Finite Group Databasebeta

← C3×#25844 #10644451

Subgroups — C3×#25844 #10644451

Order 768.

1224 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×Q8C211yesmaximal, characteristic
2C2×C3×#501C211yesmaximal, characteristic, complemented
3C2×C3×#478C211yesmaximal, characteristic
4C3×#2584C211yesmaximal, characteristic, complemented
5C3×#2058C212yesmaximal, complemented
6C3×#2057C212yesmaximal, complemented
7C3×#1987C211yesmaximal, characteristic, complemented
8C3×#1985C211yesmaximal, characteristic, complemented
9C3×#1864C212yesmaximal
10C3×#1862C212yesmaximal
11C3×#1762C211yesmaximal, characteristic
12#25844C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8—21no—
14C3×D4×Q8—24no—
15C3×D4×Q8—24no—
16C2×C3×C4⋊Q8C2212yes—
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#125C2211yescharacteristic
19C2×C3×C22⋊Q8—22no—
20C2×C3×C22⋊Q8—21no—
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×D4C2212yes—
23C2×C3×C4×D4C2211yescharacteristic
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×D4×Q8C2×C311yescharacteristic
47C2×#501C2×C311yescharacteristic, complemented
48C2×#478C2×C311yescharacteristic
49#2584C2×C311yescharacteristic, complemented
50#2058C2×C312yescomplemented
51#2057C2×C312yescomplemented
52#1987C2×C311yescharacteristic, complemented
53#1985C2×C311yescharacteristic, complemented
54#1864C2×C312yes—
55#1862C2×C312yes—
56#1762C2×C311yescharacteristic
57C22×C3×Q8—28no—
58C22×C3×Q8—21no—
59C22×C3×Q8—22no—
60C22×C3×Q8—48no—
61C22×C3×Q8D411yescharacteristic
62C22×C3×D4C2311yescharacteristic
63C23×C3×C4—22no—
64C23×C3×C4—21no—
65C22×C3×C8C2311yescharacteristic
66C22×C3×C8C2311yescharacteristic
67C22×C3×C8C2312yes—
68C3×C4⋊Q8—24no—
69C3×C4⋊Q8—24no—
70C3×C4⋊Q8—22no—
71C3×C4⋊Q8D414yes—
72C3×#125—22no—
73C3×#125—22no—
74C3×C22⋊Q8—28no—
75C3×C22⋊Q8—216no—
76C3×C4×Q8—22no—
77C3×C4×Q8—22no—
78C3×C4×D4—24no—
79C3×C4×D4—24no—
80C3×C4×D4—22no—
81C3×C4×D4D414yes—
82C2×C3×C4⋊C4—22no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—22no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—22no—
88C2×C3×C4⋊C4D412yes—
89C2×C3×C4⋊C4D411yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2312yes—
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2312yes—
95C2×C3×C22⋊C4—22no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C42—21no—
98C2×C3×C42C2312yes—
99C2×C3×C42C2311yescharacteristic
100C3×#107—22no—
101C3×#107—24no—
102C3×#106—22no—
103C3×#106—24no—
104C3×C4⋊C8—22no—
105C3×#102—22no—
106C3×#102—22no—
107C3×#102Q818yescomplemented
108C3×#97—22no—
109C3×#95—21no—
110C3×#95—22no—
111C3×#95—22no—
112C23×Q8—21no—
113D4×Q8—24no—
114D4×Q8—24no—
115C2×C4⋊Q8C22×C312yes—
116C2×C4⋊Q8C22×C311yescharacteristic
117C2×#125C22×C311yescharacteristic
118C2×C22⋊Q8—22no—
119C2×C22⋊Q8—21no—
120C2×C4×Q8C22×C311yescharacteristic
121C2×C4×D4C22×C312yes—
122C2×C4×D4C22×C311yescharacteristic
123#501—24no—
124#478—24no—
125C2×#107C22×C311yescharacteristic
126C2×#107C22×C312yes—
127C2×#106C22×C311yescharacteristic
128C2×#106C22×C312yes—
129C2×C4⋊C8C22×C311yescharacteristic
130C2×#102C22×C311yescharacteristic
131C2×#102C22×C311yescharacteristic
132C2×#102C22×C312yes—
133C2×#97C22×C311yescharacteristic
134#398—21no—
135#395—21no—
136#389—22no—
137#389C22×C312yes—
138#389C22×C312yes—
139C4×C4⋊C4C22×C311yescharacteristic
140#340C22×C312yes—
141#340C22×C312yes—
142#340C22×C312yes—
143#336C22×C312yes—
144#336C22×C312yes—
145C24×C3—21no—
146C2×C3×Q8—28no—
147C2×C3×Q8—22no—
148C2×C3×Q8—24no—
149C2×C3×Q8—24no—
150C2×C3×Q8—44no—
151C2×C3×Q8—44no—
152C2×C3×Q8—432no—
153C2×C3×Q8—44no—
154C2×C3×D4—21no—
155C2×C3×D4—22no—
156C2×C3×D4C2×D412yes—
157C2×C3×D4C2×Q814yes—
158C22×C3×C4—28no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—21no—
162C22×C3×C4—24no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—44no—
166C22×C3×C4—44no—
167C22×C3×C4—44no—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2411yescharacteristic
173C2×C3×C8—22no—
174C2×C3×C8—22no—
175C2×C3×C8—21no—
176C2×C3×C8—21no—
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C2×C3×C8C2×Q814yescomplemented
180C3×C4⋊C4—28no—
181C3×C4⋊C4—24no—
182C3×C4⋊C4—24no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4—21no—
189C3×C4⋊C4—21no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—42no—
192C3×C4⋊C4—42no—
193C3×C4⋊C4—44no—
194C3×C4⋊C4—82no—
195C3×C4⋊C4C2×D414yes—
196C3×C4⋊C4C2×D412yes—
197C3×C4⋊C4C2×Q814yescomplemented
198C3×C22⋊C4—28no—
199C3×C22⋊C4—24no—
200C3×C42—22no—
201C3×C42—22no—
202C3×C42—81no—
203C3×C42C2×D412yes—
204C3×C42C2×D412yes—
205C22×Q8—28no—
206C22×Q8—21no—
207C22×Q8—22no—
208C22×Q8—48no—
209C22×Q8C3×D411yescharacteristic
210C22×D4C23×C311yescharacteristic
211C23×C4—22no—
212C23×C4—21no—
213C22×C8C23×C311yescharacteristic
214C22×C8C23×C311yescharacteristic
215C22×C8C23×C312yes—
216C4⋊Q8—24no—
217C4⋊Q8—24no—
218C4⋊Q8—22no—
219C4⋊Q8C3×D414yes—
220#125—22no—
221#125—22no—
222C22⋊Q8—28no—
223C22⋊Q8—216no—
224C4×Q8—22no—
225C4×Q8—22no—
226C4×D4—24no—
227C4×D4—24no—
228C4×D4—22no—
229C4×D4C3×D414yes—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4C3×D412yes—
237C2×C4⋊C4C3×D411yescharacteristic
238C2×C4⋊C4C23×C311yescharacteristic
239C2×C4⋊C4C23×C311yescharacteristic
240C2×C4⋊C4C23×C312yes—
241C2×C4⋊C4C23×C311yescharacteristic
242C2×C4⋊C4C23×C312yes—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—21no—
245C2×C42—21no—
246C2×C42C23×C312yes—
247C2×C42C23×C311yescharacteristic
248#107—22no—
249#107—24no—
250#106—22no—
251#106—24no—
252C4⋊C8—22no—
253#102—22no—
254#102—22no—
255#102C3×Q818yescomplemented
256#97—22no—
257#95—21no—
258#95—22no—
259#95—22no—
260C23×C3—22no—
261C23×C3—24no—
262C23×C3—42no—
263C23×C3—41no—
264C23×C3—41no—
265C23×C3C22×D411yescharacteristic
266C3×Q8—28no—
267C3×Q8—48no—
268C3×Q8—416no—
269C3×D4—22no—
270C3×D4—24no—
271C3×D4C4⋊Q814yes—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—22no—
278C2×C3×C4—24no—
279C2×C3×C4—22no—
280C2×C3×C4—24no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—48no—
284C2×C3×C4—416no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4—42no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4#12012yes—
296C2×C3×C4#12012yes—
297C2×C3×C4AGL(1,ℤ/8)14yes—
298C2×C3×C4C22×D411yescharacteristic
299C2×C3×C4C22×D411yescharacteristic
300C2×C3×C4C22×Q812yes—
301C2×C3×C42−1+412yes—
302C3×C8—24no—
303C3×C8—41no—
304C3×C8—41no—
305C24—21no—
306C2×Q8—28no—
307C2×Q8—22no—
308C2×Q8—24no—
309C2×Q8—24no—
310C2×Q8—44no—
311C2×Q8—44no—
312C2×Q8—432no—
313C2×Q8—44no—
314C2×D4—21no—
315C2×D4—22no—
316C2×D4C2×C3×D412yes—
317C2×D4C2×C3×Q814yes—
318C22×C4—28no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—24no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4C2×C3×D412yes—
329C22×C4C2×C3×D411yescharacteristic
330C22×C4C2×C3×D411yescharacteristic
331C22×C4C2×C3×D412yes—
332C22×C4C24×C311yescharacteristic
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8C2×C3×Q814yescomplemented
340C4⋊C4—28no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—21no—
346C4⋊C4—22no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—44no—
354C4⋊C4—82no—
355C4⋊C4C2×C3×D414yes—
356C4⋊C4C2×C3×D412yes—
357C4⋊C4C2×C3×Q814yescomplemented
358C22⋊C4—28no—
359C22⋊C4—24no—
360C42—22no—
361C42—22no—
362C42—81no—
363C42C2×C3×D412yes—
364C42C2×C3×D412yes—
365C22×C3—24no—
366C22×C3—44no—
367C22×C3—44no—
368C22×C3—42no—
369C22×C3—42no—
370C22×C3C2×#12011yescharacteristic
371C22×C3D4×Q812yes—
372C22×C3C2×AGL(1,ℤ/8)12yes—
373C22×C3#57711yescharacteristic
374C22×C3#57811yescharacteristic
375C3×C4—22no—
376C3×C4—24no—
377C3×C4—22no—
378C3×C4—44no—
379C3×C4—44no—
380C3×C4—42no—
381C3×C4—42no—
382C3×C4—81no—
383C3×C4—81no—
384C3×C4C2×C4⋊Q811yescharacteristic
385C3×C4#53611yescharacteristic
386C3×C4D4×Q812yes—
387C23—22no—
388C23—24no—
389C23—42no—
390C23—41no—
391C23—41no—
392C23C22×C3×D411yescharacteristic
393Q8—28no—
394Q8—48no—
395Q8—416no—
396D4—22no—
397D4—24no—
398D4C3×C4⋊Q814yes—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—48no—
411C2×C4—416no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4C3×#12012yes—
423C2×C4C3×#12012yes—
424C2×C4C3×AGL(1,ℤ/8)14yes—
425C2×C4C22×C3×D411yescharacteristic
426C2×C4C22×C3×D411yescharacteristic
427C2×C4C22×C3×Q812yes—
428C2×C4C3×2−1+412yes—
429C8—24no—
430C8—41no—
431C8—41no—
432C2×C3—44no—
433C2×C3#243411yescharacteristic
434C2×C3#300311yescharacteristic
435C2×C3#301311yescharacteristic
436C2×C3#338412yes—
437C2×C3#338612yes—
438C22—24no—
439C22—44no—
440C22—44no—
441C22—42no—
442C22—42no—
443C22C2×C3×#12011yescharacteristic
444C22C3×D4×Q812yes—
445C22C2×C3×AGL(1,ℤ/8)12yes—
446C22C3×#57711yescharacteristic
447C22C3×#57811yescharacteristic
448C4—22no—
449C4—24no—
450C4—22no—
451C4—44no—
452C4—44no—
453C4—42no—
454C4—42no—
455C4—81no—
456C4—81no—
457C4C2×C3×C4⋊Q811yescharacteristic
458C4C3×#53611yescharacteristic
459C4C3×D4×Q812yes—
460C3#2584411yescharacteristic, direct product factor, Sylow
461C2—44no—
462C2C3×#243411yescharacteristic
463C2C3×#300311yescharacteristic
464C2C3×#301311yescharacteristic
465C2C3×#338412yes—
466C2C3×#338612yes—
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×#2058C2semidirect
6. C3×#2057C2semidirect
7. C3×#1987C2semidirect
8. C3×#1985C2semidirect
12. #25844C3direct (N × H)
47. C2×#501C2×C3semidirect
49. #2584C2×C3semidirect
50. #2058C2×C3semidirect
51. #2057C2×C3semidirect
52. #1987C2×C3semidirect
53. #1985C2×C3semidirect
107. C3×#102Q8semidirect
179. C2×C3×C8C2×Q8semidirect
197. C3×C4⋊C4C2×Q8semidirect
255. #102C3×Q8semidirect
339. C2×C8C2×C3×Q8semidirect
357. C4⋊C4C2×C3×Q8semidirect
460. C3#25844direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D4×Q82
C2×C3×#5012
C2×C3×#4782
C3×#25842
C3×#20582
C3×#20572
C3×#19872
C3×#19852
C3×#18642
C3×#18622
C3×#17622
#258443