Finite Group Databasebeta

← C2×#11785 #481635

Subgroups — C2×#11785 #481635

Order 512.

1798 conjugacy classes of subgroups.

The subgroup lattice has 477 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2409C211yesmaximal, characteristic
2C2×C4×#99C211yesmaximal, characteristic
3C2×#1466C211yesmaximal, characteristic
4#11785C218yesmaximal, direct product factor
5C2×#1351C214yesmaximal, complemented
6C24×Q8C411yescharacteristic, complemented
7C22×C22⋊Q8—21no—
8C22×C4×D4C2211yescharacteristic
9C2×#432—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×#380C411yescharacteristic, complemented
21C2×C4×C22⋊C4—21no—
22C4×#99—22no—
23C4×#99C2214yes—
24C2×#342C2212yes—
25#1466—22no—
26#1466C2214yes—
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⋊C4C2×C411yescharacteristic
55C22×C22⋊C4—22no—
56C22×C22⋊C4—21no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4D411yescharacteristic
59C22×C22⋊C4C2311yescharacteristic
60C22×C42—41no—
61C22×C42D411yescharacteristic
62#432—24no—
63#432—42no—
64C2×C4⋊C8—41no—
65C2×#99—28no—
66C2×#99—24no—
67C2×C22⋊C8—22no—
68C2×#97—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#380C2×C414yescomplemented
83C4×C22⋊C4—24no—
84C4×C22⋊C4—42no—
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—24no—
118C22×D4—24no—
119C22×D4—41no—
120C22×D4—41no—
121C22×D4C22⋊C412yes—
122C22×D4C2×D412yes—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—24no—
132C23×C4—48no—
133C23×C4—44no—
134C23×C4—44no—
135C23×C4—42no—
136C23×C4—42no—
137C23×C4—42no—
138C23×C4C22⋊C411yescharacteristic
139C23×C4C22⋊C411yescharacteristic
140C23×C4C22⋊C412yes—
141C23×C4C22⋊C412yes—
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22×C411yescharacteristic
144C23×C4C22×C412yes—
145C23×C4C2×D412yes—
146C23×C4C2×D411yescharacteristic
147C22×C8—42no—
148C22⋊Q8—416no—
149C22⋊Q8—416no—
150C4×D4—42no—
151C4×D4—42no—
152C4×D4—42no—
153C4×D4—42no—
154C4×D4—42no—
155C4×D4—42no—
156C4×D4—42no—
157C4×D4—42no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—28no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—48no—
164C2×C4⋊C4C22⋊C412yes—
165C2×C4⋊C4C22×C412yes—
166C2×C22⋊C4—28no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—48no—
176C2×C22⋊C4—42no—
177C2×C22⋊C4—44no—
178C2×C22⋊C4—44no—
179C2×C22⋊C4C22⋊C412yes—
180C2×C22⋊C4C22×C411yescharacteristic
181C2×C22⋊C4C2×D411yescharacteristic
182C2×C22⋊C4C2×D412yes—
183C2×C22⋊C4C2×D411yescharacteristic
184C2×C22⋊C4C2411yescharacteristic
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—22no—
188C2×C42—44no—
189C2×C42—84no—
190C2×C42C22⋊C412yes—
191C2×C42C2×D412yes—
192C4⋊C8—44no—
193#99—216no—
194C22⋊C8—24no—
195C22⋊C8—42no—
196#97—44no—
197#95—24no—
198#95—28no—
199#95—22no—
200#95—24no—
201#95—44no—
202#95C22×C414yes—
203C24—21no—
204C24—21no—
205C24—21no—
206C24—22no—
207C24—21no—
208C24—22no—
209C24—42no—
210C24—42no—
211C24—42no—
212C24—42no—
213C24#10414yes—
214C24C2×C22⋊C411yescharacteristic
215C24C2×C22⋊C411yescharacteristic
216C24C2×C22⋊C412yes—
217C24#13714yes—
218C24C22×D411yescharacteristic
219C2×Q8—216no—
220C2×Q8—48no—
221C2×Q8—48no—
222C2×Q8—48no—
223C2×Q8—416no—
224C2×Q8—48no—
225C2×Q8—44no—
226C2×Q8—44no—
227C2×Q8—416no—
228C2×Q8—432no—
229C2×Q8—432no—
230C2×Q8—416no—
231C2×Q8—416no—
232C2×Q8—432no—
233C2×Q8—48no—
234C2×Q8—48no—
235C2×Q8—44no—
236C2×Q8—44no—
237C2×Q8—48no—
238C2×D4—24no—
239C2×D4—24no—
240C2×D4—24no—
241C2×D4—24no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—48no—
245C2×D4—48no—
246C22×C4—24no—
247C22×C4—28no—
248C22×C4—24no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—44no—
269C22×C4—48no—
270C22×C4—416no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—48no—
277C22×C4—416no—
278C22×C4—416no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—44no—
287C22×C4—48no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—84no—
292C22×C4—82no—
293C22×C4—84no—
294C22×C4—84no—
295C22×C4#9912yes—
296C22×C4#9911yescharacteristic
297C22×C4#9911yescharacteristic
298C22×C4C2×C22⋊C412yes—
299C22×C4C2×C22⋊C412yes—
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C22⋊C412yes—
303C22×C4C23×C411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C2×C8—44no—
306C2×C8—84no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—48no—
310C4⋊C4—416no—
311C22⋊C4—28no—
312C22⋊C4—42no—
313C22⋊C4—41no—
314C22⋊C4—42no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—48no—
319C22⋊C4—48no—
320C22⋊C4—48no—
321C22⋊C4—44no—
322C22⋊C4—48no—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—88no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—21no—
332C23—21no—
333C23—22no—
334C23—24no—
335C23—24no—
336C23—24no—
337C23—21no—
338C23—21no—
339C23—24no—
340C23—42no—
341C23—48no—
342C23—42no—
343C23—42no—
344C23—48no—
345C23—42no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—48no—
350C23—41no—
351C23—42no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—81no—
356C23—84no—
357C23—81no—
358C23C2×#9911yescharacteristic
359C23C2×#9911yescharacteristic
360C23#41912yes—
361C23#41912yes—
362C23C2×#10412yes—
363C23C2×#10414yes—
364C23C22×C22⋊C411yescharacteristic
365C23C2×#13712yes—
366Q8—416no—
367Q8—48no—
368Q8—48no—
369Q8—416no—
370Q8—416no—
371D4—48no—
372D4—48no—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—28no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—48no—
389C2×C4—42no—
390C2×C4—432no—
391C2×C4—44no—
392C2×C4—48no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—48no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—88no—
416C2×C4—84no—
417C2×C4—816no—
418C2×C4—84no—
419C2×C4AΣL(1,𝔽16)14yes—
420C2×C4C2×#9912yes—
421C2×C4C2×#9912yes—
422C8—84no—
423C22—24no—
424C22—24no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—22no—
430C22—22no—
431C22—24no—
432C22—42no—
433C22—42no—
434C22—44no—
435C22—44no—
436C22—48no—
437C22—48no—
438C22—48no—
439C22—42no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—42no—
444C22—41no—
445C22—81no—
446C22—81no—
447C22—84no—
448C22#150812yes—
449C22#150812yes—
450C22C2×AΣL(1,𝔽16)12yes—
451C22#212912yes—
452C22C22×#9911yescharacteristic
453C22C2×#41911yescharacteristic
454C22C22×#10411yescharacteristic
455C4—22no—
456C4—48no—
457C4—41no—
458C4—41no—
459C4—42no—
460C4—41no—
461C4—41no—
462C4—42no—
463C4—84no—
464C4—88no—
465C4C2×AΣL(1,𝔽16)14yes—
466C2—21no—
467C2—21no—
468C2—22no—
469C2—42no—
470C2—44no—
471C2—42no—
472C2#1178514yesdirect product factor
473C2C2×#150811yescharacteristic
474C2C22×AΣL(1,𝔽16)11yescharacteristic
475C2C2×#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. #11785C2direct (N × H)
5. C2×#1351C2semidirect
6. C24×Q8C4semidirect
13. #2409C4semidirect
15. C2×#395C4semidirect
20. C2×#380C4semidirect
29. #1351C22semidirect
37. C23×Q8C2×C4semidirect
70. #395C2×C4semidirect
82. #380C2×C4semidirect
472. C2#11785direct (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×C4×#992
C2×#14662
#117852
C2×#13512