Finite Group Databasebeta

← C2×#11528 #480326

Subgroups — C2×#11528 #480326

Order 512.

1218 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2322C211yesmaximal, characteristic
2C2×#1466C211yesmaximal, characteristic
3C2×#1465C211yesmaximal, characteristic
4#11528C218yesmaximal, direct product factor
5C2×#1326C212yesmaximal, complemented
6C2×#1317C212yesmaximal, complemented
7C22×#123C411yescharacteristic
8C22×C4×D4C2211yescharacteristic
9C23×C4⋊C4C411yescharacteristic
10C2×#432—21no—
11C2×#407—21no—
12#2322—22no—
13#2322—22no—
14#2322C414yes—
15#2322C2214yes—
16C2×#389—21no—
17C2×#388—21no—
18C2×#387—21no—
19C2×#386C411yescharacteristic
20C2×#385—21no—
21C2×#385C2211yescharacteristic
22C2×#381C2211yescharacteristic
23C2×C4×C22⋊C4C411yescharacteristic
24#1466—22no—
25#1466C2214yes—
26#1465—22no—
27#1465C2214yes—
28C2×#324—21no—
29C2×#323—21no—
30C2×#323C2212yes—
31C2×#323C2212yes—
32#1326C2218yescomplemented
33#1317C2218yescomplemented
34C23×D4C2×C411yescharacteristic
35C24×C4C2×C411yescharacteristic
36C24×C4C2×C411yescharacteristic
37C24×C4C2×C411yescharacteristic
38C24×C4D411yescharacteristic
39C22×M16—41no—
40C2×#123—22no—
41C2×#123—24no—
42C2×#123—24no—
43C2×#123C2×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—22no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—42no—
57C22×C4⋊C4—41no—
58C22×C4⋊C4—41no—
59C22×C4⋊C4C2×C414yes—
60C22×C4⋊C4D411yescharacteristic
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C22⋊C4C2×C411yescharacteristic
65C22×C22⋊C4D411yescharacteristic
66C22×C22⋊C4C2311yescharacteristic
67C22×C42—21no—
68C22×C42D411yescharacteristic
69#432—24no—
70#432—42no—
71C2×C4⋊C8—41no—
72#407—24no—
73#407—42no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—41no—
77C2×#97—41no—
78#389—24no—
79#389—24no—
80#388—24no—
81#388—24no—
82#387—24no—
83#387—24no—
84#386—22no—
85#386C2×C414yes—
86#385—24no—
87#385—24no—
88#385—21no—
89#385—21no—
90#385C2×C412yes—
91#385C2312yes—
92#381—21no—
93#381—21no—
94#381C2×C412yes—
95#381C2312yes—
96C4×C22⋊C4—22no—
97C4×C22⋊C4C2×C414yes—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—21no—
103#324—24no—
104#323—24no—
105#323—22no—
106#323—22no—
107#323C2314yes—
108#323C2314yes—
109C25C22⋊C411yescharacteristic
110C25C22×C411yescharacteristic
111C22×D4—22no—
112C22×D4—22no—
113C22×D4—41no—
114C22×D4—41no—
115C22×D4—42no—
116C22×D4C22⋊C412yes—
117C22×D4C22×C412yes—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—41no—
132C23×C4—41no—
133C23×C4—42no—
134C23×C4—41no—
135C23×C4—41no—
136C23×C4—82no—
137C23×C4—82no—
138C23×C4—82no—
139C23×C4C22⋊C411yescharacteristic
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22⋊C412yes—
142C23×C4C22⋊C412yes—
143C23×C4C22⋊C411yescharacteristic
144C23×C4M1614yescomplemented
145C23×C4C22×C412yes—
146C23×C4C22×C412yes—
147C23×C4C22×C412yes—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D412yes—
151C2×M16—44no—
152C2×M16—84no—
153C22×C8—42no—
154C22×C8—42no—
155#123—216no—
156#123—28no—
157#123—28no—
158C4×D4—42no—
159C4×D4—42no—
160C4×D4—42no—
161C4×D4—42no—
162C4×D4—42no—
163C4×D4—42no—
164C4×D4—42no—
165C4×D4—42no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—82no—
184C2×C4⋊C4C22⋊C412yes—
185C2×C4⋊C4C2×D412yes—
186C2×C22⋊C4—28no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4—82no—
198C2×C22⋊C4C22⋊C412yes—
199C2×C22⋊C4M1614yescomplemented
200C2×C22⋊C4C22×C412yes—
201C2×C22⋊C4C22×C412yes—
202C2×C22⋊C4C22×C411yescharacteristic
203C2×C22⋊C4C2×D412yes—
204C2×C22⋊C4C2×D411yescharacteristic
205C2×C22⋊C4C2×D411yescharacteristic
206C2×C22⋊C4C2411yescharacteristic
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—22no—
210C2×C42—24no—
211C2×C42—82no—
212C2×C42C22⋊C412yes—
213C2×C42C2×D412yes—
214C4⋊C8—44no—
215C22⋊C8—24no—
216C22⋊C8—24no—
217C22⋊C8—42no—
218C22⋊C8—42no—
219C22⋊C8—44no—
220#97—44no—
221#95—24no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—24no—
226#95—22no—
227#95—22no—
228#95—22no—
229C24—21no—
230C24—21no—
231C24—22no—
232C24—22no—
233C24—21no—
234C24—21no—
235C24—22no—
236C24—42no—
237C24—42no—
238C24—42no—
239C24—42no—
240C24C2×C22⋊C411yescharacteristic
241C24C2×C22⋊C412yes—
242C24C2×C22⋊C411yescharacteristic
243C24C2×M1612yes—
244C24C2×M1612yes—
245C24C23×C411yescharacteristic
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—24no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—44no—
254C2×D4—44no—
255C2×D4—44no—
256C2×D4—44no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—42no—
284C22×C4—44no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—82no—
311C22×C4—82no—
312C22×C4—82no—
313C22×C4—82no—
314C22×C4—82no—
315C22×C4—84no—
316C22×C4—84no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4—84no—
322C22×C4—84no—
323C22×C4#9911yescharacteristic
324C22×C4#9911yescharacteristic
325C22×C4#9912yes—
326C22×C4C2×C22⋊C412yes—
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C412yes—
330C22×C4C2×C22⋊C412yes—
331C22×C4C2×M1614yescomplemented
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334M16—88no—
335C2×C8—44no—
336C2×C8—44no—
337C2×C8—84no—
338C2×C8—84no—
339C4⋊C4—216no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—48no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4—82no—
346C4⋊C4—82no—
347C22⋊C4—216no—
348C22⋊C4—22no—
349C22⋊C4—24no—
350C22⋊C4—42no—
351C22⋊C4—41no—
352C22⋊C4—42no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—84no—
359C22⋊C4C2×M1614yescomplemented
360C42—22no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—84no—
365C23—22no—
366C23—22no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—22no—
375C23—22no—
376C23—22no—
377C23—22no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—44no—
382C23—44no—
383C23—41no—
384C23—42no—
385C23—48no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—48no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—81no—
397C23—81no—
398C23—84no—
399C23#40712yes—
400C23#40712yes—
401C23C2×#9911yescharacteristic
402C23C2×#9911yescharacteristic
403C23#41312yes—
404C23C22×C22⋊C411yescharacteristic
405C23C22×M1611yescharacteristic
406C23C22×M1611yescharacteristic
407D4—44no—
408D4—44no—
409D4—48no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—48no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—48no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—44no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—84no—
462C2×C4—82no—
463C2×C4—84no—
464C2×C4—82no—
465C2×C4—84no—
466C2×C4C2×#9912yes—
467C2×C4C2×#9912yes—
468C8—84no—
469C8—84no—
470C22—24no—
471C22—24no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—22no—
479C22—41no—
480C22—41no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—44no—
485C22—44no—
486C22—44no—
487C22—44no—
488C22—44no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—82no—
497C22—84no—
498C22—84no—
499C22#146512yes—
500C22#146512yes—
501C22#212112yes—
502C22#212312yes—
503C22C2×#40711yescharacteristic
504C22C22×#9911yescharacteristic
505C22C2×#41311yescharacteristic
506C4—22no—
507C4—22no—
508C4—44no—
509C4—41no—
510C4—41no—
511C4—42no—
512C4—41no—
513C4—41no—
514C4—42no—
515C4—44no—
516C4—82no—
517C4—82no—
518C2—21no—
519C2—21no—
520C2—22no—
521C2—41no—
522C2—41no—
523C2—42no—
524C2—82no—
525C2#1152814yesdirect product factor
526C2C2×#146511yescharacteristic
527C2C2×#212111yescharacteristic
528C2C2×#212311yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC24C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C22 ▷ 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. #11528C2direct (N × H)
5. C2×#1326C2semidirect
6. C2×#1317C2semidirect
32. #1326C22semidirect
33. #1317C22semidirect
144. C23×C4M16semidirect
199. C2×C22⋊C4M16semidirect
331. C22×C4C2×M16semidirect
359. C22⋊C4C2×M16semidirect
525. C2#11528direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#23222
C2×#14662
C2×#14652
#115282
C2×#13262
C2×#13172