Finite Group Databasebeta

← C2×#13292 #495393

Subgroups — C2×#13292 #495393

Order 512.

1302 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2487C211yesmaximal, characteristic
2C2×#1799C211yesmaximal, characteristic
3C2×C4×#100C211yesmaximal, characteristic
4#13292C218yesmaximal, direct product factor
5C2×#1390C214yesmaximal
6C22×C4×D4C2211yescharacteristic
7C23×C4⋊C4C411yescharacteristic, complemented
8C22×#100C2212yes—
9C22×#99C2212yes—
10C2×C4×M16—21no—
11#2487—22no—
12#2487—22no—
13#2487C414yescomplemented
14#2487C2214yes—
15C2×#400—21no—
16C2×#399—21no—
17C2×#389—21no—
18C2×#386C2212yes—
19C2×#385C412yescomplemented
20C2×#380—21no—
21C2×#380—21no—
22C2×#380C411yescharacteristic, complemented
23#1799—22no—
24#1799C2214yes—
25C4×#100—22no—
26C4×#100C2214yes—
27C2×#343—21no—
28C2×#342—21no—
29#1390C22116yes—
30C23×D4C2311yescharacteristic
31C24×C4C2×C411yescharacteristic
32C24×C4C2×C412yes—
33C24×C4D411yescharacteristic
34C22×M16—22no—
35C2×C4×D4—41no—
36C2×C4×D4—41no—
37C2×C4×D4—42no—
38C2×C4×D4C2×C412yes—
39C2×C4×D4D412yes—
40C2×C4×D4D412yes—
41C2×C4×D4C2312yes—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—42no—
48C22×C4⋊C4—41no—
49C22×C4⋊C4—41no—
50C22×C4⋊C4—41no—
51C22×C4⋊C4C2×C414yescomplemented
52C22×C4⋊C4D411yescharacteristic
53C22×C22⋊C4—22no—
54C22×C22⋊C4—22no—
55C22×C22⋊C4C2×C412yes—
56C22×C22⋊C4C2×C411yescharacteristic
57C22×C22⋊C4D411yescharacteristic
58C22×C42D411yescharacteristic
59C2×#100—22no—
60C2×#100—24no—
61C2×#100Q818yes—
62C2×#100C2314yes—
63C2×#99—28no—
64C2×#99—22no—
65C2×#99C2314yes—
66C4×M16—24no—
67C4×M16—42no—
68C2×#97—41no—
69C2×C4×C8—41no—
70#400—24no—
71#400—24no—
72#399—24no—
73#399—24no—
74#389—24no—
75#389—24no—
76#386—22no—
77#386—22no—
78#386C2×C414yes—
79#386C2314yes—
80#385—24no—
81#385C2×C418yescomplemented
82#380—22no—
83#380—24no—
84#380—24no—
85#380—24no—
86#380—42no—
87#380C2×C414yescomplemented
88C2×#95—21no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—41no—
92#343—24no—
93#342—24no—
94C25C22×C411yescharacteristic
95C25C2×D411yescharacteristic
96C22×D4—41no—
97C22×D4—41no—
98C22×D4—42no—
99C22×D4C22×C411yescharacteristic
100C22×D4C2×D411yescharacteristic
101C22×D4C2×D411yescharacteristic
102C22×D4C2×Q814yes—
103C22×D4#4114yes—
104C22×D4C2411yescharacteristic
105C23×C4—22no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—44no—
115C23×C4—44no—
116C23×C4—42no—
117C23×C4—42no—
118C23×C4—42no—
119C23×C4—84no—
120C23×C4C22⋊C411yescharacteristic
121C23×C4C22⋊C411yescharacteristic
122C23×C4C22⋊C411yescharacteristic
123C23×C4C22⋊C412yes—
124C23×C4C22⋊C412yes—
125C23×C4C22⋊C411yescharacteristic
126C23×C4C22×C412yes—
127C23×C4C22×C414yes—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D412yes—
130C2×M16—28no—
131C2×M16—24no—
132C2×M16—44no—
133C2×M16—42no—
134C22×C8—42no—
135C4×D4—42no—
136C4×D4—42no—
137C4×D4—42no—
138C4×D4—42no—
139C4×D4—42no—
140C4×D4—42no—
141C4×D4—42no—
142C4×D4—42no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—28no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—44no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—84no—
161C2×C4⋊C4—84no—
162C2×C4⋊C4C22⋊C412yes—
163C2×C4⋊C4C2×D412yes—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—41no—
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—44no—
173C2×C22⋊C4—44no—
174C2×C22⋊C4—44no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4—84no—
177C2×C22⋊C4C22⋊C412yes—
178C2×C22⋊C4C22⋊C412yes—
179C2×C22⋊C4C22×C414yes—
180C2×C22⋊C4C22×C412yes—
181C2×C22⋊C4C2×D412yes—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—22no—
185C2×C42C22⋊C412yes—
186C2×C42C2×D412yes—
187#100—28no—
188#100—24no—
189#100C2×Q818yes—
190#99—28no—
191#99—44no—
192#97—44no—
193C4×C8—44no—
194#95—28no—
195#95—24no—
196#95—24no—
197#95—24no—
198#95—44no—
199C24—21no—
200C24—21no—
201C24—21no—
202C24—22no—
203C24—21no—
204C24—22no—
205C24—42no—
206C24—42no—
207C24—42no—
208C24—42no—
209C24C2×C22⋊C411yescharacteristic
210C24C2×C22⋊C411yescharacteristic
211C24C2×C22⋊C411yescharacteristic
212C24C4×Q814yes—
213C24#12412yes—
214C24C4⋊Q812yes—
215C24C23×C411yescharacteristic
216C24C22×D411yescharacteristic
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4—44no—
225C2×D4—44no—
226C2×D4—44no—
227C2×D4—44no—
228C2×D4—41no—
229C2×D4C4×Q814yes—
230C2×D4C22⋊Q814yes—
231C2×D4#12412yes—
232C2×D4C4⋊Q812yes—
233C2×D4C22×Q812yes—
234C2×D4C2×#4112yes—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—24no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—44no—
256C22×C4—48no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—84no—
277C22×C4—82no—
278C22×C4—84no—
279C22×C4—84no—
280C22×C4—88no—
281C22×C4—88no—
282C22×C4—84no—
283C22×C4—84no—
284C22×C4—84no—
285C22×C4#9912yes—
286C22×C4#9911yescharacteristic
287C22×C4#9911yescharacteristic
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C2×C22⋊C412yes—
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C412yes—
293C22×C4C22⋊Q814yes—
294C22×C4C22×D411yescharacteristic
295M16—28no—
296M16—44no—
297M16—48no—
298C2×C8—44no—
299C2×C8—48no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—48no—
303C4⋊C4—48no—
304C4⋊C4—44no—
305C4⋊C4—44no—
306C4⋊C4—88no—
307C4⋊C4—88no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—44no—
313C22⋊C4—44no—
314C22⋊C4—44no—
315C22⋊C4—44no—
316C22⋊C4—84no—
317C22⋊C4—84no—
318C22⋊C4—84no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—22no—
323C23—22no—
324C23—22no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—22no—
332C23—24no—
333C23—24no—
334C23—22no—
335C23—21no—
336C23—21no—
337C23—22no—
338C23—22no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—48no—
344C23—42no—
345C23—42no—
346C23—48no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—44no—
352C23—44no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—81no—
358C23—81no—
359C23—84no—
360C23#39112yes—
361C23#39112yes—
362C23#39112yes—
363C23C2×#9911yescharacteristic
364C23C2×#9911yescharacteristic
365C23#41012yes—
366C23C22×C22⋊C411yescharacteristic
367C23C2×C4×Q812yes—
368C23C2×#12411yescharacteristic
369C23C2×C4⋊Q811yescharacteristic
370D4—44no—
371D4—44no—
372D4—42no—
373D4—42no—
374D4—42no—
375D4—42no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—88no—
412C2×C4—84no—
413C2×C4—88no—
414C2×C4—84no—
415C2×C4—88no—
416C2×C4—84no—
417C2×C4—84no—
418C2×C4—84no—
419C2×C4—168no—
420C2×C4#35214yes—
421C2×C4#39112yes—
422C2×C4C2×#9912yes—
423C2×C4C2×#9912yes—
424C2×C4C2×C22⋊Q812yes—
425C8—48no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—24no—
431C22—24no—
432C22—21no—
433C22—22no—
434C22—21no—
435C22—22no—
436C22—22no—
437C22—42no—
438C22—41no—
439C22—41no—
440C22—42no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—48no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—44no—
454C22—48no—
455C22—44no—
456C22—81no—
457C22—81no—
458C22—82no—
459C22—82no—
460C22#199612yes—
461C22#199612yes—
462C22C2×#35212yes—
463C22#212212yes—
464C22C2×#39111yescharacteristic
465C22C22×#9911yescharacteristic
466C22C2×#41011yescharacteristic
467C4—22no—
468C4—22no—
469C4—22no—
470C4—42no—
471C4—42no—
472C4—42no—
473C4—84no—
474C4—84no—
475C4—164no—
476C4C2×#35214yes—
477C2—21no—
478C2—21no—
479C2—22no—
480C2—41no—
481C2—42no—
482C2—42no—
483C2—41no—
484C2—42no—
485C2#1329214yesdirect product factor
486C2C2×#199611yescharacteristic
487C2C22×#35211yescharacteristic
488C2C2×#212211yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC24C23×C4
Frattini, cosocleC22×D4C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #13292C2direct (N × H)
7. C23×C4⋊C4C4semidirect
13. #2487C4semidirect
19. C2×#385C4semidirect
22. C2×#380C4semidirect
51. C22×C4⋊C4C2×C4semidirect
81. #385C2×C4semidirect
87. #380C2×C4semidirect
485. C2#13292direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24872
C2×#17992
C2×C4×#1002
#132922
C2×#13902