Finite Group Databasebeta

← C2×#13295 #495403

Subgroups — C2×#13295 #495403

Order 512.

1314 conjugacy classes of subgroups.

The subgroup lattice has 485 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2487C211yesmaximal, characteristic
2C2×#1800C211yesmaximal, characteristic
3C2×C4×#99C211yesmaximal, characteristic
4#13295C218yesmaximal, direct product factor
5C2×#1390C214yesmaximal
6C22×C4×D4C2211yescharacteristic
7C23×C4⋊C4C411yescharacteristic, complemented
8C2×#423—21no—
9C22×#100C2212yes—
10C22×#99C2212yes—
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×#380C411yescharacteristic, complemented
22C2×C4×C22⋊C4—21no—
23#1800—22no—
24#1800C2214yes—
25C4×#99—22no—
26C4×#99C2214yes—
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⋊C4C2×C414yescomplemented
51C22×C4⋊C4D411yescharacteristic
52C22×C22⋊C4—22no—
53C22×C22⋊C4—22no—
54C22×C22⋊C4C2×C412yes—
55C22×C22⋊C4C2×C411yescharacteristic
56C22×C22⋊C4D411yescharacteristic
57C22×C42—41no—
58C22×C42D411yescharacteristic
59#423—24no—
60#423—42no—
61C2×C4⋊C8—41no—
62C2×C4⋊C8—41no—
63C2×#100—28no—
64C2×#100—22no—
65C2×#100C2314yes—
66C2×#99—22no—
67C2×#99—24no—
68C2×#99Q818yes—
69C2×#99C2314yes—
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#380C2×C414yescomplemented
86C4×C22⋊C4—24no—
87C4×C22⋊C4—42no—
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—24no—
131C2×M16—42no—
132C2×M16—44no—
133C2×M16—44no—
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—84no—
160C2×C4⋊C4C22⋊C412yes—
161C2×C4⋊C4C2×D412yes—
162C2×C22⋊C4—28no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—41no—
170C2×C22⋊C4—41no—
171C2×C22⋊C4—44no—
172C2×C22⋊C4—42no—
173C2×C22⋊C4—44no—
174C2×C22⋊C4—84no—
175C2×C22⋊C4C22⋊C412yes—
176C2×C22⋊C4C22⋊C412yes—
177C2×C22⋊C4C22×C414yes—
178C2×C22⋊C4C22×C412yes—
179C2×C22⋊C4C2×D412yes—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—22no—
183C2×C42—44no—
184C2×C42—84no—
185C2×C42C22⋊C412yes—
186C2×C42C2×D412yes—
187C4⋊C8—44no—
188C4⋊C8—44no—
189#100—28no—
190#100—44no—
191#99—28no—
192#99—24no—
193#99C2×Q818yes—
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—
213C24C22⋊Q814yes—
214C24C23×C411yescharacteristic
215C24C22×D411yescharacteristic
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—41no—
223C2×D4—44no—
224C2×D4—44no—
225C2×D4—44no—
226C2×D4—44no—
227C2×D4—41no—
228C2×D4C4×Q814yes—
229C2×D4C22⋊Q814yes—
230C2×D4#12412yes—
231C2×D4C4⋊Q812yes—
232C2×D4C22×Q812yes—
233C2×D4C2×#4112yes—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—44no—
255C22×C4—48no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—44no—
266C22×C4—48no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—84no—
277C22×C4—82no—
278C22×C4—84no—
279C22×C4—84no—
280C22×C4—84no—
281C22×C4—84no—
282C22×C4—88no—
283C22×C4—88no—
284C22×C4#9912yes—
285C22×C4#9911yescharacteristic
286C22×C4#9911yescharacteristic
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C412yes—
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C412yes—
292C22×C4#12412yes—
293C22×C4C4⋊Q812yes—
294C22×C4C22×D411yescharacteristic
295M16—44no—
296M16—44no—
297M16—84no—
298C2×C8—44no—
299C2×C8—84no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—48no—
303C4⋊C4—48no—
304C4⋊C4—44no—
305C4⋊C4—44no—
306C4⋊C4—88no—
307C22⋊C4—28no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—44no—
313C22⋊C4—48no—
314C22⋊C4—44no—
315C22⋊C4—44no—
316C22⋊C4—84no—
317C22⋊C4—84no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—88no—
323C23—22no—
324C23—22no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—24no—
332C23—24no—
333C23—24no—
334C23—21no—
335C23—21no—
336C23—24no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—48no—
342C23—42no—
343C23—42no—
344C23—48no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—48no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23—42no—
354C23—81no—
355C23—81no—
356C23—84no—
357C23#39112yes—
358C23#39112yes—
359C23#39112yes—
360C23C2×#9911yescharacteristic
361C23C2×#9911yescharacteristic
362C23#41012yes—
363C23C22×C22⋊C411yescharacteristic
364C23C2×C4×Q812yes—
365C23C2×C22⋊Q812yes—
366D4—44no—
367D4—44no—
368D4—42no—
369D4—42no—
370D4—42no—
371D4—42no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—48no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—88no—
411C2×C4—84no—
412C2×C4—84no—
413C2×C4—84no—
414C2×C4—816no—
415C2×C4—88no—
416C2×C4—84no—
417C2×C4—88no—
418C2×C4#35214yes—
419C2×C4#39112yes—
420C2×C4C2×#9912yes—
421C2×C4C2×#9912yes—
422C2×C4C2×#12411yescharacteristic
423C2×C4C2×C4⋊Q811yescharacteristic
424C8—84no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—24no—
430C22—24no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—22no—
435C22—42no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—48no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—48no—
451C22—48no—
452C22—81no—
453C22—81no—
454C22—84no—
455C22#199612yes—
456C22#199612yes—
457C22C2×#35212yes—
458C22#212212yes—
459C22C2×#39111yescharacteristic
460C22C22×#9911yescharacteristic
461C22C2×#41011yescharacteristic
462C4—22no—
463C4—21no—
464C4—21no—
465C4—22no—
466C4—42no—
467C4—42no—
468C4—42no—
469C4—84no—
470C4—88no—
471C4—84no—
472C4C2×#35214yes—
473C2—21no—
474C2—21no—
475C2—22no—
476C2—42no—
477C2—42no—
478C2—42no—
479C2—42no—
480C2#1329514yesdirect product factor
481C2C2×#199611yescharacteristic
482C2C22×#35211yescharacteristic
483C2C2×#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. #13295C2direct (N × H)
7. C23×C4⋊C4C4semidirect
13. #2487C4semidirect
19. C2×#385C4semidirect
21. C2×#380C4semidirect
50. C22×C4⋊C4C2×C4semidirect
81. #385C2×C4semidirect
85. #380C2×C4semidirect
480. C2#13295direct (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×#18002
C2×C4×#992
#132952
C2×#13902