Finite Group Databasebeta

← C2×#11555 #480472

Subgroups — C2×#11555 #480472

Order 512.

1106 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2502C211yesmaximal, characteristic
2C2×#1463C211yesmaximal, characteristic
3C22×#324C211yesmaximal, characteristic
4#11555C218yesmaximal, direct product factor
5C2×#1325C212yesmaximal, complemented
6C2×#1319C212yesmaximal, complemented
7C22×C22⋊Q8C411yescharacteristic
8C23×C4⋊C4C2211yescharacteristic
9C2×#407—21no—
10C22×C22⋊C8—21no—
11#2502—22no—
12#2502—22no—
13#2502C414yes—
14#2502C2214yes—
15C2×#398—21no—
16C2×#398—21no—
17C2×#391C411yescharacteristic
18C2×#391C2211yescharacteristic
19C2×#385—21no—
20C2×#385—21no—
21C2×#384C411yescharacteristic
22C2×#384C2211yescharacteristic
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#1325C2218yescomplemented
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—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—41no—
51C22×C4⋊C4—41no—
52C22×C4⋊C4—42no—
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4C2×C412yes—
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4D412yes—
59C22×C4⋊C4D412yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C411yescharacteristic
66#407—24no—
67#407—42no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—24no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—42no—
72C2×C22⋊C8—41no—
73C2×C22⋊C8—41no—
74#398—24no—
75#398—24no—
76#398—24no—
77#398—24no—
78#391—22no—
79#391—21no—
80#391—21no—
81#391C2×C412yes—
82#391C2×C414yes—
83#391C2312yes—
84#385—24no—
85#385—24no—
86#385—24no—
87#385—24no—
88#384—22no—
89#384—21no—
90#384—21no—
91#384C2×C412yes—
92#384C2×C414yes—
93#384C2312yes—
94C4×C22⋊C4—22no—
95C4×C22⋊C4C2×C414yes—
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—21no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—24no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—42no—
129C23×C4—82no—
130C23×C4—82no—
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—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4—82no—
184C2×C4⋊C4—82no—
185C2×C4⋊C4C22⋊C412yes—
186C2×C4⋊C4C22⋊C412yes—
187C2×C4⋊C4C22⋊C412yes—
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C412yes—
191C2×C4⋊C4C2×D411yescharacteristic
192C2×C4⋊C4C2×D411yescharacteristic
193C2×C4⋊C4C2×D412yes—
194C2×C4⋊C4C2×D412yes—
195C2×C4⋊C4C2411yescharacteristic
196C2×C22⋊C4—28no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C42—24no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—42no—
206C2×C42C22×C412yes—
207C2×C42C22×C412yes—
208C22⋊C8—24no—
209C22⋊C8—24no—
210C22⋊C8—42no—
211C22⋊C8—44no—
212C22⋊C8—42no—
213C22⋊C8—44no—
214#95—24no—
215#95—24no—
216#95—24no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221C24—24no—
222C24—22no—
223C24—22no—
224C24—42no—
225C24#9911yescharacteristic
226C24#9912yes—
227C24#9911yescharacteristic
228C24C2×C22⋊C412yes—
229C24C2×C22⋊C411yescharacteristic
230C2×Q8—24no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—84no—
237C2×Q8—84no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—44no—
276C22×C4—48no—
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—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—82no—
297C22×C4—82no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4—82no—
302C22×C4—84no—
303C22×C4—84no—
304C22×C4C2×C22⋊C412yes—
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4C2×C22⋊C412yes—
307C22×C4C2×C22⋊C412yes—
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4#13114yes—
311C22×C4C23×C411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314M16—88no—
315C2×C8—44no—
316C2×C8—44no—
317C2×C8—88no—
318C2×C8—84no—
319C2×C8—84no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—41no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—48no—
333C4⋊C4—84no—
334C4⋊C4—84no—
335C4⋊C4—82no—
336C4⋊C4—82no—
337C4⋊C4—82no—
338C4⋊C4—82no—
339C22⋊C4—216no—
340C22⋊C4—44no—
341C22⋊C4—44no—
342C42—24no—
343C42—42no—
344C42—42no—
345C42—42no—
346C23—22no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—24no—
354C23—24no—
355C23—22no—
356C23—24no—
357C23—22no—
358C23—42no—
359C23—42no—
360C23—48no—
361C23—48no—
362C23#35614yes—
363C23#40812yes—
364C23C2×#9912yes—
365C23C2×#9912yes—
366C23C2×#9911yescharacteristic
367C23C2×#9911yescharacteristic
368C23#41312yes—
369C23C22×C22⋊C411yescharacteristic
370Q8—84no—
371Q8—84no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—48no—
398C2×C4—48no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4#40812yes—
434C2×C4C2×#13112yes—
435C8—84no—
436C8—84no—
437C22—24no—
438C22—24no—
439C22—28no—
440C22—21no—
441C22—22no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—44no—
450C22—44no—
451C22—48no—
452C22—44no—
453C22#146812yes—
454C22#146812yes—
455C22#212712yes—
456C22C2×#35614yes—
457C22C2×#35612yes—
458C22C2×#40811yescharacteristic
459C22C22×#9911yescharacteristic
460C22C2×#41311yescharacteristic
461C4—22no—
462C4—22no—
463C4—44no—
464C4—44no—
465C4—41no—
466C4—41no—
467C4—42no—
468C4—42no—
469C4—41no—
470C4—41no—
471C4—42no—
472C4—82no—
473C4—82no—
474C2—21no—
475C2—21no—
476C2—24no—
477C2—22no—
478C2—42no—
479C2#1155514yesdirect product factor
480C2C2×#146811yescharacteristic
481C2C2×#212711yescharacteristic
482C2C22×#35611yescharacteristic
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. #11555C2direct (N × H)
5. C2×#1325C2semidirect
6. C2×#1319C2semidirect
32. #1325C22semidirect
33. #1319C22semidirect
479. C2#11555direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25022
C2×#14632
C22×#3242
#115552
C2×#13252
C2×#13192