Finite Group Databasebeta

← C2×#13217 #495145

Subgroups — C2×#13217 #495145

Order 512.

1466 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2486C211yesmaximal, characteristic
2C22×#352C214yesmaximal
3C2×#1799C211yesmaximal, characteristic
4C2×C4×#100C211yesmaximal, characteristic
5#13217C218yesmaximal, direct product factor
6C22×#123C2212yes—
7C22×C4×D4C2211yescharacteristic
8C23×C4⋊C4C411yescharacteristic, complemented
9C22×#100C2212yes—
10C22×#99C2212yes—
11C2×C4×M16—21no—
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×#380—21no—
22C2×#380C411yescharacteristic, complemented
23C2×#352—216no—
24C2×#352C22116yes—
25#1799—22no—
26#1799C2214yes—
27C4×#100—22no—
28C4×#100C2214yes—
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⋊C4—41no—
58C22×C4⋊C4C2×C414yescomplemented
59C22×C4⋊C4D411yescharacteristic
60C22×C22⋊C4—22no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4C2×C412yes—
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C22⋊C4D411yescharacteristic
65C22×C42D411yescharacteristic
66C2×#100—22no—
67C2×#100—24no—
68C2×#100D418yes—
69C2×#100C2314yes—
70C2×#99—28no—
71C2×#99—22no—
72C2×#99C2314yes—
73C4×M16—24no—
74C4×M16—42no—
75C2×#97—41no—
76C2×C4×C8—41no—
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#380—24no—
89#380—42no—
90#380C2×C414yescomplemented
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—44no—
124C23×C4—42no—
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—28no—
136C2×M16—24no—
137C2×M16—44no—
138C2×M16—42no—
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—48no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—84no—
174C2×C4⋊C4C22⋊C412yes—
175C2×C4⋊C4C2×D412yes—
176C2×C22⋊C4—28no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—24no—
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—44no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—42no—
190C2×C22⋊C4C22⋊C412yes—
191C2×C22⋊C4C22⋊C412yes—
192C2×C22⋊C4C22×C414yes—
193C2×C22⋊C4C22×C412yes—
194C2×C22⋊C4C2×D412yes—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—22no—
198C2×C42C22⋊C412yes—
199C2×C42C2×D412yes—
200#100—28no—
201#100—24no—
202#100C2×D418yes—
203#99—28no—
204#99—44no—
205#97—44no—
206C4×C8—44no—
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#12412yes—
226C24#12712yes—
227C24C23×C411yescharacteristic
228C24C22×D411yescharacteristic
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—44no—
237C2×D4—44no—
238C2×D4—44no—
239C2×D4—44no—
240C2×D4—41no—
241C2×D4C4×D414yes—
242C2×D4#12114yes—
243C2×D4#12412yes—
244C2×D4#12712yes—
245C2×D4C22×D412yes—
246C2×D4C2×#4112yes—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—28no—
268C22×C4—44no—
269C22×C4—48no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—84no—
293C22×C4—82no—
294C22×C4—88no—
295C22×C4—88no—
296C22×C4—84no—
297C22×C4—84no—
298C22×C4—84no—
299C22×C4#9912yes—
300C22×C4#9911yescharacteristic
301C22×C4#9911yescharacteristic
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C2×C22⋊C412yes—
304C22×C4C2×C22⋊C412yes—
305C22×C4C2×C22⋊C412yes—
306C22×C4C2×C22⋊C412yes—
307C22×C4#12114yes—
308C22×C4C22×D411yescharacteristic
309M16—28no—
310M16—44no—
311M16—48no—
312C2×C8—44no—
313C2×C8—48no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—48no—
317C4⋊C4—48no—
318C4⋊C4—48no—
319C4⋊C4—48no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C4⋊C4—88no—
323C22⋊C4—28no—
324C22⋊C4—42no—
325C22⋊C4—42no—
326C22⋊C4—42no—
327C22⋊C4—42no—
328C22⋊C4—44no—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C22⋊C4—48no—
332C22⋊C4—44no—
333C22⋊C4—44no—
334C22⋊C4—84no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—22no—
339C23—22no—
340C23—22no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—24no—
348C23—24no—
349C23—22no—
350C23—22no—
351C23—21no—
352C23—21no—
353C23—22no—
354C23—22no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—48no—
359C23—41no—
360C23—42no—
361C23—42no—
362C23—48no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—44no—
367C23—44no—
368C23—41no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—81no—
374C23—81no—
375C23—84no—
376C23#39012yes—
377C23#39012yes—
378C23#39012yes—
379C23C2×#9911yescharacteristic
380C23C2×#9911yescharacteristic
381C23#41012yes—
382C23C22×C22⋊C411yescharacteristic
383C23C2×C4×D412yes—
384C23C2×#12411yescharacteristic
385C23C2×#12711yescharacteristic
386D4—44no—
387D4—44no—
388D4—42no—
389D4—42no—
390D4—42no—
391D4—42no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—88no—
428C2×C4—88no—
429C2×C4—84no—
430C2×C4—84no—
431C2×C4—88no—
432C2×C4—84no—
433C2×C4—84no—
434C2×C4—84no—
435C2×C4—168no—
436C2×C4#35214yescomplemented
437C2×C4#39012yes—
438C2×C4C2×#9912yes—
439C2×C4C2×#9912yes—
440C2×C4C2×#12112yes—
441C8—48no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—24no—
447C22—24no—
448C22—22no—
449C22—21no—
450C22—22no—
451C22—21no—
452C22—22no—
453C22—42no—
454C22—41no—
455C22—41no—
456C22—42no—
457C22—42no—
458C22—42no—
459C22—42no—
460C22—48no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—44no—
470C22—48no—
471C22—44no—
472C22—81no—
473C22—81no—
474C22—82no—
475C22—82no—
476C22#196412yes—
477C22#196412yes—
478C22C2×#35212yes—
479C22#212312yes—
480C22C2×#39011yescharacteristic
481C22C22×#9911yescharacteristic
482C22C2×#41011yescharacteristic
483C4—22no—
484C4—22no—
485C4—22no—
486C4—42no—
487C4—42no—
488C4—42no—
489C4—84no—
490C4—84no—
491C4—164no—
492C4C2×#35214yescomplemented
493C2—21no—
494C2—21no—
495C2—22no—
496C2—41no—
497C2—42no—
498C2—42no—
499C2—41no—
500C2—42no—
501C2#1321714yesdirect product factor
502C2C2×#196411yescharacteristic
503C2C22×#35211yescharacteristic
504C2C2×#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. #13217C2direct (N × H)
8. C23×C4⋊C4C4semidirect
14. #2486C4semidirect
19. C2×#381C4semidirect
22. C2×#380C4semidirect
58. C22×C4⋊C4C2×C4semidirect
84. #381C2×C4semidirect
90. #380C2×C4semidirect
436. C2×C4#352semidirect
492. C4C2×#352semidirect
501. C2#13217direct (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×#17992
C2×C4×#1002
#132172