Finite Group Databasebeta

← C22×#5616 #11702205

Subgroups — C22×#5616 #11702205

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×#121C211yesmaximal, characteristic, complemented
2C22×#938C211yesmaximal, characteristic, complemented
3C22×#936C212yesmaximal, complemented
4C22×#939C212yesmaximal, complemented
5C22×D4×Dic3C211yesmaximal, characteristic, complemented
6C22×C4×#67C211yesmaximal, characteristic, complemented
7C22×#926C211yesmaximal, characteristic, complemented
8C2×#5616C2148yesmaximal, direct product factor
9C22×#888C211yesmaximal, characteristic, complemented
10C22×#892C211yesmaximal, characteristic, complemented
11C22×#885C212yesmaximal, complemented
12C22×#877C212yesmaximal, complemented
13C22×#555—31nomaximal, Sylow
14C23×C3×D4C2211yescharacteristic
15C23×C3×D4C2212yes—
16C24×C3×C4C2211yescharacteristic
17C23×#67C2211yescharacteristic
18C23×#67C2212yes—
19C24×Dic3C2212yes—
20C23×C4×D3C2211yescharacteristic
21C23×Dic6C2212yes—
22C2×C3×#121C22124yescomplemented
23C22×C3×C4⋊C4C2211yescharacteristic
24C22×C3×C22⋊C4C2212yes—
25C22×#217C2212yes—
26C22×#217C2212yes—
27C22×#217C2211yescharacteristic
28C22×#217C2212yes—
29C2×#939C22148yescomplemented
30C2×#938C22124yescomplemented
31C2×#936C22148yescomplemented
32C2×D4×Dic3C22124yescomplemented
33C2×C4×#67C22124yescomplemented
34C22×(C2×D3)⋊C4C2211yescharacteristic
35C22×(C2×D3)⋊C4C2212yes—
36C2×#926C22124yescomplemented
37C22×C4⋊Dic3C2211yescharacteristic
38C22×C4⋊Dic3C2211yescharacteristic
39C22×Dic3⋊C4C2211yescharacteristic
40C22×Dic3⋊C4C2212yes—
41C22×C4×Dic3C2212yes—
42C22×C4×Dic3C2212yes—
43C22×C4×Dic3C2211yescharacteristic
44#5616C221256yesdirect product factor
45C2×#892C22124yescomplemented
46C2×#888C22124yescomplemented
47C2×#885C22148yescomplemented
48C2×#877C22148yescomplemented
49C2×#555—348no—
50C22×C4⋊Q8—31no—
51C22×#124—32no—
52C22×#124—32no—
53C22×#121—32no—
54C22×#121—31no—
55C22×#121D311yescharacteristic, complemented
56C22×C4×D4—31no—
57C22×C4×D4—31no—
58C22×#117—31no—
59C22×#117—31no—
60C22×#117—32no—
61C25×C3C2312yes—
62C25×C3C2311yescharacteristic
63C24×D3C2311yescharacteristic
64C22×C3×D4—28no—
65C22×C3×D4—48no—
66C22×C3×D4C23112yes—
67C22×C3×D4C23124yes—
68C23×C3×C4—28no—
69C23×C3×C4C2312yes—
70C23×C3×C4C2311yescharacteristic
71C23×C3×C4C23112yes—
72C23×C3×C4C2311yescharacteristic
73C22×#67—28no—
74C22×#67—48no—
75C22×#67C23112yes—
76C22×#67C23124yes—
77C23×Dic3—216no—
78C23×Dic3C2311yescharacteristic
79C23×Dic3C2312yes—
80C23×Dic3C2311yescharacteristic
81C23×Dic3C23124yes—
82C23×Dic3C2312yes—
83C23×Dic3C2311yescharacteristic
84C22×C4×D3—28no—
85C22×C4×D3C23112yes—
86C22×Dic6—48no—
87C22×Dic6C23124yes—
88C3×#121C23164yescomplemented
89C2×C3×C4⋊C4C23112yes—
90C2×C3×C22⋊C4C23124yes—
91C2×#217C23124yes—
92C2×#217C23124yes—
93C2×#217C23124yes—
94C2×#217C23112yes—
95#939C231128yescomplemented
96#938C23164yescomplemented
97#936C231128yescomplemented
98D4×Dic3C23164yescomplemented
99C4×#67C23164yescomplemented
100C2×(C2×D3)⋊C4C23112yes—
101C2×(C2×D3)⋊C4C23124yes—
102#926C23164yescomplemented
103C2×C4⋊Dic3C23112yes—
104C2×C4⋊Dic3C23112yes—
105C2×Dic3⋊C4C23112yes—
106C2×Dic3⋊C4C23124yes—
107C2×C4×Dic3C23124yes—
108C2×C4×Dic3C23124yes—
109C2×C4×Dic3C23112yes—
110#892C23164yescomplemented
111#888C23164yescomplemented
112#885C231128yescomplemented
113#877C231128yescomplemented
114C23×Q8—32no—
115C23×D4—31no—
116C23×D4—32no—
117C23×D4C2×D311yescharacteristic
118C23×D4C2×D312yes—
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—348no—
128C2×#121C2×D3124yescomplemented
129C2×C4×D4—324no—
130C2×C4×D4—324no—
131C2×#117—324no—
132C2×#117—324no—
133C2×#117—348no—
134C22×C4⋊C4—31no—
135C22×C4⋊C4—31no—
136C22×C4⋊C4—32no—
137C22×C4⋊C4—31no—
138C22×C4⋊C4C2×D311yescharacteristic
139C22×C22⋊C4—32no—
140C22×C22⋊C4—32no—
141C22×C22⋊C4—31no—
142C22×C22⋊C4—32no—
143C22×C22⋊C4—32no—
144C22×C22⋊C4—31no—
145C22×C22⋊C4C2×D312yes—
146C22×C42—32no—
147C22×C42—32no—
148C22×C42—31no—
149C24×C3—28no—
150C24×C3—24no—
151C24×C3—24no—
152C24×C3—28no—
153C24×C3—28no—
154C24×C3—24no—
155C24×C3C2416yes—
156C24×C3C24112yes—
157C24×C3C2411yescharacteristic
158C23×D3—24no—
159C23×D3—24no—
160C23×D3—24no—
161C23×D3C2416yes—
162C2×C3×D4—248no—
163C2×C3×D4—448no—
164C2×C3×D4C24116yes—
165C2×C3×D4C24132yes—
166C22×C3×C4—24no—
167C22×C3×C4—28no—
168C22×C3×C4—248no—
169C22×C3×C4#4118yes—
170C22×C3×C4C2416yes—
171C22×C3×C4C24112yes—
172C22×C3×C4C24116yes—
173C22×C3×C4C2416yes—
174C2×#67—248no—
175C2×#67—448no—
176C2×#67C24116yes—
177C2×#67C24132yes—
178C22×Dic3—24no—
179C22×Dic3—28no—
180C22×Dic3—24no—
181C22×Dic3—28no—
182C22×Dic3—296no—
183C22×Dic3#4118yes—
184C22×Dic3C24132yes—
185C22×Dic3C2416yes—
186C22×Dic3C24112yes—
187C22×Dic3C24112yes—
188C22×Dic3C2416yes—
189C22×Dic3C2416yes—
190C2×C4×D3—248no—
191C2×C4×D3C24116yes—
192C2×Dic6—448no—
193C2×Dic6C24132yes—
194C3×C4⋊C4C24116yes—
195C3×C22⋊C4C24132yes—
196#217C24132yes—
197#217C24132yes—
198#217C24132yes—
199#217C24116yes—
200(C2×D3)⋊C4C24116yes—
201(C2×D3)⋊C4C24132yes—
202C4⋊Dic3C24116yes—
203C4⋊Dic3C24116yes—
204Dic3⋊C4C24116yes—
205Dic3⋊C4C24132yes—
206C4×Dic3C24132yes—
207C4×Dic3C24132yes—
208C4×Dic3C24116yes—
209C25—31no—
210C25C22×D311yescharacteristic
211C25C22×D312yes—
212C22×Q8—324no—
213C22×Q8—128no—
214C22×D4—28no—
215C22×D4—312no—
216C22×D4—324no—
217C22×D4—48no—
218C22×D4—68no—
219C22×D4—128no—
220C22×D4C22×D3112yes—
221C22×D4C22×D3124yes—
222C23×C4—28no—
223C23×C4—31no—
224C23×C4—324no—
225C23×C4—31no—
226C23×C4—31no—
227C23×C4—32no—
228C23×C4—32no—
229C23×C4—312no—
230C23×C4—616no—
231C23×C4—68no—
232C23×C4C22×D3112yes—
233C23×C4C22×D311yescharacteristic
234C23×C4C22×D312yes—
235C23×C4C22×D311yescharacteristic
236C4⋊Q8—364no—
237#124—3128no—
238#124—3128no—
239#121—364no—
240#121—3128no—
241#121C22×D3164yescomplemented
242C4×D4—364no—
243C4×D4—364no—
244#117—364no—
245#117—364no—
246#117—3128no—
247C2×C4⋊C4—312no—
248C2×C4⋊C4—312no—
249C2×C4⋊C4—324no—
250C2×C4⋊C4—312no—
251C2×C4⋊C4C22×D3112yes—
252C2×C22⋊C4—324no—
253C2×C22⋊C4—324no—
254C2×C22⋊C4—312no—
255C2×C22⋊C4—324no—
256C2×C22⋊C4—324no—
257C2×C22⋊C4—312no—
258C2×C22⋊C4C22×D3124yes—
259C2×C42—324no—
260C2×C42—324no—
261C2×C42—312no—
262C23×C3—224no—
263C23×C3—212no—
264C23×C3—212no—
265C23×C3—224no—
266C23×C3—224no—
267C23×C3—212no—
268C23×C3—432no—
269C23×C3—416no—
270C23×C3C2×#4114yes—
271C23×C3C2×#4114yes—
272C23×C3H(5,𝔽2)14yes—
273C23×C3C2514yes—
274C23×C3C2518yes—
275C23×C3C2513yes—
276C22×D3—212no—
277C22×D3—212no—
278C22×D3—212no—
279C22×D3—416no—
280C22×D3C2514yes—
281C3×D4—232no—
282C3×D4—432no—
283C2×C3×C4—212no—
284C2×C3×C4—224no—
285C2×C3×C4—232no—
286C2×C3×C4C2×#41124yes—
287C2×C3×C4C2514yes—
288C2×C3×C4C2518yes—
289C2×C3×C4C2514yes—
290#67—232no—
291#67—432no—
292C2×Dic3—212no—
293C2×Dic3—224no—
294C2×Dic3—212no—
295C2×Dic3—224no—
296C2×Dic3—264no—
297C2×Dic3C2×#41124yes—
298C2×Dic3C2514yes—
299C2×Dic3C2518yes—
300C2×Dic3C2518yes—
301C2×Dic3C2514yes—
302C2×Dic3C2514yes—
303C4×D3—232no—
304Dic6—432no—
305C24—24no—
306C24—28no—
307C24—24no—
308C24—28no—
309C24—28no—
310C24—24no—
311C24—36no—
312C24—64no—
313C24—64no—
314C24—64no—
315C24C23×D316yes—
316C24C23×D3112yes—
317C24C23×D311yescharacteristic
318C2×Q8—332no—
319C2×Q8—1248no—
320C2×D4—248no—
321C2×D4—316no—
322C2×D4—332no—
323C2×D4—448no—
324C2×D4—648no—
325C2×D4—1248no—
326C2×D4C23×D3116yes—
327C2×D4C23×D3132yes—
328C22×C4—24no—
329C22×C4—28no—
330C22×C4—248no—
331C22×C4—38no—
332C22×C4—36no—
333C22×C4—36no—
334C22×C4—312no—
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—68no—
344C22×C4—648no—
345C22×C4#23718yes—
346C22×C4C23×D316yes—
347C22×C4C23×D3112yes—
348C22×C4C23×D3116yes—
349C22×C4C23×D316yes—
350C4⋊C4—316no—
351C4⋊C4—316no—
352C4⋊C4—332no—
353C4⋊C4—316no—
354C4⋊C4C23×D3116yes—
355C22⋊C4—332no—
356C22⋊C4—332no—
357C22⋊C4—316no—
358C22⋊C4—332no—
359C22⋊C4—332no—
360C22⋊C4—316no—
361C22⋊C4C23×D3132yes—
362C42—332no—
363C42—332no—
364C42—316no—
365C22×C3—28no—
366C22×C3—24no—
367C22×C3—24no—
368C22×C3—28no—
369C22×C3—28no—
370C22×C3—24no—
371C22×C3—448no—
372C22×C3—424no—
373C22×C3#555116yescomplemented
374C22×C3C22×#4116yes—
375C22×C3C22×#4116yes—
376C22×C3C2×H(5,𝔽2)16yes—
377C22×C3C2611yescharacteristic
378C2×D3—24no—
379C2×D3—24no—
380C2×D3—24no—
381C2×D3—424no—
382C3×C4—24no—
383C3×C4—28no—
384C3×C4C22×#4118yes—
385Dic3—24no—
386Dic3—28no—
387Dic3—24no—
388Dic3—28no—
389Dic3C22×#4118yes—
390C23—212no—
391C23—224no—
392C23—212no—
393C23—224no—
394C23—224no—
395C23—212no—
396C23—34no—
397C23—416no—
398C23—432no—
399C23—612no—
400C23—612no—
401C23—612no—
402C23—1216no—
403C23C2×#23714yes—
404C23#100414yes—
405C23D3×#4114yes—
406C23C24×D314yes—
407C23C24×D318yes—
408C23C24×D313yes—
409Q8—1232no—
410D4—232no—
411D4—432no—
412D4—632no—
413D4—1232no—
414C2×C4—212no—
415C2×C4—224no—
416C2×C4—232no—
417C2×C4—324no—
418C2×C4—34no—
419C2×C4—34no—
420C2×C4—38no—
421C2×C4—34no—
422C2×C4—38no—
423C2×C4—664no—
424C2×C4—612no—
425C2×C4—624no—
426C2×C4—612no—
427C2×C4—624no—
428C2×C4—632no—
429C2×C4C2×#237124yes—
430C2×C4C24×D314yes—
431C2×C4C24×D318yes—
432C2×C4C24×D314yes—
433C2×C3—48no—
434C2×C3—44no—
435C2×C3C2×#555112yescomplemented
436C2×C3C23×#4111yescharacteristic
437C2×C3C23×#4111yescharacteristic
438C2×C3C22×H(5,𝔽2)11yescharacteristic
439D3—44no—
440C22—24no—
441C22—28no—
442C22—24no—
443C22—28no—
444C22—28no—
445C22—24no—
446C22—424no—
447C22—448no—
448C22—64no—
449C22—64no—
450C22—64no—
451C22—1224no—
452C22#5616116yesdirect product factor
453C22C22×#23716yes—
454C22C2×#100416yes—
455C22C2×D3×#4116yes—
456C22C25×D311yescharacteristic
457C4—24no—
458C4—28no—
459C4—38no—
460C4—64no—
461C4—68no—
462C4—64no—
463C4—68no—
464C4C22×#23718yes—
465C3C22×#55511yescharacteristic, complemented, Sylow
466C2—44no—
467C2—48no—
468C2—124no—
469C2C2×#5616112yesdirect product factor
470C2C23×#23711yescharacteristic
471C2C22×#100411yescharacteristic
472C2C22×D3×#4111yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC22×C3C26
FrattiniC22C25×D3
FittingC22×C3×#121C2
socleC24×C3C24
hypercenterC22×#121D3
hypocenterC3C22×#555
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×#121

Complements

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

NH ≅ G/Ntype
1. C22×C3×#121C2semidirect
2. C22×#938C2semidirect
3. C22×#936C2semidirect
4. C22×#939C2semidirect
5. C22×D4×Dic3C2semidirect
6. C22×C4×#67C2semidirect
7. C22×#926C2semidirect
8. C2×#5616C2direct (N × H)
9. C22×#888C2semidirect
10. C22×#892C2semidirect
11. C22×#885C2semidirect
12. C22×#877C2semidirect
22. C2×C3×#121C22semidirect
29. C2×#939C22semidirect
30. C2×#938C22semidirect
31. C2×#936C22semidirect
32. C2×D4×Dic3C22semidirect
33. C2×C4×#67C22semidirect
36. C2×#926C22semidirect
44. #5616C22direct (N × H)
45. C2×#892C22semidirect
46. C2×#888C22semidirect
47. C2×#885C22semidirect
48. C2×#877C22semidirect
55. C22×#121D3semidirect
88. C3×#121C23semidirect
95. #939C23semidirect
96. #938C23semidirect
97. #936C23semidirect
98. D4×Dic3C23semidirect
99. C4×#67C23semidirect
102. #926C23semidirect
110. #892C23semidirect
111. #888C23semidirect
112. #885C23semidirect
113. #877C23semidirect
128. C2×#121C2×D3semidirect
241. #121C22×D3semidirect
373. C22×C3#555semidirect
435. C2×C3C2×#555semidirect
452. C22#5616direct (N × H)
465. C3C22×#555semidirect
469. C2C2×#5616direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1212
C22×#9382
C22×#9362
C22×#9392
C22×D4×Dic32
C22×C4×#672
C22×#9262
C2×#56162
C22×#8882
C22×#8922
C22×#8852
C22×#8772
C22×#5553