Finite Group Databasebeta

← C3×#25703 #10644310

Subgroups — C3×#25703 #10644310

Order 768.

1318 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×#547C211yesmaximal, characteristic
2C2×C3×#500C211yesmaximal, characteristic, complemented
3C2×C3×#474C211yesmaximal, characteristic, complemented
4C3×#2547C211yesmaximal, characteristic, complemented
5C3×#2033C214yesmaximal, complemented
6C3×#1988C211yesmaximal, characteristic, complemented
7C3×#1954C211yesmaximal, characteristic, complemented
8C3×#1852C214yesmaximal, complemented
9C3×C4×#102C211yesmaximal, characteristic, complemented
10#25703C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×#41—21no—
12C3×#547—24no—
13C3×#547—24no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#124—21no—
17C2×C3×#123—22no—
18C2×C3×C22⋊Q8—21no—
19C2×C3×#121C2212yes—
20C2×C3×C4×D4C2211yescharacteristic
21C2×C3×C4×D4C2211yescharacteristic
22C22×C3×C4⋊C4C2212yes—
23C3×#500—24no—
24C3×#474—24no—
25C2×C3×#107C2214yes—
26C2×C3×#107C2212yes—
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×#547C2×C311yescharacteristic
39C2×#500C2×C311yescharacteristic, complemented
40C2×#474C2×C311yescharacteristic, complemented
41#2547C2×C311yescharacteristic, complemented
42#2033C2×C314yescomplemented
43#1988C2×C311yescharacteristic, complemented
44#1954C2×C311yescharacteristic, complemented
45#1852C2×C314yescomplemented
46C4×#102C2×C311yescharacteristic, complemented
47C2×C3×#41—28no—
48C2×C3×#41—44no—
49C2×C3×#41—44no—
50C22×C3×Q8—21no—
51C22×C3×Q8—21no—
52C22×C3×D4—22no—
53C22×C3×D4D411yescharacteristic
54C22×C3×D4C2311yescharacteristic
55C23×C3×C4—22no—
56C23×C3×C4—21no—
57C23×C3×C4C2312yes—
58C22×C3×C8C2312yes—
59C22×C3×C8C2312yes—
60C3×C4⋊Q8—22no—
61C3×C4⋊Q8—22no—
62C3×C4⋊Q8—22no—
63C3×C4⋊Q8D414yes—
64C3×#124—44no—
65C3×#123—216no—
66C3×C22⋊Q8—28no—
67C3×#121—24no—
68C3×#121—24no—
69C3×C4×D4—22no—
70C3×C4×D4—22no—
71C3×C4×D4—22no—
72C3×C4×D4D414yes—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—24no—
76C2×C3×C4⋊C4—24no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4D411yescharacteristic
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—24no—
86C2×C3×C22⋊C4D412yes—
87C2×C3×C42—21no—
88C2×C3×C42C2311yescharacteristic
89C3×#107—24no—
90C3×#107—28no—
91C3×C4⋊C8—22no—
92C3×#102—24no—
93C3×#102—24no—
94C3×C22⋊C8—28no—
95C3×C4×C8—22no—
96C3×#95—21no—
97C3×#95—24no—
98C22×#41—21no—
99#547—24no—
100#547—24no—
101C2×C4⋊Q8C22×C311yescharacteristic
102C2×C4⋊Q8C22×C311yescharacteristic
103C2×#124—21no—
104C2×#123—22no—
105C2×C22⋊Q8—21no—
106C2×#121C22×C312yes—
107C2×C4×D4C22×C311yescharacteristic
108C2×C4×D4C22×C311yescharacteristic
109C22×C4⋊C4C22×C312yes—
110#500—24no—
111#474—24no—
112C2×#107C22×C312yes—
113C2×#107C22×C314yes—
114C2×C4⋊C8C22×C311yescharacteristic
115C2×#102C22×C312yes—
116C2×#102C22×C312yes—
117C2×C22⋊C8C22×C314yes—
118C2×C4×C8C22×C311yescharacteristic
119#397—22no—
120#391—21no—
121#380C22×C314yes—
122C4×C4⋊C4C22×C311yescharacteristic
123#340C22×C312yes—
124#340C22×C314yes—
125C24×C3—21no—
126C24×C3C2×D412yes—
127C3×#41—432no—
128C2×C3×Q8—24no—
129C2×C3×Q8—42no—
130C2×C3×Q8—42no—
131C2×C3×Q8—42no—
132C2×C3×Q8—42no—
133C2×C3×Q8—42no—
134C2×C3×D4—28no—
135C2×C3×D4—22no—
136C2×C3×D4—22no—
137C2×C3×D4—21no—
138C2×C3×D4—22no—
139C2×C3×D4—22no—
140C2×C3×D4—44no—
141C2×C3×D4—44no—
142C2×C3×D4D814yes—
143C2×C3×D4C2×D412yes—
144C22×C3×C4—28no—
145C22×C3×C4—21no—
146C22×C3×C4—22no—
147C22×C3×C4—24no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—24no—
151C22×C3×C4—24no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—44no—
155C22×C3×C4—44no—
156C22×C3×C4—42no—
157C22×C3×C4—42no—
158C22×C3×C4C2×D412yes—
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2411yescharacteristic
162C2×C3×C8—22no—
163C2×C3×C8—22no—
164C2×C3×C8—22no—
165C2×C3×C8—24no—
166C2×C3×C8#4114yescomplemented
167C3×C4⋊C4—28no—
168C3×C4⋊C4—24no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—42no—
177C3×C4⋊C4C2×D412yes—
178C3×C4⋊C4C2×D414yes—
179C3×C22⋊C4—28no—
180C3×C22⋊C4—24no—
181C3×C22⋊C4—24no—
182C3×C22⋊C4—48no—
183C3×C42—42no—
184C3×C42C2×D412yes—
185C3×C42C2×D412yes—
186C2×#41—28no—
187C2×#41—44no—
188C2×#41—44no—
189C22×Q8—21no—
190C22×Q8—21no—
191C22×D4—22no—
192C22×D4C3×D411yescharacteristic
193C22×D4C23×C311yescharacteristic
194C23×C4—22no—
195C23×C4—21no—
196C23×C4C23×C312yes—
197C22×C8C23×C312yes—
198C22×C8C23×C312yes—
199C4⋊Q8—22no—
200C4⋊Q8—22no—
201C4⋊Q8—22no—
202C4⋊Q8C3×D414yes—
203#124—44no—
204#123—216no—
205C22⋊Q8—28no—
206#121—24no—
207#121—24no—
208C4×D4—22no—
209C4×D4—22no—
210C4×D4—22no—
211C4×D4C3×D414yes—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4C3×D411yescharacteristic
218C2×C4⋊C4C23×C312yes—
219C2×C4⋊C4C23×C312yes—
220C2×C4⋊C4C23×C311yescharacteristic
221C2×C4⋊C4C23×C312yes—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4C3×D412yes—
226C2×C42—21no—
227C2×C42C23×C311yescharacteristic
228#107—24no—
229#107—28no—
230C4⋊C8—22no—
231#102—24no—
232#102—24no—
233C22⋊C8—28no—
234C4×C8—22no—
235#95—21no—
236#95—24no—
237C23×C3—22no—
238C23×C3—24no—
239C23×C3—22no—
240C23×C3—24no—
241C23×C3—22no—
242C23×C3—42no—
243C23×C3—41no—
244C23×C3—41no—
245C23×C3—42no—
246C23×C3C2×D814yes—
247C23×C3C22×D411yescharacteristic
248C3×Q8—48no—
249C3×Q8—84no—
250C3×D4—22no—
251C3×D4—22no—
252C3×D4—28no—
253C3×D4—416no—
254C3×D4—42no—
255C2×C3×C4—22no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—24no—
259C2×C3×C4—24no—
260C2×C3×C4—416no—
261C2×C3×C4—41no—
262C2×C3×C4—42no—
263C2×C3×C4—41no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—44no—
268C2×C3×C4—48no—
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×C4C2×D812yes—
278C2×C3×C4AGL(1,ℤ/8)12yes—
279C2×C3×C4C22×D411yescharacteristic
280C2×C3×C4C22×D411yescharacteristic
281C2×C3×C4C2×#4112yes—
282C2×C3×C42−1+412yes—
283C3×C8—24no—
284C3×C8—42no—
285C24—21no—
286C24C2×C3×D412yes—
287#41—432no—
288C2×Q8—24no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×D4—28no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—21no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—44no—
301C2×D4—44no—
302C2×D4C3×D814yes—
303C2×D4C2×C3×D412yes—
304C22×C4—28no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4C2×C3×D412yes—
319C22×C4C2×C3×D411yescharacteristic
320C22×C4C2×C3×D411yescharacteristic
321C22×C4C24×C311yescharacteristic
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—24no—
326C2×C8C3×#4114yescomplemented
327C4⋊C4—28no—
328C4⋊C4—24no—
329C4⋊C4—22no—
330C4⋊C4—21no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—42no—
337C4⋊C4C2×C3×D412yes—
338C4⋊C4C2×C3×D414yes—
339C22⋊C4—28no—
340C22⋊C4—24no—
341C22⋊C4—24no—
342C22⋊C4—48no—
343C42—42no—
344C42C2×C3×D412yes—
345C42C2×C3×D412yes—
346C22×C3—28no—
347C22×C3—24no—
348C22×C3—44no—
349C22×C3—42no—
350C22×C3—42no—
351C22×C3—42no—
352C22×C3—42no—
353C22×C3—42no—
354C22×C3—82no—
355C22×C3—82no—
356C22×C3C2×#12011yescharacteristic
357C22×C3#54712yes—
358C22×C3C22×D811yescharacteristic
359C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
360C22×C3#57812yes—
361C3×C4—22no—
362C3×C4—42no—
363C3×C4—42no—
364C3×C4—44no—
365C3×C4—82no—
366C3×C4—82no—
367C3×C4#44714yes—
368C3×C4#53211yescharacteristic
369C3×C4#53611yescharacteristic
370C3×C4#54812yes—
371C23—22no—
372C23—24no—
373C23—22no—
374C23—24no—
375C23—22no—
376C23—41no—
377C23—41no—
378C23—42no—
379C23—42no—
380C23C2×C3×D814yes—
381C23C22×C3×D411yescharacteristic
382Q8—48no—
383Q8—84no—
384D4—28no—
385D4—22no—
386D4—22no—
387D4—416no—
388D4—42no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—416no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—48no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4C3×#12012yes—
410C2×C4C3×#12012yes—
411C2×C4C2×C3×D812yes—
412C2×C4C3×AGL(1,ℤ/8)12yes—
413C2×C4C22×C3×D411yescharacteristic
414C2×C4C22×C3×D411yescharacteristic
415C2×C4C2×C3×#4112yes—
416C2×C4C3×2−1+412yes—
417C8—24no—
418C8—42no—
419C2×C3—44no—
420C2×C3—82no—
421C2×C3#243211yescharacteristic
422C2×C3C2×#44711yescharacteristic
423C2×C3#301111yescharacteristic
424C2×C3#333412yes—
425C2×C3#335312yes—
426C22—24no—
427C22—28no—
428C22—42no—
429C22—42no—
430C22—44no—
431C22—42no—
432C22—42no—
433C22—42no—
434C22—82no—
435C22—82no—
436C22C2×C3×#12011yescharacteristic
437C22C3×#54712yes—
438C22C22×C3×D811yescharacteristic
439C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
440C22C3×#57812yes—
441C4—22no—
442C4—42no—
443C4—42no—
444C4—44no—
445C4—82no—
446C4—82no—
447C4C3×#44714yes—
448C4C3×#53211yescharacteristic
449C4C3×#53611yescharacteristic
450C4C3×#54812yes—
451C3#2570311yescharacteristic, direct product factor, Sylow
452C2—44no—
453C2—82no—
454C2C3×#243211yescharacteristic
455C2C2×C3×#44711yescharacteristic
456C2C3×#301111yescharacteristic
457C2C3×#333412yes—
458C2C3×#335312yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#547C2
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×#500C2semidirect
3. C2×C3×#474C2semidirect
4. C3×#2547C2semidirect
5. C3×#2033C2semidirect
6. C3×#1988C2semidirect
7. C3×#1954C2semidirect
8. C3×#1852C2semidirect
9. C3×C4×#102C2semidirect
10. #25703C3direct (N × H)
39. C2×#500C2×C3semidirect
40. C2×#474C2×C3semidirect
41. #2547C2×C3semidirect
42. #2033C2×C3semidirect
43. #1988C2×C3semidirect
44. #1954C2×C3semidirect
45. #1852C2×C3semidirect
46. C4×#102C2×C3semidirect
166. C2×C3×C8#41semidirect
326. C2×C8C3×#41semidirect
451. C3#25703direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5472
C2×C3×#5002
C2×C3×#4742
C3×#25472
C3×#20332
C3×#19882
C3×#19542
C3×#18522
C3×C4×#1022
#257033