Finite Group Databasebeta

← C3×#33026 #10651633

Subgroups — C3×#33026 #10651633

Order 768.

974 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#476C211yesmaximal, characteristic
2C2×C3×#461C211yesmaximal, characteristic
3C3×#2862C211yesmaximal, characteristic, complemented
4C3×C42×D4C211yesmaximal, characteristic
5C3×#1987C211yesmaximal, characteristic, complemented
6C3×#1981C211yesmaximal, characteristic, complemented
7C3×#1927C214yesmaximal, complemented
8C3×#1877C214yesmaximal
9C3×#1831C211yesmaximal, characteristic, complemented
10#33026C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×SD16C2212yes—
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C4×D4C2212yes—
15C2×C3×C4×D4C2211yescharacteristic
16C22×C3×C42—21no—
17C3×#476—24no—
18C3×#461—24no—
19C2×C3×#106C2212yes—
20C2×C3×C4⋊C8C2211yescharacteristic
21C2×C3×C4⋊C8C2211yescharacteristic
22C2×C3×#103C2214yes—
23C2×C3×#103C2212yes—
24C2×C3×#102C2212yes—
25C3×#391—22no—
26C3×#391C2214yes—
27C3×#382—21no—
28C3×#382—21no—
29C3×C4×C4⋊C4C2212yes—
30C3×C4×C4⋊C4C2211yescharacteristic
31C3×C4×C22⋊C4—21no—
32C3×C4×C22⋊C4—22no—
33C3×C43C2211yescharacteristic
34C3×#340C2214yes—
35C3×#336C2214yes—
36C2×#476C2×C311yescharacteristic
37C2×#461C2×C311yescharacteristic
38#2862C2×C311yescharacteristic, complemented
39C42×D4C2×C311yescharacteristic
40#1987C2×C311yescharacteristic, complemented
41#1981C2×C311yescharacteristic, complemented
42#1927C2×C314yescomplemented
43#1877C2×C314yes—
44#1831C2×C311yescharacteristic, complemented
45C22×C3×Q8—21no—
46C22×C3×Q8C2312yes—
47C22×C3×D4C2311yescharacteristic
48C23×C3×C4—22no—
49C23×C3×C4—21no—
50C2×C3×SD16—24no—
51C2×C3×SD16—28no—
52C22×C3×C8C2312yes—
53C22×C3×C8C2312yes—
54C3×C4⋊Q8—22no—
55C3×C4⋊Q8—22no—
56C3×C4⋊Q8—22no—
57C3×C4⋊Q8—22no—
58C3×C4×D4—24no—
59C3×C4×D4—24no—
60C3×C4×D4—22no—
61C3×C4×D4D414yes—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4C2312yes—
66C2×C3×C4⋊C4C2311yescharacteristic
67C2×C3×C4⋊C4C2312yes—
68C2×C3×C22⋊C4—22no—
69C2×C3×C22⋊C4—21no—
70C2×C3×C42—22no—
71C2×C3×C42—21no—
72C2×C3×C42—44no—
73C2×C3×C42D411yescharacteristic
74C2×C3×C42D412yes—
75C2×C3×C42D411yescharacteristic
76C2×C3×C42C2312yes—
77C2×C3×C42C2311yescharacteristic
78C3×#106—24no—
79C3×C4⋊C8—22no—
80C3×C4⋊C8—22no—
81C3×#103—24no—
82C3×#103—28no—
83C3×#102—24no—
84C3×#95—22no—
85C3×#95—21no—
86C3×#95—24no—
87C3×#95—24no—
88C22×SD16C22×C312yes—
89C2×C4⋊Q8C22×C311yescharacteristic
90C2×C4⋊Q8C22×C311yescharacteristic
91C2×C4×D4C22×C312yes—
92C2×C4×D4C22×C311yescharacteristic
93C22×C42—21no—
94#476—24no—
95#461—24no—
96C2×#106C22×C312yes—
97C2×C4⋊C8C22×C311yescharacteristic
98C2×C4⋊C8C22×C311yescharacteristic
99C2×#103C22×C314yes—
100C2×#103C22×C312yes—
101C2×#102C22×C312yes—
102#391—22no—
103#391C22×C314yes—
104#382—21no—
105#382—21no—
106C4×C4⋊C4C22×C312yes—
107C4×C4⋊C4C22×C311yescharacteristic
108C4×C22⋊C4—21no—
109C4×C22⋊C4—22no—
110C43C22×C311yescharacteristic
111#340C22×C314yes—
112#336C22×C314yes—
113C24×C3—21no—
114C2×C3×Q8—22no—
115C2×C3×Q8—24no—
116C2×C3×Q8—22no—
117C2×C3×Q8—24no—
118C2×C3×Q8—44no—
119C2×C3×Q8—42no—
120C2×C3×D4—21no—
121C2×C3×D4—22no—
122C2×C3×D4C2×D412yes—
123C2×C3×D4#4114yes—
124C22×C3×C4—28no—
125C22×C3×C4—21no—
126C22×C3×C4—22no—
127C22×C3×C4—24no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—44no—
131C22×C3×C4—44no—
132C22×C3×C4—42no—
133C22×C3×C4—42no—
134C22×C3×C4C2×D412yes—
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D412yes—
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2411yescharacteristic
139C3×SD16—48no—
140C2×C3×C8—22no—
141C2×C3×C8—22no—
142C2×C3×C8—24no—
143C2×C3×C8—24no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—22no—
146C3×C4⋊C4—22no—
147C3×C4⋊C4—22no—
148C3×C4⋊C4—42no—
149C3×C4⋊C4—42no—
150C3×C4⋊C4—42no—
151C3×C4⋊C4C2×D412yes—
152C3×C4⋊C4#4114yes—
153C3×C22⋊C4—28no—
154C3×C22⋊C4—24no—
155C3×C42—22no—
156C3×C42—22no—
157C3×C42—22no—
158C3×C42—42no—
159C3×C42—44no—
160C3×C42—42no—
161C3×C42SD1614yescomplemented
162C3×C42C2×D412yes—
163C3×C42#4112yes—
164C22×Q8—21no—
165C22×Q8C23×C312yes—
166C22×D4C23×C311yescharacteristic
167C23×C4—22no—
168C23×C4—21no—
169C2×SD16—24no—
170C2×SD16—28no—
171C22×C8C23×C312yes—
172C22×C8C23×C312yes—
173C4⋊Q8—22no—
174C4⋊Q8—22no—
175C4⋊Q8—22no—
176C4⋊Q8—22no—
177C4×D4—24no—
178C4×D4—24no—
179C4×D4—22no—
180C4×D4C3×D414yes—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4C23×C312yes—
185C2×C4⋊C4C23×C311yescharacteristic
186C2×C4⋊C4C23×C312yes—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—21no—
189C2×C42—22no—
190C2×C42—21no—
191C2×C42—44no—
192C2×C42C3×D411yescharacteristic
193C2×C42C3×D412yes—
194C2×C42C3×D411yescharacteristic
195C2×C42C23×C312yes—
196C2×C42C23×C311yescharacteristic
197#106—24no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200#103—24no—
201#103—28no—
202#102—24no—
203#95—22no—
204#95—21no—
205#95—24no—
206#95—24no—
207C23×C3—22no—
208C23×C3—24no—
209C23×C3—41no—
210C23×C3—41no—
211C23×C3—42no—
212C23×C3C22×D411yescharacteristic
213C3×Q8—42no—
214C3×Q8—44no—
215C3×Q8—42no—
216C3×Q8—84no—
217C3×D4—24no—
218C3×D4—22no—
219C3×D4#12414yes—
220C2×C3×C4—22no—
221C2×C3×C4—22no—
222C2×C3×C4—22no—
223C2×C3×C4—22no—
224C2×C3×C4—22no—
225C2×C3×C4—24no—
226C2×C3×C4—22no—
227C2×C3×C4—22no—
228C2×C3×C4—416no—
229C2×C3×C4—42no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—42no—
233C2×C3×C4—48no—
234C2×C3×C4—42no—
235C2×C3×C4—42no—
236C2×C3×C4—44no—
237C2×C3×C4—44no—
238C2×C3×C4#12112yes—
239C2×C3×C4#12112yes—
240C2×C3×C4C2×SD1614yes—
241C2×C3×C4C2×SD1612yes—
242C2×C3×C4#13712yes—
243C2×C3×C4C22×D411yescharacteristic
244C2×C3×C4C2×#4112yes—
245C2×C3×C4C2×#4111yescharacteristic
246C2×C3×C4C2×#4112yes—
247C3×C8—42no—
248C3×C8—42no—
249C24—21no—
250C2×Q8—22no—
251C2×Q8—24no—
252C2×Q8—22no—
253C2×Q8—24no—
254C2×Q8—44no—
255C2×Q8—42no—
256C2×D4—21no—
257C2×D4—22no—
258C2×D4C2×C3×D412yes—
259C2×D4C3×#4114yes—
260C22×C4—28no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4C2×C3×D412yes—
271C22×C4C2×C3×D411yescharacteristic
272C22×C4C2×C3×D412yes—
273C22×C4C2×C3×D411yescharacteristic
274C22×C4C24×C311yescharacteristic
275SD16—48no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—24no—
279C2×C8—24no—
280C4⋊C4—21no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4C2×C3×D412yes—
288C4⋊C4C3×#4114yes—
289C22⋊C4—28no—
290C22⋊C4—24no—
291C42—22no—
292C42—22no—
293C42—22no—
294C42—42no—
295C42—44no—
296C42—42no—
297C42C3×SD1614yescomplemented
298C42C2×C3×D412yes—
299C42C3×#4112yes—
300C22×C3—24no—
301C22×C3—44no—
302C22×C3—44no—
303C22×C3—42no—
304C22×C3—42no—
305C22×C3C2×#12111yescharacteristic
306C22×C3#52512yes—
307C22×C3C22×SD1611yescharacteristic
308C22×C3C2×#13711yescharacteristic
309C22×C3#57512yes—
310C3×C4—24no—
311C3×C4—22no—
312C3×C4—44no—
313C3×C4—42no—
314C3×C4—42no—
315C3×C4—82no—
316C3×C4—82no—
317C3×C4#46114yes—
318C3×C4C2×#12411yescharacteristic
319C3×C4#52912yes—
320C3×C4#53211yescharacteristic
321C23—22no—
322C23—24no—
323C23—41no—
324C23—41no—
325C23—42no—
326C23C22×C3×D411yescharacteristic
327Q8—42no—
328Q8—44no—
329Q8—42no—
330Q8—84no—
331D4—24no—
332D4—22no—
333D4C3×#12414yes—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—24no—
342C2×C4—416no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—48no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4C3×#12112yes—
353C2×C4C3×#12112yes—
354C2×C4C2×C3×SD1614yes—
355C2×C4C2×C3×SD1612yes—
356C2×C4C3×#13712yes—
357C2×C4C22×C3×D411yescharacteristic
358C2×C4C2×C3×#4112yes—
359C2×C4C2×C3×#4111yescharacteristic
360C2×C4C2×C3×#4112yes—
361C8—42no—
362C8—42no—
363C2×C3—44no—
364C2×C3#285011yescharacteristic
365C2×C3C2×#46111yescharacteristic
366C2×C3#304011yescharacteristic
367C2×C3#310112yes—
368C2×C3#311612yes—
369C22—24no—
370C22—42no—
371C22—42no—
372C22—44no—
373C22—44no—
374C22C2×C3×#12111yescharacteristic
375C22C3×#52512yes—
376C22C22×C3×SD1611yescharacteristic
377C22C2×C3×#13711yescharacteristic
378C22C3×#57512yes—
379C4—22no—
380C4—24no—
381C4—42no—
382C4—44no—
383C4—42no—
384C4—82no—
385C4—82no—
386C4C3×#46114yes—
387C4C2×C3×#12411yescharacteristic
388C4C3×#52912yes—
389C4C3×#53211yescharacteristic
390C3#3302611yescharacteristic, direct product factor, Sylow
391C2—44no—
392C2C3×#285011yescharacteristic
393C2C2×C3×#46111yescharacteristic
394C2C3×#304011yescharacteristic
395C2C3×#310112yes—
396C2C3×#311612yes—
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
3. C3×#2862C2semidirect
5. C3×#1987C2semidirect
6. C3×#1981C2semidirect
7. C3×#1927C2semidirect
9. C3×#1831C2semidirect
10. #33026C3direct (N × H)
38. #2862C2×C3semidirect
40. #1987C2×C3semidirect
41. #1981C2×C3semidirect
42. #1927C2×C3semidirect
44. #1831C2×C3semidirect
161. C3×C42SD16semidirect
297. C42C3×SD16semidirect
390. C3#33026direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4762
C2×C3×#4612
C3×#28622
C3×C42×D42
C3×#19872
C3×#19812
C3×#19272
C3×#18772
C3×#18312
#330263