Finite Group Databasebeta

← C3×#25812 #10644419

Subgroups — C3×#25812 #10644419

Order 768.

1078 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#533C211yesmaximal, characteristic
2C2×C3×#500C211yesmaximal, characteristic, complemented
3C2×C3×#498C211yesmaximal, characteristic, complemented
4C3×#2857C211yesmaximal, characteristic
5C3×#2059C214yesmaximal, complemented
6C3×#1990C211yesmaximal, characteristic, complemented
7C3×#1990C211yesmaximal, characteristic, complemented
8C3×#1854C214yesmaximal
9C3×C4×C22⋊C8C211yesmaximal, characteristic
10#25812C311yesmaximal, characteristic, direct product factor, Sylow
11C3×#533—24no—
12C3×#533—24no—
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#125C2212yes—
17C2×C3×C22⋊Q8—21no—
18C2×C3×C22⋊Q8—21no—
19C2×C3×C22⋊Q8—21no—
20C2×C3×C4×Q8—21no—
21C2×C3×C4×Q8—21no—
22C2×C3×#117C2212yes—
23C22×C3×C42C2211yescharacteristic
24C3×#500—24no—
25C3×#498—24no—
26C2×C3×#107C2214yes—
27C2×C3×#106C2214yes—
28C2×C3×C22⋊C8C2212yes—
29C2×C3×C4×C8C2211yescharacteristic
30C2×C3×C4×C8C2211yescharacteristic
31C3×#391—21no—
32C3×#389—21no—
33C3×#389C2212yes—
34C3×#384C2211yescharacteristic
35C3×#380—21no—
36C3×#380C2212yes—
37C3×#340C2214yes—
38C3×#340C2214yes—
39C3×#336C2212yes—
40C2×#533C2×C311yescharacteristic
41C2×#500C2×C311yescharacteristic, complemented
42C2×#498C2×C311yescharacteristic, complemented
43#2857C2×C311yescharacteristic
44#2059C2×C314yescomplemented
45#1990C2×C311yescharacteristic, complemented
46#1990C2×C311yescharacteristic, complemented
47#1854C2×C314yes—
48C4×C22⋊C8C2×C311yescharacteristic
49C22×C3×Q8—21no—
50C22×C3×Q8—21no—
51C22×C3×Q8—21no—
52C23×C3×C4D411yescharacteristic
53C23×C3×C4D411yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C22×C3×C8C2312yes—
56C22×C3×C8C2312yes—
57C3×C4⋊Q8—22no—
58C3×C4⋊Q8—22no—
59C3×C4⋊Q8—22no—
60C3×C4⋊Q8—22no—
61C3×C4⋊Q8D414yes—
62C3×C4⋊Q8D414yes—
63C3×#125—24no—
64C3×#125—24no—
65C3×C22⋊Q8—28no—
66C3×C22⋊Q8—28no—
67C3×C22⋊Q8—44no—
68C3×C4×Q8—28no—
69C3×C4×Q8—28no—
70C3×#117—24no—
71C3×#117—24no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C4⋊C4C2312yes—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—22no—
86C2×C3×C42—22no—
87C2×C3×C42—22no—
88C2×C3×C42—22no—
89C2×C3×C42D411yescharacteristic
90C2×C3×C42D411yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C2×C3×C42C2311yescharacteristic
93C3×#107—28no—
94C3×#106—28no—
95C3×C22⋊C8Q818yescomplemented
96C3×C4×C8—22no—
97C3×C4×C8—22no—
98C3×#95—22no—
99C3×#95—22no—
100#533—24no—
101#533—24no—
102C2×C4⋊Q8C22×C311yescharacteristic
103C2×C4⋊Q8C22×C311yescharacteristic
104C2×C4⋊Q8C22×C311yescharacteristic
105C2×#125C22×C312yes—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—21no—
109C2×C4×Q8—21no—
110C2×C4×Q8—21no—
111C2×#117C22×C312yes—
112C22×C42C22×C311yescharacteristic
113#500—24no—
114#498—24no—
115C2×#107C22×C314yes—
116C2×#106C22×C314yes—
117C2×C22⋊C8C22×C312yes—
118C2×C4×C8C22×C311yescharacteristic
119C2×C4×C8C22×C311yescharacteristic
120#391—21no—
121#389—21no—
122#389C22×C312yes—
123#384C22×C311yescharacteristic
124#380—21no—
125#380C22×C312yes—
126#340C22×C314yes—
127#340C22×C314yes—
128#336C22×C312yes—
129C24×C3C2×D411yescharacteristic
130C2×C3×Q8—24no—
131C2×C3×Q8—24no—
132C2×C3×Q8—42no—
133C2×C3×Q8—42no—
134C2×C3×Q8—42no—
135C2×C3×Q8—42no—
136C2×C3×Q8—42no—
137C2×C3×Q8—42no—
138C2×C3×Q8—42no—
139C22×C3×C4—21no—
140C22×C3×C4—21no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×Q814yes—
158C22×C3×C4C2411yescharacteristic
159C2×C3×C8—22no—
160C2×C3×C8—22no—
161C2×C3×C8—22no—
162C2×C3×C8—22no—
163C2×C3×C8C2×Q814yes—
164C2×C3×C8C2×Q814yes—
165C3×C4⋊C4—28no—
166C3×C4⋊C4—28no—
167C3×C4⋊C4—24no—
168C3×C4⋊C4—24no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—24no—
174C3×C4⋊C4—44no—
175C3×C4⋊C4—42no—
176C3×C4⋊C4—42no—
177C3×C4⋊C4—42no—
178C3×C4⋊C4C2×D414yes—
179C3×C4⋊C4C2×D414yes—
180C3×C22⋊C4—28no—
181C3×C22⋊C4—44no—
182C3×C42—28no—
183C3×C42—22no—
184C3×C42—22no—
185C3×C42—21no—
186C3×C42—21no—
187C3×C42C2×D412yes—
188C3×C42C2×D412yes—
189C22×Q8—21no—
190C22×Q8—21no—
191C22×Q8—21no—
192C23×C4C3×D411yescharacteristic
193C23×C4C3×D411yescharacteristic
194C23×C4C23×C311yescharacteristic
195C22×C8C23×C312yes—
196C22×C8C23×C312yes—
197C4⋊Q8—22no—
198C4⋊Q8—22no—
199C4⋊Q8—22no—
200C4⋊Q8—22no—
201C4⋊Q8C3×D414yes—
202C4⋊Q8C3×D414yes—
203#125—24no—
204#125—24no—
205C22⋊Q8—28no—
206C22⋊Q8—28no—
207C22⋊Q8—44no—
208C4×Q8—28no—
209C4×Q8—28no—
210#117—24no—
211#117—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4C23×C312yes—
221C2×C4⋊C4C23×C312yes—
222C2×C4⋊C4C23×C312yes—
223C2×C4⋊C4C23×C312yes—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42C3×D411yescharacteristic
230C2×C42C3×D411yescharacteristic
231C2×C42C23×C311yescharacteristic
232C2×C42C23×C311yescharacteristic
233#107—28no—
234#106—28no—
235C22⋊C8C3×Q818yescomplemented
236C4×C8—22no—
237C4×C8—22no—
238#95—22no—
239#95—22no—
240C23×C3—21no—
241C23×C3—21no—
242C23×C3—21no—
243C23×C3—21no—
244C23×C3—21no—
245C23×C3—21no—
246C23×C3C4⋊Q812yes—
247C23×C3C22×D411yescharacteristic
248C3×Q8—48no—
249C3×Q8—48no—
250C3×Q8—84no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—24no—
254C2×C3×C4—24no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—24no—
258C2×C3×C4—24no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—42no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—44no—
268C2×C3×C4—44no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4#12012yes—
276C2×C3×C4#12012yes—
277C2×C3×C4C4⋊Q812yes—
278C2×C3×C4#13512yes—
279C2×C3×C4#13512yes—
280C2×C3×C4AGL(1,ℤ/8)12yes—
281C2×C3×C4#13712yes—
282C2×C3×C4C22×D411yescharacteristic
283C2×C3×C4C22×D411yescharacteristic
284C2×C3×C4C22×Q812yes—
285C2×C3×C42−1+412yes—
286C3×C8—24no—
287C3×C8—24no—
288C24C2×C3×D411yescharacteristic
289C2×Q8—24no—
290C2×Q8—24no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4C2×C3×D411yescharacteristic
312C22×C4C2×C3×D411yescharacteristic
313C22×C4C2×C3×D411yescharacteristic
314C22×C4C2×C3×D411yescharacteristic
315C22×C4C2×C3×D411yescharacteristic
316C22×C4C2×C3×Q814yes—
317C22×C4C24×C311yescharacteristic
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8C2×C3×Q814yes—
323C2×C8C2×C3×Q814yes—
324C4⋊C4—28no—
325C4⋊C4—28no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—24no—
333C4⋊C4—44no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4C2×C3×D414yes—
338C4⋊C4C2×C3×D414yes—
339C22⋊C4—28no—
340C22⋊C4—44no—
341C42—28no—
342C42—22no—
343C42—22no—
344C42—21no—
345C42—21no—
346C42C2×C3×D412yes—
347C42C2×C3×D412yes—
348C22×C3—22no—
349C22×C3—22no—
350C22×C3—22no—
351C22×C3—41no—
352C22×C3—41no—
353C22×C3—41no—
354C22×C3—41no—
355C22×C3C2×#12011yescharacteristic
356C22×C3C2×C4⋊Q811yescharacteristic
357C22×C3#53611yescharacteristic
358C22×C3C2×#13511yescharacteristic
359C22×C3C2×#13511yescharacteristic
360C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
361C22×C3C2×#13711yescharacteristic
362C3×C4—22no—
363C3×C4—41no—
364C3×C4—44no—
365C3×C4—41no—
366C3×C4—44no—
367C3×C4—82no—
368C3×C4—82no—
369C3×C4#44914yes—
370C3×C4#45214yes—
371C3×C4D4×Q812yes—
372C3×C4D4×Q812yes—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23C3×C4⋊Q812yes—
380C23C22×C3×D411yescharacteristic
381Q8—48no—
382Q8—48no—
383Q8—84no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4C3×#12012yes—
409C2×C4C3×#12012yes—
410C2×C4C3×C4⋊Q812yes—
411C2×C4C3×#13512yes—
412C2×C4C3×#13512yes—
413C2×C4C3×AGL(1,ℤ/8)12yes—
414C2×C4C3×#13712yes—
415C2×C4C22×C3×D411yescharacteristic
416C2×C4C22×C3×D411yescharacteristic
417C2×C4C22×C3×Q812yes—
418C2×C4C3×2−1+412yes—
419C8—24no—
420C8—24no—
421C2×C3—42no—
422C2×C3#243411yescharacteristic
423C2×C3C2×#44911yescharacteristic
424C2×C3C2×#45211yescharacteristic
425C2×C3#316011yescharacteristic
426C2×C3#316511yescharacteristic
427C2×C3#322811yescharacteristic
428C2×C3#323111yescharacteristic
429C22—22no—
430C22—22no—
431C22—22no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22C2×C3×#12011yescharacteristic
437C22C2×C3×C4⋊Q811yescharacteristic
438C22C3×#53611yescharacteristic
439C22C2×C3×#13511yescharacteristic
440C22C2×C3×#13511yescharacteristic
441C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
442C22C2×C3×#13711yescharacteristic
443C4—22no—
444C4—41no—
445C4—44no—
446C4—41no—
447C4—44no—
448C4—82no—
449C4—82no—
450C4C3×#44914yes—
451C4C3×#45214yes—
452C4C3×D4×Q812yes—
453C4C3×D4×Q812yes—
454C3#2581211yescharacteristic, direct product factor, Sylow
455C2—42no—
456C2C3×#243411yescharacteristic
457C2C2×C3×#44911yescharacteristic
458C2C2×C3×#45211yescharacteristic
459C2C3×#316011yescharacteristic
460C2C3×#316511yescharacteristic
461C2C3×#322811yescharacteristic
462C2C3×#323111yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
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×#500C2semidirect
3. C2×C3×#498C2semidirect
5. C3×#2059C2semidirect
6. C3×#1990C2semidirect
7. C3×#1990C2semidirect
10. #25812C3direct (N × H)
41. C2×#500C2×C3semidirect
42. C2×#498C2×C3semidirect
44. #2059C2×C3semidirect
45. #1990C2×C3semidirect
46. #1990C2×C3semidirect
95. C3×C22⋊C8Q8semidirect
235. C22⋊C8C3×Q8semidirect
454. C3#25812direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5332
C2×C3×#5002
C2×C3×#4982
C3×#28572
C3×#20592
C3×#19902
C3×#18542
C3×C4×C22⋊C82
#258123