Finite Group Databasebeta

← C3×#25700 #10644307

Subgroups — C3×#25700 #10644307

Order 768.

1122 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#555C211yesmaximal, characteristic
2C2×C3×#498C211yesmaximal, characteristic, complemented
3C2×C3×#474C211yesmaximal, characteristic, complemented
4C3×#2547C211yesmaximal, characteristic, complemented
5C3×#2034C214yesmaximal, complemented
6C3×#1987C211yesmaximal, characteristic, complemented
7C3×#1954C211yesmaximal, characteristic, complemented
8C3×#1854C214yesmaximal, complemented
9C3×C4×#102C211yesmaximal, characteristic, complemented
10#25700C311yesmaximal, characteristic, direct product factor, Sylow
11C3×#555—24no—
12C3×#555—24no—
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×#124—21no—
16C2×C3×#124—22no—
17C2×C3×#121—21no—
18C2×C3×#121C2212yes—
19C2×C3×C4×D4C2211yescharacteristic
20C2×C3×C4×D4C2211yescharacteristic
21C2×C3×#117—21no—
22C2×C3×#117C2212yes—
23C3×#498—24no—
24C3×#474—24no—
25C2×C3×#107C2212yes—
26C2×C3×#106C2214yes—
27C2×C3×C4⋊C8C2211yescharacteristic
28C2×C3×#102C2212yes—
29C2×C3×#102C2212yes—
30C2×C3×C22⋊C8C2214yes—
31C2×C3×C4×C8C2211yescharacteristic
32C3×#397—22no—
33C3×#391—21no—
34C3×#380C2214yes—
35C3×C4×C4⋊C4C2211yescharacteristic
36C3×#340C2212yes—
37C3×#340C2214yes—
38C2×#555C2×C311yescharacteristic
39C2×#498C2×C311yescharacteristic, complemented
40C2×#474C2×C311yescharacteristic, complemented
41#2547C2×C311yescharacteristic, complemented
42#2034C2×C314yescomplemented
43#1987C2×C311yescharacteristic, complemented
44#1954C2×C311yescharacteristic, complemented
45#1854C2×C314yescomplemented
46C4×#102C2×C311yescharacteristic, complemented
47C22×C3×Q8—21no—
48C22×C3×Q8—21no—
49C22×C3×D4—22no—
50C22×C3×D4D411yescharacteristic
51C22×C3×D4C2311yescharacteristic
52C23×C3×C4—21no—
53C23×C3×C4C2312yes—
54C22×C3×C8C2312yes—
55C22×C3×C8C2312yes—
56C3×C4⋊Q8—22no—
57C3×C4⋊Q8—22no—
58C3×C4⋊Q8—22no—
59C3×C4⋊Q8D414yes—
60C3×#124—216no—
61C3×#124—44no—
62C3×#121—28no—
63C3×#121—24no—
64C3×#121—24no—
65C3×C4×D4—22no—
66C3×C4×D4—22no—
67C3×C4×D4—22no—
68C3×C4×D4D414yes—
69C3×#117—28no—
70C3×#117—24no—
71C3×#117—24no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4D411yescharacteristic
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—21no—
82C2×C3×C22⋊C4—24no—
83C2×C3×C22⋊C4D412yes—
84C2×C3×C42—22no—
85C2×C3×C42—21no—
86C2×C3×C42C2311yescharacteristic
87C3×#107—24no—
88C3×#106—28no—
89C3×C4⋊C8—22no—
90C3×#102—24no—
91C3×#102—24no—
92C3×C22⋊C8—28no—
93C3×C4×C8—22no—
94C3×#95—21no—
95C3×#95—24no—
96#555—24no—
97#555—24no—
98C2×C4⋊Q8C22×C311yescharacteristic
99C2×C4⋊Q8C22×C311yescharacteristic
100C2×#124—21no—
101C2×#124—22no—
102C2×#121—21no—
103C2×#121C22×C312yes—
104C2×C4×D4C22×C311yescharacteristic
105C2×C4×D4C22×C311yescharacteristic
106C2×#117—21no—
107C2×#117C22×C312yes—
108#498—24no—
109#474—24no—
110C2×#107C22×C312yes—
111C2×#106C22×C314yes—
112C2×C4⋊C8C22×C311yescharacteristic
113C2×#102C22×C312yes—
114C2×#102C22×C312yes—
115C2×C22⋊C8C22×C314yes—
116C2×C4×C8C22×C311yescharacteristic
117#397—22no—
118#391—21no—
119#380C22×C314yes—
120C4×C4⋊C4C22×C311yescharacteristic
121#340C22×C312yes—
122#340C22×C314yes—
123C24×C3—21no—
124C24×C3C2×D412yes—
125C2×C3×Q8—24no—
126C2×C3×Q8—42no—
127C2×C3×Q8—42no—
128C2×C3×Q8—42no—
129C2×C3×Q8—82no—
130C2×C3×D4—28no—
131C2×C3×D4—22no—
132C2×C3×D4—22no—
133C2×C3×D4—22no—
134C2×C3×D4—21no—
135C2×C3×D4—22no—
136C2×C3×D4—22no—
137C2×C3×D4—84no—
138C2×C3×D4C2×D412yes—
139C22×C3×C4—21no—
140C22×C3×C4—22no—
141C22×C3×C4—24no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—24no—
145C22×C3×C4—24no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—42no—
149C22×C3×C4—42no—
150C22×C3×C4C2×D412yes—
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2411yescharacteristic
154C2×C3×C8—22no—
155C2×C3×C8—22no—
156C2×C3×C8—22no—
157C2×C3×C8—24no—
158C2×C3×C8#4114yes—
159C3×C4⋊C4—24no—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—21no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—22no—
164C3×C4⋊C4—22no—
165C3×C4⋊C4—22no—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—42no—
168C3×C4⋊C4C2×D414yes—
169C3×C4⋊C4C2×D412yes—
170C3×C22⋊C4—28no—
171C3×C22⋊C4—28no—
172C3×C22⋊C4—24no—
173C3×C22⋊C4—24no—
174C3×C22⋊C4—48no—
175C3×C42—28no—
176C3×C42—42no—
177C3×C42C2×D412yes—
178C3×C42C2×D412yes—
179C22×Q8—21no—
180C22×Q8—21no—
181C22×D4—22no—
182C22×D4C3×D411yescharacteristic
183C22×D4C23×C311yescharacteristic
184C23×C4—21no—
185C23×C4C23×C312yes—
186C22×C8C23×C312yes—
187C22×C8C23×C312yes—
188C4⋊Q8—22no—
189C4⋊Q8—22no—
190C4⋊Q8—22no—
191C4⋊Q8C3×D414yes—
192#124—216no—
193#124—44no—
194#121—28no—
195#121—24no—
196#121—24no—
197C4×D4—22no—
198C4×D4—22no—
199C4×D4—22no—
200C4×D4C3×D414yes—
201#117—28no—
202#117—24no—
203#117—24no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4C3×D411yescharacteristic
207C2×C4⋊C4C23×C312yes—
208C2×C4⋊C4C23×C312yes—
209C2×C4⋊C4C23×C311yescharacteristic
210C2×C4⋊C4C23×C312yes—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4C3×D412yes—
216C2×C42—22no—
217C2×C42—21no—
218C2×C42C23×C311yescharacteristic
219#107—24no—
220#106—28no—
221C4⋊C8—22no—
222#102—24no—
223#102—24no—
224C22⋊C8—28no—
225C4×C8—22no—
226#95—21no—
227#95—24no—
228C23×C3—22no—
229C23×C3—22no—
230C23×C3—24no—
231C23×C3—22no—
232C23×C3—22no—
233C23×C3—24no—
234C23×C3—42no—
235C23×C3—41no—
236C23×C3—41no—
237C23×C3—42no—
238C23×C3C22×D411yescharacteristic
239C3×Q8—84no—
240C3×Q8—84no—
241C3×D4—22no—
242C3×D4—22no—
243C3×D4—42no—
244C3×D4—44no—
245C3×D4—88no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—24no—
250C2×C3×C4—24no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—44no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—44no—
257C2×C3×C4—48no—
258C2×C3×C4—42no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4#12012yes—
265C2×C3×C4#12012yes—
266C2×C3×C4#13512yes—
267C2×C3×C4AGL(1,ℤ/8)12yes—
268C2×C3×C4C22×D411yescharacteristic
269C2×C3×C4C22×D411yescharacteristic
270C2×C3×C4C2×#4112yes—
271C2×C3×C42−1+412yes—
272C3×C8—24no—
273C3×C8—42no—
274C24—21no—
275C24C2×C3×D412yes—
276C2×Q8—24no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—82no—
281C2×D4—28no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—21no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—84no—
289C2×D4C2×C3×D412yes—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4C2×C3×D412yes—
302C22×C4C2×C3×D411yescharacteristic
303C22×C4C2×C3×D411yescharacteristic
304C22×C4C24×C311yescharacteristic
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—24no—
309C2×C8C3×#4114yes—
310C4⋊C4—24no—
311C4⋊C4—22no—
312C4⋊C4—21no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—42no—
319C4⋊C4C2×C3×D414yes—
320C4⋊C4C2×C3×D412yes—
321C22⋊C4—28no—
322C22⋊C4—28no—
323C22⋊C4—24no—
324C22⋊C4—24no—
325C22⋊C4—48no—
326C42—28no—
327C42—42no—
328C42C2×C3×D412yes—
329C42C2×C3×D412yes—
330C22×C3—44no—
331C22×C3—44no—
332C22×C3—42no—
333C22×C3—42no—
334C22×C3—42no—
335C22×C3—42no—
336C22×C3—42no—
337C22×C3—42no—
338C22×C3—82no—
339C22×C3—82no—
340C22×C3C2×#12011yescharacteristic
341C22×C3#54712yes—
342C22×C3C2×#13511yescharacteristic
343C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
344C22×C3#57712yes—
345C3×C4—22no—
346C3×C4—42no—
347C3×C4—42no—
348C3×C4—44no—
349C3×C4—82no—
350C3×C4—82no—
351C3×C4#44914yes—
352C3×C4#53211yescharacteristic
353C3×C4#53611yescharacteristic
354C3×C4#54812yes—
355C23—22no—
356C23—22no—
357C23—24no—
358C23—22no—
359C23—22no—
360C23—24no—
361C23—41no—
362C23—41no—
363C23—42no—
364C23—42no—
365C23C22×C3×D411yescharacteristic
366Q8—84no—
367Q8—84no—
368D4—22no—
369D4—22no—
370D4—44no—
371D4—42no—
372D4—88no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—44no—
381C2×C4—48no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4C3×#12012yes—
392C2×C4C3×#12012yes—
393C2×C4C3×#13512yes—
394C2×C4C3×AGL(1,ℤ/8)12yes—
395C2×C4C22×C3×D411yescharacteristic
396C2×C4C22×C3×D411yescharacteristic
397C2×C4C2×C3×#4112yes—
398C2×C4C3×2−1+412yes—
399C8—24no—
400C8—42no—
401C2×C3—82no—
402C2×C3—82no—
403C2×C3#243211yescharacteristic
404C2×C3C2×#44911yescharacteristic
405C2×C3#300911yescharacteristic
406C2×C3#334012yes—
407C2×C3#335412yes—
408C22—42no—
409C22—42no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—42no—
414C22—44no—
415C22—44no—
416C22—82no—
417C22—82no—
418C22C2×C3×#12011yescharacteristic
419C22C3×#54712yes—
420C22C2×C3×#13511yescharacteristic
421C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
422C22C3×#57712yes—
423C4—22no—
424C4—42no—
425C4—42no—
426C4—44no—
427C4—82no—
428C4—82no—
429C4C3×#44914yes—
430C4C3×#53211yescharacteristic
431C4C3×#53611yescharacteristic
432C4C3×#54812yes—
433C3#2570011yescharacteristic, direct product factor, Sylow
434C2—82no—
435C2—82no—
436C2C3×#243211yescharacteristic
437C2C2×C3×#44911yescharacteristic
438C2C3×#300911yescharacteristic
439C2C3×#334012yes—
440C2C3×#335412yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#555C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×D4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#498C2semidirect
3. C2×C3×#474C2semidirect
4. C3×#2547C2semidirect
5. C3×#2034C2semidirect
6. C3×#1987C2semidirect
7. C3×#1954C2semidirect
8. C3×#1854C2semidirect
9. C3×C4×#102C2semidirect
10. #25700C3direct (N × H)
39. C2×#498C2×C3semidirect
40. C2×#474C2×C3semidirect
41. #2547C2×C3semidirect
42. #2034C2×C3semidirect
43. #1987C2×C3semidirect
44. #1954C2×C3semidirect
45. #1854C2×C3semidirect
46. C4×#102C2×C3semidirect
433. C3#25700direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5552
C2×C3×#4982
C2×C3×#4742
C3×#25472
C3×#20342
C3×#19872
C3×#19542
C3×#18542
C3×C4×#1022
#257003