Finite Group Databasebeta

← C3×#27748 #10646355

Subgroups — C3×#27748 #10646355

Order 768.

726 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
1C3×#2571C211yesmaximal, characteristic, complemented
2C3×#2547C211yesmaximal, characteristic
3C3×#2074C212yesmaximal, complemented
4C3×#2042C212yesmaximal
5C3×#1981C211yesmaximal, characteristic, complemented
6C3×#1958C212yesmaximal
7C3×#1956C211yesmaximal, characteristic, complemented
8C3×#1933C212yesmaximal, complemented
9C3×#1802C211yesmaximal, characteristic
10C3×C8×C4⋊C4C211yesmaximal, characteristic, complemented
11C3×C4×#102C211yesmaximal, characteristic
12#27748C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×#124C2212yes—
15C2×C3×C4×D4C2211yescharacteristic
16C2×C3×C4⋊C8C2211yescharacteristic
17C2×C3×#103C2212yes—
18C2×C3×#103C2212yes—
19C2×C3×#103C2212yes—
20C2×C3×#103C2212yes—
21C2×C3×#102C2211yescharacteristic
22C2×C3×#102C2211yescharacteristic
23C2×C3×#102C2212yes—
24C2×C3×C4×C8C2211yescharacteristic
25C2×C3×C4×C8C2212yes—
26C3×#400—22no—
27C3×#397—22no—
28C3×#391C2212yes—
29C3×#389—21no—
30C3×#389C2212yes—
31C3×#384C2211yescharacteristic
32C3×#382—21no—
33C3×#382C2212yes—
34C3×#380—21no—
35C3×#380—21no—
36C3×C4×C4⋊C4C2211yescharacteristic
37C3×C4×C4⋊C4C2211yescharacteristic
38C3×#340C2211yescharacteristic
39C3×#340C2211yescharacteristic
40C3×#340C2212yes—
41C3×#336C2212yes—
42#2571C2×C311yescharacteristic, complemented
43#2547C2×C311yescharacteristic
44#2074C2×C312yescomplemented
45#2042C2×C312yes—
46#1981C2×C311yescharacteristic, complemented
47#1958C2×C312yes—
48#1956C2×C311yescharacteristic, complemented
49#1933C2×C312yescomplemented
50#1802C2×C311yescharacteristic
51C8×C4⋊C4C2×C311yescharacteristic, complemented
52C4×#102C2×C311yescharacteristic
53C22×C3×Q8C2312yes—
54C22×C3×D4C2311yescharacteristic
55C23×C3×C4—21no—
56C22×C3×C8C2312yes—
57C22×C3×C8C2311yescharacteristic
58C22×C3×C8C2311yescharacteristic
59C3×C4⋊Q8—22no—
60C3×C4⋊Q8—22no—
61C3×#124—24no—
62C3×#124—42no—
63C3×C4×D4—22no—
64C3×C4×D4—22no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4D411yescharacteristic
71C2×C3×C4⋊C4D411yescharacteristic
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C22⋊C4—22no—
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4—21no—
79C2×C3×C42—21no—
80C2×C3×C42C2312yes—
81C2×C3×C42C2311yescharacteristic
82C3×C4⋊C8—22no—
83C3×#103—24no—
84C3×#103—42no—
85C3×#103—42no—
86C3×#103—42no—
87C3×#102—22no—
88C3×#102—22no—
89C3×#102—42no—
90C3×C4×C8—22no—
91C3×C4×C8—42no—
92C3×#95—21no—
93C3×#95—22no—
94C3×#95—22no—
95C3×#95—22no—
96C3×#95—22no—
97C3×#95—22no—
98C3×#95D412yes—
99C2×C4⋊Q8C22×C311yescharacteristic
100C2×#124C22×C312yes—
101C2×C4×D4C22×C311yescharacteristic
102C2×C4⋊C8C22×C311yescharacteristic
103C2×#103C22×C312yes—
104C2×#103C22×C312yes—
105C2×#103C22×C312yes—
106C2×#103C22×C312yes—
107C2×#102C22×C311yescharacteristic
108C2×#102C22×C311yescharacteristic
109C2×#102C22×C312yes—
110C2×C4×C8C22×C311yescharacteristic
111C2×C4×C8C22×C312yes—
112#400—22no—
113#397—22no—
114#391C22×C312yes—
115#389—21no—
116#389C22×C312yes—
117#384C22×C311yescharacteristic
118#382—21no—
119#382C22×C312yes—
120#380—21no—
121#380—21no—
122C4×C4⋊C4C22×C311yescharacteristic
123C4×C4⋊C4C22×C311yescharacteristic
124#340C22×C312yes—
125#340C22×C311yescharacteristic
126#340C22×C311yescharacteristic
127#336C22×C312yes—
128C24×C3—21no—
129C2×C3×Q8—22no—
130C2×C3×Q8—22no—
131C2×C3×Q8—22no—
132C2×C3×Q8—22no—
133C2×C3×Q8—22no—
134C2×C3×Q8—22no—
135C2×C3×D4—21no—
136C2×C3×D4—21no—
137C2×C3×D4—21no—
138C2×C3×D4—21no—
139C2×C3×D4—21no—
140C2×C3×D4#4112yes—
141C22×C3×C4—21no—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—42no—
147C22×C3×C4—42no—
148C22×C3×C4—42no—
149C22×C3×C4C2×D412yes—
150C22×C3×C4C2×D412yes—
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2411yescharacteristic
154C2×C3×C8—21no—
155C2×C3×C8—21no—
156C2×C3×C8—21no—
157C2×C3×C8—21no—
158C2×C3×C8—22no—
159C2×C3×C8—22no—
160C2×C3×C8—22no—
161C2×C3×C8—21no—
162C2×C3×C8#4112yes—
163C3×C4⋊C4—22no—
164C3×C4⋊C4—22no—
165C3×C4⋊C4—21no—
166C3×C4⋊C4—21no—
167C3×C4⋊C4—22no—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—21no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—21no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—42no—
174C3×C4⋊C4—42no—
175C3×C4⋊C4—42no—
176C3×C4⋊C4—81no—
177C3×C4⋊C4—82no—
178C3×C22⋊C4—44no—
179C3×C22⋊C4—44no—
180C3×C22⋊C4—42no—
181C3×C42—22no—
182C3×C42—22no—
183C3×C42—42no—
184C3×C42#4112yes—
185C3×C42#4112yes—
186C22×Q8C23×C312yes—
187C22×D4C23×C311yescharacteristic
188C23×C4—21no—
189C22×C8C23×C311yescharacteristic
190C22×C8C23×C312yes—
191C22×C8C23×C311yescharacteristic
192C4⋊Q8—22no—
193C4⋊Q8—22no—
194#124—24no—
195#124—42no—
196C4×D4—22no—
197C4×D4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4C3×D411yescharacteristic
204C2×C4⋊C4C3×D411yescharacteristic
205C2×C4⋊C4C23×C311yescharacteristic
206C2×C4⋊C4C23×C311yescharacteristic
207C2×C4⋊C4C23×C311yescharacteristic
208C2×C4⋊C4C23×C312yes—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C42—21no—
213C2×C42C23×C312yes—
214C2×C42C23×C311yescharacteristic
215C4⋊C8—22no—
216#103—24no—
217#103—42no—
218#103—42no—
219#103—42no—
220#102—22no—
221#102—22no—
222#102—42no—
223C4×C8—22no—
224C4×C8—42no—
225#95—21no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95C3×D412yes—
232C23×C3—22no—
233C23×C3—41no—
234C23×C3—41no—
235C23×C3—41no—
236C23×C3—41no—
237C23×C3—41no—
238C23×C3—41no—
239C23×C3C22×D411yescharacteristic
240C3×Q8—42no—
241C3×Q8—42no—
242C3×Q8—42no—
243C3×Q8—42no—
244C3×D4—22no—
245C3×D4—22no—
246C3×D4—41no—
247C3×D4—41no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—22no—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—41no—
257C2×C3×C4—42no—
258C2×C3×C4—42no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—42no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—42no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—84no—
272C2×C3×C4#12412yes—
273C2×C3×C4#12412yes—
274C2×C3×C4#13512yes—
275C2×C3×C4#13512yes—
276C2×C3×C4C2×#4111yescharacteristic
277C2×C3×C4C2×#4111yescharacteristic
278C2×C3×C4C2×#4111yescharacteristic
279C2×C3×C4C2×#4111yescharacteristic
280C2×C3×C4H(5,𝔽2)11yescharacteristic
281C2×C3×C4H(5,𝔽2)11yescharacteristic
282C3×C8—22no—
283C3×C8—42no—
284C3×C8—41no—
285C24—21no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4C3×#4112yes—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4C2×C3×D412yes—
307C22×C4C2×C3×D412yes—
308C22×C4C2×C3×D411yescharacteristic
309C22×C4C2×C3×D411yescharacteristic
310C22×C4C24×C311yescharacteristic
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—21no—
319C2×C8C3×#4112yes—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—22no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—22no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—81no—
334C4⋊C4—82no—
335C22⋊C4—44no—
336C22⋊C4—44no—
337C22⋊C4—42no—
338C42—22no—
339C42—22no—
340C42—42no—
341C42C3×#4112yes—
342C42C3×#4112yes—
343C22×C3—42no—
344C22×C3—42no—
345C22×C3—42no—
346C22×C3—81no—
347C22×C3—81no—
348C22×C3—81no—
349C22×C3—81no—
350C22×C3C2×#12411yescharacteristic
351C22×C3#54611yescharacteristic
352C22×C3#55011yescharacteristic
353C22×C3C2×#13511yescharacteristic
354C22×C3C2×#13511yescharacteristic
355C22×C3#57711yescharacteristic
356C22×C3#57811yescharacteristic
357C3×C4—22no—
358C3×C4—42no—
359C3×C4—42no—
360C3×C4—81no—
361C3×C4—81no—
362C3×C4—82no—
363C3×C4—82no—
364C3×C4—81no—
365C3×C4#48814yes—
366C3×C4#52911yescharacteristic
367C3×C4#54011yescharacteristic
368C3×C4#55311yescharacteristic
369C3×C4#55311yescharacteristic
370C23—22no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23C22×C3×D411yescharacteristic
378Q8—42no—
379Q8—42no—
380Q8—42no—
381Q8—42no—
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—22no—
393C2×C4—22no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—41no—
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—42no—
409C2×C4—84no—
410C2×C4C3×#12412yes—
411C2×C4C3×#12412yes—
412C2×C4C3×#13512yes—
413C2×C4C3×#13512yes—
414C2×C4C2×C3×#4111yescharacteristic
415C2×C4C2×C3×#4111yescharacteristic
416C2×C4C2×C3×#4111yescharacteristic
417C2×C4C2×C3×#4111yescharacteristic
418C2×C4C3×H(5,𝔽2)11yescharacteristic
419C2×C4C3×H(5,𝔽2)11yescharacteristic
420C8—22no—
421C8—42no—
422C8—41no—
423C2×C3—82no—
424C2×C3#249011yescharacteristic
425C2×C3C2×#48811yescharacteristic
426C2×C3#313711yescharacteristic
427C2×C3#330211yescharacteristic
428C2×C3#330311yescharacteristic
429C2×C3#339311yescharacteristic
430C2×C3#339411yescharacteristic
431C22—42no—
432C22—42no—
433C22—42no—
434C22—81no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22C2×C3×#12411yescharacteristic
439C22C3×#54611yescharacteristic
440C22C3×#55011yescharacteristic
441C22C2×C3×#13511yescharacteristic
442C22C2×C3×#13511yescharacteristic
443C22C3×#57711yescharacteristic
444C22C3×#57811yescharacteristic
445C4—22no—
446C4—42no—
447C4—42no—
448C4—81no—
449C4—81no—
450C4—82no—
451C4—82no—
452C4—81no—
453C4C3×#48814yes—
454C4C3×#52911yescharacteristic
455C4C3×#54011yescharacteristic
456C4C3×#55311yescharacteristic
457C4C3×#55311yescharacteristic
458C3#2774811yescharacteristic, direct product factor, Sylow
459C2—82no—
460C2C3×#249011yescharacteristic
461C2C2×C3×#48811yescharacteristic
462C2C3×#313711yescharacteristic
463C2C3×#330211yescharacteristic
464C2C3×#330311yescharacteristic
465C2C3×#339311yescharacteristic
466C2C3×#339411yescharacteristic
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. C3×#2571C2semidirect
3. C3×#2074C2semidirect
5. C3×#1981C2semidirect
7. C3×#1956C2semidirect
8. C3×#1933C2semidirect
10. C3×C8×C4⋊C4C2semidirect
12. #27748C3direct (N × H)
42. #2571C2×C3semidirect
44. #2074C2×C3semidirect
46. #1981C2×C3semidirect
48. #1956C2×C3semidirect
49. #1933C2×C3semidirect
51. C8×C4⋊C4C2×C3semidirect
458. C3#27748direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#25712
C3×#25472
C3×#20742
C3×#20422
C3×#19812
C3×#19582
C3×#19562
C3×#19332
C3×#18022
C3×C8×C4⋊C42
C3×C4×#1022
#277483