Finite Group Databasebeta

← C2×#10687790 #72253655

Subgroups — C2×#10687790 #72253655

Order 1536.

23164 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#72130C216yesmaximal, complemented
2#10687790C218yesmaximal, direct product factor
3C23×C3×#120C211yesmaximal, characteristic
4C2×#8541—31nomaximal, Sylow
5C27×C3C411yescharacteristic, complemented
6C24×C3×D4C2213yes—
7C22×C3×#120—212no—
8C22×C3×#120—212no—
9C22×C3×#120C414yescomplemented
10C22×C3×#120C2214yes—
11C23×C3×C22⋊C4C413yescomplemented
12C2×#5247C2214yescomplemented
13C22×#834—212no—
14#72130—212no—
15#72130C22124yescomplemented
16C23×#120D311yescharacteristic
17C2×#1781—36no—
18#8541—38no—
19C26×C3—26no—
20C26×C3—26no—
21C26×C3—424no—
22C26×C3C2×C413yes—
23C26×C3C2×C414yescomplemented
24C26×C3C2311yescharacteristic
25C23×C3×D4—212no—
26C23×C3×D4—212no—
27C23×C3×D4—424no—
28C23×C3×D4C2×C416yes—
29C23×C3×D4D4112yes—
30C23×C3×D4D4112yes—
31C23×C3×D4C2316yes—
32C24×C3×C4C2×C413yes—
33C2×C3×#120—232no—
34C2×C3×#120—296no—
35C2×C3×#120—248no—
36C2×C3×#120—248no—
37C22×C3×C22⋊C4—212no—
38C22×C3×C22⋊C4—212no—
39C22×C3×C22⋊C4—26no—
40C22×C3×C22⋊C4—26no—
41C22×C3×C22⋊C4C2×C4112yescomplemented
42C22×#217—212no—
43#5247—212no—
44#5247C2318yescomplemented
45C2×#834—248no—
46C2×#834—448no—
47C27Dic311yescharacteristic, complemented
48C24×D4C2×D313yes—
49C22×#120—212no—
50C22×#120—212no—
51C22×#120Dic314yescomplemented
52C22×#120C2×D314yes—
53C23×C22⋊C4Dic313yescomplemented
54C22×#99—612no—
55C2×#379—34no—
56#1781—324no—
57#1781—612no—
58C25×C3—224no—
59C25×C3—26no—
60C25×C3—26no—
61C25×C3—224no—
62C25×C3—28no—
63C25×C3—224no—
64C25×C3—412no—
65C25×C3—424no—
66C25×C3—448no—
67C25×C3—496no—
68C25×C3—496no—
69C25×C3—424no—
70C25×C3—448no—
71C25×C3—4256no—
72C25×C3—424no—
73C25×C3C22⋊C416yes—
74C25×C3C22⋊C416yes—
75C25×C3C22×C411yescharacteristic
76C25×C3C22×C416yes—
77C25×C3C22×C411yescharacteristic
78C25×C3C2×D416yes—
79C25×C3C2×D416yes—
80C25×C3C2411yescharacteristic
81C22×C3×D4—248no—
82C22×C3×D4—224no—
83C22×C3×D4—224no—
84C22×C3×D4—224no—
85C22×C3×D4—248no—
86C22×C3×D4—224no—
87C22×C3×D4—424no—
88C22×C3×D4—424no—
89C22×C3×D4—448no—
90C22×C3×D4—424no—
91C22×C3×D4—496no—
92C22×C3×D4—496no—
93C22×C3×D4C22⋊C4124yes—
94C22×C3×D4C2×D4124yes—
95C23×C3×C4—26no—
96C23×C3×C4—26no—
97C23×C3×C4—412no—
98C23×C3×C4C22⋊C416yes—
99C23×C3×C4C22⋊C416yes—
100C23×C3×C4C22×C416yes—
101C23×Dic3—44no—
102C3×#120—2128no—
103C3×#120—2128no—
104C2×C3×C22⋊C4—224no—
105C2×C3×C22⋊C4—248no—
106C2×C3×C22⋊C4—224no—
107C2×C3×C22⋊C4—224no—
108C2×C3×C22⋊C4—248no—
109C2×#217—224no—
110C2×#217—448no—
111C2×#217—412no—
112#834—496no—
113C26—26no—
114C26—26no—
115C26—424no—
116C26C2×Dic313yes—
117C26C2×Dic314yescomplemented
118C26C22×D311yescharacteristic
119C23×D4—212no—
120C23×D4—212no—
121C23×D4—424no—
122C23×D4C2×Dic316yes—
123C23×D4#67112yes—
124C23×D4#67112yes—
125C23×D4C22×D316yes—
126C24×C4C2×Dic313yes—
127C2×#120—296no—
128C2×#120—232no—
129C2×#120—248no—
130C2×#120—248no—
131C22×C22⋊C4—26no—
132C22×C22⋊C4—26no—
133C22×C22⋊C4—212no—
134C22×C22⋊C4—212no—
135C22×C22⋊C4—612no—
136C22×C22⋊C4C2×Dic3112yescomplemented
137C2×#99—648no—
138C2×#99—1248no—
139#379—38no—
140#379—612no—
141C24×C3—212no—
142C24×C3—248no—
143C24×C3—248no—
144C24×C3—212no—
145C24×C3—212no—
146C24×C3—26no—
147C24×C3—26no—
148C24×C3—212no—
149C24×C3—224no—
150C24×C3—224no—
151C24×C3—224no—
152C24×C3—232no—
153C24×C3—46no—
154C24×C3—412no—
155C24×C3—424no—
156C24×C3—424no—
157C24×C3—46no—
158C24×C3—424no—
159C24×C3—424no—
160C24×C3—424no—
161C24×C3—4192no—
162C24×C3—424no—
163C24×C3—424no—
164C24×C3—424no—
165C24×C3—4192no—
166C24×C3—4192no—
167C24×C3—448no—
168C24×C3—496no—
169C24×C3—496no—
170C24×C3—4384no—
171C24×C3—4256no—
172C24×C3—4768no—
173C24×C3—4384no—
174C24×C3—424no—
175C24×C3—448no—
176C24×C3—832no—
177C24×C3#99124yescomplemented
178C24×C3C2×C22⋊C413yes—
179C24×C3C2×C22⋊C413yes—
180C24×C3C2×C22⋊C416yes—
181C24×C3C2×C22⋊C4112yes—
182C24×C3#12014yes—
183C24×C3#120112yes—
184C24×C3C23×C411yescharacteristic
185C24×C3C22×D416yes—
186C2×C3×D4—248no—
187C2×C3×D4—248no—
188C2×C3×D4—248no—
189C2×C3×D4—248no—
190C2×C3×D4—424no—
191C2×C3×D4—424no—
192C2×C3×D4—496no—
193C2×C3×D4—496no—
194C2×C3×D4—496no—
195C2×C3×D4—4192no—
196C2×C3×D4—496no—
197C22×C3×C4—212no—
198C22×C3×C4—212no—
199C22×C3×C4—26no—
200C22×C3×C4—26no—
201C22×C3×C4—224no—
202C22×C3×C4—412no—
203C22×C3×C4—412no—
204C22×C3×C4—424no—
205C22×C3×C4—424no—
206C22×C3×C4C2×C22⋊C4112yes—
207C22×Dic3—48no—
208C22×Dic3—824no—
209C3×C22⋊C4—296no—
210#217—448no—
211#217—824no—
212C25—224no—
213C25—26no—
214C25—26no—
215C25—224no—
216C25—28no—
217C25—224no—
218C25—412no—
219C25—424no—
220C25—448no—
221C25—496no—
222C25—496no—
223C25—424no—
224C25—448no—
225C25—4256no—
226C25—424no—
227C25#21716yes—
228C25#21716yes—
229C25C22×Dic311yescharacteristic
230C25C22×Dic316yes—
231C25C22×Dic311yescharacteristic
232C25C2×#6716yes—
233C25C2×#6716yes—
234C25C23×D311yescharacteristic
235C22×D4—248no—
236C22×D4—224no—
237C22×D4—224no—
238C22×D4—248no—
239C22×D4—224no—
240C22×D4—224no—
241C22×D4—424no—
242C22×D4—424no—
243C22×D4—448no—
244C22×D4—424no—
245C22×D4—496no—
246C22×D4—496no—
247C22×D4#217124yes—
248C22×D4C2×#67124yes—
249C23×C4—26no—
250C23×C4—26no—
251C23×C4—412no—
252C23×C4—124no—
253C23×C4#21716yes—
254C23×C4#21716yes—
255C23×C4C22×Dic316yes—
256#120—2128no—
257#120—2128no—
258C2×C22⋊C4—224no—
259C2×C22⋊C4—248no—
260C2×C22⋊C4—224no—
261C2×C22⋊C4—224no—
262C2×C22⋊C4—248no—
263C2×C22⋊C4—624no—
264C2×C22⋊C4—1248no—
265C2×C22⋊C4—1212no—
266#99—1296no—
267C23×C3—232no—
268C23×C3—26no—
269C23×C3—26no—
270C23×C3—212no—
271C23×C3—212no—
272C23×C3—248no—
273C23×C3—248no—
274C23×C3—212no—
275C23×C3—212no—
276C23×C3—224no—
277C23×C3—224no—
278C23×C3—224no—
279C23×C3—212no—
280C23×C3—212no—
281C23×C3—224no—
282C23×C3—412no—
283C23×C3—424no—
284C23×C3—412no—
285C23×C3—448no—
286C23×C3—424no—
287C23×C3—496no—
288C23×C3—496no—
289C23×C3—4192no—
290C23×C3—4192no—
291C23×C3—448no—
292C23×C3—424no—
293C23×C3—424no—
294C23×C3—424no—
295C23×C3—4192no—
296C23×C3—4384no—
297C23×C3—4384no—
298C23×C3—4768no—
299C23×C3—4256no—
300C23×C3—424no—
301C23×C3—424no—
302C23×C3—424no—
303C23×C3—848no—
304C23×C3—88no—
305C23×C3—848no—
306C23×C3—88no—
307C23×C3#37914yes—
308C23×C3#37913yes—
309C23×C3#37911yescharacteristic
310C23×C3C2×#99112yes—
311C23×C3C2×#99124yescomplemented
312C23×C3C22×C22⋊C413yes—
313C23×C3C2×#12014yes—
314C3×D4—496no—
315C3×D4—496no—
316C2×C3×C4—224no—
317C2×C3×C4—212no—
318C2×C3×C4—424no—
319C2×C3×C4—424no—
320C2×C3×C4—412no—
321C2×C3×C4—412no—
322C2×Dic3—824no—
323C2×Dic3—1616no—
324C24—212no—
325C24—248no—
326C24—248no—
327C24—212no—
328C24—224no—
329C24—224no—
330C24—212no—
331C24—26no—
332C24—26no—
333C24—212no—
334C24—224no—
335C24—232no—
336C24—46no—
337C24—412no—
338C24—424no—
339C24—424no—
340C24—46no—
341C24—424no—
342C24—424no—
343C24—424no—
344C24—4192no—
345C24—424no—
346C24—424no—
347C24—424no—
348C24—4192no—
349C24—4192no—
350C24—448no—
351C24—496no—
352C24—496no—
353C24—4384no—
354C24—4256no—
355C24—4768no—
356C24—4384no—
357C24—424no—
358C24—448no—
359C24—832no—
360C24#834124yescomplemented
361C24C2×#21713yes—
362C24C2×#21713yes—
363C24C2×#217112yes—
364C24C2×#21716yes—
365C24#95314yes—
366C24#953112yes—
367C24C23×Dic311yescharacteristic
368C24C22×#6716yes—
369C2×D4—248no—
370C2×D4—248no—
371C2×D4—248no—
372C2×D4—248no—
373C2×D4—424no—
374C2×D4—424no—
375C2×D4—496no—
376C2×D4—496no—
377C2×D4—496no—
378C2×D4—4192no—
379C2×D4—496no—
380C22×C4—224no—
381C22×C4—212no—
382C22×C4—26no—
383C22×C4—26no—
384C22×C4—212no—
385C22×C4—412no—
386C22×C4—412no—
387C22×C4—424no—
388C22×C4—424no—
389C22×C4—128no—
390C22×C4—2424no—
391C22×C4C2×#217112yes—
392C22⋊C4—296no—
393C22⋊C4—1248no—
394C22⋊C4—2424no—
395C22×C3—28no—
396C22×C3—224no—
397C22×C3—224no—
398C22×C3—26no—
399C22×C3—26no—
400C22×C3—212no—
401C22×C3—224no—
402C22×C3—212no—
403C22×C3—424no—
404C22×C3—496no—
405C22×C3—424no—
406C22×C3—496no—
407C22×C3—448no—
408C22×C3—4256no—
409C22×C3—416no—
410C22×C3—448no—
411C22×C3—424no—
412C22×C3—812no—
413C22×C3—812no—
414C22×C3—824no—
415C22×C3#178116yes—
416C22×C3#178116yes—
417C22×C3C2×#37911yescharacteristic
418C22×C3C22×#9916yes—
419C3×C4—412no—
420Dic3—168no—
421C23—232no—
422C23—26no—
423C23—26no—
424C23—212no—
425C23—212no—
426C23—248no—
427C23—248no—
428C23—212no—
429C23—212no—
430C23—224no—
431C23—224no—
432C23—224no—
433C23—212no—
434C23—212no—
435C23—224no—
436C23—412no—
437C23—424no—
438C23—412no—
439C23—448no—
440C23—424no—
441C23—496no—
442C23—496no—
443C23—4192no—
444C23—4192no—
445C23—448no—
446C23—424no—
447C23—424no—
448C23—424no—
449C23—4192no—
450C23—4384no—
451C23—4384no—
452C23—4768no—
453C23—4256no—
454C23—424no—
455C23—424no—
456C23—424no—
457C23—848no—
458C23—88no—
459C23—848no—
460C23—88no—
461C23C2×#834112yes—
462C23C2×#834124yescomplemented
463C23#524714yes—
464C23#524713yes—
465C23#524711yescharacteristic
466C23C22×#21713yes—
467C23C2×#95314yes—
468D4—496no—
469D4—496no—
470C2×C4—224no—
471C2×C4—212no—
472C2×C4—424no—
473C2×C4—424no—
474C2×C4—412no—
475C2×C4—412no—
476C2×C4—2424no—
477C2×C4—4816no—
478C2×C3—26no—
479C2×C3—26no—
480C2×C3—44no—
481C2×C3—44no—
482C2×C3—424no—
483C2×C3#854114yescomplemented
484C2×C3C2×#178113yes—
485C22—28no—
486C22—224no—
487C22—224no—
488C22—26no—
489C22—26no—
490C22—212no—
491C22—224no—
492C22—212no—
493C22—424no—
494C22—496no—
495C22—424no—
496C22—496no—
497C22—448no—
498C22—4256no—
499C22—416no—
500C22—448no—
501C22—424no—
502C22—812no—
503C22—812no—
504C22—824no—
505C22#7213016yes—
506C22#7213016yes—
507C22C22×#83416yes—
508C22C2×#524711yescharacteristic
509C4—412no—
510C4—488no—
511C3C2×#854111yescharacteristic, complemented, Sylow
512C2—26no—
513C2—26no—
514C2—44no—
515C2—44no—
516C2—424no—
517C2#1068779014yesdirect product factor
518C2C2×#7213013yes—
subgroupG/H
centerC23#5247
commutatorC24×C3C23×C4
FrattiniC25C23×D3
Fitting, prime radicalC23×C3×#120C2
socleC23×C3#379
cosocleC25×C3C24
hypercenterC23×#120D3
hypocenterC3C2×#8541
LCSG ▷ C24×C3 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C26 ◁ C23×#120

Complements

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

NH ≅ G/Ntype
1. C2×#72130C2semidirect
2. #10687790C2direct (N × H)
5. C27×C3C4semidirect
9. C22×C3×#120C4semidirect
11. C23×C3×C22⋊C4C4semidirect
12. C2×#5247C22semidirect
15. #72130C22semidirect
23. C26×C3C2×C4semidirect
41. C22×C3×C22⋊C4C2×C4semidirect
44. #5247C23semidirect
47. C27Dic3semidirect
51. C22×#120Dic3semidirect
53. C23×C22⋊C4Dic3semidirect
117. C26C2×Dic3semidirect
136. C22×C22⋊C4C2×Dic3semidirect
177. C24×C3#99semidirect
311. C23×C3C2×#99semidirect
360. C24#834semidirect
462. C23C2×#834semidirect
483. C2×C3#8541semidirect
511. C3C2×#8541semidirect
517. C2#10687790direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#721302
#106877902
C23×C3×#1202
C2×#85413