Finite Group Databasebeta

← C3×#25689 #10644296

Subgroups — C3×#25689 #10644296

Order 768.

988 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3510C211yesmaximal, characteristic
2C3×#3166C211yesmaximal, characteristic, complemented
3C3×#3080C211yesmaximal, characteristic, complemented
4C3×#2861C211yesmaximal, characteristic, complemented
5C3×#2038C212yesmaximal, complemented
6C3×#2037C212yesmaximal
7C3×#1994C211yesmaximal, characteristic, complemented
8C3×#1967C211yesmaximal, characteristic, complemented
9C3×#1857C212yesmaximal, complemented
10C3×#1856C212yesmaximal, complemented
11C3×C4×#104C211yesmaximal, characteristic, complemented
12#25689C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#559—22no—
14C3×#556—22no—
15C3×#551—24no—
16C3×#542—22no—
17C3×#536C2211yescharacteristic
18C3×#535C2211yescharacteristic
19C3×#533C2211yescharacteristic
20C3×#529—22no—
21C2×C3×#125C2211yescharacteristic
22C2×C3×#124C2211yescharacteristic
23C3×#519C2211yescharacteristic
24C3×#518C2211yescharacteristic
25C3×C4×#41C2211yescharacteristic
26C3×#501—22no—
27C3×#499—22no—
28C3×#479—22no—
29C3×#478—22no—
30C3×#428C2212yes—
31C3×#428C2212yes—
32C3×#428C2212yes—
33C3×#423C2211yescharacteristic
34C2×C3×#104C2212yes—
35C3×#417C2212yes—
36C2×C3×#101C2212yes—
37C2×C3×#100C2212yes—
38C3×C4×M16C2211yescharacteristic
39C3×#391C2212yes—
40C3×#390C2212yes—
41C3×#388—24no—
42C3×C43C2211yescharacteristic
43C3×#339C2212yes—
44C3×#339C2212yes—
45C3×#339C2212yes—
46#3510C2×C311yescharacteristic
47#3166C2×C311yescharacteristic, complemented
48#3080C2×C311yescharacteristic, complemented
49#2861C2×C311yescharacteristic, complemented
50#2038C2×C312yescomplemented
51#2037C2×C312yes—
52#1994C2×C311yescharacteristic, complemented
53#1967C2×C311yescharacteristic, complemented
54#1857C2×C312yescomplemented
55#1856C2×C312yescomplemented
56C4×#104C2×C311yescharacteristic, complemented
57C2×C3×#41C2311yescharacteristic
58C22×C3×Q8C2311yescharacteristic
59C22×C3×D4C2311yescharacteristic
60C2×C3×M16C2312yes—
61C2×C3×M16C2312yes—
62C3×C4⋊Q8—21no—
63C3×C4⋊Q8D412yes—
64C3×#127—21no—
65C3×#126—24no—
66C3×#126—24no—
67C3×#125—22no—
68C3×#125—21no—
69C3×#125—21no—
70C3×#125D412yes—
71C3×#124—24no—
72C3×#124—21no—
73C3×#124—21no—
74C3×#124—22no—
75C3×#124—21no—
76C3×#124—21no—
77C3×#123—24no—
78C3×#123—24no—
79C3×#123—22no—
80C3×C22⋊Q8—24no—
81C3×C22⋊Q8—22no—
82C3×C22⋊Q8—22no—
83C3×#121—22no—
84C3×#120—22no—
85C3×C4×Q8—22no—
86C3×C4×Q8—22no—
87C3×C4×Q8—22no—
88C3×C4×Q8D412yes—
89C3×C4×D4—24no—
90C3×C4×D4—22no—
91C3×C4×D4—22no—
92C3×C4×D4—22no—
93C3×C4×D4D412yes—
94C3×#117—24no—
95C3×#117—22no—
96C3×#117—21no—
97C3×#117C2311yescharacteristic
98C3×#117C2311yescharacteristic
99C3×#117C2311yescharacteristic
100C2×C3×C4⋊C4—22no—
101C2×C3×C4⋊C4C2312yes—
102C2×C3×C22⋊C4—22no—
103C2×C3×C22⋊C4—24no—
104C2×C3×C22⋊C4D414yes—
105C2×C3×C42—21no—
106C2×C3×C42—22no—
107C2×C3×C42C2312yes—
108C2×C3×C42C2311yescharacteristic
109C3×#107—22no—
110C3×#107—22no—
111C3×#107—22no—
112C3×#106—22no—
113C3×#106—22no—
114C3×#106—22no—
115C3×C4⋊C8—22no—
116C3×#104—24no—
117C3×#103—22no—
118C3×#102—22no—
119C3×#101—24no—
120C3×#100—24no—
121C3×#97—21no—
122C3×C4×C8—21no—
123C3×#95—24no—
124#559—22no—
125#556—22no—
126#551—24no—
127#542—22no—
128#536C22×C311yescharacteristic
129#535C22×C311yescharacteristic
130#533C22×C311yescharacteristic
131#529—22no—
132C2×#125C22×C311yescharacteristic
133C2×#124C22×C311yescharacteristic
134#519C22×C311yescharacteristic
135#518C22×C311yescharacteristic
136C4×#41C22×C311yescharacteristic
137#501—22no—
138#499—22no—
139#479—22no—
140#478—22no—
141#428C22×C312yes—
142#428C22×C312yes—
143#428C22×C312yes—
144#423C22×C311yescharacteristic
145C2×#104C22×C312yes—
146#417C22×C312yes—
147C2×#101C22×C312yes—
148C2×#100C22×C312yes—
149C4×M16C22×C311yescharacteristic
150#391C22×C312yes—
151#390C22×C312yes—
152#388—24no—
153C43C22×C311yescharacteristic
154#339C22×C312yes—
155#339C22×C312yes—
156#339C22×C312yes—
157C24×C3C2×D412yes—
158C3×#41—22no—
159C3×#41—22no—
160C2×C3×Q8—22no—
161C2×C3×Q8—21no—
162C2×C3×Q8—21no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8C2×D411yescharacteristic
166C2×C3×D4—21no—
167C2×C3×D4—24no—
168C2×C3×D4—21no—
169C2×C3×D4—21no—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4C2×D411yescharacteristic
173C22×C3×C4—22no—
174C22×C3×C4—24no—
175C22×C3×C4—21no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2411yescharacteristic
181C3×M16—22no—
182C3×M16—22no—
183C3×M16—22no—
184C3×M16#4114yes—
185C2×C3×C8—22no—
186C2×C3×C8—22no—
187C3×C4⋊C4—24no—
188C3×C4⋊C4—24no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—42no—
196C3×C4⋊C4—42no—
197C3×C4⋊C4C2×D412yes—
198C3×C4⋊C4C2×D412yes—
199C3×C4⋊C4C2×D412yes—
200C3×C22⋊C4—24no—
201C3×C22⋊C4—24no—
202C3×C22⋊C4—24no—
203C3×C22⋊C4—21no—
204C3×C22⋊C4—21no—
205C3×C22⋊C4—21no—
206C3×C22⋊C4—24no—
207C3×C22⋊C4—24no—
208C3×C22⋊C4—22no—
209C3×C22⋊C4—48no—
210C3×C42—24no—
211C3×C42—22no—
212C3×C42—21no—
213C3×C42—21no—
214C3×C42—21no—
215C3×C42—22no—
216C3×C42—22no—
217C3×C42—44no—
218C3×C42C2×D411yescharacteristic
219C3×C42C2×D411yescharacteristic
220C3×C42C2×D412yes—
221C2×#41C23×C311yescharacteristic
222C22×Q8C23×C311yescharacteristic
223C22×D4C23×C311yescharacteristic
224C2×M16C23×C312yes—
225C2×M16C23×C312yes—
226C4⋊Q8—21no—
227C4⋊Q8C3×D412yes—
228#127—21no—
229#126—24no—
230#126—24no—
231#125—22no—
232#125—21no—
233#125—21no—
234#125C3×D412yes—
235#124—24no—
236#124—21no—
237#124—21no—
238#124—21no—
239#124—21no—
240#124—22no—
241#123—24no—
242#123—24no—
243#123—22no—
244C22⋊Q8—24no—
245C22⋊Q8—22no—
246C22⋊Q8—22no—
247#121—22no—
248#120—22no—
249C4×Q8—22no—
250C4×Q8—22no—
251C4×Q8—22no—
252C4×Q8C3×D412yes—
253C4×D4—24no—
254C4×D4—22no—
255C4×D4—22no—
256C4×D4—22no—
257C4×D4C3×D412yes—
258#117—24no—
259#117—22no—
260#117—21no—
261#117C23×C311yescharacteristic
262#117C23×C311yescharacteristic
263#117C23×C311yescharacteristic
264C2×C4⋊C4—22no—
265C2×C4⋊C4C23×C312yes—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4C3×D414yes—
269C2×C42—21no—
270C2×C42—22no—
271C2×C42C23×C312yes—
272C2×C42C23×C311yescharacteristic
273#107—22no—
274#107—22no—
275#107—22no—
276#106—22no—
277#106—22no—
278#106—22no—
279C4⋊C8—22no—
280#104—24no—
281#103—22no—
282#102—22no—
283#101—24no—
284#100—24no—
285#97—21no—
286C4×C8—21no—
287#95—24no—
288C23×C3—22no—
289C23×C3—21no—
290C23×C3—24no—
291C23×C3—44no—
292C23×C3C22×D411yescharacteristic
293C3×Q8—21no—
294C3×Q8—21no—
295C3×Q8—41no—
296C3×Q8—41no—
297C3×Q8—42no—
298C3×Q8—82no—
299C3×D4—21no—
300C3×D4—21no—
301C3×D4—42no—
302C3×D4—41no—
303C3×D4—41no—
304C3×D4—42no—
305C3×D4—84no—
306C2×C3×C4—24no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—21no—
311C2×C3×C4—22no—
312C2×C3×C4—24no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—44no—
318C2×C3×C4—42no—
319C2×C3×C4—82no—
320C2×C3×C4—84no—
321C2×C3×C4#12012yes—
322C2×C3×C4#12012yes—
323C2×C3×C4C22×D411yescharacteristic
324C2×C3×C4C22×D411yescharacteristic
325C2×C3×C4C2×#4112yes—
326C2×C3×C42−1+412yes—
327C3×C8—24no—
328C3×C8—42no—
329C24C2×C3×D412yes—
330#41—22no—
331#41—22no—
332C2×Q8—22no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8C2×C3×D411yescharacteristic
338C2×D4—21no—
339C2×D4—24no—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4C2×C3×D411yescharacteristic
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4C2×C3×D412yes—
351C22×C4C2×C3×D412yes—
352C22×C4C24×C311yescharacteristic
353M16—22no—
354M16—22no—
355M16—22no—
356M16C3×#4114yes—
357C2×C8—22no—
358C2×C8—22no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4C2×C3×D412yes—
370C4⋊C4C2×C3×D412yes—
371C4⋊C4C2×C3×D412yes—
372C22⋊C4—24no—
373C22⋊C4—24no—
374C22⋊C4—24no—
375C22⋊C4—21no—
376C22⋊C4—21no—
377C22⋊C4—21no—
378C22⋊C4—24no—
379C22⋊C4—24no—
380C22⋊C4—22no—
381C22⋊C4—48no—
382C42—24no—
383C42—22no—
384C42—21no—
385C42—21no—
386C42—21no—
387C42—22no—
388C42—22no—
389C42—44no—
390C42C2×C3×D411yescharacteristic
391C42C2×C3×D411yescharacteristic
392C42C2×C3×D412yes—
393C22×C3—22no—
394C22×C3—41no—
395C22×C3—42no—
396C22×C3—42no—
397C22×C3—44no—
398C22×C3—84no—
399C22×C3C2×#12011yescharacteristic
400C22×C3#54712yes—
401C3×C4—22no—
402C3×C4—22no—
403C3×C4—44no—
404C3×C4—42no—
405C3×C4—41no—
406C3×C4—42no—
407C3×C4—42no—
408C3×C4—42no—
409C3×C4—82no—
410C3×C4#53211yescharacteristic
411C3×C4#53611yescharacteristic
412C3×C4#54812yes—
413C23—22no—
414C23—21no—
415C23—24no—
416C23—44no—
417C23C22×C3×D411yescharacteristic
418Q8—21no—
419Q8—21no—
420Q8—41no—
421Q8—41no—
422Q8—42no—
423Q8—82no—
424D4—21no—
425D4—21no—
426D4—42no—
427D4—41no—
428D4—41no—
429D4—42no—
430D4—84no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—82no—
445C2×C4—84no—
446C2×C4C3×#12012yes—
447C2×C4C3×#12012yes—
448C2×C4C22×C3×D411yescharacteristic
449C2×C4C22×C3×D411yescharacteristic
450C2×C4C2×C3×#4112yes—
451C2×C4C3×2−1+412yes—
452C8—24no—
453C8—42no—
454C2×C3—22no—
455C2×C3—81no—
456C2×C3—82no—
457C2×C3#243211yescharacteristic
458C2×C3#301812yes—
459C22—22no—
460C22—41no—
461C22—42no—
462C22—42no—
463C22—44no—
464C22—84no—
465C22C2×C3×#12011yescharacteristic
466C22C3×#54712yes—
467C4—22no—
468C4—22no—
469C4—44no—
470C4—42no—
471C4—41no—
472C4—42no—
473C4—42no—
474C4—42no—
475C4—82no—
476C4C3×#53211yescharacteristic
477C4C3×#53611yescharacteristic
478C4C3×#54812yes—
479C3#2568911yescharacteristic, direct product factor, Sylow
480C2—22no—
481C2—82no—
482C2—81no—
483C2C3×#243211yescharacteristic
484C2C3×#301812yes—
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#3166C2semidirect
3. C3×#3080C2semidirect
4. C3×#2861C2semidirect
5. C3×#2038C2semidirect
7. C3×#1994C2semidirect
8. C3×#1967C2semidirect
9. C3×#1857C2semidirect
10. C3×#1856C2semidirect
11. C3×C4×#104C2semidirect
12. #25689C3direct (N × H)
47. #3166C2×C3semidirect
48. #3080C2×C3semidirect
49. #2861C2×C3semidirect
50. #2038C2×C3semidirect
52. #1994C2×C3semidirect
53. #1967C2×C3semidirect
54. #1857C2×C3semidirect
55. #1856C2×C3semidirect
56. C4×#104C2×C3semidirect
479. C3#25689direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35102
C3×#31662
C3×#30802
C3×#28612
C3×#20382
C3×#20372
C3×#19942
C3×#19672
C3×#18572
C3×#18562
C3×C4×#1042
#256893