Finite Group Databasebeta

← C3×#60621 #10679228

Subgroups — C3×#60621 #10679228

Order 768.

2078 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×#544C211yesmaximal, characteristic, complemented
2C3×#3449C211yesmaximal, characteristic
3C3×#3142C214yesmaximal, complemented
4C3×#3141C214yesmaximal, complemented
5C3×#3137C218yesmaximal, complemented
6C3×#3136C218yesmaximal, complemented
7C3×#2896C212yesmaximal, complemented
8C2×C3×#417C212yesmaximal, complemented
9C3×#2873C211yesmaximal, characteristic, complemented
10#60621C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×#41C2212yes—
12C22×C3×M16C2212yes—
13C3×#544—24no—
14C3×#544C2218yescomplemented
15C3×#537—21no—
16C3×#537—21no—
17C3×#536C2214yes—
18C3×#535C2214yes—
19C3×#534—22no—
20C3×#532C2214yes—
21C2×C3×C4⋊Q8—21no—
22C2×C3×C4⋊Q8C2212yescomplemented
23C2×C3×#125—21no—
24C2×C3×#125C2212yescomplemented
25C3×#525—24no—
26C2×C3×C22⋊Q8—22no—
27C2×C3×#117C2211yescharacteristic
28C2×C3×#117C2211yescharacteristic
29C22×C3×C4⋊C4C2211yescharacteristic
30C3×#491C2218yescomplemented
31C3×#490C2218yescomplemented
32C3×#489C22116yescomplemented
33C3×#488C22116yescomplemented
34C3×#487C2218yescomplemented
35C3×#486C2218yescomplemented
36C3×#419C2214yescomplemented
37C3×#418C2214yescomplemented
38C3×#417C2218yes—
39C3×#417C2214yescomplemented
40C3×#417C2218yescomplemented
41C3×#417C2214yescomplemented
42C3×#416C2218yes—
43C2×C3×#103C2214yescomplemented
44C2×C3×#102C2214yescomplemented
45C3×#405C2218yescomplemented
46C3×C4×M16C2214yescomplemented
47C2×#544C2×C311yescharacteristic, complemented
48#3449C2×C311yescharacteristic
49#3142C2×C314yescomplemented
50#3141C2×C314yescomplemented
51#3137C2×C318yescomplemented
52#3136C2×C318yescomplemented
53#2896C2×C312yescomplemented
54C2×#417C2×C312yescomplemented
55#2873C2×C311yescharacteristic, complemented
56C2×C3×#41—22no—
57C2×C3×#41—22no—
58C2×C3×#41—28no—
59C2×C3×#41—28no—
60C2×C3×#41C2314yes—
61C2×C3×#41C2314yes—
62C22×C3×Q8—22no—
63C22×C3×Q8C2312yes—
64C22×C3×D4—22no—
65C22×C3×D4C2312yes—
66C23×C3×C4—21no—
67C23×C3×C4—22no—
68C23×C3×C4C2311yescharacteristic
69C2×C3×M16—28no—
70C2×C3×M16C2314yes—
71C2×C3×M16C2314yes—
72C22×C3×C8C2314yes—
73C3×C4⋊Q8—28no—
74C3×C4⋊Q8—24no—
75C3×C4⋊Q8C2314yes—
76C3×C4⋊Q8C2318yes—
77C3×#127C2314yes—
78C3×#125—28no—
79C3×#125—24no—
80C3×#125C2318yes—
81C3×#124C2318yes—
82C3×#123—28no—
83C3×C22⋊Q8—24no—
84C3×C22⋊Q8—24no—
85C3×C22⋊Q8—216no—
86C3×#121—24no—
87C3×#121—24no—
88C3×#121—24no—
89C3×#121—24no—
90C3×#120—28no—
91C3×C4×D4—24no—
92C3×C4×D4—24no—
93C3×#117—22no—
94C3×#117D414yes—
95C3×#117C2314yes—
96C3×#117C2314yes—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—22no—
99C2×C3×C4⋊C4—22no—
100C2×C3×C4⋊C4—22no—
101C2×C3×C4⋊C4—22no—
102C2×C3×C4⋊C4D411yescharacteristic
103C2×C3×C4⋊C4D411yescharacteristic
104C2×C3×C4⋊C4C2314yes—
105C2×C3×C4⋊C4C2312yes—
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C22⋊C4—21no—
109C2×C3×C22⋊C4D412yes—
110C2×C3×C42—21no—
111C2×C3×C42C2312yes—
112C3×#103C23116yes—
113C3×#103C2318yes—
114C3×#103C2318yes—
115C3×#102C23116yes—
116C3×#102C2318yes—
117C3×#102C2318yes—
118C3×#97C2318yes—
119C3×C4×C8C2318yes—
120C22×#41C22×C312yes—
121C22×M16C22×C312yes—
122#544—24no—
123#544C22×C318yescomplemented
124#537—21no—
125#537—21no—
126#536C22×C314yes—
127#535C22×C314yes—
128#534—22no—
129#532C22×C314yes—
130C2×C4⋊Q8—21no—
131C2×C4⋊Q8C22×C312yescomplemented
132C2×#125—21no—
133C2×#125C22×C312yescomplemented
134#525—24no—
135C2×C22⋊Q8—22no—
136C2×#117C22×C311yescharacteristic
137C2×#117C22×C311yescharacteristic
138C22×C4⋊C4C22×C311yescharacteristic
139#491C22×C318yescomplemented
140#490C22×C318yescomplemented
141#489C22×C3116yescomplemented
142#488C22×C3116yescomplemented
143#487C22×C318yescomplemented
144#486C22×C318yescomplemented
145#419C22×C314yescomplemented
146#418C22×C314yescomplemented
147#417C22×C318yes—
148#417C22×C314yescomplemented
149#417C22×C318yescomplemented
150#417C22×C314yescomplemented
151#416C22×C318yes—
152C2×#103C22×C314yescomplemented
153C2×#102C22×C314yescomplemented
154#405C22×C318yescomplemented
155C4×M16C22×C314yescomplemented
156C24×C3—22no—
157C24×C3C2×D411yescharacteristic
158C3×#41—28no—
159C3×#41—28no—
160C3×#41—28no—
161C3×#41—28no—
162C3×#41—48no—
163C3×#41—48no—
164C2×C3×Q8—22no—
165C2×C3×Q8—28no—
166C2×C3×Q8—24no—
167C2×C3×Q8—24no—
168C2×C3×Q8—84no—
169C2×C3×Q8C2414yes—
170C2×C3×D4—24no—
171C2×C3×D4—24no—
172C2×C3×D4—22no—
173C2×C3×D4—24no—
174C2×C3×D4—24no—
175C2×C3×D4—24no—
176C2×C3×D4—24no—
177C2×C3×D4—48no—
178C2×C3×D4C2414yes—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—24no—
184C22×C3×C4—24no—
185C22×C3×C4—24no—
186C22×C3×C4—48no—
187C22×C3×C4—82no—
188C22×C3×C4C2×D412yes—
189C22×C3×C4C2×D412yes—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D412yes—
192C22×C3×C4#4118yes—
193C22×C3×C4C2411yescharacteristic
194C22×C3×C4C2412yes—
195C3×M16—28no—
196C3×M16—28no—
197C2×C3×C8—28no—
198C2×C3×C8C2418yes—
199C3×C4⋊C4—24no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—28no—
202C3×C4⋊C4—28no—
203C3×C4⋊C4—28no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—24no—
207C3×C4⋊C4—24no—
208C3×C4⋊C4C2×D414yes—
209C3×C4⋊C4C2×D414yes—
210C3×C4⋊C4C2414yes—
211C3×C4⋊C4C2412yes—
212C3×C4⋊C4C2412yes—
213C3×C22⋊C4—24no—
214C3×C22⋊C4—28no—
215C3×C22⋊C4—28no—
216C3×C22⋊C4C2×D418yes—
217C3×C42—24no—
218C3×C42C2×D414yes—
219C3×C42C2414yes—
220C2×#41—22no—
221C2×#41—22no—
222C2×#41—28no—
223C2×#41—28no—
224C2×#41C23×C314yes—
225C2×#41C23×C314yes—
226C22×Q8—22no—
227C22×Q8C23×C312yes—
228C22×D4—22no—
229C22×D4C23×C312yes—
230C23×C4—21no—
231C23×C4—22no—
232C23×C4C23×C311yescharacteristic
233C2×M16—28no—
234C2×M16C23×C314yes—
235C2×M16C23×C314yes—
236C22×C8C23×C314yes—
237C4⋊Q8—28no—
238C4⋊Q8—24no—
239C4⋊Q8C23×C314yes—
240C4⋊Q8C23×C318yes—
241#127C23×C314yes—
242#125—28no—
243#125—24no—
244#125C23×C318yes—
245#124C23×C318yes—
246#123—28no—
247C22⋊Q8—24no—
248C22⋊Q8—24no—
249C22⋊Q8—216no—
250#121—24no—
251#121—24no—
252#121—24no—
253#121—24no—
254#120—28no—
255C4×D4—24no—
256C4×D4—24no—
257#117—22no—
258#117C3×D414yes—
259#117C23×C314yes—
260#117C23×C314yes—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4C3×D411yescharacteristic
267C2×C4⋊C4C3×D411yescharacteristic
268C2×C4⋊C4C23×C312yes—
269C2×C4⋊C4C23×C314yes—
270C2×C4⋊C4C23×C311yescharacteristic
271C2×C4⋊C4C23×C311yescharacteristic
272C2×C22⋊C4—21no—
273C2×C22⋊C4C3×D412yes—
274C2×C42—21no—
275C2×C42C23×C312yes—
276#103C23×C3116yes—
277#103C23×C318yes—
278#103C23×C318yes—
279#102C23×C3116yes—
280#102C23×C318yes—
281#102C23×C318yes—
282#97C23×C318yes—
283C4×C8C23×C318yes—
284C23×C3—24no—
285C23×C3—41no—
286C23×C3—41no—
287C23×C3—44no—
288C23×C3—82no—
289C23×C3—82no—
290C23×C3#12414yes—
291C23×C3C22×D412yes—
292C23×C3C22×D411yescharacteristic
293C3×Q8—24no—
294C3×Q8—24no—
295C3×Q8—44no—
296C3×Q8—88no—
297C3×D4—24no—
298C3×D4—24no—
299C3×D4—48no—
300C3×D4—48no—
301C3×D4—44no—
302C3×D4—84no—
303C3×D4—84no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—24no—
307C2×C3×C4—24no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—44no—
312C2×C3×C4—42no—
313C2×C3×C4—44no—
314C2×C3×C4—48no—
315C2×C3×C4—48no—
316C2×C3×C4—84no—
317C2×C3×C4#12418yes—
318C2×C3×C4#12414yes—
319C2×C3×C4C22×D414yes—
320C2×C3×C4C22×D414yes—
321C2×C3×C4C22×D412yes—
322C2×C3×C4C22×D411yescharacteristic
323C2×C3×C4C2×#4114yes—
324C2×C3×C4C2×#4114yes—
325C2×C3×C4C2×#4114yes—
326C2×C3×C4C2511yescharacteristic
327C3×C8—28no—
328C24—22no—
329C24C2×C3×D411yescharacteristic
330#41—28no—
331#41—28no—
332#41—28no—
333#41—28no—
334#41—48no—
335#41—48no—
336C2×Q8—22no—
337C2×Q8—28no—
338C2×Q8—24no—
339C2×Q8—24no—
340C2×Q8—84no—
341C2×Q8C24×C314yes—
342C2×D4—24no—
343C2×D4—24no—
344C2×D4—22no—
345C2×D4—24no—
346C2×D4—24no—
347C2×D4—24no—
348C2×D4—24no—
349C2×D4—48no—
350C2×D4C24×C314yes—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—48no—
359C22×C4—82no—
360C22×C4C2×C3×D412yes—
361C22×C4C2×C3×D412yes—
362C22×C4C2×C3×D411yescharacteristic
363C22×C4C2×C3×D412yes—
364C22×C4C3×#4118yes—
365C22×C4C24×C311yescharacteristic
366C22×C4C24×C312yes—
367M16—28no—
368M16—28no—
369C2×C8—28no—
370C2×C8C24×C318yes—
371C4⋊C4—24no—
372C4⋊C4—22no—
373C4⋊C4—28no—
374C4⋊C4—28no—
375C4⋊C4—28no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4C2×C3×D414yes—
381C4⋊C4C2×C3×D414yes—
382C4⋊C4C24×C314yes—
383C4⋊C4C24×C312yes—
384C4⋊C4C24×C312yes—
385C22⋊C4—24no—
386C22⋊C4—28no—
387C22⋊C4—28no—
388C22⋊C4C2×C3×D418yes—
389C42—24no—
390C42C2×C3×D414yes—
391C42C24×C314yes—
392C22×C3—21no—
393C22×C3—22no—
394C22×C3—21no—
395C22×C3—22no—
396C22×C3—44no—
397C22×C3—44no—
398C22×C3—44no—
399C22×C3—44no—
400C22×C3—88no—
401C22×C3C2×#12414yes—
402C22×C3C2×#12412yes—
403C22×C3C23×D411yescharacteristic
404C3×C4—24no—
405C3×C4—24no—
406C3×C4—44no—
407C3×C4—42no—
408C3×C4—42no—
409C3×C4—44no—
410C3×C4C2×#12412yes—
411C3×C4C2×#12414yes—
412C3×C4C22×#4112yes—
413C23—24no—
414C23—41no—
415C23—41no—
416C23—44no—
417C23—82no—
418C23—82no—
419C23C3×#12414yes—
420C23C22×C3×D412yes—
421C23C22×C3×D411yescharacteristic
422Q8—24no—
423Q8—24no—
424Q8—44no—
425Q8—88no—
426D4—24no—
427D4—24no—
428D4—48no—
429D4—48no—
430D4—44no—
431D4—84no—
432D4—84no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—48no—
444C2×C4—48no—
445C2×C4—84no—
446C2×C4C3×#12418yes—
447C2×C4C3×#12414yes—
448C2×C4C22×C3×D414yes—
449C2×C4C22×C3×D414yes—
450C2×C4C22×C3×D412yes—
451C2×C4C22×C3×D411yescharacteristic
452C2×C4C2×C3×#4114yes—
453C2×C4C2×C3×#4114yes—
454C2×C4C2×C3×#4114yes—
455C2×C4C25×C311yescharacteristic
456C8—28no—
457C2×C3—24no—
458C2×C3—22no—
459C2×C3—84no—
460C2×C3C22×#12411yescharacteristic
461C2×C3#358511yescharacteristic
462C2×C3#358611yescharacteristic
463C22—21no—
464C22—22no—
465C22—21no—
466C22—22no—
467C22—44no—
468C22—44no—
469C22—44no—
470C22—44no—
471C22—88no—
472C22C2×C3×#12414yes—
473C22C2×C3×#12412yes—
474C22C23×C3×D411yescharacteristic
475C4—24no—
476C4—24no—
477C4—44no—
478C4—44no—
479C4—42no—
480C4—42no—
481C4C2×C3×#12412yes—
482C4C2×C3×#12414yes—
483C4C22×C3×#4112yes—
484C3#6062111yescharacteristic, direct product factor, Sylow
485C2—24no—
486C2—22no—
487C2—84no—
488C2C22×C3×#12411yescharacteristic
489C2C3×#358511yescharacteristic
490C2C3×#358611yescharacteristic
subgroupG/H
center, socleC22×C3C23×D4
commutator, Frattini, cosocleC2×C4C25×C3
prime radicalC3×#3449C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#544C2semidirect
3. C3×#3142C2semidirect
4. C3×#3141C2semidirect
5. C3×#3137C2semidirect
6. C3×#3136C2semidirect
7. C3×#2896C2semidirect
8. C2×C3×#417C2semidirect
9. C3×#2873C2semidirect
10. #60621C3direct (N × H)
14. C3×#544C22semidirect
22. C2×C3×C4⋊Q8C22semidirect
24. C2×C3×#125C22semidirect
30. C3×#491C22semidirect
31. C3×#490C22semidirect
32. C3×#489C22semidirect
33. C3×#488C22semidirect
34. C3×#487C22semidirect
35. C3×#486C22semidirect
36. C3×#419C22semidirect
37. C3×#418C22semidirect
39. C3×#417C22semidirect
40. C3×#417C22semidirect
41. C3×#417C22semidirect
43. C2×C3×#103C22semidirect
44. C2×C3×#102C22semidirect
45. C3×#405C22semidirect
46. C3×C4×M16C22semidirect
47. C2×#544C2×C3semidirect
49. #3142C2×C3semidirect
50. #3141C2×C3semidirect
51. #3137C2×C3semidirect
52. #3136C2×C3semidirect
53. #2896C2×C3semidirect
54. C2×#417C2×C3semidirect
55. #2873C2×C3semidirect
123. #544C22×C3semidirect
131. C2×C4⋊Q8C22×C3semidirect
133. C2×#125C22×C3semidirect
139. #491C22×C3semidirect
140. #490C22×C3semidirect
141. #489C22×C3semidirect
142. #488C22×C3semidirect
143. #487C22×C3semidirect
144. #486C22×C3semidirect
145. #419C22×C3semidirect
146. #418C22×C3semidirect
148. #417C22×C3semidirect
149. #417C22×C3semidirect
150. #417C22×C3semidirect
152. C2×#103C22×C3semidirect
153. C2×#102C22×C3semidirect
154. #405C22×C3semidirect
155. C4×M16C22×C3semidirect
484. C3#60621direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5442
C3×#34492
C3×#31422
C3×#31412
C3×#31372
C3×#31362
C3×#28962
C2×C3×#4172
C3×#28732
#606213