Finite Group Databasebeta

← C22×#5532 #11702206

Subgroups — C22×#5532 #11702206

Order 768.

8710 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
1C22×C3×#117C211yesmaximal, characteristic, complemented
2C22×#930C211yesmaximal, characteristic, complemented
3C22×C4×#67C211yesmaximal, characteristic, complemented
4C2×#5532C2148yesmaximal, direct product factor
5C22×#895C211yesmaximal, characteristic, complemented
6C22×#888C211yesmaximal, characteristic, complemented
7C22×#893C211yesmaximal, characteristic, complemented
8C22×#892C211yesmaximal, characteristic, complemented
9C22×#884C212yesmaximal, complemented
10C22×#885C212yesmaximal, complemented
11C22×#875C212yesmaximal, complemented
12C22×#872C212yesmaximal, complemented
13C22×#555—31nomaximal, Sylow
14C24×C3×C4C2211yescharacteristic
15C23×#67C2211yescharacteristic
16C23×#67C2212yes—
17C23×D12C2211yescharacteristic
18C23×D12C2212yes—
19C23×C4×D3C2212yes—
20C23×C4×D3C2211yescharacteristic
21C23×Dic6C2212yes—
22C2×C3×#117C22124yescomplemented
23C22×C3×C4⋊C4C2211yescharacteristic
24C22×C3×C4⋊C4C2211yescharacteristic
25C22×C3×C22⋊C4C2212yes—
26C22×C3×C42C2212yes—
27C22×#217C2211yescharacteristic
28C2×#930C22124yescomplemented
29C2×C4×#67C22124yescomplemented
30C22×(C2×D3)⋊C4C2212yes—
31C22×(C2×D3)⋊C4C2212yes—
32C22×(C2×D3)⋊C4C2211yescharacteristic
33C22×(C2×D3)⋊C4C2212yes—
34C22×(C2×D3)⋊C4C2212yes—
35C22×C4⋊Dic3C2211yescharacteristic
36C22×Dic3⋊C4C2211yescharacteristic
37C22×Dic3⋊C4C2212yes—
38C22×C4×Dic3C2212yes—
39C22×C4×Dic3C2211yescharacteristic
40#5532C221256yesdirect product factor
41C2×#895C22124yescomplemented
42C2×#893C22124yescomplemented
43C2×#892C22124yescomplemented
44C2×#888C22124yescomplemented
45C2×#885C22148yescomplemented
46C2×#884C22148yescomplemented
47C2×#875C22148yescomplemented
48C2×#872C22148yescomplemented
49C2×#555—348no—
50C22×C4⋊Q8—31no—
51C22×#124—32no—
52C22×#124—32no—
53C22×#121—31no—
54C22×#121—31no—
55C22×#121—32no—
56C22×C4×D4—31no—
57C22×C4×D4—31no—
58C22×#117—32no—
59C22×#117—31no—
60C22×#117D311yescharacteristic, complemented
61C25×C3C2311yescharacteristic
62C24×D3C2312yes—
63C24×D3C2311yescharacteristic
64C23×C3×C4—28no—
65C23×C3×C4C2312yes—
66C23×C3×C4C2312yes—
67C23×C3×C4C2311yescharacteristic
68C23×C3×C4C23112yes—
69C23×C3×C4C2311yescharacteristic
70C22×#67—28no—
71C22×#67—48no—
72C22×#67C23112yes—
73C22×#67C23124yes—
74C23×Dic3C2311yescharacteristic
75C23×Dic3C2312yes—
76C23×Dic3C2311yescharacteristic
77C23×Dic3C2311yescharacteristic
78C22×D12—28no—
79C22×D12—48no—
80C22×D12C23112yes—
81C22×D12C23124yes—
82C22×C4×D3—28no—
83C22×C4×D3—216no—
84C22×C4×D3C23124yes—
85C22×C4×D3C23112yes—
86C22×Dic6—48no—
87C22×Dic6C23124yes—
88C3×#117C23164yescomplemented
89C2×C3×C4⋊C4C23112yes—
90C2×C3×C4⋊C4C23112yes—
91C2×C3×C22⋊C4C23124yes—
92C2×C3×C42C23124yes—
93C2×#217C23112yes—
94#930C23164yescomplemented
95C4×#67C23164yescomplemented
96C2×(C2×D3)⋊C4C23124yes—
97C2×(C2×D3)⋊C4C23124yes—
98C2×(C2×D3)⋊C4C23112yes—
99C2×(C2×D3)⋊C4C23124yes—
100C2×(C2×D3)⋊C4C23124yes—
101C2×C4⋊Dic3C23112yes—
102C2×Dic3⋊C4C23112yes—
103C2×Dic3⋊C4C23124yes—
104C2×C4×Dic3C23124yes—
105C2×C4×Dic3C23112yes—
106#895C23164yescomplemented
107#893C23164yescomplemented
108#892C23164yescomplemented
109#888C23164yescomplemented
110#885C231128yescomplemented
111#884C231128yescomplemented
112#875C231128yescomplemented
113#872C231128yescomplemented
114C23×Q8—32no—
115C23×D4—31no—
116C23×D4—32no—
117C23×D4—31no—
118C23×D4—32no—
119C24×C4—32no—
120C24×C4—31no—
121C24×C4C2×D311yescharacteristic
122#555—3256no—
123C2×C4⋊Q8—324no—
124C2×#124—348no—
125C2×#124—348no—
126C2×#121—324no—
127C2×#121—324no—
128C2×#121—348no—
129C2×C4×D4—324no—
130C2×C4×D4—324no—
131C2×#117—324no—
132C2×#117—348no—
133C2×#117C2×D3124yescomplemented
134C22×C4⋊C4—31no—
135C22×C4⋊C4—32no—
136C22×C4⋊C4—31no—
137C22×C4⋊C4C2×D311yescharacteristic
138C22×C4⋊C4C2×D311yescharacteristic
139C22×C22⋊C4—31no—
140C22×C22⋊C4—32no—
141C22×C22⋊C4—32no—
142C22×C22⋊C4—32no—
143C22×C22⋊C4—31no—
144C22×C22⋊C4—32no—
145C22×C22⋊C4C2×D312yes—
146C22×C42—32no—
147C22×C42—31no—
148C22×C42C2×D312yes—
149C24×C3—24no—
150C24×C3—24no—
151C24×C3—24no—
152C24×C3C2416yes—
153C24×C3C2411yescharacteristic
154C23×D3—24no—
155C23×D3—28no—
156C23×D3—28no—
157C23×D3—24no—
158C23×D3—24no—
159C23×D3—28no—
160C23×D3C24112yes—
161C23×D3C2416yes—
162C22×C3×C4—28no—
163C22×C3×C4—24no—
164C22×C3×C4—28no—
165C22×C3×C4—248no—
166C22×C3×C4#4118yes—
167C22×C3×C4C24112yes—
168C22×C3×C4C2416yes—
169C22×C3×C4C24112yes—
170C22×C3×C4C24116yes—
171C22×C3×C4C2416yes—
172C2×#67—248no—
173C2×#67—448no—
174C2×#67C24116yes—
175C2×#67C24132yes—
176C22×Dic3—24no—
177C22×Dic3—24no—
178C22×Dic3—28no—
179C22×Dic3#4118yes—
180C22×Dic3C2416yes—
181C22×Dic3C24112yes—
182C22×Dic3C2416yes—
183C22×Dic3C2416yes—
184C2×D12—248no—
185C2×D12—448no—
186C2×D12C24116yes—
187C2×D12C24132yes—
188C2×C4×D3—248no—
189C2×C4×D3—296no—
190C2×C4×D3C24132yes—
191C2×C4×D3C24116yes—
192C2×Dic6—448no—
193C2×Dic6C24132yes—
194C3×C4⋊C4C24116yes—
195C3×C4⋊C4C24116yes—
196C3×C22⋊C4C24132yes—
197C3×C42C24132yes—
198#217C24116yes—
199(C2×D3)⋊C4C24132yes—
200(C2×D3)⋊C4C24132yes—
201(C2×D3)⋊C4C24116yes—
202(C2×D3)⋊C4C24132yes—
203(C2×D3)⋊C4C24132yes—
204C4⋊Dic3C24116yes—
205Dic3⋊C4C24116yes—
206Dic3⋊C4C24132yes—
207C4×Dic3C24132yes—
208C4×Dic3C24116yes—
209C25—31no—
210C25—32no—
211C25C22×D311yescharacteristic
212C22×Q8—324no—
213C22×Q8—128no—
214C22×D4—312no—
215C22×D4—324no—
216C22×D4—312no—
217C22×D4—324no—
218C22×D4—68no—
219C22×D4—68no—
220C22×D4—128no—
221C22×D4—128no—
222C23×C4—28no—
223C23×C4—31no—
224C23×C4—31no—
225C23×C4—31no—
226C23×C4—32no—
227C23×C4—324no—
228C23×C4—312no—
229C23×C4—616no—
230C23×C4—68no—
231C23×C4C22×D3112yes—
232C23×C4C22×D311yescharacteristic
233C23×C4C22×D312yes—
234C23×C4C22×D312yes—
235C23×C4C22×D311yescharacteristic
236C4⋊Q8—364no—
237#124—3128no—
238#124—3128no—
239#121—364no—
240#121—364no—
241#121—3128no—
242C4×D4—364no—
243C4×D4—364no—
244#117—364no—
245#117—3128no—
246#117C22×D3164yescomplemented
247C2×C4⋊C4—312no—
248C2×C4⋊C4—324no—
249C2×C4⋊C4—312no—
250C2×C4⋊C4C22×D3112yes—
251C2×C4⋊C4C22×D3112yes—
252C2×C22⋊C4—312no—
253C2×C22⋊C4—324no—
254C2×C22⋊C4—324no—
255C2×C22⋊C4—324no—
256C2×C22⋊C4—324no—
257C2×C22⋊C4—312no—
258C2×C22⋊C4C22×D3124yes—
259C2×C42—324no—
260C2×C42—312no—
261C2×C42C22×D3124yes—
262C23×C3—212no—
263C23×C3—212no—
264C23×C3—212no—
265C23×C3—416no—
266C23×C3C2×#4114yes—
267C23×C3C2×#4114yes—
268C23×C3H(5,𝔽2)14yes—
269C23×C3C2514yes—
270C23×C3C2513yes—
271C22×D3—212no—
272C22×D3—224no—
273C22×D3—224no—
274C22×D3—212no—
275C22×D3—212no—
276C22×D3—224no—
277C22×D3—432no—
278C22×D3—416no—
279C22×D3C2518yes—
280C22×D3C2514yes—
281C2×C3×C4—224no—
282C2×C3×C4—212no—
283C2×C3×C4—224no—
284C2×C3×C4—232no—
285C2×C3×C4C2×#41124yes—
286C2×C3×C4C2518yes—
287C2×C3×C4C2514yes—
288C2×C3×C4C2518yes—
289C2×C3×C4C2514yes—
290#67—232no—
291#67—432no—
292C2×Dic3—212no—
293C2×Dic3—212no—
294C2×Dic3—224no—
295C2×Dic3C2×#41124yes—
296C2×Dic3C2514yes—
297C2×Dic3C2518yes—
298C2×Dic3C2514yes—
299C2×Dic3C2514yes—
300D12—232no—
301D12—432no—
302C4×D3—232no—
303C4×D3—264no—
304Dic6—432no—
305C24—24no—
306C24—24no—
307C24—24no—
308C24—36no—
309C24—312no—
310C24—64no—
311C24—68no—
312C24—68no—
313C24—64no—
314C24—64no—
315C24—68no—
316C24C23×D316yes—
317C24C23×D311yescharacteristic
318C2×Q8—332no—
319C2×Q8—1248no—
320C2×D4—316no—
321C2×D4—332no—
322C2×D4—316no—
323C2×D4—332no—
324C2×D4—648no—
325C2×D4—648no—
326C2×D4—1248no—
327C2×D4—1248no—
328C22×C4—24no—
329C22×C4—28no—
330C22×C4—28no—
331C22×C4—248no—
332C22×C4—38no—
333C22×C4—36no—
334C22×C4—36no—
335C22×C4—36no—
336C22×C4—312no—
337C22×C4—332no—
338C22×C4—316no—
339C22×C4—696no—
340C22×C4—64no—
341C22×C4—68no—
342C22×C4—64no—
343C22×C4—648no—
344C22×C4#23518yes—
345C22×C4C23×D3112yes—
346C22×C4C23×D316yes—
347C22×C4C23×D3112yes—
348C22×C4C23×D3116yes—
349C22×C4C23×D316yes—
350C4⋊C4—316no—
351C4⋊C4—332no—
352C4⋊C4—316no—
353C4⋊C4C23×D3116yes—
354C4⋊C4C23×D3116yes—
355C22⋊C4—316no—
356C22⋊C4—332no—
357C22⋊C4—332no—
358C22⋊C4—332no—
359C22⋊C4—332no—
360C22⋊C4—316no—
361C22⋊C4C23×D3132yes—
362C42—332no—
363C42—316no—
364C42C23×D3132yes—
365C22×C3—24no—
366C22×C3—24no—
367C22×C3—24no—
368C22×C3—424no—
369C22×C3#555116yescomplemented
370C22×C3C22×#4116yes—
371C22×C3C22×#4116yes—
372C22×C3C2×H(5,𝔽2)16yes—
373C22×C3C2611yescharacteristic
374C2×D3—24no—
375C2×D3—28no—
376C2×D3—28no—
377C2×D3—24no—
378C2×D3—24no—
379C2×D3—28no—
380C2×D3—448no—
381C2×D3—424no—
382C3×C4—28no—
383C3×C4—24no—
384C3×C4—28no—
385C3×C4C22×#4118yes—
386Dic3—24no—
387Dic3—24no—
388Dic3—28no—
389Dic3C22×#4118yes—
390C23—212no—
391C23—212no—
392C23—212no—
393C23—34no—
394C23—38no—
395C23—416no—
396C23—612no—
397C23—624no—
398C23—624no—
399C23—612no—
400C23—612no—
401C23—624no—
402C23—1216no—
403C23—1232no—
404C23C2×#23514yes—
405C23D3×#4114yes—
406C23#100914yes—
407C23C24×D314yes—
408C23C24×D313yes—
409Q8—1232no—
410D4—632no—
411D4—632no—
412D4—1232no—
413D4—1232no—
414C2×C4—212no—
415C2×C4—224no—
416C2×C4—224no—
417C2×C4—232no—
418C2×C4—324no—
419C2×C4—34no—
420C2×C4—34no—
421C2×C4—34no—
422C2×C4—38no—
423C2×C4—664no—
424C2×C4—612no—
425C2×C4—624no—
426C2×C4—612no—
427C2×C4—632no—
428C2×C4C2×#235124yes—
429C2×C4C24×D318yes—
430C2×C4C24×D314yes—
431C2×C4C24×D318yes—
432C2×C4C24×D314yes—
433C2×C3—44no—
434C2×C3C2×#555112yescomplemented
435C2×C3C23×#4111yescharacteristic
436C2×C3C23×#4111yescharacteristic
437C2×C3C22×H(5,𝔽2)11yescharacteristic
438D3—48no—
439D3—44no—
440C22—24no—
441C22—24no—
442C22—24no—
443C22—424no—
444C22—64no—
445C22—68no—
446C22—68no—
447C22—64no—
448C22—64no—
449C22—68no—
450C22—1224no—
451C22—1248no—
452C22#5532116yesdirect product factor
453C22C22×#23516yes—
454C22C2×D3×#4116yes—
455C22C2×#100916yes—
456C22C25×D311yescharacteristic
457C4—24no—
458C4—28no—
459C4—28no—
460C4—38no—
461C4—64no—
462C4—68no—
463C4—64no—
464C4C22×#23518yes—
465C3C22×#55511yescharacteristic, complemented, Sylow
466C2—44no—
467C2—124no—
468C2—128no—
469C2C2×#5532112yesdirect product factor
470C2C23×#23511yescharacteristic
471C2C22×D3×#4111yescharacteristic
472C2C22×#100911yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC22×C3C26
FrattiniC22C25×D3
FittingC22×C3×#117C2
socleC24×C3C24
hypercenterC22×#117D3
hypocenterC3C22×#555
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
1. C22×C3×#117C2semidirect
2. C22×#930C2semidirect
3. C22×C4×#67C2semidirect
4. C2×#5532C2direct (N × H)
5. C22×#895C2semidirect
6. C22×#888C2semidirect
7. C22×#893C2semidirect
8. C22×#892C2semidirect
9. C22×#884C2semidirect
10. C22×#885C2semidirect
11. C22×#875C2semidirect
12. C22×#872C2semidirect
22. C2×C3×#117C22semidirect
28. C2×#930C22semidirect
29. C2×C4×#67C22semidirect
40. #5532C22direct (N × H)
41. C2×#895C22semidirect
42. C2×#893C22semidirect
43. C2×#892C22semidirect
44. C2×#888C22semidirect
45. C2×#885C22semidirect
46. C2×#884C22semidirect
47. C2×#875C22semidirect
48. C2×#872C22semidirect
60. C22×#117D3semidirect
88. C3×#117C23semidirect
94. #930C23semidirect
95. C4×#67C23semidirect
106. #895C23semidirect
107. #893C23semidirect
108. #892C23semidirect
109. #888C23semidirect
110. #885C23semidirect
111. #884C23semidirect
112. #875C23semidirect
113. #872C23semidirect
133. C2×#117C2×D3semidirect
246. #117C22×D3semidirect
369. C22×C3#555semidirect
434. C2×C3C2×#555semidirect
452. C22#5532direct (N × H)
465. C3C22×#555semidirect
469. C2C2×#5532direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1172
C22×#9302
C22×C4×#672
C2×#55322
C22×#8952
C22×#8882
C22×#8932
C22×#8922
C22×#8842
C22×#8852
C22×#8752
C22×#8722
C22×#5553