Finite Group Databasebeta

← C2×#85847 #11703659

Subgroups — C2×#85847 #11703659

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×#1009C211yesmaximal, characteristic, complemented
2C22×#1010C211yesmaximal, characteristic, complemented
3C2×C3×C4×#41C211yesmaximal, characteristic, complemented
4#85847C2132yesmaximal, direct product factor
5C2×#5576C214yesmaximal, complemented
6C2×#5554C2112yesmaximal, complemented
7C2×#5525C213yesmaximal, complemented
8C2×#5526C213yesmaximal, complemented
9C2×#5477C213yesmaximal, complemented
10C22×#875C213yesmaximal, complemented
11C2×#3500—31nomaximal, Sylow
12C22×C3×#41C2211yescharacteristic
13C2×#1010—216no—
14C2×#1010C22116yescomplemented
15C2×#1009—216no—
16C2×#1009C22116yescomplemented
17C22×#239C2212yes—
18C22×D3×Q8C2212yescomplemented
19C22×#237C2216yescomplemented
20C22×D3×D4C2216yes—
21C22×#235C2213yes—
22C22×#235C2213yescomplemented
23C23×D12C2213yes—
24C23×Dic6C2213yescomplemented
25C3×C4×#41C22116yescomplemented
26C2×C3×C4×Q8C2212yescomplemented
27C2×C3×C4×D4C2216yescomplemented
28C2×C3×#117C2213yescomplemented
29C22×C3×C42C2213yescomplemented
30C2×#930C22112yescomplemented
31C22×(C2×D3)⋊C4C22112yescomplemented
32#5576C22164yescomplemented
33#5554C221192yescomplemented
34#5526C22148yescomplemented
35#5525C22148yescomplemented
36C2×#897C22112yescomplemented
37C2×#895C22112yescomplemented
38C2×#886C22112yescomplemented
39C2×#882C22112yescomplemented
40#5477C22148yescomplemented
41C2×#875C22112yescomplemented
42C2×#875C2213yescomplemented
43C2×#875C2216yescomplemented
44C2×#875C2211yescharacteristic, complemented
45C2×#875C22148yescomplemented
46C2×#874C2213yescomplemented
47C2×C4×D12C22112yescomplemented
48C2×#869C2213yescomplemented
49C22×2−1+4—31no—
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×#41D311yescharacteristic, complemented
59C24×D3C2316yes—
60C2×C3×#41D4116yescomplemented
61C2×C3×#41C2318yes—
62C22×C3×Q8C2311yescharacteristic
63C22×C3×D4C2313yes—
64C23×C3×C4C2316yes—
65C23×C3×C4C2313yes—
66C22×#67C2316yes—
67C22×#67C2316yes—
68C23×Dic3C2316yes—
69#1010—2128no—
70#1009—2128no—
71C2×#239—216no—
72C2×#239C23116yes—
73C2×D3×Q8—216no—
74C2×D3×Q8C23116yescomplemented
75C2×#237—248no—
76C2×#237C23148yescomplemented
77C2×D3×D4—248no—
78C2×D3×D4C23148yes—
79C2×#235—224no—
80C2×#235—224no—
81C2×#235C23124yes—
82C2×#235C23124yescomplemented
83C22×D12—224no—
84C22×D12C2313yes—
85C22×D12C2316yes—
86C22×D12C23124yes—
87C22×D12C2311yescharacteristic
88C22×D12C23112yes—
89C22×C4×D3C2316yes—
90C22×C4×D3C2316yes—
91C22×Dic6—224no—
92C22×Dic6C2313yes—
93C22×Dic6C2316yes—
94C22×Dic6C2311yescharacteristic
95C22×Dic6C23124yescomplemented
96C3×C4×Q8C23116yescomplemented
97C3×C4×D4C23148yescomplemented
98C3×#117C23124yescomplemented
99C2×C3×C4⋊C4C2316yes—
100C2×C3×C22⋊C4C2316yes—
101C2×C3×C42C2313yes—
102C2×C3×C42C2316yes—
103C2×C3×C42C2311yescharacteristic
104C2×C3×C42C23124yescomplemented
105#930C23196yescomplemented
106C2×(C2×D3)⋊C4C2314yes—
107C2×(C2×D3)⋊C4C23112yes—
108C2×(C2×D3)⋊C4C23124yes—
109C2×(C2×D3)⋊C4C23196yescomplemented
110C2×C4⋊Dic3C23112yes—
111#897C23196yescomplemented
112#895C23196yescomplemented
113#886C23196yescomplemented
114#882C23196yescomplemented
115#875C23124yescomplemented
116#875C23148yescomplemented
117#875C23196yescomplemented
118#875C2318yescomplemented
119#874C23124yescomplemented
120C4×D12C23196yescomplemented
121#869C23124yescomplemented
122C2×2−1+4—316no—
123C2×2−1+4—616no—
124C2×H(5,𝔽2)—316no—
125C2×H(5,𝔽2)—616no—
126C22×#41—32no—
127C22×#41—36no—
128C22×#41—33no—
129C22×#41—33no—
130C22×#41C2×D311yescharacteristic
131C23×Q8—32no—
132C23×Q8—33no—
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—31no—
146C2×#124—348no—
147C2×#123—312no—
148C2×C22⋊Q8—312no—
149C2×#121—312no—
150C2×#121—312no—
151C2×#120—312no—
152C4×#41C2×D3116yescomplemented
153C2×C4×Q8C2×D312yescomplemented
154C2×C4×D4—312no—
155C2×C4×D4C2×D316yescomplemented
156C2×#117C2×D313yescomplemented
157C22×C22⋊C4—312no—
158C22×C42C2×D313yescomplemented
159C24×C3C2413yes—
160C23×D3—224no—
161C23×D3—424no—
162C23×D3C2416yes—
163C23×D3C2412yes—
164C23×D3C24124yes—
165C3×#41C2×D4164yescomplemented
166C2×C3×Q8C2×D418yes—
167C2×C3×Q8C2414yes—
168C2×C3×D4C2×D4124yes—
169C2×C3×D4C24112yes—
170C22×C3×C4—412no—
171C22×C3×C4C2×D4124yes—
172C22×C3×C4C2416yes—
173C22×C3×C4C2412yes—
174C22×C3×C4C24124yes—
175C22×C3×C4C2411yescharacteristic
176C22×C3×C4C2413yes—
177C22×C3×C4C24112yes—
178C2×#67—224no—
179C2×#67—224no—
180C2×#67C24124yes—
181C2×#67C24124yes—
182C22×Dic3—224no—
183C22×Dic3C24124yes—
184C22×Dic3C2412yes—
185C22×Dic3C2416yes—
186#239—264no—
187D3×Q8—264no—
188#237—2192no—
189D3×D4—2192no—
190#235—296no—
191#235—296no—
192C2×D12—212no—
193C2×D12—224no—
194C2×D12—296no—
195C2×D12—24no—
196C2×D12—424no—
197C2×D12C2414yes—
198C2×D12C24112yes—
199C2×D12C24124yes—
200C2×D12C24148yes—
201C2×C4×D3—224no—
202C2×C4×D3—224no—
203C2×C4×D3C24124yes—
204C2×C4×D3C24124yes—
205C2×Dic6—212no—
206C2×Dic6—224no—
207C2×Dic6—296no—
208C2×Dic6—24no—
209C2×Dic6C2414yes—
210C2×Dic6C24112yes—
211C2×Dic6C24124yes—
212C3×C4⋊C4C24124yes—
213C3×C22⋊C4C24124yes—
214C3×C42C24112yes—
215C3×C42C24124yes—
216C3×C42C2414yes—
217(C2×D3)⋊C4C24116yes—
218(C2×D3)⋊C4C24148yes—
219(C2×D3)⋊C4C24196yes—
220C4⋊Dic3C24148yes—
221C25—36no—
2222−1+4—6128no—
223H(5,𝔽2)—6128no—
224C2×#41—316no—
225C2×#41—348no—
226C2×#41—324no—
227C2×#41—324no—
228C2×#41—616no—
229C2×#41—648no—
230C2×#41—624no—
231C2×#41—624no—
232C2×#41D12116yescomplemented
233C2×#41C22×D318yes—
234C22×Q8—316no—
235C22×Q8—36no—
236C22×Q8—33no—
237C22×Q8—324no—
238C22×Q8—31no—
239C22×Q8—616no—
240C22×Q8—624no—
241C22×Q8C22×D311yescharacteristic
242C22×D4—36no—
243C22×D4—36no—
244C22×D4—348no—
245C22×D4—36no—
246C22×D4—33no—
247C22×D4—324no—
248C22×D4—31no—
249C22×D4—312no—
250C22×D4—648no—
251C22×D4—624no—
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—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×Q8C22×D3116yescomplemented
270C4×D4—396no—
271C4×D4C22×D3148yescomplemented
272#117C22×D3124yescomplemented
273C2×C4⋊C4—312no—
274C2×C4⋊C4C22×D316yes—
275C2×C22⋊C4—312no—
276C2×C22⋊C4—324no—
277C2×C22⋊C4—396no—
278C2×C22⋊C4—34no—
279C2×C22⋊C4C22×D316yes—
280C2×C42C22×D313yes—
281C2×C42C22×D316yes—
282C2×C42C22×D311yescharacteristic
283C2×C42C22×D3124yescomplemented
284C23×C3—212no—
285C23×C3C22×D4112yes—
286C23×C3C2516yes—
287C23×C3C2511yescharacteristic
288C22×D3—24no—
289C22×D3—212no—
290C22×D3—28no—
291C22×D3—224no—
292C22×D3—248no—
293C22×D3—448no—
294C22×D3—848no—
295C22×D3C25112yes—
296C22×D3C2514yes—
297C3×Q8C22×D4116yes—
298C3×D4C22×D4148yes—
299C2×C3×C4—212no—
300C2×C3×C4—24no—
301C2×C3×C4—424no—
302C2×C3×C4C22×D414yes—
303C2×C3×C4C22×D4112yes—
304C2×C3×C4C22×D4148yes—
305C2×C3×C4C25112yes—
306C2×C3×C4C2514yes—
307C2×C3×C4C2512yes—
308C2×C3×C4C2516yes—
309#67—248no—
310#67—248no—
311C2×Dic3—212no—
312C2×Dic3—248no—
313C2×Dic3—24no—
314C2×Dic3C2514yes—
315C2×Dic3C25112yes—
316D12—224no—
317D12—248no—
318D12—28no—
319D12—448no—
320C4×D3—248no—
321C4×D3—248no—
322Dic6—224no—
323Dic6—248no—
324Dic6—28no—
325C24—36no—
326C24—32no—
327C24—324no—
328C24—624no—
329C24—1224no—
330C24C23×D313yes—
331#41—664no—
332#41—6192no—
333#41—696no—
334#41—696no—
335#41C2×D12164yescomplemented
336C2×Q8—324no—
337C2×Q8—312no—
338C2×Q8—34no—
339C2×Q8—664no—
340C2×Q8—624no—
341C2×Q8—612no—
342C2×Q8—696no—
343C2×Q8—64no—
344C2×Q8C2×D1218yes—
345C2×Q8C23×D314yes—
346C2×D4—324no—
347C2×D4—324no—
348C2×D4—324no—
349C2×D4—312no—
350C2×D4—34no—
351C2×D4—348no—
352C2×D4—624no—
353C2×D4—624no—
354C2×D4—6192no—
355C2×D4—624no—
356C2×D4—612no—
357C2×D4—696no—
358C2×D4—64no—
359C2×D4—1224no—
360C2×D4C2×D12124yes—
361C2×D4C23×D3112yes—
362C22×C4—324no—
363C22×C4—36no—
364C22×C4—32no—
365C22×C4—324no—
366C22×C4—324no—
367C22×C4—412no—
368C22×C4—624no—
369C22×C4—624no—
370C22×C4—624no—
371C22×C4C2×D12124yes—
372C22×C4C23×D312yes—
373C22×C4C23×D316yes—
374C22×C4C23×D3124yes—
375C22×C4C23×D3112yes—
376C22×C4C23×D311yescharacteristic
377C22×C4C23×D313yes—
378C4⋊C4—348no—
379C4⋊C4C23×D3124yes—
380C22⋊C4—348no—
381C22⋊C4—396no—
382C22⋊C4—316no—
383C22⋊C4C23×D3124yes—
384C42C23×D314yes—
385C42C23×D3112yes—
386C42C23×D3124yes—
387C22×C3—212no—
388C22×C3—224no—
389C22×C3C23×D4112yes—
390C22×C3C23×D412yes—
391C22×C3#58514yes—
392C22×C3C2611yescharacteristic
393C2×D3—24no—
394C2×D3—212no—
395C2×D3—28no—
396C2×D3—224no—
397C2×D3—48no—
398C2×D3—424no—
399C2×D3—848no—
400C3×C4—212no—
401C3×C4—24no—
402C3×C4C23×D414yes—
403C3×C4C23×D4112yes—
404Dic3—212no—
405Dic3—24no—
406C23—212no—
407C23—312no—
408C23—34no—
409C23—64no—
410C23—612no—
411C23—68no—
412C23—624no—
413C23—648no—
414C23—1248no—
415C23—2448no—
416C23C22×D12112yes—
417C23C24×D316yes—
418C23C24×D311yescharacteristic
419Q8—648no—
420Q8—624no—
421Q8—68no—
422Q8C22×D12116yes—
423D4—648no—
424D4—648no—
425D4—648no—
426D4—624no—
427D4—68no—
428D4—1248no—
429D4C22×D12148yes—
430C2×C4—24no—
431C2×C4—212no—
432C2×C4—312no—
433C2×C4—34no—
434C2×C4—424no—
435C2×C4—64no—
436C2×C4—612no—
437C2×C4—648no—
438C2×C4—648no—
439C2×C4—648no—
440C2×C4C22×D12112yes—
441C2×C4C22×D1214yes—
442C2×C4C22×D12148yes—
443C2×C4C24×D314yes—
444C2×C4C24×D3112yes—
445C2×C4C24×D312yes—
446C2×C4C24×D316yes—
447C2×C3—212no—
448C2×C3#350014yescomplemented
449C2×C3C24×D411yescharacteristic
450C2×C3C2×#58512yes—
451D3—44no—
452D3—412no—
453C22—212no—
454C22—224no—
455C22—64no—
456C22—612no—
457C22—68no—
458C22—624no—
459C22—1224no—
460C22—128no—
461C22—2448no—
462C22C23×D12112yes—
463C22C23×D1212yes—
464C22#596414yes—
465C22C25×D311yescharacteristic
466C4—24no—
467C4—212no—
468C4—64no—
469C4—612no—
470C4C23×D12112yes—
471C4C23×D1214yes—
472C3C2×#350011yescharacteristic, complemented, Sylow
473C2—212no—
474C2—1212no—
475C2—124no—
476C2#8584714yesdirect product factor
477C2C24×D1211yescharacteristic
478C2C2×#596412yes—
subgroupG/H
centerC23C24×D3
commutator, cosocleC22×C3C26
FrattiniC22C25×D3
FittingC2×C3×C4×#41C2
socleC23×C3C25
hypercenterC2×C4×#41D3
hypocenterC3C2×#3500
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×#41

Complements

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

NH ≅ G/Ntype
1. C22×#1009C2semidirect
2. C22×#1010C2semidirect
3. C2×C3×C4×#41C2semidirect
4. #85847C2direct (N × H)
5. C2×#5576C2semidirect
6. C2×#5554C2semidirect
7. C2×#5525C2semidirect
8. C2×#5526C2semidirect
9. C2×#5477C2semidirect
10. C22×#875C2semidirect
14. C2×#1010C22semidirect
16. C2×#1009C22semidirect
18. C22×D3×Q8C22semidirect
19. C22×#237C22semidirect
22. C22×#235C22semidirect
24. C23×Dic6C22semidirect
25. C3×C4×#41C22semidirect
26. C2×C3×C4×Q8C22semidirect
27. C2×C3×C4×D4C22semidirect
28. C2×C3×#117C22semidirect
29. C22×C3×C42C22semidirect
30. C2×#930C22semidirect
31. C22×(C2×D3)⋊C4C22semidirect
32. #5576C22semidirect
33. #5554C22semidirect
34. #5526C22semidirect
35. #5525C22semidirect
36. C2×#897C22semidirect
37. C2×#895C22semidirect
38. C2×#886C22semidirect
39. C2×#882C22semidirect
40. #5477C22semidirect
41. C2×#875C22semidirect
42. C2×#875C22semidirect
43. C2×#875C22semidirect
44. C2×#875C22semidirect
45. C2×#875C22semidirect
46. C2×#874C22semidirect
47. C2×C4×D12C22semidirect
48. C2×#869C22semidirect
58. C2×C4×#41D3semidirect
60. C2×C3×#41D4semidirect
74. C2×D3×Q8C23semidirect
76. C2×#237C23semidirect
82. C2×#235C23semidirect
95. C22×Dic6C23semidirect
96. C3×C4×Q8C23semidirect
97. C3×C4×D4C23semidirect
98. C3×#117C23semidirect
104. C2×C3×C42C23semidirect
105. #930C23semidirect
109. C2×(C2×D3)⋊C4C23semidirect
111. #897C23semidirect
112. #895C23semidirect
113. #886C23semidirect
114. #882C23semidirect
115. #875C23semidirect
116. #875C23semidirect
117. #875C23semidirect
118. #875C23semidirect
119. #874C23semidirect
120. C4×D12C23semidirect
121. #869C23semidirect
152. C4×#41C2×D3semidirect
153. C2×C4×Q8C2×D3semidirect
155. C2×C4×D4C2×D3semidirect
156. C2×#117C2×D3semidirect
158. C22×C42C2×D3semidirect
165. C3×#41C2×D4semidirect
232. C2×#41D12semidirect
269. C4×Q8C22×D3semidirect
271. C4×D4C22×D3semidirect
272. #117C22×D3semidirect
283. C2×C42C22×D3semidirect
335. #41C2×D12semidirect
448. C2×C3#3500semidirect
472. C3C2×#3500semidirect
476. C2#85847direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#10092
C22×#10102
C2×C3×C4×#412
#858472
C2×#55762
C2×#55542
C2×#55252
C2×#55262
C2×#54772
C22×#8752
C2×#35003