Finite Group Databasebeta

← C2×#11900 #482110

Subgroups — C2×#11900 #482110

Order 512.

1212 conjugacy classes of subgroups.

The subgroup lattice has 463 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic
2C2×#1799C211yesmaximal, characteristic
3C2×#1459C211yesmaximal, characteristic
4#11900C218yesmaximal, direct product factor
5C2×#1351C214yesmaximal, complemented
6C22×C4⋊Q8C411yescharacteristic, complemented
7C22×C4×D4C2211yescharacteristic
8C22×#99—21no—
9C22×C22⋊C8—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×#380—21no—
21C2×C4×C4⋊C4C411yescharacteristic, complemented
22#1799—22no—
23#1799C2214yes—
24C2×#342C2212yes—
25#1459—22no—
26#1459C2214yes—
27C2×#324—21no—
28C2×#323C2212yes—
29#1351C22116yescomplemented
30C23×Q8C2×C412yes—
31C23×D4D411yescharacteristic
32C24×C4D411yescharacteristic
33C24×C4D411yescharacteristic
34C23×C8—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⋊C4—41no—
51C22×C4⋊C4D411yescharacteristic
52C22×C22⋊C4—22no—
53C22×C22⋊C4—22no—
54C22×C22⋊C4—22no—
55C22×C22⋊C4C2×C411yescharacteristic
56C22×C22⋊C4C2311yescharacteristic
57C22×C42—21no—
58C22×C42C2×C411yescharacteristic
59C2×#99—24no—
60C2×#99—44no—
61C2×C22⋊C8—22no—
62C2×C22⋊C8—24no—
63C2×C22⋊C8—42no—
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—
79#380—24no—
80#380—42no—
81C4×C4⋊C4—22no—
82C4×C4⋊C4C2×C414yescomplemented
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—41no—
100C22×D4—41no—
101C22×D4—42no—
102C22×D4C22⋊C412yes—
103C22×D4SD1614yes—
104C22×D4Q1614yes—
105C22×D4C2×D412yes—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—22no—
110C23×C4—21no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—44no—
115C23×C4—44no—
116C23×C4—42no—
117C23×C4C22⋊C411yescharacteristic
118C23×C4C22⋊C411yescharacteristic
119C23×C4C22⋊C412yes—
120C23×C4C22⋊C412yes—
121C23×C4C22⋊C411yescharacteristic
122C23×C4C22×C411yescharacteristic
123C23×C4C2×D412yes—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D412yes—
126C22×C8—42no—
127C22×C8—44no—
128C22×C8—84no—
129C4⋊Q8—44no—
130C4⋊Q8—44no—
131C4⋊Q8—44no—
132C4⋊Q8—44no—
133C4×D4—42no—
134C4×D4—42no—
135C4×D4—42no—
136C4×D4—42no—
137C4×D4—42no—
138C4×D4—42no—
139C4×D4—42no—
140C4×D4—42no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—84no—
154C2×C4⋊C4C22⋊C412yes—
155C2×C4⋊C4C2×D412yes—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—28no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—41no—
161C2×C22⋊C4—41no—
162C2×C22⋊C4—41no—
163C2×C22⋊C4—41no—
164C2×C22⋊C4—44no—
165C2×C22⋊C4—44no—
166C2×C22⋊C4—42no—
167C2×C22⋊C4—84no—
168C2×C22⋊C4—84no—
169C2×C22⋊C4C22⋊C412yes—
170C2×C22⋊C4C22×C412yes—
171C2×C22⋊C4C22×C411yescharacteristic
172C2×C22⋊C4C2×D411yescharacteristic
173C2×C22⋊C4C2×D411yescharacteristic
174C2×C22⋊C4C2411yescharacteristic
175C2×C42—22no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—24no—
179C2×C42—44no—
180C2×C42C22⋊C412yes—
181C2×C42C22×C412yes—
182#99—48no—
183C22⋊C8—24no—
184C22⋊C8—42no—
185C22⋊C8—44no—
186#95—24no—
187#95—28no—
188#95—22no—
189#95—24no—
190#95—44no—
191#95C22×C414yes—
192C24—21no—
193C24—21no—
194C24—21no—
195C24—21no—
196C24—22no—
197C24—22no—
198C24—42no—
199C24—42no—
200C24—42no—
201C24—42no—
202C24#10314yes—
203C24C2×C22⋊C411yescharacteristic
204C24C2×C22⋊C412yes—
205C24C2×C22⋊C411yescharacteristic
206C24C2×SD1612yes—
207C24C2×Q1612yes—
208C24C22×D411yescharacteristic
209C2×Q8—44no—
210C2×Q8—44no—
211C2×Q8—84no—
212C2×Q8—84no—
213C2×Q8—88no—
214C2×Q8—88no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—44no—
219C2×D4—44no—
220C2×D4—44no—
221C2×D4—44no—
222C2×D4—42no—
223C2×D4#10318yes—
224C2×D4C2×SD1614yes—
225C2×D4C2×Q1614yes—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—44no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—84no—
264C22×C4—84no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—84no—
268C22×C4—84no—
269C22×C4—84no—
270C22×C4#9912yes—
271C22×C4#9911yescharacteristic
272C22×C4#9911yescharacteristic
273C22×C4#10314yes—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C23×C411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C2×C8—44no—
282C2×C8—84no—
283C2×C8—88no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—48no—
287C4⋊C4—48no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C4⋊C4—88no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—44no—
298C22⋊C4—44no—
299C22⋊C4—84no—
300C22⋊C4—88no—
301C22⋊C4—84no—
302C22⋊C4—84no—
303C42—22no—
304C42—22no—
305C42—22no—
306C42—22no—
307C42—48no—
308C23—22no—
309C23—22no—
310C23—21no—
311C23—22no—
312C23—21no—
313C23—22no—
314C23—24no—
315C23—22no—
316C23—21no—
317C23—21no—
318C23—22no—
319C23—22no—
320C23—24no—
321C23—22no—
322C23—42no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—42no—
327C23—41no—
328C23—48no—
329C23—44no—
330C23—44no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—81no—
336C23—81no—
337C23—81no—
338C23—81no—
339C23—84no—
340C23—84no—
341C23C2×#9911yescharacteristic
342C23C2×#9911yescharacteristic
343C23C2×#10312yes—
344C23C2×#10312yes—
345C23C2×#10314yes—
346C23#42112yes—
347C23C22×C22⋊C411yescharacteristic
348C23C22×SD1611yescharacteristic
349C23C22×Q1611yescharacteristic
350Q8—88no—
351Q8—88no—
352D4—48no—
353D4—44no—
354D4—44no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—88no—
393C2×C4—84no—
394C2×C4—84no—
395C2×C4—84no—
396C2×C4—84no—
397C2×C4—84no—
398C2×C4—168no—
399C2×C4#35414yes—
400C2×C4C2×#9912yes—
401C2×C4C2×#9912yes—
402C2×C4C2×#10314yes—
403C8—84no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—24no—
409C22—24no—
410C22—21no—
411C22—21no—
412C22—22no—
413C22—24no—
414C22—41no—
415C22—41no—
416C22—44no—
417C22—42no—
418C22—41no—
419C22—41no—
420C22—42no—
421C22—41no—
422C22—42no—
423C22—41no—
424C22—44no—
425C22—48no—
426C22—44no—
427C22—81no—
428C22—81no—
429C22—82no—
430C22—82no—
431C22—84no—
432C22—82no—
433C22#152012yes—
434C22#152012yes—
435C22#212212yes—
436C22C2×#35412yes—
437C22C22×#9911yescharacteristic
438C22C22×#10311yescharacteristic
439C22C2×#42111yescharacteristic
440C4—22no—
441C4—24no—
442C4—41no—
443C4—41no—
444C4—41no—
445C4—41no—
446C4—42no—
447C4—84no—
448C4—84no—
449C4—164no—
450C4C2×#35414yes—
451C2—21no—
452C2—21no—
453C2—22no—
454C2—41no—
455C2—41no—
456C2—42no—
457C2—44no—
458C2#1190014yesdirect product factor
459C2C2×#152011yescharacteristic
460C2C2×#212211yescharacteristic
461C2C22×#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. #11900C2direct (N × H)
5. C2×#1351C2semidirect
6. C22×C4⋊Q8C4semidirect
12. #2563C4semidirect
14. C2×#393C4semidirect
21. C2×C4×C4⋊C4C4semidirect
29. #1351C22semidirect
39. C2×C4⋊Q8C2×C4semidirect
69. #393C2×C4semidirect
82. C4×C4⋊C4C2×C4semidirect
458. C2#11900direct (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×#17992
C2×#14592
#119002
C2×#13512