Finite Group Databasebeta

← C2×#87293 #11703660

Subgroups — C2×#87293 #11703660

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×C3×2−1+4C211yesmaximal, characteristic, complemented
2C22×#1009C211yesmaximal, characteristic, complemented
3#87293C2132yesmaximal, direct product factor
4C2×#5829C2112yesmaximal, complemented
5C2×#5830C213yesmaximal, complemented
6C2×#5832C213yesmaximal, complemented
7C2×Dic3×#41C211yesmaximal, characteristic, complemented
8C2×#5817C214yesmaximal, complemented
9C2×#5816C213yesmaximal, complemented
10C22×#947C213yesmaximal, complemented
11C2×#3500—31nomaximal, Sylow
12C2×C3×2−1+4—216no—
13C2×C3×2−1+4C22116yescomplemented
14C22×C3×#41C2211yescharacteristic
15C22×C3×#41C2213yescomplemented
16C22×C3×#41C2216yescomplemented
17C23×C3×Q8C2213yescomplemented
18C23×C3×Q8C2212yescomplemented
19C2×#1009—216no—
20C2×#1009C22116yescomplemented
21C22×#239C2212yes—
22C22×D3×D4C2216yes—
23C22×#235C2213yes—
24C23×D12C2213yes—
25#5832C22148yescomplemented
26#5830C22148yescomplemented
27#5829C221192yescomplemented
28Dic3×#41C22116yescomplemented
29#5817C22164yescomplemented
30#5816C22148yescomplemented
31C2×#947C2213yescomplemented
32C2×#947C2216yescomplemented
33C2×#947C22112yescomplemented
34C2×#947C22148yescomplemented
35C2×#946C22112yescomplemented
36C2×Q8×Dic3C2212yescomplemented
37C2×#944C2213yescomplemented
38C2×#940C2213yescomplemented
39C2×#939C22112yescomplemented
40C2×#937C22112yescomplemented
41C2×#936C2211yescharacteristic, complemented
42C2×D4×Dic3C2216yescomplemented
43C2×#930C22112yescomplemented
44C2×#929C22112yescomplemented
45C2×C4×#67C22112yescomplemented
46C22×(C2×D3)⋊C4C22112yescomplemented
47C2×#926C2213yescomplemented
48C22×C4×Dic3C2213yescomplemented
49C22×2−1+4D311yescharacteristic, complemented
50C22×H(5,𝔽2)—31no—
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—
59C24×D3C2316yes—
60C3×2−1+4—2128no—
61C2×C3×#41—248no—
62C2×C3×#41—224no—
63C2×C3×#41D4116yescomplemented
64C2×C3×#41C2318yes—
65C2×C3×#41C23148yescomplemented
66C2×C3×#41C23124yescomplemented
67C22×C3×Q8—216no—
68C22×C3×Q8—224no—
69C22×C3×Q8C2313yes—
70C22×C3×Q8C2316yes—
71C22×C3×Q8C2311yescharacteristic
72C22×C3×Q8C23124yescomplemented
73C22×C3×Q8C23116yescomplemented
74C22×C3×D4C2311yescharacteristic
75C22×C3×D4C2316yes—
76C22×C3×D4C2313yes—
77C23×C3×C4C2316yes—
78C23×C3×C4C2316yes—
79C23×C3×C4C2313yes—
80C22×#67C2316yes—
81C22×#67C23112yes—
82C23×Dic3C2316yes—
83#1009—2128no—
84C2×#239—216no—
85C2×#239C23116yes—
86C2×D3×D4—248no—
87C2×D3×D4C23148yes—
88C2×#235—224no—
89C2×#235C23124yes—
90C22×D12—224no—
91C22×D12C2313yes—
92C22×D12C2316yes—
93C22×D12C23124yes—
94C22×C4×D3C2316yes—
95C22×Dic6C2311yescharacteristic
96C2×#217C2316yes—
97C2×#217C2314yes—
98#947C23124yescomplemented
99#947C23148yescomplemented
100#947C23196yescomplemented
101#946C23196yescomplemented
102Q8×Dic3C23116yescomplemented
103#944C23124yescomplemented
104#940C23124yescomplemented
105#939C23196yescomplemented
106#937C23196yescomplemented
107#936C2318yescomplemented
108D4×Dic3C23148yescomplemented
109#930C23196yescomplemented
110#929C23196yescomplemented
111C4×#67C23196yescomplemented
112C2×(C2×D3)⋊C4C23112yes—
113C2×(C2×D3)⋊C4C23124yes—
114C2×(C2×D3)⋊C4C23196yescomplemented
115#926C23124yescomplemented
116C2×C4⋊Dic3C2316yes—
117C2×Dic3⋊C4C23112yes—
118C2×C4×Dic3C2311yescharacteristic
119C2×C4×Dic3C2313yes—
120C2×C4×Dic3C2316yes—
121C2×C4×Dic3C23124yescomplemented
122C2×2−1+4—216no—
123C2×2−1+4C2×D3116yescomplemented
124C2×H(5,𝔽2)—316no—
125C2×H(5,𝔽2)—616no—
126C22×#41—32no—
127C22×#41—33no—
128C22×#41C2×D311yescharacteristic
129C22×#41C2×D313yescomplemented
130C22×#41C2×D316yescomplemented
131C23×Q8C2×D313yescomplemented
132C23×Q8C2×D312yescomplemented
133C23×D4—36no—
134C23×D4—33no—
135#548—364no—
136#546—3192no—
137#536—348no—
138#535—348no—
139#532—348no—
140C2×C4⋊Q8—33no—
141C2×#127—33no—
142C2×#124—33no—
143C2×#124—36no—
144C2×#124—312no—
145C2×#124—348no—
146C2×#124—31no—
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×C3C2412yes—
160C24×C3C2413yes—
161C23×D3—224no—
162C23×D3—424no—
163C23×D3C2416yes—
164C23×D3C24124yes—
165C3×#41—2192no—
166C3×#41—296no—
167C3×#41C2×D4164yescomplemented
168C2×C3×Q8—264no—
169C2×C3×Q8—212no—
170C2×C3×Q8—224no—
171C2×C3×Q8—296no—
172C2×C3×Q8C2×D418yes—
173C2×C3×Q8C24112yes—
174C2×C3×Q8C24124yes—
175C2×C3×Q8C2414yes—
176C2×C3×D4—224no—
177C2×C3×D4—24no—
178C2×C3×D4C2×D4124yes—
179C2×C3×D4C2414yes—
180C2×C3×D4C24124yes—
181C2×C3×D4C24112yes—
182C22×C3×C4—224no—
183C22×C3×C4—224no—
184C22×C3×C4C2×D4124yes—
185C22×C3×C4C2416yes—
186C22×C3×C4C24124yes—
187C22×C3×C4C24124yes—
188C22×C3×C4C2411yescharacteristic
189C22×C3×C4C2413yes—
190C22×C3×C4C24112yes—
191C2×#67—224no—
192C2×#67—424no—
193C2×#67C24124yes—
194C2×#67C24148yes—
195C22×Dic3—412no—
196C22×Dic3C2412yes—
197C22×Dic3C2416yes—
198C22×Dic3C2412yes—
199C22×Dic3C24124yes—
200#239—264no—
201D3×D4—2192no—
202#235—296no—
203C2×D12—212no—
204C2×D12—224no—
205C2×D12—296no—
206C2×D12C24124yes—
207C2×D12C24112yes—
208C2×C4×D3—224no—
209C2×C4×D3C24124yes—
210C2×Dic6—24no—
211C2×Dic6C2414yes—
212#217C24124yes—
213#217C24116yes—
214(C2×D3)⋊C4C24148yes—
215(C2×D3)⋊C4C24196yes—
216C4⋊Dic3C24124yes—
217Dic3⋊C4C24148yes—
218C4×Dic3C2414yes—
219C4×Dic3C24112yes—
220C4×Dic3C24124yes—
221C25—36no—
2222−1+4—2128no—
223H(5,𝔽2)—6128no—
224C2×#41—248no—
225C2×#41—224no—
226C2×#41—316no—
227C2×#41—324no—
228C2×#41—616no—
229C2×#41—624no—
230C2×#41#67116yescomplemented
231C2×#41C22×D318yes—
232C2×#41C22×D3124yescomplemented
233C2×#41C22×D3148yescomplemented
234C22×Q8—216no—
235C22×Q8—224no—
236C22×Q8—31no—
237C22×Q8C22×D316yes—
238C22×Q8C22×D313yes—
239C22×Q8C22×D311yescharacteristic
240C22×Q8C22×D3124yescomplemented
241C22×Q8C22×D3116yescomplemented
242C22×D4—36no—
243C22×D4—312no—
244C22×D4—348no—
245C22×D4—36no—
246C22×D4—33no—
247C22×D4—324no—
248C22×D4—648no—
249C22×D4—624no—
250C22×D4C22×D316yes—
251C22×D4C22×D311yescharacteristic
252C22×D4C22×D313yes—
253C23×C4—36no—
254C23×C4—36no—
255C23×C4C22×D316yes—
256C23×C4C22×D316yes—
257C23×C4C22×D313yes—
258C4⋊Q8—324no—
259#127—324no—
260#124—324no—
261#124—348no—
262#124—396no—
263#124—38no—
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—36no—
276C2×C22⋊C4—34no—
277C2×C22⋊C4—312no—
278C2×C22⋊C4—324no—
279C2×C22⋊C4—396no—
280C2×C42—33no—
281C2×C42—36no—
282C2×C42—324no—
283C2×C42—31no—
284C23×C3—212no—
285C23×C3—24no—
286C23×C3—28no—
287C23×C3C22×D4112yes—
288C23×C3C2514yes—
289C23×C3C2516yes—
290C23×C3C2511yescharacteristic
291C22×D3—212no—
292C22×D3—224no—
293C22×D3—248no—
294C22×D3—448no—
295C22×D3—848no—
296C22×D3C25112yes—
297C3×Q8—224no—
298C3×Q8—248no—
299C3×Q8C22×D4116yes—
300C3×D4—248no—
301C3×D4—28no—
302C3×D4C22×D4148yes—
303C2×C3×C4—212no—
304C2×C3×C4—248no—
305C2×C3×C4—248no—
306C2×C3×C4C22×D414yes—
307C2×C3×C4C22×D4112yes—
308C2×C3×C4C22×D4148yes—
309C2×C3×C4C25112yes—
310C2×C3×C4C2512yes—
311C2×C3×C4C2516yes—
312#67—248no—
313#67—448no—
314C2×Dic3—24no—
315C2×Dic3—24no—
316C2×Dic3—212no—
317C2×Dic3—424no—
318C2×Dic3C2514yes—
319C2×Dic3C25112yes—
320C2×Dic3C2514yes—
321D12—224no—
322D12—248no—
323C4×D3—248no—
324Dic6—28no—
325C24—36no—
326C24—324no—
327C24—624no—
328C24—1224no—
329C24C23×D312yes—
330C24C23×D313yes—
331#41—2192no—
332#41—296no—
333#41—664no—
334#41—696no—
335#41C2×#67164yescomplemented
336C2×Q8—264no—
337C2×Q8—224no—
338C2×Q8—212no—
339C2×Q8—296no—
340C2×Q8—34no—
341C2×Q8—64no—
342C2×Q8C2×#6718yes—
343C2×Q8C23×D3124yes—
344C2×Q8C23×D3112yes—
345C2×Q8C23×D314yes—
346C2×D4—224no—
347C2×D4—24no—
348C2×D4—324no—
349C2×D4—348no—
350C2×D4—324no—
351C2×D4—312no—
352C2×D4—624no—
353C2×D4—6192no—
354C2×D4—624no—
355C2×D4—612no—
356C2×D4—696no—
357C2×D4—1224no—
358C2×D4C2×#67124yes—
359C2×D4C23×D3124yes—
360C2×D4C23×D314yes—
361C2×D4C23×D3112yes—
362C22×C4—224no—
363C22×C4—224no—
364C22×C4—32no—
365C22×C4—36no—
366C22×C4—32no—
367C22×C4—324no—
368C22×C4—324no—
369C22×C4—624no—
370C22×C4—1212no—
371C22×C4C2×#67124yes—
372C22×C4C23×D3124yes—
373C22×C4C23×D3124yes—
374C22×C4C23×D316yes—
375C22×C4C23×D3112yes—
376C22×C4C23×D311yescharacteristic
377C22×C4C23×D313yes—
378C4⋊C4—324no—
379C4⋊C4—348no—
380C22⋊C4—324no—
381C22⋊C4—316no—
382C22⋊C4—348no—
383C22⋊C4—396no—
384C42—34no—
385C42—312no—
386C42—324no—
387C22×C3—212no—
388C22×C3—224no—
389C22×C3—24no—
390C22×C3—28no—
391C22×C3—48no—
392C22×C3C23×D4112yes—
393C22×C3C23×D412yes—
394C22×C3#58514yes—
395C22×C3C2611yescharacteristic
396C2×D3—212no—
397C2×D3—224no—
398C2×D3—424no—
399C2×D3—848no—
400C3×C4—212no—
401C3×C4C23×D414yes—
402C3×C4C23×D4112yes—
403Dic3—24no—
404Dic3—24no—
405Dic3—212no—
406C23—212no—
407C23—28no—
408C23—24no—
409C23—312no—
410C23—624no—
411C23—612no—
412C23—648no—
413C23—1248no—
414C23—2448no—
415C23C22×#67112yes—
416C23C24×D314yes—
417C23C24×D316yes—
418C23C24×D311yescharacteristic
419Q8—248no—
420Q8—224no—
421Q8—68no—
422Q8C22×#67116yes—
423D4—248no—
424D4—28no—
425D4—648no—
426D4—648no—
427D4—624no—
428D4—1248no—
429D4C22×#67148yes—
430C2×C4—212no—
431C2×C4—248no—
432C2×C4—248no—
433C2×C4—34no—
434C2×C4—312no—
435C2×C4—34no—
436C2×C4—64no—
437C2×C4—612no—
438C2×C4—64no—
439C2×C4—648no—
440C2×C4—1224no—
441C2×C4C22×#67112yes—
442C2×C4C22×#6714yes—
443C2×C4C22×#67148yes—
444C2×C4C24×D3112yes—
445C2×C4C24×D312yes—
446C2×C4C24×D316yes—
447C2×C3—212no—
448C2×C3—44no—
449C2×C3#350014yescomplemented
450C2×C3C24×D411yescharacteristic
451C2×C3C2×#58512yes—
452D3—412no—
453C22—212no—
454C22—224no—
455C22—28no—
456C22—24no—
457C22—48no—
458C22—624no—
459C22—612no—
460C22—1224no—
461C22—2448no—
462C22#596814yes—
463C22C23×#67112yes—
464C22C23×#6712yes—
465C22C25×D311yescharacteristic
466C4—212no—
467C4—64no—
468C4—612no—
469C4—64no—
470C4C23×#67112yes—
471C4C23×#6714yes—
472C3C2×#350011yescharacteristic, complemented, Sylow
473C2—212no—
474C2—44no—
475C2—1212no—
476C2#8729314yesdirect product factor
477C2C2×#596812yes—
478C2C24×#6711yescharacteristic
subgroupG/H
centerC23C24×D3
commutator, cosocleC22×C3C26
FrattiniC22C25×D3
FittingC22×C3×2−1+4C2
socleC23×C3C25
hypercenterC22×2−1+4D3
hypocenterC3C2×#3500
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×2−1+4

Complements

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

NH ≅ G/Ntype
1. C22×C3×2−1+4C2semidirect
2. C22×#1009C2semidirect
3. #87293C2direct (N × H)
4. C2×#5829C2semidirect
5. C2×#5830C2semidirect
6. C2×#5832C2semidirect
7. C2×Dic3×#41C2semidirect
8. C2×#5817C2semidirect
9. C2×#5816C2semidirect
10. C22×#947C2semidirect
13. C2×C3×2−1+4C22semidirect
15. C22×C3×#41C22semidirect
16. C22×C3×#41C22semidirect
17. C23×C3×Q8C22semidirect
18. C23×C3×Q8C22semidirect
20. C2×#1009C22semidirect
25. #5832C22semidirect
26. #5830C22semidirect
27. #5829C22semidirect
28. Dic3×#41C22semidirect
29. #5817C22semidirect
30. #5816C22semidirect
31. C2×#947C22semidirect
32. C2×#947C22semidirect
33. C2×#947C22semidirect
34. C2×#947C22semidirect
35. C2×#946C22semidirect
36. C2×Q8×Dic3C22semidirect
37. C2×#944C22semidirect
38. C2×#940C22semidirect
39. C2×#939C22semidirect
40. C2×#937C22semidirect
41. C2×#936C22semidirect
42. C2×D4×Dic3C22semidirect
43. C2×#930C22semidirect
44. C2×#929C22semidirect
45. C2×C4×#67C22semidirect
46. C22×(C2×D3)⋊C4C22semidirect
47. C2×#926C22semidirect
48. C22×C4×Dic3C22semidirect
49. C22×2−1+4D3semidirect
63. C2×C3×#41D4semidirect
65. C2×C3×#41C23semidirect
66. C2×C3×#41C23semidirect
72. C22×C3×Q8C23semidirect
73. C22×C3×Q8C23semidirect
98. #947C23semidirect
99. #947C23semidirect
100. #947C23semidirect
101. #946C23semidirect
102. Q8×Dic3C23semidirect
103. #944C23semidirect
104. #940C23semidirect
105. #939C23semidirect
106. #937C23semidirect
107. #936C23semidirect
108. D4×Dic3C23semidirect
109. #930C23semidirect
110. #929C23semidirect
111. C4×#67C23semidirect
114. C2×(C2×D3)⋊C4C23semidirect
115. #926C23semidirect
121. C2×C4×Dic3C23semidirect
123. C2×2−1+4C2×D3semidirect
129. C22×#41C2×D3semidirect
130. C22×#41C2×D3semidirect
131. C23×Q8C2×D3semidirect
132. C23×Q8C2×D3semidirect
167. C3×#41C2×D4semidirect
230. C2×#41#67semidirect
232. C2×#41C22×D3semidirect
233. C2×#41C22×D3semidirect
240. C22×Q8C22×D3semidirect
241. C22×Q8C22×D3semidirect
335. #41C2×#67semidirect
449. C2×C3#3500semidirect
472. C3C2×#3500semidirect
476. C2#87293direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×2−1+42
C22×#10092
#872932
C2×#58292
C2×#58302
C2×#58322
C2×Dic3×#412
C2×#58172
C2×#58162
C22×#9472
C2×#35003