Finite Group Databasebeta

← C2×#11898 #482113

Subgroups — C2×#11898 #482113

Order 512.

1770 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2409C211yesmaximal, characteristic
2C2×#1799C211yesmaximal, characteristic
3C2×#1460C211yesmaximal, characteristic
4#11898C218yesmaximal, direct product factor
5C2×#1351C214yesmaximal, complemented
6C24×Q8C411yescharacteristic, complemented
7C22×C22⋊Q8—21no—
8C22×C4×D4C2211yescharacteristic
9C2×#431—21no—
10C22×#99—21no—
11#2409—22no—
12#2409—22no—
13#2409C414yescomplemented
14#2409C2214yes—
15C2×#395C412yescomplemented
16C2×#393C2212yes—
17C2×#391—21no—
18C2×#386—21no—
19C2×#385—21no—
20C2×#380—21no—
21C2×#380C411yescharacteristic, complemented
22#1799—22no—
23#1799C2214yes—
24C2×#342C2212yes—
25#1460—22no—
26#1460C2214yes—
27C2×#323—21no—
28C2×#323C2212yes—
29#1351C22116yescomplemented
30C23×Q8—22no—
31C23×Q8—21no—
32C23×Q8—41no—
33C23×Q8—42no—
34C23×Q8—41no—
35C23×Q8—48no—
36C23×Q8C2×C412yes—
37C23×Q8C2×C414yescomplemented
38C23×D4D411yescharacteristic
39C24×C4—21no—
40C24×C4C2×C411yescharacteristic
41C24×C4D411yescharacteristic
42C2×C22⋊Q8—24no—
43C2×C22⋊Q8—24no—
44C2×C22⋊Q8—48no—
45C2×C4×D4—41no—
46C2×C4×D4—41no—
47C2×C4×D4—42no—
48C2×C4×D4C2×C412yes—
49C2×C4×D4D412yes—
50C2×C4×D4D412yes—
51C2×C4×D4C2312yes—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—41no—
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C22⋊C4—21no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4—22no—
59C22×C22⋊C4D411yescharacteristic
60C22×C22⋊C4C2311yescharacteristic
61C22×C42D411yescharacteristic
62#431—24no—
63#431—42no—
64C2×C4⋊C8—41no—
65C2×#99—24no—
66C2×#99—44no—
67C2×C22⋊C8—22no—
68C2×C4×C8—41no—
69#395—24no—
70#395C2×C418yescomplemented
71#393—22no—
72#393—22no—
73#393C2×C414yes—
74#393C2314yes—
75#391—24no—
76#391—24no—
77#386—24no—
78#386—24no—
79#385—24no—
80#385—24no—
81#380—22no—
82#380—24no—
83#380—42no—
84#380C2×C414yescomplemented
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—41no—
88C2×#95C2×C412yes—
89#342—22no—
90#342C2314yes—
91#323—24no—
92#323—22no—
93#323C2314yes—
94C25C22⋊C411yescharacteristic
95C25C2×D411yescharacteristic
96C22×Q8—28no—
97C22×Q8—24no—
98C22×Q8—22no—
99C22×Q8—48no—
100C22×Q8—44no—
101C22×Q8—44no—
102C22×Q8—44no—
103C22×Q8—44no—
104C22×Q8—44no—
105C22×Q8—42no—
106C22×Q8—42no—
107C22×Q8—416no—
108C22×Q8—416no—
109C22×Q8—48no—
110C22×Q8—432no—
111C22×Q8—416no—
112C22×Q8—48no—
113C22×Q8—42no—
114C22×Q8—41no—
115C22×Q8—41no—
116C22×Q8C22×C414yes—
117C22×D4—41no—
118C22×D4—41no—
119C22×D4—42no—
120C22×D4C22⋊C412yes—
121C22×D4SD1614yes—
122C22×D4Q1614yes—
123C22×D4C2×D412yes—
124C23×C4—22no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—24no—
133C23×C4—48no—
134C23×C4—44no—
135C23×C4—44no—
136C23×C4—42no—
137C23×C4—42no—
138C23×C4—42no—
139C23×C4C22⋊C411yescharacteristic
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22⋊C412yes—
142C23×C4C22⋊C412yes—
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22×C411yescharacteristic
145C23×C4C22×C412yes—
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C22×C8—42no—
149C22⋊Q8—416no—
150C22⋊Q8—416no—
151C4×D4—42no—
152C4×D4—42no—
153C4×D4—42no—
154C4×D4—42no—
155C4×D4—42no—
156C4×D4—42no—
157C4×D4—42no—
158C4×D4—42no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—28no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—48no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—84no—
167C2×C4⋊C4C22⋊C412yes—
168C2×C4⋊C4C22×C412yes—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—48no—
178C2×C22⋊C4—44no—
179C2×C22⋊C4—44no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4—42no—
182C2×C22⋊C4C22⋊C412yes—
183C2×C22⋊C4C22×C411yescharacteristic
184C2×C22⋊C4C2×D411yescharacteristic
185C2×C22⋊C4C2×D412yes—
186C2×C22⋊C4C2×D411yescharacteristic
187C2×C22⋊C4C2411yescharacteristic
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—22no—
191C2×C42C22⋊C412yes—
192C2×C42C2×D412yes—
193C4⋊C8—44no—
194#99—48no—
195C22⋊C8—24no—
196C22⋊C8—42no—
197C4×C8—44no—
198#95—24no—
199#95—28no—
200#95—22no—
201#95—24no—
202#95—44no—
203#95C22×C414yes—
204C24—21no—
205C24—21no—
206C24—21no—
207C24—22no—
208C24—21no—
209C24—22no—
210C24—42no—
211C24—42no—
212C24—42no—
213C24—42no—
214C24#10314yes—
215C24C2×C22⋊C411yescharacteristic
216C24C2×C22⋊C411yescharacteristic
217C24C2×C22⋊C412yes—
218C24C2×SD1612yes—
219C24C2×Q1612yes—
220C24C22×D411yescharacteristic
221C2×Q8—216no—
222C2×Q8—48no—
223C2×Q8—48no—
224C2×Q8—48no—
225C2×Q8—416no—
226C2×Q8—48no—
227C2×Q8—44no—
228C2×Q8—44no—
229C2×Q8—416no—
230C2×Q8—432no—
231C2×Q8—432no—
232C2×Q8—416no—
233C2×Q8—416no—
234C2×Q8—432no—
235C2×Q8—48no—
236C2×Q8—48no—
237C2×Q8—44no—
238C2×Q8—44no—
239C2×Q8—48no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—44no—
244C2×D4—44no—
245C2×D4—44no—
246C2×D4—44no—
247C2×D4—42no—
248C2×D4#10318yes—
249C2×D4C2×SD1614yes—
250C2×D4C2×Q1614yes—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—28no—
256C22×C4—24no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—44no—
273C22×C4—48no—
274C22×C4—416no—
275C22×C4—44no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—48no—
281C22×C4—416no—
282C22×C4—416no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—84no—
295C22×C4—82no—
296C22×C4—84no—
297C22×C4—84no—
298C22×C4—84no—
299C22×C4#9912yes—
300C22×C4#9911yescharacteristic
301C22×C4#9911yescharacteristic
302C22×C4#10314yes—
303C22×C4C2×C22⋊C412yes—
304C22×C4C2×C22⋊C412yes—
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4C2×C22⋊C412yes—
308C22×C4C23×C411yescharacteristic
309C22×C4C22×D411yescharacteristic
310C2×C8—44no—
311C2×C8—84no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—48no—
315C4⋊C4—416no—
316C4⋊C4—88no—
317C22⋊C4—42no—
318C22⋊C4—41no—
319C22⋊C4—42no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—48no—
324C22⋊C4—48no—
325C22⋊C4—48no—
326C22⋊C4—44no—
327C22⋊C4—44no—
328C22⋊C4—84no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—22no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—21no—
337C23—21no—
338C23—22no—
339C23—22no—
340C23—24no—
341C23—24no—
342C23—22no—
343C23—21no—
344C23—21no—
345C23—22no—
346C23—22no—
347C23—42no—
348C23—48no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—48no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—44no—
357C23—44no—
358C23—41no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—81no—
364C23—81no—
365C23—84no—
366C23C2×#9911yescharacteristic
367C23C2×#9911yescharacteristic
368C23C2×#10312yes—
369C23C2×#10312yes—
370C23C2×#10314yes—
371C23#42112yes—
372C23C22×C22⋊C411yescharacteristic
373C23C22×SD1611yescharacteristic
374C23C22×Q1611yescharacteristic
375Q8—416no—
376Q8—48no—
377Q8—48no—
378Q8—416no—
379Q8—416no—
380D4—48no—
381D4—44no—
382D4—44no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—28no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—48no—
400C2×C4—42no—
401C2×C4—432no—
402C2×C4—44no—
403C2×C4—48no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—88no—
424C2×C4—84no—
425C2×C4—84no—
426C2×C4—84no—
427C2×C4—168no—
428C2×C4AΣL(1,𝔽16)14yes—
429C2×C4C2×#9912yes—
430C2×C4C2×#9912yes—
431C2×C4C2×#10314yes—
432C8—84no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—24no—
438C22—24no—
439C22—21no—
440C22—21no—
441C22—24no—
442C22—22no—
443C22—41no—
444C22—41no—
445C22—44no—
446C22—42no—
447C22—44no—
448C22—48no—
449C22—42no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—42no—
454C22—41no—
455C22—44no—
456C22—48no—
457C22—44no—
458C22—81no—
459C22—81no—
460C22—82no—
461C22—82no—
462C22#152012yes—
463C22#152012yes—
464C22C2×AΣL(1,𝔽16)12yes—
465C22#212912yes—
466C22C22×#9911yescharacteristic
467C22C22×#10311yescharacteristic
468C22C2×#42111yescharacteristic
469C4—22no—
470C4—24no—
471C4—48no—
472C4—41no—
473C4—41no—
474C4—42no—
475C4—41no—
476C4—41no—
477C4—84no—
478C4—164no—
479C4C2×AΣL(1,𝔽16)14yes—
480C2—21no—
481C2—21no—
482C2—22no—
483C2—41no—
484C2—44no—
485C2—41no—
486C2—42no—
487C2#1189814yesdirect product factor
488C2C2×#152011yescharacteristic
489C2C22×AΣL(1,𝔽16)11yescharacteristic
490C2C2×#212911yescharacteristic
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. #11898C2direct (N × H)
5. C2×#1351C2semidirect
6. C24×Q8C4semidirect
13. #2409C4semidirect
15. C2×#395C4semidirect
21. C2×#380C4semidirect
29. #1351C22semidirect
37. C23×Q8C2×C4semidirect
70. #395C2×C4semidirect
84. #380C2×C4semidirect
487. C2#11898direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24092
C2×#17992
C2×#14602
#118982
C2×#13512