Finite Group Databasebeta

← C3×#25450 #10644057

Subgroups — C3×#25450 #10644057

Order 768.

778 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#500C211yesmaximal, characteristic, complemented
2C2×C3×#476C211yesmaximal, characteristic
3C3×#2563C211yesmaximal, characteristic
4C3×#2369C211yesmaximal, characteristic, complemented
5C3×#2085C212yesmaximal, complemented
6C3×#2067C212yesmaximal
7C3×#1987C211yesmaximal, characteristic, complemented
8C3×#1949C212yesmaximal, complemented
9C3×#1939C211yesmaximal, characteristic, complemented
10C3×#1925C212yesmaximal
11C3×C4×#102C211yesmaximal, characteristic
12#25450C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C3×#500—24no—
17C3×#476—24no—
18C2×C3×#107C2212yes—
19C2×C3×#107C2212yes—
20C2×C3×#106C2212yes—
21C2×C3×C4⋊C8C2212yes—
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×#102C2212yes—
25C2×C3×#102C2211yescharacteristic
26C2×C3×C4×C8C2211yescharacteristic
27C3×#393—22no—
28C3×#391—22no—
29C3×#390C2212yes—
30C3×#389—21no—
31C3×#389C2212yes—
32C3×#382C2212yes—
33C3×#380—21no—
34C3×#380—21no—
35C3×C4×C4⋊C4—21no—
36C3×C4×C4⋊C4C2211yescharacteristic
37C3×C4×C4⋊C4C2212yes—
38C3×C4×C4⋊C4C2211yescharacteristic
39C3×#340C2211yescharacteristic
40C3×#340C2211yescharacteristic
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#336C2212yes—
44C2×#500C2×C311yescharacteristic, complemented
45C2×#476C2×C311yescharacteristic
46#2563C2×C311yescharacteristic
47#2369C2×C311yescharacteristic, complemented
48#2085C2×C312yescomplemented
49#2067C2×C312yes—
50#1987C2×C311yescharacteristic, complemented
51#1949C2×C312yescomplemented
52#1939C2×C311yescharacteristic, complemented
53#1925C2×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×D4—22no—
67C3×C4×D4—22no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4D411yescharacteristic
72C2×C3×C4⋊C4D411yescharacteristic
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C22⋊C4—21no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C42—21no—
82C2×C3×C42—22no—
83C2×C3×C42C2312yes—
84C2×C3×C42C2311yescharacteristic
85C3×#107—24no—
86C3×#107—24no—
87C3×#106—24no—
88C3×C4⋊C8—24no—
89C3×C4⋊C8—22no—
90C3×#102—22no—
91C3×#102—24no—
92C3×#102Q814yescomplemented
93C3×C4×C8—22no—
94C3×#95—22no—
95C3×#95—22no—
96C3×#95—21no—
97C3×#95—22no—
98C3×#95—22no—
99C3×#95D412yes—
100C2×C4⋊Q8C22×C311yescharacteristic
101C2×C4⋊Q8C22×C311yescharacteristic
102C2×C4×D4C22×C311yescharacteristic
103#500—24no—
104#476—24no—
105C2×#107C22×C312yes—
106C2×#107C22×C312yes—
107C2×#106C22×C312yes—
108C2×C4⋊C8C22×C312yes—
109C2×C4⋊C8C22×C311yescharacteristic
110C2×#102C22×C311yescharacteristic
111C2×#102C22×C312yes—
112C2×#102C22×C311yescharacteristic
113C2×C4×C8C22×C311yescharacteristic
114#393—22no—
115#391—22no—
116#390C22×C312yes—
117#389—21no—
118#389C22×C312yes—
119#382C22×C312yes—
120#380—21no—
121#380—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4C22×C312yes—
124C4×C4⋊C4C22×C311yescharacteristic
125C4×C4⋊C4C22×C311yescharacteristic
126#340C22×C312yes—
127#340C22×C311yescharacteristic
128#340C22×C312yes—
129#340C22×C311yescharacteristic
130#336C22×C312yes—
131C24×C3—21no—
132C2×C3×Q8—24no—
133C2×C3×Q8—42no—
134C2×C3×Q8—81no—
135C2×C3×Q8—81no—
136C2×C3×Q8—81no—
137C2×C3×Q8—81no—
138C2×C3×D4—21no—
139C2×C3×D4—21no—
140C2×C3×D4—21no—
141C2×C3×D4—21no—
142C2×C3×D4—21no—
143C2×C3×D4C2×Q812yes—
144C22×C3×C4—21no—
145C22×C3×C4—22no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4—42no—
150C22×C3×C4—42no—
151C22×C3×C4—42no—
152C22×C3×C4C2×D412yes—
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2411yescharacteristic
157C2×C3×C8—21no—
158C2×C3×C8—21no—
159C2×C3×C8—21no—
160C2×C3×C8—21no—
161C2×C3×C8—22no—
162C2×C3×C8—22no—
163C2×C3×C8—22no—
164C2×C3×C8C2×Q812yes—
165C2×C3×C8#4112yes—
166C3×C4⋊C4—24no—
167C3×C4⋊C4—22no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4—21no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—42no—
178C3×C4⋊C4—82no—
179C3×C4⋊C4C2×D414yes—
180C3×C4⋊C4C2×Q812yes—
181C3×C22⋊C4—42no—
182C3×C22⋊C4—82no—
183C3×C22⋊C4—82no—
184C3×C42—22no—
185C3×C42—22no—
186C3×C42—42no—
187C3×C42—82no—
188C3×C42C2×D412yes—
189C3×C42#4112yes—
190C22×Q8—21no—
191C22×Q8—21no—
192C22×D4C23×C311yescharacteristic
193C23×C4—21no—
194C22×C8C23×C312yes—
195C22×C8C23×C311yescharacteristic
196C22×C8C23×C311yescharacteristic
197C4⋊Q8—22no—
198C4⋊Q8—22no—
199C4⋊Q8—22no—
200C4⋊Q8C3×D414yes—
201C4×D4—22no—
202C4×D4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4C3×D411yescharacteristic
207C2×C4⋊C4C3×D411yescharacteristic
208C2×C4⋊C4C23×C311yescharacteristic
209C2×C4⋊C4C23×C312yes—
210C2×C4⋊C4C23×C311yescharacteristic
211C2×C4⋊C4C23×C312yes—
212C2×C4⋊C4C23×C311yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C42—21no—
217C2×C42—22no—
218C2×C42C23×C312yes—
219C2×C42C23×C311yescharacteristic
220#107—24no—
221#107—24no—
222#106—24no—
223C4⋊C8—24no—
224C4⋊C8—22no—
225#102—22no—
226#102—24no—
227#102C3×Q814yescomplemented
228C4×C8—22no—
229#95—22no—
230#95—22no—
231#95—21no—
232#95—22no—
233#95—22no—
234#95C3×D412yes—
235C23×C3—22no—
236C23×C3—41no—
237C23×C3—41no—
238C23×C3—41no—
239C23×C3—41no—
240C23×C3—41no—
241C23×C3—41no—
242C23×C3C22×D411yescharacteristic
243C3×Q8—84no—
244C3×Q8—84no—
245C3×D4—22no—
246C3×D4—22no—
247C3×D4—41no—
248C3×D4—41no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—24no—
252C2×C3×C4—22no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4—41no—
264C2×C3×C4—42no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—41no—
272C2×C3×C4—84no—
273C2×C3×C4#12112yes—
274C2×C3×C4#12112yes—
275C2×C3×C4#13512yes—
276C2×C3×C4#13712yes—
277C2×C3×C4C22×D411yescharacteristic
278C2×C3×C4C22×Q811yescharacteristic
279C2×C3×C4C2×#4111yescharacteristic
280C2×C3×C4C2×#4111yescharacteristic
281C2×C3×C42−1+411yescharacteristic
282C2×C3×C42−1+411yescharacteristic
283C3×C8—22no—
284C3×C8—22no—
285C3×C8—42no—
286C24—21no—
287C2×Q8—24no—
288C2×Q8—42no—
289C2×Q8—81no—
290C2×Q8—81no—
291C2×Q8—81no—
292C2×Q8—81no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4C2×C3×Q812yes—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4C2×C3×D412yes—
308C22×C4C2×C3×D412yes—
309C22×C4C2×C3×D411yescharacteristic
310C22×C4C2×C3×D411yescharacteristic
311C22×C4C24×C311yescharacteristic
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8C2×C3×Q812yes—
320C2×C8C3×#4112yes—
321C4⋊C4—24no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—22no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—22no—
331C4⋊C4—21no—
332C4⋊C4—42no—
333C4⋊C4—82no—
334C4⋊C4C2×C3×D414yes—
335C4⋊C4C2×C3×Q812yes—
336C22⋊C4—42no—
337C22⋊C4—82no—
338C22⋊C4—82no—
339C42—22no—
340C42—22no—
341C42—42no—
342C42—82no—
343C42C2×C3×D412yes—
344C42C3×#4112yes—
345C22×C3—42no—
346C22×C3—42no—
347C22×C3—42no—
348C22×C3—81no—
349C22×C3—81no—
350C22×C3—81no—
351C22×C3—81no—
352C22×C3C2×#12111yescharacteristic
353C22×C3#54711yescharacteristic
354C22×C3D4×Q811yescharacteristic
355C22×C3C2×#13511yescharacteristic
356C22×C3C2×#13711yescharacteristic
357C22×C3#57611yescharacteristic
358C22×C3#57611yescharacteristic
359C3×C4—22no—
360C3×C4—42no—
361C3×C4—42no—
362C3×C4—42no—
363C3×C4—44no—
364C3×C4—81no—
365C3×C4—81no—
366C3×C4—82no—
367C3×C4#46414yes—
368C3×C4#53311yescharacteristic
369C3×C4#54111yescharacteristic
370C3×C4#54811yescharacteristic
371C3×C4D4×Q811yescharacteristic
372C23—22no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23C22×C3×D411yescharacteristic
380Q8—84no—
381Q8—84no—
382D4—22no—
383D4—22no—
384D4—41no—
385D4—41no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—84no—
410C2×C4C3×#12112yes—
411C2×C4C3×#12112yes—
412C2×C4C3×#13512yes—
413C2×C4C3×#13712yes—
414C2×C4C22×C3×D411yescharacteristic
415C2×C4C22×C3×Q811yescharacteristic
416C2×C4C2×C3×#4111yescharacteristic
417C2×C4C2×C3×#4111yescharacteristic
418C2×C4C3×2−1+411yescharacteristic
419C2×C4C3×2−1+411yescharacteristic
420C8—22no—
421C8—22no—
422C8—42no—
423C2×C3—82no—
424C2×C3#242911yescharacteristic
425C2×C3C2×#46411yescharacteristic
426C2×C3#304111yescharacteristic
427C2×C3#334111yescharacteristic
428C2×C3#335711yescharacteristic
429C2×C3#338011yescharacteristic
430C2×C3#338511yescharacteristic
431C22—42no—
432C22—42no—
433C22—42no—
434C22—81no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22C2×C3×#12111yescharacteristic
439C22C3×#54711yescharacteristic
440C22C3×D4×Q811yescharacteristic
441C22C2×C3×#13511yescharacteristic
442C22C2×C3×#13711yescharacteristic
443C22C3×#57611yescharacteristic
444C22C3×#57611yescharacteristic
445C4—22no—
446C4—44no—
447C4—42no—
448C4—42no—
449C4—42no—
450C4—81no—
451C4—82no—
452C4—81no—
453C4C3×#46414yes—
454C4C3×#53311yescharacteristic
455C4C3×#54111yescharacteristic
456C4C3×#54811yescharacteristic
457C4C3×D4×Q811yescharacteristic
458C3#2545011yescharacteristic, direct product factor, Sylow
459C2—82no—
460C2C3×#242911yescharacteristic
461C2C2×C3×#46411yescharacteristic
462C2C3×#304111yescharacteristic
463C2C3×#334111yescharacteristic
464C2C3×#335711yescharacteristic
465C2C3×#338011yescharacteristic
466C2C3×#338511yescharacteristic
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×#500C2semidirect
4. C3×#2369C2semidirect
5. C3×#2085C2semidirect
7. C3×#1987C2semidirect
8. C3×#1949C2semidirect
9. C3×#1939C2semidirect
12. #25450C3direct (N × H)
44. C2×#500C2×C3semidirect
47. #2369C2×C3semidirect
48. #2085C2×C3semidirect
50. #1987C2×C3semidirect
51. #1949C2×C3semidirect
52. #1939C2×C3semidirect
92. C3×#102Q8semidirect
227. #102C3×Q8semidirect
458. C3#25450direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5002
C2×C3×#4762
C3×#25632
C3×#23692
C3×#20852
C3×#20672
C3×#19872
C3×#19492
C3×#19392
C3×#19252
C3×C4×#1022
#254503