Finite Group Databasebeta

← C3×#27660 #10646267

Subgroups — C3×#27660 #10646267

Order 768.

822 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C8×D4C211yesmaximal, characteristic
2C2×C3×#427C211yesmaximal, characteristic, complemented
3C3×#2485C212yesmaximal
4C3×#2088C212yesmaximal, complemented
5C3×#2077C212yesmaximal, complemented
6C3×#1942C211yesmaximal, characteristic, complemented
7C3×#1942C211yesmaximal, characteristic, complemented
8C3×#1825C212yesmaximal, complemented
9C3×#1808C211yesmaximal, characteristic, complemented
10C3×#1808C211yesmaximal, characteristic, complemented
11C3×#1808C211yesmaximal, characteristic, complemented
12#27660C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×C8C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×C3×#117C2212yes—
17C3×C8×D4—24no—
18C3×#427—24no—
19C2×C3×#107C2211yescharacteristic
20C2×C3×#107C2211yescharacteristic
21C2×C3×#107C2211yescharacteristic
22C2×C3×#107C2211yescharacteristic
23C2×C3×#106C2212yes—
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×C22⋊C8C2211yescharacteristic
26C2×C3×C22⋊C8C2211yescharacteristic
27C2×C3×C4×C8C2211yescharacteristic
28C3×#397—22no—
29C3×#394—22no—
30C3×#389C2212yes—
31C3×#389C2212yes—
32C3×#388—22no—
33C3×#387—22no—
34C3×#380C2212yes—
35C3×#380C2212yes—
36C3×#380C2212yes—
37C3×#340C2212yes—
38C3×#340C2212yes—
39C3×#340C2212yes—
40C3×#340C2212yes—
41C3×#340C2212yes—
42C2×C8×D4C2×C311yescharacteristic
43C2×#427C2×C311yescharacteristic, complemented
44#2485C2×C312yes—
45#2088C2×C312yescomplemented
46#2077C2×C312yescomplemented
47#1942C2×C311yescharacteristic, complemented
48#1942C2×C311yescharacteristic, complemented
49#1825C2×C312yescomplemented
50#1808C2×C311yescharacteristic, complemented
51#1808C2×C311yescharacteristic, complemented
52#1808C2×C311yescharacteristic, complemented
53C22×C3×D4D411yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C22×C3×C8—22no—
57C22×C3×C8—22no—
58C22×C3×C8—22no—
59C22×C3×C8—22no—
60C22×C3×C8—22no—
61C22×C3×C8D414yes—
62C22×C3×C8C2311yescharacteristic
63C22×C3×C8C2311yescharacteristic
64C22×C3×C8C2311yescharacteristic
65C22×C3×C8C2311yescharacteristic
66C3×C4×D4—22no—
67C3×C4×D4—22no—
68C3×#117—24no—
69C3×#117—24no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4D411yescharacteristic
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4D411yescharacteristic
82C2×C3×C22⋊C4D411yescharacteristic
83C2×C3×C42—22no—
84C2×C3×C42C2311yescharacteristic
85C3×#107—22no—
86C3×#107—22no—
87C3×#107—22no—
88C3×#107—22no—
89C3×#106—24no—
90C3×C4⋊C8—22no—
91C3×C22⋊C8—22no—
92C3×C22⋊C8—22no—
93C3×C4×C8—22no—
94C3×#95—22no—
95C3×#95—22no—
96C3×#95—22no—
97C3×#95—22no—
98C3×#95—22no—
99C23×C8C22×C311yescharacteristic
100C23×C8C22×C311yescharacteristic
101C2×C4×D4C22×C311yescharacteristic
102C2×#117C22×C312yes—
103C8×D4—24no—
104#427—24no—
105C2×#107C22×C311yescharacteristic
106C2×#107C22×C311yescharacteristic
107C2×#107C22×C311yescharacteristic
108C2×#107C22×C311yescharacteristic
109C2×#106C22×C312yes—
110C2×C4⋊C8C22×C311yescharacteristic
111C2×C22⋊C8C22×C311yescharacteristic
112C2×C22⋊C8C22×C311yescharacteristic
113C2×C4×C8C22×C311yescharacteristic
114#397—22no—
115#394—22no—
116#389C22×C312yes—
117#389C22×C312yes—
118#388—22no—
119#387—22no—
120#380C22×C312yes—
121#380C22×C312yes—
122#380C22×C312yes—
123#340C22×C312yes—
124#340C22×C312yes—
125#340C22×C312yes—
126#340C22×C312yes—
127#340C22×C312yes—
128C24×C3C2×D411yescharacteristic
129C24×C3C2×D411yescharacteristic
130C2×C3×D4—22no—
131C2×C3×D4—22no—
132C2×C3×D4—22no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—21no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4—21no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×D412yes—
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4#4112yes—
153C22×C3×C4C2411yescharacteristic
154C2×C3×C8—21no—
155C2×C3×C8—21no—
156C2×C3×C8—28no—
157C2×C3×C8—21no—
158C2×C3×C8—21no—
159C2×C3×C8—21no—
160C2×C3×C8—21no—
161C2×C3×C8—21no—
162C2×C3×C8—21no—
163C2×C3×C8—44no—
164C2×C3×C8C2×D412yes—
165C2×C3×C8C2×D412yes—
166C2×C3×C8#4112yes—
167C2×C3×C8#4112yes—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—44no—
178C3×C4⋊C4—82no—
179C3×C22⋊C4—22no—
180C3×C22⋊C4—22no—
181C3×C22⋊C4—44no—
182C3×C22⋊C4—44no—
183C3×C22⋊C4—82no—
184C3×C22⋊C4—82no—
185C3×C42—21no—
186C3×C42—21no—
187C3×C42—44no—
188C22×D4C3×D411yescharacteristic
189C23×C4C23×C311yescharacteristic
190C23×C4C23×C311yescharacteristic
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8C3×D414yes—
197C22×C8C23×C311yescharacteristic
198C22×C8C23×C311yescharacteristic
199C22×C8C23×C311yescharacteristic
200C22×C8C23×C311yescharacteristic
201C4×D4—22no—
202C4×D4—22no—
203#117—24no—
204#117—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4C3×D411yescharacteristic
208C2×C4⋊C4C23×C312yes—
209C2×C4⋊C4C23×C312yes—
210C2×C4⋊C4C23×C312yes—
211C2×C4⋊C4C23×C312yes—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4C3×D411yescharacteristic
217C2×C22⋊C4C3×D411yescharacteristic
218C2×C42—22no—
219C2×C42C23×C311yescharacteristic
220#107—22no—
221#107—22no—
222#107—22no—
223#107—22no—
224#106—24no—
225C4⋊C8—22no—
226C22⋊C8—22no—
227C22⋊C8—22no—
228C4×C8—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234C23×C3—21no—
235C23×C3—21no—
236C23×C3—21no—
237C23×C3—21no—
238C23×C3—21no—
239C23×C3—21no—
240C23×C3—21no—
241C23×C3—21no—
242C23×C3—21no—
243C23×C3—21no—
244C23×C3—21no—
245C23×C3—21no—
246C23×C3#12112yes—
247C23×C3#13512yes—
248C23×C3C22×D411yescharacteristic
249C3×D4—28no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4#12112yes—
281C2×C3×C4#13512yes—
282C2×C3×C4C22×D411yescharacteristic
283C2×C3×C4C2×#4111yescharacteristic
284C2×C3×C4C2×#4111yescharacteristic
285C2×C3×C4C2×#4111yescharacteristic
286C2×C3×C42−1+411yescharacteristic
287C2×C3×C42−1+411yescharacteristic
288C3×C8—22no—
289C3×C8—41no—
290C3×C8—41no—
291C3×C8#12314yes—
292C24C2×C3×D411yescharacteristic
293C24C2×C3×D411yescharacteristic
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4C2×C3×D411yescharacteristic
312C22×C4C2×C3×D411yescharacteristic
313C22×C4C2×C3×D411yescharacteristic
314C22×C4C2×C3×D412yes—
315C22×C4C2×C3×D411yescharacteristic
316C22×C4C3×#4112yes—
317C22×C4C24×C311yescharacteristic
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—28no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—44no—
328C2×C8C2×C3×D412yes—
329C2×C8C2×C3×D412yes—
330C2×C8C3×#4112yes—
331C2×C8C3×#4112yes—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—44no—
342C4⋊C4—82no—
343C22⋊C4—22no—
344C22⋊C4—22no—
345C22⋊C4—44no—
346C22⋊C4—44no—
347C22⋊C4—82no—
348C22⋊C4—82no—
349C42—21no—
350C42—21no—
351C42—44no—
352C22×C3—22no—
353C22×C3—22no—
354C22×C3—22no—
355C22×C3—22no—
356C22×C3—22no—
357C22×C3—22no—
358C22×C3—41no—
359C22×C3—41no—
360C22×C3—41no—
361C22×C3—41no—
362C22×C3—41no—
363C22×C3—41no—
364C22×C3—41no—
365C22×C3—41no—
366C22×C3C2×#12111yescharacteristic
367C22×C3#54711yescharacteristic
368C22×C3#54711yescharacteristic
369C22×C3C2×#13511yescharacteristic
370C22×C3C2×#13511yescharacteristic
371C22×C3#57611yescharacteristic
372C22×C3#57811yescharacteristic
373C3×C4—22no—
374C3×C4—22no—
375C3×C4—41no—
376C3×C4—41no—
377C3×C4—82no—
378C3×C4—82no—
379C3×C4—82no—
380C3×C4—82no—
381C3×C4C2×#12311yescharacteristic
382C3×C4#53611yescharacteristic
383C3×C4#55211yescharacteristic
384C3×C4#55211yescharacteristic
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23C3×#12112yes—
398C23C3×#13512yes—
399C23C22×C3×D411yescharacteristic
400D4—28no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4C3×#12112yes—
432C2×C4C3×#13512yes—
433C2×C4C22×C3×D411yescharacteristic
434C2×C4C2×C3×#4111yescharacteristic
435C2×C4C2×C3×#4111yescharacteristic
436C2×C4C2×C3×#4111yescharacteristic
437C2×C4C3×2−1+411yescharacteristic
438C2×C4C3×2−1+411yescharacteristic
439C8—22no—
440C8—41no—
441C8—41no—
442C8C3×#12314yes—
443C2×C3—42no—
444C2×C3—42no—
445C2×C3#248611yescharacteristic
446C2×C3#305011yescharacteristic
447C2×C3#306111yescharacteristic
448C2×C3#333611yescharacteristic
449C2×C3#333911yescharacteristic
450C2×C3#334111yescharacteristic
451C2×C3#334211yescharacteristic
452C22—22no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22C2×C3×#12111yescharacteristic
467C22C3×#54711yescharacteristic
468C22C3×#54711yescharacteristic
469C22C2×C3×#13511yescharacteristic
470C22C2×C3×#13511yescharacteristic
471C22C3×#57611yescharacteristic
472C22C3×#57811yescharacteristic
473C4—22no—
474C4—22no—
475C4—41no—
476C4—41no—
477C4—82no—
478C4—82no—
479C4—82no—
480C4—82no—
481C4C2×C3×#12311yescharacteristic
482C4C3×#53611yescharacteristic
483C4C3×#55211yescharacteristic
484C4C3×#55211yescharacteristic
485C3#2766011yescharacteristic, direct product factor, Sylow
486C2—42no—
487C2—42no—
488C2C3×#248611yescharacteristic
489C2C3×#305011yescharacteristic
490C2C3×#306111yescharacteristic
491C2C3×#333611yescharacteristic
492C2C3×#333911yescharacteristic
493C2C3×#334111yescharacteristic
494C2C3×#334211yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×D4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#427C2semidirect
4. C3×#2088C2semidirect
5. C3×#2077C2semidirect
6. C3×#1942C2semidirect
7. C3×#1942C2semidirect
8. C3×#1825C2semidirect
9. C3×#1808C2semidirect
10. C3×#1808C2semidirect
11. C3×#1808C2semidirect
12. #27660C3direct (N × H)
43. C2×#427C2×C3semidirect
45. #2088C2×C3semidirect
46. #2077C2×C3semidirect
47. #1942C2×C3semidirect
48. #1942C2×C3semidirect
49. #1825C2×C3semidirect
50. #1808C2×C3semidirect
51. #1808C2×C3semidirect
52. #1808C2×C3semidirect
485. C3#27660direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C8×D42
C2×C3×#4272
C3×#24852
C3×#20882
C3×#20772
C3×#19422
C3×#18252
C3×#18082
#276603