Finite Group Databasebeta

← C3×#27683 #10646290

Subgroups — C3×#27683 #10646290

Order 768.

854 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×#121C211yesmaximal, characteristic
2C3×#2285C211yesmaximal, characteristic, complemented
3C3×#2100C211yesmaximal, characteristic, complemented
4C3×#2099C211yesmaximal, characteristic, complemented
5C3×#2094C211yesmaximal, characteristic, complemented
6C3×#2093C211yesmaximal, characteristic, complemented
7C3×#1949C212yesmaximal, complemented
8C3×#1875C212yesmaximal
9C3×#1854C212yesmaximal, complemented
10C3×#1804C212yesmaximal, complemented
11C3×#1797C211yesmaximal, characteristic, complemented
12#27683C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#121C2211yescharacteristic
14C2×C3×C4×D4—21no—
15C2×C3×C4×D4C2211yescharacteristic
16C2×C3×#117C2212yes—
17C22×C3×C42C2211yescharacteristic
18C2×C3×#107C2212yes—
19C2×C3×#106C2212yes—
20C2×C3×C4⋊C8C2212yes—
21C2×C3×#102C2212yes—
22C2×C3×#102C2212yes—
23C2×C3×C22⋊C8C2212yes—
24C3×#390C2211yescharacteristic
25C3×#389C2211yescharacteristic
26C3×#388—21no—
27C3×#386—21no—
28C3×#384C2211yescharacteristic
29C3×#384C2211yescharacteristic
30C3×#382C2212yes—
31C3×#380—21no—
32C3×#380—21no—
33C3×#380C2211yescharacteristic
34C3×C4×C4⋊C4—21no—
35C3×C4×C4⋊C4—21no—
36C3×C4×C4⋊C4C2211yescharacteristic
37C3×C4×C22⋊C4—21no—
38C3×#340C2212yes—
39C3×#340C2212yes—
40C3×#340C2212yes—
41C3×#340C2212yes—
42C3×#336C2211yescharacteristic
43C3×#336C2211yescharacteristic
44C4×#121C2×C311yescharacteristic
45#2285C2×C311yescharacteristic, complemented
46#2100C2×C311yescharacteristic, complemented
47#2099C2×C311yescharacteristic, complemented
48#2094C2×C311yescharacteristic, complemented
49#2093C2×C311yescharacteristic, complemented
50#1949C2×C312yescomplemented
51#1875C2×C312yes—
52#1854C2×C312yescomplemented
53#1804C2×C312yescomplemented
54#1797C2×C311yescharacteristic, complemented
55C22×C3×D4—21no—
56C22×C3×D4C2311yescharacteristic
57C23×C3×C4—21no—
58C23×C3×C4D411yescharacteristic
59C23×C3×C4D411yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C22×C3×C8C2312yes—
62C22×C3×C8C2312yes—
63C3×#121—22no—
64C3×#121Q814yes—
65C3×C4×D4—24no—
66C3×C4×D4—44no—
67C3×#117—42no—
68C3×#117—42no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—21no—
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—21no—
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4—21no—
85C2×C3×C42—24no—
86C2×C3×C42—21no—
87C2×C3×C42—22no—
88C2×C3×C42D411yescharacteristic
89C2×C3×C42D411yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C3×#107—42no—
93C3×#106—42no—
94C3×C4⋊C8—24no—
95C3×#102—42no—
96C3×#102—42no—
97C3×C22⋊C8—24no—
98C3×#95—21no—
99C3×#95—21no—
100C3×#95—22no—
101C3×#95—22no—
102C3×#95—21no—
103C2×#121C22×C311yescharacteristic
104C2×C4×D4—21no—
105C2×C4×D4C22×C311yescharacteristic
106C2×#117C22×C312yes—
107C22×C42C22×C311yescharacteristic
108C2×#107C22×C312yes—
109C2×#106C22×C312yes—
110C2×C4⋊C8C22×C312yes—
111C2×#102C22×C312yes—
112C2×#102C22×C312yes—
113C2×C22⋊C8C22×C312yes—
114#390C22×C311yescharacteristic
115#389C22×C311yescharacteristic
116#388—21no—
117#386—21no—
118#384C22×C311yescharacteristic
119#384C22×C311yescharacteristic
120#382C22×C312yes—
121#380—21no—
122#380—21no—
123#380C22×C311yescharacteristic
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4C22×C311yescharacteristic
127C4×C22⋊C4—21no—
128#340C22×C312yes—
129#340C22×C312yes—
130#340C22×C312yes—
131#340C22×C312yes—
132#336C22×C311yescharacteristic
133#336C22×C311yescharacteristic
134C24×C3—21no—
135C24×C3C2×D411yescharacteristic
136C2×C3×D4—24no—
137C2×C3×D4—22no—
138C2×C3×D4—21no—
139C2×C3×D4—22no—
140C2×C3×D4—42no—
141C2×C3×D4—42no—
142C2×C3×D4C2×Q812yes—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—21no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—21no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—21no—
154C22×C3×C4—41no—
155C22×C3×C4—41no—
156C22×C3×C4—42no—
157C22×C3×C4—42no—
158C22×C3×C4—42no—
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×Q812yes—
165C22×C3×C4#4112yes—
166C22×C3×C4C2411yescharacteristic
167C2×C3×C8—22no—
168C2×C3×C8—22no—
169C2×C3×C8—22no—
170C2×C3×C8—22no—
171C2×C3×C8—22no—
172C2×C3×C8—22no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—42no—
181C3×C4⋊C4—42no—
182C3×C4⋊C4—42no—
183C3×C4⋊C4—42no—
184C3×C4⋊C4—42no—
185C3×C4⋊C4C2×Q812yes—
186C3×C22⋊C4—24no—
187C3×C22⋊C4—44no—
188C3×C22⋊C4—42no—
189C3×C22⋊C4—42no—
190C3×C22⋊C4—42no—
191C3×C22⋊C4—42no—
192C3×C42—22no—
193C3×C42—22no—
194C3×C42—22no—
195C3×C42—44no—
196C3×C42—42no—
197C3×C42#4114yes—
198C22×D4—21no—
199C22×D4C23×C311yescharacteristic
200C23×C4—21no—
201C23×C4C3×D411yescharacteristic
202C23×C4C3×D411yescharacteristic
203C23×C4C23×C311yescharacteristic
204C22×C8C23×C312yes—
205C22×C8C23×C312yes—
206#121—22no—
207#121C3×Q814yes—
208C4×D4—24no—
209C4×D4—44no—
210#117—42no—
211#117—42no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4C23×C312yes—
218C2×C4⋊C4C23×C312yes—
219C2×C4⋊C4C23×C311yescharacteristic
220C2×C4⋊C4C23×C311yescharacteristic
221C2×C4⋊C4C23×C311yescharacteristic
222C2×C22⋊C4—21no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C42—24no—
229C2×C42—22no—
230C2×C42—21no—
231C2×C42C3×D411yescharacteristic
232C2×C42C3×D411yescharacteristic
233C2×C42C23×C311yescharacteristic
234C2×C42C23×C311yescharacteristic
235#107—42no—
236#106—42no—
237C4⋊C8—24no—
238#102—42no—
239#102—42no—
240C22⋊C8—24no—
241#95—21no—
242#95—21no—
243#95—22no—
244#95—22no—
245#95—21no—
246C23×C3—21no—
247C23×C3—21no—
248C23×C3—21no—
249C23×C3—21no—
250C23×C3—21no—
251C23×C3—21no—
252C23×C3—22no—
253C23×C3—41no—
254C23×C3—41no—
255C23×C3—41no—
256C23×C3—41no—
257C23×C3—41no—
258C23×C3—41no—
259C23×C3C22⋊Q812yes—
260C23×C3C22×D411yescharacteristic
261C3×D4—24no—
262C3×D4—48no—
263C3×D4—42no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—24no—
275C2×C3×C4—24no—
276C2×C3×C4—22no—
277C2×C3×C4—44no—
278C2×C3×C4—44no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—42no—
287C2×C3×C4—41no—
288C2×C3×C4—41no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—41no—
293C2×C3×C4—84no—
294C2×C3×C4C22⋊Q812yes—
295C2×C3×C4#12312yes—
296C2×C3×C4#12312yes—
297C2×C3×C4#13512yes—
298C2×C3×C4AGL(1,ℤ/8)12yes—
299C2×C3×C4C22×Q811yescharacteristic
300C2×C3×C4C2×#4111yescharacteristic
301C2×C3×C4C2×#4112yes—
302C2×C3×C4H(5,𝔽2)12yes—
303C3×C8—42no—
304C3×C8—42no—
305C24—21no—
306C24C2×C3×D411yescharacteristic
307C2×D4—24no—
308C2×D4—22no—
309C2×D4—21no—
310C2×D4—22no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4C2×C3×Q812yes—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4C2×C3×D411yescharacteristic
331C22×C4C2×C3×D411yescharacteristic
332C22×C4C2×C3×D411yescharacteristic
333C22×C4C2×C3×D411yescharacteristic
334C22×C4C2×C3×D411yescharacteristic
335C22×C4C2×C3×Q812yes—
336C22×C4C3×#4112yes—
337C22×C4C24×C311yescharacteristic
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—22no—
344C4⋊C4—21no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4C2×C3×Q812yes—
357C22⋊C4—24no—
358C22⋊C4—44no—
359C22⋊C4—42no—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C42—22no—
364C42—22no—
365C42—22no—
366C42—44no—
367C42—42no—
368C42C3×#4114yes—
369C22×C3—22no—
370C22×C3—22no—
371C22×C3—22no—
372C22×C3—41no—
373C22×C3—41no—
374C22×C3—41no—
375C22×C3—41no—
376C22×C3—42no—
377C22×C3—42no—
378C22×C3—42no—
379C22×C3—81no—
380C22×C3—81no—
381C22×C3—81no—
382C22×C3—81no—
383C22×C3C2×C22⋊Q811yescharacteristic
384C22×C3C2×#12311yescharacteristic
385C22×C3#53511yescharacteristic
386C22×C3C2×#13511yescharacteristic
387C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
388C22×C3#57511yescharacteristic
389C22×C3#57511yescharacteristic
390C3×C4—22no—
391C3×C4—22no—
392C3×C4—22no—
393C3×C4—22no—
394C3×C4—22no—
395C3×C4—42no—
396C3×C4—81no—
397C3×C4—82no—
398C3×C4—81no—
399C3×C4—82no—
400C3×C4#55312yes—
401C3×C4#55412yes—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—22no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23C3×C22⋊Q812yes—
416C23C22×C3×D411yescharacteristic
417D4—24no—
418D4—48no—
419D4—42no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—84no—
450C2×C4C3×C22⋊Q812yes—
451C2×C4C3×#12312yes—
452C2×C4C3×#12312yes—
453C2×C4C3×#13512yes—
454C2×C4C3×AGL(1,ℤ/8)12yes—
455C2×C4C22×C3×Q811yescharacteristic
456C2×C4C2×C3×#4111yescharacteristic
457C2×C4C2×C3×#4112yes—
458C2×C4C3×H(5,𝔽2)12yes—
459C8—42no—
460C8—42no—
461C2×C3—42no—
462C2×C3—82no—
463C2×C3#248711yescharacteristic
464C2×C3#307611yescharacteristic
465C2×C3#307711yescharacteristic
466C2×C3#309111yescharacteristic
467C2×C3#309411yescharacteristic
468C2×C3#320711yescharacteristic
469C2×C3#321911yescharacteristic
470C22—22no—
471C22—22no—
472C22—22no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—81no—
481C22—81no—
482C22—81no—
483C22—81no—
484C22C2×C3×C22⋊Q811yescharacteristic
485C22C2×C3×#12311yescharacteristic
486C22C3×#53511yescharacteristic
487C22C2×C3×#13511yescharacteristic
488C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
489C22C3×#57511yescharacteristic
490C22C3×#57511yescharacteristic
491C4—22no—
492C4—22no—
493C4—22no—
494C4—22no—
495C4—22no—
496C4—42no—
497C4—81no—
498C4—82no—
499C4—82no—
500C4—81no—
501C4C3×#55312yes—
502C4C3×#55412yes—
503C3#2768311yescharacteristic, direct product factor, Sylow
504C2—42no—
505C2—82no—
506C2C3×#248711yescharacteristic
507C2C3×#307611yescharacteristic
508C2C3×#307711yescharacteristic
509C2C3×#309111yescharacteristic
510C2C3×#309411yescharacteristic
511C2C3×#320711yescharacteristic
512C2C3×#321911yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2285C2semidirect
3. C3×#2100C2semidirect
4. C3×#2099C2semidirect
5. C3×#2094C2semidirect
6. C3×#2093C2semidirect
7. C3×#1949C2semidirect
9. C3×#1854C2semidirect
10. C3×#1804C2semidirect
11. C3×#1797C2semidirect
12. #27683C3direct (N × H)
45. #2285C2×C3semidirect
46. #2100C2×C3semidirect
47. #2099C2×C3semidirect
48. #2094C2×C3semidirect
49. #2093C2×C3semidirect
50. #1949C2×C3semidirect
52. #1854C2×C3semidirect
53. #1804C2×C3semidirect
54. #1797C2×C3semidirect
503. C3#27683direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×#1212
C3×#22852
C3×#21002
C3×#20992
C3×#20942
C3×#20932
C3×#19492
C3×#18752
C3×#18542
C3×#18042
C3×#17972
#276833