Finite Group Databasebeta

← C3×#25622 #10644229

Subgroups — C3×#25622 #10644229

Order 768.

1294 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3506C211yesmaximal, characteristic, complemented
2C3×#3151C211yesmaximal, characteristic, complemented
3C3×#3138C211yesmaximal, characteristic, complemented
4C3×#2541C211yesmaximal, characteristic, complemented
5C3×#2002C214yesmaximal, complemented
6C3×#1986C211yesmaximal, characteristic, complemented
7C3×#1980C211yesmaximal, characteristic, complemented
8C3×#1883C214yesmaximal, complemented
9C3×C4×#100C211yesmaximal, characteristic, complemented
10#25622C311yesmaximal, characteristic, direct product factor, Sylow
11C2×C3×AGL(1,ℤ/8)C2214yes—
12C3×#559—22no—
13C3×#556—21no—
14C3×#555—21no—
15C3×#553—22no—
16C3×#550—21no—
17C3×#547—21no—
18C3×#538—22no—
19C3×#535C2212yes—
20C3×#533C2211yescharacteristic
21C3×#532C2211yescharacteristic
22C2×C3×#127C2212yes—
23C2×C3×#127C2211yescharacteristic
24C3×#525—22no—
25C2×C3×#121—22no—
26C3×#520C2212yes—
27C2×C3×C4×D4C2211yescharacteristic
28C3×#496—22no—
29C3×#493—21no—
30C3×#492—21no—
31C3×#490—22no—
32C3×#486—21no—
33C3×#486—21no—
34C2×C3×#104C2214yes—
35C3×#418C2214yes—
36C2×C3×#100C2212yes—
37C3×C4×M16C2211yescharacteristic
38C3×C4×M16C2211yescharacteristic
39C3×#392—22no—
40C3×#390C2212yes—
41C3×C4×C4⋊C4C2211yescharacteristic
42C3×#343C2212yes—
43C3×#339C2214yes—
44#3506C2×C311yescharacteristic, complemented
45#3151C2×C311yescharacteristic, complemented
46#3138C2×C311yescharacteristic, complemented
47#2541C2×C311yescharacteristic, complemented
48#2002C2×C314yescomplemented
49#1986C2×C311yescharacteristic, complemented
50#1980C2×C311yescharacteristic, complemented
51#1883C2×C314yescomplemented
52C4×#100C2×C311yescharacteristic, complemented
53C2×C3×#41—22no—
54C2×C3×#41C2312yes—
55C22×C3×D4—24no—
56C22×C3×D4—21no—
57C22×C3×D4—21no—
58C22×C3×D4C2312yes—
59C22×C3×D4C2311yescharacteristic
60C23×C3×C4D412yes—
61C3×AGL(1,ℤ/8)—28no—
62C3×AGL(1,ℤ/8)—44no—
63C2×C3×SD16—24no—
64C2×C3×D8—24no—
65C2×C3×M16C2312yes—
66C2×C3×M16C2312yes—
67C3×C4⋊Q8D411yescharacteristic, complemented
68C3×C4⋊Q8D411yescharacteristic, complemented
69C3×C4⋊Q8D411yescharacteristic, complemented
70C3×#127—22no—
71C3×#127—21no—
72C3×#127—21no—
73C3×#127—21no—
74C3×#127—21no—
75C3×#127—44no—
76C3×#127D411yescharacteristic, complemented
77C3×#126—24no—
78C3×#125D412yescomplemented
79C3×#124—24no—
80C3×#124—22no—
81C3×#124D412yescomplemented
82C3×#123—24no—
83C3×#123—24no—
84C3×C22⋊Q8—22no—
85C3×C22⋊Q8—21no—
86C3×C22⋊Q8—21no—
87C3×#121—22no—
88C3×#121—24no—
89C3×#121—21no—
90C3×#121—21no—
91C3×#121—22no—
92C3×#121—22no—
93C3×#121—48no—
94C3×#120—24no—
95C3×C4×Q8—22no—
96C3×C4×Q8—21no—
97C3×C4×Q8—21no—
98C3×C4×D4—24no—
99C3×C4×D4—22no—
100C3×C4×D4—24no—
101C3×C4×D4—21no—
102C3×C4×D4—21no—
103C3×C4×D4—21no—
104C3×C4×D4—21no—
105C3×C4×D4—21no—
106C3×C4×D4—21no—
107C3×#117—22no—
108C3×#117—22no—
109C3×#117C2312yes—
110C2×C3×C4⋊C4—21no—
111C2×C3×C4⋊C4—22no—
112C2×C3×C4⋊C4C2311yescharacteristic
113C2×C3×C22⋊C4—24no—
114C2×C3×C22⋊C4D412yes—
115C2×C3×C42C2312yes—
116C2×C3×C42C2311yescharacteristic
117C3×#104—44no—
118C3×#102—24no—
119C3×#102—24no—
120C3×#100—24no—
121C3×#97—21no—
122C3×#97—21no—
123C3×C4×C8—21no—
124C3×C4×C8—21no—
125C3×#95—21no—
126C2×AGL(1,ℤ/8)C22×C314yes—
127#559—22no—
128#556—21no—
129#555—21no—
130#553—22no—
131#550—21no—
132#547—21no—
133#538—22no—
134#535C22×C312yes—
135#533C22×C311yescharacteristic
136#532C22×C311yescharacteristic
137C2×#127C22×C311yescharacteristic
138C2×#127C22×C312yes—
139#525—22no—
140C2×#121—22no—
141#520C22×C312yes—
142C2×C4×D4C22×C311yescharacteristic
143#496—22no—
144#493—21no—
145#492—21no—
146#490—22no—
147#486—21no—
148#486—21no—
149C2×#104C22×C314yes—
150#418C22×C314yes—
151C2×#100C22×C312yes—
152C4×M16C22×C311yescharacteristic
153C4×M16C22×C311yescharacteristic
154#392—22no—
155#390C22×C312yes—
156C4×C4⋊C4C22×C311yescharacteristic
157#343C22×C312yes—
158#339C22×C314yes—
159C24×C3—22no—
160C24×C3C2×D412yes—
161C3×#41—24no—
162C3×#41—24no—
163C3×#41—84no—
164C2×C3×Q8—21no—
165C2×C3×Q8—21no—
166C2×C3×Q8C2×D412yes—
167C2×C3×D4—22no—
168C2×C3×D4—24no—
169C2×C3×D4—24no—
170C2×C3×D4—21no—
171C2×C3×D4—21no—
172C2×C3×D4—21no—
173C2×C3×D4—21no—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—24no—
178C2×C3×D4—22no—
179C2×C3×D4—24no—
180C2×C3×D4—44no—
181C2×C3×D4—48no—
182C2×C3×D4—48no—
183C2×C3×D4—41no—
184C2×C3×D4—41no—
185C2×C3×D4—41no—
186C2×C3×D4—41no—
187C2×C3×D4—82no—
188C2×C3×D4—84no—
189C2×C3×D4C2×D412yes—
190C22×C3×C4—22no—
191C22×C3×C4—24no—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4—22no—
195C22×C3×C4—44no—
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D412yes—
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2411yescharacteristic
200C3×SD16—48no—
201C3×D8—48no—
202C3×M16—22no—
203C3×M16—22no—
204C3×M16—22no—
205C3×M16#4114yescomplemented
206C2×C3×C8—22no—
207C2×C3×C8—22no—
208C3×C4⋊C4—24no—
209C3×C4⋊C4—24no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4C2×D412yes—
219C3×C4⋊C4C2×D412yes—
220C3×C22⋊C4—24no—
221C3×C22⋊C4—24no—
222C3×C22⋊C4—24no—
223C3×C22⋊C4—22no—
224C3×C22⋊C4—22no—
225C3×C22⋊C4—22no—
226C3×C22⋊C4—22no—
227C3×C22⋊C4—22no—
228C3×C22⋊C4—48no—
229C3×C42—24no—
230C3×C42—22no—
231C3×C42—24no—
232C3×C42C2×D411yescharacteristic
233C3×C42C2×D411yescharacteristic
234C3×C42C2×D411yescharacteristic
235C3×C42C2×D411yescharacteristic
236C2×#41—22no—
237C2×#41C23×C312yes—
238C22×D4—24no—
239C22×D4—21no—
240C22×D4—21no—
241C22×D4C23×C312yes—
242C22×D4C23×C311yescharacteristic
243C23×C4C3×D412yes—
244AGL(1,ℤ/8)—28no—
245AGL(1,ℤ/8)—44no—
246C2×SD16—24no—
247C2×D8—24no—
248C2×M16C23×C312yes—
249C2×M16C23×C312yes—
250C4⋊Q8C3×D411yescharacteristic, complemented
251C4⋊Q8C3×D411yescharacteristic, complemented
252C4⋊Q8C3×D411yescharacteristic, complemented
253#127—22no—
254#127—21no—
255#127—21no—
256#127—21no—
257#127—21no—
258#127—44no—
259#127C3×D411yescharacteristic, complemented
260#126—24no—
261#125C3×D412yescomplemented
262#124—24no—
263#124—22no—
264#124C3×D412yescomplemented
265#123—24no—
266#123—24no—
267C22⋊Q8—22no—
268C22⋊Q8—21no—
269C22⋊Q8—21no—
270#121—22no—
271#121—24no—
272#121—21no—
273#121—21no—
274#121—22no—
275#121—22no—
276#121—48no—
277#120—24no—
278C4×Q8—22no—
279C4×Q8—21no—
280C4×Q8—21no—
281C4×D4—24no—
282C4×D4—22no—
283C4×D4—24no—
284C4×D4—21no—
285C4×D4—21no—
286C4×D4—21no—
287C4×D4—21no—
288C4×D4—21no—
289C4×D4—21no—
290#117—22no—
291#117—22no—
292#117C23×C312yes—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4C23×C311yescharacteristic
296C2×C22⋊C4—24no—
297C2×C22⋊C4C3×D412yes—
298C2×C42C23×C312yes—
299C2×C42C23×C311yescharacteristic
300#104—44no—
301#102—24no—
302#102—24no—
303#100—24no—
304#97—21no—
305#97—21no—
306C4×C8—21no—
307C4×C8—21no—
308#95—21no—
309C23×C3—22no—
310C23×C3—22no—
311C23×C3—22no—
312C23×C3—22no—
313C23×C3—42no—
314C23×C3—42no—
315C23×C3—44no—
316C23×C3—44no—
317C23×C3—42no—
318C23×C3—42no—
319C23×C3—42no—
320C23×C3C22×D411yescharacteristic
321C3×Q8—24no—
322C3×Q8—42no—
323C3×Q8—81no—
324C3×D4—24no—
325C3×D4—42no—
326C3×D4—42no—
327C3×D4—42no—
328C3×D4—44no—
329C3×D4—42no—
330C3×D4—44no—
331C3×D4—42no—
332C3×D4—81no—
333C3×D4—82no—
334C3×D4—82no—
335C3×D4—82no—
336C3×D4—84no—
337C3×D4—84no—
338C3×D4—816no—
339C3×D4—82no—
340C2×C3×C4—22no—
341C2×C3×C4—24no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—41no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—44no—
350C2×C3×C4—44no—
351C2×C3×C4—42no—
352C2×C3×C4—82no—
353C2×C3×C4—84no—
354C2×C3×C4#12011yescharacteristic
355C2×C3×C4#12011yescharacteristic
356C2×C3×C4#12011yescharacteristic
357C2×C3×C4#12011yescharacteristic
358C2×C3×C4C22×D411yescharacteristic
359C2×C3×C4C22×D411yescharacteristic
360C2×C3×C4C2×#4112yes—
361C2×C3×C4H(5,𝔽2)12yes—
362C3×C8—24no—
363C3×C8—42no—
364C24—22no—
365C24C2×C3×D412yes—
366#41—24no—
367#41—24no—
368#41—84no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8C2×C3×D412yes—
372C2×D4—22no—
373C2×D4—24no—
374C2×D4—24no—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—24no—
383C2×D4—22no—
384C2×D4—24no—
385C2×D4—44no—
386C2×D4—48no—
387C2×D4—48no—
388C2×D4—41no—
389C2×D4—41no—
390C2×D4—41no—
391C2×D4—41no—
392C2×D4—82no—
393C2×D4—84no—
394C2×D4C2×C3×D412yes—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—44no—
401C22×C4C2×C3×D411yescharacteristic
402C22×C4C2×C3×D412yes—
403C22×C4C2×C3×D411yescharacteristic
404C22×C4C24×C311yescharacteristic
405SD16—48no—
406D8—48no—
407M16—22no—
408M16—22no—
409M16—22no—
410M16C3×#4114yescomplemented
411C2×C8—22no—
412C2×C8—22no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—22no—
418C4⋊C4—21no—
419C4⋊C4—22no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4C2×C3×D412yes—
424C4⋊C4C2×C3×D412yes—
425C22⋊C4—24no—
426C22⋊C4—24no—
427C22⋊C4—24no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—48no—
434C42—24no—
435C42—22no—
436C42—24no—
437C42C2×C3×D411yescharacteristic
438C42C2×C3×D411yescharacteristic
439C42C2×C3×D411yescharacteristic
440C42C2×C3×D411yescharacteristic
441C22×C3—21no—
442C22×C3—21no—
443C22×C3—44no—
444C22×C3—42no—
445C22×C3—42no—
446C22×C3—42no—
447C22×C3—42no—
448C22×C3—42no—
449C22×C3—42no—
450C22×C3—82no—
451C22×C3—84no—
452C22×C3—84no—
453C22×C3—82no—
454C22×C3—82no—
455C22×C3—82no—
456C22×C3C2×#12011yescharacteristic
457C22×C3#53211yescharacteristic
458C22×C3#53511yescharacteristic
459C3×C4—21no—
460C3×C4—21no—
461C3×C4—21no—
462C3×C4—44no—
463C3×C4—42no—
464C3×C4—42no—
465C3×C4—42no—
466C3×C4—42no—
467C3×C4#45312yes—
468C3×C4#54612yes—
469C3×C4#55012yes—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—42no—
475C23—42no—
476C23—44no—
477C23—44no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23C22×C3×D411yescharacteristic
482Q8—24no—
483Q8—42no—
484Q8—81no—
485D4—24no—
486D4—42no—
487D4—42no—
488D4—42no—
489D4—42no—
490D4—44no—
491D4—42no—
492D4—44no—
493D4—81no—
494D4—82no—
495D4—82no—
496D4—82no—
497D4—84no—
498D4—84no—
499D4—816no—
500D4—82no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—82no—
514C2×C4—84no—
515C2×C4C3×#12011yescharacteristic
516C2×C4C3×#12011yescharacteristic
517C2×C4C3×#12011yescharacteristic
518C2×C4C3×#12011yescharacteristic
519C2×C4C22×C3×D411yescharacteristic
520C2×C4C22×C3×D411yescharacteristic
521C2×C4C2×C3×#4112yes—
522C2×C4C3×H(5,𝔽2)12yes—
523C8—24no—
524C8—42no—
525C2×C3—21no—
526C2×C3—21no—
527C2×C3—82no—
528C2×C3—82no—
529C2×C3—162no—
530C2×C3#243111yescharacteristic
531C2×C3C2×#45311yescharacteristic
532C2×C3#302111yescharacteristic
533C22—21no—
534C22—21no—
535C22—44no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—82no—
543C22—84no—
544C22—84no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22C2×C3×#12011yescharacteristic
549C22C3×#53211yescharacteristic
550C22C3×#53511yescharacteristic
551C4—21no—
552C4—21no—
553C4—21no—
554C4—44no—
555C4—42no—
556C4—42no—
557C4—42no—
558C4—42no—
559C4C3×#45312yes—
560C4C3×#54612yes—
561C4C3×#55012yes—
562C3#2562211yescharacteristic, direct product factor, Sylow
563C2—21no—
564C2—21no—
565C2—82no—
566C2—82no—
567C2—162no—
568C2C3×#243111yescharacteristic
569C2C2×C3×#45311yescharacteristic
570C2C3×#302111yescharacteristic
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×#3506C2semidirect
2. C3×#3151C2semidirect
3. C3×#3138C2semidirect
4. C3×#2541C2semidirect
5. C3×#2002C2semidirect
6. C3×#1986C2semidirect
7. C3×#1980C2semidirect
8. C3×#1883C2semidirect
9. C3×C4×#100C2semidirect
10. #25622C3direct (N × H)
44. #3506C2×C3semidirect
45. #3151C2×C3semidirect
46. #3138C2×C3semidirect
47. #2541C2×C3semidirect
48. #2002C2×C3semidirect
49. #1986C2×C3semidirect
50. #1980C2×C3semidirect
51. #1883C2×C3semidirect
52. C4×#100C2×C3semidirect
67. C3×C4⋊Q8D4semidirect
68. C3×C4⋊Q8D4semidirect
69. C3×C4⋊Q8D4semidirect
76. C3×#127D4semidirect
78. C3×#125D4semidirect
81. C3×#124D4semidirect
205. C3×M16#41semidirect
250. C4⋊Q8C3×D4semidirect
251. C4⋊Q8C3×D4semidirect
252. C4⋊Q8C3×D4semidirect
259. #127C3×D4semidirect
261. #125C3×D4semidirect
264. #124C3×D4semidirect
410. M16C3×#41semidirect
562. C3#25622direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35062
C3×#31512
C3×#31382
C3×#25412
C3×#20022
C3×#19862
C3×#19802
C3×#18832
C3×C4×#1002
#256223