Finite Group Databasebeta

← C3×#25595 #10644202

Subgroups — C3×#25595 #10644202

Order 768.

1180 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3510C211yesmaximal, characteristic, complemented
2C3×#3138C211yesmaximal, characteristic, complemented
3C3×#3038C211yesmaximal, characteristic, complemented
4C3×#2657C211yesmaximal, characteristic, complemented
5C3×#2003C212yesmaximal, complemented
6C3×#2002C212yesmaximal, complemented
7C3×#1995C211yesmaximal, characteristic, complemented
8C3×#1967C211yesmaximal, characteristic, complemented
9C3×#1888C212yesmaximal, complemented
10C3×#1887C212yesmaximal, complemented
11C3×#1759C211yesmaximal, characteristic, complemented
12#25595C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×AGL(1,ℤ/8)C2212yes—
14C3×#559—22no—
15C3×#556—22no—
16C3×#551—24no—
17C3×#542—22no—
18C3×#536C2211yescharacteristic
19C3×#535C2211yescharacteristic
20C3×#533C2211yescharacteristic
21C2×C3×#127C2212yes—
22C2×C3×#127C2211yescharacteristic
23C3×#529—22no—
24C2×C3×#124C2212yes—
25C2×C3×#124C2211yescharacteristic
26C3×#519C2211yescharacteristic
27C3×#518C2211yescharacteristic
28C3×C4×#41C2211yescharacteristic
29C3×#490—22no—
30C3×#486—22no—
31C3×#460—22no—
32C3×#459—22no—
33C3×#423C2211yescharacteristic
34C2×C3×#104C2212yes—
35C3×#418C2212yes—
36C3×#418C2212yes—
37C3×#418C2212yes—
38C2×C3×#101C2212yes—
39C2×C3×#100C2212yes—
40C3×C4×M16C2211yescharacteristic
41C3×#392—24no—
42C3×#376C2211yescharacteristic
43C3×#339C2212yes—
44C3×#339C2212yes—
45C3×#339C2212yes—
46#3510C2×C311yescharacteristic, complemented
47#3138C2×C311yescharacteristic, complemented
48#3038C2×C311yescharacteristic, complemented
49#2657C2×C311yescharacteristic, complemented
50#2003C2×C312yescomplemented
51#2002C2×C312yescomplemented
52#1995C2×C311yescharacteristic, complemented
53#1967C2×C311yescharacteristic, complemented
54#1888C2×C312yescomplemented
55#1887C2×C312yescomplemented
56#1759C2×C311yescharacteristic, complemented
57C2×C3×#41C2311yescharacteristic
58C22×C3×Q8C2311yescharacteristic
59C22×C3×D4—24no—
60C22×C3×D4—22no—
61C22×C3×D4C2312yes—
62C22×C3×D4C2311yescharacteristic
63C3×AGL(1,ℤ/8)—42no—
64C3×AGL(1,ℤ/8)—42no—
65C2×C3×SD16—22no—
66C2×C3×D8—22no—
67C2×C3×M16C2312yes—
68C2×C3×M16C2312yes—
69C3×C4⋊Q8—21no—
70C3×C4⋊Q8D412yescomplemented
71C3×#127—21no—
72C3×#127—22no—
73C3×#127—21no—
74C3×#127—21no—
75C3×#127—24no—
76C3×#127—42no—
77C3×#126—24no—
78C3×#126—24no—
79C3×#125D412yescomplemented
80C3×#124—24no—
81C3×#124—21no—
82C3×#124—21no—
83C3×#124—22no—
84C3×#124—21no—
85C3×#124—21no—
86C3×#124—24no—
87C3×#124—42no—
88C3×#123—24no—
89C3×#123—24no—
90C3×#123—22no—
91C3×C22⋊Q8—24no—
92C3×C22⋊Q8—22no—
93C3×C22⋊Q8—22no—
94C3×#121—22no—
95C3×#120—22no—
96C3×C4×Q8—22no—
97C3×C4×Q8—22no—
98C3×C4×Q8—22no—
99C3×C4×Q8D412yescomplemented
100C3×C4×D4—24no—
101C3×C4×D4—22no—
102C3×C4×D4—22no—
103C3×C4×D4—22no—
104C3×C4×D4D412yescomplemented
105C3×#117—24no—
106C3×#117—22no—
107C3×#117—21no—
108C3×#117C2311yescharacteristic
109C3×#117C2311yescharacteristic
110C3×#117C2311yescharacteristic
111C2×C3×C22⋊C4—22no—
112C2×C3×C22⋊C4—24no—
113C2×C3×C22⋊C4D414yes—
114C2×C3×C42—21no—
115C2×C3×C42C2312yes—
116C2×C3×C42C2311yescharacteristic
117C3×C4⋊C8—22no—
118C3×#104—24no—
119C3×#102—24no—
120C3×#102—24no—
121C3×#102—24no—
122C3×#101—42no—
123C3×#100—42no—
124C3×#97—21no—
125C3×C4×C8—21no—
126C3×#95—22no—
127C2×AGL(1,ℤ/8)C22×C312yes—
128#559—22no—
129#556—22no—
130#551—24no—
131#542—22no—
132#536C22×C311yescharacteristic
133#535C22×C311yescharacteristic
134#533C22×C311yescharacteristic
135C2×#127C22×C311yescharacteristic
136C2×#127C22×C312yes—
137#529—22no—
138C2×#124C22×C312yes—
139C2×#124C22×C311yescharacteristic
140#519C22×C311yescharacteristic
141#518C22×C311yescharacteristic
142C4×#41C22×C311yescharacteristic
143#490—22no—
144#486—22no—
145#460—22no—
146#459—22no—
147#423C22×C311yescharacteristic
148C2×#104C22×C312yes—
149#418C22×C312yes—
150#418C22×C312yes—
151#418C22×C312yes—
152C2×#101C22×C312yes—
153C2×#100C22×C312yes—
154C4×M16C22×C311yescharacteristic
155#392—24no—
156#376C22×C311yescharacteristic
157#339C22×C312yes—
158#339C22×C312yes—
159#339C22×C312yes—
160C24×C3—22no—
161C24×C3C2×D412yes—
162C3×#41—22no—
163C3×#41—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8—21no—
166C2×C3×Q8—21no—
167C2×C3×Q8—22no—
168C2×C3×Q8—22no—
169C2×C3×Q8C2×D411yescharacteristic
170C2×C3×D4—21no—
171C2×C3×D4—24no—
172C2×C3×D4—21no—
173C2×C3×D4—21no—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—48no—
183C2×C3×D4—42no—
184C2×C3×D4—42no—
185C2×C3×D4—84no—
186C2×C3×D4—88no—
187C2×C3×D4C2×D411yescharacteristic
188C22×C3×C4—22no—
189C22×C3×C4—24no—
190C22×C3×C4—21no—
191C22×C3×C4—22no—
192C22×C3×C4C2×D412yes—
193C22×C3×C4C2×D412yes—
194C22×C3×C4C2411yescharacteristic
195C3×SD16—44no—
196C3×D8—44no—
197C3×M16—22no—
198C3×M16—22no—
199C3×M16—22no—
200C3×M16—22no—
201C2×C3×C8—22no—
202C2×C3×C8—22no—
203C3×C4⋊C4—24no—
204C3×C4⋊C4—24no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4C2×D412yes—
210C3×C4⋊C4C2×D412yes—
211C3×C4⋊C4C2×D412yes—
212C3×C22⋊C4—24no—
213C3×C22⋊C4—24no—
214C3×C22⋊C4—24no—
215C3×C22⋊C4—21no—
216C3×C22⋊C4—21no—
217C3×C22⋊C4—21no—
218C3×C22⋊C4—24no—
219C3×C22⋊C4—24no—
220C3×C22⋊C4—22no—
221C3×C22⋊C4—48no—
222C3×C42—24no—
223C3×C42—22no—
224C3×C42—21no—
225C3×C42—21no—
226C3×C42—21no—
227C3×C42—22no—
228C3×C42C2×D411yescharacteristic
229C3×C42C2×D411yescharacteristic
230C3×C42C2×D412yes—
231C3×C42#4114yes—
232C2×#41C23×C311yescharacteristic
233C22×Q8C23×C311yescharacteristic
234C22×D4—24no—
235C22×D4—22no—
236C22×D4C23×C312yes—
237C22×D4C23×C311yescharacteristic
238AGL(1,ℤ/8)—42no—
239AGL(1,ℤ/8)—42no—
240C2×SD16—22no—
241C2×D8—22no—
242C2×M16C23×C312yes—
243C2×M16C23×C312yes—
244C4⋊Q8—21no—
245C4⋊Q8C3×D412yescomplemented
246#127—21no—
247#127—22no—
248#127—21no—
249#127—21no—
250#127—24no—
251#127—42no—
252#126—24no—
253#126—24no—
254#125C3×D412yescomplemented
255#124—24no—
256#124—21no—
257#124—21no—
258#124—21no—
259#124—21no—
260#124—22no—
261#124—24no—
262#124—42no—
263#123—24no—
264#123—24no—
265#123—22no—
266C22⋊Q8—24no—
267C22⋊Q8—22no—
268C22⋊Q8—22no—
269#121—22no—
270#120—22no—
271C4×Q8—22no—
272C4×Q8—22no—
273C4×Q8—22no—
274C4×Q8C3×D412yescomplemented
275C4×D4—24no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4C3×D412yescomplemented
280#117—24no—
281#117—22no—
282#117—21no—
283#117C23×C311yescharacteristic
284#117C23×C311yescharacteristic
285#117C23×C311yescharacteristic
286C2×C22⋊C4—22no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4C3×D414yes—
289C2×C42—21no—
290C2×C42C23×C312yes—
291C2×C42C23×C311yescharacteristic
292C4⋊C8—22no—
293#104—24no—
294#102—24no—
295#102—24no—
296#102—24no—
297#101—42no—
298#100—42no—
299#97—21no—
300C4×C8—21no—
301#95—22no—
302C23×C3—22no—
303C23×C3—21no—
304C23×C3—24no—
305C23×C3—44no—
306C23×C3—48no—
307C23×C3—42no—
308C23×C3—42no—
309C23×C3—42no—
310C23×C3C22×D411yescharacteristic
311C3×Q8—21no—
312C3×Q8—21no—
313C3×Q8—41no—
314C3×Q8—41no—
315C3×Q8—42no—
316C3×Q8—82no—
317C3×D4—21no—
318C3×D4—21no—
319C3×D4—42no—
320C3×D4—41no—
321C3×D4—42no—
322C3×D4—41no—
323C3×D4—42no—
324C3×D4—42no—
325C3×D4—42no—
326C3×D4—42no—
327C3×D4—84no—
328C3×D4—84no—
329C3×D4—84no—
330C3×D4—816no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—21no—
335C2×C3×C4—22no—
336C2×C3×C4—24no—
337C2×C3×C4—24no—
338C2×C3×C4—42no—
339C2×C3×C4—44no—
340C2×C3×C4—42no—
341C2×C3×C4—82no—
342C2×C3×C4—84no—
343C2×C3×C4#12012yes—
344C2×C3×C4#12012yes—
345C2×C3×C4C22×D411yescharacteristic
346C2×C3×C4C22×D411yescharacteristic
347C2×C3×C4C2×#4112yes—
348C2×C3×C4H(5,𝔽2)12yes—
349C3×C8—42no—
350C3×C8—42no—
351C24—22no—
352C24C2×C3×D412yes—
353#41—22no—
354#41—22no—
355C2×Q8—22no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8C2×C3×D411yescharacteristic
361C2×D4—21no—
362C2×D4—24no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—48no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—84no—
377C2×D4—88no—
378C2×D4C2×C3×D411yescharacteristic
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4C2×C3×D412yes—
384C22×C4C2×C3×D412yes—
385C22×C4C24×C311yescharacteristic
386SD16—44no—
387D8—44no—
388M16—22no—
389M16—22no—
390M16—22no—
391M16—22no—
392C2×C8—22no—
393C2×C8—22no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4C2×C3×D412yes—
401C4⋊C4C2×C3×D412yes—
402C4⋊C4C2×C3×D412yes—
403C22⋊C4—24no—
404C22⋊C4—24no—
405C22⋊C4—24no—
406C22⋊C4—21no—
407C22⋊C4—21no—
408C22⋊C4—21no—
409C22⋊C4—24no—
410C22⋊C4—24no—
411C22⋊C4—22no—
412C22⋊C4—48no—
413C42—24no—
414C42—22no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—22no—
419C42C2×C3×D411yescharacteristic
420C42C2×C3×D411yescharacteristic
421C42C2×C3×D412yes—
422C42C3×#4114yes—
423C22×C3—22no—
424C22×C3—41no—
425C22×C3—42no—
426C22×C3—42no—
427C22×C3—44no—
428C22×C3—84no—
429C22×C3—84no—
430C22×C3—84no—
431C22×C3—82no—
432C22×C3—84no—
433C22×C3C2×#12011yescharacteristic
434C22×C3#54612yes—
435C3×C4—22no—
436C3×C4—22no—
437C3×C4—44no—
438C3×C4—42no—
439C3×C4—41no—
440C3×C4—42no—
441C3×C4—42no—
442C3×C4—42no—
443C3×C4#53211yescharacteristic
444C3×C4#53511yescharacteristic
445C3×C4#55012yes—
446C23—22no—
447C23—21no—
448C23—24no—
449C23—44no—
450C23—48no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23C22×C3×D411yescharacteristic
455Q8—21no—
456Q8—21no—
457Q8—41no—
458Q8—41no—
459Q8—42no—
460Q8—82no—
461D4—21no—
462D4—21no—
463D4—42no—
464D4—41no—
465D4—42no—
466D4—41no—
467D4—42no—
468D4—42no—
469D4—42no—
470D4—42no—
471D4—84no—
472D4—84no—
473D4—84no—
474D4—816no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—82no—
486C2×C4—84no—
487C2×C4C3×#12012yes—
488C2×C4C3×#12012yes—
489C2×C4C22×C3×D411yescharacteristic
490C2×C4C22×C3×D411yescharacteristic
491C2×C4C2×C3×#4112yes—
492C2×C4C3×H(5,𝔽2)12yes—
493C8—42no—
494C8—42no—
495C2×C3—22no—
496C2×C3—81no—
497C2×C3—82no—
498C2×C3—162no—
499C2×C3#243111yescharacteristic
500C2×C3#302112yes—
501C22—22no—
502C22—41no—
503C22—42no—
504C22—42no—
505C22—44no—
506C22—84no—
507C22—84no—
508C22—84no—
509C22—82no—
510C22—84no—
511C22C2×C3×#12011yescharacteristic
512C22C3×#54612yes—
513C4—22no—
514C4—22no—
515C4—44no—
516C4—42no—
517C4—41no—
518C4—42no—
519C4—42no—
520C4—42no—
521C4C3×#53211yescharacteristic
522C4C3×#53511yescharacteristic
523C4C3×#55012yes—
524C3#2559511yescharacteristic, direct product factor, Sylow
525C2—22no—
526C2—82no—
527C2—81no—
528C2—162no—
529C2C3×#243111yescharacteristic
530C2C3×#302112yes—
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
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
1. C3×#3510C2semidirect
2. C3×#3138C2semidirect
3. C3×#3038C2semidirect
4. C3×#2657C2semidirect
5. C3×#2003C2semidirect
6. C3×#2002C2semidirect
7. C3×#1995C2semidirect
8. C3×#1967C2semidirect
9. C3×#1888C2semidirect
10. C3×#1887C2semidirect
11. C3×#1759C2semidirect
12. #25595C3direct (N × H)
46. #3510C2×C3semidirect
47. #3138C2×C3semidirect
48. #3038C2×C3semidirect
49. #2657C2×C3semidirect
50. #2003C2×C3semidirect
51. #2002C2×C3semidirect
52. #1995C2×C3semidirect
53. #1967C2×C3semidirect
54. #1888C2×C3semidirect
55. #1887C2×C3semidirect
56. #1759C2×C3semidirect
70. C3×C4⋊Q8D4semidirect
79. C3×#125D4semidirect
99. C3×C4×Q8D4semidirect
104. C3×C4×D4D4semidirect
245. C4⋊Q8C3×D4semidirect
254. #125C3×D4semidirect
274. C4×Q8C3×D4semidirect
279. C4×D4C3×D4semidirect
524. C3#25595direct (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×#31382
C3×#30382
C3×#26572
C3×#20032
C3×#20022
C3×#19952
C3×#19672
C3×#18882
C3×#18872
C3×#17592
#255953