Finite Group Databasebeta

← C2×#11827 #481895

Subgroups — C2×#11827 #481895

Order 512.

1108 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2633C211yesmaximal, characteristic
2C2×#1799C211yesmaximal, characteristic
3C2×#1465C211yesmaximal, characteristic
4#11827C218yesmaximal, direct product factor
5C2×#1352C212yesmaximal, complemented
6C2×#1344C212yesmaximal, complemented
7C22×#125C411yescharacteristic, complemented
8C22×C4×D4C2211yescharacteristic
9C22×#99—21no—
10C2×#407—21no—
11#2633—22no—
12#2633—22no—
13#2633C414yescomplemented
14#2633C2214yes—
15C2×#399C411yescharacteristic, complemented
16C2×#399C2211yescharacteristic
17C2×#398—21no—
18C2×#397C411yescharacteristic, complemented
19C2×#397C2211yescharacteristic
20C2×#394—21no—
21C2×#385—21no—
22C2×#385—21no—
23C2×#383C411yescharacteristic, complemented
24C2×#380—21no—
25#1799—22no—
26#1799C2214yes—
27C2×#342C2212yes—
28#1465—22no—
29#1465C2214yes—
30C2×#324—21no—
31C2×#323C2212yes—
32#1352C2218yescomplemented
33#1344C2218yescomplemented
34C23×D4D411yescharacteristic
35C24×C4D411yescharacteristic
36C24×C4D411yescharacteristic
37C22×M16—41no—
38C2×#125—22no—
39C2×#125—42no—
40C2×#125—41no—
41C2×#125—41no—
42C2×#125C2×C414yescomplemented
43C2×C4×D4—41no—
44C2×C4×D4—41no—
45C2×C4×D4—42no—
46C2×C4×D4C2×C412yes—
47C2×C4×D4D412yes—
48C2×C4×D4D412yes—
49C2×C4×D4C2312yes—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—41no—
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4D411yescharacteristic
58C22×C22⋊C4—21no—
59C22×C22⋊C4—21no—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C22⋊C4C2311yescharacteristic
65C22×C42C2×C411yescharacteristic
66C2×#99—24no—
67C2×#99—44no—
68#407—24no—
69#407—42no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—41no—
72#399—22no—
73#399—21no—
74#399—21no—
75#399C2×C412yes—
76#399C2×C414yescomplemented
77#399C2312yes—
78#398—24no—
79#398—24no—
80#397—22no—
81#397—21no—
82#397—21no—
83#397C2×C412yes—
84#397C2×C414yescomplemented
85#397C2312yes—
86#394—24no—
87#394—24no—
88#385—24no—
89#385—24no—
90#385—24no—
91#385—24no—
92#383—22no—
93#383C2×C414yescomplemented
94#380—24no—
95#380—42no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—41no—
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C411yescharacteristic
103#342—22no—
104#342C2314yes—
105#324—24no—
106#323—22no—
107#323C2314yes—
108C25C22⋊C411yescharacteristic
109C25C2×D411yescharacteristic
110C22×D4—22no—
111C22×D4—22no—
112C22×D4—41no—
113C22×D4—41no—
114C22×D4—42no—
115C22×D4C22⋊C412yes—
116C22×D4C2×D412yes—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—22no—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—42no—
128C23×C4—82no—
129C23×C4—82no—
130C23×C4C22⋊C411yescharacteristic
131C23×C4C22⋊C411yescharacteristic
132C23×C4C22⋊C412yes—
133C23×C4C22⋊C412yes—
134C23×C4C22⋊C411yescharacteristic
135C23×C4C22×C411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D412yes—
138C23×C4C2×D412yes—
139C2×M16—44no—
140C2×M16—84no—
141C22×C8—42no—
142#125—44no—
143#125—44no—
144#125—44no—
145#125—44no—
146C4×D4—42no—
147C4×D4—42no—
148C4×D4—42no—
149C4×D4—42no—
150C4×D4—42no—
151C4×D4—42no—
152C4×D4—42no—
153C4×D4—42no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—82no—
174C2×C4⋊C4—82no—
175C2×C4⋊C4—84no—
176C2×C4⋊C4C22⋊C412yes—
177C2×C4⋊C4C22×C412yes—
178C2×C4⋊C4C22×C412yes—
179C2×C4⋊C4C2×D412yes—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4—42no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4—82no—
198C2×C22⋊C4—82no—
199C2×C22⋊C4C22⋊C412yes—
200C2×C22⋊C4C22×C412yes—
201C2×C22⋊C4C22×C411yescharacteristic
202C2×C22⋊C4C2×D411yescharacteristic
203C2×C22⋊C4C2×D411yescharacteristic
204C2×C22⋊C4C2411yescharacteristic
205C2×C42—22no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42C22⋊C412yes—
209C2×C42C22×C412yes—
210#99—48no—
211C22⋊C8—24no—
212C22⋊C8—42no—
213C22⋊C8—44no—
214#95—24no—
215#95—24no—
216#95—22no—
217#95—24no—
218#95—21no—
219#95—21no—
220#95—22no—
221#95—44no—
222#95C22×C412yes—
223#95C22×C412yes—
224C24—21no—
225C24—21no—
226C24—22no—
227C24—21no—
228C24—22no—
229C24—21no—
230C24—22no—
231C24—42no—
232C24—42no—
233C24—42no—
234C24—42no—
235C24C2×C22⋊C411yescharacteristic
236C24C2×C22⋊C412yes—
237C24C2×C22⋊C411yescharacteristic
238C24AGL(1,ℤ/8)12yes—
239C24#13712yes—
240C24C22×D411yescharacteristic
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—24no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—44no—
249C2×D4—44no—
250C2×D4—44no—
251C2×D4—44no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—42no—
275C22×C4—44no—
276C22×C4—44no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—44no—
295C22×C4—82no—
296C22×C4—82no—
297C22×C4—82no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4—84no—
302C22×C4—84no—
303C22×C4—84no—
304C22×C4—84no—
305C22×C4—84no—
306C22×C4#9911yescharacteristic
307C22×C4#9911yescharacteristic
308C22×C4#9912yes—
309C22×C4C2×C22⋊C412yes—
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C412yes—
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C412yes—
314C22×C4C23×C411yescharacteristic
315C22×C4C22×D411yescharacteristic
316M16—88no—
317C2×C8—44no—
318C2×C8—84no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—48no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C4⋊C4—82no—
329C4⋊C4—84no—
330C4⋊C4—82no—
331C4⋊C4—88no—
332C22⋊C4—42no—
333C22⋊C4—41no—
334C22⋊C4—42no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—48no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—44no—
342C22⋊C4—44no—
343C22⋊C4—44no—
344C22⋊C4—82no—
345C22⋊C4—84no—
346C22⋊C4—82no—
347C22⋊C4—84no—
348C42—22no—
349C42—22no—
350C42—22no—
351C42—22no—
352C23—22no—
353C23—22no—
354C23—21no—
355C23—21no—
356C23—22no—
357C23—22no—
358C23—21no—
359C23—21no—
360C23—22no—
361C23—22no—
362C23—22no—
363C23—22no—
364C23—22no—
365C23—41no—
366C23—42no—
367C23—41no—
368C23—41no—
369C23—42no—
370C23—44no—
371C23—44no—
372C23—41no—
373C23—48no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—81no—
381C23—81no—
382C23—84no—
383C23—81no—
384C23—81no—
385C23—84no—
386C23C2×#9911yescharacteristic
387C23C2×#9911yescharacteristic
388C23#41712yes—
389C23#41712yes—
390C23#42112yes—
391C23C22×C22⋊C411yescharacteristic
392C23C2×AGL(1,ℤ/8)11yescharacteristic
393C23C2×#13711yescharacteristic
394D4—44no—
395D4—44no—
396D4—48no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—84no—
443C2×C4—84no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—84no—
449C2×C4—84no—
450C2×C4—84no—
451C2×C4—168no—
452C2×C4C2×#9912yes—
453C2×C4C2×#9912yes—
454C8—84no—
455C22—24no—
456C22—24no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—22no—
464C22—41no—
465C22—41no—
466C22—42no—
467C22—44no—
468C22—44no—
469C22—44no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—41no—
474C22—42no—
475C22—42no—
476C22—42no—
477C22—81no—
478C22—81no—
479C22—82no—
480C22—82no—
481C22—82no—
482C22—84no—
483C22—84no—
484C22#151312yes—
485C22#151312yes—
486C22#212312yes—
487C22#212712yes—
488C22C22×#9911yescharacteristic
489C22C2×#41711yescharacteristic
490C22C2×#42111yescharacteristic
491C4—22no—
492C4—22no—
493C4—41no—
494C4—41no—
495C4—41no—
496C4—42no—
497C4—41no—
498C4—42no—
499C4—82no—
500C4—82no—
501C4—82no—
502C4—82no—
503C4—164no—
504C2—21no—
505C2—21no—
506C2—22no—
507C2—41no—
508C2—41no—
509C2—42no—
510C2—82no—
511C2#1182714yesdirect product factor
512C2C2×#151311yescharacteristic
513C2C2×#212311yescharacteristic
514C2C2×#212711yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11827C2direct (N × H)
5. C2×#1352C2semidirect
6. C2×#1344C2semidirect
7. C22×#125C4semidirect
13. #2633C4semidirect
15. C2×#399C4semidirect
18. C2×#397C4semidirect
23. C2×#383C4semidirect
32. #1352C22semidirect
33. #1344C22semidirect
42. C2×#125C2×C4semidirect
76. #399C2×C4semidirect
84. #397C2×C4semidirect
93. #383C2×C4semidirect
511. C2#11827direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#26332
C2×#17992
C2×#14652
#118272
C2×#13522
C2×#13442