Finite Group Databasebeta

← C22×#5568 #11702199

Subgroups — C22×#5568 #11702199

Order 768.

7846 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×Q8C211yesmaximal, characteristic, complemented
2C22×#944C211yesmaximal, characteristic, complemented
3C22×#947C211yesmaximal, characteristic
4C2×#5568C2148yesmaximal, direct product factor
5C22×#896C212yesmaximal, complemented
6C22×#898C212yesmaximal, complemented
7C22×#887C211yesmaximal, characteristic, complemented
8C22×#893C211yesmaximal, characteristic
9C22×#876C212yesmaximal, complemented
10C22×#875C211yesmaximal, characteristic
11C22×#872C212yesmaximal, complemented
12C22×C4×Dic6C211yesmaximal, characteristic, complemented
13C22×#556—31nomaximal, Sylow
14C23×C3×Q8C2211yescharacteristic
15C23×D12C2211yescharacteristic
16C23×C4×D3C2212yes—
17C23×Dic6C2211yescharacteristic
18C23×Dic6C2211yescharacteristic
19C2×C3×C4×Q8C22124yescomplemented
20C22×C3×C4⋊C4C2211yescharacteristic
21C22×C3×C4⋊C4C2212yes—
22C22×C3×C42C2211yescharacteristic
23C22×C3×C42C2212yes—
24C2×#947C22124yes—
25C2×#944C22124yescomplemented
26C22×(C2×D3)⋊C4C2212yes—
27C22×(C2×D3)⋊C4C2212yes—
28C22×(C2×D3)⋊C4C2212yes—
29C22×(C2×D3)⋊C4C2212yes—
30C22×(C2×D3)⋊C4C2212yes—
31C22×C4⋊Dic3C2211yescharacteristic
32C22×Dic3⋊C4C2211yescharacteristic
33C22×Dic3⋊C4C2211yescharacteristic
34C22×Dic3⋊C4C2212yes—
35C22×Dic3⋊C4C2212yes—
36C22×Dic3⋊C4C2212yes—
37C22×C4×Dic3C2211yescharacteristic
38C22×C4×Dic3C2211yescharacteristic
39C22×C4×Dic3C2212yes—
40#5568C221256yesdirect product factor
41C2×#898C22148yescomplemented
42C2×#896C22148yescomplemented
43C2×#893C22124yes—
44C2×#887C22124yescomplemented
45C2×#876C22148yescomplemented
46C2×#875C22124yes—
47C2×#872C22148yescomplemented
48C2×C4×Dic6C22124yescomplemented
49C2×#556—348no—
50C22×C4⋊Q8—31no—
51C22×#126—32no—
52C22×#126—32no—
53C22×#124—31no—
54C22×#124—31no—
55C22×C22⋊Q8—32no—
56C22×C4×Q8—31no—
57C22×C4×Q8—31no—
58C22×C4×Q8D311yescharacteristic, complemented
59C22×C4×D4—31no—
60C22×#117—32no—
61C24×D3C2312yes—
62C22×C3×Q8—28no—
63C22×C3×Q8C23112yes—
64C23×C3×C4C2311yescharacteristic
65C23×C3×C4C2312yes—
66C23×C3×C4C2312yes—
67C23×C3×C4C2311yescharacteristic
68C23×C3×C4C2311yescharacteristic
69C23×Dic3C2311yescharacteristic
70C23×Dic3C2312yes—
71C23×Dic3C2311yescharacteristic
72C23×Dic3C2311yescharacteristic
73C23×Dic3C2311yescharacteristic
74C22×D12—28no—
75C22×D12C23112yes—
76C22×C4×D3—216no—
77C22×C4×D3C23124yes—
78C22×Dic6—28no—
79C22×Dic6—28no—
80C22×Dic6C23112yes—
81C22×Dic6C23112yes—
82C3×C4×Q8C23164yescomplemented
83C2×C3×C4⋊C4C23112yes—
84C2×C3×C4⋊C4C23124yes—
85C2×C3×C42C23112yes—
86C2×C3×C42C23124yes—
87#947C23164yes—
88#944C23164yescomplemented
89C2×(C2×D3)⋊C4C23124yes—
90C2×(C2×D3)⋊C4C23124yes—
91C2×(C2×D3)⋊C4C23124yes—
92C2×(C2×D3)⋊C4C23124yes—
93C2×(C2×D3)⋊C4C23124yes—
94C2×C4⋊Dic3C23112yes—
95C2×Dic3⋊C4C23112yes—
96C2×Dic3⋊C4C23112yes—
97C2×Dic3⋊C4C23124yes—
98C2×Dic3⋊C4C23124yes—
99C2×Dic3⋊C4C23124yes—
100C2×C4×Dic3C23112yes—
101C2×C4×Dic3C23112yes—
102C2×C4×Dic3C23124yes—
103#898C231128yescomplemented
104#896C231128yescomplemented
105#893C23164yes—
106#887C23164yescomplemented
107#876C231128yescomplemented
108#875C23164yes—
109#872C231128yescomplemented
110C4×Dic6C23164yescomplemented
111C23×Q8—31no—
112C23×Q8—31no—
113C23×Q8C2×D311yescharacteristic
114C23×D4—31no—
115C24×C4—32no—
116#556—3256no—
117C2×C4⋊Q8—324no—
118C2×#126—348no—
119C2×#126—348no—
120C2×#124—324no—
121C2×#124—324no—
122C2×C22⋊Q8—348no—
123C2×C4×Q8—324no—
124C2×C4×Q8—324no—
125C2×C4×Q8C2×D3124yescomplemented
126C2×C4×D4—324no—
127C2×#117—348no—
128C22×C4⋊C4—31no—
129C22×C4⋊C4—32no—
130C22×C4⋊C4—32no—
131C22×C4⋊C4—31no—
132C22×C4⋊C4—31no—
133C22×C4⋊C4—32no—
134C22×C4⋊C4C2×D311yescharacteristic
135C22×C4⋊C4C2×D312yes—
136C22×C22⋊C4—32no—
137C22×C22⋊C4—32no—
138C22×C22⋊C4—32no—
139C22×C22⋊C4—32no—
140C22×C22⋊C4—32no—
141C22×C42—32no—
142C22×C42—31no—
143C22×C42—31no—
144C22×C42C2×D312yes—
145C22×C42C2×D311yescharacteristic
146C24×C3C2411yescharacteristic
147C23×D3—28no—
148C23×D3—28no—
149C23×D3—28no—
150C23×D3C24112yes—
151C2×C3×Q8—248no—
152C2×C3×Q8C24116yes—
153C22×C3×C4—24no—
154C22×C3×C4—28no—
155C22×C3×C4—28no—
156C22×C3×C4—24no—
157C22×C3×C4#4118yes—
158C22×C3×C4C24112yes—
159C22×C3×C4C2416yes—
160C22×C3×C4C24112yes—
161C22×C3×C4C2416yes—
162C22×C3×C4C2416yes—
163C22×Dic3—28no—
164C22×Dic3—24no—
165C22×Dic3—24no—
166C22×Dic3—24no—
167C22×Dic3#4118yes—
168C22×Dic3C2416yes—
169C22×Dic3C2416yes—
170C22×Dic3C2416yes—
171C22×Dic3C24112yes—
172C22×Dic3C2416yes—
173C2×D12—248no—
174C2×D12C24116yes—
175C2×C4×D3—296no—
176C2×C4×D3C24132yes—
177C2×Dic6—248no—
178C2×Dic6—248no—
179C2×Dic6C24116yes—
180C2×Dic6C24116yes—
181C3×C4⋊C4C24116yes—
182C3×C4⋊C4C24132yes—
183C3×C42C24116yes—
184C3×C42C24132yes—
185(C2×D3)⋊C4C24132yes—
186(C2×D3)⋊C4C24132yes—
187(C2×D3)⋊C4C24132yes—
188(C2×D3)⋊C4C24132yes—
189(C2×D3)⋊C4C24132yes—
190C4⋊Dic3C24116yes—
191Dic3⋊C4C24116yes—
192Dic3⋊C4C24116yes—
193Dic3⋊C4C24132yes—
194Dic3⋊C4C24132yes—
195Dic3⋊C4C24132yes—
196C4×Dic3C24116yes—
197C4×Dic3C24116yes—
198C4×Dic3C24132yes—
199C25—32no—
200C22×Q8—28no—
201C22×Q8—312no—
202C22×Q8—312no—
203C22×Q8—68no—
204C22×Q8—68no—
205C22×Q8C22×D3112yes—
206C22×D4—312no—
207C22×D4—68no—
208C23×C4—31no—
209C23×C4—31no—
210C23×C4—32no—
211C23×C4—31no—
212C23×C4—31no—
213C23×C4—324no—
214C23×C4—616no—
215C23×C4C22×D312yes—
216C23×C4C22×D311yescharacteristic
217C23×C4C22×D311yescharacteristic
218C23×C4C22×D312yes—
219C23×C4C22×D311yescharacteristic
220C4⋊Q8—364no—
221#126—3128no—
222#126—3128no—
223#124—364no—
224#124—364no—
225C22⋊Q8—3128no—
226C4×Q8—364no—
227C4×Q8—364no—
228C4×Q8C22×D3164yescomplemented
229C4×D4—364no—
230#117—3128no—
231C2×C4⋊C4—312no—
232C2×C4⋊C4—324no—
233C2×C4⋊C4—324no—
234C2×C4⋊C4—312no—
235C2×C4⋊C4—312no—
236C2×C4⋊C4—324no—
237C2×C4⋊C4C22×D3112yes—
238C2×C4⋊C4C22×D3124yes—
239C2×C22⋊C4—324no—
240C2×C22⋊C4—324no—
241C2×C22⋊C4—324no—
242C2×C22⋊C4—324no—
243C2×C22⋊C4—324no—
244C2×C42—324no—
245C2×C42—312no—
246C2×C42—312no—
247C2×C42C22×D3124yes—
248C2×C42C22×D3112yes—
249C23×C3C2×#4114yes—
250C23×C3C2×#4114yes—
251C23×C32−1+414yes—
252C23×C3C2513yes—
253C22×D3—224no—
254C22×D3—224no—
255C22×D3—224no—
256C22×D3—432no—
257C22×D3C2518yes—
258C3×Q8—232no—
259C2×C3×C4—212no—
260C2×C3×C4—224no—
261C2×C3×C4—224no—
262C2×C3×C4—212no—
263C2×C3×C4C2×#41124yes—
264C2×C3×C4C2518yes—
265C2×C3×C4C2514yes—
266C2×C3×C4C2518yes—
267C2×C3×C4C2514yes—
268C2×C3×C4C2514yes—
269C2×Dic3—224no—
270C2×Dic3—212no—
271C2×Dic3—212no—
272C2×Dic3—212no—
273C2×Dic3C2×#41124yes—
274C2×Dic3C2514yes—
275C2×Dic3C2514yes—
276C2×Dic3C2514yes—
277C2×Dic3C2518yes—
278C2×Dic3C2514yes—
279D12—232no—
280C4×D3—264no—
281Dic6—232no—
282Dic6—232no—
283C24—312no—
284C24—68no—
285C24—68no—
286C24—68no—
287C24C23×D311yescharacteristic
288C2×Q8—248no—
289C2×Q8—316no—
290C2×Q8—316no—
291C2×Q8—648no—
292C2×Q8—648no—
293C2×Q8C23×D3116yes—
294C2×D4—316no—
295C2×D4—648no—
296C22×C4—28no—
297C22×C4—24no—
298C22×C4—28no—
299C22×C4—24no—
300C22×C4—38no—
301C22×C4—36no—
302C22×C4—36no—
303C22×C4—36no—
304C22×C4—312no—
305C22×C4—36no—
306C22×C4—332no—
307C22×C4—696no—
308C22×C4—68no—
309C22×C4—64no—
310C22×C4—64no—
311C22×C4—64no—
312C22×C4#23518yes—
313C22×C4C23×D3112yes—
314C22×C4C23×D316yes—
315C22×C4C23×D3112yes—
316C22×C4C23×D316yes—
317C22×C4C23×D316yes—
318C4⋊C4—316no—
319C4⋊C4—332no—
320C4⋊C4—332no—
321C4⋊C4—316no—
322C4⋊C4—316no—
323C4⋊C4—332no—
324C4⋊C4C23×D3116yes—
325C4⋊C4C23×D3132yes—
326C22⋊C4—332no—
327C22⋊C4—332no—
328C22⋊C4—332no—
329C22⋊C4—332no—
330C22⋊C4—332no—
331C42—332no—
332C42—316no—
333C42—316no—
334C42C23×D3132yes—
335C42C23×D3116yes—
336C22×C3#556116yescomplemented
337C22×C3C22×#4116yes—
338C22×C3C22×#4116yes—
339C22×C3C2×2−1+416yes—
340C22×C3C2611yescharacteristic
341C2×D3—28no—
342C2×D3—28no—
343C2×D3—28no—
344C2×D3—448no—
345C3×C4—24no—
346C3×C4—28no—
347C3×C4—28no—
348C3×C4—24no—
349C3×C4C22×#4118yes—
350Dic3—28no—
351Dic3—24no—
352Dic3—24no—
353Dic3—24no—
354Dic3C22×#4118yes—
355C23—38no—
356C23—624no—
357C23—624no—
358C23—624no—
359C23—1232no—
360C23C2×#23514yes—
361C23#100714yes—
362C23D3×#4114yes—
363C23C24×D313yes—
364Q8—232no—
365Q8—632no—
366Q8—632no—
367D4—632no—
368C2×C4—224no—
369C2×C4—212no—
370C2×C4—224no—
371C2×C4—212no—
372C2×C4—324no—
373C2×C4—34no—
374C2×C4—34no—
375C2×C4—34no—
376C2×C4—38no—
377C2×C4—34no—
378C2×C4—664no—
379C2×C4—624no—
380C2×C4—612no—
381C2×C4—612no—
382C2×C4—612no—
383C2×C4C2×#235124yes—
384C2×C4C24×D318yes—
385C2×C4C24×D314yes—
386C2×C4C24×D318yes—
387C2×C4C24×D314yes—
388C2×C4C24×D314yes—
389C2×C3C2×#556112yescomplemented
390C2×C3C23×#4111yescharacteristic
391C2×C3C23×#4111yescharacteristic
392C2×C3C22×2−1+411yescharacteristic
393D3—48no—
394C22—68no—
395C22—68no—
396C22—68no—
397C22—1248no—
398C22#5568116yesdirect product factor
399C22C22×#23516yes—
400C22C2×#100716yes—
401C22C2×D3×#4116yes—
402C22C25×D311yescharacteristic
403C4—28no—
404C4—24no—
405C4—28no—
406C4—24no—
407C4—38no—
408C4—68no—
409C4—64no—
410C4—64no—
411C4—64no—
412C4C22×#23518yes—
413C3C22×#55611yescharacteristic, complemented, Sylow
414C2—128no—
415C2C2×#5568112yesdirect product factor
416C2C23×#23511yescharacteristic
417C2C22×#100711yescharacteristic
418C2C22×D3×#4111yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC22×C3C26
FrattiniC22C25×D3
FittingC22×C3×C4×Q8C2
socleC24×C3C24
prime radicalC23×D12C22
hypercenterC22×C4×Q8D3
hypocenterC3C22×#556
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C22×C3×C4×Q8C2semidirect
2. C22×#944C2semidirect
4. C2×#5568C2direct (N × H)
5. C22×#896C2semidirect
6. C22×#898C2semidirect
7. C22×#887C2semidirect
9. C22×#876C2semidirect
11. C22×#872C2semidirect
12. C22×C4×Dic6C2semidirect
19. C2×C3×C4×Q8C22semidirect
25. C2×#944C22semidirect
40. #5568C22direct (N × H)
41. C2×#898C22semidirect
42. C2×#896C22semidirect
44. C2×#887C22semidirect
45. C2×#876C22semidirect
47. C2×#872C22semidirect
48. C2×C4×Dic6C22semidirect
58. C22×C4×Q8D3semidirect
82. C3×C4×Q8C23semidirect
88. #944C23semidirect
103. #898C23semidirect
104. #896C23semidirect
106. #887C23semidirect
107. #876C23semidirect
109. #872C23semidirect
110. C4×Dic6C23semidirect
125. C2×C4×Q8C2×D3semidirect
228. C4×Q8C22×D3semidirect
336. C22×C3#556semidirect
389. C2×C3C2×#556semidirect
398. C22#5568direct (N × H)
413. C3C22×#556semidirect
415. C2C2×#5568direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×Q82
C22×#9442
C22×#9472
C2×#55682
C22×#8962
C22×#8982
C22×#8872
C22×#8932
C22×#8762
C22×#8752
C22×#8722
C22×C4×Dic62
C22×#5563