Finite Group Databasebeta

← C3×#25568 #10644175

Subgroups — C3×#25568 #10644175

Order 768.

1190 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#555C211yesmaximal, characteristic
2C2×C3×#487C211yesmaximal, characteristic, complemented
3C2×C3×#461C211yesmaximal, characteristic, complemented
4C3×#2542C211yesmaximal, characteristic, complemented
5C3×#2000C214yesmaximal, complemented
6C3×#1987C211yesmaximal, characteristic, complemented
7C3×#1954C211yesmaximal, characteristic, complemented
8C3×#1873C214yesmaximal, complemented
9C3×C4×#102C211yesmaximal, characteristic, complemented
10#25568C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×SD16C2212yes—
12C3×#555—24no—
13C3×#555—24no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#124—22no—
17C2×C3×C22⋊Q8C2214yes—
18C2×C3×#121—21no—
19C2×C3×#121C2212yes—
20C2×C3×C4×D4C2211yescharacteristic
21C2×C3×C4×D4C2211yescharacteristic
22C2×C3×#117—21no—
23C2×C3×#117C2212yes—
24C3×#487—24no—
25C3×#461—24no—
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#103C2214yes—
28C2×C3×#103C2212yes—
29C2×C3×#102C2212yes—
30C2×C3×C22⋊C8C2214yes—
31C2×C3×C4×C8C2211yescharacteristic
32C3×#399—22no—
33C3×#390—21no—
34C3×#384—21no—
35C3×C4×C4⋊C4C2211yescharacteristic
36C3×#340C2214yes—
37C3×#340C2212yes—
38C2×#555C2×C311yescharacteristic
39C2×#487C2×C311yescharacteristic, complemented
40C2×#461C2×C311yescharacteristic, complemented
41#2542C2×C311yescharacteristic, complemented
42#2000C2×C314yescomplemented
43#1987C2×C311yescharacteristic, complemented
44#1954C2×C311yescharacteristic, complemented
45#1873C2×C314yescomplemented
46C4×#102C2×C311yescharacteristic, complemented
47C22×C3×Q8—21no—
48C22×C3×Q8C2312yes—
49C22×C3×D4—22no—
50C22×C3×D4D411yescharacteristic
51C22×C3×D4C2311yescharacteristic
52C23×C3×C4—21no—
53C23×C3×C4C2312yes—
54C2×C3×SD16—24no—
55C2×C3×SD16—44no—
56C22×C3×C8C2312yes—
57C22×C3×C8C2312yes—
58C3×C4⋊Q8—22no—
59C3×C4⋊Q8—22no—
60C3×C4⋊Q8—22no—
61C3×C4⋊Q8D414yes—
62C3×#124—216no—
63C3×C22⋊Q8—28no—
64C3×C22⋊Q8—44no—
65C3×#121—28no—
66C3×#121—24no—
67C3×#121—24no—
68C3×C4×D4—22no—
69C3×C4×D4—22no—
70C3×C4×D4—22no—
71C3×C4×D4D414yes—
72C3×#117—28no—
73C3×#117—24no—
74C3×#117—24no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—24no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4D411yescharacteristic
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4—24no—
87C2×C3×C22⋊C4D412yes—
88C2×C3×C42—22no—
89C2×C3×C42—21no—
90C2×C3×C42C2311yescharacteristic
91C3×C4⋊C8—22no—
92C3×#103—24no—
93C3×#103—28no—
94C3×#102—24no—
95C3×C22⋊C8—44no—
96C3×C4×C8—22no—
97C3×#95—21no—
98C22×SD16C22×C312yes—
99#555—24no—
100#555—24no—
101C2×C4⋊Q8C22×C311yescharacteristic
102C2×C4⋊Q8C22×C311yescharacteristic
103C2×#124—22no—
104C2×C22⋊Q8C22×C314yes—
105C2×#121—21no—
106C2×#121C22×C312yes—
107C2×C4×D4C22×C311yescharacteristic
108C2×C4×D4C22×C311yescharacteristic
109C2×#117—21no—
110C2×#117C22×C312yes—
111#487—24no—
112#461—24no—
113C2×C4⋊C8C22×C311yescharacteristic
114C2×#103C22×C314yes—
115C2×#103C22×C312yes—
116C2×#102C22×C312yes—
117C2×C22⋊C8C22×C314yes—
118C2×C4×C8C22×C311yescharacteristic
119#399—22no—
120#390—21no—
121#384—21no—
122C4×C4⋊C4C22×C311yescharacteristic
123#340C22×C312yes—
124#340C22×C314yes—
125C24×C3—21no—
126C24×C3C2×D412yes—
127C2×C3×Q8—24no—
128C2×C3×Q8—22no—
129C2×C3×Q8—22no—
130C2×C3×Q8—24no—
131C2×C3×Q8—24no—
132C2×C3×Q8—82no—
133C2×C3×D4—28no—
134C2×C3×D4—22no—
135C2×C3×D4—22no—
136C2×C3×D4—22no—
137C2×C3×D4—21no—
138C2×C3×D4—22no—
139C2×C3×D4—22no—
140C2×C3×D4—84no—
141C2×C3×D4C2×D412yes—
142C22×C3×C4—22no—
143C22×C3×C4—21no—
144C22×C3×C4—22no—
145C22×C3×C4—24no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—24no—
149C22×C3×C4—24no—
150C22×C3×C4—22no—
151C22×C3×C4—42no—
152C22×C3×C4—42no—
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2411yescharacteristic
157C3×SD16—48no—
158C2×C3×C8—22no—
159C2×C3×C8—22no—
160C2×C3×C8—22no—
161C2×C3×C8—22no—
162C2×C3×C8—24no—
163C3×C4⋊C4—24no—
164C3×C4⋊C4—22no—
165C3×C4⋊C4—21no—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—22no—
168C3×C4⋊C4—48no—
169C3×C4⋊C4—42no—
170C3×C4⋊C4—42no—
171C3×C4⋊C4C2×D412yes—
172C3×C4⋊C4C2×D414yes—
173C3×C4⋊C4#4114yes—
174C3×C22⋊C4—28no—
175C3×C22⋊C4—28no—
176C3×C22⋊C4—24no—
177C3×C22⋊C4—24no—
178C3×C22⋊C4—48no—
179C3×C42—28no—
180C3×C42—42no—
181C3×C42C2×D412yes—
182C3×C42C2×D412yes—
183C22×Q8—21no—
184C22×Q8C23×C312yes—
185C22×D4—22no—
186C22×D4C3×D411yescharacteristic
187C22×D4C23×C311yescharacteristic
188C23×C4—21no—
189C23×C4C23×C312yes—
190C2×SD16—24no—
191C2×SD16—44no—
192C22×C8C23×C312yes—
193C22×C8C23×C312yes—
194C4⋊Q8—22no—
195C4⋊Q8—22no—
196C4⋊Q8—22no—
197C4⋊Q8C3×D414yes—
198#124—216no—
199C22⋊Q8—28no—
200C22⋊Q8—44no—
201#121—28no—
202#121—24no—
203#121—24no—
204C4×D4—22no—
205C4×D4—22no—
206C4×D4—22no—
207C4×D4C3×D414yes—
208#117—28no—
209#117—24no—
210#117—24no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4C3×D411yescharacteristic
216C2×C4⋊C4C23×C311yescharacteristic
217C2×C4⋊C4C23×C312yes—
218C2×C4⋊C4C23×C312yes—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4C3×D412yes—
224C2×C42—22no—
225C2×C42—21no—
226C2×C42C23×C311yescharacteristic
227C4⋊C8—22no—
228#103—24no—
229#103—28no—
230#102—24no—
231C22⋊C8—44no—
232C4×C8—22no—
233#95—21no—
234C23×C3—22no—
235C23×C3—22no—
236C23×C3—24no—
237C23×C3—22no—
238C23×C3—22no—
239C23×C3—24no—
240C23×C3—42no—
241C23×C3—41no—
242C23×C3—41no—
243C23×C3—42no—
244C23×C3C22×D411yescharacteristic
245C3×Q8—42no—
246C3×Q8—42no—
247C3×Q8—44no—
248C3×Q8—84no—
249C3×D4—22no—
250C3×D4—22no—
251C3×D4—42no—
252C3×D4—44no—
253C3×D4—88no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—22no—
257C2×C3×C4—24no—
258C2×C3×C4—24no—
259C2×C3×C4—24no—
260C2×C3×C4—42no—
261C2×C3×C4—41no—
262C2×C3×C4—41no—
263C2×C3×C4—42no—
264C2×C3×C4—44no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—44no—
268C2×C3×C4—48no—
269C2×C3×C4—42no—
270C2×C3×C4—44no—
271C2×C3×C4#12012yes—
272C2×C3×C4#12012yes—
273C2×C3×C4#13512yes—
274C2×C3×C4#13712yes—
275C2×C3×C4C22×D411yescharacteristic
276C2×C3×C4C22×D411yescharacteristic
277C2×C3×C4C2×#4112yes—
278C2×C3×C4H(5,𝔽2)12yes—
279C3×C8—42no—
280C3×C8—42no—
281C24—21no—
282C24C2×C3×D412yes—
283C2×Q8—24no—
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—24no—
287C2×Q8—24no—
288C2×Q8—82no—
289C2×D4—28no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—21no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—84no—
297C2×D4C2×C3×D412yes—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4C2×C3×D412yes—
310C22×C4C2×C3×D411yescharacteristic
311C22×C4C2×C3×D411yescharacteristic
312C22×C4C24×C311yescharacteristic
313SD16—48no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—24no—
319C4⋊C4—24no—
320C4⋊C4—22no—
321C4⋊C4—21no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—48no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4C2×C3×D412yes—
328C4⋊C4C2×C3×D414yes—
329C4⋊C4C3×#4114yes—
330C22⋊C4—28no—
331C22⋊C4—28no—
332C22⋊C4—24no—
333C22⋊C4—24no—
334C22⋊C4—48no—
335C42—28no—
336C42—42no—
337C42C2×C3×D412yes—
338C42C2×C3×D412yes—
339C22×C3—44no—
340C22×C3—42no—
341C22×C3—42no—
342C22×C3—42no—
343C22×C3—42no—
344C22×C3—42no—
345C22×C3—42no—
346C22×C3—44no—
347C22×C3—82no—
348C22×C3—82no—
349C22×C3C2×#12011yescharacteristic
350C22×C3#54612yes—
351C22×C3C2×#13511yescharacteristic
352C22×C3C2×#13711yescharacteristic
353C22×C3#57712yes—
354C3×C4—22no—
355C3×C4—42no—
356C3×C4—42no—
357C3×C4—44no—
358C3×C4—44no—
359C3×C4—82no—
360C3×C4#45214yes—
361C3×C4#53211yescharacteristic
362C3×C4#53511yescharacteristic
363C3×C4#55012yes—
364C23—22no—
365C23—22no—
366C23—24no—
367C23—22no—
368C23—22no—
369C23—24no—
370C23—41no—
371C23—41no—
372C23—42no—
373C23—42no—
374C23C22×C3×D411yescharacteristic
375Q8—42no—
376Q8—42no—
377Q8—44no—
378Q8—84no—
379D4—22no—
380D4—22no—
381D4—44no—
382D4—42no—
383D4—88no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—48no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4C3×#12012yes—
402C2×C4C3×#12012yes—
403C2×C4C3×#13512yes—
404C2×C4C3×#13712yes—
405C2×C4C22×C3×D411yescharacteristic
406C2×C4C22×C3×D411yescharacteristic
407C2×C4C2×C3×#4112yes—
408C2×C4C3×H(5,𝔽2)12yes—
409C8—42no—
410C8—42no—
411C2×C3—82no—
412C2×C3—82no—
413C2×C3#243111yescharacteristic
414C2×C3C2×#45211yescharacteristic
415C2×C3#300911yescharacteristic
416C2×C3#330312yes—
417C2×C3#332912yes—
418C22—42no—
419C22—42no—
420C22—42no—
421C22—42no—
422C22—42no—
423C22—42no—
424C22—44no—
425C22—44no—
426C22—82no—
427C22—82no—
428C22C2×C3×#12011yescharacteristic
429C22C3×#54612yes—
430C22C2×C3×#13511yescharacteristic
431C22C2×C3×#13711yescharacteristic
432C22C3×#57712yes—
433C4—22no—
434C4—42no—
435C4—42no—
436C4—44no—
437C4—44no—
438C4—82no—
439C4C3×#45214yes—
440C4C3×#53211yescharacteristic
441C4C3×#53511yescharacteristic
442C4C3×#55012yes—
443C3#2556811yescharacteristic, direct product factor, Sylow
444C2—82no—
445C2—82no—
446C2C3×#243111yescharacteristic
447C2C2×C3×#45211yescharacteristic
448C2C3×#300911yescharacteristic
449C2C3×#330312yes—
450C2C3×#332912yes—
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×#487C2semidirect
3. C2×C3×#461C2semidirect
4. C3×#2542C2semidirect
5. C3×#2000C2semidirect
6. C3×#1987C2semidirect
7. C3×#1954C2semidirect
8. C3×#1873C2semidirect
9. C3×C4×#102C2semidirect
10. #25568C3direct (N × H)
39. C2×#487C2×C3semidirect
40. C2×#461C2×C3semidirect
41. #2542C2×C3semidirect
42. #2000C2×C3semidirect
43. #1987C2×C3semidirect
44. #1954C2×C3semidirect
45. #1873C2×C3semidirect
46. C4×#102C2×C3semidirect
443. C3#25568direct (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×#4872
C2×C3×#4612
C3×#25422
C3×#20002
C3×#19872
C3×#19542
C3×#18732
C3×C4×#1022
#255683