Finite Group Databasebeta

← C2×#73747 #10961990

Subgroups — C2×#73747 #10961990

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×#876C211yesmaximal, characteristic
2C22×#870C211yesmaximal, characteristic, complemented
3C2×C3×C4×C4⋊C4C211yesmaximal, characteristic, complemented
4C2×#4994C212yesmaximal
5C2×#4981C212yesmaximal, complemented
6C2×#4928C212yesmaximal, complemented
7C2×#4940C212yesmaximal
8#73747C2116yesmaximal, direct product factor
9C2×#4677C211yesmaximal, characteristic
10C2×#4659C211yesmaximal, characteristic, complemented
11C2×#4661C211yesmaximal, characteristic, complemented
12C2×#4674C211yesmaximal, characteristic
13C2×#2664—31nomaximal, Sylow
14C22×C3×C4⋊C4C2212yes—
15C22×C3×C42C2212yes—
16C22×C3×C42C2211yescharacteristic
17C22×(C2×D3)⋊C4C2211yescharacteristic
18C22×(C2×D3)⋊C4C2212yes—
19C22×(C2×D3)⋊C4C2211yescharacteristic
20C22×(C2×D3)⋊C4C2211yescharacteristic
21C22×(C2×D3)⋊C4C2212yes—
22C22×C4⋊Dic3C2211yescharacteristic
23C22×C4⋊Dic3C2211yescharacteristic
24C22×C4⋊Dic3C2212yes—
25C22×C4⋊Dic3C2212yes—
26C22×Dic3⋊C4C2212yes—
27C22×Dic3⋊C4C2211yescharacteristic
28C22×Dic3⋊C4C2212yes—
29C22×Dic3⋊C4C2211yescharacteristic
30C22×Dic3⋊C4C2211yescharacteristic
31C2×#876—28no—
32C2×#876C2218yes—
33C2×#870—28no—
34C2×#870C2218yescomplemented
35C3×C4×C4⋊C4C2218yescomplemented
36C2×C3×#95C2211yescharacteristic
37C2×C3×#95C2211yescharacteristic
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2212yes—
40C2×C3⋊#95C2212yes—
41C2×C3⋊#95C2212yes—
42#4994C22116yes—
43#4981C22116yescomplemented
44#4940C22116yes—
45#4928C22116yescomplemented
46#4677C2218yes—
47#4674C2218yes—
48#4661C2218yescomplemented
49#4659C2218yescomplemented
50C22×#126—31no—
51C22×#125—31no—
52#2664—316no—
53C2×#400—31no—
54C2×#399—31no—
55C2×#398—31no—
56C2×#398—32no—
57C2×#397—31no—
58C2×#396—32no—
59C2×#389—32no—
60C2×#388—32no—
61C2×C4×C4⋊C4D311yescharacteristic, complemented
62C24×D3C2311yescharacteristic
63C23×C3×C4C2312yes—
64C23×C3×C4C2311yescharacteristic
65C23×C3×C4C2311yescharacteristic
66C23×C3×C4C2311yescharacteristic
67C23×C3×C4C2312yes—
68C23×Dic3C2311yescharacteristic
69C23×Dic3C2312yes—
70C23×Dic3C2312yes—
71C23×Dic3C2311yescharacteristic
72C23×Dic3C2311yescharacteristic
73C2×C3×C4⋊C4—28no—
74C2×C3×C4⋊C4C2318yes—
75C2×C3×C42—44no—
76C2×C3×C42D418yes—
77C2×C3×C42C2318yes—
78C2×C3×C42C2314yes—
79C2×(C2×D3)⋊C4—24no—
80C2×(C2×D3)⋊C4—24no—
81C2×(C2×D3)⋊C4—24no—
82C2×(C2×D3)⋊C4—28no—
83C2×(C2×D3)⋊C4—44no—
84C2×(C2×D3)⋊C4C2314yes—
85C2×(C2×D3)⋊C4C2318yes—
86C2×(C2×D3)⋊C4C2314yes—
87C2×(C2×D3)⋊C4C2314yes—
88C2×(C2×D3)⋊C4C2318yes—
89C2×C4⋊Dic3—24no—
90C2×C4⋊Dic3—24no—
91C2×C4⋊Dic3—28no—
92C2×C4⋊Dic3—44no—
93C2×C4⋊Dic3C2314yes—
94C2×C4⋊Dic3C2314yes—
95C2×C4⋊Dic3C2318yes—
96C2×C4⋊Dic3C2318yes—
97C2×Dic3⋊C4—28no—
98C2×Dic3⋊C4—28no—
99C2×Dic3⋊C4—24no—
100C2×Dic3⋊C4—24no—
101C2×Dic3⋊C4—24no—
102C2×Dic3⋊C4C2314yes—
103C2×Dic3⋊C4C2314yes—
104C2×Dic3⋊C4C2318yes—
105C2×Dic3⋊C4C2318yes—
106C2×Dic3⋊C4C2314yes—
107#876—232no—
108#870—232no—
109C3×#95C2314yes—
110C3×#95C2314yes—
111C3⋊#95C2318yes—
112C3⋊#95C2318yes—
113C3⋊#95C2318yes—
114C3⋊#95C2318yes—
115C2×#126—38no—
116C2×#126—68no—
117C2×#125—38no—
118C2×#125—68no—
119C22×C4⋊C4—32no—
120C22×C4⋊C4—31no—
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×D312yes—
129C22×C22⋊C4—31no—
130C22×C22⋊C4—32no—
131C22×C22⋊C4—31no—
132C22×C22⋊C4—31no—
133C22×C22⋊C4—32no—
134C22×C42C2×D312yes—
135C22×C42C2×D311yescharacteristic
136#400—38no—
137#399—38no—
138#398—316no—
139#398—38no—
140#397—38no—
141#396—316no—
142#389—316no—
143#388—316no—
144C4×C4⋊C4C2×D318yescomplemented
145C2×#95—32no—
146C2×#95—32no—
147C2×#95—32no—
148C2×#95—32no—
149C2×#95C2×D311yescharacteristic
150C2×#95C2×D311yescharacteristic
151C24×C3C2411yescharacteristic
152C23×D3—22no—
153C23×D3—24no—
154C23×D3—22no—
155C23×D3—22no—
156C23×D3—24no—
157C23×D3C2412yes—
158C22×C3×C4—24no—
159C22×C3×C4—24no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
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×C4C2×D414yes—
169C22×C3×C4C2×D414yes—
170C22×C3×C4#4118yes—
171C22×C3×C4C2414yes—
172C22×C3×C4C2412yes—
173C22×C3×C4C2412yes—
174C22×C3×C4C2412yes—
175C22×C3×C4C2414yes—
176C22×Dic3—24no—
177C22×Dic3—24no—
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—24no—
182C22×Dic3—24no—
183C22×Dic3—24no—
184C22×Dic3—22no—
185C22×Dic3—24no—
186C22×Dic3—24no—
187C22×Dic3—22no—
188C22×Dic3—24no—
189C22×Dic3C2412yes—
190C22×Dic3C2414yes—
191C22×Dic3C2414yes—
192C22×Dic3C2412yes—
193C22×Dic3C2412yes—
194C3×C4⋊C4—216no—
195C3×C42—48no—
196C3×C42C2×D4116yes—
197(C2×D3)⋊C4—28no—
198(C2×D3)⋊C4—28no—
199(C2×D3)⋊C4—28no—
200(C2×D3)⋊C4—216no—
201(C2×D3)⋊C4—48no—
202C4⋊Dic3—28no—
203C4⋊Dic3—28no—
204C4⋊Dic3—216no—
205C4⋊Dic3—48no—
206Dic3⋊C4—216no—
207Dic3⋊C4—216no—
208Dic3⋊C4—28no—
209Dic3⋊C4—28no—
210Dic3⋊C4—28no—
211C25—31no—
212C23×C4—31no—
213C23×C4—31no—
214C23×C4—32no—
215C23×C4—32no—
216C23×C4—31no—
217C23×C4C22×D312yes—
218C23×C4C22×D311yescharacteristic
219C23×C4C22×D311yescharacteristic
220C23×C4C22×D311yescharacteristic
221C23×C4C22×D312yes—
222#126—632no—
223#125—632no—
224C2×C4⋊C4—28no—
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—34no—
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—64no—
239C2×C4⋊C4—64no—
240C2×C4⋊C4—64no—
241C2×C4⋊C4—68no—
242C2×C4⋊C4—124no—
243C2×C4⋊C4C22×D318yes—
244C2×C22⋊C4—34no—
245C2×C22⋊C4—34no—
246C2×C22⋊C4—34no—
247C2×C22⋊C4—38no—
248C2×C22⋊C4—38no—
249C2×C22⋊C4—64no—
250C2×C22⋊C4—64no—
251C2×C22⋊C4—64no—
252C2×C22⋊C4—68no—
253C2×C22⋊C4—124no—
254C2×C42—44no—
255C2×C42D1218yes—
256C2×C42C22×D318yes—
257C2×C42C22×D314yes—
258#95—38no—
259#95—38no—
260#95—38no—
261#95—38no—
262#95C22×D314yes—
263#95C22×D314yes—
264C23×C3C22×D412yes—
265C23×C3C2×#4114yes—
266C23×C3H(5,𝔽2)12yes—
267C23×C32−1+412yes—
268C23×C32−1+414yes—
269C23×C3C2511yescharacteristic
270C22×D3—22no—
271C22×D3—24no—
272C22×D3—22no—
273C22×D3—22no—
274C22×D3—24no—
275C22×D3—44no—
276C22×D3—44no—
277C22×D3—44no—
278C22×D3—48no—
279C22×D3—48no—
280C2×C3×C4—28no—
281C2×C3×C4—28no—
282C2×C3×C4—28no—
283C2×C3×C4—216no—
284C2×C3×C4—24no—
285C2×C3×C4—24no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—24no—
289C2×C3×C4—24no—
290C2×C3×C4—24no—
291C2×C3×C4—24no—
292C2×C3×C4—24no—
293C2×C3×C4—48no—
294C2×C3×C4C22×D414yes—
295C2×C3×C4C22×D414yes—
296C2×C3×C4C22×D414yes—
297C2×C3×C4C2×#4118yes—
298C2×Dic3—24no—
299C2×Dic3—24no—
300C2×Dic3—22no—
301C2×Dic3—22no—
302C2×Dic3—22no—
303C2×Dic3—24no—
304C2×Dic3—24no—
305C2×Dic3—24no—
306C2×Dic3—22no—
307C2×Dic3—24no—
308C2×Dic3—24no—
309C2×Dic3—22no—
310C2×Dic3—24no—
311C2×Dic3—48no—
312C2×Dic3—44no—
313C2×Dic3—44no—
314C2×Dic3—48no—
315C2×Dic3—44no—
316C24—32no—
317C24—62no—
318C24—64no—
319C24—62no—
320C24—62no—
321C24—64no—
322C24C23×D311yescharacteristic
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—32no—
333C22×C4—32no—
334C22×C4—32no—
335C22×C4—34no—
336C22×C4—34no—
337C22×C4—62no—
338C22×C4—62no—
339C22×C4—62no—
340C22×C4—64no—
341C22×C4—64no—
342C22×C4—64no—
343C22×C4—64no—
344C22×C4—64no—
345C22×C4—64no—
346C22×C4—62no—
347C22×C4—62no—
348C22×C4—64no—
349C22×C4—64no—
350C22×C4C2×D1214yes—
351C22×C4C2×D1214yes—
352C22×C4C2×D1214yes—
353C22×C4#23518yes—
354C22×C4C23×D314yes—
355C22×C4C23×D312yes—
356C22×C4C23×D312yes—
357C22×C4C23×D312yes—
358C22×C4C23×D314yes—
359C4⋊C4—216no—
360C4⋊C4—616no—
361C4⋊C4—616no—
362C4⋊C4—68no—
363C4⋊C4—68no—
364C4⋊C4—68no—
365C4⋊C4—68no—
366C4⋊C4—68no—
367C4⋊C4—616no—
368C4⋊C4—128no—
369C22⋊C4—68no—
370C22⋊C4—68no—
371C22⋊C4—68no—
372C22⋊C4—616no—
373C22⋊C4—128no—
374C42—48no—
375C42C2×D12116yes—
376C22×C3#53214yes—
377C22×C3#53614yes—
378C22×C3#53714yes—
379C22×C3#53918yes—
380C22×C3#54118yes—
381C22×C3C23×D411yescharacteristic
382C22×C3C22×#4112yes—
383C22×C3C2×H(5,𝔽2)11yescharacteristic
384C22×C3C2×2−1+411yescharacteristic
385C22×C3C2×2−1+412yes—
386C2×D3—42no—
387C2×D3—42no—
388C2×D3—42no—
389C2×D3—44no—
390C2×D3—44no—
391C2×D3—88no—
392C3×C4—24no—
393C3×C4—24no—
394C3×C4—24no—
395C3×C4—28no—
396C3×C4—44no—
397Dic3—44no—
398Dic3—42no—
399Dic3—42no—
400Dic3—44no—
401Dic3—42no—
402C23—62no—
403C23—64no—
404C23—62no—
405C23—62no—
406C23—64no—
407C23—124no—
408C23—124no—
409C23—124no—
410C23—128no—
411C23—128no—
412C23C22×D1212yes—
413C23C2×#23514yes—
414C23#100412yes—
415C23#100712yes—
416C23#101014yes—
417C23C24×D311yescharacteristic
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—28no—
425C2×C4—28no—
426C2×C4—28no—
427C2×C4—216no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—48no—
432C2×C4—62no—
433C2×C4—62no—
434C2×C4—62no—
435C2×C4—64no—
436C2×C4—64no—
437C2×C4—64no—
438C2×C4—64no—
439C2×C4—64no—
440C2×C4—64no—
441C2×C4—62no—
442C2×C4—62no—
443C2×C4—64no—
444C2×C4—64no—
445C2×C4—124no—
446C2×C4—128no—
447C2×C4—124no—
448C2×C4—128no—
449C2×C4—124no—
450C2×C4C22×D1214yes—
451C2×C4C22×D1214yes—
452C2×C4C22×D1214yes—
453C2×C4C2×#23518yes—
454C2×C3#266418yescomplemented
455C2×C3C2×#53211yescharacteristic
456C2×C3C2×#53611yescharacteristic
457C2×C3C2×#53711yescharacteristic
458C2×C3C2×#53912yes—
459C2×C3C2×#54112yes—
460D3—82no—
461C22—124no—
462C22—122no—
463C22—122no—
464C22—122no—
465C22—124no—
466C22—248no—
467C22#547714yes—
468C22#551014yes—
469C22#552614yes—
470C22#556418yes—
471C22#558318yes—
472C22C23×D1211yescharacteristic
473C22C22×#23512yes—
474C22C2×#100411yescharacteristic
475C22C2×#100711yescharacteristic
476C22C2×#101012yes—
477C4—24no—
478C4—24no—
479C4—24no—
480C4—28no—
481C4—44no—
482C4—122no—
483C4—124no—
484C4—122no—
485C4—124no—
486C4—122no—
487C3C2×#266411yescharacteristic, complemented, Sylow
488C2—242no—
489C2#7374718yesdirect product factor
490C2C2×#547711yescharacteristic
491C2C2×#551011yescharacteristic
492C2C2×#552611yescharacteristic
493C2C2×#556412yes—
494C2C2×#558312yes—
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×C4×C4⋊C4C2
socleC24×C3C24
prime radicalC24×D3C23
hypercenterC2×C4×C4⋊C4D3
hypocenterC3C2×#2664
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×#870C2semidirect
3. C2×C3×C4×C4⋊C4C2semidirect
5. C2×#4981C2semidirect
6. C2×#4928C2semidirect
8. #73747C2direct (N × H)
10. C2×#4659C2semidirect
11. C2×#4661C2semidirect
34. C2×#870C22semidirect
35. C3×C4×C4⋊C4C22semidirect
43. #4981C22semidirect
45. #4928C22semidirect
48. #4661C22semidirect
49. #4659C22semidirect
61. C2×C4×C4⋊C4D3semidirect
144. C4×C4⋊C4C2×D3semidirect
454. C2×C3#2664semidirect
487. C3C2×#2664semidirect
489. C2#73747direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#8762
C22×#8702
C2×C3×C4×C4⋊C42
C2×#49942
C2×#49812
C2×#49282
C2×#49402
#737472
C2×#46772
C2×#46592
C2×#46612
C2×#46742
C2×#26643