Finite Group Databasebeta

← C3×#32921 #10651528

Subgroups — C3×#32921 #10651528

Order 768.

1402 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×2−1+4C211yesmaximal, characteristic, complemented
2C3×#3038C212yesmaximal, complemented
3C2×C3×#454C212yesmaximal, complemented
4C3×#2548C211yesmaximal, characteristic, complemented
5C3×#1888C212yesmaximal, complemented
6C3×#1883C214yesmaximal, complemented
7C3×#1806C212yesmaximal, complemented
8C3×#1801C211yesmaximal, characteristic, complemented
9#32921C311yesmaximal, characteristic, direct product factor, Sylow
10C2×C3×2−1+4C2211yescharacteristic
11C2×C3×AGL(1,ℤ/8)C2214yes—
12C2×C3×#127C2211yescharacteristic
13C2×C3×#124C2212yes—
14C3×#520—22no—
15C3×#520—22no—
16C3×#520C2212yes—
17C3×#519—22no—
18C3×#519C2211yescharacteristic
19C3×C4×#41—22no—
20C3×C4×#41—22no—
21C3×C4×#41C2212yes—
22C2×C3×C4×Q8—22no—
23C2×C3×C4×Q8C2211yescharacteristic
24C3×#460—22no—
25C3×#460—22no—
26C3×#459—22no—
27C3×#459—22no—
28C3×#454—28no—
29C3×#423C2212yes—
30C2×C3×#104C2214yes—
31C3×#418C2214yes—
32C2×C3×#101C2212yes—
33C3×#394—22no—
34C3×#390—22no—
35C3×#390C2212yes—
36C3×C4×C4⋊C4C2211yescharacteristic
37C3×#343C2212yes—
38C3×#339C2214yes—
39C4×2−1+4C2×C311yescharacteristic, complemented
40#3038C2×C312yescomplemented
41C2×#454C2×C312yescomplemented
42#2548C2×C311yescharacteristic, complemented
43#1888C2×C312yescomplemented
44#1883C2×C314yescomplemented
45#1806C2×C312yescomplemented
46#1801C2×C311yescharacteristic, complemented
47C3×2−1+4—24no—
48C3×2−1+4—22no—
49C3×2−1+4D414yescomplemented
50C2×C3×#41—22no—
51C2×C3×#41—22no—
52C2×C3×#41C2312yes—
53C22×C3×Q8—22no—
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)—28no—
60C2×C3×SD16—24no—
61C2×C3×D8—24no—
62C2×C3×M16C2314yes—
63C3×#127—22no—
64C3×#127—22no—
65C3×#124—24no—
66C3×#124—24no—
67C3×C4×Q8—22no—
68C3×C4×Q8—24no—
69C3×C4×Q8—22no—
70C3×C4×Q8—22no—
71C3×C4×Q8—22no—
72C3×C4×Q8—22no—
73C3×C4×Q8—22no—
74C3×C4×Q8—22no—
75C3×C4×Q8D412yes—
76C3×C4×Q8D412yes—
77C3×C4×D4—22no—
78C3×C4×D4—24no—
79C3×C4×D4—22no—
80C3×C4×D4—22no—
81C3×C4×D4—22no—
82C3×C4×D4—24no—
83C3×C4×D4—22no—
84C3×C4×D4D412yes—
85C3×C4×D4D412yes—
86C3×#117—24no—
87C3×#117—24no—
88C3×#117—22no—
89C3×#117—22no—
90C3×#117—22no—
91C3×#117C2312yes—
92C2×C3×C4⋊C4—24no—
93C2×C3×C4⋊C4—22no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4D412yes—
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C22⋊C4—24no—
98C2×C3×C22⋊C4—24no—
99C2×C3×C42—24no—
100C2×C3×C42—22no—
101C2×C3×C42D412yes—
102C2×C3×C42C2312yes—
103C2×C3×C42C2311yescharacteristic
104C3×C4⋊C8—22no—
105C3×C4⋊C8—22no—
106C3×#104—28no—
107C3×#102—24no—
108C3×#102—24no—
109C3×#101—24no—
110C3×#95—21no—
111C2×2−1+4C22×C311yescharacteristic
112C2×AGL(1,ℤ/8)C22×C314yes—
113C2×#127C22×C311yescharacteristic
114C2×#124C22×C312yes—
115#520—22no—
116#520—22no—
117#520C22×C312yes—
118#519—22no—
119#519C22×C311yescharacteristic
120C4×#41—22no—
121C4×#41—22no—
122C4×#41C22×C312yes—
123C2×C4×Q8—22no—
124C2×C4×Q8C22×C311yescharacteristic
125#460—22no—
126#460—22no—
127#459—22no—
128#459—22no—
129#454—28no—
130#423C22×C312yes—
131C2×#104C22×C314yes—
132#418C22×C314yes—
133C2×#101C22×C312yes—
134#394—22no—
135#390—22no—
136#390C22×C312yes—
137C4×C4⋊C4C22×C311yescharacteristic
138#343C22×C312yes—
139#339C22×C314yes—
140C24×C3—22no—
141C3×#41—28no—
142C3×#41—28no—
143C3×#41—28no—
144C3×#41—28no—
145C3×#41—22no—
146C3×#41—22no—
147C3×#41C2×D414yes—
148C3×#41#4114yes—
149C2×C3×Q8—22no—
150C2×C3×Q8—28no—
151C2×C3×Q8—28no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8C2×D412yes—
155C2×C3×Q8C2×D412yes—
156C2×C3×Q8#4112yes—
157C2×C3×D4—22no—
158C2×C3×D4—22no—
159C2×C3×D4—24no—
160C2×C3×D4—22no—
161C2×C3×D4—22no—
162C2×C3×D4—24no—
163C2×C3×D4—44no—
164C2×C3×D4—41no—
165C2×C3×D4—42no—
166C2×C3×D4—41no—
167C2×C3×D4—82no—
168C2×C3×D4C2×D412yes—
169C22×C3×C4—24no—
170C22×C3×C4—22no—
171C22×C3×C4—24no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2411yescharacteristic
180C3×SD16—48no—
181C3×D8—48no—
182C3×M16—24no—
183C3×M16—24no—
184C2×C3×C8—24no—
185C3×C4⋊C4—24no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—24no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—42no—
193C3×C4⋊C4C2×D412yes—
194C3×C4⋊C4C2×D412yes—
195C3×C22⋊C4—24no—
196C3×C22⋊C4—22no—
197C3×C22⋊C4—22no—
198C3×C22⋊C4—22no—
199C3×C22⋊C4—48no—
200C3×C22⋊C4—84no—
201C3×C42—24no—
202C3×C42—24no—
203C3×C42—22no—
204C3×C42—22no—
205C3×C42—22no—
206C3×C42—22no—
207C3×C42—22no—
208C3×C42—22no—
209C3×C42—22no—
210C3×C42C2×D412yes—
211C3×C42C2×D412yes—
2122−1+4—24no—
2132−1+4—22no—
2142−1+4C3×D414yescomplemented
215C2×#41—22no—
216C2×#41—22no—
217C2×#41C23×C312yes—
218C22×Q8—22no—
219C22×Q8C23×C311yescharacteristic
220C22×D4—21no—
221C22×D4—21no—
222C22×D4C23×C312yes—
223AGL(1,ℤ/8)—28no—
224AGL(1,ℤ/8)—28no—
225C2×SD16—24no—
226C2×D8—24no—
227C2×M16C23×C314yes—
228#127—22no—
229#127—22no—
230#124—24no—
231#124—24no—
232C4×Q8—22no—
233C4×Q8—24no—
234C4×Q8—22no—
235C4×Q8—22no—
236C4×Q8—22no—
237C4×Q8—22no—
238C4×Q8—22no—
239C4×Q8—22no—
240C4×Q8C3×D412yes—
241C4×Q8C3×D412yes—
242C4×D4—22no—
243C4×D4—24no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—22no—
247C4×D4—24no—
248C4×D4—22no—
249C4×D4C3×D412yes—
250C4×D4C3×D412yes—
251#117—24no—
252#117—24no—
253#117—22no—
254#117—22no—
255#117—22no—
256#117C23×C312yes—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4C3×D412yes—
261C2×C4⋊C4C23×C311yescharacteristic
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C42—24no—
265C2×C42—22no—
266C2×C42C3×D412yes—
267C2×C42C23×C312yes—
268C2×C42C23×C311yescharacteristic
269C4⋊C8—22no—
270C4⋊C8—22no—
271#104—28no—
272#102—24no—
273#102—24no—
274#101—24no—
275#95—21no—
276C23×C3—22no—
277C23×C3—44no—
278C23×C3—44no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3—42no—
282C23×C3C22×D411yescharacteristic
283C3×Q8—28no—
284C3×Q8—24no—
285C3×Q8—22no—
286C3×Q8—24no—
287C3×Q8#12114yes—
288C3×D4—28no—
289C3×D4—24no—
290C3×D4—24no—
291C3×D4—22no—
292C3×D4—44no—
293C3×D4—44no—
294C3×D4—82no—
295C3×D4—84no—
296C3×D4—82no—
297C3×D4#12114yes—
298C2×C3×C4—216no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—24no—
303C2×C3×C4—24no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—41no—
309C2×C3×C4—44no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—82no—
315C2×C3×C4—84no—
316C2×C3×C4#12011yescharacteristic
317C2×C3×C4#12012yes—
318C2×C3×C4#12011yescharacteristic
319C2×C3×C4#12114yes—
320C2×C3×C4C22×D411yescharacteristic
321C2×C3×C4C22×D412yes—
322C2×C3×C4C2×#4112yes—
323C2×C3×C4C2×#4111yescharacteristic
324C3×C8—44no—
325C24—22no—
326#41—28no—
327#41—28no—
328#41—28no—
329#41—28no—
330#41—22no—
331#41—22no—
332#41C2×C3×D414yes—
333#41C3×#4114yes—
334C2×Q8—22no—
335C2×Q8—28no—
336C2×Q8—28no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8C2×C3×D412yes—
340C2×Q8C2×C3×D412yes—
341C2×Q8C3×#4112yes—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—24no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—24no—
348C2×D4—44no—
349C2×D4—41no—
350C2×D4—42no—
351C2×D4—41no—
352C2×D4—82no—
353C2×D4C2×C3×D412yes—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4C2×C3×D411yescharacteristic
361C22×C4C2×C3×D412yes—
362C22×C4C2×C3×D412yes—
363C22×C4C2×C3×D411yescharacteristic
364C22×C4C24×C311yescharacteristic
365SD16—48no—
366D8—48no—
367M16—24no—
368M16—24no—
369C2×C8—24no—
370C4⋊C4—24no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—24no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—42no—
378C4⋊C4C2×C3×D412yes—
379C4⋊C4C2×C3×D412yes—
380C22⋊C4—24no—
381C22⋊C4—22no—
382C22⋊C4—22no—
383C22⋊C4—22no—
384C22⋊C4—48no—
385C22⋊C4—84no—
386C42—24no—
387C42—24no—
388C42—22no—
389C42—22no—
390C42—22no—
391C42—22no—
392C42—22no—
393C42—22no—
394C42—22no—
395C42C2×C3×D412yes—
396C42C2×C3×D412yes—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—24no—
400C22×C3—42no—
401C22×C3—42no—
402C22×C3—84no—
403C22×C3—84no—
404C22×C3—82no—
405C22×C3—82no—
406C22×C3—82no—
407C22×C3C2×#12011yescharacteristic
408C22×C3C2×#12111yescharacteristic
409C22×C3#52511yescharacteristic
410C3×C4—24no—
411C3×C4—24no—
412C3×C4—21no—
413C3×C4—22no—
414C3×C4—42no—
415C3×C4—42no—
416C3×C4—42no—
417C3×C4—82no—
418C3×C4#45412yescomplemented
419C3×C4C2×#12112yes—
420C3×C4#52512yes—
421C23—22no—
422C23—44no—
423C23—44no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23C22×C3×D411yescharacteristic
428Q8—28no—
429Q8—24no—
430Q8—22no—
431Q8—24no—
432Q8C3×#12114yes—
433D4—28no—
434D4—24no—
435D4—24no—
436D4—22no—
437D4—44no—
438D4—44no—
439D4—82no—
440D4—84no—
441D4—82no—
442D4C3×#12114yes—
443C2×C4—216no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—41no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—82no—
460C2×C4—84no—
461C2×C4C3×#12011yescharacteristic
462C2×C4C3×#12012yes—
463C2×C4C3×#12011yescharacteristic
464C2×C4C3×#12114yes—
465C2×C4C22×C3×D411yescharacteristic
466C2×C4C22×C3×D412yes—
467C2×C4C2×C3×#4112yes—
468C2×C4C2×C3×#4111yescharacteristic
469C8—44no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—44no—
473C2×C3—162no—
474C2×C3#284711yescharacteristic
475C2×C3C2×#45411yescharacteristic
476C2×C3#302111yescharacteristic
477C22—21no—
478C22—21no—
479C22—24no—
480C22—42no—
481C22—42no—
482C22—84no—
483C22—84no—
484C22—82no—
485C22—82no—
486C22—82no—
487C22C2×C3×#12011yescharacteristic
488C22C2×C3×#12111yescharacteristic
489C22C3×#52511yescharacteristic
490C4—21no—
491C4—22no—
492C4—24no—
493C4—24no—
494C4—42no—
495C4—42no—
496C4—42no—
497C4—82no—
498C4C3×#45412yescomplemented
499C4C2×C3×#12112yes—
500C4C3×#52512yes—
501C3#3292111yescharacteristic, direct product factor, Sylow
502C2—21no—
503C2—21no—
504C2—44no—
505C2—162no—
506C2C3×#284711yescharacteristic
507C2C2×C3×#45411yescharacteristic
508C2C3×#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×C4×2−1+4C2semidirect
2. C3×#3038C2semidirect
3. C2×C3×#454C2semidirect
4. C3×#2548C2semidirect
5. C3×#1888C2semidirect
6. C3×#1883C2semidirect
7. C3×#1806C2semidirect
8. C3×#1801C2semidirect
9. #32921C3direct (N × H)
39. C4×2−1+4C2×C3semidirect
40. #3038C2×C3semidirect
41. C2×#454C2×C3semidirect
42. #2548C2×C3semidirect
43. #1888C2×C3semidirect
44. #1883C2×C3semidirect
45. #1806C2×C3semidirect
46. #1801C2×C3semidirect
49. C3×2−1+4D4semidirect
214. 2−1+4C3×D4semidirect
418. C3×C4#454semidirect
498. C4C3×#454semidirect
501. C3#32921direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×2−1+42
C3×#30382
C2×C3×#4542
C3×#25482
C3×#18882
C3×#18832
C3×#18062
C3×#18012
#329213