Finite Group Databasebeta

← C3×#27768 #10646375

Subgroups — C3×#27768 #10646375

Order 768.

802 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#488C212yesmaximal
2C3×#2552C211yesmaximal, characteristic, complemented
3C3×#2547C211yesmaximal, characteristic
4C3×#2042C212yesmaximal
5C3×#2026C212yesmaximal, complemented
6C3×#1981C211yesmaximal, characteristic, complemented
7C3×#1956C211yesmaximal, characteristic, complemented
8C3×#1933C212yesmaximal, complemented
9C3×#1802C211yesmaximal, characteristic
10C3×C8×C4⋊C4C211yesmaximal, characteristic, complemented
11C3×C4×#102C211yesmaximal, characteristic
12#27768C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×#125C2212yes—
15C2×C3×#124C2212yes—
16C2×C3×C4×D4C2211yescharacteristic
17C3×#488—28no—
18C2×C3×C4⋊C8C2211yescharacteristic
19C2×C3×#103C2212yes—
20C2×C3×#103C2212yes—
21C2×C3×#103C2212yes—
22C2×C3×#103C2212yes—
23C2×C3×#102C2212yes—
24C2×C3×#102C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×C4×C8C2212yes—
27C2×C3×C4×C8C2211yescharacteristic
28C3×#398—22no—
29C3×#397—22no—
30C3×#391C2212yes—
31C3×#389—21no—
32C3×#389C2212yes—
33C3×#384—21no—
34C3×#384C2211yescharacteristic
35C3×#380—21no—
36C3×#380—21no—
37C3×C4×C4⋊C4C2211yescharacteristic
38C3×C4×C4⋊C4C2211yescharacteristic
39C3×#340C2211yescharacteristic
40C3×#340C2211yescharacteristic
41C3×#340C2212yes—
42C3×#336C2212yes—
43C2×#488C2×C312yes—
44#2552C2×C311yescharacteristic, complemented
45#2547C2×C311yescharacteristic
46#2042C2×C312yes—
47#2026C2×C312yescomplemented
48#1981C2×C311yescharacteristic, complemented
49#1956C2×C311yescharacteristic, complemented
50#1933C2×C312yescomplemented
51#1802C2×C311yescharacteristic
52C8×C4⋊C4C2×C311yescharacteristic, complemented
53C4×#102C2×C311yescharacteristic
54C22×C3×Q8C2312yes—
55C22×C3×D4C2311yescharacteristic
56C23×C3×C4—21no—
57C22×C3×C8C2312yes—
58C22×C3×C8C2311yescharacteristic
59C22×C3×C8C2311yescharacteristic
60C3×C4⋊Q8—22no—
61C3×C4⋊Q8—22no—
62C3×#125—24no—
63C3×#125—42no—
64C3×#124—24no—
65C3×#124—42no—
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⋊C4—22no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4D411yescharacteristic
76C2×C3×C4⋊C4D411yescharacteristic
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4—21no—
84C2×C3×C42—21no—
85C2×C3×C42C2312yes—
86C2×C3×C42C2311yescharacteristic
87C3×C4⋊C8—22no—
88C3×#103—24no—
89C3×#103—24no—
90C3×#103—24no—
91C3×#103—42no—
92C3×#102—24no—
93C3×#102—22no—
94C3×#102D414yes—
95C3×C4×C8—24no—
96C3×C4×C8—22no—
97C3×#95—21no—
98C3×#95—22no—
99C3×#95—22no—
100C3×#95—22no—
101C3×#95D412yes—
102C2×C4⋊Q8C22×C311yescharacteristic
103C2×#125C22×C312yes—
104C2×#124C22×C312yes—
105C2×C4×D4C22×C311yescharacteristic
106#488—28no—
107C2×C4⋊C8C22×C311yescharacteristic
108C2×#103C22×C312yes—
109C2×#103C22×C312yes—
110C2×#103C22×C312yes—
111C2×#103C22×C312yes—
112C2×#102C22×C312yes—
113C2×#102C22×C311yescharacteristic
114C2×#102C22×C311yescharacteristic
115C2×C4×C8C22×C312yes—
116C2×C4×C8C22×C311yescharacteristic
117#398—22no—
118#397—22no—
119#391C22×C312yes—
120#389—21no—
121#389C22×C312yes—
122#384—21no—
123#384C22×C311yescharacteristic
124#380—21no—
125#380—21no—
126C4×C4⋊C4C22×C311yescharacteristic
127C4×C4⋊C4C22×C311yescharacteristic
128#340C22×C312yes—
129#340C22×C311yescharacteristic
130#340C22×C311yescharacteristic
131#336C22×C312yes—
132C24×C3—21no—
133C2×C3×Q8—22no—
134C2×C3×Q8—22no—
135C2×C3×Q8—22no—
136C2×C3×Q8—22no—
137C2×C3×Q8—22no—
138C2×C3×Q8—22no—
139C2×C3×D4—21no—
140C2×C3×D4—21no—
141C2×C3×D4—21no—
142C2×C3×D4—21no—
143C2×C3×D4—21no—
144C2×C3×D4C2×D412yes—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—22no—
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—22no—
161C2×C3×C8—21no—
162C2×C3×C8—21no—
163C2×C3×C8—22no—
164C2×C3×C8—22no—
165C2×C3×C8—21no—
166C2×C3×C8C2×D412yes—
167C3×C4⋊C4—22no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—21no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—44no—
176C3×C4⋊C4—44no—
177C3×C4⋊C4—42no—
178C3×C4⋊C4—42no—
179C3×C4⋊C4—42no—
180C3×C4⋊C4—81no—
181C3×C4⋊C4—82no—
182C3×C4⋊C4C2×D412yes—
183C3×C4⋊C4#4112yes—
184C3×C22⋊C4—44no—
185C3×C22⋊C4—44no—
186C3×C22⋊C4—42no—
187C3×C42—22no—
188C3×C42—22no—
189C3×C42—42no—
190C3×C42#4112yes—
191C3×C42#4112yes—
192C22×Q8C23×C312yes—
193C22×D4C23×C311yescharacteristic
194C23×C4—21no—
195C22×C8C23×C311yescharacteristic
196C22×C8C23×C312yes—
197C22×C8C23×C311yescharacteristic
198C4⋊Q8—22no—
199C4⋊Q8—22no—
200#125—24no—
201#125—42no—
202#124—24no—
203#124—42no—
204C4×D4—22no—
205C4×D4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4C3×D411yescharacteristic
214C2×C4⋊C4C3×D411yescharacteristic
215C2×C4⋊C4C23×C311yescharacteristic
216C2×C4⋊C4C23×C312yes—
217C2×C4⋊C4C23×C311yescharacteristic
218C2×C4⋊C4C23×C311yescharacteristic
219C2×C22⋊C4—21no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C42—21no—
223C2×C42C23×C312yes—
224C2×C42C23×C311yescharacteristic
225C4⋊C8—22no—
226#103—24no—
227#103—24no—
228#103—24no—
229#103—42no—
230#102—24no—
231#102—22no—
232#102C3×D414yes—
233C4×C8—24no—
234C4×C8—22no—
235#95—21no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95C3×D412yes—
240C23×C3—22no—
241C23×C3—41no—
242C23×C3—41no—
243C23×C3—41no—
244C23×C3—41no—
245C23×C3—41no—
246C23×C3—41no—
247C23×C3C22×D411yescharacteristic
248C3×Q8—42no—
249C3×Q8—42no—
250C3×Q8—42no—
251C3×Q8—42no—
252C3×D4—22no—
253C3×D4—22no—
254C3×D4—41no—
255C3×D4—41no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—41no—
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—42no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—41no—
277C2×C3×C4—41no—
278C2×C3×C4—42no—
279C2×C3×C4—84no—
280C2×C3×C4#12412yes—
281C2×C3×C4#12412yes—
282C2×C3×C4#13512yes—
283C2×C3×C4#13512yes—
284C2×C3×C4C22×D411yescharacteristic
285C2×C3×C4C2×#4111yescharacteristic
286C2×C3×C4C2×#4111yescharacteristic
287C2×C3×C4C2×#4111yescharacteristic
288C2×C3×C4H(5,𝔽2)11yescharacteristic
289C2×C3×C42−1+411yescharacteristic
290C3×C8—22no—
291C3×C8—42no—
292C3×C8—41no—
293C24—21no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4C2×C3×D412yes—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4C2×C3×D412yes—
315C22×C4C2×C3×D412yes—
316C22×C4C2×C3×D411yescharacteristic
317C22×C4C2×C3×D411yescharacteristic
318C22×C4C24×C311yescharacteristic
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—22no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—21no—
327C2×C8C2×C3×D412yes—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—22no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—22no—
336C4⋊C4—44no—
337C4⋊C4—44no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—81no—
342C4⋊C4—82no—
343C4⋊C4C2×C3×D412yes—
344C4⋊C4C3×#4112yes—
345C22⋊C4—44no—
346C22⋊C4—44no—
347C22⋊C4—42no—
348C42—22no—
349C42—22no—
350C42—42no—
351C42C3×#4112yes—
352C42C3×#4112yes—
353C22×C3—42no—
354C22×C3—42no—
355C22×C3—42no—
356C22×C3—81no—
357C22×C3—81no—
358C22×C3—81no—
359C22×C3—81no—
360C22×C3C2×#12411yescharacteristic
361C22×C3D4211yescharacteristic
362C22×C3#54811yescharacteristic
363C22×C3C2×#13511yescharacteristic
364C22×C3C2×#13511yescharacteristic
365C22×C3#57711yescharacteristic
366C22×C3#57811yescharacteristic
367C3×C4—22no—
368C3×C4—42no—
369C3×C4—42no—
370C3×C4—42no—
371C3×C4—42no—
372C3×C4—81no—
373C3×C4—82no—
374C3×C4—82no—
375C3×C4#48814yescomplemented
376C3×C4#53211yescharacteristic
377C3×C4#54211yescharacteristic
378C3×C4#54711yescharacteristic
379C3×C4#55511yescharacteristic
380C23—22no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23C22×C3×D411yescharacteristic
388Q8—42no—
389Q8—42no—
390Q8—42no—
391Q8—42no—
392D4—22no—
393D4—22no—
394D4—41no—
395D4—41no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—84no—
420C2×C4C3×#12412yes—
421C2×C4C3×#12412yes—
422C2×C4C3×#13512yes—
423C2×C4C3×#13512yes—
424C2×C4C22×C3×D411yescharacteristic
425C2×C4C2×C3×#4111yescharacteristic
426C2×C4C2×C3×#4111yescharacteristic
427C2×C4C2×C3×#4111yescharacteristic
428C2×C4C3×H(5,𝔽2)11yescharacteristic
429C2×C4C3×2−1+411yescharacteristic
430C8—22no—
431C8—42no—
432C8—41no—
433C2×C3—82no—
434C2×C3#249111yescharacteristic
435C2×C3C2×#48811yescharacteristic
436C2×C3#313711yescharacteristic
437C2×C3#328311yescharacteristic
438C2×C3#328411yescharacteristic
439C2×C3#336011yescharacteristic
440C2×C3#336411yescharacteristic
441C22—42no—
442C22—42no—
443C22—42no—
444C22—81no—
445C22—81no—
446C22—81no—
447C22—81no—
448C22C2×C3×#12411yescharacteristic
449C22C3×D4211yescharacteristic
450C22C3×#54811yescharacteristic
451C22C2×C3×#13511yescharacteristic
452C22C2×C3×#13511yescharacteristic
453C22C3×#57711yescharacteristic
454C22C3×#57811yescharacteristic
455C4—22no—
456C4—42no—
457C4—42no—
458C4—42no—
459C4—42no—
460C4—81no—
461C4—82no—
462C4—82no—
463C4C3×#48814yescomplemented
464C4C3×#53211yescharacteristic
465C4C3×#54211yescharacteristic
466C4C3×#54711yescharacteristic
467C4C3×#55511yescharacteristic
468C3#2776811yescharacteristic, direct product factor, Sylow
469C2—82no—
470C2C3×#249111yescharacteristic
471C2C2×C3×#48811yescharacteristic
472C2C3×#313711yescharacteristic
473C2C3×#328311yescharacteristic
474C2C3×#328411yescharacteristic
475C2C3×#336011yescharacteristic
476C2C3×#336411yescharacteristic
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
2. C3×#2552C2semidirect
5. C3×#2026C2semidirect
6. C3×#1981C2semidirect
7. C3×#1956C2semidirect
8. C3×#1933C2semidirect
10. C3×C8×C4⋊C4C2semidirect
12. #27768C3direct (N × H)
44. #2552C2×C3semidirect
47. #2026C2×C3semidirect
48. #1981C2×C3semidirect
49. #1956C2×C3semidirect
50. #1933C2×C3semidirect
52. C8×C4⋊C4C2×C3semidirect
375. C3×C4#488semidirect
463. C4C3×#488semidirect
468. C3#27768direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4882
C3×#25522
C3×#25472
C3×#20422
C3×#20262
C3×#19812
C3×#19562
C3×#19332
C3×#18022
C3×C8×C4⋊C42
C3×C4×#1022
#277683