—
32C2×#540C2×C311yescharacteristic
33C2×#461C2×C312yescomplemented
34#2621C2×C311yescharacteristic, complemented
35#2026C2×C312yescomplemented
36#2014C2×C312yescomplemented
37#2000C2×C314yescomplemented
38#1804C2×C312yescomplemented
39#1798C2×C311yescharacteristic, complemented
40C22×C3×Q8C2312yes—
41C22×C3×D4—22no—
42C22×C3×D4—22no—
43C22×C3×D4C2312yes—
44C23×C3×C4—22no—
45C23×C3×C4C2311yescharacteristic
46C2×C3×SD16—28no—
47C2×C3×SD16—44no—
48C2×C3×SD16—44no—
49C22×C3×C8C2314yes—
50C3×C4⋊Q8—24no—
51C3×#127—24no—
52C3×#125—24no—
53C3×#123—216no—
54C3×C22⋊Q8—44no—
55C3×#121—216no—
56C3×#121—24no—
57C3×#121—24no—
58C3×C4×D4—24no—
59C3×C4×D4D418yes—
60C3×#117—24no—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4D412yes—
65C2×C3×C4⋊C4C2312yes—
66C2×C3×C4⋊C4C2312yes—
67C2×C3×C22⋊C4—22no—
68C2×C3×C22⋊C4—22no—
69C2×C3×C22⋊C4—22no—
70C2×C3×C22⋊C4D412yes—
71C2×C3×C42C2311yescharacteristic
72C2×C3×C42C2311yescharacteristic
73C3×C4⋊C8—24no—
74C3×#103—28no—
75C3×#102—44no—
76C3×C22⋊C8—42no—
77C3×#95—22no—
78C3×#95—24no—
79C22×SD16C22×C314yes—
80#540—28no—
81C2×C4⋊Q8C22×C311yescharacteristic
82C2×#127C22×C311yescharacteristic
83C2×#125C22×C311yescharacteristic
84C2×#123—22no—
85C2×C22⋊Q8C22×C312yes—
86C2×#121—22no—
87C2×#121C22×C312yes—
88C2×C4×D4C22×C312yes—
89C2×#117C22×C311yescharacteristic
90#461—28no—
91C2×C4⋊C8C22×C312yes—
92C2×#103C22×C314yes—
93C2×#102C22×C314yes—
94C2×C22⋊C8C22×C312yes—
95#398—22no—
96#397—22no—
97#391C22×C312yes—
98#382C22×C311yescharacteristic
99#340C22×C314yes—
100#336C22×C312yes—
101C24×C3—22no—
102C24×C3C2×D411yescharacteristic
103C2×C3×Q8—24no—
104C2×C3×Q8—24no—
105C2×C3×Q8—22no—
106C2×C3×Q8—22no—
107C2×C3×D4—28no—
108C2×C3×D4—28no—
109C2×C3×D4—22no—
110C2×C3×D4—22no—
111C2×C3×D4—22no—
112C2×C3×D4—22no—
113C2×C3×D4—22no—
114C2×C3×D4—84no—
115C2×C3×D4—84no—
116C2×C3×D4C2×D414yes—
117C22×C3×C4—22no—
118C22×C3×C4—22no—
119C22×C3×C4—21no—
120C22×C3×C4—22no—
121C22×C3×C4—21no—
122C22×C3×C4—28no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—84no—
126C22×C3×C4C2×D412yes—
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D412yes—
129C22×C3×C4C2411yescharacteristic
130C3×SD16—416no—
131C2×C3×C8—24no—
132C2×C3×C8—24no—
133C2×C3×C8—24no—
134C3×C4⋊C4—28no—
135C3×C4⋊C4—24no—
136C3×C4⋊C4—22no—
137C3×C4⋊C4—24no—
138C3×C4⋊C4—44no—
139C3×C4⋊C4—44no—
140C3×C4⋊C4C2×D414yes—
141C3×C22⋊C4—28no—
142C3×C22⋊C4—28no—
143C3×C22⋊C4—24no—
144C3×C22⋊C4—44no—
145C3×C42—22no—
146C3×C42C2×D414yes—
147C22×Q8C23×C312yes—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4C23×C312yes—
151C23×C4—22no—
152C23×C4C23×C311yescharacteristic
153C2×SD16—28no—
154C2×SD16—44no—
155C2×SD16—44no—
156C22×C8C23×C314yes—
157C4⋊Q8—24no—
158#127—24no—
159#125—24no—
160#123—216no—
161C22⋊Q8—44no—
162#121—216no—
163#121—24no—
164#121—24no—
165C4×D4—24no—
166C4×D4C3×D418yes—
167#117—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4C3×D412yes—
172C2×C4⋊C4C23×C312yes—
173C2×C4⋊C4C23×C312yes—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4C3×D412yes—
178C2×C42C23×C311yescharacteristic
179C2×C42C23×C311yescharacteristic
180C4⋊C8—24no—
181#103—28no—
182#102—44no—
183C22⋊C8—42no—
184#95—22no—
185#95—24no—
186C23×C3—21no—
187C23×C3—24no—
188C23×C3—21no—
189C23×C3—21no—
190C23×C3—24no—
191C23×C3—42no—
192C23×C3—42no—
193C23×C3—44no—
194C23×C3—44no—
195C23×C3C22×D411yescharacteristic
196C3×Q8—44no—
197C3×Q8—44no—
198C3×D4—24no—
199C3×D4—24no—
200C3×D4—42no—
201C3×D4—42no—
202C3×D4—88no—
203C3×D4—88no—
204C2×C3×C4—24no—
205C2×C3×C4—22no—
206C2×C3×C4—21no—
207C2×C3×C4—24no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—24no—
211C2×C3×C4—44no—
212C2×C3×C4—42no—
213C2×C3×C4—42no—
214C2×C3×C4—42no—
215C2×C3×C4—42no—
216C2×C3×C4—42no—
217C2×C3×C4—42no—
218C2×C3×C4—44no—
219C2×C3×C4—88no—
220C2×C3×C4#12014yes—
221C2×C3×C4C22×D412yes—
222C2×C3×C4H(5,𝔽2)11yescharacteristic
223C2×C3×C4H(5,𝔽2)12yes—
224C2×C3×C42−1+411yescharacteristic
225C3×C8—44no—
226C24—22no—
227C24C2×C3×D411yescharacteristic
228C2×Q8—24no—
229C2×Q8—24no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×D4—28no—
233C2×D4—28no—
234C2×D4—22no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—22no—
239C2×D4—84no—
240C2×D4—84no—
241C2×D4C2×C3×D414yes—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—28no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—84no—
251C22×C4C2×C3×D412yes—
252C22×C4C2×C3×D411yescharacteristic
253C22×C4C2×C3×D412yes—
254C22×C4C24×C311yescharacteristic
255SD16—416no—
256C2×C8—24no—
257C2×C8—24no—
258C2×C8—24no—
259C4⋊C4—28no—
260C4⋊C4—24no—
261C4⋊C4—22no—
262C4⋊C4—24no—
263C4⋊C4—44no—
264C4⋊C4—44no—
265C4⋊C4C2×C3×D414yes—
266C22⋊C4—28no—
267C22⋊C4—28no—
268C22⋊C4—24no—
269C22⋊C4—44no—
270C42—22no—
271C42C2×C3×D414yes—
272C22×C3—41no—
273C22×C3—44no—
274C22×C3—42no—
275C22×C3—42no—
276C22×C3—44no—
277C22×C3—42no—
278C22×C3—44no—
279C22×C3—84no—
280C22×C3—84no—
281C22×C3C2×#12011yescharacteristic
282C22×C3#53511yescharacteristic
283C22×C3#53711yescharacteristic
284C22×C3#57714yes—
285C3×C4—24no—
286C3×C4—41no—
287C3×C4—42no—
288C3×C4—44no—
289C3×C4—82no—
290C3×C4—82no—
291C3×C4#53512yes—
292C3×C4#53712yes—
293C23—24no—
294C23—21no—
295C23—24no—
296C23—21no—
297C23—21no—
298C23—44no—
299C23—44no—
300C23—42no—
301C23—42no—
302C23C22×C3×D411yescharacteristic
303Q8—44no—
304Q8—44no—
305D4—24no—
306D4—24no—
307D4—42no—
308D4—42no—
309D4—88no—
310D4—88no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—24no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—44no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—44no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—88no—
327C2×C4C3×#12014yes—
328C2×C4C22×C3×D412yes—
329C2×C4C3×H(5,𝔽2)11yescharacteristic
330C2×C4C3×H(5,𝔽2)12yes—
331C2×C4C3×2−1+411yescharacteristic
332C8—44no—
333C2×C3—81no—
334C2×C3—84no—
335C2×C3#261511yescharacteristic
336C2×C3#300912yes—
337C2×C3#322112yes—
338C2×C3#326012yes—
339C22—42no—
340C22—42no—
341C22—44no—
342C22—42no—
343C22—44no—
344C22—41no—
345C22—44no—
346C22—84no—
347C22—84no—
348C22C2×C3×#12011yescharacteristic
349C22C3×#53511yescharacteristic
350C22C3×#53711yescharacteristic
351C22C3×#57714yes—
352C4—24no—
353C4—42no—
354C4—44no—
355C4—41no—
356C4—82no—
357C4—82no—
358C4C3×#53512yes—
359C4C3×#53712yes—
360C3#2967311yescharacteristic, direct product factor, Sylow
361C2—81no—
362C2—84no—
363C2C3×#261511yescharacteristic
364C2C3×#300912yes—
365C2C3×#322112yes—
366C2C3×#326012yes—
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×#461C2semidirect
3. C3×#2621C2semidirect
4. C3×#2026C2semidirect
5. C3×#2014C2semidirect
6. C3×#2000C2semidirect
7. C3×#1804C2semidirect
8. C3×#1798C2semidirect
9. #29673C3direct (N × H)
33. C2×#461C2×C3semidirect
34. #2621C2×C3semidirect
35. #2026C2×C3semidirect
36. #2014C2×C3semidirect
37. #2000C2×C3semidirect
38. #1804C2×C3semidirect
39. #1798C2×C3semidirect
360. C3#29673direct (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×#4612
C3×#26212
C3×#20262
C3×#20142
C3×#20002
C3×#18042
C3×#17982
#296733