Finite Group Databasebeta

← C3×#20984 #10639591

Subgroups — C3×#20984 #10639591

Order 768.

2102 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#534C211yesmaximal, characteristic
2C2×C3×#518C211yesmaximal, characteristic, complemented
3C22×C3×#99C212yesmaximal, complemented
4C3×#2308C211yesmaximal, characteristic, complemented
5C3×#1899C214yesmaximal, complemented
6C3×#1898C214yesmaximal, complemented
7C2×C3×#342C212yesmaximal, complemented
8#20984C311yesmaximal, characteristic, direct product factor, Sylow
9C23×C3×D4C2211yescharacteristic
10C23×C3×D4C2211yescharacteristic
11C3×#534—24no—
12C3×#534—24no—
13C2×C3×#123C412yescomplemented
14C2×C3×#123C2212yes—
15C2×C3×#121C414yescomplemented
16C2×C3×#120C412yescomplemented
17C2×C3×#120C2212yes—
18C3×#518—28no—
19C2×C3×C4×D4—24no—
20C2×C3×#117C2212yescomplemented
21C22×C3×C22⋊C4C2212yes—
22C22×C3×C22⋊C4C2211yescharacteristic
23C22×C3×C22⋊C4C2212yes—
24C22×C3×C22⋊C4C2212yescomplemented
25C2×C3×#99—28no—
26C2×C3×#99C2218yes—
27C3×#386—24no—
28C3×#381C2212yescomplemented
29C3×#379C2212yescomplemented
30C3×#342C2218yescomplemented
31C2×#534C2×C311yescharacteristic
32C2×#518C2×C311yescharacteristic, complemented
33C22×#99C2×C312yescomplemented
34#2308C2×C311yescharacteristic, complemented
35#1899C2×C314yescomplemented
36#1898C2×C314yescomplemented
37C2×#342C2×C312yescomplemented
38C25×C3C2×C411yescharacteristic, complemented
39C25×C3C2311yescharacteristic
40C25×C3C2311yescharacteristic
41C22×C3×D4—24no—
42C22×C3×D4—24no—
43C22×C3×D4—24no—
44C22×C3×D4—22no—
45C22×C3×D4—22no—
46C22×C3×D4—24no—
47C22×C3×D4C2×C412yes—
48C22×C3×D4C2×C412yes—
49C22×C3×D4C2×C412yescomplemented
50C22×C3×D4C2×C414yescomplemented
51C22×C3×D4D414yes—
52C22×C3×D4D414yes—
53C22×C3×D4C2312yes—
54C23×C3×C4—22no—
55C23×C3×C4—22no—
56C23×C3×C4—22no—
57C23×C3×C4C2×C411yescharacteristic, complemented
58C23×C3×C4C2×C412yescomplemented
59C23×C3×C4C2311yescharacteristic
60C3×#123—28no—
61C3×#123—24no—
62C3×#123—24no—
63C3×#121—216no—
64C3×#120—24no—
65C3×#120—24no—
66C3×#120—28no—
67C3×C4×D4—232no—
68C3×#117—28no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4C2×C414yescomplemented
71C2×C3×C22⋊C4—22no—
72C2×C3×C22⋊C4—24no—
73C2×C3×C22⋊C4—22no—
74C2×C3×C22⋊C4—22no—
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4—28no—
77C2×C3×C22⋊C4—24no—
78C2×C3×C22⋊C4—24no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—24no—
81C2×C3×C22⋊C4C2×C412yescomplemented
82C2×C3×C22⋊C4C2×C414yescomplemented
83C2×C3×C22⋊C4C2×C414yescomplemented
84C2×C3×C22⋊C4D418yescomplemented
85C2×C3×C22⋊C4C2314yes—
86C2×C3×C22⋊C4C2312yes—
87C2×C3×C22⋊C4C2314yes—
88C2×C3×C42—22no—
89C3×#99—216no—
90C3×#95—24no—
91C23×D4C22×C311yescharacteristic
92C23×D4C22×C311yescharacteristic
93#534—24no—
94#534—24no—
95C2×#123C3×C412yescomplemented
96C2×#123C22×C312yes—
97C2×#121C3×C414yescomplemented
98C2×#120C3×C412yescomplemented
99C2×#120C22×C312yes—
100#518—28no—
101C2×C4×D4—24no—
102C2×#117C22×C312yescomplemented
103C22×C22⋊C4C22×C312yes—
104C22×C22⋊C4C22×C311yescharacteristic
105C22×C22⋊C4C22×C312yes—
106C22×C22⋊C4C22×C312yescomplemented
107C2×#99—28no—
108C2×#99C22×C318yes—
109#386—24no—
110#381C22×C312yescomplemented
111#379C22×C312yescomplemented
112#342C22×C318yescomplemented
113C24×C3—22no—
114C24×C3—22no—
115C24×C3—22no—
116C24×C3—24no—
117C24×C3—82no—
118C24×C3—82no—
119C24×C3—82no—
120C24×C3C22×C412yes—
121C24×C3C22×C412yes—
122C24×C3C22×C411yescharacteristic
123C24×C3C22×C411yescharacteristic
124C24×C3C2×D412yes—
125C24×C3C2×D412yes—
126C24×C3C2×D412yes—
127C24×C3C2×D412yes—
128C24×C3C2×D414yes—
129C24×C3#4114yes—
130C24×C3C2411yescharacteristic
131C2×C3×D4—28no—
132C2×C3×D4—28no—
133C2×C3×D4—28no—
134C2×C3×D4—28no—
135C2×C3×D4—28no—
136C2×C3×D4—28no—
137C2×C3×D4—24no—
138C2×C3×D4—28no—
139C2×C3×D4—24no—
140C2×C3×D4—22no—
141C2×C3×D4—22no—
142C2×C3×D4—24no—
143C2×C3×D4—28no—
144C2×C3×D4—44no—
145C2×C3×D4—44no—
146C2×C3×D4—44no—
147C2×C3×D4—44no—
148C2×C3×D4—84no—
149C2×C3×D4C22⋊C418yes—
150C2×C3×D4C22⋊C418yes—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—24no—
155C22×C3×C4—24no—
156C22×C3×C4—28no—
157C22×C3×C4—24no—
158C22×C3×C4—28no—
159C22×C3×C4—24no—
160C22×C3×C4—44no—
161C22×C3×C4—84no—
162C22×C3×C4—84no—
163C22×C3×C4—84no—
164C22×C3×C4—82no—
165C22×C3×C4C22×C412yes—
166C22×C3×C4C22×C412yes—
167C22×C3×C4C22×C414yes—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D412yes—
170C3×C4⋊C4—28no—
171C3×C4⋊C4—28no—
172C3×C22⋊C4—28no—
173C3×C22⋊C4—24no—
174C3×C22⋊C4—28no—
175C3×C22⋊C4—28no—
176C3×C22⋊C4—24no—
177C3×C22⋊C4—22no—
178C3×C22⋊C4—22no—
179C3×C22⋊C4—28no—
180C3×C22⋊C4—24no—
181C3×C22⋊C4—24no—
182C3×C22⋊C4—44no—
183C3×C22⋊C4—84no—
184C3×C22⋊C4C2×D418yes—
185C3×C42—28no—
186C25C2×C3×C411yescharacteristic, complemented
187C25C23×C311yescharacteristic
188C25C23×C311yescharacteristic
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4C2×C3×C412yes—
196C22×D4C2×C3×C412yes—
197C22×D4C2×C3×C412yescomplemented
198C22×D4C2×C3×C414yescomplemented
199C22×D4C3×D414yes—
200C22×D4C3×D414yes—
201C22×D4C23×C312yes—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4C2×C3×C411yescharacteristic, complemented
206C23×C4C2×C3×C412yescomplemented
207C23×C4C23×C311yescharacteristic
208#123—28no—
209#123—24no—
210#123—24no—
211#121—216no—
212#120—24no—
213#120—24no—
214#120—28no—
215C4×D4—232no—
216#117—28no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4C2×C3×C414yescomplemented
219C2×C22⋊C4—22no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4C2×C3×C412yescomplemented
230C2×C22⋊C4C2×C3×C414yescomplemented
231C2×C22⋊C4C2×C3×C414yescomplemented
232C2×C22⋊C4C3×D418yescomplemented
233C2×C22⋊C4C23×C314yes—
234C2×C22⋊C4C23×C312yes—
235C2×C22⋊C4C23×C314yes—
236C2×C42—22no—
237#99—216no—
238#95—24no—
239C23×C3—22no—
240C23×C3—22no—
241C23×C3—24no—
242C23×C3—24no—
243C23×C3—22no—
244C23×C3—24no—
245C23×C3—42no—
246C23×C3—44no—
247C23×C3—44no—
248C23×C3—44no—
249C23×C3—42no—
250C23×C3—42no—
251C23×C3—44no—
252C23×C3—44no—
253C23×C3—44no—
254C23×C3—44no—
255C23×C3—84no—
256C23×C3—84no—
257C23×C3—84no—
258C23×C3—84no—
259C23×C3—84no—
260C23×C3—88no—
261C23×C3C2×C22⋊C414yes—
262C23×C3C2×C22⋊C412yes—
263C23×C3C2×C22⋊C412yes—
264C23×C3C4×D414yes—
265C23×C3C4×D414yes—
266C23×C3#12014yes—
267C23×C3#12114yes—
268C23×C3#12114yes—
269C23×C3#12314yes—
270C23×C3C23×C411yescharacteristic
271C23×C3C22×D411yescharacteristic
272C23×C3C22×D411yescharacteristic
273C23×C3C22×D412yes—
274C23×C3C2×#4112yes—
275C3×D4—216no—
276C3×D4—216no—
277C3×D4—42no—
278C3×D4—44no—
279C3×D4—42no—
280C3×D4—48no—
281C3×D4—88no—
282C2×C3×C4—24no—
283C2×C3×C4—22no—
284C2×C3×C4—24no—
285C2×C3×C4—22no—
286C2×C3×C4—24no—
287C2×C3×C4—22no—
288C2×C3×C4—24no—
289C2×C3×C4—28no—
290C2×C3×C4—48no—
291C2×C3×C4—44no—
292C2×C3×C4—44no—
293C2×C3×C4—48no—
294C2×C3×C4—44no—
295C2×C3×C4—84no—
296C2×C3×C4—88no—
297C2×C3×C4—88no—
298C2×C3×C4—88no—
299C2×C3×C4C2×C22⋊C414yes—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—24no—
304C24—82no—
305C24—82no—
306C24—82no—
307C24C22×C3×C412yes—
308C24C22×C3×C412yes—
309C24C22×C3×C411yescharacteristic
310C24C22×C3×C411yescharacteristic
311C24C2×C3×D412yes—
312C24C2×C3×D412yes—
313C24C2×C3×D412yes—
314C24C2×C3×D412yes—
315C24C2×C3×D414yes—
316C24C3×#4114yes—
317C24C24×C311yescharacteristic
318C2×D4—28no—
319C2×D4—28no—
320C2×D4—28no—
321C2×D4—28no—
322C2×D4—28no—
323C2×D4—28no—
324C2×D4—24no—
325C2×D4—28no—
326C2×D4—24no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—24no—
330C2×D4—28no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—84no—
336C2×D4C3×C22⋊C418yes—
337C2×D4C3×C22⋊C418yes—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—28no—
344C22×C4—24no—
345C22×C4—28no—
346C22×C4—24no—
347C22×C4—44no—
348C22×C4—84no—
349C22×C4—84no—
350C22×C4—84no—
351C22×C4—82no—
352C22×C4C22×C3×C412yes—
353C22×C4C22×C3×C412yes—
354C22×C4C22×C3×C414yes—
355C22×C4C2×C3×D412yes—
356C22×C4C2×C3×D412yes—
357C4⋊C4—28no—
358C4⋊C4—28no—
359C22⋊C4—28no—
360C22⋊C4—24no—
361C22⋊C4—28no—
362C22⋊C4—28no—
363C22⋊C4—24no—
364C22⋊C4—22no—
365C22⋊C4—22no—
366C22⋊C4—28no—
367C22⋊C4—24no—
368C22⋊C4—24no—
369C22⋊C4—44no—
370C22⋊C4—84no—
371C22⋊C4C2×C3×D418yes—
372C42—28no—
373C22×C3—22no—
374C22×C3—22no—
375C22×C3—22no—
376C22×C3—24no—
377C22×C3—24no—
378C22×C3—28no—
379C22×C3—24no—
380C22×C3—44no—
381C22×C3—44no—
382C22×C3—44no—
383C22×C3—42no—
384C22×C3—42no—
385C22×C3—44no—
386C22×C3—44no—
387C22×C3—44no—
388C22×C3—44no—
389C22×C3—44no—
390C22×C3—44no—
391C22×C3—42no—
392C22×C3—84no—
393C22×C3—84no—
394C22×C3—84no—
395C22×C3—88no—
396C22×C3#38614yes—
397C22×C3#38614yes—
398C22×C3C22×C22⋊C411yescharacteristic
399C22×C3C2×C4×D412yes—
400C22×C3C2×#12011yescharacteristic
401C22×C3C2×#12112yes—
402C22×C3C2×#12311yescharacteristic
403C3×C4—24no—
404C3×C4—48no—
405C3×C4—44no—
406C3×C4—42no—
407C3×C4—84no—
408C23—22no—
409C23—22no—
410C23—24no—
411C23—24no—
412C23—22no—
413C23—24no—
414C23—42no—
415C23—44no—
416C23—44no—
417C23—44no—
418C23—42no—
419C23—42no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23—84no—
425C23—84no—
426C23—84no—
427C23—84no—
428C23—84no—
429C23—88no—
430C23C2×C3×C22⋊C414yes—
431C23C2×C3×C22⋊C412yes—
432C23C2×C3×C22⋊C412yes—
433C23C3×C4×D414yes—
434C23C3×C4×D414yes—
435C23C3×#12014yes—
436C23C3×#12114yes—
437C23C3×#12114yes—
438C23C3×#12314yes—
439C23C23×C3×C411yescharacteristic
440C23C22×C3×D411yescharacteristic
441C23C22×C3×D411yescharacteristic
442C23C22×C3×D412yes—
443C23C2×C3×#4112yes—
444D4—216no—
445D4—216no—
446D4—48no—
447D4—42no—
448D4—44no—
449D4—42no—
450D4—88no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—28no—
459C2×C4—48no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—48no—
463C2×C4—44no—
464C2×C4—84no—
465C2×C4—88no—
466C2×C4—88no—
467C2×C4—88no—
468C2×C4C2×C3×C22⋊C414yes—
469C2×C3—21no—
470C2×C3—22no—
471C2×C3—21no—
472C2×C3—22no—
473C2×C3—24no—
474C2×C3—24no—
475C2×C3—44no—
476C2×C3—82no—
477C2×C3C2×#38611yescharacteristic
478C2×C3#288312yes—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—24no—
483C22—24no—
484C22—28no—
485C22—24no—
486C22—44no—
487C22—44no—
488C22—44no—
489C22—42no—
490C22—42no—
491C22—44no—
492C22—44no—
493C22—44no—
494C22—44no—
495C22—44no—
496C22—44no—
497C22—42no—
498C22—84no—
499C22—84no—
500C22—84no—
501C22—88no—
502C22C3×#38614yes—
503C22C3×#38614yes—
504C22C22×C3×C22⋊C411yescharacteristic
505C22C2×C3×C4×D412yes—
506C22C2×C3×#12011yescharacteristic
507C22C2×C3×#12112yes—
508C22C2×C3×#12311yescharacteristic
509C4—24no—
510C4—48no—
511C4—44no—
512C4—42no—
513C4—84no—
514C3#2098411yescharacteristic, direct product factor, Sylow
515C2—21no—
516C2—22no—
517C2—21no—
518C2—22no—
519C2—24no—
520C2—24no—
521C2—44no—
522C2—82no—
523C2C2×C3×#38611yescharacteristic
524C2C3×#288312yes—
subgroupG/H
center, socleC22×C3C22×C22⋊C4
commutatorC23C23×C3×C4
Frattini, cosocleC24C24×C3
prime radicalC2×C3×#534C2
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×D4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#518C2semidirect
3. C22×C3×#99C2semidirect
4. C3×#2308C2semidirect
5. C3×#1899C2semidirect
6. C3×#1898C2semidirect
7. C2×C3×#342C2semidirect
8. #20984C3direct (N × H)
13. C2×C3×#123C4semidirect
15. C2×C3×#121C4semidirect
16. C2×C3×#120C4semidirect
20. C2×C3×#117C22semidirect
24. C22×C3×C22⋊C4C22semidirect
28. C3×#381C22semidirect
29. C3×#379C22semidirect
30. C3×#342C22semidirect
32. C2×#518C2×C3semidirect
33. C22×#99C2×C3semidirect
34. #2308C2×C3semidirect
35. #1899C2×C3semidirect
36. #1898C2×C3semidirect
37. C2×#342C2×C3semidirect
38. C25×C3C2×C4semidirect
49. C22×C3×D4C2×C4semidirect
50. C22×C3×D4C2×C4semidirect
57. C23×C3×C4C2×C4semidirect
58. C23×C3×C4C2×C4semidirect
70. C2×C3×C4⋊C4C2×C4semidirect
81. C2×C3×C22⋊C4C2×C4semidirect
82. C2×C3×C22⋊C4C2×C4semidirect
83. C2×C3×C22⋊C4C2×C4semidirect
84. C2×C3×C22⋊C4D4semidirect
95. C2×#123C3×C4semidirect
97. C2×#121C3×C4semidirect
98. C2×#120C3×C4semidirect
102. C2×#117C22×C3semidirect
106. C22×C22⋊C4C22×C3semidirect
110. #381C22×C3semidirect
111. #379C22×C3semidirect
112. #342C22×C3semidirect
186. C25C2×C3×C4semidirect
197. C22×D4C2×C3×C4semidirect
198. C22×D4C2×C3×C4semidirect
205. C23×C4C2×C3×C4semidirect
206. C23×C4C2×C3×C4semidirect
218. C2×C4⋊C4C2×C3×C4semidirect
229. C2×C22⋊C4C2×C3×C4semidirect
230. C2×C22⋊C4C2×C3×C4semidirect
231. C2×C22⋊C4C2×C3×C4semidirect
232. C2×C22⋊C4C3×D4semidirect
514. C3#20984direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5342
C2×C3×#5182
C22×C3×#992
C3×#23082
C3×#18992
C3×#18982
C2×C3×#3422
#209843