Finite Group Databasebeta

← C3×#27684 #10646291

Subgroups — C3×#27684 #10646291

Order 768.

882 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×#121C211yesmaximal, characteristic
2C2×C3×#382C211yesmaximal, characteristic, complemented
3C3×#2101C212yesmaximal, complemented
4C3×#2092C212yesmaximal, complemented
5C3×#1949C212yesmaximal, complemented
6C3×#1874C211yesmaximal, characteristic
7C3×#1874C211yesmaximal, characteristic
8C3×#1853C211yesmaximal, characteristic, complemented
9C3×#1852C211yesmaximal, characteristic, complemented
10C3×#1804C212yesmaximal, complemented
11C3×#1797C211yesmaximal, characteristic, complemented
12#27684C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#121C2211yescharacteristic
14C2×C3×C4×D4—21no—
15C2×C3×C4×D4C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C42C2211yescharacteristic
19C2×C3×#107C2212yes—
20C2×C3×#106C2212yes—
21C2×C3×C4⋊C8C2212yes—
22C2×C3×#102C2212yes—
23C2×C3×#102C2212yes—
24C2×C3×C22⋊C8C2211yescharacteristic
25C2×C3×C22⋊C8C2211yescharacteristic
26C3×#390C2211yescharacteristic
27C3×#389C2211yescharacteristic
28C3×#388—21no—
29C3×#386—21no—
30C3×#382—21no—
31C3×#382—21no—
32C3×#382C2212yes—
33C3×#382C2212yes—
34C3×#380C2211yescharacteristic
35C3×C4×C4⋊C4C2211yescharacteristic
36C3×C4×C22⋊C4—21no—
37C2×C3×#95—21no—
38C2×C3×#95—21no—
39C3×#340C2212yes—
40C3×#340C2212yes—
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#336C2212yes—
44C4×#121C2×C311yescharacteristic
45C2×#382C2×C311yescharacteristic, complemented
46#2101C2×C312yescomplemented
47#2092C2×C312yescomplemented
48#1949C2×C312yescomplemented
49#1874C2×C311yescharacteristic
50#1874C2×C311yescharacteristic
51#1853C2×C311yescharacteristic, complemented
52#1852C2×C311yescharacteristic, complemented
53#1804C2×C312yescomplemented
54#1797C2×C311yescharacteristic, complemented
55C22×C3×D4—21no—
56C22×C3×D4C2311yescharacteristic
57C23×C3×C4—21no—
58C23×C3×C4—21no—
59C23×C3×C4—21no—
60C23×C3×C4D411yescharacteristic
61C23×C3×C4D411yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C22×C3×C8C2312yes—
64C22×C3×C8C2312yes—
65C3×#121—22no—
66C3×#121Q814yes—
67C3×C4×D4—24no—
68C3×C4×D4—44no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—42no—
73C2×C3×C4⋊C4—42no—
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—21no—
81C2×C3×C22⋊C4—21no—
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—21no—
84C2×C3×C42—24no—
85C2×C3×C42—21no—
86C2×C3×C42D411yescharacteristic
87C2×C3×C42D411yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C3×#107—42no—
91C3×#106—42no—
92C3×C4⋊C8—24no—
93C3×#102—42no—
94C3×#102—42no—
95C3×C22⋊C8—22no—
96C3×C22⋊C8—22no—
97C3×#95—22no—
98C3×#95—22no—
99C3×#95—22no—
100C3×#95—22no—
101C3×#95—21no—
102C3×#95—21no—
103C3×#95—21no—
104C2×#121C22×C311yescharacteristic
105C2×C4×D4—21no—
106C2×C4×D4C22×C311yescharacteristic
107C22×C4⋊C4C22×C311yescharacteristic
108C22×C4⋊C4C22×C311yescharacteristic
109C22×C42C22×C311yescharacteristic
110C2×#107C22×C312yes—
111C2×#106C22×C312yes—
112C2×C4⋊C8C22×C312yes—
113C2×#102C22×C312yes—
114C2×#102C22×C312yes—
115C2×C22⋊C8C22×C311yescharacteristic
116C2×C22⋊C8C22×C311yescharacteristic
117#390C22×C311yescharacteristic
118#389C22×C311yescharacteristic
119#388—21no—
120#386—21no—
121#382—21no—
122#382—21no—
123#382C22×C312yes—
124#382C22×C312yes—
125#380C22×C311yescharacteristic
126C4×C4⋊C4C22×C311yescharacteristic
127C4×C22⋊C4—21no—
128C2×#95—21no—
129C2×#95—21no—
130#340C22×C312yes—
131#340C22×C312yes—
132#340C22×C312yes—
133#340C22×C312yes—
134#336C22×C312yes—
135C24×C3—21no—
136C24×C3C2×D411yescharacteristic
137C2×C3×D4—24no—
138C2×C3×D4—22no—
139C2×C3×D4—21no—
140C2×C3×D4—22no—
141C2×C3×D4—42no—
142C2×C3×D4—42no—
143C2×C3×D4C2×Q812yes—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—21no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—41no—
156C22×C3×C4—41no—
157C22×C3×C4—42no—
158C22×C3×C4—42no—
159C22×C3×C4—42no—
160C22×C3×C4—42no—
161C22×C3×C4—42no—
162C22×C3×C4—42no—
163C22×C3×C4—42no—
164C22×C3×C4—42no—
165C22×C3×C4—42no—
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×Q812yes—
172C22×C3×C4#4112yes—
173C22×C3×C4C2411yescharacteristic
174C2×C3×C8—22no—
175C2×C3×C8—22no—
176C2×C3×C8—22no—
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C2×C3×C8—22no—
180C3×C4⋊C4—21no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—42no—
188C3×C4⋊C4—42no—
189C3×C4⋊C4—42no—
190C3×C4⋊C4C2×Q812yes—
191C3×C22⋊C4—24no—
192C3×C22⋊C4—42no—
193C3×C22⋊C4—42no—
194C3×C22⋊C4—42no—
195C3×C22⋊C4—42no—
196C3×C42—22no—
197C3×C42—22no—
198C3×C42—22no—
199C3×C42—42no—
200C3×C42#4114yes—
201C22×D4—21no—
202C22×D4C23×C311yescharacteristic
203C23×C4—21no—
204C23×C4—21no—
205C23×C4—21no—
206C23×C4C3×D411yescharacteristic
207C23×C4C3×D411yescharacteristic
208C23×C4C23×C311yescharacteristic
209C22×C8C23×C312yes—
210C22×C8C23×C312yes—
211#121—22no—
212#121C3×Q814yes—
213C4×D4—24no—
214C4×D4—44no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4C23×C312yes—
221C2×C4⋊C4C23×C312yes—
222C2×C4⋊C4C23×C311yescharacteristic
223C2×C4⋊C4C23×C311yescharacteristic
224C2×C4⋊C4C23×C311yescharacteristic
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C42—24no—
231C2×C42—21no—
232C2×C42C3×D411yescharacteristic
233C2×C42C3×D411yescharacteristic
234C2×C42C23×C311yescharacteristic
235C2×C42C23×C311yescharacteristic
236#107—42no—
237#106—42no—
238C4⋊C8—24no—
239#102—42no—
240#102—42no—
241C22⋊C8—22no—
242C22⋊C8—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—21no—
248#95—21no—
249#95—21no—
250C23×C3—21no—
251C23×C3—21no—
252C23×C3—21no—
253C23×C3—21no—
254C23×C3—22no—
255C23×C3—41no—
256C23×C3—41no—
257C23×C3—41no—
258C23×C3—41no—
259C23×C3—41no—
260C23×C3—41no—
261C23×C3C22⋊Q812yes—
262C23×C3#13512yes—
263C23×C3AGL(1,ℤ/8)12yes—
264C23×C3C22×D411yescharacteristic
265C3×D4—24no—
266C3×D4—48no—
267C3×D4—42no—
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—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—44no—
284C2×C3×C4—44no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—41no—
291C2×C3×C4—42no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—42no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—84no—
300C2×C3×C4—84no—
301C2×C3×C4—84no—
302C2×C3×C4C22⋊Q812yes—
303C2×C3×C4#12312yes—
304C2×C3×C4#12312yes—
305C2×C3×C4C22×Q811yescharacteristic
306C2×C3×C4C2×#4111yescharacteristic
307C2×C3×C4C2×#4112yes—
308C2×C3×C4H(5,𝔽2)12yes—
309C3×C8—42no—
310C3×C8—42no—
311C24—21no—
312C24C2×C3×D411yescharacteristic
313C2×D4—24no—
314C2×D4—22no—
315C2×D4—21no—
316C2×D4—22no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4C2×C3×Q812yes—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4C2×C3×D411yescharacteristic
343C22×C4C2×C3×D411yescharacteristic
344C22×C4C2×C3×D411yescharacteristic
345C22×C4C2×C3×D411yescharacteristic
346C22×C4C2×C3×D411yescharacteristic
347C22×C4C2×C3×Q812yes—
348C22×C4C3×#4112yes—
349C22×C4C24×C311yescharacteristic
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—22no—
356C4⋊C4—21no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4C2×C3×Q812yes—
367C22⋊C4—24no—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C22⋊C4—42no—
371C22⋊C4—42no—
372C42—22no—
373C42—22no—
374C42—22no—
375C42—42no—
376C42C3×#4114yes—
377C22×C3—22no—
378C22×C3—22no—
379C22×C3—22no—
380C22×C3—22no—
381C22×C3—22no—
382C22×C3—22no—
383C22×C3—42no—
384C22×C3—42no—
385C22×C3—42no—
386C22×C3—81no—
387C22×C3—81no—
388C22×C3—81no—
389C22×C3—81no—
390C22×C3#47814yes—
391C22×C3C2×C22⋊Q811yescharacteristic
392C22×C3C2×#12311yescharacteristic
393C22×C3#53511yescharacteristic
394C22×C3C2×#13511yescharacteristic
395C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
396C22×C3#57511yescharacteristic
397C22×C3#57511yescharacteristic
398C3×C4—22no—
399C3×C4—22no—
400C3×C4—22no—
401C3×C4—42no—
402C3×C4—41no—
403C3×C4—41no—
404C3×C4—81no—
405C3×C4—82no—
406C3×C4—81no—
407C3×C4—82no—
408C3×C4#55312yes—
409C3×C4#55412yes—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—22no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23C3×C22⋊Q812yes—
422C23C3×#13512yes—
423C23C3×AGL(1,ℤ/8)12yes—
424C23C22×C3×D411yescharacteristic
425D4—24no—
426D4—48no—
427D4—42no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—84no—
460C2×C4—84no—
461C2×C4—84no—
462C2×C4C3×C22⋊Q812yes—
463C2×C4C3×#12312yes—
464C2×C4C3×#12312yes—
465C2×C4C22×C3×Q811yescharacteristic
466C2×C4C2×C3×#4111yescharacteristic
467C2×C4C2×C3×#4112yes—
468C2×C4C3×H(5,𝔽2)12yes—
469C8—42no—
470C8—42no—
471C2×C3—24no—
472C2×C3—82no—
473C2×C3#248711yescharacteristic
474C2×C3C2×#47811yescharacteristic
475C2×C3#308011yescharacteristic
476C2×C3#309111yescharacteristic
477C2×C3#309411yescharacteristic
478C2×C3#321111yescharacteristic
479C2×C3#321311yescharacteristic
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—81no—
490C22—81no—
491C22—81no—
492C22—81no—
493C22C3×#47814yes—
494C22C2×C3×C22⋊Q811yescharacteristic
495C22C2×C3×#12311yescharacteristic
496C22C3×#53511yescharacteristic
497C22C2×C3×#13511yescharacteristic
498C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
499C22C3×#57511yescharacteristic
500C22C3×#57511yescharacteristic
501C4—22no—
502C4—22no—
503C4—22no—
504C4—42no—
505C4—41no—
506C4—41no—
507C4—82no—
508C4—82no—
509C4—81no—
510C4—81no—
511C4C3×#55312yes—
512C4C3×#55412yes—
513C3#2768411yescharacteristic, direct product factor, Sylow
514C2—24no—
515C2—82no—
516C2C3×#248711yescharacteristic
517C2C2×C3×#47811yescharacteristic
518C2C3×#308011yescharacteristic
519C2C3×#309111yescharacteristic
520C2C3×#309411yescharacteristic
521C2C3×#321111yescharacteristic
522C2C3×#321311yescharacteristic
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. C2×C3×#382C2semidirect
3. C3×#2101C2semidirect
4. C3×#2092C2semidirect
5. C3×#1949C2semidirect
8. C3×#1853C2semidirect
9. C3×#1852C2semidirect
10. C3×#1804C2semidirect
11. C3×#1797C2semidirect
12. #27684C3direct (N × H)
45. C2×#382C2×C3semidirect
46. #2101C2×C3semidirect
47. #2092C2×C3semidirect
48. #1949C2×C3semidirect
51. #1853C2×C3semidirect
52. #1852C2×C3semidirect
53. #1804C2×C3semidirect
54. #1797C2×C3semidirect
513. C3#27684direct (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
C2×C3×#3822
C3×#21012
C3×#20922
C3×#19492
C3×#18742
C3×#18532
C3×#18522
C3×#18042
C3×#17972
#276843