Finite Group Databasebeta

← C3×#25699 #10644306

Subgroups — C3×#25699 #10644306

Order 768.

1298 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#547C211yesmaximal, characteristic
2C2×C3×#498C211yesmaximal, characteristic, complemented
3C2×C3×#476C211yesmaximal, characteristic, complemented
4C3×#2563C211yesmaximal, characteristic, complemented
5C3×#2034C214yesmaximal, complemented
6C3×#1988C211yesmaximal, characteristic, complemented
7C3×#1954C211yesmaximal, characteristic, complemented
8C3×#1853C214yesmaximal, complemented
9C3×C4×#102C211yesmaximal, characteristic, complemented
10#25699C311yesmaximal, 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×#123—22no—
17C2×C3×C22⋊Q8—21no—
18C2×C3×#121C2212yes—
19C2×C3×C4×D4C2211yescharacteristic
20C2×C3×C4×D4C2211yescharacteristic
21C22×C3×C4⋊C4C2212yes—
22C3×#498—24no—
23C3×#476—24no—
24C2×C3×#106C2214yes—
25C2×C3×#106C2212yes—
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#102C2212yes—
28C2×C3×#102C2212yes—
29C2×C3×C22⋊C8C2214yes—
30C2×C3×C4×C8C2211yescharacteristic
31C3×#393—22no—
32C3×#390—21no—
33C3×#382—21no—
34C3×#380C2214yes—
35C3×C4×C4⋊C4C2211yescharacteristic
36C3×#340C2212yes—
37C3×#340C2214yes—
38C2×#547C2×C311yescharacteristic
39C2×#498C2×C311yescharacteristic, complemented
40C2×#476C2×C311yescharacteristic, complemented
41#2563C2×C311yescharacteristic, complemented
42#2034C2×C314yescomplemented
43#1988C2×C311yescharacteristic, complemented
44#1954C2×C311yescharacteristic, complemented
45#1853C2×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×#123—216no—
65C3×C22⋊Q8—28no—
66C3×#121—24no—
67C3×#121—24no—
68C3×C4×D4—22no—
69C3×C4×D4—22no—
70C3×C4×D4—22no—
71C3×C4×D4D414yes—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—24no—
75C2×C3×C4⋊C4—24no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4D411yescharacteristic
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—24no—
85C2×C3×C22⋊C4D412yes—
86C2×C3×C42—21no—
87C2×C3×C42C2311yescharacteristic
88C3×#106—24no—
89C3×#106—28no—
90C3×C4⋊C8—22no—
91C3×#102—24no—
92C3×#102—24no—
93C3×C22⋊C8—28no—
94C3×C4×C8—22no—
95C3×#95—21no—
96C3×#95—24no—
97C22×#41—21no—
98#547—24no—
99#547—24no—
100C2×C4⋊Q8C22×C311yescharacteristic
101C2×C4⋊Q8C22×C311yescharacteristic
102C2×#123—22no—
103C2×C22⋊Q8—21no—
104C2×#121C22×C312yes—
105C2×C4×D4C22×C311yescharacteristic
106C2×C4×D4C22×C311yescharacteristic
107C22×C4⋊C4C22×C312yes—
108#498—24no—
109#476—24no—
110C2×#106C22×C312yes—
111C2×#106C22×C314yes—
112C2×C4⋊C8C22×C311yescharacteristic
113C2×#102C22×C312yes—
114C2×#102C22×C312yes—
115C2×C22⋊C8C22×C314yes—
116C2×C4×C8C22×C311yescharacteristic
117#393—22no—
118#390—21no—
119#382—21no—
120#380C22×C314yes—
121C4×C4⋊C4C22×C311yescharacteristic
122#340C22×C312yes—
123#340C22×C314yes—
124C24×C3—21no—
125C24×C3C2×D412yes—
126C3×#41—432no—
127C2×C3×Q8—24no—
128C2×C3×Q8—42no—
129C2×C3×Q8—42no—
130C2×C3×Q8—42no—
131C2×C3×Q8—82no—
132C2×C3×D4—28no—
133C2×C3×D4—22no—
134C2×C3×D4—22no—
135C2×C3×D4—21no—
136C2×C3×D4—22no—
137C2×C3×D4—22no—
138C2×C3×D4—44no—
139C2×C3×D4—44no—
140C2×C3×D4SD1614yes—
141C2×C3×D4C2×D412yes—
142C22×C3×C4—28no—
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—22no—
152C22×C3×C4—44no—
153C22×C3×C4—44no—
154C22×C3×C4—42no—
155C22×C3×C4—42no—
156C22×C3×C4C2×D412yes—
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2411yescharacteristic
160C2×C3×C8—22no—
161C2×C3×C8—22no—
162C2×C3×C8—22no—
163C2×C3×C8—24no—
164C2×C3×C8#4114yescomplemented
165C3×C4⋊C4—28no—
166C3×C4⋊C4—24no—
167C3×C4⋊C4—22no—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—42no—
175C3×C4⋊C4C2×D412yes—
176C3×C4⋊C4C2×D414yes—
177C3×C22⋊C4—28no—
178C3×C22⋊C4—24no—
179C3×C22⋊C4—24no—
180C3×C22⋊C4—84no—
181C3×C42—42no—
182C3×C42C2×D412yes—
183C3×C42C2×D412yes—
184C2×#41—28no—
185C2×#41—44no—
186C2×#41—44no—
187C22×Q8—21no—
188C22×Q8—21no—
189C22×D4—22no—
190C22×D4C3×D411yescharacteristic
191C22×D4C23×C311yescharacteristic
192C23×C4—22no—
193C23×C4—21no—
194C23×C4C23×C312yes—
195C22×C8C23×C312yes—
196C22×C8C23×C312yes—
197C4⋊Q8—22no—
198C4⋊Q8—22no—
199C4⋊Q8—22no—
200C4⋊Q8C3×D414yes—
201#123—216no—
202C22⋊Q8—28no—
203#121—24no—
204#121—24no—
205C4×D4—22no—
206C4×D4—22no—
207C4×D4—22no—
208C4×D4C3×D414yes—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4C3×D411yescharacteristic
215C2×C4⋊C4C23×C312yes—
216C2×C4⋊C4C23×C312yes—
217C2×C4⋊C4C23×C311yescharacteristic
218C2×C4⋊C4C23×C312yes—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4C3×D412yes—
223C2×C42—21no—
224C2×C42C23×C311yescharacteristic
225#106—24no—
226#106—28no—
227C4⋊C8—22no—
228#102—24no—
229#102—24no—
230C22⋊C8—28no—
231C4×C8—22no—
232#95—21no—
233#95—24no—
234C23×C3—22no—
235C23×C3—24no—
236C23×C3—22no—
237C23×C3—24no—
238C23×C3—22no—
239C23×C3—42no—
240C23×C3—41no—
241C23×C3—41no—
242C23×C3—42no—
243C23×C3C2×SD1614yes—
244C23×C3C22×D411yescharacteristic
245C3×Q8—48no—
246C3×Q8—84no—
247C3×D4—22no—
248C3×D4—22no—
249C3×D4—28no—
250C3×D4—416no—
251C3×D4—42no—
252C2×C3×C4—22no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—24no—
256C2×C3×C4—24no—
257C2×C3×C4—416no—
258C2×C3×C4—41no—
259C2×C3×C4—42no—
260C2×C3×C4—41no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4—44no—
265C2×C3×C4—48no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—44no—
271C2×C3×C4#12012yes—
272C2×C3×C4#12012yes—
273C2×C3×C4C2×SD1612yes—
274C2×C3×C4AGL(1,ℤ/8)12yes—
275C2×C3×C4C22×D411yescharacteristic
276C2×C3×C4C22×D411yescharacteristic
277C2×C3×C4C2×#4112yes—
278C2×C3×C42−1+412yes—
279C3×C8—24no—
280C3×C8—42no—
281C24—21no—
282C24C2×C3×D412yes—
283#41—432no—
284C2×Q8—24no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—82no—
289C2×D4—28no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—21no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—44no—
296C2×D4—44no—
297C2×D4C3×SD1614yes—
298C2×D4C2×C3×D412yes—
299C22×C4—28no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4C2×C3×D412yes—
314C22×C4C2×C3×D411yescharacteristic
315C22×C4C2×C3×D411yescharacteristic
316C22×C4C24×C311yescharacteristic
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—24no—
321C2×C8C3×#4114yescomplemented
322C4⋊C4—28no—
323C4⋊C4—24no—
324C4⋊C4—22no—
325C4⋊C4—21no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—42no—
332C4⋊C4C2×C3×D412yes—
333C4⋊C4C2×C3×D414yes—
334C22⋊C4—28no—
335C22⋊C4—24no—
336C22⋊C4—24no—
337C22⋊C4—84no—
338C42—42no—
339C42C2×C3×D412yes—
340C42C2×C3×D412yes—
341C22×C3—28no—
342C22×C3—24no—
343C22×C3—44no—
344C22×C3—42no—
345C22×C3—42no—
346C22×C3—42no—
347C22×C3—42no—
348C22×C3—42no—
349C22×C3—82no—
350C22×C3—82no—
351C22×C3C2×#12011yescharacteristic
352C22×C3#54712yes—
353C22×C3C22×SD1611yescharacteristic
354C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
355C22×C3#57712yes—
356C3×C4—22no—
357C3×C4—42no—
358C3×C4—42no—
359C3×C4—44no—
360C3×C4—82no—
361C3×C4—82no—
362C3×C4#45014yes—
363C3×C4#53211yescharacteristic
364C3×C4#53611yescharacteristic
365C3×C4#54812yes—
366C23—22no—
367C23—24no—
368C23—22no—
369C23—24no—
370C23—22no—
371C23—41no—
372C23—41no—
373C23—42no—
374C23—42no—
375C23C2×C3×SD1614yes—
376C23C22×C3×D411yescharacteristic
377Q8—48no—
378Q8—84no—
379D4—28no—
380D4—22no—
381D4—22no—
382D4—416no—
383D4—42no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—416no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—48no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4C3×#12012yes—
404C2×C4C3×#12012yes—
405C2×C4C2×C3×SD1612yes—
406C2×C4C3×AGL(1,ℤ/8)12yes—
407C2×C4C22×C3×D411yescharacteristic
408C2×C4C22×C3×D411yescharacteristic
409C2×C4C2×C3×#4112yes—
410C2×C4C3×2−1+412yes—
411C8—24no—
412C8—42no—
413C2×C3—44no—
414C2×C3—82no—
415C2×C3#243211yescharacteristic
416C2×C3C2×#45011yescharacteristic
417C2×C3#301011yescharacteristic
418C2×C3#333512yes—
419C2×C3#335412yes—
420C22—24no—
421C22—28no—
422C22—42no—
423C22—42no—
424C22—44no—
425C22—42no—
426C22—42no—
427C22—42no—
428C22—82no—
429C22—82no—
430C22C2×C3×#12011yescharacteristic
431C22C3×#54712yes—
432C22C22×C3×SD1611yescharacteristic
433C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
434C22C3×#57712yes—
435C4—22no—
436C4—42no—
437C4—42no—
438C4—44no—
439C4—82no—
440C4—82no—
441C4C3×#45014yes—
442C4C3×#53211yescharacteristic
443C4C3×#53611yescharacteristic
444C4C3×#54812yes—
445C3#2569911yescharacteristic, direct product factor, Sylow
446C2—44no—
447C2—82no—
448C2C3×#243211yescharacteristic
449C2C2×C3×#45011yescharacteristic
450C2C3×#301011yescharacteristic
451C2C3×#333512yes—
452C2C3×#335412yes—
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×#498C2semidirect
3. C2×C3×#476C2semidirect
4. C3×#2563C2semidirect
5. C3×#2034C2semidirect
6. C3×#1988C2semidirect
7. C3×#1954C2semidirect
8. C3×#1853C2semidirect
9. C3×C4×#102C2semidirect
10. #25699C3direct (N × H)
39. C2×#498C2×C3semidirect
40. C2×#476C2×C3semidirect
41. #2563C2×C3semidirect
42. #2034C2×C3semidirect
43. #1988C2×C3semidirect
44. #1954C2×C3semidirect
45. #1853C2×C3semidirect
46. C4×#102C2×C3semidirect
164. C2×C3×C8#41semidirect
321. C2×C8C3×#41semidirect
445. C3#25699direct (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×#4982
C2×C3×#4762
C3×#25632
C3×#20342
C3×#19882
C3×#19542
C3×#18532
C3×C4×#1022
#256993