Finite Group Databasebeta

← C3×#25845 #10644452

Subgroups — C3×#25845 #10644452

Order 768.

1040 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#554C211yesmaximal, characteristic
2C2×C3×#501C211yesmaximal, characteristic, complemented
3C2×C3×#478C211yesmaximal, characteristic
4C3×#2584C211yesmaximal, characteristic, complemented
5C3×#2059C212yesmaximal, complemented
6C3×#2059C212yesmaximal, complemented
7C3×#1988C211yesmaximal, characteristic, complemented
8C3×#1985C211yesmaximal, characteristic, complemented
9C3×#1864C212yesmaximal
10C3×#1862C212yesmaximal
11C3×#1762C211yesmaximal, characteristic
12#25845C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#554—24no—
14C3×#554—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#125C2211yescharacteristic
17C2×C3×#125C2212yes—
18C2×C3×C22⋊Q8—21no—
19C2×C3×C22⋊Q8—22no—
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×D4C2212yes—
22C2×C3×C4×D4C2211yescharacteristic
23C22×C3×C4⋊C4—21no—
24C3×#501—24no—
25C3×#478—24no—
26C2×C3×#107C2212yes—
27C2×C3×#107C2211yescharacteristic
28C2×C3×#106C2212yes—
29C2×C3×#106C2211yescharacteristic
30C2×C3×C4⋊C8C2211yescharacteristic
31C2×C3×#102C2211yescharacteristic
32C2×C3×#102C2211yescharacteristic
33C2×C3×#102C2212yes—
34C2×C3×#97C2211yescharacteristic
35C3×#398—21no—
36C3×#395—21no—
37C3×#389—22no—
38C3×#389C2212yes—
39C3×#389C2212yes—
40C3×C4×C4⋊C4C2211yescharacteristic
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#340C2212yes—
44C3×#336C2212yes—
45C3×#336C2212yes—
46C2×#554C2×C311yescharacteristic
47C2×#501C2×C311yescharacteristic, complemented
48C2×#478C2×C311yescharacteristic
49#2584C2×C311yescharacteristic, complemented
50#2059C2×C312yescomplemented
51#2059C2×C312yescomplemented
52#1988C2×C311yescharacteristic, complemented
53#1985C2×C311yescharacteristic, complemented
54#1864C2×C312yes—
55#1862C2×C312yes—
56#1762C2×C311yescharacteristic
57C22×C3×Q8—21no—
58C22×C3×Q8D411yescharacteristic
59C22×C3×D4C2311yescharacteristic
60C23×C3×C4—21no—
61C23×C3×C4—22no—
62C22×C3×C8C2311yescharacteristic
63C22×C3×C8C2311yescharacteristic
64C22×C3×C8C2312yes—
65C3×C4⋊Q8—22no—
66C3×C4⋊Q8D414yes—
67C3×#125—24no—
68C3×#125—24no—
69C3×#125—22no—
70C3×#125—22no—
71C3×C22⋊Q8—28no—
72C3×C22⋊Q8—216no—
73C3×C4×Q8—22no—
74C3×C4×Q8—22no—
75C3×C4×D4—24no—
76C3×C4×D4—24no—
77C3×C4×D4—22no—
78C3×C4×D4D414yes—
79C2×C3×C4⋊C4—28no—
80C2×C3×C4⋊C4—22no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—22no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—22no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—22no—
87C2×C3×C4⋊C4D412yes—
88C2×C3×C4⋊C4D411yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2312yes—
93C2×C3×C4⋊C4C2312yes—
94C2×C3×C22⋊C4—22no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C42—21no—
97C2×C3×C42C2312yes—
98C2×C3×C42C2311yescharacteristic
99C3×#107—22no—
100C3×#107—24no—
101C3×#106—22no—
102C3×#106—24no—
103C3×C4⋊C8—22no—
104C3×#102—22no—
105C3×#102—22no—
106C3×#102Q818yescomplemented
107C3×#97—22no—
108C3×#95—21no—
109C3×#95—22no—
110C3×#95—22no—
111#554—24no—
112#554—24no—
113C2×C4⋊Q8C22×C311yescharacteristic
114C2×#125C22×C311yescharacteristic
115C2×#125C22×C312yes—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8—22no—
118C2×C4×Q8C22×C311yescharacteristic
119C2×C4×D4C22×C312yes—
120C2×C4×D4C22×C311yescharacteristic
121C22×C4⋊C4—21no—
122#501—24no—
123#478—24no—
124C2×#107C22×C311yescharacteristic
125C2×#107C22×C312yes—
126C2×#106C22×C311yescharacteristic
127C2×#106C22×C312yes—
128C2×C4⋊C8C22×C311yescharacteristic
129C2×#102C22×C311yescharacteristic
130C2×#102C22×C311yescharacteristic
131C2×#102C22×C312yes—
132C2×#97C22×C311yescharacteristic
133#398—21no—
134#395—21no—
135#389—22no—
136#389C22×C312yes—
137#389C22×C312yes—
138C4×C4⋊C4C22×C311yescharacteristic
139#340C22×C312yes—
140#340C22×C312yes—
141#340C22×C312yes—
142#336C22×C312yes—
143#336C22×C312yes—
144C24×C3—21no—
145C2×C3×Q8—22no—
146C2×C3×Q8—24no—
147C2×C3×Q8—42no—
148C2×C3×Q8—81no—
149C2×C3×Q8—81no—
150C2×C3×D4—21no—
151C2×C3×D4—22no—
152C2×C3×D4C2×D412yes—
153C2×C3×D4C2×Q814yes—
154C22×C3×C4—22no—
155C22×C3×C4—21no—
156C22×C3×C4—22no—
157C22×C3×C4—28no—
158C22×C3×C4—24no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—44no—
162C22×C3×C4—82no—
163C22×C3×C4—82no—
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D412yes—
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2411yescharacteristic
169C2×C3×C8—22no—
170C2×C3×C8—22no—
171C2×C3×C8—21no—
172C2×C3×C8—21no—
173C2×C3×C8—22no—
174C2×C3×C8—22no—
175C2×C3×C8C2×Q814yes—
176C3×C4⋊C4—28no—
177C3×C4⋊C4—28no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—24no—
180C3×C4⋊C4—24no—
181C3×C4⋊C4—21no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—21no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4—44no—
189C3×C4⋊C4—42no—
190C3×C4⋊C4—42no—
191C3×C4⋊C4—82no—
192C3×C4⋊C4C2×D414yes—
193C3×C4⋊C4C2×D412yes—
194C3×C4⋊C4C2×Q814yes—
195C3×C22⋊C4—28no—
196C3×C22⋊C4—24no—
197C3×C42—22no—
198C3×C42—22no—
199C3×C42—81no—
200C3×C42C2×D412yes—
201C3×C42C2×D412yes—
202C22×Q8—21no—
203C22×Q8C3×D411yescharacteristic
204C22×D4C23×C311yescharacteristic
205C23×C4—21no—
206C23×C4—22no—
207C22×C8C23×C311yescharacteristic
208C22×C8C23×C311yescharacteristic
209C22×C8C23×C312yes—
210C4⋊Q8—22no—
211C4⋊Q8C3×D414yes—
212#125—24no—
213#125—24no—
214#125—22no—
215#125—22no—
216C22⋊Q8—28no—
217C22⋊Q8—216no—
218C4×Q8—22no—
219C4×Q8—22no—
220C4×D4—24no—
221C4×D4—24no—
222C4×D4—22no—
223C4×D4C3×D414yes—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4C3×D411yescharacteristic
233C2×C4⋊C4C3×D412yes—
234C2×C4⋊C4C23×C311yescharacteristic
235C2×C4⋊C4C23×C311yescharacteristic
236C2×C4⋊C4C23×C311yescharacteristic
237C2×C4⋊C4C23×C312yes—
238C2×C4⋊C4C23×C312yes—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—21no—
241C2×C42—21no—
242C2×C42C23×C312yes—
243C2×C42C23×C311yescharacteristic
244#107—22no—
245#107—24no—
246#106—22no—
247#106—24no—
248C4⋊C8—22no—
249#102—22no—
250#102—22no—
251#102C3×Q818yescomplemented
252#97—22no—
253#95—21no—
254#95—22no—
255#95—22no—
256C23×C3—22no—
257C23×C3—24no—
258C23×C3—42no—
259C23×C3—41no—
260C23×C3—41no—
261C23×C3C22×D411yescharacteristic
262C3×Q8—44no—
263C3×Q8—84no—
264C3×D4—22no—
265C3×D4—24no—
266C3×D4C4⋊Q814yes—
267C2×C3×C4—22no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—24no—
274C2×C3×C4—24no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—48no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—42no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—88no—
291C2×C3×C4#12012yes—
292C2×C3×C4#12012yes—
293C2×C3×C4C22×D411yescharacteristic
294C2×C3×C4C22×D411yescharacteristic
295C2×C3×C4C22×Q812yes—
296C2×C3×C42−1+412yes—
297C3×C8—24no—
298C3×C8—41no—
299C3×C8—41no—
300C24—21no—
301C2×Q8—22no—
302C2×Q8—24no—
303C2×Q8—42no—
304C2×Q8—81no—
305C2×Q8—81no—
306C2×D4—21no—
307C2×D4—22no—
308C2×D4C2×C3×D412yes—
309C2×D4C2×C3×Q814yes—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—28no—
314C22×C4—24no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—44no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4C2×C3×D412yes—
321C22×C4C2×C3×D412yes—
322C22×C4C2×C3×D411yescharacteristic
323C22×C4C2×C3×D411yescharacteristic
324C22×C4C24×C311yescharacteristic
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8C2×C3×Q814yes—
332C4⋊C4—28no—
333C4⋊C4—28no—
334C4⋊C4—22no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—21no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—44no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—82no—
348C4⋊C4C2×C3×D414yes—
349C4⋊C4C2×C3×D412yes—
350C4⋊C4C2×C3×Q814yes—
351C22⋊C4—28no—
352C22⋊C4—24no—
353C42—22no—
354C42—22no—
355C42—81no—
356C42C2×C3×D412yes—
357C42C2×C3×D412yes—
358C22×C3—24no—
359C22×C3—44no—
360C22×C3—44no—
361C22×C3—42no—
362C22×C3—42no—
363C22×C3C2×#12011yescharacteristic
364C22×C3D4×Q812yes—
365C22×C3#57512yes—
366C22×C3#57611yescharacteristic
367C22×C3#57711yescharacteristic
368C3×C4—22no—
369C3×C4—22no—
370C3×C4—44no—
371C3×C4—42no—
372C3×C4—42no—
373C3×C4—42no—
374C3×C4—44no—
375C3×C4—81no—
376C3×C4—81no—
377C3×C4C2×C4⋊Q811yescharacteristic
378C3×C4#53611yescharacteristic
379C3×C4D4×Q812yes—
380C23—22no—
381C23—24no—
382C23—42no—
383C23—41no—
384C23—41no—
385C23C22×C3×D411yescharacteristic
386Q8—44no—
387Q8—84no—
388D4—22no—
389D4—24no—
390D4C3×C4⋊Q814yes—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—48no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—88no—
415C2×C4C3×#12012yes—
416C2×C4C3×#12012yes—
417C2×C4C22×C3×D411yescharacteristic
418C2×C4C22×C3×D411yescharacteristic
419C2×C4C22×C3×Q812yes—
420C2×C4C3×2−1+412yes—
421C8—24no—
422C8—41no—
423C8—41no—
424C2×C3—44no—
425C2×C3#243411yescharacteristic
426C2×C3#300511yescharacteristic
427C2×C3#301211yescharacteristic
428C2×C3#338812yes—
429C2×C3#338912yes—
430C22—24no—
431C22—44no—
432C22—44no—
433C22—42no—
434C22—42no—
435C22C2×C3×#12011yescharacteristic
436C22C3×D4×Q812yes—
437C22C3×#57512yes—
438C22C3×#57611yescharacteristic
439C22C3×#57711yescharacteristic
440C4—22no—
441C4—22no—
442C4—44no—
443C4—42no—
444C4—42no—
445C4—42no—
446C4—44no—
447C4—81no—
448C4—81no—
449C4C2×C3×C4⋊Q811yescharacteristic
450C4C3×#53611yescharacteristic
451C4C3×D4×Q812yes—
452C3#2584511yescharacteristic, direct product factor, Sylow
453C2—44no—
454C2C3×#243411yescharacteristic
455C2C3×#300511yescharacteristic
456C2C3×#301211yescharacteristic
457C2C3×#338812yes—
458C2C3×#338912yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#501C2semidirect
4. C3×#2584C2semidirect
5. C3×#2059C2semidirect
6. C3×#2059C2semidirect
7. C3×#1988C2semidirect
8. C3×#1985C2semidirect
12. #25845C3direct (N × H)
47. C2×#501C2×C3semidirect
49. #2584C2×C3semidirect
50. #2059C2×C3semidirect
51. #2059C2×C3semidirect
52. #1988C2×C3semidirect
53. #1985C2×C3semidirect
106. C3×#102Q8semidirect
251. #102C3×Q8semidirect
452. C3#25845direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5542
C2×C3×#5012
C2×C3×#4782
C3×#25842
C3×#20592
C3×#19882
C3×#19852
C3×#18642
C3×#18622
C3×#17622
#258453