Finite Group Databasebeta

← C3×#18319 #10636926

Subgroups — C3×#18319 #10636926

Order 768.

1058 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#543C211yesmaximal, characteristic
2C3×#3195C212yesmaximal, complemented
3C3×#2561C211yesmaximal, characteristic
4C3×#1608C214yesmaximal, complemented
5C3×#1602C214yesmaximal, complemented
6C3×#1508C212yesmaximal, complemented
7C3×#1462C211yesmaximal, characteristic
8#18319C311yesmaximal, characteristic, direct product factor, Sylow
9C3×#543—24no—
10C3×#543—22no—
11C3×#543C2214yes—
12C2×C3×C22⋊Q8—22no—
13C2×C3×C22⋊Q8—21no—
14C2×C3×C22⋊Q8—22no—
15C2×C3×C22⋊Q8C2212yes—
16C2×C3×C22⋊Q8C2211yescharacteristic
17C22×C3×C22⋊C4—21no—
18C22×C3×C22⋊C4C2211yescharacteristic
19C3×#483—22no—
20C3×#483C2214yescomplemented
21C3×#470—24no—
22C3×#451—22no—
23C3×#451C2214yescomplemented
24C3×#407C2212yes—
25C3×#397C2212yes—
26C3×#393C2212yes—
27C3×#388—22no—
28C3×#381—21no—
29C3×#379—21no—
30C3×C4×C22⋊C4C2211yescharacteristic
31C3×#328C2218yescomplemented
32C3×#323C2214yes—
33C2×#543C2×C311yescharacteristic
34#3195C2×C312yescomplemented
35#2561C2×C311yescharacteristic
36#1608C2×C314yescomplemented
37#1602C2×C314yescomplemented
38#1508C2×C312yescomplemented
39#1462C2×C311yescharacteristic
40C25×C3D411yescharacteristic
41C22×C3×Q8—21no—
42C22×C3×Q8D412yes—
43C22×C3×Q8C2311yescharacteristic
44C23×C3×C4—21no—
45C23×C3×C4—21no—
46C23×C3×C4D411yescharacteristic
47C23×C3×C4C2311yescharacteristic
48C2×C3×Q16—24no—
49C2×C3×M16—22no—
50C3×C22⋊Q8—28no—
51C3×C22⋊Q8—28no—
52C3×C22⋊Q8—24no—
53C3×C22⋊Q8—28no—
54C3×C22⋊Q8—24no—
55C3×C22⋊Q8—22no—
56C3×C22⋊Q8—22no—
57C3×C22⋊Q8—28no—
58C3×C22⋊Q8—22no—
59C3×C22⋊Q8—22no—
60C3×C22⋊Q8D414yes—
61C3×C22⋊Q8C2314yes—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4D412yes—
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C22⋊C4—22no—
69C2×C3×C22⋊C4—24no—
70C2×C3×C22⋊C4—22no—
71C2×C3×C22⋊C4—24no—
72C2×C3×C22⋊C4—22no—
73C2×C3×C22⋊C4—22no—
74C2×C3×C22⋊C4—21no—
75C2×C3×C22⋊C4—21no—
76C2×C3×C22⋊C4—22no—
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4D412yes—
79C2×C3×C22⋊C4C2312yes—
80C2×C3×C42—21no—
81C3×#106—24no—
82C3×#103—24no—
83C3×#103—24no—
84C3×C22⋊C8—22no—
85C3×C22⋊C8C2314yes—
86C3×#95—22no—
87C3×#95—22no—
88C3×#95C2312yes—
89#543—24no—
90#543—22no—
91#543C22×C314yes—
92C2×C22⋊Q8—22no—
93C2×C22⋊Q8—21no—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8C22×C312yes—
96C2×C22⋊Q8C22×C311yescharacteristic
97C22×C22⋊C4—21no—
98C22×C22⋊C4C22×C311yescharacteristic
99#483—22no—
100#483C22×C314yescomplemented
101#470—24no—
102#451—22no—
103#451C22×C314yescomplemented
104#407C22×C312yes—
105#397C22×C312yes—
106#393C22×C312yes—
107#388—22no—
108#381—21no—
109#379—21no—
110C4×C22⋊C4C22×C311yescharacteristic
111#328C22×C318yescomplemented
112#323C22×C314yes—
113C24×C3—21no—
114C24×C3—21no—
115C24×C3—24no—
116C24×C3—42no—
117C24×C3—81no—
118C24×C3C2×D411yescharacteristic
119C24×C3C2×D412yes—
120C2×C3×Q8—24no—
121C2×C3×Q8—22no—
122C2×C3×Q8—22no—
123C2×C3×Q8—81no—
124C2×C3×Q8—82no—
125C2×C3×Q8—82no—
126C2×C3×Q8C2×D414yes—
127C22×C3×C4—22no—
128C22×C3×C4—21no—
129C22×C3×C4—22no—
130C22×C3×C4—24no—
131C22×C3×C4—24no—
132C22×C3×C4—21no—
133C22×C3×C4—22no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—42no—
137C22×C3×C4—81no—
138C22×C3×C4—82no—
139C22×C3×C4—82no—
140C22×C3×C4C2×D412yes—
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D412yes—
143C22×C3×C4C2×D412yes—
144C22×C3×C4C2411yescharacteristic
145C3×Q16—84no—
146C3×M16—82no—
147C2×C3×C8—24no—
148C3×C4⋊C4—24no—
149C3×C4⋊C4—28no—
150C3×C4⋊C4—24no—
151C3×C4⋊C4—22no—
152C3×C4⋊C4—24no—
153C3×C4⋊C4—22no—
154C3×C4⋊C4—42no—
155C3×C4⋊C4C2×D414yes—
156C3×C22⋊C4—28no—
157C3×C22⋊C4—24no—
158C3×C22⋊C4—24no—
159C3×C22⋊C4—24no—
160C3×C22⋊C4—24no—
161C3×C22⋊C4—42no—
162C3×C22⋊C4—41no—
163C3×C22⋊C4—41no—
164C3×C22⋊C4—44no—
165C3×C22⋊C4—82no—
166C3×C42—81no—
167C25C3×D411yescharacteristic
168C22×Q8—21no—
169C22×Q8C3×D412yes—
170C22×Q8C23×C311yescharacteristic
171C23×C4—21no—
172C23×C4—21no—
173C23×C4C3×D411yescharacteristic
174C23×C4C23×C311yescharacteristic
175C2×Q16—24no—
176C2×M16—22no—
177C22⋊Q8—28no—
178C22⋊Q8—28no—
179C22⋊Q8—24no—
180C22⋊Q8—28no—
181C22⋊Q8—24no—
182C22⋊Q8—22no—
183C22⋊Q8—22no—
184C22⋊Q8—28no—
185C22⋊Q8—22no—
186C22⋊Q8—22no—
187C22⋊Q8C3×D414yes—
188C22⋊Q8C23×C314yes—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4C3×D412yes—
194C2×C4⋊C4C23×C311yescharacteristic
195C2×C22⋊C4—22no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4C3×D412yes—
206C2×C22⋊C4C23×C312yes—
207C2×C42—21no—
208#106—24no—
209#103—24no—
210#103—24no—
211C22⋊C8—22no—
212C22⋊C8C23×C314yes—
213#95—22no—
214#95—22no—
215#95C23×C312yes—
216C23×C3—24no—
217C23×C3—24no—
218C23×C3—22no—
219C23×C3—22no—
220C23×C3—22no—
221C23×C3—22no—
222C23×C3—41no—
223C23×C3—42no—
224C23×C3—42no—
225C23×C3—42no—
226C23×C3—42no—
227C23×C3—81no—
228C23×C3—81no—
229C23×C3—88no—
230C23×C3#12011yescharacteristic
231C23×C3#12011yescharacteristic
232C23×C3#13714yes—
233C23×C3C22×D411yescharacteristic
234C3×Q8—44no—
235C3×Q8—44no—
236C3×Q8—82no—
237C3×Q8—84no—
238C2×C3×C4—24no—
239C2×C3×C4—22no—
240C2×C3×C4—24no—
241C2×C3×C4—41no—
242C2×C3×C4—42no—
243C2×C3×C4—42no—
244C2×C3×C4—41no—
245C2×C3×C4—44no—
246C2×C3×C4—42no—
247C2×C3×C4—44no—
248C2×C3×C4—42no—
249C2×C3×C4—42no—
250C2×C3×C4—42no—
251C2×C3×C4—42no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—81no—
257C2×C3×C4—84no—
258C2×C3×C4—84no—
259C2×C3×C4#12012yes—
260C2×C3×C4C22×D412yes—
261C3×C8—44no—
262C24—21no—
263C24—21no—
264C24—24no—
265C24—42no—
266C24—81no—
267C24C2×C3×D411yescharacteristic
268C24C2×C3×D412yes—
269C2×Q8—24no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—81no—
273C2×Q8—82no—
274C2×Q8—82no—
275C2×Q8C2×C3×D414yes—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—42no—
286C22×C4—81no—
287C22×C4—82no—
288C22×C4—82no—
289C22×C4C2×C3×D412yes—
290C22×C4C2×C3×D411yescharacteristic
291C22×C4C2×C3×D412yes—
292C22×C4C2×C3×D412yes—
293C22×C4C24×C311yescharacteristic
294Q16—84no—
295M16—82no—
296C2×C8—24no—
297C4⋊C4—24no—
298C4⋊C4—28no—
299C4⋊C4—24no—
300C4⋊C4—22no—
301C4⋊C4—24no—
302C4⋊C4—22no—
303C4⋊C4—42no—
304C4⋊C4C2×C3×D414yes—
305C22⋊C4—28no—
306C22⋊C4—24no—
307C22⋊C4—24no—
308C22⋊C4—24no—
309C22⋊C4—24no—
310C22⋊C4—42no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—44no—
314C22⋊C4—82no—
315C42—81no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—24no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—44no—
325C22×C3—44no—
326C22×C3—48no—
327C22×C3—42no—
328C22×C3—42no—
329C22×C3—82no—
330C22×C3—84no—
331C22×C3#45512yes—
332C22×C3C2×#12011yescharacteristic
333C22×C3C2×#13712yes—
334C3×C4—22no—
335C3×C4—22no—
336C3×C4—42no—
337C3×C4—44no—
338C3×C4—44no—
339C3×C4—82no—
340C3×C4—81no—
341C3×C4—81no—
342C23—24no—
343C23—24no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—22no—
348C23—41no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23—81no—
354C23—81no—
355C23—88no—
356C23C3×#12011yescharacteristic
357C23C3×#12011yescharacteristic
358C23C3×#13714yes—
359C23C22×C3×D411yescharacteristic
360Q8—44no—
361Q8—44no—
362Q8—82no—
363Q8—84no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—81no—
383C2×C4—84no—
384C2×C4—84no—
385C2×C4C3×#12012yes—
386C2×C4C22×C3×D412yes—
387C8—44no—
388C2×C3—22no—
389C2×C3—22no—
390C2×C3—41no—
391C2×C3—44no—
392C2×C3#300411yescharacteristic
393C2×C3C2×#45511yescharacteristic
394C2×C3#302611yescharacteristic
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—24no—
401C22—21no—
402C22—21no—
403C22—44no—
404C22—44no—
405C22—48no—
406C22—42no—
407C22—42no—
408C22—82no—
409C22—84no—
410C22C3×#45512yes—
411C22C2×C3×#12011yescharacteristic
412C22C2×C3×#13712yes—
413C4—22no—
414C4—22no—
415C4—42no—
416C4—44no—
417C4—44no—
418C4—82no—
419C4—81no—
420C4—81no—
421C3#1831911yescharacteristic, direct product factor, Sylow
422C2—22no—
423C2—22no—
424C2—41no—
425C2—44no—
426C2C3×#300411yescharacteristic
427C2C2×C3×#45511yescharacteristic
428C2C3×#302611yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC25×C3D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#3195C2semidirect
4. C3×#1608C2semidirect
5. C3×#1602C2semidirect
6. C3×#1508C2semidirect
8. #18319C3direct (N × H)
20. C3×#483C22semidirect
23. C3×#451C22semidirect
31. C3×#328C22semidirect
34. #3195C2×C3semidirect
36. #1608C2×C3semidirect
37. #1602C2×C3semidirect
38. #1508C2×C3semidirect
100. #483C22×C3semidirect
103. #451C22×C3semidirect
111. #328C22×C3semidirect
421. C3#18319direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5432
C3×#31952
C3×#25612
C3×#16082
C3×#16022
C3×#15082
C3×#14622
#183193