Finite Group Databasebeta

← C3×#23930 #10642537

Subgroups — C3×#23930 #10642537

Order 768.

1768 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#532C211yesmaximal, characteristic, complemented
2C2×C3×#447C212yesmaximal, complemented
3C2×C3×#390C211yesmaximal, characteristic, complemented
4C3×#2023C212yesmaximal, complemented
5C3×#2011C212yesmaximal, complemented
6C3×#1879C214yesmaximal, complemented
7C3×#1874C212yesmaximal, complemented
8C3×#1817C211yesmaximal, characteristic, complemented
9#23930C311yesmaximal, characteristic, direct product factor, Sylow
10C22×C3×#41—21no—
11C23×C3×D4C2211yescharacteristic
12C22×C3×D8C2214yes—
13C3×#532—28no—
14C2×C3×C4⋊Q8C2211yescharacteristic, complemented
15C2×C3×#127C2211yescharacteristic, complemented
16C2×C3×#124—21no—
17C2×C3×#121—21no—
18C2×C3×#121C2212yes—
19C2×C3×C4×D4—21no—
20C2×C3×C4×D4C2212yescomplemented
21C22×C3×C4⋊C4C2211yescharacteristic
22C22×C3×C22⋊C4—21no—
23C22×C3×C22⋊C4—21no—
24C22×C3×C42C2211yescharacteristic
25C3×#447—28no—
26C2×C3×#102C2214yes—
27C2×C3×#102C2214yescomplemented
28C2×C3×C22⋊C8C2212yes—
29C3×#390—21no—
30C3×#390—21no—
31C3×#390C2212yes—
32C3×#390C2212yes—
33C3×#340C2214yescomplemented
34C3×#336C2212yescomplemented
35C3×#336C2212yescomplemented
36C2×#532C2×C311yescharacteristic, complemented
37C2×#447C2×C312yescomplemented
38C2×#390C2×C311yescharacteristic, complemented
39#2023C2×C312yescomplemented
40#2011C2×C312yescomplemented
41#1879C2×C314yescomplemented
42#1874C2×C312yescomplemented
43#1817C2×C311yescharacteristic, complemented
44C25×C3—21no—
45C2×C3×#41—28no—
46C2×C3×#41—44no—
47C2×C3×#41—44no—
48C22×C3×Q8—21no—
49C22×C3×D4—22no—
50C22×C3×D4—24no—
51C22×C3×D4—21no—
52C22×C3×D4—21no—
53C22×C3×D4—21no—
54C22×C3×D4—24no—
55C22×C3×D4—42no—
56C22×C3×D4C2312yes—
57C22×C3×D4C2312yes—
58C23×C3×C4—22no—
59C23×C3×C4—21no—
60C23×C3×C4D411yescharacteristic
61C23×C3×C4D411yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C2×C3×D8—28no—
64C2×C3×D8—28no—
65C2×C3×D8—44no—
66C22×C3×C8C2314yes—
67C3×C4⋊Q8—24no—
68C3×#127—24no—
69C3×#124—28no—
70C3×#121—28no—
71C3×#121—24no—
72C3×#121D418yes—
73C3×C4×D4—28no—
74C3×C4×D4—24no—
75C3×C4×D4—24no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4—42no—
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4—22no—
87C2×C3×C22⋊C4—44no—
88C2×C3×C22⋊C4—44no—
89C2×C3×C42—24no—
90C2×C3×C42D411yescharacteristic
91C2×C3×C42D411yescharacteristic
92C2×C3×C42C2311yescharacteristic
93C2×C3×C42C2311yescharacteristic
94C3×#102—28no—
95C3×#102—44no—
96C3×C22⋊C8—24no—
97C22×#41—21no—
98C23×D4C22×C311yescharacteristic
99C22×D8C22×C314yes—
100#532—28no—
101C2×C4⋊Q8C22×C311yescharacteristic, complemented
102C2×#127C22×C311yescharacteristic, complemented
103C2×#124—21no—
104C2×#121—21no—
105C2×#121C22×C312yes—
106C2×C4×D4—21no—
107C2×C4×D4C22×C312yescomplemented
108C22×C4⋊C4C22×C311yescharacteristic
109C22×C22⋊C4—21no—
110C22×C22⋊C4—21no—
111C22×C42C22×C311yescharacteristic
112#447—28no—
113C2×#102C22×C314yes—
114C2×#102C22×C314yescomplemented
115C2×C22⋊C8C22×C312yes—
116#390—21no—
117#390—21no—
118#390C22×C312yes—
119#390C22×C312yes—
120#340C22×C314yescomplemented
121#336C22×C312yescomplemented
122#336C22×C312yescomplemented
123C24×C3—24no—
124C24×C3—22no—
125C24×C3—22no—
126C24×C3—42no—
127C24×C3—42no—
128C24×C3—42no—
129C24×C3—42no—
130C24×C3—42no—
131C24×C3—42no—
132C24×C3C2×D411yescharacteristic
133C3×#41—432no—
134C2×C3×Q8—24no—
135C2×C3×Q8—42no—
136C2×C3×Q8—42no—
137C2×C3×D4—24no—
138C2×C3×D4—24no—
139C2×C3×D4—24no—
140C2×C3×D4—28no—
141C2×C3×D4—24no—
142C2×C3×D4—22no—
143C2×C3×D4—24no—
144C2×C3×D4—22no—
145C2×C3×D4—24no—
146C2×C3×D4—24no—
147C2×C3×D4—416no—
148C2×C3×D4—42no—
149C2×C3×D4—42no—
150C2×C3×D4—42no—
151C2×C3×D4—44no—
152C2×C3×D4—44no—
153C2×C3×D4—42no—
154C2×C3×D4—42no—
155C2×C3×D4—42no—
156C2×C3×D4C2×D414yes—
157C2×C3×D4#4114yescomplemented
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—28no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—41no—
168C22×C3×C4—44no—
169C22×C3×C4—44no—
170C22×C3×C4—42no—
171C22×C3×C4—42no—
172C22×C3×C4—42no—
173C22×C3×C4D814yescomplemented
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D414yes—
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2411yescharacteristic
181C3×D8—416no—
182C2×C3×C8—24no—
183C2×C3×C8—24no—
184C2×C3×C8—24no—
185C3×C4⋊C4—24no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—24no—
188C3×C4⋊C4—42no—
189C3×C4⋊C4C2×D414yes—
190C3×C22⋊C4—28no—
191C3×C22⋊C4—28no—
192C3×C22⋊C4—44no—
193C3×C22⋊C4—44no—
194C3×C22⋊C4—44no—
195C3×C22⋊C4—44no—
196C3×C42—22no—
197C3×C42—22no—
198C3×C42—22no—
199C3×C42—22no—
200C25—21no—
201C2×#41—28no—
202C2×#41—44no—
203C2×#41—44no—
204C22×Q8—21no—
205C22×D4—22no—
206C22×D4—24no—
207C22×D4—21no—
208C22×D4—21no—
209C22×D4—21no—
210C22×D4—24no—
211C22×D4—42no—
212C22×D4C23×C312yes—
213C22×D4C23×C312yes—
214C23×C4—22no—
215C23×C4—21no—
216C23×C4C3×D411yescharacteristic
217C23×C4C3×D411yescharacteristic
218C23×C4C23×C311yescharacteristic
219C2×D8—28no—
220C2×D8—28no—
221C2×D8—44no—
222C22×C8C23×C314yes—
223C4⋊Q8—24no—
224#127—24no—
225#124—28no—
226#121—28no—
227#121—24no—
228#121C3×D418yes—
229C4×D4—28no—
230C4×D4—24no—
231C4×D4—24no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4C23×C312yes—
236C2×C4⋊C4C23×C312yes—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C42—24no—
246C2×C42C3×D411yescharacteristic
247C2×C42C3×D411yescharacteristic
248C2×C42C23×C311yescharacteristic
249C2×C42C23×C311yescharacteristic
250#102—28no—
251#102—44no—
252C22⋊C8—24no—
253C23×C3—21no—
254C23×C3—21no—
255C23×C3—21no—
256C23×C3—21no—
257C23×C3—24no—
258C23×C3—24no—
259C23×C3—48no—
260C23×C3—48no—
261C23×C3—48no—
262C23×C3—42no—
263C23×C3—42no—
264C23×C3—42no—
265C23×C3—42no—
266C23×C3—42no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3—42no—
272C23×C3—42no—
273C23×C3—42no—
274C23×C3—84no—
275C23×C3—84no—
276C23×C3—84no—
277C23×C3—84no—
278C23×C3#12012yes—
279C23×C3C2×D812yes—
280C23×C3AGL(1,ℤ/8)12yes—
281C23×C3C22×D411yescharacteristic
282C3×Q8—48no—
283C3×D4—28no—
284C3×D4—28no—
285C3×D4—48no—
286C3×D4—48no—
287C3×D4—48no—
288C3×D4—416no—
289C3×D4—44no—
290C3×D4—44no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—28no—
300C2×C3×C4—28no—
301C2×C3×C4—21no—
302C2×C3×C4—24no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—44no—
306C2×C3×C4—416no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—84no—
313C2×C3×C4#12012yes—
314C2×C3×C4C2×D812yes—
315C2×C3×C4C2×D812yes—
316C2×C3×C4AGL(1,ℤ/8)12yes—
317C2×C3×C4C22×D412yes—
318C2×C3×C4C2×#4112yes—
319C2×C3×C4H(5,𝔽2)12yes—
320C3×C8—44no—
321C24—24no—
322C24—22no—
323C24—22no—
324C24—42no—
325C24—42no—
326C24—42no—
327C24—42no—
328C24—42no—
329C24—42no—
330C24C2×C3×D411yescharacteristic
331#41—432no—
332C2×Q8—24no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×D4—24no—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—28no—
339C2×D4—24no—
340C2×D4—22no—
341C2×D4—24no—
342C2×D4—22no—
343C2×D4—24no—
344C2×D4—24no—
345C2×D4—416no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—44no—
350C2×D4—44no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4C2×C3×D414yes—
355C2×D4C3×#4114yescomplemented
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—28no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—41no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4C3×D814yescomplemented
372C22×C4C2×C3×D411yescharacteristic
373C22×C4C2×C3×D411yescharacteristic
374C22×C4C2×C3×D414yes—
375C22×C4C2×C3×D411yescharacteristic
376C22×C4C2×C3×D411yescharacteristic
377C22×C4C2×C3×D411yescharacteristic
378C22×C4C24×C311yescharacteristic
379D8—416no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C4⋊C4—24no—
384C4⋊C4—22no—
385C4⋊C4—24no—
386C4⋊C4—42no—
387C4⋊C4C2×C3×D414yes—
388C22⋊C4—28no—
389C22⋊C4—28no—
390C22⋊C4—44no—
391C22⋊C4—44no—
392C22⋊C4—44no—
393C22⋊C4—44no—
394C42—22no—
395C42—22no—
396C42—22no—
397C42—22no—
398C22×C3—22no—
399C22×C3—22no—
400C22×C3—22no—
401C22×C3—22no—
402C22×C3—22no—
403C22×C3—22no—
404C22×C3—44no—
405C22×C3—44no—
406C22×C3—44no—
407C22×C3—44no—
408C22×C3—44no—
409C22×C3—44no—
410C22×C3—816no—
411C22×C3—82no—
412C22×C3—82no—
413C22×C3—82no—
414C22×C3—82no—
415C22×C3—82no—
416C22×C3—82no—
417C22×C3—82no—
418C22×C3—82no—
419C22×C3#44714yes—
420C22×C3C2×#12011yescharacteristic
421C22×C3#52511yescharacteristic
422C22×C3#53511yescharacteristic
423C22×C3C22×D811yescharacteristic
424C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
425C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
426C22×C3#57511yescharacteristic
427C3×C4—22no—
428C3×C4—22no—
429C3×C4—22no—
430C3×C4—22no—
431C3×C4—44no—
432C3×C4—41no—
433C3×C4—82no—
434C3×C4#54614yes—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—24no—
440C23—24no—
441C23—48no—
442C23—48no—
443C23—48no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—84no—
457C23—84no—
458C23—84no—
459C23—84no—
460C23C3×#12012yes—
461C23C2×C3×D812yes—
462C23C3×AGL(1,ℤ/8)12yes—
463C23C22×C3×D411yescharacteristic
464Q8—48no—
465D4—28no—
466D4—28no—
467D4—48no—
468D4—48no—
469D4—48no—
470D4—416no—
471D4—44no—
472D4—44no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—28no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—28no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—24no—
487C2×C4—44no—
488C2×C4—416no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—84no—
495C2×C4C3×#12012yes—
496C2×C4C2×C3×D812yes—
497C2×C4C2×C3×D812yes—
498C2×C4C3×AGL(1,ℤ/8)12yes—
499C2×C4C22×C3×D412yes—
500C2×C4C2×C3×#4112yes—
501C2×C4C3×H(5,𝔽2)12yes—
502C8—44no—
503C2×C3—24no—
504C2×C3—84no—
505C2×C3—84no—
506C2×C3#240411yescharacteristic
507C2×C3C2×#44711yescharacteristic
508C2×C3#300611yescharacteristic
509C2×C3#310011yescharacteristic
510C2×C3#310911yescharacteristic
511C2×C3#320511yescharacteristic
512C2×C3#321311yescharacteristic
513C22—22no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—44no—
520C22—44no—
521C22—44no—
522C22—44no—
523C22—44no—
524C22—44no—
525C22—816no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22—82no—
530C22—82no—
531C22—82no—
532C22—82no—
533C22—82no—
534C22C3×#44714yes—
535C22C2×C3×#12011yescharacteristic
536C22C3×#52511yescharacteristic
537C22C3×#53511yescharacteristic
538C22C22×C3×D811yescharacteristic
539C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
540C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
541C22C3×#57511yescharacteristic
542C4—22no—
543C4—22no—
544C4—22no—
545C4—22no—
546C4—44no—
547C4—41no—
548C4—82no—
549C4C3×#54614yes—
550C3#2393011yescharacteristic, direct product factor, Sylow
551C2—24no—
552C2—84no—
553C2—84no—
554C2C3×#240411yescharacteristic
555C2C2×C3×#44711yescharacteristic
556C2C3×#300611yescharacteristic
557C2C3×#310011yescharacteristic
558C2C3×#310911yescharacteristic
559C2C3×#320511yescharacteristic
560C2C3×#321311yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#532C2semidirect
2. C2×C3×#447C2semidirect
3. C2×C3×#390C2semidirect
4. C3×#2023C2semidirect
5. C3×#2011C2semidirect
6. C3×#1879C2semidirect
7. C3×#1874C2semidirect
8. C3×#1817C2semidirect
9. #23930C3direct (N × H)
14. C2×C3×C4⋊Q8C22semidirect
15. C2×C3×#127C22semidirect
20. C2×C3×C4×D4C22semidirect
27. C2×C3×#102C22semidirect
33. C3×#340C22semidirect
34. C3×#336C22semidirect
35. C3×#336C22semidirect
36. C2×#532C2×C3semidirect
37. C2×#447C2×C3semidirect
38. C2×#390C2×C3semidirect
39. #2023C2×C3semidirect
40. #2011C2×C3semidirect
41. #1879C2×C3semidirect
42. #1874C2×C3semidirect
43. #1817C2×C3semidirect
101. C2×C4⋊Q8C22×C3semidirect
102. C2×#127C22×C3semidirect
107. C2×C4×D4C22×C3semidirect
114. C2×#102C22×C3semidirect
120. #340C22×C3semidirect
121. #336C22×C3semidirect
122. #336C22×C3semidirect
157. C2×C3×D4#41semidirect
173. C22×C3×C4D8semidirect
355. C2×D4C3×#41semidirect
371. C22×C4C3×D8semidirect
550. C3#23930direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5322
C2×C3×#4472
C2×C3×#3902
C3×#20232
C3×#20112
C3×#18792
C3×#18742
C3×#18172
#239303