Finite Group Databasebeta

← C22×#5507 #11702131

Subgroups — C22×#5507 #11702131

Order 768.

12230 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#235C212yesmaximal, complemented
2C23×C3×C4⋊C4C211yesmaximal, characteristic, complemented
3C22×#929C212yesmaximal, complemented
4C22×#930C211yesmaximal, characteristic, complemented
5C22×C4×#67C211yesmaximal, characteristic, complemented
6C2×#5507C2148yesmaximal, direct product factor
7C22×#894C212yesmaximal, complemented
8C22×#896C212yesmaximal, complemented
9C22×#895C211yesmaximal, characteristic, complemented
10C22×D3×C4⋊C4C211yesmaximal, characteristic, complemented
11C22×#888C211yesmaximal, characteristic, complemented
12C22×#893C211yesmaximal, characteristic, complemented
13C22×#547—31nomaximal, Sylow
14C24×C3×C4C2212yes—
15C24×C3×C4C2211yescharacteristic
16C23×#67C2212yes—
17C23×#67C2211yescharacteristic
18C22×#235—232no—
19C22×#235C22148yescomplemented
20C23×D12C2212yes—
21C23×D12C2211yescharacteristic
22C23×C4×D3C2212yes—
23C23×C4×D3C2212yes—
24C23×C4×D3C2211yescharacteristic
25C23×Dic6C2212yes—
26C22×C3×C4⋊C4C2211yescharacteristic
27C22×C3×C4⋊C4C2211yescharacteristic
28C22×C3×C4⋊C4C2212yes—
29C22×C3×C4⋊C4C22124yescomplemented
30C22×#217C2211yescharacteristic
31C2×#930C22124yescomplemented
32C2×#929C22148yescomplemented
33C2×C4×#67C22124yescomplemented
34C22×(C2×D3)⋊C4C2211yescharacteristic
35C22×(C2×D3)⋊C4C2212yes—
36C22×(C2×D3)⋊C4C2212yes—
37C22×(C2×D3)⋊C4C2212yes—
38C22×C4⋊Dic3C2211yescharacteristic
39C22×Dic3⋊C4C2211yescharacteristic
40C22×Dic3⋊C4C2212yes—
41C22×Dic3⋊C4C2212yes—
42C22×C4×Dic3C2211yescharacteristic
43C2×#896C22148yescomplemented
44#5507C221256yesdirect product factor
45C2×#895C22124yescomplemented
46C2×#894C22148yescomplemented
47C2×#893C22124yescomplemented
48C2×D3×C4⋊C4C22124yescomplemented
49C2×#888C22124yescomplemented
50C23×#41—32no—
51C2×#547—348no—
52C22×C4⋊Q8—31no—
53C22×#123—32no—
54C22×#123—32no—
55C22×C22⋊Q8—32no—
56C22×#121—31no—
57C22×#121—31no—
58C22×C4×D4—31no—
59C22×C4×D4—31no—
60C23×C4⋊C4—31no—
61C23×C4⋊C4D311yescharacteristic, complemented
62C25×C3C2311yescharacteristic
63C24×D3C2312yes—
64C24×D3C2311yescharacteristic
65C23×C3×C4—216no—
66C23×C3×C4—28no—
67C23×C3×C4C2312yes—
68C23×C3×C4C2312yes—
69C23×C3×C4C23124yes—
70C23×C3×C4C2311yescharacteristic
71C23×C3×C4C23112yes—
72C23×C3×C4C2311yescharacteristic
73C22×#67—216no—
74C22×#67D4116yescomplemented
75C22×#67C23124yes—
76C22×#67C23112yes—
77C23×Dic3C2312yes—
78C23×Dic3C2311yescharacteristic
79C23×Dic3C2311yescharacteristic
80C23×Dic3C2311yescharacteristic
81C2×#235—2384no—
82C2×#235C231128yescomplemented
83C22×D12—216no—
84C22×D12—28no—
85C22×D12C23124yes—
86C22×D12C23112yes—
87C22×C4×D3—216no—
88C22×C4×D3—28no—
89C22×C4×D3—216no—
90C22×C4×D3C23124yes—
91C22×C4×D3C23124yes—
92C22×C4×D3C23112yes—
93C22×Dic6—216no—
94C22×Dic6C23124yes—
95C2×C3×C4⋊C4C23112yes—
96C2×C3×C4⋊C4C23112yes—
97C2×C3×C4⋊C4C23124yes—
98C2×C3×C4⋊C4C23164yescomplemented
99C2×#217C23112yes—
100#930C23164yescomplemented
101#929C231128yescomplemented
102C4×#67C23164yescomplemented
103C2×(C2×D3)⋊C4C23124yes—
104C2×(C2×D3)⋊C4C23112yes—
105C2×(C2×D3)⋊C4C23124yes—
106C2×(C2×D3)⋊C4C23124yes—
107C2×C4⋊Dic3C23112yes—
108C2×Dic3⋊C4C23124yes—
109C2×Dic3⋊C4C23124yes—
110C2×Dic3⋊C4C23112yes—
111C2×C4×Dic3C23112yes—
112#896C231128yescomplemented
113#895C23164yescomplemented
114#894C231128yescomplemented
115#893C23164yescomplemented
116D3×C4⋊C4C23164yescomplemented
117#888C23164yescomplemented
118C22×#41—348no—
119C22×#41—632no—
120C23×Q8—32no—
121C23×D4—31no—
122C23×D4—32no—
123C23×D4—32no—
124C23×D4—31no—
125C24×C4—32no—
126C24×C4—32no—
127C24×C4—31no—
128C24×C4C2×D312yes—
129C24×C4C2×D311yescharacteristic
130#547—3256no—
131C2×C4⋊Q8—324no—
132C2×#123—348no—
133C2×#123—348no—
134C2×C22⋊Q8—348no—
135C2×#121—324no—
136C2×#121—324no—
137C2×C4×D4—324no—
138C2×C4×D4—324no—
139C22×C4⋊C4—31no—
140C22×C4⋊C4—32no—
141C22×C4⋊C4—31no—
142C22×C4⋊C4—32no—
143C22×C4⋊C4—324no—
144C22×C4⋊C4C2×D311yescharacteristic
145C22×C4⋊C4C2×D311yescharacteristic
146C22×C4⋊C4C2×D312yes—
147C22×C4⋊C4C2×D3124yescomplemented
148C22×C22⋊C4—31no—
149C22×C22⋊C4—31no—
150C22×C22⋊C4—32no—
151C22×C22⋊C4—32no—
152C22×C22⋊C4—32no—
153C22×C42—31no—
154C24×C3—24no—
155C24×C3—24no—
156C24×C3C2×D418yes—
157C24×C3C2416yes—
158C24×C3C2411yescharacteristic
159C23×D3—24no—
160C23×D3—24no—
161C23×D3—28no—
162C23×D3—28no—
163C23×D3—28no—
164C23×D3C2×D418yes—
165C23×D3C24112yes—
166C23×D3C2416yes—
167C22×C3×C4—28no—
168C22×C3×C4—28no—
169C22×C3×C4—296no—
170C22×C3×C4—24no—
171C22×C3×C4—248no—
172C22×C3×C4#4118yescomplemented
173C22×C3×C4C2416yes—
174C22×C3×C4C24112yes—
175C22×C3×C4C24112yes—
176C22×C3×C4C24116yes—
177C22×C3×C4C24132yes—
178C22×C3×C4C2416yes—
179C2×#67—296no—
180C2×#67C2×D4196yescomplemented
181C2×#67C24132yes—
182C2×#67C24116yes—
183C22×Dic3—28no—
184C22×Dic3—24no—
185C22×Dic3—24no—
186C22×Dic3C2×D418yes—
187C22×Dic3C2416yes—
188C22×Dic3C2416yes—
189C22×Dic3C24112yes—
190C22×Dic3C2416yes—
191#235—2512no—
192C2×D12—296no—
193C2×D12—248no—
194C2×D12C24116yes—
195C2×D12C24132yes—
196C2×C4×D3—296no—
197C2×C4×D3—248no—
198C2×C4×D3—296no—
199C2×C4×D3C24132yes—
200C2×C4×D3C24116yes—
201C2×C4×D3C24132yes—
202C2×Dic6—296no—
203C2×Dic6C24132yes—
204C3×C4⋊C4C24116yes—
205C3×C4⋊C4C24116yes—
206C3×C4⋊C4C24132yes—
207#217C24116yes—
208(C2×D3)⋊C4C24132yes—
209(C2×D3)⋊C4C24116yes—
210(C2×D3)⋊C4C24132yes—
211(C2×D3)⋊C4C24132yes—
212C4⋊Dic3C24116yes—
213Dic3⋊C4C24132yes—
214Dic3⋊C4C24132yes—
215Dic3⋊C4C24116yes—
216C4×Dic3C24116yes—
217C25—31no—
218C25—32no—
219C25C22×D311yescharacteristic
220C2×#41—3128no—
221C2×#41—6384no—
222C22×Q8—324no—
223C22×Q8—616no—
224C22×D4—316no—
225C22×D4—312no—
226C22×D4—324no—
227C22×D4—312no—
228C22×D4—324no—
229C22×D4—616no—
230C22×D4—616no—
231C22×D4—68no—
232C23×C4—216no—
233C23×C4—28no—
234C23×C4—31no—
235C23×C4—31no—
236C23×C4—31no—
237C23×C4—32no—
238C23×C4—312no—
239C23×C4—324no—
240C23×C4—324no—
241C23×C4—616no—
242C23×C4—616no—
243C23×C4—68no—
244C23×C4C22×D3112yes—
245C23×C4C22×D3124yes—
246C23×C4C22×D311yescharacteristic
247C23×C4C22×D312yes—
248C23×C4C22×D312yes—
249C23×C4C22×D311yescharacteristic
250C4⋊Q8—364no—
251#123—3128no—
252#123—3128no—
253C22⋊Q8—3128no—
254#121—364no—
255#121—364no—
256C4×D4—364no—
257C4×D4—364no—
258C2×C4⋊C4—312no—
259C2×C4⋊C4—324no—
260C2×C4⋊C4—312no—
261C2×C4⋊C4—324no—
262C2×C4⋊C4—364no—
263C2×C4⋊C4C22×D3112yes—
264C2×C4⋊C4C22×D3112yes—
265C2×C4⋊C4C22×D3124yes—
266C2×C4⋊C4C22×D3164yescomplemented
267C2×C22⋊C4—312no—
268C2×C22⋊C4—312no—
269C2×C22⋊C4—324no—
270C2×C22⋊C4—324no—
271C2×C22⋊C4—324no—
272C2×C42—312no—
273C23×C3—232no—
274C23×C3—212no—
275C23×C3—212no—
276C23×C3C22×D4124yes—
277C23×C3C22×D414yes—
278C23×C3C2×#4114yes—
279C23×C32−1+414yes—
280C23×C3C2514yes—
281C23×C3C2513yes—
282C22×D3—232no—
283C22×D3—212no—
284C22×D3—212no—
285C22×D3—224no—
286C22×D3—224no—
287C22×D3—224no—
288C22×D3—432no—
289C22×D3C22×D4124yes—
290C22×D3C2518yes—
291C22×D3C2514yes—
292C2×C3×C4—224no—
293C2×C3×C4—224no—
294C2×C3×C4—264no—
295C2×C3×C4—212no—
296C2×C3×C4—232no—
297C2×C3×C4C2×#41124yescomplemented
298C2×C3×C4C2514yes—
299C2×C3×C4C2518yes—
300C2×C3×C4C2518yes—
301C2×C3×C4C2514yes—
302#67—264no—
303#67C22×D4164yescomplemented
304C2×Dic3—224no—
305C2×Dic3—212no—
306C2×Dic3—212no—
307C2×Dic3C22×D4124yes—
308C2×Dic3C2514yes—
309C2×Dic3C2514yes—
310C2×Dic3C2518yes—
311C2×Dic3C2514yes—
312D12—264no—
313D12—232no—
314C4×D3—264no—
315C4×D3—232no—
316C4×D3—264no—
317Dic6—264no—
318C24—24no—
319C24—24no—
320C24—38no—
321C24—36no—
322C24—312no—
323C24—64no—
324C24—64no—
325C24—68no—
326C24—68no—
327C24—68no—
328C24D3×D418yes—
329C24C23×D316yes—
330C24C23×D311yescharacteristic
331#41—6512no—
332C2×Q8—332no—
333C2×Q8—696no—
334C2×D4—396no—
335C2×D4—316no—
336C2×D4—332no—
337C2×D4—316no—
338C2×D4—332no—
339C2×D4—696no—
340C2×D4—696no—
341C2×D4—648no—
342C22×C4—24no—
343C22×C4—28no—
344C22×C4—28no—
345C22×C4—296no—
346C22×C4—248no—
347C22×C4—38no—
348C22×C4—36no—
349C22×C4—36no—
350C22×C4—36no—
351C22×C4—312no—
352C22×C4—316no—
353C22×C4—332no—
354C22×C4—332no—
355C22×C4—696no—
356C22×C4—696no—
357C22×C4—64no—
358C22×C4—68no—
359C22×C4—64no—
360C22×C4—648no—
361C22×C4#23518yescomplemented
362C22×C4C23×D316yes—
363C22×C4C23×D3112yes—
364C22×C4C23×D3112yes—
365C22×C4C23×D3116yes—
366C22×C4C23×D3132yes—
367C22×C4C23×D316yes—
368C4⋊C4—316no—
369C4⋊C4—332no—
370C4⋊C4—316no—
371C4⋊C4—332no—
372C4⋊C4C23×D3116yes—
373C4⋊C4C23×D3116yes—
374C4⋊C4C23×D3132yes—
375C22⋊C4—316no—
376C22⋊C4—316no—
377C22⋊C4—332no—
378C22⋊C4—332no—
379C22⋊C4—332no—
380C42—316no—
381C22×C3—24no—
382C22×C3—24no—
383C22×C3—248no—
384C22×C3#547116yescomplemented
385C22×C3C23×D418yes—
386C22×C3C23×D416yes—
387C22×C3C22×#4116yes—
388C22×C3C2×2−1+416yes—
389C22×C3C2611yescharacteristic
390C2×D3—24no—
391C2×D3—24no—
392C2×D3—248no—
393C2×D3—28no—
394C2×D3—28no—
395C2×D3—28no—
396C2×D3—448no—
397C2×D3C23×D418yes—
398C3×C4—28no—
399C3×C4—28no—
400C3×C4—24no—
401C3×C4C22×#4118yescomplemented
402Dic3—28no—
403Dic3—24no—
404Dic3—24no—
405Dic3C23×D418yes—
406C23—212no—
407C23—212no—
408C23—232no—
409C23—324no—
410C23—34no—
411C23—38no—
412C23—612no—
413C23—612no—
414C23—632no—
415C23—624no—
416C23—624no—
417C23—624no—
418C23—1232no—
419C23C2×#23514yes—
420C23C2×D3×D4124yes—
421C23C2×D3×D414yes—
422C23#100714yes—
423C23C24×D314yes—
424C23C24×D313yes—
425Q8—664no—
426D4—364no—
427D4—664no—
428D4—664no—
429D4—632no—
430C2×C4—212no—
431C2×C4—224no—
432C2×C4—224no—
433C2×C4—264no—
434C2×C4—232no—
435C2×C4—324no—
436C2×C4—34no—
437C2×C4—34no—
438C2×C4—34no—
439C2×C4—38no—
440C2×C4—664no—
441C2×C4—664no—
442C2×C4—612no—
443C2×C4—624no—
444C2×C4—612no—
445C2×C4—632no—
446C2×C4C2×#235124yescomplemented
447C2×C4C24×D314yes—
448C2×C4C24×D318yes—
449C2×C4C24×D318yes—
450C2×C4C24×D314yes—
451C2×C3—28no—
452C2×C3C2×#547112yescomplemented
453C2×C3C24×D411yescharacteristic
454C2×C3C23×#4111yescharacteristic
455C2×C3C22×2−1+411yescharacteristic
456D3—28no—
457D3—48no—
458C22—24no—
459C22—24no—
460C22—248no—
461C22—38no—
462C22—64no—
463C22—64no—
464C22—648no—
465C22—68no—
466C22—68no—
467C22—68no—
468C22—1248no—
469C22#5507116yesdirect product factor
470C22C22×#23516yes—
471C22C22×D3×D418yes—
472C22C22×D3×D416yes—
473C22C2×#100716yes—
474C22C25×D311yescharacteristic
475C4—24no—
476C4—28no—
477C4—28no—
478C4—38no—
479C4—64no—
480C4—68no—
481C4—64no—
482C4C22×#23518yescomplemented
483C3C22×#54711yescharacteristic, complemented, Sylow
484C2—28no—
485C2—68no—
486C2—128no—
487C2C2×#5507112yesdirect product factor
488C2C23×#23511yescharacteristic
489C2C23×D3×D411yescharacteristic
490C2C22×#100711yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC22×C3C26
FrattiniC22C25×D3
FittingC23×C3×C4⋊C4C2
socleC24×C3C24
hypercenterC23×C4⋊C4D3
hypocenterC3C22×#547
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C24 ◁ C23×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C23×#235C2semidirect
2. C23×C3×C4⋊C4C2semidirect
3. C22×#929C2semidirect
4. C22×#930C2semidirect
5. C22×C4×#67C2semidirect
6. C2×#5507C2direct (N × H)
7. C22×#894C2semidirect
8. C22×#896C2semidirect
9. C22×#895C2semidirect
10. C22×D3×C4⋊C4C2semidirect
11. C22×#888C2semidirect
12. C22×#893C2semidirect
19. C22×#235C22semidirect
29. C22×C3×C4⋊C4C22semidirect
31. C2×#930C22semidirect
32. C2×#929C22semidirect
33. C2×C4×#67C22semidirect
43. C2×#896C22semidirect
44. #5507C22direct (N × H)
45. C2×#895C22semidirect
46. C2×#894C22semidirect
47. C2×#893C22semidirect
48. C2×D3×C4⋊C4C22semidirect
49. C2×#888C22semidirect
61. C23×C4⋊C4D3semidirect
74. C22×#67D4semidirect
82. C2×#235C23semidirect
98. C2×C3×C4⋊C4C23semidirect
100. #930C23semidirect
101. #929C23semidirect
102. C4×#67C23semidirect
112. #896C23semidirect
113. #895C23semidirect
114. #894C23semidirect
115. #893C23semidirect
116. D3×C4⋊C4C23semidirect
117. #888C23semidirect
147. C22×C4⋊C4C2×D3semidirect
172. C22×C3×C4#41semidirect
180. C2×#67C2×D4semidirect
266. C2×C4⋊C4C22×D3semidirect
297. C2×C3×C4C2×#41semidirect
303. #67C22×D4semidirect
361. C22×C4#235semidirect
384. C22×C3#547semidirect
401. C3×C4C22×#41semidirect
446. C2×C4C2×#235semidirect
452. C2×C3C2×#547semidirect
469. C22#5507direct (N × H)
482. C4C22×#235semidirect
483. C3C22×#547semidirect
487. C2C2×#5507direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#2352
C23×C3×C4⋊C42
C22×#9292
C22×#9302
C22×C4×#672
C2×#55072
C22×#8942
C22×#8962
C22×#8952
C22×D3×C4⋊C42
C22×#8882
C22×#8932
C22×#5473