Finite Group Databasebeta

← C22×#5558 #11702194

Subgroups — C22×#5558 #11702194

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
1C22×C3×C4×D4C211yesmaximal, characteristic, complemented
2C22×#938C211yesmaximal, characteristic, complemented
3C22×#938C211yesmaximal, characteristic, complemented
4C22×#929C212yesmaximal, complemented
5C22×C4×#67C212yesmaximal, complemented
6C2×#5558C2148yesmaximal, direct product factor
7C22×D3×C4⋊C4C211yesmaximal, characteristic, complemented
8C22×#892C211yesmaximal, characteristic, complemented
9C22×#884C212yesmaximal, complemented
10C22×#879C212yesmaximal, complemented
11C22×C4×D12C211yesmaximal, characteristic, complemented
12C22×#870C211yesmaximal, characteristic, complemented
13C22×#553—31nomaximal, Sylow
14C23×C3×D4C2211yescharacteristic
15C24×C3×C4C2212yes—
16C23×#67C2212yes—
17C23×#67C2212yes—
18C23×D12C2211yescharacteristic
19C23×C4×D3C2211yescharacteristic
20C23×C4×D3C2211yescharacteristic
21C23×C4×D3C2212yes—
22C2×C3×C4×D4C22124yescomplemented
23C22×C3×C4⋊C4C2211yescharacteristic
24C22×C3×C22⋊C4C2212yes—
25C22×C3×C42C2211yescharacteristic
26C22×#217C2212yes—
27C22×#217C2212yes—
28C2×#938C22124yescomplemented
29C2×#938C22124yescomplemented
30C2×#929C22148yescomplemented
31C2×C4×#67C22148yescomplemented
32C22×(C2×D3)⋊C4C2211yescharacteristic
33C22×(C2×D3)⋊C4C2211yescharacteristic
34C22×(C2×D3)⋊C4C2212yes—
35C22×C4⋊Dic3C2211yescharacteristic
36C22×C4⋊Dic3C2211yescharacteristic
37C22×C4⋊Dic3C2211yescharacteristic
38C22×Dic3⋊C4C2212yes—
39C22×Dic3⋊C4C2212yes—
40C22×Dic3⋊C4C2212yes—
41C22×C4×Dic3C2212yes—
42#5558C221256yesdirect product factor
43C2×#892C22124yescomplemented
44C2×D3×C4⋊C4C22124yescomplemented
45C2×#884C22148yescomplemented
46C2×#879C22148yescomplemented
47C2×C4×D12C22124yescomplemented
48C2×#870C22124yescomplemented
49C2×#553—348no—
50C22×#126—32no—
51C22×#125—31no—
52C22×#123—32no—
53C22×#121—31no—
54C22×#121—31no—
55C22×#121—32no—
56C22×C4×D4—31no—
57C22×C4×D4—32no—
58C22×C4×D4D311yescharacteristic, complemented
59C22×#117—31no—
60C23×C4⋊C4—31no—
61C25×C3C2312yes—
62C24×D3C2311yescharacteristic
63C24×D3C2311yescharacteristic
64C22×C3×D4—28no—
65C22×C3×D4C23112yes—
66C23×C3×C4—48no—
67C23×C3×C4C2312yes—
68C23×C3×C4C2311yescharacteristic
69C23×C3×C4C2311yescharacteristic
70C23×C3×C4C23124yes—
71C23×C3×C4C2311yescharacteristic
72C22×#67—216no—
73C22×#67—48no—
74C22×#67C23124yes—
75C22×#67C23124yes—
76C23×Dic3C2312yes—
77C23×Dic3C2311yescharacteristic
78C23×Dic3C2311yescharacteristic
79C23×Dic3C2312yes—
80C22×D12—28no—
81C22×D12C23112yes—
82C22×C4×D3—28no—
83C22×C4×D3—28no—
84C22×C4×D3—48no—
85C22×C4×D3C23112yes—
86C22×C4×D3C23112yes—
87C22×C4×D3C23124yes—
88C3×C4×D4C23164yescomplemented
89C2×C3×C4⋊C4C23112yes—
90C2×C3×C22⋊C4C23124yes—
91C2×C3×C42C23112yes—
92C2×#217C23124yes—
93C2×#217C23124yes—
94#938C23164yescomplemented
95#938C23164yescomplemented
96#929C231128yescomplemented
97C4×#67C231128yescomplemented
98C2×(C2×D3)⋊C4C23112yes—
99C2×(C2×D3)⋊C4C23112yes—
100C2×(C2×D3)⋊C4C23124yes—
101C2×C4⋊Dic3C23112yes—
102C2×C4⋊Dic3C23112yes—
103C2×C4⋊Dic3C23112yes—
104C2×Dic3⋊C4C23124yes—
105C2×Dic3⋊C4C23124yes—
106C2×Dic3⋊C4C23124yes—
107C2×C4×Dic3C23124yes—
108#892C23164yescomplemented
109D3×C4⋊C4C23164yescomplemented
110#884C231128yescomplemented
111#879C231128yescomplemented
112C4×D12C23164yescomplemented
113#870C23164yescomplemented
114C23×D4—32no—
115C23×D4—32no—
116C23×D4—31no—
117C23×D4C2×D311yescharacteristic
118C24×C4—31no—
119C24×C4—31no—
120C24×C4—32no—
121C24×C4C2×D312yes—
122#553—3256no—
123C2×#126—348no—
124C2×#125—324no—
125C2×#123—348no—
126C2×#121—324no—
127C2×#121—324no—
128C2×#121—348no—
129C2×C4×D4—348no—
130C2×C4×D4—324no—
131C2×C4×D4C2×D3124yescomplemented
132C2×#117—324no—
133C22×C4⋊C4—31no—
134C22×C4⋊C4—32no—
135C22×C4⋊C4—31no—
136C22×C4⋊C4—32no—
137C22×C4⋊C4—31no—
138C22×C4⋊C4—32no—
139C22×C4⋊C4—324no—
140C22×C4⋊C4C2×D311yescharacteristic
141C22×C22⋊C4—32no—
142C22×C22⋊C4—32no—
143C22×C22⋊C4—32no—
144C22×C22⋊C4—31no—
145C22×C22⋊C4—31no—
146C22×C22⋊C4C2×D312yes—
147C22×C42—32no—
148C22×C42C2×D311yescharacteristic
149C24×C3—28no—
150C24×C3—28no—
151C24×C3—28no—
152C24×C3C24112yes—
153C24×C3C2411yescharacteristic
154C23×D3—24no—
155C23×D3—24no—
156C23×D3—24no—
157C23×D3—24no—
158C23×D3—24no—
159C23×D3#4118yes—
160C23×D3C2416yes—
161C23×D3C2416yes—
162C2×C3×D4—248no—
163C2×C3×D4C24116yes—
164C22×C3×C4—24no—
165C22×C3×C4—28no—
166C22×C3×C4—24no—
167C22×C3×C4—448no—
168C22×C3×C4#4118yes—
169C22×C3×C4C24112yes—
170C22×C3×C4C2416yes—
171C22×C3×C4C2416yes—
172C22×C3×C4C24132yes—
173C22×C3×C4C2416yes—
174C2×#67—296no—
175C2×#67—448no—
176C2×#67C24132yes—
177C2×#67C24132yes—
178C22×Dic3—24no—
179C22×Dic3—24no—
180C22×Dic3—28no—
181C22×Dic3—28no—
182C22×Dic3C2416yes—
183C22×Dic3C24112yes—
184C22×Dic3C24112yes—
185C22×Dic3C2416yes—
186C2×D12—248no—
187C2×D12C24116yes—
188C2×C4×D3—248no—
189C2×C4×D3—248no—
190C2×C4×D3—448no—
191C2×C4×D3C24116yes—
192C2×C4×D3C24116yes—
193C2×C4×D3C24132yes—
194C3×C4⋊C4C24116yes—
195C3×C22⋊C4C24132yes—
196C3×C42C24116yes—
197#217C24132yes—
198#217C24132yes—
199(C2×D3)⋊C4C24116yes—
200(C2×D3)⋊C4C24116yes—
201(C2×D3)⋊C4C24132yes—
202C4⋊Dic3C24116yes—
203C4⋊Dic3C24116yes—
204C4⋊Dic3C24116yes—
205Dic3⋊C4C24132yes—
206Dic3⋊C4C24132yes—
207Dic3⋊C4C24132yes—
208C4×Dic3C24132yes—
209C25—31no—
210C25—31no—
211C25C22×D312yes—
212C22×D4—28no—
213C22×D4—324no—
214C22×D4—324no—
215C22×D4—312no—
216C22×D4—68no—
217C22×D4—616no—
218C22×D4—128no—
219C22×D4C22×D3112yes—
220C23×C4—31no—
221C23×C4—31no—
222C23×C4—32no—
223C23×C4—32no—
224C23×C4—312no—
225C23×C4—312no—
226C23×C4—324no—
227C23×C4—48no—
228C23×C4—68no—
229C23×C4—68no—
230C23×C4—128no—
231C23×C4C22×D311yescharacteristic
232C23×C4C22×D312yes—
233C23×C4C22×D311yescharacteristic
234C23×C4C22×D3124yes—
235C23×C4C22×D311yescharacteristic
236#126—3128no—
237#125—364no—
238#123—3128no—
239#121—364no—
240#121—364no—
241#121—3128no—
242C4×D4—3128no—
243C4×D4—364no—
244C4×D4C22×D3164yescomplemented
245#117—364no—
246C2×C4⋊C4—312no—
247C2×C4⋊C4—324no—
248C2×C4⋊C4—312no—
249C2×C4⋊C4—324no—
250C2×C4⋊C4—312no—
251C2×C4⋊C4—324no—
252C2×C4⋊C4—364no—
253C2×C4⋊C4C22×D3112yes—
254C2×C22⋊C4—324no—
255C2×C22⋊C4—324no—
256C2×C22⋊C4—324no—
257C2×C22⋊C4—312no—
258C2×C22⋊C4—312no—
259C2×C22⋊C4C22×D3124yes—
260C2×C42—324no—
261C2×C42C22×D3112yes—
262C23×C3—224no—
263C23×C3—224no—
264C23×C3—224no—
265C23×C3—432no—
266C23×C3C2×#4114yes—
267C23×C3C2×#4114yes—
268C23×C3H(5,𝔽2)14yes—
269C23×C3C2518yes—
270C23×C3C2513yes—
271C22×D3—232no—
272C22×D3—212no—
273C22×D3—212no—
274C22×D3—212no—
275C22×D3—212no—
276C22×D3—212no—
277C22×D3—416no—
278C22×D3C2×#41124yes—
279C22×D3C2514yes—
280C22×D3C2514yes—
281C3×D4—232no—
282C2×C3×C4—212no—
283C2×C3×C4—224no—
284C2×C3×C4—212no—
285C2×C3×C4—432no—
286C2×C3×C4C2×#41124yes—
287C2×C3×C4C2518yes—
288C2×C3×C4C2514yes—
289C2×C3×C4C2514yes—
290C2×C3×C4C2514yes—
291#67—264no—
292#67—432no—
293C2×Dic3—212no—
294C2×Dic3—212no—
295C2×Dic3—224no—
296C2×Dic3—224no—
297C2×Dic3C2514yes—
298C2×Dic3C2518yes—
299C2×Dic3C2518yes—
300C2×Dic3C2514yes—
301D12—232no—
302C4×D3—232no—
303C4×D3—232no—
304C4×D3—432no—
305C24—28no—
306C24—28no—
307C24—28no—
308C24—38no—
309C24—36no—
310C24—36no—
311C24—64no—
312C24—64no—
313C24—64no—
314C24—64no—
315C24—64no—
316C24C23×D3112yes—
317C24C23×D311yescharacteristic
318C2×D4—248no—
319C2×D4—332no—
320C2×D4—332no—
321C2×D4—316no—
322C2×D4—648no—
323C2×D4—696no—
324C2×D4—1248no—
325C2×D4C23×D3116yes—
326C22×C4—24no—
327C22×C4—28no—
328C22×C4—24no—
329C22×C4—36no—
330C22×C4—312no—
331C22×C4—36no—
332C22×C4—312no—
333C22×C4—316no—
334C22×C4—316no—
335C22×C4—332no—
336C22×C4—448no—
337C22×C4—64no—
338C22×C4—64no—
339C22×C4—68no—
340C22×C4—68no—
341C22×C4—648no—
342C22×C4—648no—
343C22×C4—1248no—
344C22×C4#23518yes—
345C22×C4C23×D3112yes—
346C22×C4C23×D316yes—
347C22×C4C23×D316yes—
348C22×C4C23×D3132yes—
349C22×C4C23×D316yes—
350C4⋊C4—316no—
351C4⋊C4—332no—
352C4⋊C4—316no—
353C4⋊C4—332no—
354C4⋊C4—316no—
355C4⋊C4—332no—
356C4⋊C4C23×D3116yes—
357C22⋊C4—332no—
358C22⋊C4—332no—
359C22⋊C4—332no—
360C22⋊C4—316no—
361C22⋊C4—316no—
362C22⋊C4C23×D3132yes—
363C42—332no—
364C42C23×D3116yes—
365C22×C3—28no—
366C22×C3—28no—
367C22×C3—28no—
368C22×C3—448no—
369C22×C3#553116yescomplemented
370C22×C3C22×#4116yes—
371C22×C3C22×#4116yes—
372C22×C3C2×H(5,𝔽2)16yes—
373C22×C3C2611yescharacteristic
374C2×D3—24no—
375C2×D3—24no—
376C2×D3—248no—
377C2×D3—24no—
378C2×D3—24no—
379C2×D3—24no—
380C2×D3—424no—
381C2×D3C22×#4118yes—
382C3×C4—24no—
383C3×C4—28no—
384C3×C4—24no—
385C3×C4C22×#4118yes—
386Dic3—24no—
387Dic3—24no—
388Dic3—28no—
389Dic3—28no—
390C23—224no—
391C23—224no—
392C23—224no—
393C23—324no—
394C23—34no—
395C23—34no—
396C23—432no—
397C23—612no—
398C23—612no—
399C23—632no—
400C23—612no—
401C23—612no—
402C23—612no—
403C23—1216no—
404C23C2×#23514yes—
405C23#100414yes—
406C23D3×#4114yes—
407C23C24×D318yes—
408C23C24×D313yes—
409D4—232no—
410D4—632no—
411D4—664no—
412D4—1232no—
413C2×C4—212no—
414C2×C4—224no—
415C2×C4—212no—
416C2×C4—34no—
417C2×C4—38no—
418C2×C4—34no—
419C2×C4—38no—
420C2×C4—432no—
421C2×C4—612no—
422C2×C4—612no—
423C2×C4—624no—
424C2×C4—624no—
425C2×C4—632no—
426C2×C4—632no—
427C2×C4—1232no—
428C2×C4C2×#235124yes—
429C2×C4C24×D318yes—
430C2×C4C24×D314yes—
431C2×C4C24×D314yes—
432C2×C4C24×D314yes—
433C2×C3—48no—
434C2×C3C2×#553112yescomplemented
435C2×C3C23×#4111yescharacteristic
436C2×C3C23×#4111yescharacteristic
437C2×C3C22×H(5,𝔽2)11yescharacteristic
438D3—28no—
439D3—44no—
440C22—28no—
441C22—28no—
442C22—28no—
443C22—38no—
444C22—448no—
445C22—64no—
446C22—64no—
447C22—648no—
448C22—64no—
449C22—64no—
450C22—64no—
451C22—1224no—
452C22#5558116yesdirect product factor
453C22C22×#23516yes—
454C22C2×#100416yes—
455C22C2×D3×#4116yes—
456C22C25×D311yescharacteristic
457C4—24no—
458C4—28no—
459C4—24no—
460C4—64no—
461C4—64no—
462C4—68no—
463C4—68no—
464C4C22×#23518yes—
465C3C22×#55311yescharacteristic, complemented, Sylow
466C2—48no—
467C2—68no—
468C2—124no—
469C2C2×#5558112yesdirect product factor
470C2C23×#23511yescharacteristic
471C2C22×#100411yescharacteristic
472C2C22×D3×#4111yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC22×C3C26
FrattiniC22C25×D3
FittingC22×C3×C4×D4C2
socleC24×C3C24
hypercenterC22×C4×D4D3
hypocenterC3C22×#553
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
1. C22×C3×C4×D4C2semidirect
2. C22×#938C2semidirect
3. C22×#938C2semidirect
4. C22×#929C2semidirect
5. C22×C4×#67C2semidirect
6. C2×#5558C2direct (N × H)
7. C22×D3×C4⋊C4C2semidirect
8. C22×#892C2semidirect
9. C22×#884C2semidirect
10. C22×#879C2semidirect
11. C22×C4×D12C2semidirect
12. C22×#870C2semidirect
22. C2×C3×C4×D4C22semidirect
28. C2×#938C22semidirect
29. C2×#938C22semidirect
30. C2×#929C22semidirect
31. C2×C4×#67C22semidirect
42. #5558C22direct (N × H)
43. C2×#892C22semidirect
44. C2×D3×C4⋊C4C22semidirect
45. C2×#884C22semidirect
46. C2×#879C22semidirect
47. C2×C4×D12C22semidirect
48. C2×#870C22semidirect
58. C22×C4×D4D3semidirect
88. C3×C4×D4C23semidirect
94. #938C23semidirect
95. #938C23semidirect
96. #929C23semidirect
97. C4×#67C23semidirect
108. #892C23semidirect
109. D3×C4⋊C4C23semidirect
110. #884C23semidirect
111. #879C23semidirect
112. C4×D12C23semidirect
113. #870C23semidirect
131. C2×C4×D4C2×D3semidirect
244. C4×D4C22×D3semidirect
369. C22×C3#553semidirect
434. C2×C3C2×#553semidirect
452. C22#5558direct (N × H)
465. C3C22×#553semidirect
469. C2C2×#5558direct (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×D42
C22×#9382
C22×#9292
C22×C4×#672
C2×#55582
C22×D3×C4⋊C42
C22×#8922
C22×#8842
C22×#8792
C22×C4×D122
C22×#8702
C22×#5533