Finite Group Databasebeta

← C2×#11569 #480538

Subgroups — C2×#11569 #480538

Order 512.

1138 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic
2C2×#1465C211yesmaximal, characteristic
3C2×#1459C211yesmaximal, characteristic
4#11569C218yesmaximal, direct product factor
5C2×#1324C214yesmaximal, complemented
6C22×C4⋊Q8C411yescharacteristic
7C22×C4×D4C2211yescharacteristic
8C2×#407—21no—
9C22×C22⋊C8—21no—
10#2563—22no—
11#2563—22no—
12#2563C414yes—
13#2563C2214yes—
14C2×#393—21no—
15C2×#393—21no—
16C2×#390C412yes—
17C2×#382C2212yes—
18C2×#380—21no—
19C2×#380—21no—
20C2×C4×C4⋊C4C411yescharacteristic
21#1465—22no—
22#1465C2214yes—
23#1459—22no—
24#1459C2214yes—
25C2×#324C2212yes—
26C2×#324C2212yes—
27C2×#323—21no—
28C2×#323—21no—
29#1324C22116yescomplemented
30C23×Q8—21no—
31C23×D4C2×C411yescharacteristic
32C24×C4D411yescharacteristic
33C24×C4D411yescharacteristic
34C22×M16—41no—
35C23×C8—41no—
36C2×C4⋊Q8—22no—
37C2×C4⋊Q8—42no—
38C2×C4⋊Q8—41no—
39C2×C4⋊Q8—41no—
40C2×C4⋊Q8C2×C414yes—
41C2×C4×D4—41no—
42C2×C4×D4—41no—
43C2×C4×D4—42no—
44C2×C4×D4C2×C412yes—
45C2×C4×D4D412yes—
46C2×C4×D4D412yes—
47C2×C4×D4C2312yes—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4C2×C412yes—
51C22×C4⋊C4C2311yescharacteristic
52C22×C22⋊C4—22no—
53C22×C22⋊C4—22no—
54C22×C22⋊C4D411yescharacteristic
55C22×C22⋊C4D411yescharacteristic
56C22×C42C2×C412yes—
57C22×C42C2×C411yescharacteristic
58#407—24no—
59#407—42no—
60C2×C22⋊C8—22no—
61C2×C22⋊C8—24no—
62C2×C22⋊C8—22no—
63C2×C22⋊C8—42no—
64C2×C22⋊C8—41no—
65C2×C22⋊C8—41no—
66#393—24no—
67#393—24no—
68#393—24no—
69#393—24no—
70#390—24no—
71#390C2×C418yes—
72#382—22no—
73#382—22no—
74#382C2×C414yes—
75#382C2314yes—
76#380—24no—
77#380—24no—
78#380—24no—
79#380—24no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4C2×C414yes—
82C2×#95—21no—
83C2×#95—22no—
84C2×#95—22no—
85#324—22no—
86#324—22no—
87#324C2314yes—
88#324C2314yes—
89#323—24no—
90#323—24no—
91C25C22⋊C411yescharacteristic
92C25C22⋊C411yescharacteristic
93C22×Q8—24no—
94C22×Q8—42no—
95C22×Q8—42no—
96C22×Q8—84no—
97C22×D4—24no—
98C22×D4—24no—
99C22×D4—41no—
100C22×D4—41no—
101C22×D4C22⋊C412yes—
102C22×D4C22×C412yes—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—44no—
112C23×C4—44no—
113C23×C4C22⋊C411yescharacteristic
114C23×C4C22⋊C412yes—
115C23×C4C22⋊C412yes—
116C23×C4C22⋊C411yescharacteristic
117C23×C4C22×C411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D412yes—
121C23×C4C2×D412yes—
122C2×M16—44no—
123C2×M16—84no—
124C22×C8—42no—
125C22×C8—44no—
126C22×C8—42no—
127C22×C8—84no—
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—22no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—44no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4C22×C414yes—
152C2×C4⋊C4C22×C411yescharacteristic
153C2×C4⋊C4C2×D411yescharacteristic
154C2×C4⋊C4C2×D411yescharacteristic
155C2×C4⋊C4C2411yescharacteristic
156C2×C22⋊C4—28no—
157C2×C22⋊C4—28no—
158C2×C22⋊C4—41no—
159C2×C22⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C22⋊C4—41no—
162C2×C22⋊C4—84no—
163C2×C22⋊C4—84no—
164C2×C22⋊C4C22⋊C412yes—
165C2×C22⋊C4C22⋊C412yes—
166C2×C22⋊C4C2×D412yes—
167C2×C22⋊C4C2×D412yes—
168C2×C42—22no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—28no—
172C2×C42—22no—
173C2×C42C22⋊C412yes—
174C2×C42C22×C414yes—
175C2×C42C22×C412yes—
176C22⋊C8—24no—
177C22⋊C8—24no—
178C22⋊C8—42no—
179C22⋊C8—44no—
180C22⋊C8—42no—
181C22⋊C8—44no—
182#95—24no—
183#95—24no—
184#95—24no—
185#95—24no—
186#95—24no—
187C24—21no—
188C24—22no—
189C24—21no—
190C24—21no—
191C24—22no—
192C24—22no—
193C24—21no—
194C24—42no—
195C24—42no—
196C24—42no—
197C24—42no—
198C24#10114yes—
199C24C2×C22⋊C411yescharacteristic
200C24C2×C22⋊C412yes—
201C24C2×C22⋊C412yes—
202C2×Q8—48no—
203C2×Q8—88no—
204C2×Q8—88no—
205C2×Q8—88no—
206C2×D4—24no—
207C2×D4—24no—
208C2×D4—24no—
209C2×D4—24no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4—48no—
213C2×D4—48no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—44no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—84no—
251C22×C4—84no—
252C22×C4—82no—
253C22×C4—82no—
254C22×C4#9912yes—
255C22×C4#9911yescharacteristic
256C22×C4#9911yescharacteristic
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C22⋊C412yes—
262C22×C4#13114yes—
263C22×C4C23×C411yescharacteristic
264C22×C4C22×D411yescharacteristic
265C22×C4C22×D411yescharacteristic
266M16—88no—
267C2×C8—44no—
268C2×C8—44no—
269C2×C8—84no—
270C2×C8—88no—
271C2×C8—84no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—44no—
277C4⋊C4—48no—
278C4⋊C4—44no—
279C4⋊C4—44no—
280C4⋊C4—44no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C22⋊C4—42no—
284C22⋊C4—42no—
285C22⋊C4—88no—
286C22⋊C4—88no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—28no—
292C42—44no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—24no—
297C23—22no—
298C23—24no—
299C23—22no—
300C23—22no—
301C23—22no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—42no—
306C23—48no—
307C23—48no—
308C23—42no—
309C23—42no—
310C23—42no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—81no—
315C23—81no—
316C23—84no—
317C23—81no—
318C23—81no—
319C23—84no—
320C23#40812yes—
321C23C2×#9911yescharacteristic
322C23C2×#9911yescharacteristic
323C23C2×#10112yes—
324C23C2×#10114yes—
325C23C22×C22⋊C411yescharacteristic
326Q8—816no—
327D4—48no—
328D4—48no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—88no—
369C2×C4—84no—
370C2×C4—84no—
371C2×C4#35414yes—
372C2×C4#40812yes—
373C2×C4C2×#9912yes—
374C2×C4C2×#9912yes—
375C2×C4C2×#13112yes—
376C8—84no—
377C8—84no—
378C22—24no—
379C22—24no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—24no—
385C22—44no—
386C22—44no—
387C22—48no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—82no—
394C22—84no—
395C22—82no—
396C22—84no—
397C22—84no—
398C22#147212yes—
399C22#147212yes—
400C22C2×#35412yes—
401C22#213412yes—
402C22C2×#40811yescharacteristic
403C22C22×#9911yescharacteristic
404C22C22×#10111yescharacteristic
405C4—22no—
406C4—22no—
407C4—22no—
408C4—41no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4—42no—
413C4—84no—
414C4—84no—
415C4C2×#35414yes—
416C2—21no—
417C2—21no—
418C2—22no—
419C2—44no—
420C2—82no—
421C2#1156914yesdirect product factor
422C2C2×#147211yescharacteristic
423C2C22×#35411yescharacteristic
424C2C2×#213411yescharacteristic
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. #11569C2direct (N × H)
5. C2×#1324C2semidirect
29. #1324C22semidirect
421. C2#11569direct (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×#14652
C2×#14592
#115692
C2×#13242