Finite Group Databasebeta

← C22×#5514 #11702185

Subgroups — C22×#5514 #11702185

Order 768.

8798 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C4⋊C4C211yesmaximal, characteristic, complemented
2C22×#930C211yesmaximal, characteristic, complemented
3C22×#930C212yesmaximal, complemented
4C22×C4×#67C211yesmaximal, characteristic, complemented
5C22×C4×#67C212yesmaximal, complemented
6C2×#5514C2148yesmaximal, direct product factor
7C22×#894C212yesmaximal, complemented
8C22×#895C211yesmaximal, characteristic, complemented
9C22×#898C212yesmaximal, complemented
10C22×#890C211yesmaximal, characteristic, complemented
11C22×#893C211yesmaximal, characteristic, complemented
12C22×#892C211yesmaximal, characteristic, complemented
13C22×#553—31nomaximal, Sylow
14C24×C3×C4C2211yescharacteristic
15C24×C3×C4C2212yes—
16C23×#67C2212yes—
17C23×#67C2211yescharacteristic
18C23×D12C2211yescharacteristic
19C23×D12C2212yes—
20C23×C4×D3C2211yescharacteristic
21C23×C4×D3C2212yes—
22C22×C3×C4⋊C4C2211yescharacteristic
23C22×C3×C4⋊C4C2211yescharacteristic
24C22×C3×C4⋊C4C2212yes—
25C22×C3×C4⋊C4C22124yescomplemented
26C22×#217C2211yescharacteristic
27C2×#930C22124yescomplemented
28C2×#930C22148yescomplemented
29C2×C4×#67C22124yescomplemented
30C2×C4×#67C22148yescomplemented
31C22×(C2×D3)⋊C4C2212yes—
32C22×(C2×D3)⋊C4C2211yescharacteristic
33C22×(C2×D3)⋊C4C2212yes—
34C22×(C2×D3)⋊C4C2212yes—
35C22×(C2×D3)⋊C4C2212yes—
36C22×C4⋊Dic3C2211yescharacteristic
37C22×C4⋊Dic3C2212yes—
38C22×Dic3⋊C4C2212yes—
39C22×Dic3⋊C4C2211yescharacteristic
40C22×C4×Dic3C2212yes—
41C22×C4×Dic3C2211yescharacteristic
42#5514C221256yesdirect product factor
43C2×#898C22148yescomplemented
44C2×#895C22124yescomplemented
45C2×#894C22148yescomplemented
46C2×#893C22124yescomplemented
47C2×#892C22124yescomplemented
48C2×#890C22124yescomplemented
49C2×#553—348no—
50C22×#126—32no—
51C22×#125—31no—
52C22×#123—32no—
53C22×#121—32no—
54C22×#121—31no—
55C22×#121—31no—
56C22×C4×D4—32no—
57C22×C4×D4—31no—
58C22×C4×D4—31no—
59C22×#117—31no—
60C23×C4⋊C4D311yescharacteristic, complemented
61C25×C3C2311yescharacteristic
62C24×D3C2311yescharacteristic
63C24×D3C2312yes—
64C23×C3×C4—28no—
65C23×C3×C4—48no—
66C23×C3×C4C2312yes—
67C23×C3×C4C2312yes—
68C23×C3×C4C2311yescharacteristic
69C23×C3×C4C23112yes—
70C23×C3×C4C23124yes—
71C23×C3×C4C2311yescharacteristic
72C22×#67—216no—
73C22×#67—28no—
74C22×#67C23112yes—
75C22×#67C23124yes—
76C23×Dic3C2311yescharacteristic
77C23×Dic3C2311yescharacteristic
78C23×Dic3C2312yes—
79C23×Dic3C2311yescharacteristic
80C22×D12—28no—
81C22×D12—48no—
82C22×D12C23112yes—
83C22×D12C23124yes—
84C22×C4×D3—28no—
85C22×C4×D3—48no—
86C22×C4×D3C23112yes—
87C22×C4×D3C23124yes—
88C2×C3×C4⋊C4C23112yes—
89C2×C3×C4⋊C4C23112yes—
90C2×C3×C4⋊C4C23124yes—
91C2×C3×C4⋊C4C23164yescomplemented
92C2×#217C23112yes—
93#930C23164yescomplemented
94#930C231128yescomplemented
95C4×#67C23164yescomplemented
96C4×#67C231128yescomplemented
97C2×(C2×D3)⋊C4C23124yes—
98C2×(C2×D3)⋊C4C23124yes—
99C2×(C2×D3)⋊C4C23112yes—
100C2×(C2×D3)⋊C4C23124yes—
101C2×(C2×D3)⋊C4C23124yes—
102C2×C4⋊Dic3C23112yes—
103C2×C4⋊Dic3C23124yes—
104C2×Dic3⋊C4C23124yes—
105C2×Dic3⋊C4C23112yes—
106C2×C4×Dic3C23112yes—
107C2×C4×Dic3C23124yes—
108#898C231128yescomplemented
109#895C23164yescomplemented
110#894C231128yescomplemented
111#893C23164yescomplemented
112#892C23164yescomplemented
113#890C23164yescomplemented
114C23×D4—32no—
115C23×D4—31no—
116C23×D4—31no—
117C23×D4—32no—
118C24×C4—31no—
119C24×C4—32no—
120C24×C4C2×D311yescharacteristic
121C24×C4C2×D312yes—
122#553—3256no—
123C2×#126—348no—
124C2×#125—324no—
125C2×#123—348no—
126C2×#121—348no—
127C2×#121—324no—
128C2×#121—324no—
129C2×C4×D4—348no—
130C2×C4×D4—324no—
131C2×C4×D4—324no—
132C2×#117—324no—
133C22×C4⋊C4—31no—
134C22×C4⋊C4—32no—
135C22×C4⋊C4—31no—
136C22×C4⋊C4—32no—
137C22×C4⋊C4C2×D311yescharacteristic
138C22×C4⋊C4C2×D311yescharacteristic
139C22×C4⋊C4C2×D312yes—
140C22×C4⋊C4C2×D3124yescomplemented
141C22×C22⋊C4—31no—
142C22×C22⋊C4—32no—
143C22×C22⋊C4—32no—
144C22×C22⋊C4—32no—
145C22×C22⋊C4—31no—
146C22×C22⋊C4—32no—
147C22×C42—32no—
148C22×C42—31no—
149C24×C3—24no—
150C24×C3—24no—
151C24×C3#4118yes—
152C24×C3C2416yes—
153C24×C3C2411yescharacteristic
154C23×D3—24no—
155C23×D3—28no—
156C23×D3—24no—
157C23×D3—28no—
158C23×D3—28no—
159C23×D3—24no—
160C23×D3C2416yes—
161C23×D3C24112yes—
162C22×C3×C4—24no—
163C22×C3×C4—28no—
164C22×C3×C4—28no—
165C22×C3×C4—248no—
166C22×C3×C4—448no—
167C22×C3×C4#4118yes—
168C22×C3×C4C24112yes—
169C22×C3×C4C24112yes—
170C22×C3×C4C2416yes—
171C22×C3×C4C24116yes—
172C22×C3×C4C24132yes—
173C22×C3×C4C2416yes—
174C2×#67—248no—
175C2×#67—296no—
176C2×#67C24116yes—
177C2×#67C24132yes—
178C22×Dic3—24no—
179C22×Dic3—28no—
180C22×Dic3—24no—
181C22×Dic3—24no—
182C22×Dic3C2416yes—
183C22×Dic3C2416yes—
184C22×Dic3C2416yes—
185C22×Dic3C24112yes—
186C2×D12—248no—
187C2×D12—448no—
188C2×D12C24116yes—
189C2×D12C24132yes—
190C2×C4×D3—248no—
191C2×C4×D3—448no—
192C2×C4×D3C24116yes—
193C2×C4×D3C24132yes—
194C3×C4⋊C4C24116yes—
195C3×C4⋊C4C24116yes—
196C3×C4⋊C4C24132yes—
197#217C24116yes—
198(C2×D3)⋊C4C24132yes—
199(C2×D3)⋊C4C24132yes—
200(C2×D3)⋊C4C24116yes—
201(C2×D3)⋊C4C24132yes—
202(C2×D3)⋊C4C24132yes—
203C4⋊Dic3C24116yes—
204C4⋊Dic3C24132yes—
205Dic3⋊C4C24132yes—
206Dic3⋊C4C24116yes—
207C4×Dic3C24116yes—
208C4×Dic3C24132yes—
209C25—31no—
210C25—32no—
211C25C22×D311yescharacteristic
212C22×D4—324no—
213C22×D4—312no—
214C22×D4—312no—
215C22×D4—324no—
216C22×D4—616no—
217C22×D4—68no—
218C22×D4—68no—
219C22×D4—128no—
220C23×C4—28no—
221C23×C4—31no—
222C23×C4—32no—
223C23×C4—31no—
224C23×C4—31no—
225C23×C4—312no—
226C23×C4—324no—
227C23×C4—48no—
228C23×C4—68no—
229C23×C4—128no—
230C23×C4C22×D3112yes—
231C23×C4C22×D311yescharacteristic
232C23×C4C22×D312yes—
233C23×C4C22×D312yes—
234C23×C4C22×D3124yes—
235C23×C4C22×D311yescharacteristic
236#126—3128no—
237#125—364no—
238#123—3128no—
239#121—3128no—
240#121—364no—
241#121—364no—
242C4×D4—3128no—
243C4×D4—364no—
244C4×D4—364no—
245#117—364no—
246C2×C4⋊C4—312no—
247C2×C4⋊C4—324no—
248C2×C4⋊C4—312no—
249C2×C4⋊C4—324no—
250C2×C4⋊C4C22×D3112yes—
251C2×C4⋊C4C22×D3112yes—
252C2×C4⋊C4C22×D3124yes—
253C2×C4⋊C4C22×D3164yescomplemented
254C2×C22⋊C4—312no—
255C2×C22⋊C4—324no—
256C2×C22⋊C4—324no—
257C2×C22⋊C4—324no—
258C2×C22⋊C4—312no—
259C2×C22⋊C4—324no—
260C2×C42—324no—
261C2×C42—312no—
262C23×C3—232no—
263C23×C3—212no—
264C23×C3—212no—
265C23×C3C2×#4114yes—
266C23×C3C2×#41124yes—
267C23×C3C2×#4114yes—
268C23×C3H(5,𝔽2)14yes—
269C23×C3C2514yes—
270C23×C3C2513yes—
271C22×D3—212no—
272C22×D3—224no—
273C22×D3—212no—
274C22×D3—224no—
275C22×D3—224no—
276C22×D3—212no—
277C22×D3—416no—
278C22×D3—432no—
279C22×D3C2514yes—
280C22×D3C2518yes—
281C2×C3×C4—212no—
282C2×C3×C4—224no—
283C2×C3×C4—224no—
284C2×C3×C4—232no—
285C2×C3×C4—432no—
286C2×C3×C4C2×#41124yes—
287C2×C3×C4C2518yes—
288C2×C3×C4C2518yes—
289C2×C3×C4C2514yes—
290C2×C3×C4C2514yes—
291#67—232no—
292#67—264no—
293C2×Dic3—212no—
294C2×Dic3—212no—
295C2×Dic3—224no—
296C2×Dic3—212no—
297C2×Dic3C2514yes—
298C2×Dic3C2514yes—
299C2×Dic3C2514yes—
300C2×Dic3C2518yes—
301D12—232no—
302D12—432no—
303C4×D3—232no—
304C4×D3—432no—
305C24—24no—
306C24—24no—
307C24—36no—
308C24—312no—
309C24—64no—
310C24—68no—
311C24—68no—
312C24—64no—
313C24—64no—
314C24—68no—
315C24#23918yes—
316C24C23×D316yes—
317C24C23×D311yescharacteristic
318C2×D4—332no—
319C2×D4—316no—
320C2×D4—316no—
321C2×D4—332no—
322C2×D4—696no—
323C2×D4—648no—
324C2×D4—648no—
325C2×D4—1248no—
326C22×C4—24no—
327C22×C4—28no—
328C22×C4—28no—
329C22×C4—248no—
330C22×C4—36no—
331C22×C4—312no—
332C22×C4—36no—
333C22×C4—36no—
334C22×C4—316no—
335C22×C4—332no—
336C22×C4—448no—
337C22×C4—64no—
338C22×C4—64no—
339C22×C4—68no—
340C22×C4—64no—
341C22×C4—648no—
342C22×C4—1248no—
343C22×C4#23518yes—
344C22×C4C23×D3112yes—
345C22×C4C23×D316yes—
346C22×C4C23×D3112yes—
347C22×C4C23×D3116yes—
348C22×C4C23×D3132yes—
349C22×C4C23×D316yes—
350C4⋊C4—316no—
351C4⋊C4—332no—
352C4⋊C4—316no—
353C4⋊C4—332no—
354C4⋊C4C23×D3116yes—
355C4⋊C4C23×D3116yes—
356C4⋊C4C23×D3132yes—
357C22⋊C4—316no—
358C22⋊C4—332no—
359C22⋊C4—332no—
360C22⋊C4—332no—
361C22⋊C4—316no—
362C22⋊C4—332no—
363C42—332no—
364C42—316no—
365C22×C3—24no—
366C22×C3—24no—
367C22×C3—248no—
368C22×C3#553116yescomplemented
369C22×C3C22×#4116yes—
370C22×C3C22×#4118yes—
371C22×C3C22×#4116yes—
372C22×C3C2×H(5,𝔽2)16yes—
373C22×C3C2611yescharacteristic
374C2×D3—24no—
375C2×D3—28no—
376C2×D3—24no—
377C2×D3—28no—
378C2×D3—28no—
379C2×D3—24no—
380C2×D3—424no—
381C2×D3—448no—
382C3×C4—24no—
383C3×C4—28no—
384C3×C4—28no—
385C3×C4C22×#4118yes—
386Dic3—24no—
387Dic3—24no—
388Dic3—28no—
389Dic3—24no—
390C23—212no—
391C23—212no—
392C23—232no—
393C23—34no—
394C23—38no—
395C23—612no—
396C23—624no—
397C23—624no—
398C23—612no—
399C23—612no—
400C23—624no—
401C23—1216no—
402C23—1232no—
403C23C2×#23514yes—
404C23#100414yes—
405C23C2×#239124yes—
406C23C2×#23914yes—
407C23C24×D314yes—
408C23C24×D313yes—
409D4—664no—
410D4—632no—
411D4—632no—
412D4—1232no—
413C2×C4—212no—
414C2×C4—224no—
415C2×C4—224no—
416C2×C4—232no—
417C2×C4—34no—
418C2×C4—38no—
419C2×C4—34no—
420C2×C4—34no—
421C2×C4—432no—
422C2×C4—612no—
423C2×C4—612no—
424C2×C4—624no—
425C2×C4—612no—
426C2×C4—632no—
427C2×C4—1232no—
428C2×C4C2×#235124yes—
429C2×C4C24×D318yes—
430C2×C4C24×D314yes—
431C2×C4C24×D318yes—
432C2×C4C24×D314yes—
433C2×C3—28no—
434C2×C3C2×#553112yescomplemented
435C2×C3C23×#4111yescharacteristic
436C2×C3C23×#4111yescharacteristic
437C2×C3C22×H(5,𝔽2)11yescharacteristic
438D3—44no—
439D3—48no—
440C22—24no—
441C22—24no—
442C22—248no—
443C22—64no—
444C22—68no—
445C22—68no—
446C22—64no—
447C22—64no—
448C22—68no—
449C22—1224no—
450C22—1248no—
451C22#5514116yesdirect product factor
452C22C22×#23516yes—
453C22C2×#100416yes—
454C22C22×#23918yes—
455C22C22×#23916yes—
456C22C25×D311yescharacteristic
457C4—24no—
458C4—28no—
459C4—28no—
460C4—64no—
461C4—64no—
462C4—68no—
463C4—64no—
464C4C22×#23518yes—
465C3C22×#55311yescharacteristic, complemented, Sylow
466C2—28no—
467C2—124no—
468C2—128no—
469C2C2×#5514112yesdirect product factor
470C2C23×#23511yescharacteristic
471C2C22×#100411yescharacteristic
472C2C23×#23911yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC22×C3C26
FrattiniC22C25×D3
FittingC23×C3×C4⋊C4C2
socleC24×C3C24
hypercenterC23×C4⋊C4D3
hypocenterC3C22×#553
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C24 ◁ C23×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C23×C3×C4⋊C4C2semidirect
2. C22×#930C2semidirect
3. C22×#930C2semidirect
4. C22×C4×#67C2semidirect
5. C22×C4×#67C2semidirect
6. C2×#5514C2direct (N × H)
7. C22×#894C2semidirect
8. C22×#895C2semidirect
9. C22×#898C2semidirect
10. C22×#890C2semidirect
11. C22×#893C2semidirect
12. C22×#892C2semidirect
25. C22×C3×C4⋊C4C22semidirect
27. C2×#930C22semidirect
28. C2×#930C22semidirect
29. C2×C4×#67C22semidirect
30. C2×C4×#67C22semidirect
42. #5514C22direct (N × H)
43. C2×#898C22semidirect
44. C2×#895C22semidirect
45. C2×#894C22semidirect
46. C2×#893C22semidirect
47. C2×#892C22semidirect
48. C2×#890C22semidirect
60. C23×C4⋊C4D3semidirect
91. C2×C3×C4⋊C4C23semidirect
93. #930C23semidirect
94. #930C23semidirect
95. C4×#67C23semidirect
96. C4×#67C23semidirect
108. #898C23semidirect
109. #895C23semidirect
110. #894C23semidirect
111. #893C23semidirect
112. #892C23semidirect
113. #890C23semidirect
140. C22×C4⋊C4C2×D3semidirect
253. C2×C4⋊C4C22×D3semidirect
368. C22×C3#553semidirect
434. C2×C3C2×#553semidirect
451. C22#5514direct (N × H)
465. C3C22×#553semidirect
469. C2C2×#5514direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C4⋊C42
C22×#9302
C22×C4×#672
C2×#55142
C22×#8942
C22×#8952
C22×#8982
C22×#8902
C22×#8932
C22×#8922
C22×#5533