Finite Group Databasebeta

← C2×#11551 #480467

Subgroups — C2×#11551 #480467

Order 512.

1062 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2503C211yesmaximal, characteristic
2C2×#1463C211yesmaximal, characteristic
3C22×#324C211yesmaximal, characteristic
4#11551C218yesmaximal, direct product factor
5C2×#1324C212yesmaximal, complemented
6C2×#1319C212yesmaximal, complemented
7C22×C22⋊Q8C411yescharacteristic
8C23×C4⋊C4C2211yescharacteristic
9C2×#407—21no—
10C22×C22⋊C8—21no—
11#2503—22no—
12#2503—22no—
13#2503C414yes—
14#2503C2214yes—
15C2×#400—21no—
16C2×#400—21no—
17C2×#391C411yescharacteristic
18C2×#391C2211yescharacteristic
19C2×#382C411yescharacteristic
20C2×#382C2211yescharacteristic
21C2×#381—21no—
22C2×#381—21no—
23C2×C4×C22⋊C4C411yescharacteristic
24#1463—22no—
25#1463C2214yes—
26C2×#324—21no—
27C2×#324—22no—
28C2×#324—21no—
29C2×#324C2214yes—
30C2×#324C2212yes—
31C2×#324C2212yes—
32#1324C2218yescomplemented
33#1319C2218yescomplemented
34C23×Q8C2×C411yescharacteristic
35C24×C4C2×C411yescharacteristic
36C24×C4D411yescharacteristic
37C24×C4D411yescharacteristic
38C22×M16—41no—
39C23×C8—41no—
40C2×C22⋊Q8—22no—
41C2×C22⋊Q8—42no—
42C2×C22⋊Q8—41no—
43C2×C22⋊Q8—41no—
44C2×C22⋊Q8C2×C414yes—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—41no—
47C22×C4⋊C4—41no—
48C22×C4⋊C4—42no—
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C412yes—
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4D411yescharacteristic
53C22×C4⋊C4D411yescharacteristic
54C22×C4⋊C4D412yes—
55C22×C4⋊C4D412yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2312yes—
58C22×C22⋊C4—21no—
59C22×C22⋊C4—21no—
60C22×C42C2×C411yescharacteristic
61C22×C42C2×C411yescharacteristic
62#407—24no—
63#407—42no—
64C2×C22⋊C8—22no—
65C2×C22⋊C8—24no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—42no—
68C2×C22⋊C8—41no—
69C2×C22⋊C8—41no—
70#400—24no—
71#400—24no—
72#400—24no—
73#400—24no—
74#391—22no—
75#391—21no—
76#391—21no—
77#391C2×C412yes—
78#391C2×C414yes—
79#391C2312yes—
80#382—22no—
81#382—21no—
82#382—21no—
83#382C2×C412yes—
84#382C2×C414yes—
85#382C2312yes—
86#381—24no—
87#381—24no—
88#381—24no—
89#381—24no—
90C4×C22⋊C4—22no—
91C4×C22⋊C4C2×C414yes—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101#324—22no—
102#324—24no—
103#324—24no—
104#324—22no—
105#324C2314yes—
106#324C2314yes—
107C25C22⋊C411yescharacteristic
108C22×Q8—21no—
109C22×Q8—24no—
110C22×Q8—41no—
111C22×Q8—41no—
112C22×Q8—42no—
113C22×Q8C22×C412yes—
114C23×C4—22no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—22no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4—24no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—42no—
129C23×C4—44no—
130C23×C4—44no—
131C23×C4C22⋊C411yescharacteristic
132C23×C4C22⋊C412yes—
133C23×C4C22⋊C411yescharacteristic
134C23×C4C22⋊C411yescharacteristic
135C23×C4C22⋊C412yes—
136C23×C4C22⋊C412yes—
137C23×C4C22×C412yes—
138C23×C4C22×C411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D412yes—
142C23×C4C2×D412yes—
143C2×M16—44no—
144C2×M16—84no—
145C22×C8—44no—
146C22×C8—42no—
147C22×C8—42no—
148C22×C8—84no—
149C22⋊Q8—44no—
150C22⋊Q8—44no—
151C22⋊Q8—44no—
152C22⋊Q8—44no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4C22⋊C412yes—
176C2×C4⋊C4C22⋊C412yes—
177C2×C4⋊C4C22⋊C412yes—
178C2×C4⋊C4C22×C412yes—
179C2×C4⋊C4C22×C411yescharacteristic
180C2×C4⋊C4C22×C412yes—
181C2×C4⋊C4C2×D411yescharacteristic
182C2×C4⋊C4C2×D411yescharacteristic
183C2×C4⋊C4C2×D412yes—
184C2×C4⋊C4C2×D412yes—
185C2×C4⋊C4C2411yescharacteristic
186C2×C22⋊C4—24no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—42no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C42—24no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—42no—
196C2×C42C22×C412yes—
197C2×C42C22×C412yes—
198C22⋊C8—24no—
199C22⋊C8—24no—
200C22⋊C8—44no—
201C22⋊C8—42no—
202C22⋊C8—42no—
203C22⋊C8—44no—
204#95—24no—
205#95—24no—
206#95—24no—
207#95—22no—
208#95—22no—
209#95—24no—
210#95—24no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217C24—24no—
218C24—22no—
219C24—22no—
220C24—42no—
221C24#9911yescharacteristic
222C24#9912yes—
223C24#9911yescharacteristic
224C24C2×C22⋊C412yes—
225C24C2×C22⋊C411yescharacteristic
226C2×Q8—24no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—84no—
233C2×Q8—84no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—44no—
270C22×C4—48no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—44no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—84no—
296C22×C4—82no—
297C22×C4—82no—
298C22×C4—84no—
299C22×C4C2×C22⋊C412yes—
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C22⋊C412yes—
302C22×C4C2×C22⋊C412yes—
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4#13114yes—
306C22×C4C23×C411yescharacteristic
307C22×C4C22×D411yescharacteristic
308C22×C4C22×D411yescharacteristic
309M16—88no—
310C2×C8—44no—
311C2×C8—44no—
312C2×C8—88no—
313C2×C8—84no—
314C2×C8—84no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—41no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—48no—
328C22⋊C4—48no—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C42—24no—
332C42—42no—
333C42—42no—
334C42—42no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—24no—
342C23—24no—
343C23—22no—
344C23—22no—
345C23—24no—
346C23—22no—
347C23—42no—
348C23—42no—
349C23—48no—
350C23—48no—
351C23#35414yes—
352C23#40812yes—
353C23C2×#9912yes—
354C23C2×#9912yes—
355C23C2×#9911yescharacteristic
356C23C2×#9911yescharacteristic
357C23#41312yes—
358C23C22×C22⋊C411yescharacteristic
359Q8—84no—
360Q8—84no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—84no—
416C2×C4—84no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4#40812yes—
422C2×C4C2×#13112yes—
423C8—84no—
424C8—84no—
425C22—24no—
426C22—24no—
427C22—28no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—22no—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—44no—
438C22—44no—
439C22—48no—
440C22—44no—
441C22#146812yes—
442C22#146812yes—
443C22C2×#35414yes—
444C22C2×#35412yes—
445C22#213312yes—
446C22C2×#40811yescharacteristic
447C22C22×#9911yescharacteristic
448C22C2×#41311yescharacteristic
449C4—22no—
450C4—22no—
451C4—41no—
452C4—41no—
453C4—42no—
454C4—41no—
455C4—41no—
456C4—42no—
457C4—42no—
458C4—82no—
459C4—82no—
460C4—82no—
461C4—82no—
462C2—21no—
463C2—21no—
464C2—24no—
465C2—22no—
466C2—42no—
467C2#1155114yesdirect product factor
468C2C2×#146811yescharacteristic
469C2C22×#35411yescharacteristic
470C2C2×#213311yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ 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. #11551C2direct (N × H)
5. C2×#1324C2semidirect
6. C2×#1319C2semidirect
32. #1324C22semidirect
33. #1319C22semidirect
467. C2#11551direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25032
C2×#14632
C22×#3242
#115512
C2×#13242
C2×#13192