Finite Group Databasebeta

← C2×#87283 #11703656

Subgroups — C2×#87283 #11703656

Order 768.

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×C3×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2C22×#1010C211yesmaximal, characteristic, complemented
3#87283C2132yesmaximal, direct product factor
4C2×#5831C214yesmaximal, complemented
5C2×#5825C213yesmaximal, complemented
6C2×#5832C213yesmaximal, complemented
7C2×Dic3×#41C211yesmaximal, characteristic, complemented
8C2×#5805C213yesmaximal, complemented
9C2×#5806C2112yesmaximal, complemented
10C22×#936C213yesmaximal, complemented
11C2×#3500—31nomaximal, Sylow
12C2×C3×H(5,𝔽2)—216no—
13C2×C3×H(5,𝔽2)C22116yescomplemented
14C22×C3×#41C2213yes—
15C22×C3×#41C2212yes—
16C22×C3×#41C2211yescharacteristic
17C23×C3×D4C2213yes—
18C23×C3×D4C2216yes—
19C2×#1010—216no—
20C2×#1010C22116yescomplemented
21C22×D3×Q8C2212yescomplemented
22C22×#237C2216yescomplemented
23C22×#235C2213yescomplemented
24C23×Dic6C2213yescomplemented
25C2×#953C22112yescomplemented
26C22×#217C22112yescomplemented
27#5832C22148yescomplemented
28#5831C22164yescomplemented
29Dic3×#41C22116yescomplemented
30#5825C22148yescomplemented
31C2×#947C2211yescharacteristic, complemented
32C2×Q8×Dic3C2212yescomplemented
33C2×#944C2213yescomplemented
34#5806C221192yescomplemented
35#5805C22148yescomplemented
36C2×#940C2213yescomplemented
37C2×#939C22112yescomplemented
38C2×#938C22112yescomplemented
39C2×#936C2213yescomplemented
40C2×#936C22112yescomplemented
41C2×#936C2216yescomplemented
42C2×#936C22148yescomplemented
43C2×#935C22112yescomplemented
44C2×D4×Dic3C2216yescomplemented
45C2×C4×#67C22112yescomplemented
46C2×#926C2213yescomplemented
47C2×#924C22112yescomplemented
48C22×C4×Dic3C2213yescomplemented
49C22×2−1+4—31no—
50C22×H(5,𝔽2)D311yescharacteristic, complemented
51#3500—332no—
52C2×#548—34no—
53C2×#546—312no—
54C2×#536—33no—
55C2×#535—33no—
56C2×#532—33no—
57C22×#124—33no—
58C2×C4×#41—31no—
59C25×C3C2316yes—
60C3×H(5,𝔽2)—2128no—
61C2×C3×#41—216no—
62C2×C3×#41—224no—
63C2×C3×#41D4116yescomplemented
64C2×C3×#41C23124yes—
65C2×C3×#41C23116yes—
66C2×C3×#41C2318yes—
67C22×C3×Q8C2311yescharacteristic
68C22×C3×Q8C2311yescharacteristic
69C22×C3×D4—248no—
70C22×C3×D4—224no—
71C22×C3×D4C2316yes—
72C22×C3×D4C2313yes—
73C22×C3×D4C2316yes—
74C22×C3×D4C23124yes—
75C22×C3×D4C23148yes—
76C22×C3×D4C2313yes—
77C23×C3×C4C2316yes—
78C23×C3×C4C2313yes—
79C22×#67C2316yes—
80C22×#67C23112yes—
81C23×Dic3C2316yes—
82C23×Dic3C2316yes—
83#1010—2128no—
84C2×D3×Q8—216no—
85C2×D3×Q8C23116yescomplemented
86C2×#237—248no—
87C2×#237C23148yescomplemented
88C2×#235—224no—
89C2×#235C23124yescomplemented
90C22×D12C2311yescharacteristic
91C22×C4×D3C2316yes—
92C22×Dic6—224no—
93C22×Dic6C2313yes—
94C22×Dic6C2316yes—
95C22×Dic6C23124yescomplemented
96#953C23196yescomplemented
97C2×#217C23124yes—
98C2×#217C2316yes—
99C2×#217C23112yes—
100C2×#217C23196yescomplemented
101#947C2318yescomplemented
102Q8×Dic3C23116yescomplemented
103#944C23124yescomplemented
104#940C23124yescomplemented
105#939C23196yescomplemented
106#938C23196yescomplemented
107#936C23124yescomplemented
108#936C23148yescomplemented
109#936C23196yescomplemented
110#935C23196yescomplemented
111D4×Dic3C23148yescomplemented
112C4×#67C23196yescomplemented
113C2×(C2×D3)⋊C4C2314yes—
114#926C23124yescomplemented
115C2×C4⋊Dic3C2316yes—
116#924C23196yescomplemented
117C2×Dic3⋊C4C23112yes—
118C2×C4×Dic3C2311yescharacteristic
119C2×C4×Dic3C2313yes—
120C2×C4×Dic3C2316yes—
121C2×C4×Dic3C23124yescomplemented
122C2×2−1+4—316no—
123C2×2−1+4—616no—
124C2×H(5,𝔽2)—216no—
125C2×H(5,𝔽2)C2×D3116yescomplemented
126C22×#41—36no—
127C22×#41—33no—
128C22×#41C2×D313yes—
129C22×#41C2×D312yes—
130C22×#41C2×D311yescharacteristic
131C23×Q8—32no—
132C23×Q8—33no—
133C23×D4C2×D313yes—
134C23×D4C2×D316yes—
135#548—364no—
136#546—3192no—
137#536—348no—
138#535—348no—
139#532—348no—
140C2×C4⋊Q8—33no—
141C2×#127—33no—
142C2×#124—31no—
143C2×#124—33no—
144C2×#124—36no—
145C2×#124—312no—
146C2×#124—348no—
147C2×#123—312no—
148C2×C22⋊Q8—312no—
149C2×#121—312no—
150C2×#121—312no—
151C2×#120—312no—
152C4×#41—316no—
153C2×C4×Q8—32no—
154C2×C4×D4—36no—
155C2×C4×D4—312no—
156C2×#117—33no—
157C22×C22⋊C4—312no—
158C22×C42—33no—
159C24×C3—224no—
160C24×C3—424no—
161C24×C3C2416yes—
162C24×C3C2413yes—
163C24×C3C24124yes—
164C23×D3C2412yes—
165C3×#41—264no—
166C3×#41—296no—
167C3×#41C2×D4164yescomplemented
168C2×C3×Q8—24no—
169C2×C3×Q8C2×D418yes—
170C2×C3×Q8C2414yes—
171C2×C3×Q8C2414yes—
172C2×C3×D4—224no—
173C2×C3×D4—2192no—
174C2×C3×D4—212no—
175C2×C3×D4—224no—
176C2×C3×D4—296no—
177C2×C3×D4C2×D4124yes—
178C2×C3×D4C24124yes—
179C2×C3×D4C24112yes—
180C2×C3×D4C24124yes—
181C2×C3×D4C24112yes—
182C22×C3×C4—224no—
183C22×C3×C4C2×D4124yes—
184C22×C3×C4C2412yes—
185C22×C3×C4C24124yes—
186C22×C3×C4C2411yescharacteristic
187C22×C3×C4C2413yes—
188C22×C3×C4C24112yes—
189C2×#67—224no—
190C2×#67—424no—
191C2×#67C24124yes—
192C2×#67C24148yes—
193C22×Dic3—224no—
194C22×Dic3—412no—
195C22×Dic3C2412yes—
196C22×Dic3C2416yes—
197C22×Dic3C2416yes—
198C22×Dic3C24124yes—
199C22×Dic3C24124yes—
200D3×Q8—264no—
201#237—2192no—
202#235—296no—
203C2×D12—24no—
204C2×D12C2414yes—
205C2×C4×D3—224no—
206C2×C4×D3C24124yes—
207C2×Dic6—212no—
208C2×Dic6—224no—
209C2×Dic6—296no—
210C2×Dic6C24124yes—
211C2×Dic6C24112yes—
212#217C24196yes—
213#217C24124yes—
214#217C24148yes—
215(C2×D3)⋊C4C24116yes—
216C4⋊Dic3C24124yes—
217Dic3⋊C4C24148yes—
218C4×Dic3C2414yes—
219C4×Dic3C24112yes—
220C4×Dic3C24124yes—
221C25C22×D316yes—
2222−1+4—6128no—
223H(5,𝔽2)—2128no—
224C2×#41—216no—
225C2×#41—224no—
226C2×#41—348no—
227C2×#41—324no—
228C2×#41—648no—
229C2×#41—624no—
230C2×#41#67116yescomplemented
231C2×#41C22×D3124yes—
232C2×#41C22×D3116yes—
233C2×#41C22×D318yes—
234C22×Q8—316no—
235C22×Q8—36no—
236C22×Q8—33no—
237C22×Q8—324no—
238C22×Q8—616no—
239C22×Q8—624no—
240C22×Q8C22×D311yescharacteristic
241C22×Q8C22×D311yescharacteristic
242C22×D4—248no—
243C22×D4—224no—
244C22×D4—36no—
245C22×D4—312no—
246C22×D4—31no—
247C22×D4C22×D316yes—
248C22×D4C22×D316yes—
249C22×D4C22×D313yes—
250C22×D4C22×D3124yes—
251C22×D4C22×D3148yes—
252C22×D4C22×D313yes—
253C23×C4—36no—
254C23×C4—36no—
255C23×C4—36no—
256C23×C4C22×D316yes—
257C23×C4C22×D313yes—
258C4⋊Q8—324no—
259#127—324no—
260#124—38no—
261#124—324no—
262#124—348no—
263#124—396no—
264#123—396no—
265C22⋊Q8—396no—
266#121—396no—
267#121—396no—
268#120—396no—
269C4×Q8—316no—
270C4×D4—348no—
271C4×D4—396no—
272#117—324no—
273C2×C4⋊C4—36no—
274C2×C4⋊C4—312no—
275C2×C22⋊C4—312no—
276C2×C22⋊C4—324no—
277C2×C22⋊C4—396no—
278C2×C22⋊C4—36no—
279C2×C22⋊C4—34no—
280C2×C42—33no—
281C2×C42—36no—
282C2×C42—324no—
283C2×C42—31no—
284C23×C3—212no—
285C23×C3—212no—
286C23×C3—224no—
287C23×C3—248no—
288C23×C3—448no—
289C23×C3—848no—
290C23×C3C22×D4112yes—
291C23×C3C25112yes—
292C23×C3C2516yes—
293C23×C3C2511yescharacteristic
294C22×D3—24no—
295C22×D3—28no—
296C22×D3C2514yes—
297C3×Q8—28no—
298C3×Q8C22×D4116yes—
299C3×D4—248no—
300C3×D4—224no—
301C3×D4—248no—
302C3×D4C22×D4148yes—
303C2×C3×C4—24no—
304C2×C3×C4—248no—
305C2×C3×C4C22×D414yes—
306C2×C3×C4C22×D4112yes—
307C2×C3×C4C22×D4148yes—
308C2×C3×C4C2514yes—
309C2×C3×C4C2512yes—
310C2×C3×C4C2516yes—
311#67—248no—
312#67—448no—
313C2×Dic3—212no—
314C2×Dic3—248no—
315C2×Dic3—24no—
316C2×Dic3—212no—
317C2×Dic3—424no—
318C2×Dic3C2514yes—
319C2×Dic3C25112yes—
320C2×Dic3C25112yes—
321D12—28no—
322C4×D3—248no—
323Dic6—224no—
324Dic6—248no—
325C24—224no—
326C24—32no—
327C24—424no—
328C24C23×D316yes—
329C24C23×D313yes—
330C24C23×D3124yes—
331#41—264no—
332#41—296no—
333#41—6192no—
334#41—696no—
335#41C2×#67164yescomplemented
336C2×Q8—24no—
337C2×Q8—324no—
338C2×Q8—312no—
339C2×Q8—664no—
340C2×Q8—624no—
341C2×Q8—612no—
342C2×Q8—696no—
343C2×Q8C2×#6718yes—
344C2×Q8C23×D314yes—
345C2×Q8C23×D314yes—
346C2×D4—224no—
347C2×D4—2192no—
348C2×D4—224no—
349C2×D4—212no—
350C2×D4—296no—
351C2×D4—324no—
352C2×D4—348no—
353C2×D4—34no—
354C2×D4—624no—
355C2×D4—64no—
356C2×D4—1224no—
357C2×D4C2×#67124yes—
358C2×D4C23×D3124yes—
359C2×D4C23×D3124yes—
360C2×D4C23×D3112yes—
361C2×D4C23×D3112yes—
362C22×C4—224no—
363C22×C4—324no—
364C22×C4—32no—
365C22×C4—36no—
366C22×C4—36no—
367C22×C4—324no—
368C22×C4—324no—
369C22×C4—624no—
370C22×C4—624no—
371C22×C4—1212no—
372C22×C4C2×#67124yes—
373C22×C4C23×D3124yes—
374C22×C4C23×D312yes—
375C22×C4C23×D3112yes—
376C22×C4C23×D311yescharacteristic
377C22×C4C23×D313yes—
378C4⋊C4—324no—
379C4⋊C4—348no—
380C22⋊C4—348no—
381C22⋊C4—396no—
382C22⋊C4—324no—
383C22⋊C4—316no—
384C42—34no—
385C42—312no—
386C42—324no—
387C22×C3—212no—
388C22×C3—224no—
389C22×C3—212no—
390C22×C3—224no—
391C22×C3—424no—
392C22×C3—848no—
393C22×C3C23×D4112yes—
394C22×C3C23×D412yes—
395C22×C3#58514yes—
396C22×C3C2611yescharacteristic
397C2×D3—24no—
398C2×D3—28no—
399C2×D3—48no—
400C3×C4—24no—
401C3×C4C23×D414yes—
402C3×C4C23×D4112yes—
403Dic3—212no—
404Dic3—24no—
405Dic3—212no—
406C23—212no—
407C23—224no—
408C23—212no—
409C23—248no—
410C23—34no—
411C23—448no—
412C23—68no—
413C23—64no—
414C23—848no—
415C23C22×#67112yes—
416C23C24×D3112yes—
417C23C24×D316yes—
418C23C24×D311yescharacteristic
419Q8—28no—
420Q8—648no—
421Q8—624no—
422Q8C22×#67116yes—
423D4—248no—
424D4—248no—
425D4—224no—
426D4—648no—
427D4—68no—
428D4—1248no—
429D4C22×#67148yes—
430C2×C4—24no—
431C2×C4—248no—
432C2×C4—34no—
433C2×C4—312no—
434C2×C4—312no—
435C2×C4—64no—
436C2×C4—612no—
437C2×C4—612no—
438C2×C4—648no—
439C2×C4—648no—
440C2×C4—1224no—
441C2×C4C22×#67112yes—
442C2×C4C22×#6714yes—
443C2×C4C22×#67148yes—
444C2×C4C24×D314yes—
445C2×C4C24×D312yes—
446C2×C4C24×D316yes—
447C2×C3—212no—
448C2×C3—412no—
449C2×C3#350014yescomplemented
450C2×C3C24×D411yescharacteristic
451C2×C3C2×#58512yes—
452D3—44no—
453C22—212no—
454C22—224no—
455C22—224no—
456C22—212no—
457C22—424no—
458C22—68no—
459C22—64no—
460C22—848no—
461C22—128no—
462C22#596614yes—
463C22C23×#67112yes—
464C22C23×#6712yes—
465C22C25×D311yescharacteristic
466C4—24no—
467C4—64no—
468C4—612no—
469C4—612no—
470C4C23×#67112yes—
471C4C23×#6714yes—
472C3C2×#350011yescharacteristic, complemented, Sylow
473C2—212no—
474C2—412no—
475C2—124no—
476C2#8728314yesdirect product factor
477C2C2×#596612yes—
478C2C24×#6711yescharacteristic
subgroupG/H
centerC23C24×D3
commutator, cosocleC22×C3C26
FrattiniC22C25×D3
FittingC22×C3×H(5,𝔽2)C2
socleC23×C3C25
hypercenterC22×H(5,𝔽2)D3
hypocenterC3C2×#3500
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C22×C3×H(5,𝔽2)C2semidirect
2. C22×#1010C2semidirect
3. #87283C2direct (N × H)
4. C2×#5831C2semidirect
5. C2×#5825C2semidirect
6. C2×#5832C2semidirect
7. C2×Dic3×#41C2semidirect
8. C2×#5805C2semidirect
9. C2×#5806C2semidirect
10. C22×#936C2semidirect
13. C2×C3×H(5,𝔽2)C22semidirect
20. C2×#1010C22semidirect
21. C22×D3×Q8C22semidirect
22. C22×#237C22semidirect
23. C22×#235C22semidirect
24. C23×Dic6C22semidirect
25. C2×#953C22semidirect
26. C22×#217C22semidirect
27. #5832C22semidirect
28. #5831C22semidirect
29. Dic3×#41C22semidirect
30. #5825C22semidirect
31. C2×#947C22semidirect
32. C2×Q8×Dic3C22semidirect
33. C2×#944C22semidirect
34. #5806C22semidirect
35. #5805C22semidirect
36. C2×#940C22semidirect
37. C2×#939C22semidirect
38. C2×#938C22semidirect
39. C2×#936C22semidirect
40. C2×#936C22semidirect
41. C2×#936C22semidirect
42. C2×#936C22semidirect
43. C2×#935C22semidirect
44. C2×D4×Dic3C22semidirect
45. C2×C4×#67C22semidirect
46. C2×#926C22semidirect
47. C2×#924C22semidirect
48. C22×C4×Dic3C22semidirect
50. C22×H(5,𝔽2)D3semidirect
63. C2×C3×#41D4semidirect
85. C2×D3×Q8C23semidirect
87. C2×#237C23semidirect
89. C2×#235C23semidirect
95. C22×Dic6C23semidirect
96. #953C23semidirect
100. C2×#217C23semidirect
101. #947C23semidirect
102. Q8×Dic3C23semidirect
103. #944C23semidirect
104. #940C23semidirect
105. #939C23semidirect
106. #938C23semidirect
107. #936C23semidirect
108. #936C23semidirect
109. #936C23semidirect
110. #935C23semidirect
111. D4×Dic3C23semidirect
112. C4×#67C23semidirect
114. #926C23semidirect
116. #924C23semidirect
121. C2×C4×Dic3C23semidirect
125. C2×H(5,𝔽2)C2×D3semidirect
167. C3×#41C2×D4semidirect
230. C2×#41#67semidirect
335. #41C2×#67semidirect
449. C2×C3#3500semidirect
472. C3C2×#3500semidirect
476. C2#87283direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×H(5,𝔽2)2
C22×#10102
#872832
C2×#58312
C2×#58252
C2×#58322
C2×Dic3×#412
C2×#58052
C2×#58062
C22×#9362
C2×#35003