Finite Group Databasebeta

← C22×#5691 #11701869

Subgroups — C22×#5691 #11701869

Order 768.

8670 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#125C211yesmaximal, characteristic, complemented
2C2×#5691C2148yesmaximal, direct product factor
3C22×#894C212yesmaximal, complemented
4C22×#897C211yesmaximal, characteristic, complemented
5C22×#895C211yesmaximal, characteristic
6C22×#898C212yesmaximal, complemented
7C22×#887C211yesmaximal, characteristic, complemented
8C22×#893C212yesmaximal, complemented
9C22×#893C211yesmaximal, characteristic, complemented
10C22×#892C212yesmaximal, complemented
11C22×#875C211yesmaximal, characteristic, complemented
12C22×C42×D3C211yesmaximal, characteristic, complemented
13C22×#529—31nomaximal, Sylow
14C23×D12C2211yescharacteristic
15C23×D12C2212yes—
16C23×C4×D3C2211yescharacteristic
17C23×C4×D3C2211yescharacteristic
18C23×C4×D3C2211yescharacteristic
19C23×C4×D3C2212yes—
20C23×Dic6C2211yescharacteristic
21C2×C3×#125C22124yescomplemented
22C22×C3×C4⋊C4C2212yes—
23C22×C3×C4⋊C4C2212yes—
24C22×C3×C4⋊C4C2211yescharacteristic
25C22×C3×C4⋊C4C2211yescharacteristic
26C22×C3×C42C2211yescharacteristic
27C22×(C2×D3)⋊C4C2212yes—
28C22×(C2×D3)⋊C4C2212yes—
29C22×(C2×D3)⋊C4C2212yes—
30C22×(C2×D3)⋊C4C2212yes—
31C22×(C2×D3)⋊C4C2212yes—
32C22×C4⋊Dic3C2212yes—
33C22×Dic3⋊C4C2212yes—
34C22×C4×Dic3C2211yescharacteristic
35C22×C4×Dic3C2211yescharacteristic
36C22×C4×Dic3C2211yescharacteristic
37C22×C4×Dic3C2212yes—
38#5691C221256yesdirect product factor
39C2×#898C22148yescomplemented
40C2×#897C22124yescomplemented
41C2×#895C22124yes—
42C2×#894C22148yescomplemented
43C2×#893C22124yescomplemented
44C2×#893C22148yescomplemented
45C2×#892C22148yescomplemented
46C2×#887C22124yescomplemented
47C2×#875C22124yescomplemented
48C2×C42×D3C22124yescomplemented
49C2×#529—348no—
50C22×#126—32no—
51C22×#125D311yescharacteristic, complemented
52C22×#124—31no—
53C22×#123—32no—
54C22×C22⋊Q8—31no—
55C22×#121—31no—
56C22×C4×Q8—31no—
57C22×C4×D4—32no—
58C22×C4×D4—31no—
59C22×#117—32no—
60C23×C42—31no—
61C24×D3C2312yes—
62C24×D3C2311yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C23×C3×C4C2311yescharacteristic
65C23×C3×C4C2312yes—
66C23×C3×C4C2312yes—
67C23×C3×C4C2311yescharacteristic
68C23×Dic3C2312yes—
69C23×Dic3C2311yescharacteristic
70C23×Dic3C2311yescharacteristic
71C23×Dic3C2311yescharacteristic
72C22×D12—216no—
73C22×D12—28no—
74C22×D12C23112yes—
75C22×D12C23124yes—
76C22×C4×D3—28no—
77C22×C4×D3—28no—
78C22×C4×D3—28no—
79C22×C4×D3—216no—
80C22×C4×D3C23112yes—
81C22×C4×D3C23112yes—
82C22×C4×D3C23112yes—
83C22×C4×D3C23124yes—
84C22×Dic6—28no—
85C22×Dic6C23112yes—
86C3×#125C23164yescomplemented
87C2×C3×C4⋊C4C23112yes—
88C2×C3×C4⋊C4C23124yes—
89C2×C3×C4⋊C4C23112yes—
90C2×C3×C4⋊C4C23124yes—
91C2×C3×C42C23112yes—
92C2×(C2×D3)⋊C4C23124yes—
93C2×(C2×D3)⋊C4C23124yes—
94C2×(C2×D3)⋊C4C23124yes—
95C2×(C2×D3)⋊C4C23124yes—
96C2×(C2×D3)⋊C4C23124yes—
97C2×C4⋊Dic3C23124yes—
98C2×Dic3⋊C4C23124yes—
99C2×C4×Dic3C23112yes—
100C2×C4×Dic3C23112yes—
101C2×C4×Dic3C23112yes—
102C2×C4×Dic3C23124yes—
103#898C231128yescomplemented
104#897C23164yescomplemented
105#895C23164yes—
106#894C231128yescomplemented
107#893C23164yescomplemented
108#893C231128yescomplemented
109#892C231128yescomplemented
110#887C23164yescomplemented
111#875C23164yescomplemented
112C42×D3C23164yescomplemented
113C23×Q8—31no—
114C23×D4—32no—
115C23×D4—31no—
116C24×C4—31no—
117C24×C4—31no—
118C24×C4—31no—
119C24×C4—32no—
120#529—3256no—
121C2×#126—348no—
122C2×#125C2×D3124yescomplemented
123C2×#124—324no—
124C2×#123—348no—
125C2×C22⋊Q8—324no—
126C2×#121—324no—
127C2×C4×Q8—324no—
128C2×C4×D4—348no—
129C2×C4×D4—324no—
130C2×#117—348no—
131C22×C4⋊C4—32no—
132C22×C4⋊C4—32no—
133C22×C4⋊C4C2×D312yes—
134C22×C4⋊C4C2×D311yescharacteristic
135C22×C4⋊C4C2×D312yes—
136C22×C4⋊C4C2×D311yescharacteristic
137C22×C22⋊C4—32no—
138C22×C22⋊C4—32no—
139C22×C22⋊C4—32no—
140C22×C22⋊C4—32no—
141C22×C22⋊C4—32no—
142C22×C42—31no—
143C22×C42—31no—
144C22×C42—31no—
145C22×C42—32no—
146C22×C42—324no—
147C22×C42C2×D311yescharacteristic
148C24×C3C2411yescharacteristic
149C23×D3—24no—
150C23×D3—24no—
151C23×D3—28no—
152C23×D3—28no—
153C23×D3—28no—
154C23×D3#4118yes—
155C23×D3C24112yes—
156C23×D3C2416yes—
157C22×C3×C4—24no—
158C22×C3×C4—24no—
159C22×C3×C4—28no—
160C22×C3×C4—28no—
161C22×C3×C4#4118yes—
162C22×C3×C4C24112yes—
163C22×C3×C4C2416yes—
164C22×C3×C4C2416yes—
165C22×C3×C4C24112yes—
166C22×C3×C4C2416yes—
167C22×Dic3—28no—
168C22×Dic3—24no—
169C22×Dic3—24no—
170C22×Dic3#4118yes—
171C22×Dic3C2416yes—
172C22×Dic3C2416yes—
173C22×Dic3C24112yes—
174C22×Dic3C2416yes—
175C2×D12—296no—
176C2×D12—248no—
177C2×D12C24116yes—
178C2×D12C24132yes—
179C2×C4×D3—248no—
180C2×C4×D3—248no—
181C2×C4×D3—248no—
182C2×C4×D3—296no—
183C2×C4×D3C24116yes—
184C2×C4×D3C24116yes—
185C2×C4×D3C24116yes—
186C2×C4×D3C24132yes—
187C2×Dic6—248no—
188C2×Dic6C24116yes—
189C3×C4⋊C4C24116yes—
190C3×C4⋊C4C24132yes—
191C3×C4⋊C4C24116yes—
192C3×C4⋊C4C24132yes—
193C3×C42C24116yes—
194(C2×D3)⋊C4C24132yes—
195(C2×D3)⋊C4C24132yes—
196(C2×D3)⋊C4C24132yes—
197(C2×D3)⋊C4C24132yes—
198(C2×D3)⋊C4C24132yes—
199C4⋊Dic3C24132yes—
200Dic3⋊C4C24132yes—
201C4×Dic3C24116yes—
202C4×Dic3C24116yes—
203C4×Dic3C24116yes—
204C4×Dic3C24132yes—
205C25—31no—
206C25—32no—
207C22×Q8—312no—
208C22×Q8—68no—
209C22×D4—324no—
210C22×D4—312no—
211C22×D4—616no—
212C22×D4—68no—
213C23×C4—31no—
214C23×C4—32no—
215C23×C4—31no—
216C23×C4—31no—
217C23×C4—324no—
218C23×C4—312no—
219C23×C4—312no—
220C23×C4—312no—
221C23×C4—68no—
222C23×C4—616no—
223C23×C4—68no—
224C23×C4—68no—
225C23×C4C22×D312yes—
226C23×C4C22×D311yescharacteristic
227C23×C4C22×D311yescharacteristic
228C23×C4C22×D312yes—
229C23×C4C22×D311yescharacteristic
230#126—3128no—
231#125C22×D3164yescomplemented
232#124—364no—
233#123—3128no—
234C22⋊Q8—364no—
235#121—364no—
236C4×Q8—364no—
237C4×D4—364no—
238C4×D4—3128no—
239#117—3128no—
240C2×C4⋊C4—324no—
241C2×C4⋊C4—324no—
242C2×C4⋊C4C22×D3124yes—
243C2×C4⋊C4C22×D3112yes—
244C2×C4⋊C4C22×D3124yes—
245C2×C4⋊C4C22×D3112yes—
246C2×C22⋊C4—324no—
247C2×C22⋊C4—324no—
248C2×C22⋊C4—324no—
249C2×C22⋊C4—324no—
250C2×C22⋊C4—324no—
251C2×C42—312no—
252C2×C42—324no—
253C2×C42—312no—
254C2×C42—312no—
255C2×C42—364no—
256C2×C42C22×D3112yes—
257C23×C3C2×#4114yes—
258C23×C3C2×#4114yes—
259C23×C3C2×#4114yes—
260C23×C3C2513yes—
261C22×D3—232no—
262C22×D3—212no—
263C22×D3—212no—
264C22×D3—224no—
265C22×D3—224no—
266C22×D3—224no—
267C22×D3—432no—
268C22×D3C2×#41124yes—
269C22×D3C2518yes—
270C22×D3C2514yes—
271C2×C3×C4—212no—
272C2×C3×C4—212no—
273C2×C3×C4—224no—
274C2×C3×C4—224no—
275C2×C3×C4C2×#41124yes—
276C2×C3×C4C2518yes—
277C2×C3×C4C2514yes—
278C2×C3×C4C2514yes—
279C2×C3×C4C2518yes—
280C2×C3×C4C2514yes—
281C2×Dic3—224no—
282C2×Dic3—212no—
283C2×Dic3—212no—
284C2×Dic3C2×#41124yes—
285C2×Dic3C2514yes—
286C2×Dic3C2514yes—
287C2×Dic3C2518yes—
288C2×Dic3C2514yes—
289D12—264no—
290D12—232no—
291C4×D3—232no—
292C4×D3—232no—
293C4×D3—232no—
294C4×D3—264no—
295Dic6—232no—
296C24—38no—
297C24—36no—
298C24—312no—
299C24—64no—
300C24—64no—
301C24—68no—
302C24—68no—
303C24—68no—
304C24C23×D311yescharacteristic
305C2×Q8—316no—
306C2×Q8—648no—
307C2×D4—332no—
308C2×D4—316no—
309C2×D4—696no—
310C2×D4—648no—
311C22×C4—28no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—28no—
315C22×C4—38no—
316C22×C4—36no—
317C22×C4—36no—
318C22×C4—36no—
319C22×C4—312no—
320C22×C4—316no—
321C22×C4—316no—
322C22×C4—316no—
323C22×C4—332no—
324C22×C4—648no—
325C22×C4—696no—
326C22×C4—648no—
327C22×C4—64no—
328C22×C4—68no—
329C22×C4—64no—
330C22×C4—648no—
331C22×C4#23918yes—
332C22×C4C23×D316yes—
333C22×C4C23×D316yes—
334C22×C4C23×D3112yes—
335C22×C4C23×D3112yes—
336C22×C4C23×D316yes—
337C4⋊C4—332no—
338C4⋊C4—332no—
339C4⋊C4C23×D3132yes—
340C4⋊C4C23×D3116yes—
341C4⋊C4C23×D3132yes—
342C4⋊C4C23×D3116yes—
343C22⋊C4—332no—
344C22⋊C4—332no—
345C22⋊C4—332no—
346C22⋊C4—332no—
347C22⋊C4—332no—
348C42—316no—
349C42—316no—
350C42—316no—
351C42—332no—
352C42C23×D3116yes—
353C22×C3#529116yescomplemented
354C22×C3C22×#4116yes—
355C22×C3C22×#4116yes—
356C22×C3C22×#4116yes—
357C22×C3C2611yescharacteristic
358C2×D3—24no—
359C2×D3—24no—
360C2×D3—248no—
361C2×D3—28no—
362C2×D3—28no—
363C2×D3—28no—
364C2×D3—448no—
365C2×D3C22×#4118yes—
366C3×C4—24no—
367C3×C4—24no—
368C3×C4—28no—
369C3×C4—28no—
370C3×C4C22×#4118yes—
371Dic3—28no—
372Dic3—24no—
373Dic3—24no—
374Dic3C22×#4118yes—
375C23—324no—
376C23—34no—
377C23—38no—
378C23—632no—
379C23—612no—
380C23—612no—
381C23—624no—
382C23—624no—
383C23—624no—
384C23—1232no—
385C23C2×#23914yes—
386C23D3×#4114yes—
387C23D3×#4114yes—
388C23C24×D313yes—
389Q8—632no—
390D4—664no—
391D4—632no—
392C2×C4—212no—
393C2×C4—224no—
394C2×C4—212no—
395C2×C4—224no—
396C2×C4—324no—
397C2×C4—34no—
398C2×C4—34no—
399C2×C4—34no—
400C2×C4—38no—
401C2×C4—632no—
402C2×C4—664no—
403C2×C4—632no—
404C2×C4—612no—
405C2×C4—624no—
406C2×C4—612no—
407C2×C4—632no—
408C2×C4C2×#239124yes—
409C2×C4C24×D314yes—
410C2×C4C24×D314yes—
411C2×C4C24×D318yes—
412C2×C4C24×D318yes—
413C2×C4C24×D314yes—
414C2×C3C2×#529112yescomplemented
415C2×C3C23×#4111yescharacteristic
416C2×C3C23×#4111yescharacteristic
417C2×C3C23×#4111yescharacteristic
418D3—28no—
419D3—48no—
420C22—38no—
421C22—64no—
422C22—64no—
423C22—648no—
424C22—68no—
425C22—68no—
426C22—68no—
427C22—1248no—
428C22#5691116yesdirect product factor
429C22C22×#23916yes—
430C22C2×D3×#4116yes—
431C22C2×D3×#4116yes—
432C22C25×D311yescharacteristic
433C4—24no—
434C4—28no—
435C4—24no—
436C4—28no—
437C4—38no—
438C4—64no—
439C4—68no—
440C4—64no—
441C4C22×#23918yes—
442C3C22×#52911yescharacteristic, complemented, Sylow
443C2—68no—
444C2—128no—
445C2C2×#5691112yesdirect product factor
446C2C23×#23911yescharacteristic
447C2C22×D3×#4111yescharacteristic
448C2C22×D3×#4111yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC22×C3C26
FrattiniC22C25×D3
FittingC22×C3×#125C2
socleC24×C3C24
prime radicalC22×#895C2
hypercenterC22×#125D3
hypocenterC3C22×#529
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×#125

Complements

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

NH ≅ G/Ntype
1. C22×C3×#125C2semidirect
2. C2×#5691C2direct (N × H)
3. C22×#894C2semidirect
4. C22×#897C2semidirect
6. C22×#898C2semidirect
7. C22×#887C2semidirect
8. C22×#893C2semidirect
9. C22×#893C2semidirect
10. C22×#892C2semidirect
11. C22×#875C2semidirect
12. C22×C42×D3C2semidirect
21. C2×C3×#125C22semidirect
38. #5691C22direct (N × H)
39. C2×#898C22semidirect
40. C2×#897C22semidirect
42. C2×#894C22semidirect
43. C2×#893C22semidirect
44. C2×#893C22semidirect
45. C2×#892C22semidirect
46. C2×#887C22semidirect
47. C2×#875C22semidirect
48. C2×C42×D3C22semidirect
51. C22×#125D3semidirect
86. C3×#125C23semidirect
103. #898C23semidirect
104. #897C23semidirect
106. #894C23semidirect
107. #893C23semidirect
108. #893C23semidirect
109. #892C23semidirect
110. #887C23semidirect
111. #875C23semidirect
112. C42×D3C23semidirect
122. C2×#125C2×D3semidirect
231. #125C22×D3semidirect
353. C22×C3#529semidirect
414. C2×C3C2×#529semidirect
428. C22#5691direct (N × H)
442. C3C22×#529semidirect
445. C2C2×#5691direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1252
C2×#56912
C22×#8942
C22×#8972
C22×#8952
C22×#8982
C22×#8872
C22×#8932
C22×#8922
C22×#8752
C22×C42×D32
C22×#5293