Finite Group Databasebeta

← C22×#5139 #11658779

Subgroups — C22×#5139 #11658779

Order 768.

5198 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#235C211yesmaximal, characteristic
2C23×C3×M16C211yesmaximal, characteristic, complemented
3C22×#926C211yesmaximal, characteristic, complemented
4C2×#5139C2124yesmaximal, direct product factor
5C22×#825C214yesmaximal, complemented
6C22×#421—31nomaximal, Sylow
7C24×C3×C4C2211yescharacteristic
8C23×#67—21no—
9C22×#235—21no—
10C22×#235—26no—
11C22×#235—21no—
12C22×#235—26no—
13C22×#235C412yescomplemented
14C22×#235C4112yescomplemented
15C22×#235C2212yes—
16C22×#235C22112yes—
17C23×D12C411yescharacteristic, complemented
18C23×C4×D3—21no—
19C23×Dic6C411yescharacteristic, complemented
20C22×C3×M16—21no—
21C22×C3×M16—26no—
22C22×C3×M16C2212yescomplemented
23C22×C3×M16C22112yescomplemented
24C23×C3×C8—21no—
25C22×#217—21no—
26C2×#926—26no—
27C2×#926C22112yescomplemented
28C22×C4⋊Dic3—21no—
29C22×C4×Dic3C2212yescomplemented
30#5139C22164yesdirect product factor
31C2×#825C22148yescomplemented
32C23×#41—31no—
33C23×M16D311yescharacteristic, complemented
34C22×#117—31no—
35C2×#421—324no—
36C22×#104—34no—
37C25×C3D411yescharacteristic
38C24×D3—21no—
39C23×C3×C4C2×C411yescharacteristic
40C23×C3×C4C2×C411yescharacteristic
41C23×C3×C4C2×C416yes—
42C23×C3×C4D411yescharacteristic
43C23×C3×C4D416yes—
44C23×C3×C4D411yescharacteristic
45C23×C3×C4D411yescharacteristic
46C23×C3×C4D416yes—
47C23×C3×C4C2311yescharacteristic
48C23×C3×C4C2316yes—
49C22×#67—22no—
50C22×#67—212no—
51C22×#67—22no—
52C22×#67—212no—
53C23×Dic3—21no—
54C23×Dic3—22no—
55C2×#235—212no—
56C2×#235—212no—
57C2×#235—248no—
58C2×#235—212no—
59C2×#235—26no—
60C2×#235—212no—
61C2×#235—224no—
62C2×#235—26no—
63C2×#235—212no—
64C2×#235—224no—
65C2×#235C2×C4112yes—
66C2×#235C2×C4124yescomplemented
67C2×#235C2×C4148yescomplemented
68C2×#235C23112yes—
69C2×#235C23116yes—
70C22×D12—26no—
71C22×D12—21no—
72C22×D12C2×C412yes—
73C22×D12C2×C4112yescomplemented
74C22×C4×D3—212no—
75C22×C4×D3—22no—
76C22×C4×D3—212no—
77C22×C4×D3—22no—
78C22×Dic6—26no—
79C22×Dic6—21no—
80C22×Dic6C2×C412yes—
81C22×Dic6C2×C4112yescomplemented
82C2×C3×M16—212no—
83C2×C3×M16—26no—
84C2×C3×M16—224no—
85C2×C3×M16C23112yescomplemented
86C2×C3×M16C23116yescomplemented
87C22×C3×C8—212no—
88C22×C3×C8—22no—
89C2×#217—212no—
90#926—224no—
91#926C23116yescomplemented
92C2×C4⋊Dic3—212no—
93C2×C4×Dic3—26no—
94C2×C4×Dic3C23112yescomplemented
95#825C23164yescomplemented
96C22×#41—32no—
97C22×#41—312no—
98C22×#41—32no—
99C22×#41—312no—
100C22×#41—61no—
101C22×#41—66no—
102C22×#41—66no—
103C22×#41—61no—
104C23×Q8—31no—
105C23×D4—31no—
106C23×D4—61no—
107C24×C4—61no—
108C24×C4C2×D311yescharacteristic
109C22×M16—26no—
110C22×M16—21no—
111C22×M16C2×D312yescomplemented
112C22×M16C2×D3112yescomplemented
113C23×C8—21no—
114C2×#117—312no—
115C2×#117—66no—
116C22×C4⋊C4—61no—
117C22×C22⋊C4—61no—
118C22×C42—32no—
119#421—364no—
120C2×#104—348no—
121C24×C3—44no—
122C24×C3C22⋊C411yescharacteristic
123C24×C3C22⋊C411yescharacteristic
124C24×C3C22⋊C416yes—
125C24×C3C2×D411yescharacteristic
126C24×C3C2×D416yes—
127C23×D3—22no—
128C23×D3—212no—
129C23×D3—48no—
130C22×C3×C4C22⋊C416yes—
131C22×C3×C4C22⋊C411yescharacteristic
132C22×C3×C4C22⋊C416yes—
133C22×C3×C4C22⋊C411yescharacteristic
134C22×C3×C4C22⋊C416yes—
135C22×C3×C4C22⋊C416yes—
136C22×C3×C4C22⋊C416yes—
137C22×C3×C4C22⋊C4124yes—
138C22×C3×C4C22×C411yescharacteristic
139C22×C3×C4C22×C416yes—
140C22×C3×C4C22×C413yes—
141C22×C3×C4C22×C416yes—
142C22×C3×C4C22×C4112yes—
143C22×C3×C4C2×D413yes—
144C22×C3×C4C2×D416yes—
145C22×C3×C4C2×D4112yes—
146C22×C3×C4C2×D416yes—
147C22×C3×C4C2×D416yes—
148C22×C3×C4C2×D413yes—
149C22×C3×C4C2×D4112yes—
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2413yes—
152C22×C3×C4C2414yes—
153C2×#67—212no—
154C2×#67—224no—
155C2×#67—248no—
156C2×#67—212no—
157C2×#67—216no—
158C22×Dic3—22no—
159C22×Dic3—212no—
160C22×Dic3—212no—
161C22×Dic3—48no—
162#235—224no—
163#235—248no—
164#235—248no—
165#235—248no—
166#235—212no—
167#235—216no—
168#235—212no—
169#235—216no—
170#235C22×C4124yes—
171#235C22×C4132yescomplemented
172#235C2418yes—
173C2×D12—26no—
174C2×D12—224no—
175C2×D12—212no—
176C2×D12C22×C4112yes—
177C2×D12C22×C4116yescomplemented
178C2×C4×D3—212no—
179C2×C4×D3—248no—
180C2×C4×D3—224no—
181C2×C4×D3—212no—
182C2×C4×D3—216no—
183C2×Dic6—26no—
184C2×Dic6—224no—
185C2×Dic6—212no—
186C2×Dic6C22×C4112yes—
187C2×Dic6C22×C4116yescomplemented
188C3×M16—216no—
189C3×M16—212no—
190C3×M16C2418yescomplemented
191C2×C3×C8—212no—
192C2×C3×C8—216no—
193#217—216no—
194C4⋊Dic3—216no—
195C4×Dic3—212no—
196C4×Dic3C2418yescomplemented
197C25—61no—
198C25D1211yescharacteristic
199C2×#41—312no—
200C2×#41—324no—
201C2×#41—348no—
202C2×#41—312no—
203C2×#41—316no—
204C2×#41—612no—
205C2×#41—612no—
206C2×#41—648no—
207C2×#41—612no—
208C2×#41—66no—
209C2×#41—612no—
210C2×#41—624no—
211C2×#41—612no—
212C2×#41—66no—
213C2×#41—624no—
214C22×Q8—32no—
215C22×Q8—312no—
216C22×Q8—66no—
217C22×Q8—61no—
218C22×D4—32no—
219C22×D4—312no—
220C22×D4—62no—
221C22×D4—612no—
222C22×D4—62no—
223C22×D4—612no—
224C22×D4—66no—
225C22×D4—61no—
226C23×C4—61no—
227C23×C4—612no—
228C23×C4—62no—
229C23×C4—612no—
230C23×C4—62no—
231C23×C4—62no—
232C23×C4C4×D311yescharacteristic
233C23×C4C4×D311yescharacteristic
234C23×C4C4×D316yes—
235C23×C4D1211yescharacteristic
236C23×C4D1216yes—
237C23×C4#6716yes—
238C23×C4#6711yescharacteristic
239C23×C4#6711yescharacteristic
240C23×C4C22×D316yes—
241C23×C4C22×D311yescharacteristic
242C2×M16—212no—
243C2×M16—26no—
244C2×M16—224no—
245C2×M16C22×D3112yescomplemented
246C2×M16C22×D3116yescomplemented
247C22×C8—212no—
248C22×C8—22no—
249#117—316no—
250#117—624no—
251C2×C4⋊C4—612no—
252C2×C22⋊C4—612no—
253C2×C42—312no—
254C2×C42—66no—
255#104—364no—
256C23×C3—24no—
257C23×C3—24no—
258C23×C3—24no—
259C23×C3—412no—
260C23×C3—412no—
261C23×C3C2×C22⋊C416yes—
262C23×C3C2×C22⋊C413yes—
263C23×C3C2×C22⋊C416yes—
264C23×C3C2×C22⋊C4112yes—
265C23×C3C2×C22⋊C411yescharacteristic
266C23×C3C22×D413yes—
267C23×C3C22×D414yes—
268C22×D3—212no—
269C22×D3—216no—
270C22×D3—448no—
271C22×D3—48no—
272C2×C3×C4C2×C22⋊C4112yes—
273C2×C3×C4C2×C22⋊C413yes—
274C2×C3×C4C2×C22⋊C416yes—
275C2×C3×C4C2×C22⋊C4112yes—
276C2×C3×C4C2×C22⋊C416yes—
277C2×C3×C4C2×C22⋊C413yes—
278C2×C3×C4C2×C22⋊C416yes—
279C2×C3×C4C2×C22⋊C4112yes—
280C2×C3×C4C2×C22⋊C4112yes—
281C2×C3×C4C2×C22⋊C4112yes—
282C2×C3×C4C23×C413yes—
283C2×C3×C4C23×C414yes—
284C2×C3×C4C23×C413yes—
285C2×C3×C4C23×C414yes—
286C2×C3×C4C22×D413yes—
287C2×C3×C4C22×D414yes—
288C2×C3×C4C22×D413yes—
289C2×C3×C4C22×D414yes—
290C2×C3×C4C22×D414yes—
291C2×C3×C4C22×D413yes—
292C2×C3×C4C2511yescharacteristic
293#67—224no—
294#67—232no—
295#67—28no—
296C2×Dic3—212no—
297C2×Dic3—216no—
298C2×Dic3—28no—
299C2×Dic3—424no—
300D12—212no—
301D12—216no—
302D12C23×C418yes—
303C4×D3—224no—
304C4×D3—232no—
305C4×D3—28no—
306Dic6—212no—
307Dic6—216no—
308Dic6C23×C418yes—
309C3×C8—28no—
310C24—44no—
311C24—612no—
312C24—62no—
313C24—128no—
314C24(C2×D3)⋊C411yescharacteristic
315C24(C2×D3)⋊C411yescharacteristic
316C24(C2×D3)⋊C416yes—
317C24C2×D1211yescharacteristic
318C24C2×D1216yes—
319#41—324no—
320#41—332no—
321#41—38no—
322#41—624no—
323#41—648no—
324#41—648no—
325#41—648no—
326#41—612no—
327#41—616no—
328#41—616no—
329#41—612no—
330C2×Q8—312no—
331C2×Q8—316no—
332C2×Q8—66no—
333C2×Q8—624no—
334C2×Q8—612no—
335C2×D4—312no—
336C2×D4—316no—
337C2×D4—612no—
338C2×D4—624no—
339C2×D4—648no—
340C2×D4—612no—
341C2×D4—616no—
342C2×D4—66no—
343C2×D4—624no—
344C2×D4—612no—
345C22×C4—612no—
346C22×C4—62no—
347C22×C4—648no—
348C22×C4—624no—
349C22×C4—612no—
350C22×C4—612no—
351C22×C4—616no—
352C22×C4—612no—
353C22×C4—128no—
354C22×C4(C2×D3)⋊C411yescharacteristic
355C22×C4(C2×D3)⋊C416yes—
356C22×C4(C2×D3)⋊C416yes—
357C22×C4(C2×D3)⋊C411yescharacteristic
358C22×C4(C2×D3)⋊C416yes—
359C22×C4(C2×D3)⋊C416yes—
360C22×C4(C2×D3)⋊C4124yes—
361C22×C4(C2×D3)⋊C416yes—
362C22×C4C2×C4×D316yes—
363C22×C4C2×C4×D311yescharacteristic
364C22×C4C2×C4×D316yes—
365C22×C4C2×C4×D3112yes—
366C22×C4C2×C4×D313yes—
367C22×C4C2×D1213yes—
368C22×C4C2×D1216yes—
369C22×C4C2×D12112yes—
370C22×C4C2×#6716yes—
371C22×C4C2×#6716yes—
372C22×C4C2×#6713yes—
373C22×C4C2×#67112yes—
374C22×C4C2×#6711yescharacteristic
375C22×C4C23×D313yes—
376C22×C4C23×D314yes—
377M16—216no—
378M16—212no—
379M16C23×D318yescomplemented
380C2×C8—212no—
381C2×C8—216no—
382C4⋊C4—616no—
383C22⋊C4—616no—
384C42—38no—
385C42—612no—
386C22×C3—26no—
387C22×C3—212no—
388C22×C3—26no—
389C22×C3—212no—
390C22×C3—412no—
391C22×C3—44no—
392C22×C3#42114yescomplemented
393C22×C3C22×C22⋊C414yes—
394C22×C3C22×C22⋊C413yes—
395C22×C3C22×C22⋊C414yes—
396C22×C3C22×C22⋊C413yes—
397C22×C3C23×D411yescharacteristic
398C2×D3—28no—
399C2×D3—432no—
400C2×D3—424no—
401C3×C4C22×C22⋊C413yes—
402C3×C4C22×C22⋊C414yes—
403C3×C4C22×C22⋊C413yes—
404C3×C4C22×C22⋊C414yes—
405C3×C4C24×C411yescharacteristic
406C3×C4C23×D411yescharacteristic
407Dic3—28no—
408Dic3—48no—
409C23—24no—
410C23—24no—
411C23—24no—
412C23—412no—
413C23—412no—
414C23—616no—
415C23—612no—
416C23—1248no—
417C23—128no—
418C23C2×(C2×D3)⋊C416yes—
419C23C2×(C2×D3)⋊C416yes—
420C23C2×(C2×D3)⋊C413yes—
421C23C2×(C2×D3)⋊C4112yes—
422C23C2×(C2×D3)⋊C411yescharacteristic
423C23C22×D1213yes—
424C23C22×D1214yes—
425Q8—38no—
426Q8—612no—
427Q8—616no—
428D4—38no—
429D4—68no—
430D4—624no—
431D4—632no—
432D4—612no—
433D4—616no—
434C2×C4—616no—
435C2×C4—612no—
436C2×C4—68no—
437C2×C4—624no—
438C2×C4—632no—
439C2×C4—68no—
440C2×C4—1224no—
441C2×C4C2×(C2×D3)⋊C4112yes—
442C2×C4C2×(C2×D3)⋊C416yes—
443C2×C4C2×(C2×D3)⋊C413yes—
444C2×C4C2×(C2×D3)⋊C4112yes—
445C2×C4C2×(C2×D3)⋊C413yes—
446C2×C4C2×(C2×D3)⋊C416yes—
447C2×C4C2×(C2×D3)⋊C416yes—
448C2×C4C2×(C2×D3)⋊C4112yes—
449C2×C4C2×(C2×D3)⋊C4112yes—
450C2×C4C2×(C2×D3)⋊C4112yes—
451C2×C4C22×C4×D314yes—
452C2×C4C22×C4×D313yes—
453C2×C4C22×C4×D313yes—
454C2×C4C22×C4×D314yes—
455C2×C4C22×D1213yes—
456C2×C4C22×D1214yes—
457C2×C4C22×#6713yes—
458C2×C4C22×#6714yes—
459C2×C4C22×#6714yes—
460C2×C4C22×#6713yes—
461C2×C4C24×D311yescharacteristic
462C8—28no—
463C2×C3—21no—
464C2×C3—24no—
465C2×C3—23no—
466C2×C3—24no—
467C2×C3C2×#42116yescomplemented
468C2×C3C23×C22⋊C411yescharacteristic
469D3—48no—
470C22—26no—
471C22—212no—
472C22—26no—
473C22—212no—
474C22—412no—
475C22—44no—
476C22—68no—
477C22—1232no—
478C22—1224no—
479C22#513914yesdirect product factor
480C22C22×(C2×D3)⋊C414yes—
481C22C22×(C2×D3)⋊C414yes—
482C22C22×(C2×D3)⋊C413yes—
483C22C22×(C2×D3)⋊C413yes—
484C22C23×D1211yescharacteristic
485C4—68no—
486C4—128no—
487C4C22×(C2×D3)⋊C414yes—
488C4C22×(C2×D3)⋊C413yes—
489C4C22×(C2×D3)⋊C413yes—
490C4C22×(C2×D3)⋊C414yes—
491C4C23×C4×D311yescharacteristic
492C4C23×#6711yescharacteristic
493C3C22×#42111yescharacteristic, complemented, Sylow
494C2—21no—
495C2—24no—
496C2—23no—
497C2—24no—
498C2—128no—
499C2C2×#513916yesdirect product factor
500C2C23×(C2×D3)⋊C411yescharacteristic
subgroupG/H
centerC22×C4C2×#67
commutatorC3×C4C24×C4
FrattiniC2×C4C24×D3
FittingC23×C3×M16C2
socleC23×C3C2×C22⋊C4
prime radicalC23×#235C2
cosocleC2×C3×C4C25
hypercenterC23×M16D3
hypocenterC3C22×#421
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ C23×M16

Complements

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

NH ≅ G/Ntype
2. C23×C3×M16C2semidirect
3. C22×#926C2semidirect
4. C2×#5139C2direct (N × H)
5. C22×#825C2semidirect
13. C22×#235C4semidirect
14. C22×#235C4semidirect
17. C23×D12C4semidirect
19. C23×Dic6C4semidirect
22. C22×C3×M16C22semidirect
23. C22×C3×M16C22semidirect
27. C2×#926C22semidirect
29. C22×C4×Dic3C22semidirect
30. #5139C22direct (N × H)
31. C2×#825C22semidirect
33. C23×M16D3semidirect
66. C2×#235C2×C4semidirect
67. C2×#235C2×C4semidirect
73. C22×D12C2×C4semidirect
81. C22×Dic6C2×C4semidirect
85. C2×C3×M16C23semidirect
86. C2×C3×M16C23semidirect
91. #926C23semidirect
94. C2×C4×Dic3C23semidirect
95. #825C23semidirect
111. C22×M16C2×D3semidirect
112. C22×M16C2×D3semidirect
171. #235C22×C4semidirect
177. C2×D12C22×C4semidirect
187. C2×Dic6C22×C4semidirect
190. C3×M16C24semidirect
196. C4×Dic3C24semidirect
245. C2×M16C22×D3semidirect
246. C2×M16C22×D3semidirect
379. M16C23×D3semidirect
392. C22×C3#421semidirect
467. C2×C3C2×#421semidirect
479. C22#5139direct (N × H)
493. C3C22×#421semidirect
499. C2C2×#5139direct (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×M162
C22×#9262
C2×#51392
C22×#8252
C22×#4213