Finite Group Databasebeta

← C3×#25624 #10644231

Subgroups — C3×#25624 #10644231

Order 768.

1126 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3507C211yesmaximal, characteristic, complemented
2C3×#3151C211yesmaximal, characteristic
3C3×#3138C211yesmaximal, characteristic, complemented
4C3×#2548C211yesmaximal, characteristic, complemented
5C3×#2003C214yesmaximal, complemented
6C3×#1986C211yesmaximal, characteristic, complemented
7C3×#1980C211yesmaximal, characteristic, complemented
8C3×#1883C214yesmaximal, complemented
9C3×C4×#101C211yesmaximal, characteristic, complemented
10#25624C311yesmaximal, characteristic, direct product factor, Sylow
11C2×C3×AGL(1,ℤ/8)C2214yes—
12C3×#559—21no—
13C3×#557—21no—
14C3×#556—21no—
15C3×#556—22no—
16C3×#554—21no—
17C3×#552—22no—
18C3×#539—22no—
19C3×#536C2212yes—
20C3×#533C2211yescharacteristic
21C3×#532C2211yescharacteristic
22C2×C3×#127C2211yescharacteristic
23C3×#529—22no—
24C2×C3×#124C2212yes—
25C3×#520C2212yes—
26C2×C3×C4×Q8C2211yescharacteristic
27C3×#496—22no—
28C3×#493—21no—
29C3×#492—21no—
30C3×#490—22no—
31C3×#486—21no—
32C3×#486—21no—
33C2×C3×#104C2214yes—
34C3×#418C2214yes—
35C2×C3×#101C2212yes—
36C3×C4×M16C2211yescharacteristic
37C3×C4×M16C2211yescharacteristic
38C3×#394—22no—
39C3×#390—22no—
40C3×#390C2212yes—
41C3×C4×C4⋊C4C2211yescharacteristic
42C3×#343C2212yes—
43C3×#339C2214yes—
44#3507C2×C311yescharacteristic, complemented
45#3151C2×C311yescharacteristic
46#3138C2×C311yescharacteristic, complemented
47#2548C2×C311yescharacteristic, complemented
48#2003C2×C314yescomplemented
49#1986C2×C311yescharacteristic, complemented
50#1980C2×C311yescharacteristic, complemented
51#1883C2×C314yescomplemented
52C4×#101C2×C311yescharacteristic, complemented
53C2×C3×#41C2312yes—
54C22×C3×Q8C2311yescharacteristic
55C22×C3×D4—21no—
56C22×C3×D4—21no—
57C22×C3×D4C2312yes—
58C3×AGL(1,ℤ/8)—28no—
59C3×AGL(1,ℤ/8)—44no—
60C2×C3×SD16—24no—
61C2×C3×D8—24no—
62C2×C3×M16C2312yes—
63C2×C3×M16C2312yes—
64C3×C4⋊Q8—22no—
65C3×C4⋊Q8D411yescharacteristic
66C3×C4⋊Q8D411yescharacteristic
67C3×C4⋊Q8D411yescharacteristic
68C3×#127—21no—
69C3×#127—21no—
70C3×#127—21no—
71C3×#127—21no—
72C3×#127D411yescharacteristic
73C3×#126—24no—
74C3×#126—24no—
75C3×#125—22no—
76C3×#125—24no—
77C3×#125D412yes—
78C3×#124—22no—
79C3×#124—22no—
80C3×#124—44no—
81C3×#124D412yes—
82C3×#123—24no—
83C3×#123—24no—
84C3×C22⋊Q8—24no—
85C3×C22⋊Q8—21no—
86C3×C22⋊Q8—21no—
87C3×C22⋊Q8—22no—
88C3×C22⋊Q8—22no—
89C3×#121—21no—
90C3×#121—21no—
91C3×C4×Q8—22no—
92C3×C4×Q8—22no—
93C3×C4×Q8—21no—
94C3×C4×Q8—21no—
95C3×C4×Q8—21no—
96C3×C4×Q8—21no—
97C3×C4×Q8—21no—
98C3×C4×Q8—21no—
99C3×C4×D4—22no—
100C3×C4×D4—22no—
101C3×C4×D4—24no—
102C3×C4×D4—21no—
103C3×C4×D4—21no—
104C3×#117—24no—
105C3×#117—22no—
106C3×#117C2312yes—
107C2×C3×C4⋊C4—22no—
108C2×C3×C4⋊C4—21no—
109C2×C3×C4⋊C4D412yes—
110C2×C3×C4⋊C4C2311yescharacteristic
111C2×C3×C22⋊C4—24no—
112C2×C3×C22⋊C4—24no—
113C2×C3×C42D412yes—
114C2×C3×C42C2312yes—
115C2×C3×C42C2311yescharacteristic
116C3×#104—44no—
117C3×#102—24no—
118C3×#102—24no—
119C3×#101—24no—
120C3×#97—21no—
121C3×#97—21no—
122C3×C4×C8—21no—
123C3×C4×C8—21no—
124C3×#95—21no—
125C2×AGL(1,ℤ/8)C22×C314yes—
126#559—21no—
127#557—21no—
128#556—21no—
129#556—22no—
130#554—21no—
131#552—22no—
132#539—22no—
133#536C22×C312yes—
134#533C22×C311yescharacteristic
135#532C22×C311yescharacteristic
136C2×#127C22×C311yescharacteristic
137#529—22no—
138C2×#124C22×C312yes—
139#520C22×C312yes—
140C2×C4×Q8C22×C311yescharacteristic
141#496—22no—
142#493—21no—
143#492—21no—
144#490—22no—
145#486—21no—
146#486—21no—
147C2×#104C22×C314yes—
148#418C22×C314yes—
149C2×#101C22×C312yes—
150C4×M16C22×C311yescharacteristic
151C4×M16C22×C311yescharacteristic
152#394—22no—
153#390—22no—
154#390C22×C312yes—
155C4×C4⋊C4C22×C311yescharacteristic
156#343C22×C312yes—
157#339C22×C314yes—
158C24×C3—22no—
159C3×#41—24no—
160C3×#41—24no—
161C2×C3×Q8—21no—
162C2×C3×Q8—21no—
163C2×C3×Q8—21no—
164C2×C3×Q8—21no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8C2×D412yes—
168C2×C3×D4—22no—
169C2×C3×D4—21no—
170C2×C3×D4—21no—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C2×C3×D4—24no—
174C2×C3×D4—24no—
175C2×C3×D4—44no—
176C2×C3×D4—41no—
177C2×C3×D4—41no—
178C2×C3×D4—41no—
179C2×C3×D4—41no—
180C2×C3×D4—82no—
181C2×C3×D4C2×D412yes—
182C22×C3×C4—24no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2411yescharacteristic
190C3×SD16—48no—
191C3×D8—48no—
192C3×M16—22no—
193C3×M16—22no—
194C3×M16—22no—
195C3×M16#4114yes—
196C2×C3×C8—22no—
197C2×C3×C8—22no—
198C3×C4⋊C4—24no—
199C3×C4⋊C4—24no—
200C3×C4⋊C4—24no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—42no—
212C3×C4⋊C4C2×D412yes—
213C3×C4⋊C4C2×D412yes—
214C3×C22⋊C4—24no—
215C3×C22⋊C4—24no—
216C3×C22⋊C4—24no—
217C3×C22⋊C4—22no—
218C3×C22⋊C4—22no—
219C3×C22⋊C4—22no—
220C3×C22⋊C4—48no—
221C3×C22⋊C4—84no—
222C3×C42—24no—
223C3×C42—22no—
224C3×C42—22no—
225C3×C42—22no—
226C3×C42—24no—
227C3×C42C2×D411yescharacteristic
228C3×C42C2×D411yescharacteristic
229C3×C42C2×D411yescharacteristic
230C3×C42C2×D411yescharacteristic
231C2×#41C23×C312yes—
232C22×Q8C23×C311yescharacteristic
233C22×D4—21no—
234C22×D4—21no—
235C22×D4C23×C312yes—
236AGL(1,ℤ/8)—28no—
237AGL(1,ℤ/8)—44no—
238C2×SD16—24no—
239C2×D8—24no—
240C2×M16C23×C312yes—
241C2×M16C23×C312yes—
242C4⋊Q8—22no—
243C4⋊Q8C3×D411yescharacteristic
244C4⋊Q8C3×D411yescharacteristic
245C4⋊Q8C3×D411yescharacteristic
246#127—21no—
247#127—21no—
248#127—21no—
249#127—21no—
250#127C3×D411yescharacteristic
251#126—24no—
252#126—24no—
253#125—22no—
254#125—24no—
255#125C3×D412yes—
256#124—22no—
257#124—22no—
258#124—44no—
259#124C3×D412yes—
260#123—24no—
261#123—24no—
262C22⋊Q8—24no—
263C22⋊Q8—21no—
264C22⋊Q8—21no—
265C22⋊Q8—22no—
266C22⋊Q8—22no—
267#121—21no—
268#121—21no—
269C4×Q8—22no—
270C4×Q8—22no—
271C4×Q8—21no—
272C4×Q8—21no—
273C4×Q8—21no—
274C4×Q8—21no—
275C4×Q8—21no—
276C4×Q8—21no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—24no—
280C4×D4—21no—
281C4×D4—21no—
282#117—24no—
283#117—22no—
284#117C23×C312yes—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4C3×D412yes—
288C2×C4⋊C4C23×C311yescharacteristic
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C42C3×D412yes—
292C2×C42C23×C312yes—
293C2×C42C23×C311yescharacteristic
294#104—44no—
295#102—24no—
296#102—24no—
297#101—24no—
298#97—21no—
299#97—21no—
300C4×C8—21no—
301C4×C8—21no—
302#95—21no—
303C23×C3—22no—
304C23×C3—44no—
305C23×C3—44no—
306C23×C3—42no—
307C23×C3—42no—
308C23×C3—42no—
309C23×C3C22×D411yescharacteristic
310C3×Q8—24no—
311C3×Q8—42no—
312C3×Q8—42no—
313C3×Q8—42no—
314C3×Q8—81no—
315C3×D4—24no—
316C3×D4—42no—
317C3×D4—42no—
318C3×D4—44no—
319C3×D4—44no—
320C3×D4—42no—
321C3×D4—81no—
322C3×D4—82no—
323C3×D4—82no—
324C3×D4—82no—
325C3×D4—82no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—41no—
335C2×C3×C4—44no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—84no—
341C2×C3×C4#12011yescharacteristic
342C2×C3×C4#12011yescharacteristic
343C2×C3×C4#12011yescharacteristic
344C2×C3×C4#12011yescharacteristic
345C2×C3×C4C22×D411yescharacteristic
346C2×C3×C4C22×D411yescharacteristic
347C2×C3×C4C2×#4112yes—
348C2×C3×C4H(5,𝔽2)12yes—
349C3×C8—24no—
350C3×C8—42no—
351C24—22no—
352#41—24no—
353#41—24no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8C2×C3×D412yes—
361C2×D4—22no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—24no—
367C2×D4—24no—
368C2×D4—44no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—82no—
374C2×D4C2×C3×D412yes—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4C2×C3×D412yes—
379C22×C4C2×C3×D411yescharacteristic
380C22×C4C2×C3×D412yes—
381C22×C4C2×C3×D411yescharacteristic
382C22×C4C24×C311yescharacteristic
383SD16—48no—
384D8—48no—
385M16—22no—
386M16—22no—
387M16—22no—
388M16C3×#4114yes—
389C2×C8—22no—
390C2×C8—22no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—22no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—42no—
405C4⋊C4C2×C3×D412yes—
406C4⋊C4C2×C3×D412yes—
407C22⋊C4—24no—
408C22⋊C4—24no—
409C22⋊C4—24no—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C22⋊C4—22no—
413C22⋊C4—48no—
414C22⋊C4—84no—
415C42—24no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—24no—
420C42C2×C3×D411yescharacteristic
421C42C2×C3×D411yescharacteristic
422C42C2×C3×D411yescharacteristic
423C42C2×C3×D411yescharacteristic
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—42no—
427C22×C3—42no—
428C22×C3—42no—
429C22×C3—84no—
430C22×C3—84no—
431C22×C3—82no—
432C22×C3—82no—
433C22×C3—82no—
434C22×C3C2×#12011yescharacteristic
435C22×C3#53211yescharacteristic
436C22×C3#53511yescharacteristic
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—21no—
440C3×C4—42no—
441C3×C4—42no—
442C3×C4—42no—
443C3×C4—42no—
444C3×C4—42no—
445C3×C4—82no—
446C3×C4#45412yes—
447C3×C4#54612yes—
448C3×C4#55012yes—
449C23—22no—
450C23—44no—
451C23—44no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23C22×C3×D411yescharacteristic
456Q8—24no—
457Q8—42no—
458Q8—42no—
459Q8—42no—
460Q8—81no—
461D4—24no—
462D4—42no—
463D4—42no—
464D4—42no—
465D4—44no—
466D4—44no—
467D4—81no—
468D4—82no—
469D4—82no—
470D4—82no—
471D4—82no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—41no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—84no—
487C2×C4C3×#12011yescharacteristic
488C2×C4C3×#12011yescharacteristic
489C2×C4C3×#12011yescharacteristic
490C2×C4C3×#12011yescharacteristic
491C2×C4C22×C3×D411yescharacteristic
492C2×C4C22×C3×D411yescharacteristic
493C2×C4C2×C3×#4112yes—
494C2×C4C3×H(5,𝔽2)12yes—
495C8—24no—
496C8—42no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—82no—
500C2×C3—162no—
501C2×C3#243111yescharacteristic
502C2×C3C2×#45411yescharacteristic
503C2×C3#302111yescharacteristic
504C22—21no—
505C22—21no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22—84no—
510C22—84no—
511C22—82no—
512C22—82no—
513C22—82no—
514C22C2×C3×#12011yescharacteristic
515C22C3×#53211yescharacteristic
516C22C3×#53511yescharacteristic
517C4—21no—
518C4—21no—
519C4—21no—
520C4—42no—
521C4—42no—
522C4—42no—
523C4—42no—
524C4—42no—
525C4—82no—
526C4C3×#45412yes—
527C4C3×#54612yes—
528C4C3×#55012yes—
529C3#2562411yescharacteristic, direct product factor, Sylow
530C2—21no—
531C2—21no—
532C2—82no—
533C2—162no—
534C2C3×#243111yescharacteristic
535C2C2×C3×#45411yescharacteristic
536C2C3×#302111yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#3151C2
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×#3507C2semidirect
3. C3×#3138C2semidirect
4. C3×#2548C2semidirect
5. C3×#2003C2semidirect
6. C3×#1986C2semidirect
7. C3×#1980C2semidirect
8. C3×#1883C2semidirect
9. C3×C4×#101C2semidirect
10. #25624C3direct (N × H)
44. #3507C2×C3semidirect
46. #3138C2×C3semidirect
47. #2548C2×C3semidirect
48. #2003C2×C3semidirect
49. #1986C2×C3semidirect
50. #1980C2×C3semidirect
51. #1883C2×C3semidirect
52. C4×#101C2×C3semidirect
529. C3#25624direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35072
C3×#31512
C3×#31382
C3×#25482
C3×#20032
C3×#19862
C3×#19802
C3×#18832
C3×C4×#1012
#256243