Finite Group Databasebeta

← C2×#13082404 #15662242

Subgroups — C2×#13082404 #15662242

Order 2816.

11534 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×#67043C211yesmaximal, characteristic, complemented
2#13082404C2132yesmaximal, direct product factor
3C2×#10623362C214yesmaximal, complemented
4C2×#10623337C2112yesmaximal, complemented
5C2×#10623356C213yesmaximal, complemented
6C2×#10623336C213yesmaximal, complemented
7C2×#10623363C213yesmaximal, complemented
8C22×#66989C213yesmaximal, complemented
9C2×#41×Dic11C211yesmaximal, characteristic, complemented
10C22×C11×H(5,𝔽2)C211yesmaximal, characteristic, complemented
11C2×C11×H(5,𝔽2)—216no—
12C2×C11×H(5,𝔽2)C22116yescomplemented
13C22×C11×#41C2213yes—
14C22×C11×#41C2212yes—
15C22×C11×#41C2211yescharacteristic
16C23×D4×C11C2213yes—
17C23×D4×C11C2216yes—
18C2×#67043—216no—
19C2×#67043C22116yescomplemented
20C22×Q8×D11C2212yescomplemented
21C22×#4341C2216yescomplemented
22C22×#4339C2213yescomplemented
23C23×Dic22C2213yescomplemented
24C2×#67006C22112yescomplemented
25C22×C22⋊Dic11C22112yescomplemented
26#10623363C22148yescomplemented
27#10623362C22164yescomplemented
28#41×Dic11C22116yescomplemented
29#10623356C22148yescomplemented
30C2×#67000C2211yescharacteristic, complemented
31C2×Q8×Dic11C2212yescomplemented
32C2×#66997C2213yescomplemented
33#10623337C221192yescomplemented
34#10623336C22148yescomplemented
35C2×#66993C2213yescomplemented
36C2×#66992C22112yescomplemented
37C2×#66991C22112yescomplemented
38C2×#66989C2213yescomplemented
39C2×#66989C2216yescomplemented
40C2×#66989C22112yescomplemented
41C2×#66989C22148yescomplemented
42C2×#66988C22112yescomplemented
43C2×D4×Dic11C2216yescomplemented
44C2×C4×C22⋊D11C22112yescomplemented
45C2×#66979C2213yescomplemented
46C2×#66977C22112yescomplemented
47C22×C4×Dic11C2213yescomplemented
48C25×C11C2316yes—
49C11×H(5,𝔽2)—2128no—
50C2×C11×#41—216no—
51C2×C11×#41—224no—
52C2×C11×#41D4116yescomplemented
53C2×C11×#41C23124yes—
54C2×C11×#41C23116yes—
55C2×C11×#41C2318yes—
56C22×Q8×C11C2311yescharacteristic
57C22×Q8×C11C2311yescharacteristic
58C22×D4×C11—248no—
59C22×D4×C11—224no—
60C22×D4×C11C2316yes—
61C22×D4×C11C2313yes—
62C22×D4×C11C2316yes—
63C22×D4×C11C23124yes—
64C22×D4×C11C23148yes—
65C22×D4×C11C2313yes—
66C23×C4×C11C2316yes—
67C23×C4×C11C2313yes—
68C22×C22⋊D11C2316yes—
69C22×C22⋊D11C23112yes—
70C23×Dic11C2316yes—
71C23×Dic11C2316yes—
72#67043—2128no—
73C2×Q8×D11—216no—
74C2×Q8×D11C23116yescomplemented
75C2×#4341—248no—
76C2×#4341C23148yescomplemented
77C2×#4339—224no—
78C2×#4339C23124yescomplemented
79C22×D44C2311yescharacteristic
80C22×C4×D11C2316yes—
81C22×Dic22—224no—
82C22×Dic22C2316yes—
83C22×Dic22C2313yes—
84C22×Dic22C23124yescomplemented
85#67006C23196yescomplemented
86C2×C22⋊Dic11C23124yes—
87C2×C22⋊Dic11C2316yes—
88C2×C22⋊Dic11C23112yes—
89C2×C22⋊Dic11C23196yescomplemented
90#67000C2318yescomplemented
91Q8×Dic11C23116yescomplemented
92#66997C23124yescomplemented
93#66993C23124yescomplemented
94#66992C23196yescomplemented
95#66991C23196yescomplemented
96#66989C23124yescomplemented
97#66989C23148yescomplemented
98#66989C23196yescomplemented
99#66988C23196yescomplemented
100D4×Dic11C23148yescomplemented
101C4×C22⋊D11C23196yescomplemented
102C2×(C2×D11)⋊C4C2314yes—
103#66979C23124yescomplemented
104C2×C4⋊Dic11C2316yes—
105#66977C23196yescomplemented
106C2×Dic11⋊C4C23112yes—
107C2×C4×Dic11C2311yescharacteristic
108C2×C4×Dic11C2313yes—
109C2×C4×Dic11C2316yes—
110C2×C4×Dic11C23124yescomplemented
111C2×#3500—111nomaximal, Sylow
112C24×C11—224no—
113C24×C11—424no—
114C24×C11C2416yes—
115C24×C11C2413yes—
116C24×C11C24124yes—
117C23×D11C2412yes—
118C11×#41—264no—
119C11×#41—296no—
120C11×#41C2×D4164yescomplemented
121C2×Q8×C11—24no—
122C2×Q8×C11C2×D418yes—
123C2×Q8×C11C2414yes—
124C2×Q8×C11C2414yes—
125C2×D4×C11—224no—
126C2×D4×C11—2192no—
127C2×D4×C11—212no—
128C2×D4×C11—224no—
129C2×D4×C11—296no—
130C2×D4×C11C2×D4124yes—
131C2×D4×C11C24124yes—
132C2×D4×C11C24112yes—
133C2×D4×C11C24124yes—
134C2×D4×C11C24112yes—
135C22×C4×C11—224no—
136C22×C4×C11C2×D4124yes—
137C22×C4×C11C2412yes—
138C22×C4×C11C24124yes—
139C22×C4×C11C2411yescharacteristic
140C22×C4×C11C2413yes—
141C22×C4×C11C24112yes—
142C2×C22⋊D11—224no—
143C2×C22⋊D11—424no—
144C2×C22⋊D11C24124yes—
145C2×C22⋊D11C24148yes—
146C22×Dic11—224no—
147C22×Dic11—412no—
148C22×Dic11C2412yes—
149C22×Dic11C2416yes—
150C22×Dic11C2416yes—
151C22×Dic11C24124yes—
152C22×Dic11C24124yes—
153Q8×D11—264no—
154#4341—2192no—
155#4339—296no—
156C2×D44—24no—
157C2×D44C2414yes—
158C2×C4×D11—224no—
159C2×C4×D11C24124yes—
160C2×Dic22—212no—
161C2×Dic22—224no—
162C2×Dic22—296no—
163C2×Dic22C24124yes—
164C2×Dic22C24112yes—
165C22⋊Dic11C24196yes—
166C22⋊Dic11C24124yes—
167C22⋊Dic11C24148yes—
168(C2×D11)⋊C4C24116yes—
169C4⋊Dic11C24124yes—
170Dic11⋊C4C24148yes—
171C4×Dic11C2414yes—
172C4×Dic11C24112yes—
173C4×Dic11C24124yes—
174C22×2−1+4—111no—
175C22×H(5,𝔽2)D1111yescharacteristic, complemented
176#3500—1132no—
177C2×#548—114no—
178C2×#546—1112no—
179C2×#536—113no—
180C2×#535—113no—
181C2×#532—113no—
182C22×#124—113no—
183C2×C4×#41—111no—
184C23×C11—212no—
185C23×C11—212no—
186C23×C11—224no—
187C23×C11—248no—
188C23×C11—448no—
189C23×C11—848no—
190C23×C11C22×D4112yes—
191C23×C11C25112yes—
192C23×C11C2516yes—
193C23×C11C2511yescharacteristic
194C22×D11—24no—
195C22×D11—28no—
196C22×D11C2514yes—
197Q8×C11—28no—
198Q8×C11C22×D4116yes—
199D4×C11—248no—
200D4×C11—248no—
201D4×C11—224no—
202D4×C11C22×D4148yes—
203C2×C4×C11—24no—
204C2×C4×C11—248no—
205C2×C4×C11C22×D414yes—
206C2×C4×C11C22×D4112yes—
207C2×C4×C11C22×D4148yes—
208C2×C4×C11C2514yes—
209C2×C4×C11C2512yes—
210C2×C4×C11C2516yes—
211C22⋊D11—248no—
212C22⋊D11—448no—
213C2×Dic11—212no—
214C2×Dic11—248no—
215C2×Dic11—24no—
216C2×Dic11—212no—
217C2×Dic11—424no—
218C2×Dic11C2514yes—
219C2×Dic11C25112yes—
220C2×Dic11C25112yes—
221D44—28no—
222C4×D11—248no—
223Dic22—224no—
224Dic22—248no—
225C2×2−1+4—1116no—
226C2×2−1+4—2216no—
227C2×H(5,𝔽2)—216no—
228C2×H(5,𝔽2)C2×D11116yescomplemented
229C22×#41—116no—
230C22×#41—113no—
231C22×#41C2×D1113yes—
232C22×#41C2×D1112yes—
233C22×#41C2×D1111yescharacteristic
234C23×Q8—112no—
235C23×Q8—113no—
236C23×D4C2×D1113yes—
237C23×D4C2×D1116yes—
238#548—1164no—
239#546—11192no—
240#536—1148no—
241#535—1148no—
242#532—1148no—
243C2×C4⋊Q8—113no—
244C2×#127—113no—
245C2×#124—111no—
246C2×#124—113no—
247C2×#124—116no—
248C2×#124—1112no—
249C2×#124—1148no—
250C2×#123—1112no—
251C2×C22⋊Q8—1112no—
252C2×#121—1112no—
253C2×#121—1112no—
254C2×#120—1112no—
255C4×#41—1116no—
256C2×C4×Q8—112no—
257C2×C4×D4—116no—
258C2×C4×D4—1112no—
259C2×#117—113no—
260C22×C22⋊C4—1112no—
261C22×C42—113no—
262C22×C11—212no—
263C22×C11—224no—
264C22×C11—212no—
265C22×C11—224no—
266C22×C11—424no—
267C22×C11—848no—
268C22×C11C23×D4112yes—
269C22×C11C23×D412yes—
270C22×C11#58514yes—
271C22×C11C2611yescharacteristic
272C2×D11—24no—
273C2×D11—28no—
274C2×D11—48no—
275C4×C11—24no—
276C4×C11C23×D4112yes—
277C4×C11C23×D414yes—
278Dic11—212no—
279Dic11—24no—
280Dic11—212no—
281C25C22×D1116yes—
2822−1+4—22128no—
283H(5,𝔽2)—2128no—
284C2×#41—216no—
285C2×#41—224no—
286C2×#41—1148no—
287C2×#41—1124no—
288C2×#41—2248no—
289C2×#41—2224no—
290C2×#41C22⋊D11116yescomplemented
291C2×#41C22×D11124yes—
292C2×#41C22×D11116yes—
293C2×#41C22×D1118yes—
294C22×Q8—1116no—
295C22×Q8—116no—
296C22×Q8—113no—
297C22×Q8—1124no—
298C22×Q8—2216no—
299C22×Q8—2224no—
300C22×Q8C22×D1111yescharacteristic
301C22×Q8C22×D1111yescharacteristic
302C22×D4—248no—
303C22×D4—224no—
304C22×D4—116no—
305C22×D4—1112no—
306C22×D4—111no—
307C22×D4C22×D1116yes—
308C22×D4C22×D1116yes—
309C22×D4C22×D1113yes—
310C22×D4C22×D11124yes—
311C22×D4C22×D11148yes—
312C22×D4C22×D1113yes—
313C23×C4—116no—
314C23×C4—116no—
315C23×C4—116no—
316C23×C4C22×D1116yes—
317C23×C4C22×D1113yes—
318C4⋊Q8—1124no—
319#127—1124no—
320#124—118no—
321#124—1124no—
322#124—1148no—
323#124—1196no—
324#123—1196no—
325C22⋊Q8—1196no—
326#121—1196no—
327#121—1196no—
328#120—1196no—
329C4×Q8—1116no—
330C4×D4—1148no—
331C4×D4—1196no—
332#117—1124no—
333C2×C4⋊C4—116no—
334C2×C4⋊C4—1112no—
335C2×C22⋊C4—1112no—
336C2×C22⋊C4—1124no—
337C2×C22⋊C4—1196no—
338C2×C22⋊C4—116no—
339C2×C22⋊C4—114no—
340C2×C42—113no—
341C2×C42—116no—
342C2×C42—1124no—
343C2×C42—111no—
344C2×C11—212no—
345C2×C11—412no—
346C2×C11#350014yescomplemented
347C2×C11C24×D411yescharacteristic
348C2×C11C2×#58512yes—
349D11—44no—
350C24—224no—
351C24—424no—
352C24—112no—
353C24C23×D1116yes—
354C24C23×D1113yes—
355C24C23×D11124yes—
356#41—264no—
357#41—296no—
358#41—22192no—
359#41—2296no—
360#41C2×C22⋊D11164yescomplemented
361C2×Q8—24no—
362C2×Q8—1124no—
363C2×Q8—1112no—
364C2×Q8—2264no—
365C2×Q8—2224no—
366C2×Q8—2212no—
367C2×Q8—2296no—
368C2×Q8C2×C22⋊D1118yes—
369C2×Q8C23×D1114yes—
370C2×Q8C23×D1114yes—
371C2×D4—224no—
372C2×D4—2192no—
373C2×D4—224no—
374C2×D4—212no—
375C2×D4—296no—
376C2×D4—1124no—
377C2×D4—1148no—
378C2×D4—114no—
379C2×D4—2224no—
380C2×D4—224no—
381C2×D4—4424no—
382C2×D4C2×C22⋊D11124yes—
383C2×D4C23×D11124yes—
384C2×D4C23×D11124yes—
385C2×D4C23×D11112yes—
386C2×D4C23×D11112yes—
387C22×C4—224no—
388C22×C4—1124no—
389C22×C4—112no—
390C22×C4—116no—
391C22×C4—116no—
392C22×C4—1124no—
393C22×C4—1124no—
394C22×C4—2224no—
395C22×C4—2224no—
396C22×C4—4412no—
397C22×C4C2×C22⋊D11124yes—
398C22×C4C23×D11124yes—
399C22×C4C23×D1112yes—
400C22×C4C23×D11112yes—
401C22×C4C23×D1111yescharacteristic
402C22×C4C23×D1113yes—
403C4⋊C4—1124no—
404C4⋊C4—1148no—
405C22⋊C4—1148no—
406C22⋊C4—1196no—
407C22⋊C4—1124no—
408C22⋊C4—1116no—
409C42—114no—
410C42—1112no—
411C42—1124no—
412C11C2×#350011yescharacteristic, complemented, Sylow
413C23—212no—
414C23—224no—
415C23—212no—
416C23—248no—
417C23—448no—
418C23—848no—
419C23—114no—
420C23—228no—
421C23—224no—
422C23C22×C22⋊D11112yes—
423C23C24×D11112yes—
424C23C24×D1116yes—
425C23C24×D1111yescharacteristic
426Q8—28no—
427Q8—2248no—
428Q8—2224no—
429Q8C22×C22⋊D11116yes—
430D4—248no—
431D4—248no—
432D4—224no—
433D4—2248no—
434D4—228no—
435D4—4448no—
436D4C22×C22⋊D11148yes—
437C2×C4—24no—
438C2×C4—248no—
439C2×C4—114no—
440C2×C4—1112no—
441C2×C4—1112no—
442C2×C4—224no—
443C2×C4—2212no—
444C2×C4—2212no—
445C2×C4—2248no—
446C2×C4—2248no—
447C2×C4—4424no—
448C2×C4C22×C22⋊D11112yes—
449C2×C4C22×C22⋊D1114yes—
450C2×C4C22×C22⋊D11148yes—
451C2×C4C24×D1114yes—
452C2×C4C24×D1112yes—
453C2×C4C24×D1116yes—
454C22—212no—
455C22—224no—
456C22—212no—
457C22—224no—
458C22—424no—
459C22—848no—
460C22—224no—
461C22—228no—
462C22—448no—
463C22#1062345514yes—
464C22C23×C22⋊D11112yes—
465C22C23×C22⋊D1112yes—
466C22C25×D1111yescharacteristic
467C4—24no—
468C4—2212no—
469C4—224no—
470C4—2212no—
471C4C23×C22⋊D11112yes—
472C4C23×C22⋊D1114yes—
473C2—212no—
474C2—412no—
475C2—444no—
476C2#1308240414yesdirect product factor
477C2C24×C22⋊D1111yescharacteristic
478C2C2×#1062345512yes—
subgroupG/H
centerC23C24×D11
commutator, cosocleC22×C11C26
FrattiniC22C25×D11
FittingC22×C11×H(5,𝔽2)C2
socleC23×C11C25
hypercenterC22×H(5,𝔽2)D11
hypocenterC11C2×#3500
LCSG ▷ C22×C11 ▷ C11
UCSC1 ◁ C23 ◁ C22×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C22×#67043C2semidirect
2. #13082404C2direct (N × H)
3. C2×#10623362C2semidirect
4. C2×#10623337C2semidirect
5. C2×#10623356C2semidirect
6. C2×#10623336C2semidirect
7. C2×#10623363C2semidirect
8. C22×#66989C2semidirect
9. C2×#41×Dic11C2semidirect
10. C22×C11×H(5,𝔽2)C2semidirect
12. C2×C11×H(5,𝔽2)C22semidirect
19. C2×#67043C22semidirect
20. C22×Q8×D11C22semidirect
21. C22×#4341C22semidirect
22. C22×#4339C22semidirect
23. C23×Dic22C22semidirect
24. C2×#67006C22semidirect
25. C22×C22⋊Dic11C22semidirect
26. #10623363C22semidirect
27. #10623362C22semidirect
28. #41×Dic11C22semidirect
29. #10623356C22semidirect
30. C2×#67000C22semidirect
31. C2×Q8×Dic11C22semidirect
32. C2×#66997C22semidirect
33. #10623337C22semidirect
34. #10623336C22semidirect
35. C2×#66993C22semidirect
36. C2×#66992C22semidirect
37. C2×#66991C22semidirect
38. C2×#66989C22semidirect
39. C2×#66989C22semidirect
40. C2×#66989C22semidirect
41. C2×#66989C22semidirect
42. C2×#66988C22semidirect
43. C2×D4×Dic11C22semidirect
44. C2×C4×C22⋊D11C22semidirect
45. C2×#66979C22semidirect
46. C2×#66977C22semidirect
47. C22×C4×Dic11C22semidirect
52. C2×C11×#41D4semidirect
74. C2×Q8×D11C23semidirect
76. C2×#4341C23semidirect
78. C2×#4339C23semidirect
84. C22×Dic22C23semidirect
85. #67006C23semidirect
89. C2×C22⋊Dic11C23semidirect
90. #67000C23semidirect
91. Q8×Dic11C23semidirect
92. #66997C23semidirect
93. #66993C23semidirect
94. #66992C23semidirect
95. #66991C23semidirect
96. #66989C23semidirect
97. #66989C23semidirect
98. #66989C23semidirect
99. #66988C23semidirect
100. D4×Dic11C23semidirect
101. C4×C22⋊D11C23semidirect
103. #66979C23semidirect
105. #66977C23semidirect
110. C2×C4×Dic11C23semidirect
120. C11×#41C2×D4semidirect
175. C22×H(5,𝔽2)D11semidirect
228. C2×H(5,𝔽2)C2×D11semidirect
290. C2×#41C22⋊D11semidirect
346. C2×C11#3500semidirect
360. #41C2×C22⋊D11semidirect
412. C11C2×#3500semidirect
476. C2#13082404direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#670432
#130824042
C2×#106233622
C2×#106233372
C2×#106233562
C2×#106233362
C2×#106233632
C22×#669892
C2×#41×Dic112
C22×C11×H(5,𝔽2)2
C2×#350011