Finite Group Databasebeta

← C2×#11572 #480542

Subgroups — C2×#11572 #480542

Order 512.

1106 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2566C211yesmaximal, characteristic
2C2×#1465C211yesmaximal, characteristic
3C2×#1459C211yesmaximal, characteristic
4#11572C218yesmaximal, direct product factor
5C2×#1325C212yesmaximal, complemented
6C2×#1324C212yesmaximal, complemented
7C22×#125C411yescharacteristic
8C22×C4×D4C2211yescharacteristic
9C2×#407—21no—
10C22×C22⋊C8—21no—
11#2566—22no—
12#2566—22no—
13#2566C414yes—
14#2566C2214yes—
15C2×#396—21no—
16C2×#396—21no—
17C2×#390C411yescharacteristic
18C2×#390C411yescharacteristic
19C2×#385—21no—
20C2×#385—21no—
21C2×#384C2211yescharacteristic
22C2×#382C2211yescharacteristic
23C2×C4×C4⋊C4C411yescharacteristic
24#1465—22no—
25#1465C2214yes—
26#1459—22no—
27#1459C2214yes—
28C2×#324C2212yes—
29C2×#324C2212yes—
30C2×#323—21no—
31C2×#323—21no—
32#1325C2218yescomplemented
33#1324C2218yescomplemented
34C23×D4C2×C411yescharacteristic
35C24×C4D411yescharacteristic
36C24×C4D411yescharacteristic
37C22×M16—41no—
38C23×C8—41no—
39C2×#125—22no—
40C2×#125—42no—
41C2×#125—41no—
42C2×#125—41no—
43C2×#125C2×C414yes—
44C2×C4×D4—41no—
45C2×C4×D4—41no—
46C2×C4×D4—42no—
47C2×C4×D4C2×C412yes—
48C2×C4×D4D412yes—
49C2×C4×D4D412yes—
50C2×C4×D4C2312yes—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C22⋊C4—21no—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4D411yescharacteristic
64C22×C22⋊C4D411yescharacteristic
65C22×C42C2×C411yescharacteristic
66C22×C42C2×C411yescharacteristic
67C22×C42C2×C411yescharacteristic
68#407—24no—
69#407—42no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—24no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8—42no—
74C2×C22⋊C8—41no—
75C2×C22⋊C8—41no—
76#396—24no—
77#396—24no—
78#396—24no—
79#396—24no—
80#390—22no—
81#390—22no—
82#390C2×C414yes—
83#390C2×C414yes—
84#385—24no—
85#385—24no—
86#385—24no—
87#385—24no—
88#384—21no—
89#384—21no—
90#384C2×C412yes—
91#384C2312yes—
92#382—21no—
93#382—21no—
94#382C2×C412yes—
95#382C2312yes—
96C4×C4⋊C4—22no—
97C4×C4⋊C4C2×C414yes—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101#324—22no—
102#324—22no—
103#324C2314yes—
104#324C2314yes—
105#323—24no—
106#323—24no—
107C25C22⋊C411yescharacteristic
108C25C22⋊C411yescharacteristic
109C22×D4—24no—
110C22×D4—24no—
111C22×D4—41no—
112C22×D4—41no—
113C22×D4C22⋊C412yes—
114C22×D4C22×C412yes—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—82no—
126C23×C4—82no—
127C23×C4C22⋊C411yescharacteristic
128C23×C4C22⋊C412yes—
129C23×C4C22⋊C412yes—
130C23×C4C22⋊C411yescharacteristic
131C23×C4C22×C411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D412yes—
135C23×C4C2×D412yes—
136C2×M16—44no—
137C2×M16—84no—
138C22×C8—42no—
139C22×C8—44no—
140C22×C8—42no—
141C22×C8—84no—
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—24no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—82no—
176C2×C4⋊C4—82no—
177C2×C4⋊C4C22×C412yes—
178C2×C4⋊C4C22×C412yes—
179C2×C4⋊C4C22×C411yescharacteristic
180C2×C4⋊C4C2×D411yescharacteristic
181C2×C4⋊C4C2×D411yescharacteristic
182C2×C4⋊C4C2411yescharacteristic
183C2×C22⋊C4—24no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—82no—
194C2×C22⋊C4—82no—
195C2×C22⋊C4C22⋊C412yes—
196C2×C22⋊C4C22⋊C412yes—
197C2×C22⋊C4C2×D412yes—
198C2×C22⋊C4C2×D412yes—
199C2×C42—22no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—24no—
203C2×C42—24no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42C22⋊C412yes—
207C2×C42C22×C412yes—
208C2×C42C22×C412yes—
209C2×C42C22×C412yes—
210C22⋊C8—24no—
211C22⋊C8—24no—
212C22⋊C8—42no—
213C22⋊C8—44no—
214C22⋊C8—42no—
215C22⋊C8—44no—
216#95—24no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221C24—21no—
222C24—22no—
223C24—21no—
224C24—21no—
225C24—22no—
226C24—22no—
227C24—21no—
228C24—42no—
229C24—42no—
230C24—42no—
231C24—42no—
232C24#10114yes—
233C24C2×C22⋊C411yescharacteristic
234C24C2×C22⋊C412yes—
235C24C2×C22⋊C412yes—
236C2×D4—24no—
237C2×D4—24no—
238C2×D4—24no—
239C2×D4—24no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—48no—
243C2×D4—48no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—42no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—82no—
286C22×C4—82no—
287C22×C4—82no—
288C22×C4—82no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4—84no—
292C22×C4—84no—
293C22×C4#9911yescharacteristic
294C22×C4#9911yescharacteristic
295C22×C4#9912yes—
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C22⋊C412yes—
300C22×C4C2×C22⋊C412yes—
301C22×C4#13114yes—
302C22×C4C23×C411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4C22×D411yescharacteristic
305M16—88no—
306C2×C8—44no—
307C2×C8—44no—
308C2×C8—88no—
309C2×C8—84no—
310C2×C8—84no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—48no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—48no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C4⋊C4—82no—
324C4⋊C4—82no—
325C4⋊C4—82no—
326C4⋊C4—82no—
327C22⋊C4—42no—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C22⋊C4—42no—
331C22⋊C4—48no—
332C22⋊C4—48no—
333C22⋊C4—84no—
334C22⋊C4—84no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—24no—
340C42—24no—
341C42—42no—
342C42—42no—
343C23—22no—
344C23—22no—
345C23—22no—
346C23—24no—
347C23—22no—
348C23—24no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—48no—
357C23—48no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—81no—
365C23—81no—
366C23—84no—
367C23—81no—
368C23—81no—
369C23—84no—
370C23#40812yes—
371C23C2×#9911yescharacteristic
372C23C2×#9911yescharacteristic
373C23C2×#10112yes—
374C23C2×#10114yes—
375C23C22×C22⋊C411yescharacteristic
376D4—48no—
377D4—48no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—42no—
399C2×C4—48no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—84no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4#40812yes—
431C2×C4C2×#9912yes—
432C2×C4C2×#9912yes—
433C2×C4C2×#13112yes—
434C8—84no—
435C8—84no—
436C22—24no—
437C22—24no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—24no—
443C22—44no—
444C22—44no—
445C22—48no—
446C22—42no—
447C22—42no—
448C22—42no—
449C22—42no—
450C22—42no—
451C22—82no—
452C22—84no—
453C22—82no—
454C22—84no—
455C22—84no—
456C22#147212yes—
457C22#147212yes—
458C22#212712yes—
459C22#213212yes—
460C22C2×#40811yescharacteristic
461C22C22×#9911yescharacteristic
462C22C22×#10111yescharacteristic
463C4—22no—
464C4—22no—
465C4—22no—
466C4—22no—
467C4—44no—
468C4—41no—
469C4—41no—
470C4—41no—
471C4—41no—
472C4—42no—
473C4—82no—
474C4—82no—
475C4—82no—
476C2—21no—
477C2—21no—
478C2—22no—
479C2—44no—
480C2—82no—
481C2#1157214yesdirect product factor
482C2C2×#147211yescharacteristic
483C2C2×#212711yescharacteristic
484C2C2×#213211yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11572C2direct (N × H)
5. C2×#1325C2semidirect
6. C2×#1324C2semidirect
32. #1325C22semidirect
33. #1324C22semidirect
481. C2#11572direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25662
C2×#14652
C2×#14592
#115722
C2×#13252
C2×#13242