Finite Group Databasebeta

← C2×#11764 #481692

Subgroups — C2×#11764 #481692

Order 512.

1226 conjugacy classes of subgroups.

The subgroup lattice has 487 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2484C211yesmaximal, characteristic
2C22×#343C211yesmaximal, characteristic
3C2×#1463C211yesmaximal, characteristic
4#11764C218yesmaximal, direct product factor
5C2×#1353C212yesmaximal, complemented
6C2×#1346C212yesmaximal, complemented
7C23×M16—21no—
8C22×#117C411yescharacteristic
9C23×C4⋊C4C2211yescharacteristic
10C2×#407—21no—
11#2484—22no—
12#2484—22no—
13#2484C414yes—
14#2484C2214yes—
15C2×#400C411yescharacteristic
16C2×#400C2211yescharacteristic
17C2×#398C411yescharacteristic
18C2×#398C2211yescharacteristic
19C2×#387—21no—
20C2×#387—21no—
21C2×#385—21no—
22C2×#381—21no—
23C2×#381C411yescharacteristic
24C2×#343—22no—
25C2×#343—21no—
26C2×#343C2212yes—
27C2×#343C2214yes—
28#1463—22no—
29#1463C2214yes—
30C2×#324—21no—
31C2×#324C2212yes—
32#1353C2218yescomplemented
33#1346C2218yescomplemented
34C24×C4C2×C411yescharacteristic
35C24×C4D411yescharacteristic
36C24×C4D411yescharacteristic
37C22×M16—28no—
38C22×M16—24no—
39C22×M16—22no—
40C22×M16—42no—
41C22×M16—41no—
42C22×M16—44no—
43C22×M16—41no—
44C23×C8—41no—
45C2×#117—22no—
46C2×#117—42no—
47C2×#117—41no—
48C2×#117—41no—
49C2×#117C2×C414yes—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—41no—
53C22×C4⋊C4—41no—
54C22×C4⋊C4—42no—
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2×C412yes—
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D412yes—
61C22×C4⋊C4D412yes—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C42—21no—
68#407—24no—
69#407—42no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—41no—
72#400—22no—
73#400—21no—
74#400—21no—
75#400C2×C414yes—
76#400C2×C412yes—
77#400C2312yes—
78#398—22no—
79#398—21no—
80#398—21no—
81#398C2×C414yes—
82#398C2×C412yes—
83#398C2312yes—
84#387—24no—
85#387—24no—
86#387—24no—
87#387—24no—
88#385—24no—
89#385—24no—
90#381—22no—
91#381—24no—
92#381—24no—
93#381C2×C414yes—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C411yescharacteristic
103#343—24no—
104#343—22no—
105#343C2314yes—
106#324—24no—
107#324—22no—
108#324C2314yes—
109C25C22⋊C411yescharacteristic
110C23×C4—21no—
111C23×C4—22no—
112C23×C4—21no—
113C23×C4—24no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—44no—
127C23×C4—44no—
128C23×C4C22⋊C412yes—
129C23×C4C22⋊C411yescharacteristic
130C23×C4C22⋊C411yescharacteristic
131C23×C4C22⋊C412yes—
132C23×C4C22⋊C412yes—
133C23×C4C22⋊C411yescharacteristic
134C23×C4C22×C411yescharacteristic
135C23×C4C22×C412yes—
136C23×C4C2×D412yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D412yes—
140C2×M16—28no—
141C2×M16—216no—
142C2×M16—24no—
143C2×M16—416no—
144C2×M16—48no—
145C2×M16—44no—
146C2×M16—42no—
147C2×M16—48no—
148C2×M16—44no—
149C2×M16—44no—
150C2×M16—84no—
151C22×C8—48no—
152C22×C8—44no—
153C22×C8—42no—
154C22×C8—42no—
155#117—44no—
156#117—44no—
157#117—44no—
158#117—44no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4C22⋊C412yes—
188C2×C4⋊C4C22⋊C412yes—
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22×C412yes—
191C2×C4⋊C4C22×C412yes—
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C4⋊C4C22×C412yes—
194C2×C4⋊C4C2×D412yes—
195C2×C4⋊C4C2×D411yescharacteristic
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×D411yescharacteristic
198C2×C4⋊C4C2411yescharacteristic
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—41no—
204C2×C22⋊C4—41no—
205C2×C42—24no—
206C2×C42—42no—
207C2×C42—42no—
208C22⋊C8—24no—
209C22⋊C8—42no—
210C22⋊C8—44no—
211#95—24no—
212#95—22no—
213#95—24no—
214#95—22no—
215#95—24no—
216#95—22no—
217#95—24no—
218#95—21no—
219#95—21no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95C22×C412yes—
224#95C22×C412yes—
225C24—24no—
226C24—22no—
227C24—22no—
228C24—42no—
229C24#9911yescharacteristic
230C24#9912yes—
231C24#9911yescharacteristic
232C24C2×C22⋊C412yes—
233C24C2×C22⋊C411yescharacteristic
234C22×C4—24no—
235C22×C4—28no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—28no—
239C22×C4—24no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—48no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—42no—
280C22×C4—44no—
281C22×C4—42no—
282C22×C4—44no—
283C22×C4—42no—
284C22×C4—44no—
285C22×C4—42no—
286C22×C4—44no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—44no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—82no—
301C22×C4—84no—
302C22×C4#10414yes—
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C2×C22⋊C412yes—
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4C2×C22⋊C412yes—
307C22×C4C2×C22⋊C412yes—
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4AGL(1,ℤ/8)12yes—
310C22×C4#13712yes—
311C22×C4C23×C411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314M16—28no—
315M16—416no—
316M16—48no—
317M16—44no—
318M16—88no—
319C2×C8—416no—
320C2×C8—44no—
321C2×C8—48no—
322C2×C8—44no—
323C2×C8—84no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—41no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—41no—
331C4⋊C4—42no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—44no—
341C4⋊C4—48no—
342C22⋊C4—48no—
343C22⋊C4—44no—
344C22⋊C4—44no—
345C42—48no—
346C23—22no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—24no—
351C23—24no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—24no—
357C23—22no—
358C23—42no—
359C23—42no—
360C23—48no—
361C23—48no—
362C23#35214yes—
363C23C2×#9912yes—
364C23C2×#9912yes—
365C23C2×#9911yescharacteristic
366C23C2×#9911yescharacteristic
367C23#41712yes—
368C23C2×#10412yes—
369C23C22×C22⋊C411yescharacteristic
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—28no—
375C2×C4—28no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—21no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—42no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—81no—
424C2×C4—82no—
425C2×C4—81no—
426C2×C4—82no—
427C2×C4—84no—
428C2×C4—84no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4#41712yes—
434C2×C4C2×#10414yes—
435C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
436C2×C4C2×#13711yescharacteristic
437C8—48no—
438C8—84no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—22no—
444C22—22no—
445C22—24no—
446C22—24no—
447C22—28no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—44no—
452C22—44no—
453C22—48no—
454C22—44no—
455C22#150412yes—
456C22#150412yes—
457C22C2×#35214yes—
458C22C2×#35212yes—
459C22#213312yes—
460C22C22×#9911yescharacteristic
461C22C2×#41711yescharacteristic
462C22C22×#10411yescharacteristic
463C4—21no—
464C4—21no—
465C4—22no—
466C4—24no—
467C4—41no—
468C4—42no—
469C4—41no—
470C4—41no—
471C4—41no—
472C4—42no—
473C4—82no—
474C4—82no—
475C4—82no—
476C4—82no—
477C2—21no—
478C2—21no—
479C2—24no—
480C2—22no—
481C2—42no—
482C2#1176414yesdirect product factor
483C2C2×#150411yescharacteristic
484C2C22×#35211yescharacteristic
485C2C2×#213311yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11764C2direct (N × H)
5. C2×#1353C2semidirect
6. C2×#1346C2semidirect
32. #1353C22semidirect
33. #1346C22semidirect
482. C2#11764direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24842
C22×#3432
C2×#14632
#117642
C2×#13532
C2×#13462