Finite Group Databasebeta

← C2×#13220 #495157

Subgroups — C2×#13220 #495157

Order 512.

1478 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2486C211yesmaximal, characteristic
2C22×#352C214yesmaximal
3C2×#1800C211yesmaximal, characteristic
4C2×C4×#99C211yesmaximal, characteristic
5#13220C218yesmaximal, direct product factor
6C22×#123C2212yes—
7C22×C4×D4C2211yescharacteristic
8C23×C4⋊C4C411yescharacteristic, complemented
9C2×#423—21no—
10C22×#100C2212yes—
11C22×#99C2212yes—
12#2486—22no—
13#2486—22no—
14#2486C414yescomplemented
15#2486C2214yes—
16C2×#398—21no—
17C2×#397—21no—
18C2×#389—21no—
19C2×#381C412yescomplemented
20C2×#380—21no—
21C2×#380C411yescharacteristic, complemented
22C2×C4×C22⋊C4—21no—
23C2×#352—216no—
24C2×#352C22116yes—
25#1800—22no—
26#1800C2214yes—
27C4×#99—22no—
28C4×#99C2214yes—
29C2×#343—21no—
30C2×#342—21no—
31C23×D4C2311yescharacteristic
32C24×C4C2×C412yes—
33C24×C4C2×C411yescharacteristic
34C24×C4D411yescharacteristic
35C22×M16—22no—
36C2×#123—22no—
37C2×#123—22no—
38C2×#123—28no—
39C2×#123—28no—
40C2×#123C2×C414yes—
41C2×#123C2314yes—
42C2×C4×D4—41no—
43C2×C4×D4—41no—
44C2×C4×D4—42no—
45C2×C4×D4C2×C412yes—
46C2×C4×D4D412yes—
47C2×C4×D4D412yes—
48C2×C4×D4C2312yes—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—42no—
55C22×C4⋊C4—41no—
56C22×C4⋊C4—41no—
57C22×C4⋊C4C2×C414yescomplemented
58C22×C4⋊C4D411yescharacteristic
59C22×C22⋊C4—22no—
60C22×C22⋊C4—22no—
61C22×C22⋊C4C2×C412yes—
62C22×C22⋊C4C2×C411yescharacteristic
63C22×C22⋊C4D411yescharacteristic
64C22×C42—41no—
65C22×C42D411yescharacteristic
66#423—24no—
67#423—42no—
68C2×C4⋊C8—41no—
69C2×C4⋊C8—41no—
70C2×#100—28no—
71C2×#100—22no—
72C2×#100C2314yes—
73C2×#99—22no—
74C2×#99—24no—
75C2×#99D418yes—
76C2×#99C2314yes—
77#398—24no—
78#398—24no—
79#397—24no—
80#397—24no—
81#389—24no—
82#389—24no—
83#381—24no—
84#381C2×C418yescomplemented
85#380—22no—
86#380—24no—
87#380—24no—
88#380C2×C414yescomplemented
89C4×C22⋊C4—24no—
90C4×C22⋊C4—42no—
91C2×#95—21no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—41no—
95#352—232no—
96#343—24no—
97#342—24no—
98C25C22×C411yescharacteristic
99C25C2×D411yescharacteristic
100C22×D4—41no—
101C22×D4—41no—
102C22×D4—42no—
103C22×D4C22×C411yescharacteristic
104C22×D4C2×D411yescharacteristic
105C22×D4C2×D414yes—
106C22×D4C2×D411yescharacteristic
107C22×D4#4114yes—
108C22×D4C2411yescharacteristic
109C23×C4—22no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—28no—
119C23×C4—44no—
120C23×C4—44no—
121C23×C4—42no—
122C23×C4—42no—
123C23×C4—42no—
124C23×C4—44no—
125C23×C4C22⋊C411yescharacteristic
126C23×C4C22⋊C411yescharacteristic
127C23×C4C22⋊C411yescharacteristic
128C23×C4C22⋊C412yes—
129C23×C4C22⋊C412yes—
130C23×C4C22⋊C411yescharacteristic
131C23×C4C22×C414yes—
132C23×C4C22×C412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D412yes—
135C2×M16—24no—
136C2×M16—42no—
137C2×M16—44no—
138C2×M16—44no—
139C22×C8—42no—
140#123—216no—
141#123—28no—
142#123—28no—
143#123—48no—
144#123—44no—
145#123—44no—
146C4×D4—42no—
147C4×D4—42no—
148C4×D4—42no—
149C4×D4—42no—
150C4×D4—42no—
151C4×D4—42no—
152C4×D4—42no—
153C4×D4—42no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4C22⋊C412yes—
174C2×C4⋊C4C2×D412yes—
175C2×C22⋊C4—28no—
176C2×C22⋊C4—28no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—41no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4—41no—
184C2×C22⋊C4—41no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—42no—
187C2×C22⋊C4—48no—
188C2×C22⋊C4C22⋊C412yes—
189C2×C22⋊C4C22⋊C412yes—
190C2×C22⋊C4C22×C414yes—
191C2×C22⋊C4C22×C412yes—
192C2×C22⋊C4C2×D412yes—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—22no—
196C2×C42—44no—
197C2×C42—84no—
198C2×C42C22⋊C412yes—
199C2×C42C2×D412yes—
200C4⋊C8—44no—
201C4⋊C8—44no—
202#100—28no—
203#100—44no—
204#99—28no—
205#99—24no—
206#99C2×D418yes—
207#95—28no—
208#95—24no—
209#95—28no—
210#95—44no—
211C24—21no—
212C24—21no—
213C24—21no—
214C24—22no—
215C24—21no—
216C24—22no—
217C24—42no—
218C24—42no—
219C24—42no—
220C24—42no—
221C24C2×C22⋊C411yescharacteristic
222C24C2×C22⋊C411yescharacteristic
223C24C2×C22⋊C411yescharacteristic
224C24C4×D414yes—
225C24#12114yes—
226C24C23×C411yescharacteristic
227C24C22×D411yescharacteristic
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—44no—
236C2×D4—44no—
237C2×D4—44no—
238C2×D4—44no—
239C2×D4—41no—
240C2×D4C4×D414yes—
241C2×D4#12114yes—
242C2×D4#12412yes—
243C2×D4#12712yes—
244C2×D4C22×D412yes—
245C2×D4C2×#4112yes—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—28no—
267C22×C4—44no—
268C22×C4—48no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—44no—
279C22×C4—48no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—84no—
293C22×C4—82no—
294C22×C4—84no—
295C22×C4—84no—
296C22×C4—88no—
297C22×C4—88no—
298C22×C4#9912yes—
299C22×C4#9911yescharacteristic
300C22×C4#9911yescharacteristic
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C22⋊C412yes—
303C22×C4C2×C22⋊C412yes—
304C22×C4C2×C22⋊C412yes—
305C22×C4C2×C22⋊C412yes—
306C22×C4#12412yes—
307C22×C4#12712yes—
308C22×C4C22×D411yescharacteristic
309M16—44no—
310M16—44no—
311M16—84no—
312C2×C8—44no—
313C2×C8—84no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—48no—
317C4⋊C4—48no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—48no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C22⋊C4—28no—
324C22⋊C4—28no—
325C22⋊C4—42no—
326C22⋊C4—42no—
327C22⋊C4—42no—
328C22⋊C4—42no—
329C22⋊C4—44no—
330C22⋊C4—48no—
331C22⋊C4—48no—
332C22⋊C4—44no—
333C22⋊C4—48no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—88no—
339C23—22no—
340C23—22no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—24no—
348C23—24no—
349C23—24no—
350C23—21no—
351C23—21no—
352C23—24no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—48no—
357C23—41no—
358C23—42no—
359C23—42no—
360C23—48no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—48no—
365C23—41no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—81no—
371C23—81no—
372C23—84no—
373C23#39012yes—
374C23#39012yes—
375C23#39012yes—
376C23C2×#9911yescharacteristic
377C23C2×#9911yescharacteristic
378C23#41012yes—
379C23C22×C22⋊C411yescharacteristic
380C23C2×C4×D412yes—
381C23C2×#12112yes—
382D4—44no—
383D4—44no—
384D4—42no—
385D4—42no—
386D4—42no—
387D4—42no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—48no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—88no—
427C2×C4—88no—
428C2×C4—84no—
429C2×C4—84no—
430C2×C4—816no—
431C2×C4—84no—
432C2×C4—84no—
433C2×C4—88no—
434C2×C4#35214yescomplemented
435C2×C4#39012yes—
436C2×C4C2×#9912yes—
437C2×C4C2×#9912yes—
438C2×C4C2×#12411yescharacteristic
439C2×C4C2×#12711yescharacteristic
440C8—84no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—24no—
446C22—24no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—42no—
452C22—42no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22—48no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—48no—
467C22—48no—
468C22—81no—
469C22—81no—
470C22—84no—
471C22#196412yes—
472C22#196412yes—
473C22C2×#35212yes—
474C22#212312yes—
475C22C2×#39011yescharacteristic
476C22C22×#9911yescharacteristic
477C22C2×#41011yescharacteristic
478C4—22no—
479C4—21no—
480C4—21no—
481C4—22no—
482C4—42no—
483C4—42no—
484C4—42no—
485C4—84no—
486C4—88no—
487C4—84no—
488C4C2×#35214yescomplemented
489C2—21no—
490C2—21no—
491C2—22no—
492C2—42no—
493C2—42no—
494C2—42no—
495C2—42no—
496C2#1322014yesdirect product factor
497C2C2×#196411yescharacteristic
498C2C22×#35211yescharacteristic
499C2C2×#212311yescharacteristic
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
5. #13220C2direct (N × H)
8. C23×C4⋊C4C4semidirect
14. #2486C4semidirect
19. C2×#381C4semidirect
21. C2×#380C4semidirect
57. C22×C4⋊C4C2×C4semidirect
84. #381C2×C4semidirect
88. #380C2×C4semidirect
434. C2×C4#352semidirect
488. C4C2×#352semidirect
496. C2#13220direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24862
C22×#3522
C2×#18002
C2×C4×#992
#132202