Finite Group Databasebeta

← C2×#11787 #481632

Subgroups — C2×#11787 #481632

Order 512.

1234 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic
2C2×C4×#99C211yesmaximal, characteristic
3C2×#1465C211yesmaximal, characteristic
4#11787C218yesmaximal, direct product factor
5C2×#1351C214yesmaximal, complemented
6C22×C4⋊Q8C411yescharacteristic, complemented
7C22×C4×D4C2211yescharacteristic
8C22×#99—21no—
9C2×#407—21no—
10#2563—22no—
11#2563—22no—
12#2563C414yescomplemented
13#2563C2214yes—
14C2×#393C412yescomplemented
15C2×#393C2212yes—
16C2×#390—21no—
17C2×#382—21no—
18C2×#380—21no—
19C2×#380—21no—
20C2×C4×C4⋊C4C411yescharacteristic, complemented
21C2×C4×C22⋊C4—21no—
22C4×#99—22no—
23C4×#99C2214yes—
24C2×#342C2212yes—
25#1465—22no—
26#1465C2214yes—
27C2×#324—21no—
28C2×#323C2212yes—
29#1351C22116yescomplemented
30C23×Q8C2×C412yes—
31C23×D4D411yescharacteristic
32C24×C4D411yescharacteristic
33C24×C4D411yescharacteristic
34C22×M16—41no—
35C2×C4⋊Q8—22no—
36C2×C4⋊Q8—42no—
37C2×C4⋊Q8—41no—
38C2×C4⋊Q8—41no—
39C2×C4⋊Q8C2×C414yescomplemented
40C2×C4×D4—41no—
41C2×C4×D4—41no—
42C2×C4×D4—42no—
43C2×C4×D4C2×C412yes—
44C2×C4×D4D412yes—
45C2×C4×D4D412yes—
46C2×C4×D4C2312yes—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4D411yescharacteristic
51C22×C22⋊C4—22no—
52C22×C22⋊C4—22no—
53C22×C22⋊C4—22no—
54C22×C22⋊C4C2×C411yescharacteristic
55C22×C22⋊C4C2311yescharacteristic
56C22×C42—21no—
57C22×C42—41no—
58C22×C42C2×C411yescharacteristic
59C2×#99—28no—
60C2×#99—24no—
61#407—24no—
62#407—42no—
63C2×C22⋊C8—22no—
64C2×C22⋊C8—41no—
65#393—24no—
66#393—22no—
67#393—22no—
68#393C2×C414yes—
69#393C2×C418yescomplemented
70#393C2314yes—
71#390—24no—
72#390—24no—
73#382—24no—
74#382—24no—
75#380—24no—
76#380—24no—
77#380—24no—
78#380—24no—
79C4×C4⋊C4—22no—
80C4×C4⋊C4C2×C414yescomplemented
81C4×C22⋊C4—24no—
82C4×C22⋊C4—42no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—41no—
86C2×#95C2×C412yes—
87#342—22no—
88#342C2314yes—
89#324—24no—
90#323—22no—
91#323C2314yes—
92C25C22⋊C411yescharacteristic
93C25C2×D411yescharacteristic
94C22×Q8—22no—
95C22×Q8—42no—
96C22×Q8—42no—
97C22×Q8—84no—
98C22×Q8C22×C414yes—
99C22×D4—24no—
100C22×D4—24no—
101C22×D4—41no—
102C22×D4—41no—
103C22×D4C22⋊C412yes—
104C22×D4C2×D412yes—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—22no—
109C23×C4—21no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—44no—
114C23×C4—44no—
115C23×C4—42no—
116C23×C4C22⋊C411yescharacteristic
117C23×C4C22⋊C411yescharacteristic
118C23×C4C22⋊C412yes—
119C23×C4C22⋊C412yes—
120C23×C4C22⋊C411yescharacteristic
121C23×C4C22×C411yescharacteristic
122C23×C4C2×D412yes—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D412yes—
125C2×M16—44no—
126C2×M16—84no—
127C22×C8—42no—
128C4⋊Q8—44no—
129C4⋊Q8—44no—
130C4⋊Q8—44no—
131C4⋊Q8—44no—
132C4×D4—42no—
133C4×D4—42no—
134C4×D4—42no—
135C4×D4—42no—
136C4×D4—42no—
137C4×D4—42no—
138C4×D4—42no—
139C4×D4—42no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4C22⋊C412yes—
152C2×C4⋊C4C2×D412yes—
153C2×C22⋊C4—28no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—28no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—41no—
159C2×C22⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C22⋊C4—41no—
162C2×C22⋊C4—42no—
163C2×C22⋊C4—44no—
164C2×C22⋊C4—84no—
165C2×C22⋊C4—84no—
166C2×C22⋊C4C22⋊C412yes—
167C2×C22⋊C4C22×C412yes—
168C2×C22⋊C4C22×C411yescharacteristic
169C2×C22⋊C4C2×D411yescharacteristic
170C2×C22⋊C4C2×D411yescharacteristic
171C2×C22⋊C4C2411yescharacteristic
172C2×C42—22no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—24no—
176C2×C42—44no—
177C2×C42—44no—
178C2×C42—84no—
179C2×C42C22⋊C412yes—
180C2×C42C22×C412yes—
181#99—216no—
182C22⋊C8—24no—
183C22⋊C8—42no—
184C22⋊C8—44no—
185#95—24no—
186#95—28no—
187#95—22no—
188#95—24no—
189#95—44no—
190#95C22×C414yes—
191C24—21no—
192C24—21no—
193C24—22no—
194C24—21no—
195C24—22no—
196C24—21no—
197C24—22no—
198C24—42no—
199C24—42no—
200C24—42no—
201C24—42no—
202C24C2×C22⋊C411yescharacteristic
203C24C2×C22⋊C412yes—
204C24C2×C22⋊C411yescharacteristic
205C24#13714yes—
206C24C22×D411yescharacteristic
207C2×Q8—44no—
208C2×Q8—44no—
209C2×Q8—84no—
210C2×Q8—84no—
211C2×Q8—88no—
212C2×Q8—88no—
213C2×D4—24no—
214C2×D4—24no—
215C2×D4—24no—
216C2×D4—24no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—48no—
220C2×D4—48no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—24no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—44no—
251C22×C4—48no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—44no—
259C22×C4—84no—
260C22×C4—84no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4—84no—
264C22×C4—84no—
265C22×C4#9912yes—
266C22×C4#9911yescharacteristic
267C22×C4#9911yescharacteristic
268C22×C4#10414yes—
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C23×C411yescharacteristic
275C22×C4C22×D411yescharacteristic
276M16—88no—
277C2×C8—44no—
278C2×C8—84no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—48no—
282C4⋊C4—48no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C22⋊C4—28no—
286C22⋊C4—42no—
287C22⋊C4—41no—
288C22⋊C4—42no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—48no—
293C22⋊C4—44no—
294C22⋊C4—84no—
295C22⋊C4—88no—
296C22⋊C4—84no—
297C42—22no—
298C42—22no—
299C42—22no—
300C42—22no—
301C42—48no—
302C42—88no—
303C23—22no—
304C23—22no—
305C23—21no—
306C23—21no—
307C23—24no—
308C23—21no—
309C23—21no—
310C23—24no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—41no—
315C23—42no—
316C23—41no—
317C23—41no—
318C23—42no—
319C23—41no—
320C23—48no—
321C23—48no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23—81no—
329C23—81no—
330C23—84no—
331C23—81no—
332C23—81no—
333C23—84no—
334C23C2×#9911yescharacteristic
335C23C2×#9911yescharacteristic
336C23#41912yes—
337C23#41912yes—
338C23C2×#10412yes—
339C23C22×C22⋊C411yescharacteristic
340C23C2×#13712yes—
341Q8—88no—
342Q8—88no—
343D4—48no—
344D4—48no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—48no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—88no—
384C2×C4—84no—
385C2×C4—84no—
386C2×C4—816no—
387C2×C4—84no—
388C2×C4—84no—
389C2×C4#35414yes—
390C2×C4C2×#9912yes—
391C2×C4C2×#9912yes—
392C2×C4C2×#10414yes—
393C8—84no—
394C22—24no—
395C22—24no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—22no—
401C22—22no—
402C22—42no—
403C22—42no—
404C22—44no—
405C22—48no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—81no—
414C22—81no—
415C22—84no—
416C22—82no—
417C22—84no—
418C22—84no—
419C22#150812yes—
420C22#150812yes—
421C22#212212yes—
422C22C2×#35412yes—
423C22C22×#9911yescharacteristic
424C22C2×#41911yescharacteristic
425C22C22×#10411yescharacteristic
426C4—22no—
427C4—24no—
428C4—41no—
429C4—41no—
430C4—41no—
431C4—41no—
432C4—42no—
433C4—84no—
434C4—88no—
435C4—84no—
436C4C2×#35414yes—
437C2—21no—
438C2—21no—
439C2—22no—
440C2—42no—
441C2—42no—
442C2—82no—
443C2#1178714yesdirect product factor
444C2C2×#150811yescharacteristic
445C2C2×#212211yescharacteristic
446C2C22×#35411yescharacteristic
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. #11787C2direct (N × H)
5. C2×#1351C2semidirect
6. C22×C4⋊Q8C4semidirect
12. #2563C4semidirect
14. C2×#393C4semidirect
20. C2×C4×C4⋊C4C4semidirect
29. #1351C22semidirect
39. C2×C4⋊Q8C2×C4semidirect
69. #393C2×C4semidirect
80. C4×C4⋊C4C2×C4semidirect
443. C2#11787direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25632
C2×C4×#992
C2×#14652
#117872
C2×#13512