Finite Group Databasebeta

← C2×C4×#6194 #13150439

Subgroups — C2×C4×#6194 #13150439

Order 1728.

1386 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D3×C9C211yesmaximal, characteristic, complemented
2C22×C3×C4×D9C211yesmaximal, characteristic, complemented
3C22×#6194C211yesmaximal, characteristic
4C2×C4×C3⋊Dic9C211yesmaximal, characteristic, complemented
5C2×(C2×D3)⋊Dic9C211yesmaximal, characteristic, complemented
6C2×(C2×D9)⋊Dic3C211yesmaximal, characteristic, complemented
7C2×#88442C211yesmaximal, characteristic, complemented
8C4×#6194C218yesmaximal, direct product factor
9C2×C4×#625—31nomaximal
10C2×C4×#611—31nomaximal
11C23×D3×C9C2211yescharacteristic
12C23×C3×D9C2211yescharacteristic
13C22×C3×C4×C9C2211yescharacteristic
14C22×C3⋊Dic9C2211yescharacteristic
15C22×C3⋊Dic9C2211yescharacteristic
16C22×C9×Dic3C2211yescharacteristic
17C22×C3×Dic9C2211yescharacteristic
18C2×C4×D3×C9—24no—
19C2×C4×D3×C9C2214yescomplemented
20C2×C3×C4×D9—24no—
21C2×C3×C4×D9C2214yescomplemented
22C2×#6194C418yesdirect product factor
23C2×#6194C2214yes—
24C4×C3⋊Dic9C2214yescomplemented
25(C2×D3)⋊Dic9C2214yescomplemented
26(C2×D9)⋊Dic3C2214yescomplemented
27#88442C2214yescomplemented
28C22×C3×C4×D3—31no—
29C22×C3×C4×D3D311yescharacteristic
30C22×#625—31no—
31C2×C4×C3⋊Dic3—31no—
32C2×#3724—31no—
33C2×(C2×D3)⋊Dic3—31no—
34C2×(C2×D3)⋊Dic3—31no—
35C4×#625—38no—
36C23×C4×C9—31no—
37C22×#611—31no—
38C22×C4×D9D311yescharacteristic, complemented
39C2×#3676—31no—
40C4×#611—38no—
41C2×(C2×D9)⋊C4—31no—
42C2×Dic9⋊C4—31no—
43C2×C4×Dic9—31no—
44C2×C4×Dic9—61no—
45C23×C3×C9C2311yescharacteristic
46C22×D3×C9—22no—
47C22×D3×C9—22no—
48C22×D3×C9C2×C414yescomplemented
49C22×D3×C9C2312yes—
50C22×C3×D9—22no—
51C22×C3×D9—22no—
52C22×C3×D9C2×C414yescomplemented
53C22×C3×D9C2312yes—
54C2×C3×C4×C9D414yescomplemented
55C2×C3×C4×C9C2312yes—
56C2×C3⋊Dic9—22no—
57C2×C3⋊Dic9C2×C414yescomplemented
58C2×C3⋊Dic9C2312yes—
59C2×C3⋊Dic9C2312yes—
60C2×C9×Dic3—22no—
61C2×C9×Dic3C2312yes—
62C2×C3×Dic9—22no—
63C2×C3×Dic9C2312yes—
64C4×D3×C9—28no—
65C3×C4×D9—28no—
66#6194C2×C4116yesdirect product factor
67C2×C4×#67—91no—
68C2×C4×#67—91no—
69C23×C3×D3—31no—
70C23×C3×D3C2×D311yescharacteristic
71C22×C32×C4C2×D311yescharacteristic
72C22×C3⋊Dic3—31no—
73C22×C3⋊Dic3—31no—
74C22×C3×Dic3—31no—
75C22×C3×Dic3C2×D311yescharacteristic
76C2×C3×C4×D3—24no—
77C2×C3×C4×D3—34no—
78C2×C3×C4×D3—64no—
79C2×C3×C4×D3C2×D314yes—
80C2×#625—38no—
81C2×#625—34no—
82C24×C9—31no—
83C23×D9C2×D311yescharacteristic
84C4×C3⋊Dic3—34no—
85#3724—34no—
86(C2×D3)⋊Dic3—34no—
87(C2×D3)⋊Dic3—34no—
88C22×C4×C9—21no—
89C22×C4×C9—31no—
90C22×C4×C9—34no—
91C22×C4×C9—64no—
92C22×C4×C9C2×D311yescharacteristic
93C2×#611—38no—
94C2×#611—34no—
95C22×Dic9—31no—
96C22×Dic9—31no—
97C22×Dic9—61no—
98C22×Dic9—61no—
99C22×Dic9C2×D311yescharacteristic
100C2×C4×D9—24no—
101C2×C4×D9C2×D314yescomplemented
102#3676—34no—
103(C2×D9)⋊C4—34no—
104Dic9⋊C4—34no—
105C4×Dic9—34no—
106C4×Dic9—64no—
107C22×C3×C9C22×C412yes—
108C22×C3×C9C2×D412yes—
109C22×C3×C9#4112yes—
110C22×C3×C9C2411yescharacteristic
111C2×D3×C9—28no—
112C2×D3×C9—22no—
113C2×D3×C9—22no—
114C2×D3×C9C22×C414yescomplemented
115C2×C3×D9—28no—
116C2×C3×D9—22no—
117C2×C3×D9—22no—
118C2×C3×D9C22×C414yescomplemented
119C3×C4×C9C2×D414yescomplemented
120C3⋊Dic9—22no—
121C3⋊Dic9C22×C414yescomplemented
122C9×Dic3—22no—
123C3×Dic9—22no—
124C23×C3×C4—31no—
125C23×C3×C4—91no—
126C22×#67—91no—
127C22×#67—91no—
128C22×C4×D3—31no—
129C22×C4×D3D911yescharacteristic, complemented
130C2×#217—91no—
131C2×#217—91no—
132C4×#67—98no—
133C4×#67—98no—
134C2×(C2×D3)⋊C4—91no—
135C2×(C2×D3)⋊C4—91no—
136C2×Dic3⋊C4—91no—
137C2×Dic3⋊C4—91no—
138C2×C4×Dic3—91no—
139C2×C4×Dic3—91no—
140C2×C4×Dic3—181no—
141C23×C32C22×D311yescharacteristic
142C22×C3×D3—22no—
143C22×C3×D3—22no—
144C22×C3×D3—34no—
145C22×C3×D3—32no—
146C22×C3×D3—62no—
147C22×C3×D3—62no—
148C22×C3×D3C4×D314yes—
149C22×C3×D3C22×D312yes—
150C2×C32×C4#6714yes—
151C2×C32×C4C22×D312yes—
152C2×C3⋊Dic3—34no—
153C2×C3⋊Dic3—32no—
154C2×C3⋊Dic3—32no—
155C2×C3⋊Dic3—62no—
156C2×C3×Dic3—22no—
157C2×C3×Dic3—32no—
158C2×C3×Dic3—62no—
159C2×C3×Dic3C22×D312yes—
160C3×C4×D3—28no—
161C3×C4×D3—68no—
162#625—316no—
163C23×C9—21no—
164C23×C9—34no—
165C23×C9—32no—
166C23×C9—62no—
167C23×C9—62no—
168C23×C9C22×D311yescharacteristic
169C22×D9—22no—
170C22×D9—22no—
171C22×D9C4×D314yescomplemented
172C22×D9C22×D312yes—
173C2×C4×C9—24no—
174C2×C4×C9—22no—
175C2×C4×C9—32no—
176C2×C4×C9—62no—
177C2×C4×C9—68no—
178C2×C4×C9#6714yescomplemented
179C2×C4×C9C22×D312yes—
180#611—316no—
181C2×Dic9—22no—
182C2×Dic9—34no—
183C2×Dic9—32no—
184C2×Dic9—32no—
185C2×Dic9—62no—
186C2×Dic9—62no—
187C2×Dic9—64no—
188C2×Dic9—64no—
189C2×Dic9—62no—
190C2×Dic9C22×D312yes—
191C4×D9—28no—
192C2×C4×D4—271noSylow
193C2×C3×C9C4×D414yescomplemented
194C2×C3×C9C23×C411yescharacteristic
195C2×C3×C9C22×D411yescharacteristic
196C2×C3×C9C2×#4111yescharacteristic
197D3×C9—24no—
198C3×D9—24no—
199C24×C3—31no—
200C24×C3—91no—
201C23×D3—31no—
202C23×D3C2×D911yescharacteristic
203C22×C3×C4—21no—
204C22×C3×C4—31no—
205C22×C3×C4—34no—
206C22×C3×C4—64no—
207C22×C3×C4—91no—
208C22×C3×C4—94no—
209C22×C3×C4—184no—
210C22×C3×C4C2×D911yescharacteristic
211C22×C3×C4D3211yescharacteristic
212C2×#67—94no—
213C2×#67—98no—
214C2×#67—98no—
215C2×#67—94no—
216C22×Dic3—31no—
217C22×Dic3—91no—
218C22×Dic3—91no—
219C22×Dic3—91no—
220C22×Dic3—91no—
221C22×Dic3—181no—
222C22×Dic3—181no—
223C22×Dic3C2×D911yescharacteristic
224C2×C4×D3—24no—
225C2×C4×D3—34no—
226C2×C4×D3—64no—
227C2×C4×D3C2×D914yescomplemented
228#217—94no—
229#217—94no—
230(C2×D3)⋊C4—94no—
231(C2×D3)⋊C4—94no—
232Dic3⋊C4—94no—
233Dic3⋊C4—94no—
234C4×Dic3—94no—
235C4×Dic3—94no—
236C4×Dic3—184no—
237C22×C32C2×C4×D312yes—
238C22×C32#23512yes—
239C22×C32C2×#6712yes—
240C22×C32C23×D311yescharacteristic
241C2×C3×D3—28no—
242C2×C3×D3—22no—
243C2×C3×D3—22no—
244C2×C3×D3—34no—
245C2×C3×D3—68no—
246C2×C3×D3—62no—
247C2×C3×D3—62no—
248C2×C3×D3C2×C4×D314yes—
249C32×C4C2×#6714yes—
250C3⋊Dic3—34no—
251C3⋊Dic3—62no—
252C3×Dic3—22no—
253C3×Dic3—62no—
254C22×C9—21no—
255C22×C9—22no—
256C22×C9—22no—
257C22×C9—22no—
258C22×C9—34no—
259C22×C9—68no—
260C22×C9—62no—
261C22×C9—62no—
262C22×C9C2×C4×D312yes—
263C22×C9#23512yes—
264C22×C9C2×#6712yes—
265C22×C9C23×D311yescharacteristic
266C2×D9—28no—
267C2×D9—22no—
268C2×D9—22no—
269C2×D9C2×C4×D314yescomplemented
270C4×C9—24no—
271C4×C9—62no—
272C4×C9C2×#6714yescomplemented
273Dic9—22no—
274Dic9—34no—
275Dic9—64no—
276Dic9—64no—
277Dic9—62no—
278C22×D4—271no—
279C23×C4—31no—
280C23×C4—91no—
281C4×D4—278no—
282C2×C4⋊C4—271no—
283C2×C22⋊C4—271no—
284C2×C22⋊C4—271no—
285C2×C42—271no—
286C3×C9C2×C4×D411yescharacteristic, complemented, Sylow
287C23×C3—21no—
288C23×C3—32no—
289C23×C3—34no—
290C23×C3—62no—
291C23×C3—62no—
292C23×C3—94no—
293C23×C3—92no—
294C23×C3—182no—
295C23×C3—182no—
296C23×C3C22×D911yescharacteristic
297C23×C3C2×D3211yescharacteristic
298C22×D3—22no—
299C22×D3—22no—
300C22×D3—32no—
301C22×D3—34no—
302C22×D3—62no—
303C22×D3—62no—
304C22×D3C4×D914yescomplemented
305C22×D3C22×D912yes—
306C2×C3×C4—24no—
307C2×C3×C4—22no—
308C2×C3×C4—32no—
309C2×C3×C4—62no—
310C2×C3×C4—68no—
311C2×C3×C4—92no—
312C2×C3×C4—182no—
313C2×C3×C4—188no—
314C2×C3×C4#61114yescomplemented
315C2×C3×C4C22×D912yes—
316C2×C3×C4#62514yes—
317C2×C3×C4C2×D3212yes—
318#67—916no—
319#67—916no—
320C2×Dic3—22no—
321C2×Dic3—32no—
322C2×Dic3—62no—
323C2×Dic3—92no—
324C2×Dic3—94no—
325C2×Dic3—92no—
326C2×Dic3—94no—
327C2×Dic3—92no—
328C2×Dic3—92no—
329C2×Dic3—182no—
330C2×Dic3—182no—
331C2×Dic3—184no—
332C2×Dic3—184no—
333C2×Dic3—182no—
334C2×Dic3—182no—
335C2×Dic3C22×D912yes—
336C4×D3—28no—
337C4×D3—68no—
338C2×C32C4×#6714yes—
339C2×C32C22×C4×D311yescharacteristic
340C2×C32C2×#23511yescharacteristic
341C2×C32C22×#6711yescharacteristic
342C3×D3—24no—
343C3×D3—64no—
344C2×C9—21no—
345C2×C9—21no—
346C2×C9—21no—
347C2×C9—24no—
348C2×C9—64no—
349C2×C9C4×#6714yescomplemented
350C2×C9C22×C4×D311yescharacteristic
351C2×C9C2×#23511yescharacteristic
352C2×C9C22×#6711yescharacteristic
353D9—24no—
354C24—31no—
355C24—91no—
356C2×D4—274no—
357C2×D4—278no—
358C22×C4—31no—
359C22×C4—34no—
360C22×C4—64no—
361C22×C4—91no—
362C22×C4—94no—
363C22×C4—184no—
364C22×C4—271no—
365C22×C4—271no—
366C22×C4D3×D911yescharacteristic
367C4⋊C4—274no—
368C22⋊C4—274no—
369C22⋊C4—274no—
370C42—274no—
371C22×C3—22no—
372C22×C3—21no—
373C22×C3—22no—
374C22×C3—22no—
375C22×C3—34no—
376C22×C3—62no—
377C22×C3—62no—
378C22×C3—68no—
379C22×C3—94no—
380C22×C3—188no—
381C22×C3—182no—
382C22×C3—182no—
383C22×C3C2×C4×D912yes—
384C22×C3#369712yes—
385C22×C3C2×#61112yes—
386C22×C3C23×D911yescharacteristic
387C22×C3#379812yes—
388C22×C3C4×D3212yes—
389C22×C3C2×#62512yes—
390C22×C3C22×D3211yescharacteristic
391C2×D3—28no—
392C2×D3—22no—
393C2×D3—22no—
394C2×D3—34no—
395C2×D3—62no—
396C2×D3—62no—
397C2×D3—68no—
398C2×D3C2×C4×D914yescomplemented
399C3×C4—24no—
400C3×C4—62no—
401C3×C4—182no—
402C3×C4C2×#61114yescomplemented
403C3×C4C2×#62514yes—
404Dic3—22no—
405Dic3—62no—
406Dic3—94no—
407Dic3—94no—
408Dic3—184no—
409Dic3—184no—
410Dic3—182no—
411Dic3—182no—
412C32C2×C4×#6711yescharacteristic
413C9—21no—
414C9C2×C4×#6711yescharacteristic, complemented
415C23—32no—
416C23—34no—
417C23—62no—
418C23—62no—
419C23—92no—
420C23—94no—
421C23—182no—
422C23—182no—
423C23C2×D3×D911yescharacteristic
424D4—2716no—
425C2×C4—32no—
426C2×C4—62no—
427C2×C4—68no—
428C2×C4—92no—
429C2×C4—182no—
430C2×C4—188no—
431C2×C4—272no—
432C2×C4—274no—
433C2×C4—272no—
434C2×C4—542no—
435C2×C4#619414yesdirect product factor
436C2×C4C2×D3×D912yes—
437C2×C3—21no—
438C2×C3—24no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—64no—
442C2×C3—184no—
443C2×C3C4×#61114yescomplemented
444C2×C3C22×C4×D911yescharacteristic
445C2×C3C2×#369711yescharacteristic
446C2×C3C22×#61111yescharacteristic
447C2×C3C4×#62514yes—
448C2×C3C2×#379811yescharacteristic
449C2×C3C2×C4×D3211yescharacteristic
450C2×C3C22×#62511yescharacteristic
451D3—24no—
452D3—64no—
453C22—34no—
454C22—62no—
455C22—68no—
456C22—62no—
457C22—94no—
458C22—182no—
459C22—188no—
460C22—182no—
461C22#8864012yes—
462C22C4×D3×D912yes—
463C22C2×#619412yes—
464C22C22×D3×D911yescharacteristic
465C4—62no—
466C4—182no—
467C4—274no—
468C4—542no—
469C4C2×#619414yesdirect product factor
470C3—21no—
471C3C2×C4×#61111yescharacteristic, complemented
472C3C2×C4×#62511yescharacteristic
473C2—64no—
474C2—184no—
475C2C4×#619414yesdirect product factor
476C2C2×#8864011yescharacteristic
477C2C2×C4×D3×D911yescharacteristic
478C2C22×#619411yescharacteristic
subgroupG/H
center, hypercenterC22×C4D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC23×C32C22×D3
prime radicalC22×#6194C2
cosocleC22×C3×C9C24
hypocenterC3×C9C2×C4×D4
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C4×D3×C9C2semidirect
2. C22×C3×C4×D9C2semidirect
4. C2×C4×C3⋊Dic9C2semidirect
5. C2×(C2×D3)⋊Dic9C2semidirect
6. C2×(C2×D9)⋊Dic3C2semidirect
7. C2×#88442C2semidirect
8. C4×#6194C2direct (N × H)
19. C2×C4×D3×C9C22semidirect
21. C2×C3×C4×D9C22semidirect
22. C2×#6194C4direct (N × H)
24. C4×C3⋊Dic9C22semidirect
25. (C2×D3)⋊Dic9C22semidirect
26. (C2×D9)⋊Dic3C22semidirect
27. #88442C22semidirect
38. C22×C4×D9D3semidirect
48. C22×D3×C9C2×C4semidirect
52. C22×C3×D9C2×C4semidirect
54. C2×C3×C4×C9D4semidirect
57. C2×C3⋊Dic9C2×C4semidirect
66. #6194C2×C4direct (N × H)
101. C2×C4×D9C2×D3semidirect
114. C2×D3×C9C22×C4semidirect
118. C2×C3×D9C22×C4semidirect
119. C3×C4×C9C2×D4semidirect
121. C3⋊Dic9C22×C4semidirect
129. C22×C4×D3D9semidirect
171. C22×D9C4×D3semidirect
178. C2×C4×C9#67semidirect
193. C2×C3×C9C4×D4semidirect
227. C2×C4×D3C2×D9semidirect
269. C2×D9C2×C4×D3semidirect
272. C4×C9C2×#67semidirect
286. C3×C9C2×C4×D4semidirect
304. C22×D3C4×D9semidirect
314. C2×C3×C4#611semidirect
349. C2×C9C4×#67semidirect
398. C2×D3C2×C4×D9semidirect
402. C3×C4C2×#611semidirect
414. C9C2×C4×#67semidirect
435. C2×C4#6194direct (N × H)
443. C2×C3C4×#611semidirect
469. C4C2×#6194direct (N × H)
471. C3C2×C4×#611semidirect
475. C2C4×#6194direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×D3×C92
C22×C3×C4×D92
C22×#61942
C2×C4×C3⋊Dic92
C2×(C2×D3)⋊Dic92
C2×(C2×D9)⋊Dic32
C2×#884422
C4×#61942
C2×C4×#6253
C2×C4×#6113