Finite Group Databasebeta

← C2×#11567 #480518

Subgroups — C2×#11567 #480518

Order 512.

1272 conjugacy classes of subgroups.

The subgroup lattice has 421 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2548C211yesmaximal, characteristic
2C2×#1467C211yesmaximal, characteristic, complemented
3C2×#1461C211yesmaximal, characteristic, complemented
4#11567C218yesmaximal, direct product factor
5C2×#1318C214yesmaximal, complemented
6C22×#127C411yescharacteristic
7C22×#124C412yes—
8C22×C4×Q8C2211yescharacteristic
9C2×#432—21no—
10C2×#431—21no—
11#2548—22no—
12#2548—22no—
13#2548C414yes—
14#2548C2214yes—
15C2×#394—21no—
16C2×#394—21no—
17C2×#390—21no—
18C2×#390—21no—
19C2×#390C2212yes—
20C2×C4×C4⋊C4C411yescharacteristic
21#1467—22no—
22#1467C2214yescomplemented
23#1461—22no—
24#1461C2214yescomplemented
25C2×#324—21no—
26C2×#324—21no—
27C2×#324C2212yes—
28C2×#324C2212yes—
29#1318C22116yescomplemented
30C23×Q8C2×C411yescharacteristic
31C23×D4—21no—
32C23×D4—21no—
33C23×D4C2×C412yes—
34C2×#127—22no—
35C2×#127—42no—
36C2×#127—41no—
37C2×#127—41no—
38C2×#127C2×C414yes—
39C2×#124—24no—
40C2×#124—48no—
41C2×#124C2×C418yes—
42C2×C4×Q8—41no—
43C2×C4×Q8—41no—
44C2×C4×Q8—42no—
45C2×C4×Q8C2×C412yes—
46C2×C4×Q8D412yes—
47C2×C4×Q8D412yes—
48C2×C4×Q8C2312yes—
49C22×C4⋊C4—21no—
50C22×C4⋊C4D411yescharacteristic
51C22×C4⋊C4D411yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C22⋊C4—22no—
54C22×C22⋊C4—22no—
55C22×C22⋊C4—22no—
56C22×C22⋊C4—22no—
57C22×C42C2×C411yescharacteristic
58C22×C42C2×C412yes—
59C22×C42D411yescharacteristic
60C22×C42D411yescharacteristic
61#432—24no—
62#432—42no—
63#431—24no—
64#431—42no—
65C2×C4⋊C8—41no—
66C2×C4⋊C8—41no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8—22no—
69C2×#97—41no—
70C2×C4×C8—41no—
71#394—24no—
72#394—24no—
73#394—24no—
74#394—24no—
75#390—22no—
76#390—22no—
77#390—24no—
78#390—24no—
79#390—24no—
80#390—24no—
81#390C2×C414yes—
82#390C2314yes—
83C4×C4⋊C4—22no—
84C4×C4⋊C4C2×C414yes—
85C2×#95—21no—
86#324—24no—
87#324—24no—
88#324—22no—
89#324—22no—
90#324C2314yes—
91#324C2314yes—
92C25—22no—
93C22×Q8—24no—
94C22×Q8—24no—
95C22×Q8—41no—
96C22×Q8—41no—
97C22×Q8C22⋊C412yes—
98C22×Q8C22×C412yes—
99C22×D4—22no—
100C22×D4—24no—
101C22×D4—24no—
102C22×D4—28no—
103C22×D4—44no—
104C22×D4—42no—
105C22×D4—42no—
106C22×D4—42no—
107C22×D4—42no—
108C22×D4—44no—
109C22×D4—44no—
110C22×D4—42no—
111C22×D4—41no—
112C22×D4—41no—
113C22×D4—84no—
114C22×D4C22×C414yes—
115C23×C4—22no—
116C23×C4C22⋊C411yescharacteristic
117C23×C4C22⋊C411yescharacteristic
118C23×C4C22⋊C411yescharacteristic
119C23×C4C22⋊C411yescharacteristic
120C23×C4C22×C411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C22×C8—42no—
124C22×C8—42no—
125#127—44no—
126#127—44no—
127#127—44no—
128#127—44no—
129#124—416no—
130#124—416no—
131C4×Q8—42no—
132C4×Q8—42no—
133C4×Q8—42no—
134C4×Q8—42no—
135C4×Q8—42no—
136C4×Q8—42no—
137C4×Q8—42no—
138C4×Q8—42no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4C22⋊C412yes—
148C2×C4⋊C4C22⋊C412yes—
149C2×C4⋊C4C22×C411yescharacteristic
150C2×C4⋊C4C2×D412yes—
151C2×C4⋊C4C2×D412yes—
152C2×C4⋊C4C2×D411yescharacteristic
153C2×C4⋊C4C2×D411yescharacteristic
154C2×C4⋊C4C2411yescharacteristic
155C2×C22⋊C4—28no—
156C2×C22⋊C4—28no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—48no—
162C2×C22⋊C4—48no—
163C2×C22⋊C4—84no—
164C2×C22⋊C4—84no—
165C2×C42—22no—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—44no—
176C2×C42C22⋊C412yes—
177C2×C42C22⋊C412yes—
178C2×C42C22⋊C412yes—
179C2×C42C22×C412yes—
180C2×C42C22×C414yes—
181C2×C42C2×D412yes—
182C2×C42C2×D412yes—
183C4⋊C8—44no—
184C4⋊C8—44no—
185C22⋊C8—24no—
186C22⋊C8—24no—
187C22⋊C8—42no—
188C22⋊C8—42no—
189#97—44no—
190C4×C8—44no—
191#95—24no—
192C24—24no—
193C24—44no—
194C24—44no—
195C24—44no—
196C24—44no—
197C24—44no—
198C24—42no—
199C24—42no—
200C24—44no—
201C24C2×C22⋊C411yescharacteristic
202C2×Q8—24no—
203C2×Q8—24no—
204C2×Q8—24no—
205C2×Q8—24no—
206C2×Q8—42no—
207C2×Q8—42no—
208C2×Q8—48no—
209C2×Q8—48no—
210C2×D4—28no—
211C2×D4—44no—
212C2×D4—44no—
213C2×D4—44no—
214C2×D4—44no—
215C2×D4—48no—
216C2×D4—48no—
217C2×D4—48no—
218C2×D4—44no—
219C2×D4—44no—
220C2×D4—44no—
221C2×D4—88no—
222C2×D4—88no—
223C2×D4—84no—
224C2×D4—88no—
225C2×D4—88no—
226C2×D4—88no—
227C2×D4—88no—
228C2×D4—84no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—24no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4#9912yes—
254C22×C4#9911yescharacteristic
255C22×C4#9911yescharacteristic
256C22×C4#10014yes—
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4#13114yes—
265C22×C4C23×C411yescharacteristic
266C22×C4C22×D411yescharacteristic
267C22×C4C22×D411yescharacteristic
268C2×C8—44no—
269C2×C8—44no—
270C2×C8—84no—
271C2×C8—84no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—41no—
280C4⋊C4—48no—
281C22⋊C4—416no—
282C22⋊C4—416no—
283C22⋊C4—84no—
284C22⋊C4—84no—
285C22⋊C4—84no—
286C22⋊C4—84no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—22no—
293C42—22no—
294C42—22no—
295C42—22no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42—44no—
300C42—44no—
301C23—22no—
302C23—22no—
303C23—44no—
304C23—44no—
305C23—44no—
306C23—44no—
307C23—44no—
308C23—44no—
309C23—44no—
310C23—84no—
311C23—84no—
312C23—88no—
313C23—88no—
314C23—88no—
315C23—88no—
316C23—84no—
317C23—84no—
318C23—84no—
319C23—84no—
320C23#40812yes—
321C23C2×#9911yescharacteristic
322C23C2×#9911yescharacteristic
323C23C2×#10012yes—
324C23C22×C22⋊C411yescharacteristic
325Q8—48no—
326Q8—48no—
327D4—88no—
328D4—88no—
329D4—88no—
330D4—816no—
331D4—88no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—88no—
370C2×C4#35514yes—
371C2×C4#40812yes—
372C2×C4C2×#9912yes—
373C2×C4C2×#9912yes—
374C2×C4C2×#10014yes—
375C2×C4C2×#13112yes—
376C8—84no—
377C8—84no—
378C22—24no—
379C22—24no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—84no—
385C22—84no—
386C22—84no—
387C22—84no—
388C22—84no—
389C22—84no—
390C22—84no—
391C22—168no—
392C22—168no—
393C22#147112yes—
394C22#147112yes—
395C22#212812yes—
396C22C2×#35512yes—
397C22C2×#40811yescharacteristic
398C22C22×#9911yescharacteristic
399C22C22×#10011yescharacteristic
400C4—22no—
401C4—22no—
402C4—22no—
403C4—24no—
404C4—41no—
405C4—41no—
406C4—42no—
407C4—41no—
408C4—41no—
409C4—42no—
410C4—84no—
411C4C2×#35514yes—
412C2—21no—
413C2—21no—
414C2—22no—
415C2—164no—
416C2#1156714yesdirect product factor
417C2C2×#147111yescharacteristic
418C2C2×#212811yescharacteristic
419C2C22×#35511yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC22×#127C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1467C2semidirect
3. C2×#1461C2semidirect
4. #11567C2direct (N × H)
5. C2×#1318C2semidirect
22. #1467C22semidirect
24. #1461C22semidirect
29. #1318C22semidirect
416. C2#11567direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25482
C2×#14672
C2×#14612
#115672
C2×#13182