Finite Group Databasebeta

← C2×#11776 #481598

Subgroups — C2×#11776 #481598

Order 512.

1894 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2541C211yesmaximal, characteristic
2C2×C4×#99C211yesmaximal, characteristic, complemented
3C2×#1465C211yesmaximal, characteristic, complemented
4#11776C218yesmaximal, direct product factor
5C2×#1343C214yesmaximal, complemented
6C22×#127—21no—
7C22×#127C411yescharacteristic, complemented
8C22×#121—21no—
9C22×#121—21no—
10C22×C4×D4C2211yescharacteristic
11C22×#99—21no—
12C2×#407—21no—
13#2541—22no—
14#2541—22no—
15#2541C414yescomplemented
16#2541C2214yes—
17C2×#392C412yescomplemented
18C2×#392C2212yes—
19C2×#390—21no—
20C2×C4×C4⋊C4C411yescharacteristic, complemented
21C2×C4×C22⋊C4—21no—
22C4×#99—22no—
23C4×#99C2214yescomplemented
24C2×#342C2212yes—
25#1465—22no—
26#1465C2214yescomplemented
27C2×#324—21no—
28C2×#323C2212yes—
29#1343C22116yescomplemented
30C23×D4—22no—
31C23×D4—22no—
32C23×D4—21no—
33C23×D4—21no—
34C23×D4C2×C412yes—
35C23×D4D411yescharacteristic
36C24×C4D411yescharacteristic
37C24×C4D411yescharacteristic
38C22×M16—41no—
39C2×#127—24no—
40C2×#127—24no—
41C2×#127—22no—
42C2×#127—42no—
43C2×#127—41no—
44C2×#127—41no—
45C2×#127—48no—
46C2×#127C2×C414yescomplemented
47C2×#121—24no—
48C2×#121—24no—
49C2×#121—24no—
50C2×#121—24no—
51C2×#121—48no—
52C2×#121—48no—
53C2×C4×D4—41no—
54C2×C4×D4—41no—
55C2×C4×D4—42no—
56C2×C4×D4C2×C412yes—
57C2×C4×D4D412yes—
58C2×C4×D4D412yes—
59C2×C4×D4C2312yes—
60C22×C4⋊C4—21no—
61C22×C4⋊C4D411yescharacteristic
62C22×C22⋊C4—22no—
63C22×C22⋊C4—22no—
64C22×C22⋊C4—22no—
65C22×C22⋊C4C2×C411yescharacteristic
66C22×C22⋊C4C2311yescharacteristic
67C22×C42—21no—
68C22×C42—41no—
69C22×C42C2×C411yescharacteristic
70C2×#99—28no—
71C2×#99—24no—
72#407—24no—
73#407—42no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—41no—
76#392—24no—
77#392—22no—
78#392—22no—
79#392C2×C414yes—
80#392C2×C418yescomplemented
81#392C2314yes—
82#390—24no—
83#390—24no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4C2×C414yescomplemented
86C4×C22⋊C4—24no—
87C4×C22⋊C4—42no—
88C2×#95—41no—
89C2×#95C2×C412yes—
90#342—22no—
91#342C2314yes—
92#324—24no—
93#323—22no—
94#323C2314yes—
95C25—22no—
96C25C22⋊C411yescharacteristic
97C25C2×D411yescharacteristic
98C22×D4—24no—
99C22×D4—24no—
100C22×D4—28no—
101C22×D4—24no—
102C22×D4—24no—
103C22×D4—24no—
104C22×D4—24no—
105C22×D4—22no—
106C22×D4—41no—
107C22×D4—41no—
108C22×D4—42no—
109C22×D4—42no—
110C22×D4—44no—
111C22×D4—44no—
112C22×D4—48no—
113C22×D4—48no—
114C22×D4—42no—
115C22×D4—42no—
116C22×D4—44no—
117C22×D4—48no—
118C22×D4—48no—
119C22×D4—42no—
120C22×D4—41no—
121C22×D4—41no—
122C22×D4—44no—
123C22×D4—84no—
124C22×D4—84no—
125C22×D4—84no—
126C22×D4C22⋊C412yes—
127C22×D4C22×C414yes—
128C22×D4C2×D412yes—
129C23×C4—22no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—44no—
137C23×C4—44no—
138C23×C4—42no—
139C23×C4C22⋊C411yescharacteristic
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22⋊C412yes—
142C23×C4C22⋊C412yes—
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22×C411yescharacteristic
145C23×C4C2×D412yes—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D412yes—
148C2×M16—44no—
149C2×M16—84no—
150C22×C8—42no—
151#127—44no—
152#127—44no—
153#127—44no—
154#127—44no—
155#127—416no—
156#127—416no—
157#121—416no—
158#121—416no—
159#121—416no—
160#121—416no—
161C4×D4—42no—
162C4×D4—42no—
163C4×D4—42no—
164C4×D4—42no—
165C4×D4—42no—
166C4×D4—42no—
167C4×D4—42no—
168C4×D4—42no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4C22⋊C412yes—
174C2×C4⋊C4C2×D412yes—
175C2×C22⋊C4—28no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—28no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4—41no—
184C2×C22⋊C4—48no—
185C2×C22⋊C4—48no—
186C2×C22⋊C4—42no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4C22⋊C412yes—
189C2×C22⋊C4C22×C412yes—
190C2×C22⋊C4C22×C411yescharacteristic
191C2×C22⋊C4C2×D411yescharacteristic
192C2×C22⋊C4C2×D411yescharacteristic
193C2×C22⋊C4C2411yescharacteristic
194C2×C42—22no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—24no—
198C2×C42—44no—
199C2×C42—44no—
200C2×C42—84no—
201C2×C42C22⋊C412yes—
202C2×C42C22×C412yes—
203#99—216no—
204C22⋊C8—24no—
205C22⋊C8—42no—
206C22⋊C8—44no—
207#95—22no—
208#95—44no—
209#95C22×C414yes—
210C24—21no—
211C24—21no—
212C24—22no—
213C24—21no—
214C24—22no—
215C24—21no—
216C24—22no—
217C24—24no—
218C24—42no—
219C24—42no—
220C24—42no—
221C24—42no—
222C24—44no—
223C24—44no—
224C24—44no—
225C24—44no—
226C24—42no—
227C24—42no—
228C24—44no—
229C24—44no—
230C24C2×C22⋊C411yescharacteristic
231C24C2×C22⋊C412yes—
232C24C2×C22⋊C411yescharacteristic
233C24AGL(1,ℤ/8)14yes—
234C24C22×D411yescharacteristic
235C2×D4—24no—
236C2×D4—24no—
237C2×D4—24no—
238C2×D4—24no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—48no—
242C2×D4—48no—
243C2×D4—44no—
244C2×D4—48no—
245C2×D4—48no—
246C2×D4—48no—
247C2×D4—416no—
248C2×D4—48no—
249C2×D4—48no—
250C2×D4—48no—
251C2×D4—44no—
252C2×D4—416no—
253C2×D4—48no—
254C2×D4—48no—
255C2×D4—44no—
256C2×D4—44no—
257C2×D4—48no—
258C2×D4—84no—
259C2×D4—88no—
260C2×D4—88no—
261C2×D4—88no—
262C2×D4—84no—
263C2×D4—88no—
264C2×D4—84no—
265C2×D4—84no—
266C2×D4—832no—
267C2×D4—832no—
268C2×D4—88no—
269C2×D4—88no—
270C2×D4—84no—
271C2×D4—84no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—28no—
277C22×C4—24no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4—44no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—44no—
304C22×C4—48no—
305C22×C4—84no—
306C22×C4—84no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4—84no—
310C22×C4—84no—
311C22×C4#9912yes—
312C22×C4#9911yescharacteristic
313C22×C4#9911yescharacteristic
314C22×C4#10414yes—
315C22×C4C2×C22⋊C412yes—
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C2×C22⋊C412yes—
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C23×C411yescharacteristic
321C22×C4C22×D411yescharacteristic
322M16—88no—
323C2×C8—44no—
324C2×C8—84no—
325C4⋊C4—216no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C22⋊C4—28no—
329C22⋊C4—42no—
330C22⋊C4—41no—
331C22⋊C4—42no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—48no—
336C22⋊C4—416no—
337C22⋊C4—48no—
338C22⋊C4—48no—
339C22⋊C4—44no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—48no—
345C42—88no—
346C23—22no—
347C23—22no—
348C23—21no—
349C23—21no—
350C23—24no—
351C23—21no—
352C23—21no—
353C23—24no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—41no—
358C23—42no—
359C23—41no—
360C23—41no—
361C23—42no—
362C23—48no—
363C23—41no—
364C23—48no—
365C23—44no—
366C23—44no—
367C23—44no—
368C23—44no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—44no—
376C23—44no—
377C23—44no—
378C23—81no—
379C23—81no—
380C23—84no—
381C23—81no—
382C23—81no—
383C23—84no—
384C23—88no—
385C23—88no—
386C23—84no—
387C23—84no—
388C23—88no—
389C23—88no—
390C23—84no—
391C23—84no—
392C23—84no—
393C23—84no—
394C23C2×#9911yescharacteristic
395C23C2×#9911yescharacteristic
396C23#41812yes—
397C23#41812yes—
398C23C2×#10412yes—
399C23C22×C22⋊C411yescharacteristic
400C23C2×AGL(1,ℤ/8)12yes—
401D4—48no—
402D4—48no—
403D4—88no—
404D4—816no—
405D4—816no—
406D4—816no—
407D4—832no—
408D4—88no—
409D4—88no—
410D4—88no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—28no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—21no—
424C2×C4—416no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—48no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—84no—
448C2×C4—816no—
449C2×C4—84no—
450C2×C4#35314yes—
451C2×C4C2×#9912yes—
452C2×C4C2×#9912yes—
453C2×C4C2×#10414yes—
454C8—84no—
455C22—24no—
456C22—24no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—22no—
462C22—22no—
463C22—42no—
464C22—42no—
465C22—44no—
466C22—48no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—42no—
472C22—42no—
473C22—42no—
474C22—81no—
475C22—81no—
476C22—84no—
477C22—82no—
478C22—84no—
479C22—84no—
480C22—84no—
481C22—84no—
482C22—84no—
483C22—84no—
484C22—84no—
485C22—84no—
486C22—84no—
487C22—168no—
488C22—168no—
489C22#150712yes—
490C22#150712yes—
491C22#212212yes—
492C22C2×#35312yes—
493C22C22×#9911yescharacteristic
494C22C2×#41811yescharacteristic
495C22C22×#10411yescharacteristic
496C4—22no—
497C4—24no—
498C4—48no—
499C4—41no—
500C4—41no—
501C4—41no—
502C4—41no—
503C4—42no—
504C4—88no—
505C4C2×#35314yes—
506C2—21no—
507C2—21no—
508C2—22no—
509C2—42no—
510C2—42no—
511C2—82no—
512C2—164no—
513C2#1177614yesdirect product factor
514C2C2×#150711yescharacteristic
515C2C2×#212211yescharacteristic
516C2C22×#35311yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC2×#2541C2
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
2. C2×C4×#99C2semidirect
3. C2×#1465C2semidirect
4. #11776C2direct (N × H)
5. C2×#1343C2semidirect
7. C22×#127C4semidirect
15. #2541C4semidirect
17. C2×#392C4semidirect
20. C2×C4×C4⋊C4C4semidirect
23. C4×#99C22semidirect
26. #1465C22semidirect
29. #1343C22semidirect
46. C2×#127C2×C4semidirect
80. #392C2×C4semidirect
85. C4×C4⋊C4C2×C4semidirect
513. C2#11776direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25412
C2×C4×#992
C2×#14652
#117762
C2×#13432