Finite Group Databasebeta

← C3×#18133 #10636740

Subgroups — C3×#18133 #10636740

Order 768.

1310 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#543C211yesmaximal, characteristic, complemented
2C3×#3007C212yesmaximal, complemented
3C3×#2404C211yesmaximal, characteristic
4C3×#1602C214yesmaximal, complemented
5C3×#1596C214yesmaximal, complemented
6C3×#1511C212yesmaximal, complemented
7C3×#1462C211yesmaximal, characteristic, complemented
8#18133C311yesmaximal, characteristic, direct product factor, Sylow
9C22×C3×#41C2211yescharacteristic
10C2×C3×#137—22no—
11C3×#543—24no—
12C3×#543—22no—
13C3×#543C2214yes—
14C2×C3×C22⋊Q8—22no—
15C2×C3×C22⋊Q8—21no—
16C2×C3×C22⋊Q8—22no—
17C2×C3×C22⋊Q8C2212yes—
18C2×C3×#120—21no—
19C22×C3×C22⋊C4—21no—
20C22×C3×C22⋊C4C2211yescharacteristic
21C3×#452—24no—
22C3×#451C2214yescomplemented
23C3×#450C2214yescomplemented
24C3×#417—22no—
25C3×#407C2212yes—
26C3×#393C2212yes—
27C3×#392C2212yes—
28C3×#386—21no—
29C3×#386—22no—
30C3×#381C2211yescharacteristic
31C3×#328C2218yescomplemented
32C3×#323C2214yes—
33C2×#543C2×C311yescharacteristic, complemented
34#3007C2×C312yescomplemented
35#2404C2×C311yescharacteristic
36#1602C2×C314yescomplemented
37#1596C2×C314yescomplemented
38#1511C2×C312yescomplemented
39#1462C2×C311yescharacteristic, complemented
40C25×C3D411yescharacteristic
41C2×C3×#41—22no—
42C2×C3×#41—22no—
43C2×C3×#41—44no—
44C22×C3×Q8—21no—
45C22×C3×Q8D412yescomplemented
46C22×C3×Q8C2311yescharacteristic
47C22×C3×D4—21no—
48C22×C3×D4—22no—
49C22×C3×D4C2311yescharacteristic
50C23×C3×C4—21no—
51C23×C3×C4—21no—
52C23×C3×C4—21no—
53C23×C3×C4D411yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C3×#137—48no—
56C2×C3×Q16—24no—
57C2×C3×SD16—24no—
58C2×C3×M16—22no—
59C3×C22⋊Q8—28no—
60C3×C22⋊Q8—28no—
61C3×C22⋊Q8—24no—
62C3×C22⋊Q8—28no—
63C3×C22⋊Q8—24no—
64C3×C22⋊Q8—22no—
65C3×C22⋊Q8—22no—
66C3×C22⋊Q8—28no—
67C3×C22⋊Q8D414yescomplemented
68C3×C22⋊Q8C2314yes—
69C3×#120—44no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4D412yescomplemented
74C2×C3×C22⋊C4—22no—
75C2×C3×C22⋊C4—24no—
76C2×C3×C22⋊C4—22no—
77C2×C3×C22⋊C4—24no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—21no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4D412yes—
84C2×C3×C22⋊C4C2312yes—
85C3×#103—24no—
86C3×#102—24no—
87C3×C22⋊C8—22no—
88C3×C22⋊C8C2314yes—
89C3×#95—21no—
90C3×#95—22no—
91C3×#95C2312yes—
92C22×#41C22×C311yescharacteristic
93C2×#137—22no—
94#543—24no—
95#543—22no—
96#543C22×C314yes—
97C2×C22⋊Q8—22no—
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8—22no—
100C2×C22⋊Q8C22×C312yes—
101C2×#120—21no—
102C22×C22⋊C4—21no—
103C22×C22⋊C4C22×C311yescharacteristic
104#452—24no—
105#451C22×C314yescomplemented
106#450C22×C314yescomplemented
107#417—22no—
108#407C22×C312yes—
109#393C22×C312yes—
110#392C22×C312yes—
111#386—21no—
112#386—22no—
113#381C22×C311yescharacteristic
114#328C22×C318yescomplemented
115#323C22×C314yes—
116C24×C3—21no—
117C24×C3—21no—
118C24×C3—24no—
119C24×C3—21no—
120C24×C3—42no—
121C24×C3—81no—
122C24×C3C2×D411yescharacteristic
123C24×C3C2×D412yes—
124C3×#41—28no—
125C3×#41—44no—
126C3×#41—44no—
127C3×#41—44no—
128C2×C3×Q8—24no—
129C2×C3×Q8—22no—
130C2×C3×Q8—22no—
131C2×C3×Q8—42no—
132C2×C3×Q8—44no—
133C2×C3×Q8—82no—
134C2×C3×Q8C2×D414yes—
135C2×C3×D4—24no—
136C2×C3×D4—22no—
137C2×C3×D4—42no—
138C2×C3×D4—42no—
139C2×C3×D4—41no—
140C2×C3×D4—41no—
141C2×C3×D4—44no—
142C2×C3×D4—44no—
143C2×C3×D4—82no—
144C22×C3×C4—22no—
145C22×C3×C4—21no—
146C22×C3×C4—22no—
147C22×C3×C4—21no—
148C22×C3×C4—24no—
149C22×C3×C4—24no—
150C22×C3×C4—21no—
151C22×C3×C4—22no—
152C22×C3×C4—42no—
153C22×C3×C4—42no—
154C22×C3×C4—42no—
155C22×C3×C4—42no—
156C22×C3×C4—81no—
157C22×C3×C4—82no—
158C22×C3×C4—82no—
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D412yes—
163C22×C3×C4C2411yescharacteristic
164C3×Q16—48no—
165C3×SD16—48no—
166C3×M16—44no—
167C2×C3×C8—24no—
168C3×C4⋊C4—24no—
169C3×C4⋊C4—28no—
170C3×C4⋊C4—24no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—24no—
173C3×C4⋊C4C2×D414yes—
174C3×C22⋊C4—28no—
175C3×C22⋊C4—24no—
176C3×C22⋊C4—24no—
177C3×C22⋊C4—24no—
178C3×C22⋊C4—24no—
179C3×C22⋊C4—44no—
180C3×C22⋊C4—42no—
181C3×C22⋊C4—82no—
182C25C3×D411yescharacteristic
183C2×#41—22no—
184C2×#41—22no—
185C2×#41—44no—
186C22×Q8—21no—
187C22×Q8C3×D412yescomplemented
188C22×Q8C23×C311yescharacteristic
189C22×D4—21no—
190C22×D4—22no—
191C22×D4C23×C311yescharacteristic
192C23×C4—21no—
193C23×C4—21no—
194C23×C4—21no—
195C23×C4C3×D411yescharacteristic
196C23×C4C23×C311yescharacteristic
197#137—48no—
198C2×Q16—24no—
199C2×SD16—24no—
200C2×M16—22no—
201C22⋊Q8—28no—
202C22⋊Q8—28no—
203C22⋊Q8—24no—
204C22⋊Q8—28no—
205C22⋊Q8—24no—
206C22⋊Q8—22no—
207C22⋊Q8—22no—
208C22⋊Q8—28no—
209C22⋊Q8C3×D414yescomplemented
210C22⋊Q8C23×C314yes—
211#120—44no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4C3×D412yescomplemented
216C2×C22⋊C4—22no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4C3×D412yes—
226C2×C22⋊C4C23×C312yes—
227#103—24no—
228#102—24no—
229C22⋊C8—22no—
230C22⋊C8C23×C314yes—
231#95—21no—
232#95—22no—
233#95C23×C312yes—
234C23×C3—24no—
235C23×C3—24no—
236C23×C3—22no—
237C23×C3—22no—
238C23×C3—22no—
239C23×C3—22no—
240C23×C3—22no—
241C23×C3—41no—
242C23×C3—42no—
243C23×C3—42no—
244C23×C3—42no—
245C23×C3—42no—
246C23×C3—41no—
247C23×C3—42no—
248C23×C3—41no—
249C23×C3—42no—
250C23×C3—81no—
251C23×C3—81no—
252C23×C3—88no—
253C23×C3#12011yescharacteristic
254C23×C3#12011yescharacteristic
255C23×C3#13714yescomplemented
256C23×C3C22×D411yescharacteristic
257C3×Q8—24no—
258C3×Q8—44no—
259C3×Q8—42no—
260C3×Q8—82no—
261C3×Q8—84no—
262C3×D4—24no—
263C3×D4—42no—
264C3×D4—44no—
265C3×D4—44no—
266C3×D4—88no—
267C2×C3×C4—24no—
268C2×C3×C4—22no—
269C2×C3×C4—24no—
270C2×C3×C4—22no—
271C2×C3×C4—24no—
272C2×C3×C4—41no—
273C2×C3×C4—42no—
274C2×C3×C4—41no—
275C2×C3×C4—48no—
276C2×C3×C4—41no—
277C2×C3×C4—42no—
278C2×C3×C4—44no—
279C2×C3×C4—41no—
280C2×C3×C4—44no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—81no—
286C2×C3×C4—84no—
287C2×C3×C4—84no—
288C2×C3×C4#12012yes—
289C2×C3×C4C22×D412yes—
290C3×C8—44no—
291C24—21no—
292C24—21no—
293C24—24no—
294C24—21no—
295C24—42no—
296C24—81no—
297C24C2×C3×D411yescharacteristic
298C24C2×C3×D412yes—
299#41—28no—
300#41—44no—
301#41—44no—
302#41—44no—
303C2×Q8—24no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—42no—
307C2×Q8—44no—
308C2×Q8—82no—
309C2×Q8C2×C3×D414yes—
310C2×D4—24no—
311C2×D4—22no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—44no—
317C2×D4—44no—
318C2×D4—82no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—81no—
332C22×C4—82no—
333C22×C4—82no—
334C22×C4C2×C3×D412yes—
335C22×C4C2×C3×D412yes—
336C22×C4C2×C3×D411yescharacteristic
337C22×C4C2×C3×D412yes—
338C22×C4C24×C311yescharacteristic
339Q16—48no—
340SD16—48no—
341M16—44no—
342C2×C8—24no—
343C4⋊C4—24no—
344C4⋊C4—28no—
345C4⋊C4—24no—
346C4⋊C4—22no—
347C4⋊C4—24no—
348C4⋊C4C2×C3×D414yes—
349C22⋊C4—28no—
350C22⋊C4—24no—
351C22⋊C4—24no—
352C22⋊C4—24no—
353C22⋊C4—24no—
354C22⋊C4—44no—
355C22⋊C4—42no—
356C22⋊C4—82no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—21no—
361C22×C3—21no—
362C22×C3—24no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—44no—
366C22×C3—44no—
367C22×C3—48no—
368C22×C3—44no—
369C22×C3—42no—
370C22×C3—42no—
371C22×C3—42no—
372C22×C3—82no—
373C22×C3—82no—
374C22×C3—84no—
375C22×C3—81no—
376C22×C3—81no—
377C22×C3B(4,𝔽2)12yes—
378C22×C3C2×#12011yescharacteristic
379C22×C3C2×#13712yes—
380C3×C4—22no—
381C3×C4—22no—
382C3×C4—42no—
383C3×C4—44no—
384C3×C4—42no—
385C3×C4—44no—
386C3×C4—82no—
387C23—24no—
388C23—24no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—22no—
393C23—22no—
394C23—41no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—41no—
400C23—42no—
401C23—42no—
402C23—41no—
403C23—81no—
404C23—81no—
405C23—88no—
406C23C3×#12011yescharacteristic
407C23C3×#12011yescharacteristic
408C23C3×#13714yescomplemented
409C23C22×C3×D411yescharacteristic
410Q8—24no—
411Q8—44no—
412Q8—42no—
413Q8—82no—
414Q8—84no—
415D4—24no—
416D4—42no—
417D4—44no—
418D4—44no—
419D4—88no—
420C2×C4—24no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—48no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—81no—
439C2×C4—84no—
440C2×C4—84no—
441C2×C4C3×#12012yes—
442C2×C4C22×C3×D412yes—
443C8—44no—
444C2×C3—22no—
445C2×C3—22no—
446C2×C3—41no—
447C2×C3—44no—
448C2×C3—82no—
449C2×C3#300411yescharacteristic
450C2×C3#301911yescharacteristic
451C2×C3C2×B(4,𝔽2)11yescharacteristic
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—24no—
458C22—21no—
459C22—21no—
460C22—44no—
461C22—44no—
462C22—48no—
463C22—44no—
464C22—42no—
465C22—42no—
466C22—42no—
467C22—82no—
468C22—82no—
469C22—84no—
470C22—81no—
471C22—81no—
472C22C3×B(4,𝔽2)12yes—
473C22C2×C3×#12011yescharacteristic
474C22C2×C3×#13712yes—
475C4—22no—
476C4—22no—
477C4—42no—
478C4—44no—
479C4—42no—
480C4—44no—
481C4—82no—
482C3#1813311yescharacteristic, direct product factor, Sylow
483C2—22no—
484C2—22no—
485C2—41no—
486C2—44no—
487C2—82no—
488C2C3×#300411yescharacteristic
489C2C3×#301911yescharacteristic
490C2C2×C3×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#2404C2
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
1. C2×C3×#543C2semidirect
2. C3×#3007C2semidirect
4. C3×#1602C2semidirect
5. C3×#1596C2semidirect
6. C3×#1511C2semidirect
7. C3×#1462C2semidirect
8. #18133C3direct (N × H)
22. C3×#451C22semidirect
23. C3×#450C22semidirect
31. C3×#328C22semidirect
33. C2×#543C2×C3semidirect
34. #3007C2×C3semidirect
36. #1602C2×C3semidirect
37. #1596C2×C3semidirect
38. #1511C2×C3semidirect
39. #1462C2×C3semidirect
45. C22×C3×Q8D4semidirect
67. C3×C22⋊Q8D4semidirect
73. C2×C3×C4⋊C4D4semidirect
105. #451C22×C3semidirect
106. #450C22×C3semidirect
114. #328C22×C3semidirect
187. C22×Q8C3×D4semidirect
209. C22⋊Q8C3×D4semidirect
215. C2×C4⋊C4C3×D4semidirect
255. C23×C3#137semidirect
408. C23C3×#137semidirect
482. C3#18133direct (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×#30072
C3×#24042
C3×#16022
C3×#15962
C3×#15112
C3×#14622
#181333