Finite Group Databasebeta

← C3×#23931 #10642538

Subgroups — C3×#23931 #10642538

Order 768.

1552 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#532C211yesmaximal, characteristic
2C2×C3×#448C212yesmaximal, complemented
3C2×C3×#391C211yesmaximal, characteristic, complemented
4C3×#2024C212yesmaximal, complemented
5C3×#2014C212yesmaximal, complemented
6C3×#1877C214yesmaximal, complemented
7C3×#1874C212yesmaximal, complemented
8C3×#1817C211yesmaximal, characteristic, complemented
9#23931C311yesmaximal, characteristic, direct product factor, Sylow
10C22×C3×#41—21no—
11C23×C3×Q8C2211yescharacteristic
12C22×C3×SD16C2214yes—
13C3×#532—28no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×#127C2211yescharacteristic
16C2×C3×#124—21no—
17C2×C3×#121—21no—
18C2×C3×#121C2212yes—
19C2×C3×C4×D4—21no—
20C2×C3×C4×D4C2212yes—
21C22×C3×C4⋊C4C2211yescharacteristic
22C22×C3×C42C2211yescharacteristic
23C3×#448—28no—
24C2×C3×#103C2214yes—
25C2×C3×#102C2214yes—
26C2×C3×C22⋊C8C2212yes—
27C3×#391—21no—
28C3×#391—21no—
29C3×#391C2212yes—
30C3×#391C2212yes—
31C2×C3×#95—21no—
32C2×C3×#95—21no—
33C3×#340C2214yes—
34C3×#336C2212yes—
35C3×#336C2212yes—
36C2×#532C2×C311yescharacteristic
37C2×#448C2×C312yescomplemented
38C2×#391C2×C311yescharacteristic, complemented
39#2024C2×C312yescomplemented
40#2014C2×C312yescomplemented
41#1877C2×C314yescomplemented
42#1874C2×C312yescomplemented
43#1817C2×C311yescharacteristic, complemented
44C2×C3×#41—28no—
45C2×C3×#41—44no—
46C2×C3×#41—44no—
47C22×C3×Q8—24no—
48C22×C3×Q8—21no—
49C22×C3×Q8—21no—
50C22×C3×Q8—24no—
51C22×C3×Q8—42no—
52C22×C3×Q8C2312yes—
53C22×C3×D4—22no—
54C22×C3×D4—21no—
55C22×C3×D4—21no—
56C22×C3×D4C2312yes—
57C23×C3×C4—21no—
58C23×C3×C4—22no—
59C23×C3×C4—21no—
60C23×C3×C4D411yescharacteristic
61C23×C3×C4D411yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C2×C3×SD16—28no—
64C2×C3×SD16—28no—
65C2×C3×SD16—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×C42—24no—
84C2×C3×C42D411yescharacteristic
85C2×C3×C42D411yescharacteristic
86C2×C3×C42C2311yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C3×#103—28no—
89C3×#102—44no—
90C3×C22⋊C8—24no—
91C3×#95—22no—
92C3×#95—22no—
93C3×#95—22no—
94C3×#95—22no—
95C22×#41—21no—
96C23×Q8C22×C311yescharacteristic
97C22×SD16C22×C314yes—
98#532—28no—
99C2×C4⋊Q8C22×C311yescharacteristic
100C2×#127C22×C311yescharacteristic
101C2×#124—21no—
102C2×#121—21no—
103C2×#121C22×C312yes—
104C2×C4×D4—21no—
105C2×C4×D4C22×C312yes—
106C22×C4⋊C4C22×C311yescharacteristic
107C22×C42C22×C311yescharacteristic
108#448—28no—
109C2×#103C22×C314yes—
110C2×#102C22×C314yes—
111C2×C22⋊C8C22×C312yes—
112#391—21no—
113#391—21no—
114#391C22×C312yes—
115#391C22×C312yes—
116C2×#95—21no—
117C2×#95—21no—
118#340C22×C314yes—
119#336C22×C312yes—
120#336C22×C312yes—
121C24×C3—22no—
122C24×C3C2×D411yescharacteristic
123C3×#41—432no—
124C2×C3×Q8—24no—
125C2×C3×Q8—24no—
126C2×C3×Q8—22no—
127C2×C3×Q8—24no—
128C2×C3×Q8—24no—
129C2×C3×Q8—416no—
130C2×C3×Q8—42no—
131C2×C3×Q8—42no—
132C2×C3×Q8—42no—
133C2×C3×Q8—42no—
134C2×C3×Q8#4114yescomplemented
135C2×C3×D4—24no—
136C2×C3×D4—24no—
137C2×C3×D4—28no—
138C2×C3×D4—22no—
139C2×C3×D4—24no—
140C2×C3×D4—24no—
141C2×C3×D4—42no—
142C2×C3×D4—42no—
143C2×C3×D4—44no—
144C2×C3×D4—44no—
145C2×C3×D4—42no—
146C2×C3×D4—42no—
147C2×C3×D4C2×D414yes—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—28no—
153C22×C3×C4—24no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—41no—
160C22×C3×C4—44no—
161C22×C3×C4—42no—
162C22×C3×C4—42no—
163C22×C3×C4—44no—
164C22×C3×C4—42no—
165C22×C3×C4—42no—
166C22×C3×C4—42no—
167C22×C3×C4SD1614yescomplemented
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D414yes—
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2411yescharacteristic
175C3×SD16—416no—
176C2×C3×C8—24no—
177C2×C3×C8—24no—
178C2×C3×C8—24no—
179C3×C4⋊C4—24no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—24no—
182C3×C4⋊C4—42no—
183C3×C4⋊C4C2×D414yes—
184C3×C22⋊C4—28no—
185C3×C22⋊C4—28no—
186C3×C42—22no—
187C3×C42—22no—
188C3×C42—22no—
189C3×C42—22no—
190C2×#41—28no—
191C2×#41—44no—
192C2×#41—44no—
193C22×Q8—24no—
194C22×Q8—21no—
195C22×Q8—21no—
196C22×Q8—24no—
197C22×Q8—42no—
198C22×Q8C23×C312yes—
199C22×D4—22no—
200C22×D4—21no—
201C22×D4—21no—
202C22×D4C23×C312yes—
203C23×C4—21no—
204C23×C4—22no—
205C23×C4—21no—
206C23×C4C3×D411yescharacteristic
207C23×C4C3×D411yescharacteristic
208C23×C4C23×C311yescharacteristic
209C2×SD16—28no—
210C2×SD16—28no—
211C2×SD16—44no—
212C22×C8C23×C314yes—
213C4⋊Q8—24no—
214#127—24no—
215#124—28no—
216#121—28no—
217#121—24no—
218#121C3×D418yes—
219C4×D4—28no—
220C4×D4—24no—
221C4×D4—24no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4C23×C312yes—
226C2×C4⋊C4C23×C312yes—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C42—24no—
230C2×C42C3×D411yescharacteristic
231C2×C42C3×D411yescharacteristic
232C2×C42C23×C311yescharacteristic
233C2×C42C23×C311yescharacteristic
234#103—28no—
235#102—44no—
236C22⋊C8—24no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241C23×C3—21no—
242C23×C3—21no—
243C23×C3—21no—
244C23×C3—21no—
245C23×C3—24no—
246C23×C3—42no—
247C23×C3—42no—
248C23×C3—42no—
249C23×C3—42no—
250C23×C3—42no—
251C23×C3—42no—
252C23×C3#12012yes—
253C23×C3C2×SD1612yes—
254C23×C3AGL(1,ℤ/8)12yes—
255C23×C3C22×D411yescharacteristic
256C3×Q8—28no—
257C3×Q8—48no—
258C3×Q8—48no—
259C3×Q8—44no—
260C3×D4—28no—
261C3×D4—48no—
262C3×D4—48no—
263C3×D4—416no—
264C3×D4—44no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—28no—
274C2×C3×C4—28no—
275C2×C3×C4—21no—
276C2×C3×C4—24no—
277C2×C3×C4—24no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—44no—
281C2×C3×C4—416no—
282C2×C3×C4—48no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—84no—
291C2×C3×C4#12012yes—
292C2×C3×C4C2×SD1612yes—
293C2×C3×C4C2×SD1612yes—
294C2×C3×C4#13712yes—
295C2×C3×C4C22×D412yes—
296C2×C3×C4C2×#4112yes—
297C2×C3×C4H(5,𝔽2)12yes—
298C3×C8—44no—
299C24—22no—
300C24C2×C3×D411yescharacteristic
301#41—432no—
302C2×Q8—24no—
303C2×Q8—24no—
304C2×Q8—22no—
305C2×Q8—24no—
306C2×Q8—24no—
307C2×Q8—416no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8C3×#4114yescomplemented
313C2×D4—24no—
314C2×D4—24no—
315C2×D4—28no—
316C2×D4—22no—
317C2×D4—24no—
318C2×D4—24no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—44no—
322C2×D4—44no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4C2×C3×D414yes—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—28no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—41no—
338C22×C4—44no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—44no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4C3×SD1614yescomplemented
346C22×C4C2×C3×D411yescharacteristic
347C22×C4C2×C3×D411yescharacteristic
348C22×C4C2×C3×D414yes—
349C22×C4C2×C3×D411yescharacteristic
350C22×C4C2×C3×D411yescharacteristic
351C22×C4C2×C3×D411yescharacteristic
352C22×C4C24×C311yescharacteristic
353SD16—416no—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—24no—
357C4⋊C4—24no—
358C4⋊C4—22no—
359C4⋊C4—24no—
360C4⋊C4—42no—
361C4⋊C4C2×C3×D414yes—
362C22⋊C4—28no—
363C22⋊C4—28no—
364C42—22no—
365C42—22no—
366C42—22no—
367C42—22no—
368C22×C3—22no—
369C22×C3—22no—
370C22×C3—22no—
371C22×C3—22no—
372C22×C3—22no—
373C22×C3—22no—
374C22×C3—44no—
375C22×C3—44no—
376C22×C3—44no—
377C22×C3—82no—
378C22×C3—82no—
379C22×C3—82no—
380C22×C3—82no—
381C22×C3#45014yes—
382C22×C3C2×#12011yescharacteristic
383C22×C3#52511yescharacteristic
384C22×C3#53511yescharacteristic
385C22×C3C22×SD1611yescharacteristic
386C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
387C22×C3C2×#13711yescharacteristic
388C22×C3#57511yescharacteristic
389C3×C4—22no—
390C3×C4—22no—
391C3×C4—22no—
392C3×C4—22no—
393C3×C4—44no—
394C3×C4—44no—
395C3×C4—41no—
396C3×C4—82no—
397C3×C4#54614yes—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—24no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23C3×#12012yes—
410C23C2×C3×SD1612yes—
411C23C3×AGL(1,ℤ/8)12yes—
412C23C22×C3×D411yescharacteristic
413Q8—28no—
414Q8—48no—
415Q8—48no—
416Q8—44no—
417D4—28no—
418D4—48no—
419D4—48no—
420D4—416no—
421D4—44no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—28no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—28no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—44no—
438C2×C4—416no—
439C2×C4—48no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—84no—
448C2×C4C3×#12012yes—
449C2×C4C2×C3×SD1612yes—
450C2×C4C2×C3×SD1612yes—
451C2×C4C3×#13712yes—
452C2×C4C22×C3×D412yes—
453C2×C4C2×C3×#4112yes—
454C2×C4C3×H(5,𝔽2)12yes—
455C8—44no—
456C2×C3—24no—
457C2×C3—84no—
458C2×C3#240411yescharacteristic
459C2×C3C2×#45011yescharacteristic
460C2×C3#300711yescharacteristic
461C2×C3#310311yescharacteristic
462C2×C3#311211yescharacteristic
463C2×C3#320611yescharacteristic
464C2×C3#321311yescharacteristic
465C22—22no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—44no—
472C22—44no—
473C22—44no—
474C22—82no—
475C22—82no—
476C22—82no—
477C22—82no—
478C22C3×#45014yes—
479C22C2×C3×#12011yescharacteristic
480C22C3×#52511yescharacteristic
481C22C3×#53511yescharacteristic
482C22C22×C3×SD1611yescharacteristic
483C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
484C22C2×C3×#13711yescharacteristic
485C22C3×#57511yescharacteristic
486C4—22no—
487C4—22no—
488C4—22no—
489C4—22no—
490C4—44no—
491C4—44no—
492C4—41no—
493C4—82no—
494C4C3×#54614yes—
495C3#2393111yescharacteristic, direct product factor, Sylow
496C2—24no—
497C2—84no—
498C2C3×#240411yescharacteristic
499C2C2×C3×#45011yescharacteristic
500C2C3×#300711yescharacteristic
501C2C3×#310311yescharacteristic
502C2C3×#311211yescharacteristic
503C2C3×#320611yescharacteristic
504C2C3×#321311yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#532C2
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
2. C2×C3×#448C2semidirect
3. C2×C3×#391C2semidirect
4. C3×#2024C2semidirect
5. C3×#2014C2semidirect
6. C3×#1877C2semidirect
7. C3×#1874C2semidirect
8. C3×#1817C2semidirect
9. #23931C3direct (N × H)
37. C2×#448C2×C3semidirect
38. C2×#391C2×C3semidirect
39. #2024C2×C3semidirect
40. #2014C2×C3semidirect
41. #1877C2×C3semidirect
42. #1874C2×C3semidirect
43. #1817C2×C3semidirect
134. C2×C3×Q8#41semidirect
167. C22×C3×C4SD16semidirect
312. C2×Q8C3×#41semidirect
345. C22×C4C3×SD16semidirect
495. C3#23931direct (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×#4482
C2×C3×#3912
C3×#20242
C3×#20142
C3×#18772
C3×#18742
C3×#18172
#239313