Finite Group Databasebeta

← C3×#25452 #10644059

Subgroups — C3×#25452 #10644059

Order 768.

806 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#498C211yesmaximal, characteristic, complemented
2C2×C3×#474C211yesmaximal, characteristic
3C3×#2563C211yesmaximal, characteristic
4C3×#2365C211yesmaximal, characteristic, complemented
5C3×#2085C212yesmaximal, complemented
6C3×#2065C212yesmaximal
7C3×#1987C211yesmaximal, characteristic, complemented
8C3×#1949C212yesmaximal, complemented
9C3×#1939C211yesmaximal, characteristic, complemented
10C3×#1926C212yesmaximal
11C3×C4×#102C211yesmaximal, characteristic
12#25452C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4×Q8—21no—
16C2×C3×C4×D4C2211yescharacteristic
17C3×#498—24no—
18C3×#474—24no—
19C2×C3×#107C2212yes—
20C2×C3×#106C2212yes—
21C2×C3×#106C2212yes—
22C2×C3×C4⋊C8C2212yes—
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×#102C2212yes—
26C2×C3×#102C2211yescharacteristic
27C2×C3×C4×C8C2211yescharacteristic
28C3×#393—22no—
29C3×#391—21no—
30C3×#390C2212yes—
31C3×#389C2212yes—
32C3×#387—22no—
33C3×#382C2212yes—
34C3×#380—21no—
35C3×#380—21no—
36C3×C4×C4⋊C4C2211yescharacteristic
37C3×C4×C4⋊C4C2212yes—
38C3×C4×C4⋊C4C2211yescharacteristic
39C3×#340C2211yescharacteristic
40C3×#340C2211yescharacteristic
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#336C2212yes—
44C2×#498C2×C311yescharacteristic, complemented
45C2×#474C2×C311yescharacteristic
46#2563C2×C311yescharacteristic
47#2365C2×C311yescharacteristic, complemented
48#2085C2×C312yescomplemented
49#2065C2×C312yes—
50#1987C2×C311yescharacteristic, complemented
51#1949C2×C312yescomplemented
52#1939C2×C311yescharacteristic, complemented
53#1926C2×C312yes—
54C4×#102C2×C311yescharacteristic
55C22×C3×Q8—21no—
56C22×C3×Q8—21no—
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—21no—
59C22×C3×C8C2312yes—
60C22×C3×C8C2311yescharacteristic
61C22×C3×C8C2311yescharacteristic
62C3×C4⋊Q8—22no—
63C3×C4⋊Q8—22no—
64C3×C4⋊Q8—22no—
65C3×C4⋊Q8D414yes—
66C3×C4×Q8—44no—
67C3×C4×D4—22no—
68C3×C4×D4—22no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4D411yescharacteristic
73C2×C3×C4⋊C4D411yescharacteristic
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C22⋊C4—21no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C42—22no—
83C2×C3×C42—21no—
84C2×C3×C42C2312yes—
85C2×C3×C42C2311yescharacteristic
86C3×#107—24no—
87C3×#106—24no—
88C3×#106—24no—
89C3×C4⋊C8—24no—
90C3×C4⋊C8—22no—
91C3×#102—22no—
92C3×#102—24no—
93C3×#102Q814yescomplemented
94C3×C4×C8—22no—
95C3×#95—21no—
96C3×#95—22no—
97C3×#95—22no—
98C3×#95—22no—
99C3×#95—22no—
100C3×#95D412yes—
101C2×C4⋊Q8C22×C311yescharacteristic
102C2×C4⋊Q8C22×C311yescharacteristic
103C2×C4×Q8—21no—
104C2×C4×D4C22×C311yescharacteristic
105#498—24no—
106#474—24no—
107C2×#107C22×C312yes—
108C2×#106C22×C312yes—
109C2×#106C22×C312yes—
110C2×C4⋊C8C22×C312yes—
111C2×C4⋊C8C22×C311yescharacteristic
112C2×#102C22×C311yescharacteristic
113C2×#102C22×C312yes—
114C2×#102C22×C311yescharacteristic
115C2×C4×C8C22×C311yescharacteristic
116#393—22no—
117#391—21no—
118#390C22×C312yes—
119#389C22×C312yes—
120#387—22no—
121#382C22×C312yes—
122#380—21no—
123#380—21no—
124C4×C4⋊C4C22×C312yes—
125C4×C4⋊C4C22×C311yescharacteristic
126C4×C4⋊C4C22×C311yescharacteristic
127#340C22×C312yes—
128#340C22×C311yescharacteristic
129#340C22×C312yes—
130#340C22×C311yescharacteristic
131#336C22×C312yes—
132C24×C3—21no—
133C2×C3×Q8—24no—
134C2×C3×Q8—42no—
135C2×C3×Q8—42no—
136C2×C3×Q8—42no—
137C2×C3×Q8—81no—
138C2×C3×Q8—81no—
139C2×C3×D4—21no—
140C2×C3×D4—21no—
141C2×C3×D4—21no—
142C2×C3×D4—21no—
143C2×C3×D4—21no—
144C2×C3×D4C2×Q812yes—
145C22×C3×C4—21no—
146C22×C3×C4—22no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—22no—
150C22×C3×C4—42no—
151C22×C3×C4—42no—
152C22×C3×C4—42no—
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2×D412yes—
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2411yescharacteristic
158C2×C3×C8—21no—
159C2×C3×C8—21no—
160C2×C3×C8—21no—
161C2×C3×C8—21no—
162C2×C3×C8—22no—
163C2×C3×C8—22no—
164C2×C3×C8—22no—
165C2×C3×C8C2×Q812yes—
166C2×C3×C8#4112yes—
167C3×C4⋊C4—24no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—21no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—21no—
178C3×C4⋊C4—44no—
179C3×C4⋊C4—42no—
180C3×C4⋊C4C2×D414yes—
181C3×C4⋊C4C2×Q812yes—
182C3×C22⋊C4—42no—
183C3×C22⋊C4—82no—
184C3×C22⋊C4—82no—
185C3×C42—22no—
186C3×C42—22no—
187C3×C42—44no—
188C3×C42—42no—
189C3×C42C2×D412yes—
190C3×C42#4112yes—
191C22×Q8—21no—
192C22×Q8—21no—
193C22×D4C23×C311yescharacteristic
194C23×C4—21no—
195C22×C8C23×C312yes—
196C22×C8C23×C311yescharacteristic
197C22×C8C23×C311yescharacteristic
198C4⋊Q8—22no—
199C4⋊Q8—22no—
200C4⋊Q8—22no—
201C4⋊Q8C3×D414yes—
202C4×Q8—44no—
203C4×D4—22no—
204C4×D4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4C3×D411yescharacteristic
209C2×C4⋊C4C3×D411yescharacteristic
210C2×C4⋊C4C23×C311yescharacteristic
211C2×C4⋊C4C23×C312yes—
212C2×C4⋊C4C23×C311yescharacteristic
213C2×C4⋊C4C23×C312yes—
214C2×C4⋊C4C23×C311yescharacteristic
215C2×C22⋊C4—21no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C42—22no—
219C2×C42—21no—
220C2×C42C23×C312yes—
221C2×C42C23×C311yescharacteristic
222#107—24no—
223#106—24no—
224#106—24no—
225C4⋊C8—24no—
226C4⋊C8—22no—
227#102—22no—
228#102—24no—
229#102C3×Q814yescomplemented
230C4×C8—22no—
231#95—21no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95C3×D412yes—
237C23×C3—22no—
238C23×C3—41no—
239C23×C3—41no—
240C23×C3—41no—
241C23×C3—41no—
242C23×C3—41no—
243C23×C3—41no—
244C23×C3C22×D411yescharacteristic
245C3×Q8—48no—
246C3×Q8—84no—
247C3×D4—22no—
248C3×D4—22no—
249C3×D4—41no—
250C3×D4—41no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—24no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—42no—
262C2×C3×C4—41no—
263C2×C3×C4—42no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—42no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—41no—
274C2×C3×C4—84no—
275C2×C3×C4#12112yes—
276C2×C3×C4#12112yes—
277C2×C3×C4#13512yes—
278C2×C3×C4AGL(1,ℤ/8)12yes—
279C2×C3×C4C22×D411yescharacteristic
280C2×C3×C4C22×Q811yescharacteristic
281C2×C3×C4C2×#4111yescharacteristic
282C2×C3×C4C2×#4111yescharacteristic
283C2×C3×C42−1+411yescharacteristic
284C2×C3×C42−1+411yescharacteristic
285C3×C8—22no—
286C3×C8—22no—
287C3×C8—42no—
288C24—21no—
289C2×Q8—24no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—81no—
294C2×Q8—81no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4C2×C3×Q812yes—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4C2×C3×D412yes—
310C22×C4C2×C3×D412yes—
311C22×C4C2×C3×D411yescharacteristic
312C22×C4C2×C3×D411yescharacteristic
313C22×C4C24×C311yescharacteristic
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8C2×C3×Q812yes—
322C2×C8C3×#4112yes—
323C4⋊C4—24no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—21no—
328C4⋊C4—22no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—22no—
333C4⋊C4—21no—
334C4⋊C4—44no—
335C4⋊C4—42no—
336C4⋊C4C2×C3×D414yes—
337C4⋊C4C2×C3×Q812yes—
338C22⋊C4—42no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C42—22no—
342C42—22no—
343C42—44no—
344C42—42no—
345C42C2×C3×D412yes—
346C42C3×#4112yes—
347C22×C3—42no—
348C22×C3—42no—
349C22×C3—42no—
350C22×C3—81no—
351C22×C3—81no—
352C22×C3—81no—
353C22×C3—81no—
354C22×C3C2×#12111yescharacteristic
355C22×C3#54711yescharacteristic
356C22×C3D4×Q811yescharacteristic
357C22×C3C2×#13511yescharacteristic
358C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
359C22×C3#57711yescharacteristic
360C22×C3#57711yescharacteristic
361C3×C4—22no—
362C3×C4—42no—
363C3×C4—42no—
364C3×C4—42no—
365C3×C4—44no—
366C3×C4—81no—
367C3×C4—81no—
368C3×C4—82no—
369C3×C4#46314yes—
370C3×C4#53311yescharacteristic
371C3×C4#54111yescharacteristic
372C3×C4#54811yescharacteristic
373C3×C4D4×Q811yescharacteristic
374C23—22no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23C22×C3×D411yescharacteristic
382Q8—48no—
383Q8—84no—
384D4—22no—
385D4—22no—
386D4—41no—
387D4—41no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—24no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—84no—
412C2×C4C3×#12112yes—
413C2×C4C3×#12112yes—
414C2×C4C3×#13512yes—
415C2×C4C3×AGL(1,ℤ/8)12yes—
416C2×C4C22×C3×D411yescharacteristic
417C2×C4C22×C3×Q811yescharacteristic
418C2×C4C2×C3×#4111yescharacteristic
419C2×C4C2×C3×#4111yescharacteristic
420C2×C4C3×2−1+411yescharacteristic
421C2×C4C3×2−1+411yescharacteristic
422C8—22no—
423C8—22no—
424C8—42no—
425C2×C3—82no—
426C2×C3#242911yescharacteristic
427C2×C3C2×#46311yescharacteristic
428C2×C3#304311yescharacteristic
429C2×C3#334011yescharacteristic
430C2×C3#335411yescharacteristic
431C2×C3#338111yescharacteristic
432C2×C3#338411yescharacteristic
433C22—42no—
434C22—42no—
435C22—42no—
436C22—81no—
437C22—81no—
438C22—81no—
439C22—81no—
440C22C2×C3×#12111yescharacteristic
441C22C3×#54711yescharacteristic
442C22C3×D4×Q811yescharacteristic
443C22C2×C3×#13511yescharacteristic
444C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
445C22C3×#57711yescharacteristic
446C22C3×#57711yescharacteristic
447C4—22no—
448C4—42no—
449C4—44no—
450C4—42no—
451C4—42no—
452C4—81no—
453C4—82no—
454C4—81no—
455C4C3×#46314yes—
456C4C3×#53311yescharacteristic
457C4C3×#54111yescharacteristic
458C4C3×#54811yescharacteristic
459C4C3×D4×Q811yescharacteristic
460C3#2545211yescharacteristic, direct product factor, Sylow
461C2—82no—
462C2C3×#242911yescharacteristic
463C2C2×C3×#46311yescharacteristic
464C2C3×#304311yescharacteristic
465C2C3×#334011yescharacteristic
466C2C3×#335411yescharacteristic
467C2C3×#338111yescharacteristic
468C2C3×#338411yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C4×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#498C2semidirect
4. C3×#2365C2semidirect
5. C3×#2085C2semidirect
7. C3×#1987C2semidirect
8. C3×#1949C2semidirect
9. C3×#1939C2semidirect
12. #25452C3direct (N × H)
44. C2×#498C2×C3semidirect
47. #2365C2×C3semidirect
48. #2085C2×C3semidirect
50. #1987C2×C3semidirect
51. #1949C2×C3semidirect
52. #1939C2×C3semidirect
93. C3×#102Q8semidirect
229. #102C3×Q8semidirect
460. C3#25452direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4982
C2×C3×#4742
C3×#25632
C3×#23652
C3×#20852
C3×#20652
C3×#19872
C3×#19492
C3×#19392
C3×#19262
C3×C4×#1022
#254523