Finite Group Databasebeta

← C3×#27665 #10646272

Subgroups — C3×#27665 #10646272

Order 768.

796 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#435C211yesmaximal, characteristic
2C2×C3×#427C211yesmaximal, characteristic, complemented
3C3×#2485C212yesmaximal
4C3×#2088C212yesmaximal, complemented
5C3×#2077C212yesmaximal, complemented
6C3×#1944C211yesmaximal, characteristic, complemented
7C3×#1944C211yesmaximal, characteristic, complemented
8C3×#1825C212yesmaximal, complemented
9C3×#1809C211yesmaximal, characteristic, complemented
10C3×#1809C211yesmaximal, characteristic, complemented
11C3×#1807C211yesmaximal, characteristic, complemented
12#27665C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×M16C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×C3×#117C2212yes—
17C3×#435—24no—
18C3×#427—24no—
19C2×C3×#107C2211yescharacteristic
20C2×C3×#107C2211yescharacteristic
21C2×C3×#106C2212yes—
22C2×C3×#106C2211yescharacteristic
23C2×C3×#106C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×C22⋊C8C2211yescharacteristic
26C2×C3×C22⋊C8C2211yescharacteristic
27C2×C3×#97C2211yescharacteristic
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×#435C2×C311yescharacteristic
43C2×#427C2×C311yescharacteristic, complemented
44#2485C2×C312yes—
45#2088C2×C312yescomplemented
46#2077C2×C312yescomplemented
47#1944C2×C311yescharacteristic, complemented
48#1944C2×C311yescharacteristic, complemented
49#1825C2×C312yescomplemented
50#1809C2×C311yescharacteristic, complemented
51#1809C2×C311yescharacteristic, complemented
52#1807C2×C311yescharacteristic, complemented
53C22×C3×D4D411yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C2×C3×M16—22no—
57C2×C3×M16—22no—
58C2×C3×M16—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×#106—24no—
88C3×#106—22no—
89C3×#106—22no—
90C3×C4⋊C8—22no—
91C3×C22⋊C8—22no—
92C3×C22⋊C8—22no—
93C3×#97—22no—
94C3×#95—22no—
95C3×#95—22no—
96C3×#95—22no—
97C3×#95—22no—
98C3×#95—22no—
99C22×M16C22×C311yescharacteristic
100C23×C8C22×C311yescharacteristic
101C2×C4×D4C22×C311yescharacteristic
102C2×#117C22×C312yes—
103#435—24no—
104#427—24no—
105C2×#107C22×C311yescharacteristic
106C2×#107C22×C311yescharacteristic
107C2×#106C22×C311yescharacteristic
108C2×#106C22×C311yescharacteristic
109C2×#106C22×C312yes—
110C2×C4⋊C8C22×C311yescharacteristic
111C2×C22⋊C8C22×C311yescharacteristic
112C2×C22⋊C8C22×C311yescharacteristic
113C2×#97C22×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—42no—
132C22×C3×C4—21no—
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—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×D412yes—
151C22×C3×C4#4112yes—
152C22×C3×C4C2411yescharacteristic
153C3×M16—44no—
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×C8C2×D412yes—
164C2×C3×C8C2×D412yes—
165C2×C3×C8#4112yes—
166C2×C3×C8#4112yes—
167C3×C4⋊C4—22no—
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—44no—
177C3×C4⋊C4—82no—
178C3×C22⋊C4—22no—
179C3×C22⋊C4—22no—
180C3×C22⋊C4—44no—
181C3×C22⋊C4—44no—
182C3×C22⋊C4—82no—
183C3×C22⋊C4—82no—
184C3×C42—21no—
185C3×C42—21no—
186C3×C42—44no—
187C22×D4C3×D411yescharacteristic
188C23×C4C23×C311yescharacteristic
189C23×C4C23×C311yescharacteristic
190C2×M16—22no—
191C2×M16—22no—
192C2×M16—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8C3×D414yes—
196C22×C8C23×C311yescharacteristic
197C22×C8C23×C311yescharacteristic
198C22×C8C23×C311yescharacteristic
199C22×C8C23×C311yescharacteristic
200C4×D4—22no—
201C4×D4—22no—
202#117—24no—
203#117—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4C3×D411yescharacteristic
207C2×C4⋊C4C23×C312yes—
208C2×C4⋊C4C23×C312yes—
209C2×C4⋊C4C23×C312yes—
210C2×C4⋊C4C23×C312yes—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4C3×D411yescharacteristic
216C2×C22⋊C4C3×D411yescharacteristic
217C2×C42—22no—
218C2×C42C23×C311yescharacteristic
219#107—22no—
220#107—22no—
221#106—24no—
222#106—22no—
223#106—22no—
224C4⋊C8—22no—
225C22⋊C8—22no—
226C22⋊C8—22no—
227#97—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233C23×C3—21no—
234C23×C3—21no—
235C23×C3—22no—
236C23×C3—22no—
237C23×C3—21no—
238C23×C3—21no—
239C23×C3—21no—
240C23×C3—22no—
241C23×C3—22no—
242C23×C3#12112yes—
243C23×C3C22×D411yescharacteristic
244C3×D4—44no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—22no—
250C2×C3×C4—22no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
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#12112yes—
275C2×C3×C4C22×D411yescharacteristic
276C2×C3×C4C2×#4111yescharacteristic
277C2×C3×C4C2×#4111yescharacteristic
278C2×C3×C4C2×#4111yescharacteristic
279C2×C3×C42−1+411yescharacteristic
280C2×C3×C42−1+411yescharacteristic
281C3×C8—22no—
282C3×C8—41no—
283C3×C8—41no—
284C3×C8#12314yes—
285C24C2×C3×D411yescharacteristic
286C24C2×C3×D411yescharacteristic
287C2×D4—22no—
288C2×D4—42no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4C2×C3×D411yescharacteristic
304C22×C4C2×C3×D411yescharacteristic
305C22×C4C2×C3×D411yescharacteristic
306C22×C4C2×C3×D411yescharacteristic
307C22×C4C2×C3×D412yes—
308C22×C4C3×#4112yes—
309C22×C4C24×C311yescharacteristic
310M16—44no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—28no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8C2×C3×D412yes—
321C2×C8C2×C3×D412yes—
322C2×C8C3×#4112yes—
323C2×C8C3×#4112yes—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—44no—
334C4⋊C4—82no—
335C22⋊C4—22no—
336C22⋊C4—22no—
337C22⋊C4—44no—
338C22⋊C4—44no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C42—21no—
342C42—21no—
343C42—44no—
344C22×C3—22no—
345C22×C3—22no—
346C22×C3—22no—
347C22×C3—42no—
348C22×C3—42no—
349C22×C3—42no—
350C22×C3—42no—
351C22×C3—41no—
352C22×C3—41no—
353C22×C3—41no—
354C22×C3C2×#12111yescharacteristic
355C22×C3#54711yescharacteristic
356C22×C3#54711yescharacteristic
357C22×C3#57512yes—
358C22×C3#57712yes—
359C3×C4—22no—
360C3×C4—41no—
361C3×C4—41no—
362C3×C4—41no—
363C3×C4—82no—
364C3×C4—82no—
365C3×C4—82no—
366C3×C4—82no—
367C3×C4C2×#12311yescharacteristic
368C3×C4#53611yescharacteristic
369C3×C4#55211yescharacteristic
370C3×C4#55211yescharacteristic
371C23—22no—
372C23—22no—
373C23—21no—
374C23—21no—
375C23—22no—
376C23—22no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23C3×#12112yes—
381C23C22×C3×D411yescharacteristic
382D4—44no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4C3×#12112yes—
413C2×C4C22×C3×D411yescharacteristic
414C2×C4C2×C3×#4111yescharacteristic
415C2×C4C2×C3×#4111yescharacteristic
416C2×C4C2×C3×#4111yescharacteristic
417C2×C4C3×2−1+411yescharacteristic
418C2×C4C3×2−1+411yescharacteristic
419C8—22no—
420C8—41no—
421C8—41no—
422C8C3×#12314yes—
423C2×C3—42no—
424C2×C3—81no—
425C2×C3#248611yescharacteristic
426C2×C3#305411yescharacteristic
427C2×C3#306011yescharacteristic
428C2×C3#334812yes—
429C2×C3#335012yes—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—42no—
434C22—42no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—42no—
439C22—42no—
440C22C2×C3×#12111yescharacteristic
441C22C3×#54711yescharacteristic
442C22C3×#54711yescharacteristic
443C22C3×#57512yes—
444C22C3×#57712yes—
445C4—22no—
446C4—41no—
447C4—41no—
448C4—41no—
449C4—82no—
450C4—82no—
451C4—82no—
452C4—82no—
453C4C2×C3×#12311yescharacteristic
454C4C3×#53611yescharacteristic
455C4C3×#55211yescharacteristic
456C4C3×#55211yescharacteristic
457C3#2766511yescharacteristic, direct product factor, Sylow
458C2—42no—
459C2—81no—
460C2C3×#248611yescharacteristic
461C2C3×#305411yescharacteristic
462C2C3×#306011yescharacteristic
463C2C3×#334812yes—
464C2C3×#335012yes—
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×#1944C2semidirect
7. C3×#1944C2semidirect
8. C3×#1825C2semidirect
9. C3×#1809C2semidirect
10. C3×#1809C2semidirect
11. C3×#1807C2semidirect
12. #27665C3direct (N × H)
43. C2×#427C2×C3semidirect
45. #2088C2×C3semidirect
46. #2077C2×C3semidirect
47. #1944C2×C3semidirect
48. #1944C2×C3semidirect
49. #1825C2×C3semidirect
50. #1809C2×C3semidirect
51. #1809C2×C3semidirect
52. #1807C2×C3semidirect
457. C3#27665direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4352
C2×C3×#4272
C3×#24852
C3×#20882
C3×#20772
C3×#19442
C3×#18252
C3×#18092
C3×#18072
#276653