Finite Group Databasebeta

← C2×#77566 #10961987

Subgroups — C2×#77566 #10961987

Order 768.

2354 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#890C211yesmaximal, characteristic, complemented
2C22×#879C211yesmaximal, characteristic
3C2×C3×#397C211yesmaximal, characteristic
4C2×#4983C211yesmaximal, characteristic, complemented
5C2×#4949C211yesmaximal, characteristic
6C2×#4948C212yesmaximal
7C2×#4944C212yesmaximal
8#77566C2116yesmaximal, direct product factor
9C2×#4659C211yesmaximal, characteristic, complemented
10C2×#4657C212yesmaximal, complemented
11C2×#4647C212yesmaximal, complemented
12C2×#4649C211yesmaximal, characteristic, complemented
13C2×#2664—31nomaximal, Sylow
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C22⋊C4C2211yescharacteristic
16C22×C3×C22⋊C4C2212yes—
17C22×#217C2211yescharacteristic
18C22×#217C2212yes—
19C22×#217C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C22×C4⋊Dic3C2212yes—
22C22×C4⋊Dic3C2211yescharacteristic
23C22×Dic3⋊C4C2212yes—
24C22×Dic3⋊C4C2212yes—
25C22×Dic3⋊C4C2212yes—
26C22×Dic3⋊C4C2211yescharacteristic
27C22×Dic3⋊C4C2211yescharacteristic
28C22×Dic3⋊C4C2212yes—
29C22×C4×Dic3C2212yes—
30C22×C4×Dic3C2211yescharacteristic
31C2×#890—28no—
32C2×#890C2218yescomplemented
33C2×#879—28no—
34C2×#879C2218yes—
35C3×#397C2218yes—
36C2×C3×#95C2211yescharacteristic
37C2×C3×#95C2212yes—
38C2×C3⋊#95C2211yescharacteristic
39C2×C3⋊#95C2212yes—
40C2×C3⋊#95C2212yes—
41C2×C3⋊#95C2212yes—
42#4983C2218yescomplemented
43#4949C2218yes—
44#4948C22116yes—
45#4944C22116yes—
46#4659C2218yescomplemented
47#4657C22116yescomplemented
48#4649C2218yescomplemented
49#4647C22116yescomplemented
50C22×#126—31no—
51C22×#125—31no—
52#2664—316no—
53C2×#400—31no—
54C2×#399—31no—
55C2×#398—32no—
56C2×#398—31no—
57C2×#397D311yescharacteristic
58C2×#396—32no—
59C2×#389—32no—
60C2×#388—32no—
61C2×C4×C4⋊C4—31no—
62C25×C3C2311yescharacteristic
63C23×C3×C4C2312yes—
64C23×C3×C4C2312yes—
65C23×C3×C4C2311yescharacteristic
66C23×C3×C4C2311yescharacteristic
67C23×Dic3C2311yescharacteristic
68C23×Dic3C2311yescharacteristic
69C23×Dic3C2312yes—
70C23×Dic3C2311yescharacteristic
71C23×Dic3C2311yescharacteristic
72C23×Dic3C2312yes—
73C2×C3×C4⋊C4—24no—
74C2×C3×C4⋊C4C2314yes—
75C2×C3×C22⋊C4—24no—
76C2×C3×C22⋊C4—44no—
77C2×C3×C22⋊C4C2314yes—
78C2×C3×C22⋊C4C2318yes—
79C2×#217—24no—
80C2×#217—24no—
81C2×#217—28no—
82C2×#217C2314yes—
83C2×#217C2318yes—
84C2×#217C2314yes—
85C2×C4⋊Dic3—24no—
86C2×C4⋊Dic3—24no—
87C2×C4⋊Dic3—28no—
88C2×C4⋊Dic3C2314yes—
89C2×C4⋊Dic3C2318yes—
90C2×C4⋊Dic3C2314yes—
91C2×Dic3⋊C4—24no—
92C2×Dic3⋊C4—24no—
93C2×Dic3⋊C4—28no—
94C2×Dic3⋊C4—28no—
95C2×Dic3⋊C4—28no—
96C2×Dic3⋊C4—44no—
97C2×Dic3⋊C4C2318yes—
98C2×Dic3⋊C4C2318yes—
99C2×Dic3⋊C4C2318yes—
100C2×Dic3⋊C4C2314yes—
101C2×Dic3⋊C4C2314yes—
102C2×Dic3⋊C4C2318yes—
103C2×C4×Dic3—44no—
104C2×C4×Dic3D418yes—
105C2×C4×Dic3C2318yes—
106C2×C4×Dic3C2314yes—
107#890—232no—
108#879—232no—
109C3×#95C2318yes—
110C3×#95C2314yes—
111C3⋊#95C2318yes—
112C3⋊#95C2314yes—
113C3⋊#95C2318yes—
114C3⋊#95C2318yes—
115C2×#126—38no—
116C2×#126—68no—
117C2×#125—38no—
118C2×#125—68no—
119C22×C4⋊C4—32no—
120C22×C4⋊C4—32no—
121C22×C4⋊C4—31no—
122C22×C4⋊C4—32no—
123C22×C4⋊C4—31no—
124C22×C4⋊C4—31no—
125C22×C4⋊C4—32no—
126C22×C4⋊C4—31no—
127C22×C4⋊C4—32no—
128C22×C4⋊C4C2×D311yescharacteristic
129C22×C22⋊C4—31no—
130C22×C22⋊C4—32no—
131C22×C22⋊C4—31no—
132C22×C22⋊C4C2×D311yescharacteristic
133C22×C22⋊C4C2×D312yes—
134C22×C42—31no—
135C22×C42—32no—
136#400—38no—
137#399—38no—
138#398—38no—
139#398—316no—
140#397C2×D318yes—
141#396—316no—
142#389—316no—
143#388—316no—
144C4×C4⋊C4—38no—
145C2×#95—32no—
146C2×#95—32no—
147C2×#95—31no—
148C2×#95—32no—
149C2×#95C2×D312yes—
150C2×#95C2×D311yescharacteristic
151C24×C3—22no—
152C24×C3—24no—
153C24×C3—22no—
154C24×C3—24no—
155C24×C3—22no—
156C24×C3C2412yes—
157C24×C3C2411yescharacteristic
158C22×C3×C4—24no—
159C22×C3×C4—22no—
160C22×C3×C4—24no—
161C22×C3×C4—24no—
162C22×C3×C4—24no—
163C22×C3×C4—24no—
164C22×C3×C4—24no—
165C22×C3×C4—24no—
166C22×C3×C4—24no—
167C22×C3×C4C2×D414yes—
168C22×C3×C4C2414yes—
169C22×C3×C4C2414yes—
170C22×C3×C4C2412yes—
171C22×C3×C4C2412yes—
172C22×Dic3—24no—
173C22×Dic3—22no—
174C22×Dic3—24no—
175C22×Dic3—22no—
176C22×Dic3—24no—
177C22×Dic3—22no—
178C22×Dic3—24no—
179C22×Dic3—22no—
180C22×Dic3—24no—
181C22×Dic3—24no—
182C22×Dic3—22no—
183C22×Dic3—24no—
184C22×Dic3—22no—
185C22×Dic3C2×D414yes—
186C22×Dic3C2×D414yes—
187C22×Dic3#4118yes—
188C22×Dic3C2412yes—
189C22×Dic3C2412yes—
190C22×Dic3C2414yes—
191C22×Dic3C2412yes—
192C22×Dic3C2412yes—
193C22×Dic3C2414yes—
194C3×C4⋊C4—28no—
195C3×C22⋊C4—28no—
196C3×C22⋊C4—48no—
197#217—28no—
198#217—28no—
199#217—216no—
200C4⋊Dic3—28no—
201C4⋊Dic3—28no—
202C4⋊Dic3—216no—
203Dic3⋊C4—28no—
204Dic3⋊C4—28no—
205Dic3⋊C4—216no—
206Dic3⋊C4—216no—
207Dic3⋊C4—216no—
208Dic3⋊C4—48no—
209C4×Dic3—48no—
210C4×Dic3C2×D4116yes—
211C25C22×D311yescharacteristic
212C23×C4—31no—
213C23×C4—32no—
214C23×C4—31no—
215C23×C4—31no—
216C23×C4—32no—
217C23×C4—31no—
218C23×C4C22×D311yescharacteristic
219C23×C4C22×D312yes—
220C23×C4C22×D312yes—
221C23×C4C22×D311yescharacteristic
222#126—632no—
223#125—632no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—38no—
226C2×C4⋊C4—38no—
227C2×C4⋊C4—34no—
228C2×C4⋊C4—34no—
229C2×C4⋊C4—34no—
230C2×C4⋊C4—34no—
231C2×C4⋊C4—38no—
232C2×C4⋊C4—38no—
233C2×C4⋊C4—38no—
234C2×C4⋊C4—68no—
235C2×C4⋊C4—68no—
236C2×C4⋊C4—64no—
237C2×C4⋊C4—64no—
238C2×C4⋊C4—68no—
239C2×C4⋊C4—68no—
240C2×C4⋊C4—64no—
241C2×C4⋊C4—64no—
242C2×C4⋊C4—124no—
243C2×C4⋊C4C22×D314yes—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—34no—
246C2×C22⋊C4—34no—
247C2×C22⋊C4—38no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—64no—
250C2×C22⋊C4—64no—
251C2×C22⋊C4—68no—
252C2×C22⋊C4C22×D314yes—
253C2×C22⋊C4C22×D318yes—
254C2×C42—38no—
255C2×C42—34no—
256C2×C42—38no—
257C2×C42—124no—
258#95—34no—
259#95—38no—
260#95—38no—
261#95—38no—
262#95C22×D314yes—
263#95C22×D318yes—
264C23×C3—22no—
265C23×C3—24no—
266C23×C3—22no—
267C23×C3—24no—
268C23×C3—22no—
269C23×C3—44no—
270C23×C3—44no—
271C23×C3—44no—
272C23×C3—48no—
273C23×C3—48no—
274C23×C3C22×D412yes—
275C23×C3C2×#4114yes—
276C23×C3H(5,𝔽2)12yes—
277C23×C32−1+414yes—
278C23×C32−1+412yes—
279C23×C3C2511yescharacteristic
280C2×C3×C4—28no—
281C2×C3×C4—24no—
282C2×C3×C4—22no—
283C2×C3×C4—24no—
284C2×C3×C4—24no—
285C2×C3×C4—24no—
286C2×C3×C4—24no—
287C2×C3×C4—24no—
288C2×C3×C4—24no—
289C2×C3×C4—24no—
290C2×C3×C4—44no—
291C2×C3×C4—48no—
292C2×C3×C4—48no—
293C2×C3×C4C22×D414yes—
294C2×Dic3—28no—
295C2×Dic3—28no—
296C2×Dic3—216no—
297C2×Dic3—24no—
298C2×Dic3—22no—
299C2×Dic3—24no—
300C2×Dic3—22no—
301C2×Dic3—24no—
302C2×Dic3—22no—
303C2×Dic3—24no—
304C2×Dic3—22no—
305C2×Dic3—24no—
306C2×Dic3—24no—
307C2×Dic3—22no—
308C2×Dic3—24no—
309C2×Dic3—22no—
310C2×Dic3—48no—
311C2×Dic3—44no—
312C2×Dic3—44no—
313C2×Dic3C22×D414yes—
314C2×Dic3C22×D414yes—
315C2×Dic3C2×#4118yes—
316C24—22no—
317C24—22no—
318C24—24no—
319C24—24no—
320C24—22no—
321C24C23×D312yes—
322C24C23×D311yescharacteristic
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—34no—
333C22×C4—34no—
334C22×C4—38no—
335C22×C4—34no—
336C22×C4—32no—
337C22×C4—32no—
338C22×C4—32no—
339C22×C4—32no—
340C22×C4—34no—
341C22×C4—62no—
342C22×C4—64no—
343C22×C4—64no—
344C22×C4—64no—
345C22×C4—62no—
346C22×C4—62no—
347C22×C4—62no—
348C22×C4—64no—
349C22×C4—62no—
350C22×C4—64no—
351C22×C4—64no—
352C22×C4—64no—
353C22×C4—62no—
354C22×C4D3×D414yes—
355C22×C4C23×D314yes—
356C22×C4C23×D314yes—
357C22×C4C23×D312yes—
358C22×C4C23×D312yes—
359C4⋊C4—28no—
360C4⋊C4—616no—
361C4⋊C4—616no—
362C4⋊C4—68no—
363C4⋊C4—68no—
364C4⋊C4—616no—
365C4⋊C4—616no—
366C4⋊C4—68no—
367C4⋊C4—68no—
368C4⋊C4—128no—
369C22⋊C4—28no—
370C22⋊C4—48no—
371C22⋊C4—68no—
372C22⋊C4—68no—
373C22⋊C4—616no—
374C42—316no—
375C42—128no—
376C22×C3—42no—
377C22×C3—42no—
378C22×C3—42no—
379C22×C3—44no—
380C22×C3—44no—
381C22×C3—88no—
382C22×C3#53214yes—
383C22×C3#53614yes—
384C22×C3#53714yes—
385C22×C3#53918yes—
386C22×C3#54118yes—
387C22×C3C23×D411yescharacteristic
388C22×C3C22×#4112yes—
389C22×C3C2×H(5,𝔽2)11yescharacteristic
390C22×C3C2×2−1+412yes—
391C22×C3C2×2−1+411yescharacteristic
392C3×C4—24no—
393C3×C4—42no—
394C3×C4—44no—
395C3×C4—44no—
396Dic3—24no—
397Dic3—24no—
398Dic3—28no—
399Dic3—44no—
400Dic3—42no—
401Dic3—42no—
402C23—22no—
403C23—22no—
404C23—24no—
405C23—22no—
406C23—24no—
407C23—44no—
408C23—48no—
409C23—44no—
410C23—48no—
411C23—44no—
412C23C2×D3×D412yes—
413C23#100412yes—
414C23D3×#4114yes—
415C23#101014yes—
416C23#101012yes—
417C23C24×D311yescharacteristic
418C2×C4—28no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—34no—
429C2×C4—34no—
430C2×C4—38no—
431C2×C4—48no—
432C2×C4—44no—
433C2×C4—48no—
434C2×C4—616no—
435C2×C4—68no—
436C2×C4—62no—
437C2×C4—64no—
438C2×C4—64no—
439C2×C4—62no—
440C2×C4—64no—
441C2×C4—68no—
442C2×C4—62no—
443C2×C4—62no—
444C2×C4—64no—
445C2×C4—62no—
446C2×C4—64no—
447C2×C4—64no—
448C2×C4—62no—
449C2×C4—64no—
450C2×C4—124no—
451C2×C4—128no—
452C2×C4—124no—
453C2×C4C2×D3×D414yes—
454C2×C3—82no—
455C2×C3#266418yescomplemented
456C2×C3C2×#53211yescharacteristic
457C2×C3C2×#53611yescharacteristic
458C2×C3C2×#53711yescharacteristic
459C2×C3C2×#53912yes—
460C2×C3C2×#54112yes—
461C22—42no—
462C22—44no—
463C22—42no—
464C22—42no—
465C22—44no—
466C22—88no—
467C22#560814yes—
468C22#564814yes—
469C22#566018yes—
470C22#566814yes—
471C22#570118yes—
472C22C22×D3×D411yescharacteristic
473C22C2×#100411yescharacteristic
474C22C2×D3×#4112yes—
475C22C2×#101011yescharacteristic
476C22C2×#101012yes—
477C4—24no—
478C4—42no—
479C4—44no—
480C4—44no—
481C4—64no—
482C4—64no—
483C4—68no—
484C4—124no—
485C4—122no—
486C4—122no—
487C3C2×#266411yescharacteristic, complemented, Sylow
488C2—82no—
489C2#7756618yesdirect product factor
490C2C2×#560811yescharacteristic
491C2C2×#564811yescharacteristic
492C2C2×#566012yes—
493C2C2×#566811yescharacteristic
494C2C2×#570112yes—
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#397C2
socleC24×C3C24
prime radicalC25×C3C23
hypercenterC2×#397D3
hypocenterC3C2×#2664
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#397

Complements

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

NH ≅ G/Ntype
1. C22×#890C2semidirect
4. C2×#4983C2semidirect
8. #77566C2direct (N × H)
9. C2×#4659C2semidirect
10. C2×#4657C2semidirect
11. C2×#4647C2semidirect
12. C2×#4649C2semidirect
32. C2×#890C22semidirect
42. #4983C22semidirect
46. #4659C22semidirect
47. #4657C22semidirect
48. #4649C22semidirect
49. #4647C22semidirect
455. C2×C3#2664semidirect
487. C3C2×#2664semidirect
489. C2#77566direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#8902
C22×#8792
C2×C3×#3972
C2×#49832
C2×#49492
C2×#49482
C2×#49442
#775662
C2×#46592
C2×#46572
C2×#46472
C2×#46492
C2×#26643