Finite Group Databasebeta

← C2×D4×#4183 #72276131

Subgroups — C2×D4×#4183 #72276131

Order 2592.

2360 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 2360 conjugacy classes of subgroups — too many to draw. Only its 211 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#4183C212yesmaximal, complemented
2C2×D4×AΣL(1,𝔽27)C211yesmaximal, characteristic, complemented
3C2×#10619679C212yesmaximal, complemented
4D4×#4183C218yesmaximal, direct product factor
5C2×C4⋊#4183C211yesmaximal, characteristic, complemented
6C2×C4×#4183C211yesmaximal, characteristic, complemented
7C2×C3×D4×C3⋊D3—31nomaximal
8C2×D4×AGL(1,ℤ/9)—32nomaximal
9C2×D4×(C3⋊D3)⋊C3—31nomaximal
10C23×AΣL(1,𝔽27)C2212yescomplemented
11C22×#4183—28no—
12C22×#4183C2211yescharacteristic
13C22×#4183C2218yescomplemented
14C22×#4183C2212yescomplemented
15D4×AΣL(1,𝔽27)C2214yescomplemented
16C2×C4×AΣL(1,𝔽27)C2211yescharacteristic, complemented
17#10619679C2218yescomplemented
18C2×#66330C2211yescharacteristic, complemented
19C4⋊#4183C2214yescomplemented
20C4×#4183C2214yescomplemented
21C23×C3×C3⋊D3—32no—
22C2×C33×D4D311yescharacteristic, complemented
23C2×C3×#638—32no—
24C3×D4×C3⋊D3—38no—
25C2×C3×C4⋊(C3⋊D3)—31no—
26C2×C3×C4×C3⋊D3—31no—
27C23×AGL(1,ℤ/9)—34no—
28C23×(C3⋊D3)⋊C3—32no—
29C2×D4×3−1+2D312yescomplemented
30C2×D4×H(3,𝔽3)D311yescharacteristic, complemented
31C2×#611⋊C3—34no—
32C2×#638⋊C3—32no—
33D4×AGL(1,ℤ/9)—316no—
34D4×(C3⋊D3)⋊C3—38no—
35C2×C4⋊AGL(1,ℤ/9)—32no—
36C2×C4×AGL(1,ℤ/9)—32no—
37C2×C4⋊((C3⋊D3)⋊C3)—31no—
38C2×C4×(C3⋊D3)⋊C3—31no—
39C22×AΣL(1,𝔽27)—24no—
40C22×AΣL(1,𝔽27)C2311yescharacteristic
41C22×AΣL(1,𝔽27)C2314yescomplemented
42C2×#4183—24no—
43C2×#4183—216no—
44C2×#4183D414yesdirect product factor
45C2×#4183C2312yes—
46C2×#4183C2314yescomplemented
47C4×AΣL(1,𝔽27)C2312yescomplemented
48#66330C2312yescomplemented
49C2×D4×C3⋊D3—31no—
50C2×D4×C3⋊D3—31no—
51C2×C3×D3×D4—93no—
52C2×C3×D3×D4—91no—
53C2×D4×D9—32no—
54C23×C33C2×D312yescomplemented
55C22×C3×C3⋊D3—31no—
56C22×C3×C3⋊D3—32no—
57C22×C3×C3⋊D3—38no—
58C22×C3×C3⋊D3—68no—
59C33×D4C2×D314yescomplemented
60C2×C33×C4C2×D311yescharacteristic, complemented
61C3×#638—38no—
62C2×C3×C3⋊Dic3—31no—
63C3×C4⋊(C3⋊D3)—34no—
64C3×C4×C3⋊D3—34no—
65C23×3−1+2C2×D314yescomplemented
66C23×H(3,𝔽3)C2×D312yescomplemented
67C22×AGL(1,ℤ/9)—32no—
68C22×AGL(1,ℤ/9)—34no—
69C22×AGL(1,ℤ/9)—316no—
70C22×AGL(1,ℤ/9)—616no—
71C22×(C3⋊D3)⋊C3—31no—
72C22×(C3⋊D3)⋊C3—32no—
73C22×(C3⋊D3)⋊C3—38no—
74C22×(C3⋊D3)⋊C3—68no—
75D4×3−1+2C2×D318yescomplemented
76D4×H(3,𝔽3)C2×D314yescomplemented
77C2×C4×3−1+2C2×D312yescomplemented
78C2×C4×H(3,𝔽3)C2×D311yescharacteristic, complemented
79#611⋊C3—316no—
80C2×3−1+2⋊C4—32no—
81#638⋊C3—38no—
82C2×(C3⋊Dic3)⋊C3—31no—
83C4⋊AGL(1,ℤ/9)—38no—
84C4×AGL(1,ℤ/9)—38no—
85C4⋊((C3⋊D3)⋊C3)—34no—
86C4×(C3⋊D3)⋊C3—34no—
87C2×AΣL(1,𝔽27)—24no—
88C2×AΣL(1,𝔽27)C2×D412yescomplemented
89C2×AΣL(1,𝔽27)C2411yescharacteristic
90#4183—24no—
91#4183C2×D414yesdirect product factor
92C23×C3⋊D3—32no—
93C23×C3⋊D3—32no—
94C23×C3×D3—96no—
95C23×C3×D3—92no—
96C2×C32×D4—33no—
97C2×C32×D4—31no—
98C2×C32×D4—31no—
99C2×C32×D4C3⋊D311yescharacteristic
100C2×#638—32no—
101C2×#638—32no—
102D4×C3⋊D3—38no—
103D4×C3⋊D3—38no—
104C2×C4⋊(C3⋊D3)—31no—
105C2×C4⋊(C3⋊D3)—31no—
106C2×C4×C3⋊D3—31no—
107C2×C4×C3⋊D3—31no—
108C2×C3×#67—96no—
109C2×C3×#67—92no—
110C3×D3×D4—924no—
111C3×D3×D4—98no—
112C2×C3×D12—93no—
113C2×C3×D12—91no—
114C2×C3×C4×D3—93no—
115C2×C3×C4×D3—91no—
116C23×D9—34no—
117C2×D4×C9—32no—
118C2×#611—34no—
119D4×D9—316no—
120C2×D36—32no—
121C2×C4×D9—32no—
122C22×C33—24no—
123C22×C33C22×D311yescharacteristic
124C22×C33C22×D314yescomplemented
125C2×C3×C3⋊D3—34no—
126C2×C3×C3⋊D3—32no—
127C2×C3×C3⋊D3—34no—
128C2×C3×C3⋊D3—616no—
129C2×C3×C3⋊D3—64no—
130C33×C4C22×D312yescomplemented
131C3×C3⋊Dic3—32no—
132C22×3−1+2—28no—
133C22×3−1+2C22×D312yes—
134C22×3−1+2C22×D318yescomplemented
135C22×H(3,𝔽3)—24no—
136C22×H(3,𝔽3)C22×D311yescharacteristic
137C22×H(3,𝔽3)C22×D314yescomplemented
138C2×AGL(1,ℤ/9)—38no—
139C2×AGL(1,ℤ/9)—34no—
140C2×AGL(1,ℤ/9)—38no—
141C2×AGL(1,ℤ/9)—632no—
142C2×AGL(1,ℤ/9)—68no—
143C2×(C3⋊D3)⋊C3—34no—
144C2×(C3⋊D3)⋊C3—32no—
145C2×(C3⋊D3)⋊C3—34no—
146C2×(C3⋊D3)⋊C3—616no—
147C2×(C3⋊D3)⋊C3—64no—
148C4×3−1+2C22×D314yescomplemented
149C4×H(3,𝔽3)C22×D312yescomplemented
1503−1+2⋊C4—34no—
151(C3⋊Dic3)⋊C3—32no—
152C22×C3×D4—271no—
153C2×D3×D4—93no—
154C2×D3×D4—91no—
155C2×D3×D4—271no—
156AΣL(1,𝔽27)C22×D411yescharacteristic, complemented, Sylow
157C23×C32—32no—
158C23×C32—36no—
159C23×C32—32no—
160C23×C32C2×C3⋊D312yes—
161C22×C3⋊D3—31no—
162C22×C3⋊D3—32no—
163C22×C3⋊D3—38no—
164C22×C3⋊D3—31no—
165C22×C3⋊D3—32no—
166C22×C3⋊D3—38no—
167C22×C3⋊D3—68no—
168C22×C3⋊D3—68no—
169C22×C3×D3—93no—
170C22×C3×D3—91no—
171C22×C3×D3—96no—
172C22×C3×D3—924no—
173C22×C3×D3—92no—
174C22×C3×D3—98no—
175C22×C3×D3—188no—
176C22×C3×D3—1824no—
177C32×D4—34no—
178C32×D4—312no—
179C32×D4—34no—
180C32×D4C2×C3⋊D314yes—
181C2×C32×C4—31no—
182C2×C32×C4—33no—
183C2×C32×C4—31no—
184C2×C32×C4C2×C3⋊D311yescharacteristic
185#638—38no—
186#638—38no—
187C2×C3⋊Dic3—31no—
188C2×C3⋊Dic3—31no—
189C4⋊(C3⋊D3)—34no—
190C4⋊(C3⋊D3)—34no—
191C4×C3⋊D3—34no—
192C4×C3⋊D3—34no—
193C3×#67—924no—
194C3×#67—98no—
195C2×C3×Dic3—93no—
196C2×C3×Dic3—91no—
197C3×D12—912no—
198C3×D12—94no—
199C3×C4×D3—912no—
200C3×C4×D3—94no—
201C23×C9—34no—
202C22×D9—32no—
203C22×D9—34no—
204C22×D9—316no—
205C22×D9—616no—
206D4×C9—38no—
207C2×C4×C9—32no—
208#611—316no—
209C2×Dic9—32no—
210D36—38no—
211C4×D9—38no—
212C2×C33—24no—
213C2×C33D3×D412yescomplemented
214C2×C33C23×D311yescharacteristic
215C3×C3⋊D3—34no—
216C3×C3⋊D3—64no—
217C2×3−1+2—28no—
218C2×3−1+2D3×D414yescomplemented
219C2×3−1+2C23×D312yes—
220C2×H(3,𝔽3)—24no—
221C2×H(3,𝔽3)D3×D412yescomplemented
222C2×H(3,𝔽3)C23×D311yescharacteristic
223AGL(1,ℤ/9)—38no—
224AGL(1,ℤ/9)—68no—
225(C3⋊D3)⋊C3—34no—
226(C3⋊D3)⋊C3—64no—
227C24×C3—272no—
228C23×D3—96no—
229C23×D3—92no—
230C23×D3—272no—
231C2×C3×D4—33no—
232C2×C3×D4—31no—
233C2×C3×D4—91no—
234C2×C3×D4—271no—
235C2×C3×D4—278no—
236C2×C3×D4—272no—
237C2×C3×D4#27211yescharacteristic
238C22×C3×C4—271no—
239C2×#67—96no—
240C2×#67—92no—
241C2×#67—272no—
242D3×D4—924no—
243D3×D4—98no—
244D3×D4—278no—
245C2×D12—93no—
246C2×D12—91no—
247C2×D12—271no—
248C2×C4×D3—93no—
249C2×C4×D3—91no—
250C2×C4×D3—271no—
251C22×C32—24no—
252C22×C32—31no—
253C22×C32—34no—
254C22×C32—33no—
255C22×C32—312no—
256C22×C32—31no—
257C22×C32—34no—
258C22×C32—64no—
259C22×C32—64no—
260C22×C32—612no—
261C22×C32C22×C3⋊D314yes—
262C22×C32C22×C3⋊D311yescharacteristic
263C2×C3⋊D3—32no—
264C2×C3⋊D3—34no—
265C2×C3⋊D3—34no—
266C2×C3⋊D3—32no—
267C2×C3⋊D3—34no—
268C2×C3⋊D3—34no—
269C2×C3⋊D3—616no—
270C2×C3⋊D3—64no—
271C2×C3⋊D3—616no—
272C2×C3⋊D3—64no—
273C2×C3×D3—96no—
274C2×C3×D3—912no—
275C2×C3×D3—92no—
276C2×C3×D3—94no—
277C2×C3×D3—912no—
278C2×C3×D3—94no—
279C2×C3×D3—1816no—
280C2×C3×D3—1848no—
281C2×C3×D3—1812no—
282C2×C3×D3—184no—
283C32×C4—32no—
284C32×C4—36no—
285C32×C4—32no—
286C32×C4C22×C3⋊D312yes—
287C3⋊Dic3—32no—
288C3⋊Dic3—32no—
289C3×Dic3—96no—
290C3×Dic3—92no—
291C22×C9—32no—
292C22×C9—38no—
293C22×C9—68no—
294C2×D9—34no—
295C2×D9—38no—
296C2×D9—38no—
297C2×D9—632no—
298C2×D9—68no—
299C4×C9—34no—
300Dic9—34no—
301C22×D4—271noSylow
302C33C2×D3×D411yescharacteristic, complemented
3033−1+2C2×D3×D412yescomplemented
304H(3,𝔽3)C2×D3×D411yescharacteristic, complemented
305C23×C3—32no—
306C23×C3—36no—
307C23×C3—92no—
308C23×C3—271no—
309C23×C3—272no—
310C23×C3—278no—
311C23×C3—548no—
312C23×C3C2×#27212yes—
313C22×D3—93no—
314C22×D3—91no—
315C22×D3—96no—
316C22×D3—924no—
317C22×D3—92no—
318C22×D3—98no—
319C22×D3—188no—
320C22×D3—1824no—
321C22×D3—271no—
322C22×D3—272no—
323C22×D3—278no—
324C22×D3—548no—
325C3×D4—34no—
326C3×D4—312no—
327C3×D4—94no—
328C3×D4—274no—
329C3×D4—278no—
330C3×D4C2×#27214yes—
331C2×C3×C4—31no—
332C2×C3×C4—33no—
333C2×C3×C4—91no—
334C2×C3×C4—274no—
335C2×C3×C4—271no—
336C2×C3×C4C2×#27211yescharacteristic
337#67—924no—
338#67—98no—
339#67—278no—
340C2×Dic3—93no—
341C2×Dic3—91no—
342C2×Dic3—271no—
343D12—912no—
344D12—94no—
345D12—274no—
346C4×D3—912no—
347C4×D3—94no—
348C4×D3—274no—
349C2×C32—24no—
350C2×C32—31no—
351C2×C32—32no—
352C2×C32—36no—
353C2×C32—33no—
354C2×C32—31no—
355C2×C32—32no—
356C2×C32—64no—
357C2×C32—64no—
358C2×C32—612no—
359C2×C32D4×C3⋊D312yes—
360C2×C32C23×C3⋊D311yescharacteristic
361C3⋊D3—34no—
362C3⋊D3—34no—
363C3⋊D3—64no—
364C3⋊D3—64no—
365C3×D3—912no—
366C3×D3—94no—
367C3×D3—1812no—
368C3×D3—184no—
369C2×C9—32no—
370C2×C9—34no—
371C2×C9—68no—
372D9—38no—
373D9—68no—
374C24—272no—
375C2×D4—272no—
376C2×D4—278no—
377C2×D4—271no—
378C2×D4#418311yescharacteristic, direct product factor
379C22×C4—271no—
380C22×C3—24no—
381C22×C3—33no—
382C22×C3—31no—
383C22×C3—312no—
384C22×C3—34no—
385C22×C3—612no—
386C22×C3—64no—
387C22×C3—91no—
388C22×C3—94no—
389C22×C3—184no—
390C22×C3—272no—
391C22×C3—274no—
392C22×C3—274no—
393C22×C3—544no—
394C22×C3—5416no—
395C22×C3C22×#27211yescharacteristic
396C22×C3C22×#27214yes—
397C2×D3—96no—
398C2×D3—912no—
399C2×D3—92no—
400C2×D3—94no—
401C2×D3—912no—
402C2×D3—94no—
403C2×D3—1816no—
404C2×D3—1848no—
405C2×D3—1812no—
406C2×D3—184no—
407C2×D3—272no—
408C2×D3—274no—
409C2×D3—274no—
410C2×D3—5416no—
411C2×D3—544no—
412C3×C4—36no—
413C3×C4—32no—
414C3×C4—92no—
415C3×C4—272no—
416C3×C4C22×#27212yes—
417Dic3—96no—
418Dic3—92no—
419Dic3—272no—
420C32—33no—
421C32—31no—
422C32—31no—
423C32C2×D4×C3⋊D311yescharacteristic
424C9—32no—
425C23—271no—
426C23—272no—
427C23—278no—
428C23—548no—
429C23C2×#418312yescomplemented
430D4—278no—
431D4—274no—
432D4C2×#418314yesdirect product factor
433C2×C4—271no—
434C2×C4—274no—
435C2×C4C2×#418311yescharacteristic, complemented
436C2×C3—24no—
437C2×C3—33no—
438C2×C3—31no—
439C2×C3—36no—
440C2×C3—32no—
441C2×C3—612no—
442C2×C3—64no—
443C2×C3—91no—
444C2×C3—92no—
445C2×C3—184no—
446C2×C3—274no—
447C2×C3—544no—
448C2×C3D4×#27212yes—
449C2×C3C23×#27211yescharacteristic
450D3—912no—
451D3—94no—
452D3—1812no—
453D3—184no—
454D3—274no—
455D3—544no—
456C22—24no—
457C22—272no—
458C22—274no—
459C22—274no—
460C22—5416no—
461C22—544no—
462C22C22×#418311yescharacteristic
463C22C22×#418314yescomplemented
464C4—272no—
465C4C22×#418312yescomplemented
466C3—33no—
467C3—31no—
468C3—91no—
469C3C2×D4×#27211yescharacteristic
470C2—24no—
471C2—274no—
472C2—544no—
473C2D4×#418312yesdirect product factor
474C2C23×#418311yescharacteristic
subgroupG/H
centerC22C22×#4183
commutator, cosocleC2×AΣL(1,𝔽27)C24
FrattiniC2×C32C23×C3⋊D3
FittingC2×D4×AΣL(1,𝔽27)C2
socleC22×C3C22×#272
hypercenterC2×D4#4183
hypocenterAΣL(1,𝔽27)C22×D4
LCSG ▷ C2×AΣL(1,𝔽27) ▷ AΣL(1,𝔽27)
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C23×#4183C2semidirect
2. C2×D4×AΣL(1,𝔽27)C2semidirect
3. C2×#10619679C2semidirect
4. D4×#4183C2direct (N × H)
5. C2×C4⋊#4183C2semidirect
6. C2×C4×#4183C2semidirect
10. C23×AΣL(1,𝔽27)C22semidirect
13. C22×#4183C22semidirect
14. C22×#4183C22semidirect
15. D4×AΣL(1,𝔽27)C22semidirect
16. C2×C4×AΣL(1,𝔽27)C22semidirect
17. #10619679C22semidirect
18. C2×#66330C22semidirect
19. C4⋊#4183C22semidirect
20. C4×#4183C22semidirect
22. C2×C33×D4D3semidirect
29. C2×D4×3−1+2D3semidirect
30. C2×D4×H(3,𝔽3)D3semidirect
41. C22×AΣL(1,𝔽27)C23semidirect
44. C2×#4183D4direct (N × H)
46. C2×#4183C23semidirect
47. C4×AΣL(1,𝔽27)C23semidirect
48. #66330C23semidirect
54. C23×C33C2×D3semidirect
59. C33×D4C2×D3semidirect
60. C2×C33×C4C2×D3semidirect
65. C23×3−1+2C2×D3semidirect
66. C23×H(3,𝔽3)C2×D3semidirect
75. D4×3−1+2C2×D3semidirect
76. D4×H(3,𝔽3)C2×D3semidirect
77. C2×C4×3−1+2C2×D3semidirect
78. C2×C4×H(3,𝔽3)C2×D3semidirect
88. C2×AΣL(1,𝔽27)C2×D4semidirect
91. #4183C2×D4direct (N × H)
124. C22×C33C22×D3semidirect
130. C33×C4C22×D3semidirect
134. C22×3−1+2C22×D3semidirect
137. C22×H(3,𝔽3)C22×D3semidirect
148. C4×3−1+2C22×D3semidirect
149. C4×H(3,𝔽3)C22×D3semidirect
156. AΣL(1,𝔽27)C22×D4semidirect
213. C2×C33D3×D4semidirect
218. C2×3−1+2D3×D4semidirect
221. C2×H(3,𝔽3)D3×D4semidirect
302. C33C2×D3×D4semidirect
303. 3−1+2C2×D3×D4semidirect
304. H(3,𝔽3)C2×D3×D4semidirect
378. C2×D4#4183direct (N × H)
429. C23C2×#4183semidirect
432. D4C2×#4183direct (N × H)
435. C2×C4C2×#4183semidirect
463. C22C22×#4183semidirect
465. C4C22×#4183semidirect
473. C2D4×#4183direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#41832
C2×D4×AΣL(1,𝔽27)2
C2×#106196792
D4×#41832
C2×C4⋊#41832
C2×C4×#41832
C2×C3×D4×C3⋊D33
C2×D4×AGL(1,ℤ/9)3
C2×D4×(C3⋊D3)⋊C33