Finite Group Databasebeta

← C3×#32699 #10651306

Subgroups — C3×#32699 #10651306

Order 768.

538 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2102C211yesmaximal, characteristic, complemented
2C3×#2081C211yesmaximal, characteristic, complemented
3C3×#2081C211yesmaximal, characteristic, complemented
4C3×#2070C211yesmaximal, characteristic, complemented
5C3×#2048C211yesmaximal, characteristic
6C3×#2003C211yesmaximal, characteristic
7C3×#1967C211yesmaximal, characteristic, complemented
8C3×#1932C211yesmaximal, characteristic, complemented
9C3×#1915C211yesmaximal, characteristic, complemented
10C3×#1912C211yesmaximal, characteristic, complemented
11C3×#1903C211yesmaximal, characteristic
12C3×#1888C211yesmaximal, characteristic, complemented
13C3×#1861C211yesmaximal, characteristic, complemented
14C3×#1841C211yesmaximal, characteristic, complemented
15C3×#1830C211yesmaximal, characteristic, complemented
16#32699C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×AGL(1,ℤ/8)C2211yescharacteristic
18C3×#568C2211yescharacteristic
19C3×#536C2211yescharacteristic
20C2×C3×#124C2211yescharacteristic
21C3×#519C2211yescharacteristic
22C3×#430C2211yescharacteristic
23C3×#430C2211yescharacteristic
24C3×#430C2211yescharacteristic
25C3×#423C2211yescharacteristic
26C3×#421C2211yescharacteristic
27C3×#421C2211yescharacteristic
28C2×C3×#104C2211yescharacteristic
29C3×#418C2211yescharacteristic
30C3×#418C2211yescharacteristic
31C3×#418C2211yescharacteristic
32C3×#413C2211yescharacteristic
33C2×C3×#101C2211yescharacteristic
34C2×C3×#101C2211yescharacteristic
35C2×C3×#101C2211yescharacteristic
36C2×C3×#101C2211yescharacteristic
37C2×C3×#100C2211yescharacteristic
38C2×C3×#100C2211yescharacteristic
39C3×C4×M16C2211yescharacteristic
40C3×#344C2211yescharacteristic
41C3×#344C2211yescharacteristic
42C3×#344C2211yescharacteristic
43C3×#344C2211yescharacteristic
44C3×#344C2211yescharacteristic
45C3×#344C2211yescharacteristic
46C3×#339C2211yescharacteristic
47C3×#339C2211yescharacteristic
48C3×#339C2211yescharacteristic
49C3×#338C2211yescharacteristic
50C3×H(3,ℤ/4)C2211yescharacteristic
51C3×H(3,ℤ/4)C2211yescharacteristic
52#2102C2×C311yescharacteristic, complemented
53#2081C2×C311yescharacteristic, complemented
54#2081C2×C311yescharacteristic, complemented
55#2070C2×C311yescharacteristic, complemented
56#2048C2×C311yescharacteristic
57#2003C2×C311yescharacteristic
58#1967C2×C311yescharacteristic, complemented
59#1932C2×C311yescharacteristic, complemented
60#1915C2×C311yescharacteristic, complemented
61#1912C2×C311yescharacteristic, complemented
62#1903C2×C311yescharacteristic
63#1888C2×C311yescharacteristic, complemented
64#1861C2×C311yescharacteristic, complemented
65#1841C2×C311yescharacteristic, complemented
66#1830C2×C311yescharacteristic, complemented
67C2×C3×#41C2311yescharacteristic
68C22×C3×Q8C2311yescharacteristic
69C22×C3×D4C2311yescharacteristic
70C3×AGL(1,ℤ/8)—41no—
71C3×AGL(1,ℤ/8)—41no—
72C2×C3×SD16—21no—
73C2×C3×D8—21no—
74C3×#131—41no—
75C3×#131—41no—
76C2×C3×M16—41no—
77C2×C3×M16C2311yescharacteristic
78C2×C3×M16C2311yescharacteristic
79C2×C3×M16C2311yescharacteristic
80C2×C3×M16C2311yescharacteristic
81C2×C3×M16C2311yescharacteristic
82C2×C3×M16C2311yescharacteristic
83C2×C3×M16C2311yescharacteristic
84C2×C3×M16C2311yescharacteristic
85C3×C4⋊Q8—21no—
86C3×#124—21no—
87C3×#124—21no—
88C3×#124—21no—
89C3×#124—41no—
90C3×#123—22no—
91C3×C22⋊Q8—41no—
92C3×C4×Q8—41no—
93C3×C4×Q8—41no—
94C3×#117—22no—
95C3×#117C2311yescharacteristic
96C3×#117C2311yescharacteristic
97C3×#117C2311yescharacteristic
98C2×C3×C22⋊C4—22no—
99C2×C3×C42C2311yescharacteristic
100C3×#108—41no—
101C3×#108—41no—
102C3×#108—41no—
103C3×C4⋊C8—41no—
104C3×#104—21no—
105C3×#104—21no—
106C3×#104—41no—
107C3×#104—41no—
108C3×#102—41no—
109C3×#102—41no—
110C3×#102—41no—
111C3×#101—21no—
112C3×#101—41no—
113C3×#101—41no—
114C3×#101—41no—
115C3×#101—41no—
116C3×#100—21no—
117C3×#100—41no—
118C3×#100—41no—
119C3×#97—21no—
120C3×C4×C8—21no—
121C2×AGL(1,ℤ/8)C22×C311yescharacteristic
122#568C22×C311yescharacteristic
123#536C22×C311yescharacteristic
124C2×#124C22×C311yescharacteristic
125#519C22×C311yescharacteristic
126#430C22×C311yescharacteristic
127#430C22×C311yescharacteristic
128#430C22×C311yescharacteristic
129#423C22×C311yescharacteristic
130#421C22×C311yescharacteristic
131#421C22×C311yescharacteristic
132C2×#104C22×C311yescharacteristic
133#418C22×C311yescharacteristic
134#418C22×C311yescharacteristic
135#418C22×C311yescharacteristic
136#413C22×C311yescharacteristic
137C2×#101C22×C311yescharacteristic
138C2×#101C22×C311yescharacteristic
139C2×#101C22×C311yescharacteristic
140C2×#101C22×C311yescharacteristic
141C2×#100C22×C311yescharacteristic
142C2×#100C22×C311yescharacteristic
143C4×M16C22×C311yescharacteristic
144#344C22×C311yescharacteristic
145#344C22×C311yescharacteristic
146#344C22×C311yescharacteristic
147#344C22×C311yescharacteristic
148#344C22×C311yescharacteristic
149#344C22×C311yescharacteristic
150#339C22×C311yescharacteristic
151#339C22×C311yescharacteristic
152#339C22×C311yescharacteristic
153#338C22×C311yescharacteristic
154H(3,ℤ/4)C22×C311yescharacteristic
155H(3,ℤ/4)C22×C311yescharacteristic
156C24×C3—21no—
157C3×#41—21no—
158C3×#41—21no—
159C3×#41—21no—
160C3×#41—21no—
161C2×C3×Q8—21no—
162C2×C3×Q8—21no—
163C2×C3×Q8—21no—
164C2×C3×Q8—21no—
165C2×C3×Q8—21no—
166C2×C3×Q8—21no—
167C2×C3×Q8#4111yescharacteristic
168C2×C3×D4—21no—
169C2×C3×D4—21no—
170C2×C3×D4—21no—
171C2×C3×D4—21no—
172C2×C3×D4—21no—
173C2×C3×D4—21no—
174C2×C3×D4—21no—
175C2×C3×D4#4111yescharacteristic
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4#4112yes—
179C22×C3×C4C2411yescharacteristic
180C3×SD16—42no—
181C3×D8—42no—
182C3×M16—21no—
183C3×M16—21no—
184C3×M16—21no—
185C3×M16—21no—
186C3×M16—21no—
187C3×M16—21no—
188C3×M16—21no—
189C3×M16—21no—
190C3×M16—21no—
191C3×M16—21no—
192C3×M16—21no—
193C3×M16—21no—
194C3×M16—21no—
195C3×M16—21no—
196C3×M16—21no—
197C3×M16—21no—
198C3×M16—42no—
199C2×C3×C8—21no—
200C2×C3×C8—21no—
201C2×C3×C8—21no—
202C2×C3×C8—21no—
203C2×C3×C8—21no—
204C2×C3×C8—21no—
205C2×C3×C8—21no—
206C2×C3×C8—21no—
207C2×C3×C8—42no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—42no—
212C3×C4⋊C4—42no—
213C3×C22⋊C4—21no—
214C3×C22⋊C4—21no—
215C3×C22⋊C4—21no—
216C3×C22⋊C4—42no—
217C3×C22⋊C4—42no—
218C3×C22⋊C4—42no—
219C3×C42—21no—
220C3×C42—21no—
221C3×C42—21no—
222C3×C42—21no—
223C3×C42—42no—
224C3×C42#4111yescharacteristic
225C3×C42#4111yescharacteristic
226C2×#41C23×C311yescharacteristic
227C22×Q8C23×C311yescharacteristic
228C22×D4C23×C311yescharacteristic
229AGL(1,ℤ/8)—41no—
230AGL(1,ℤ/8)—41no—
231C2×SD16—21no—
232C2×D8—21no—
233#131—41no—
234#131—41no—
235C2×M16—41no—
236C2×M16C23×C311yescharacteristic
237C2×M16C23×C311yescharacteristic
238C2×M16C23×C311yescharacteristic
239C2×M16C23×C311yescharacteristic
240C2×M16C23×C311yescharacteristic
241C2×M16C23×C311yescharacteristic
242C2×M16C23×C311yescharacteristic
243C2×M16C23×C311yescharacteristic
244C4⋊Q8—21no—
245#124—21no—
246#124—21no—
247#124—21no—
248#124—41no—
249#123—22no—
250C22⋊Q8—41no—
251C4×Q8—41no—
252C4×Q8—41no—
253#117—22no—
254#117C23×C311yescharacteristic
255#117C23×C311yescharacteristic
256#117C23×C311yescharacteristic
257C2×C22⋊C4—22no—
258C2×C42C23×C311yescharacteristic
259#108—41no—
260#108—41no—
261#108—41no—
262C4⋊C8—41no—
263#104—21no—
264#104—21no—
265#104—41no—
266#104—41no—
267#102—41no—
268#102—41no—
269#102—41no—
270#101—21no—
271#101—41no—
272#101—41no—
273#101—41no—
274#101—41no—
275#100—21no—
276#100—41no—
277#100—41no—
278#97—21no—
279C4×C8—21no—
280C23×C3—21no—
281C23×C3—41no—
282C23×C3—41no—
283C23×C3—41no—
284C23×C3—82no—
285C23×C3C2×#4111yescharacteristic
286C3×Q8—21no—
287C3×Q8—21no—
288C3×Q8—41no—
289C3×Q8—41no—
290C3×Q8—41no—
291C3×Q8—41no—
292C3×D4—21no—
293C3×D4—21no—
294C3×D4—41no—
295C3×D4—41no—
296C3×D4—41no—
297C3×D4—41no—
298C3×D4—41no—
299C3×D4—41no—
300C2×C3×C4—22no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—21no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—81no—
312C2×C3×C4C2×#4111yescharacteristic
313C2×C3×C4C2×#4111yescharacteristic
314C2×C3×C4H(5,𝔽2)11yescharacteristic
315C2×C3×C4H(5,𝔽2)11yescharacteristic
316C2×C3×C4H(5,𝔽2)11yescharacteristic
317C2×C3×C4H(5,𝔽2)11yescharacteristic
318C3×C8—41no—
319C3×C8—41no—
320C3×C8—41no—
321C3×C8—41no—
322C3×C8—41no—
323C3×C8—41no—
324C3×C8—41no—
325C3×C8—41no—
326C24—21no—
327#41—21no—
328#41—21no—
329#41—21no—
330#41—21no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8C3×#4111yescharacteristic
338C2×D4—21no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4C3×#4111yescharacteristic
346C22×C4—21no—
347C22×C4—21no—
348C22×C4C3×#4112yes—
349C22×C4C24×C311yescharacteristic
350SD16—42no—
351D8—42no—
352M16—21no—
353M16—21no—
354M16—21no—
355M16—21no—
356M16—21no—
357M16—21no—
358M16—21no—
359M16—21no—
360M16—21no—
361M16—21no—
362M16—21no—
363M16—21no—
364M16—21no—
365M16—21no—
366M16—21no—
367M16—21no—
368M16—42no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—42no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C22⋊C4—21no—
384C22⋊C4—21no—
385C22⋊C4—21no—
386C22⋊C4—42no—
387C22⋊C4—42no—
388C22⋊C4—42no—
389C42—21no—
390C42—21no—
391C42—21no—
392C42—21no—
393C42—42no—
394C42C3×#4111yescharacteristic
395C42C3×#4111yescharacteristic
396C22×C3—41no—
397C22×C3—41no—
398C22×C3—81no—
399C22×C3—82no—
400C22×C3—81no—
401C22×C3—161no—
402C22×C3—161no—
403C22×C3#53811yescharacteristic
404C22×C3#55111yescharacteristic
405C22×C3#55111yescharacteristic
406C3×C4—41no—
407C3×C4—41no—
408C3×C4—42no—
409C3×C4—41no—
410C3×C4—81no—
411C3×C4—81no—
412C3×C4—81no—
413C3×C4#53811yescharacteristic
414C3×C4#54011yescharacteristic
415C3×C4#55311yescharacteristic
416C3×C4#56011yescharacteristic
417C23—21no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—82no—
422C23C2×C3×#4111yescharacteristic
423Q8—21no—
424Q8—21no—
425Q8—41no—
426Q8—41no—
427Q8—41no—
428Q8—41no—
429D4—21no—
430D4—21no—
431D4—41no—
432D4—41no—
433D4—41no—
434D4—41no—
435D4—41no—
436D4—41no—
437C2×C4—22no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—21no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—81no—
449C2×C4C2×C3×#4111yescharacteristic
450C2×C4C2×C3×#4111yescharacteristic
451C2×C4C3×H(5,𝔽2)11yescharacteristic
452C2×C4C3×H(5,𝔽2)11yescharacteristic
453C2×C4C3×H(5,𝔽2)11yescharacteristic
454C2×C4C3×H(5,𝔽2)11yescharacteristic
455C8—41no—
456C8—41no—
457C8—41no—
458C8—41no—
459C8—41no—
460C8—41no—
461C8—41no—
462C8—41no—
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—81no—
467C2×C3—161no—
468C2×C3#274911yescharacteristic
469C22—41no—
470C22—41no—
471C22—81no—
472C22—82no—
473C22—81no—
474C22—161no—
475C22—161no—
476C22C3×#53811yescharacteristic
477C22C3×#55111yescharacteristic
478C22C3×#55111yescharacteristic
479C4—41no—
480C4—41no—
481C4—42no—
482C4—41no—
483C4—81no—
484C4—81no—
485C4—81no—
486C4C3×#53811yescharacteristic
487C4C3×#54011yescharacteristic
488C4C3×#55311yescharacteristic
489C4C3×#56011yescharacteristic
490C3#3269911yescharacteristic, direct product factor, Sylow
491C2—21no—
492C2—21no—
493C2—21no—
494C2—81no—
495C2—161no—
496C2C3×#274911yescharacteristic
subgroupG/H
center, socleC2×C3#2749
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×AGL(1,ℤ/8)C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#2102C2semidirect
2. C3×#2081C2semidirect
3. C3×#2081C2semidirect
4. C3×#2070C2semidirect
7. C3×#1967C2semidirect
8. C3×#1932C2semidirect
9. C3×#1915C2semidirect
10. C3×#1912C2semidirect
12. C3×#1888C2semidirect
13. C3×#1861C2semidirect
14. C3×#1841C2semidirect
15. C3×#1830C2semidirect
16. #32699C3direct (N × H)
52. #2102C2×C3semidirect
53. #2081C2×C3semidirect
54. #2081C2×C3semidirect
55. #2070C2×C3semidirect
58. #1967C2×C3semidirect
59. #1932C2×C3semidirect
60. #1915C2×C3semidirect
61. #1912C2×C3semidirect
63. #1888C2×C3semidirect
64. #1861C2×C3semidirect
65. #1841C2×C3semidirect
66. #1830C2×C3semidirect
490. C3#32699direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#21022
C3×#20812
C3×#20702
C3×#20482
C3×#20032
C3×#19672
C3×#19322
C3×#19152
C3×#19122
C3×#19032
C3×#18882
C3×#18612
C3×#18412
C3×#18302
#326993