Finite Group Databasebeta

← C2×#11731 #481513

Subgroups — C2×#11731 #481513

Order 512.

1162 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×#2638C211yesmaximal, characteristic
2C2×C4×#99C211yesmaximal, characteristic
3C2×#1460C211yesmaximal, characteristic
4#11731C218yesmaximal, direct product factor
5C2×#1351C212yesmaximal, complemented
6C2×#1344C212yesmaximal, complemented
7C22×C22⋊Q8C411yescharacteristic, complemented
8C22×C4×D4C2211yescharacteristic
9C2×#431—21no—
10C22×#99—21no—
11#2638—22no—
12#2638—22no—
13#2638C414yescomplemented
14#2638C2214yes—
15C2×#398C411yescharacteristic, complemented
16C2×#397—21no—
17C2×#397C2211yescharacteristic
18C2×#395C411yescharacteristic, complemented
19C2×#394—21no—
20C2×#393C2211yescharacteristic
21C2×#387—21no—
22C2×#385—21no—
23C2×#381C411yescharacteristic, complemented
24C2×C4×C22⋊C4—21no—
25C4×#99—22no—
26C4×#99C2214yes—
27C2×#342C2212yes—
28#1460—22no—
29#1460C2214yes—
30C2×#323—21no—
31C2×#323C2212yes—
32#1351C2218yescomplemented
33#1344C2218yescomplemented
34C23×Q8C2×C411yescharacteristic
35C23×D4D411yescharacteristic
36C24×C4C2×C411yescharacteristic
37C24×C4D411yescharacteristic
38C2×C22⋊Q8—22no—
39C2×C22⋊Q8—42no—
40C2×C22⋊Q8—41no—
41C2×C22⋊Q8—41no—
42C2×C22⋊Q8C2×C414yescomplemented
43C2×C4×D4—41no—
44C2×C4×D4—41no—
45C2×C4×D4—42no—
46C2×C4×D4C2×C412yes—
47C2×C4×D4D412yes—
48C2×C4×D4D412yes—
49C2×C4×D4C2312yes—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C22⋊C4—22no—
56C22×C22⋊C4—21no—
57C22×C22⋊C4—21no—
58C22×C22⋊C4—21no—
59C22×C22⋊C4—21no—
60C22×C22⋊C4D411yescharacteristic
61C22×C22⋊C4C2311yescharacteristic
62C22×C42—41no—
63C22×C42D411yescharacteristic
64#431—24no—
65#431—42no—
66C2×C4⋊C8—41no—
67C2×#99—28no—
68C2×#99—24no—
69C2×C22⋊C8—22no—
70C2×C4×C8—41no—
71#398—22no—
72#398C2×C414yescomplemented
73#397—24no—
74#397—21no—
75#397—21no—
76#397—24no—
77#397C2×C412yes—
78#397C2312yes—
79#395—22no—
80#395C2×C414yescomplemented
81#394—24no—
82#394—24no—
83#393—21no—
84#393—21no—
85#393C2×C412yes—
86#393C2312yes—
87#387—24no—
88#387—24no—
89#385—24no—
90#385—24no—
91#381—22no—
92#381C2×C414yescomplemented
93C4×C22⋊C4—24no—
94C4×C22⋊C4—42no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—41no—
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C411yescharacteristic
103#342—22no—
104#342C2314yes—
105#323—24no—
106#323—22no—
107#323C2314yes—
108C25C22⋊C411yescharacteristic
109C25C2×D411yescharacteristic
110C22×Q8—21no—
111C22×Q8—41no—
112C22×Q8—41no—
113C22×Q8—82no—
114C22×Q8C22×C412yes—
115C22×D4—24no—
116C22×D4—24no—
117C22×D4—41no—
118C22×D4—41no—
119C22×D4C22⋊C412yes—
120C22×D4C2×D412yes—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—42no—
132C23×C4—82no—
133C23×C4—82no—
134C23×C4C22⋊C411yescharacteristic
135C23×C4C22⋊C411yescharacteristic
136C23×C4C22⋊C412yes—
137C23×C4C22⋊C412yes—
138C23×C4C22⋊C411yescharacteristic
139C23×C4C22×C411yescharacteristic
140C23×C4C22×C412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D412yes—
143C22×C8—42no—
144C22⋊Q8—44no—
145C22⋊Q8—44no—
146C22⋊Q8—44no—
147C22⋊Q8—44no—
148C4×D4—42no—
149C4×D4—42no—
150C4×D4—42no—
151C4×D4—42no—
152C4×D4—42no—
153C4×D4—42no—
154C4×D4—42no—
155C4×D4—42no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—82no—
169C2×C4⋊C4C22⋊C412yes—
170C2×C4⋊C4C22×C412yes—
171C2×C4⋊C4C22×C412yes—
172C2×C22⋊C4—28no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4—41no—
184C2×C22⋊C4—42no—
185C2×C22⋊C4—42no—
186C2×C22⋊C4—42no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—82no—
190C2×C22⋊C4—82no—
191C2×C22⋊C4C22⋊C412yes—
192C2×C22⋊C4C22×C411yescharacteristic
193C2×C22⋊C4C2×D411yescharacteristic
194C2×C22⋊C4C2×D412yes—
195C2×C22⋊C4C2×D411yescharacteristic
196C2×C22⋊C4C2411yescharacteristic
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—22no—
200C2×C42—44no—
201C2×C42—84no—
202C2×C42C22⋊C412yes—
203C2×C42C2×D412yes—
204C4⋊C8—44no—
205#99—216no—
206C22⋊C8—24no—
207C22⋊C8—42no—
208C4×C8—44no—
209#95—24no—
210#95—24no—
211#95—22no—
212#95—24no—
213#95—22no—
214#95—21no—
215#95—21no—
216#95—22no—
217#95—22no—
218#95—44no—
219#95C22×C412yes—
220#95C22×C412yes—
221C24—21no—
222C24—21no—
223C24—22no—
224C24—21no—
225C24—21no—
226C24—22no—
227C24—22no—
228C24—42no—
229C24—42no—
230C24—42no—
231C24—42no—
232C24C2×C22⋊C411yescharacteristic
233C24C2×C22⋊C412yes—
234C24C2×C22⋊C411yescharacteristic
235C24#13514yes—
236C24C22×D411yescharacteristic
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—82no—
240C2×Q8—82no—
241C2×Q8—84no—
242C2×Q8—84no—
243C2×D4—24no—
244C2×D4—24no—
245C2×D4—24no—
246C2×D4—24no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—48no—
250C2×D4—48no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—42no—
273C22×C4—44no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—44no—
278C22×C4—48no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—82no—
294C22×C4—82no—
295C22×C4—82no—
296C22×C4—82no—
297C22×C4—82no—
298C22×C4—84no—
299C22×C4—84no—
300C22×C4—82no—
301C22×C4—82no—
302C22×C4—84no—
303C22×C4—84no—
304C22×C4#9911yescharacteristic
305C22×C4#9911yescharacteristic
306C22×C4#9912yes—
307C22×C4#10414yes—
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C412yes—
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C2×C22⋊C412yes—
313C22×C4C23×C411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C2×C8—44no—
316C2×C8—84no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C4⋊C4—48no—
324C4⋊C4—84no—
325C22⋊C4—28no—
326C22⋊C4—42no—
327C22⋊C4—41no—
328C22⋊C4—41no—
329C22⋊C4—42no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—48no—
333C22⋊C4—48no—
334C22⋊C4—44no—
335C22⋊C4—44no—
336C22⋊C4—44no—
337C22⋊C4—84no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—88no—
345C23—22no—
346C23—22no—
347C23—21no—
348C23—21no—
349C23—24no—
350C23—21no—
351C23—21no—
352C23—24no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—48no—
360C23—41no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—81no—
372C23—81no—
373C23—84no—
374C23—84no—
375C23—84no—
376C23C2×#9911yescharacteristic
377C23C2×#9911yescharacteristic
378C23#41612yes—
379C23#41612yes—
380C23C2×#10412yes—
381C23C22×C22⋊C411yescharacteristic
382C23C2×#13512yes—
383Q8—84no—
384Q8—84no—
385D4—48no—
386D4—48no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—48no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—84no—
433C2×C4—84no—
434C2×C4—82no—
435C2×C4—84no—
436C2×C4—84no—
437C2×C4—816no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4C2×#9912yes—
442C2×C4C2×#9912yes—
443C2×C4C2×#10414yes—
444C8—84no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—24no—
450C22—24no—
451C22—22no—
452C22—22no—
453C22—42no—
454C22—42no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—48no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—81no—
464C22—81no—
465C22—84no—
466C22—82no—
467C22—82no—
468C22—84no—
469C22—84no—
470C22#150112yes—
471C22#150112yes—
472C22#212112yes—
473C22#212912yes—
474C22C22×#9911yescharacteristic
475C22C2×#41611yescharacteristic
476C22C22×#10411yescharacteristic
477C4—22no—
478C4—22no—
479C4—24no—
480C4—42no—
481C4—41no—
482C4—41no—
483C4—41no—
484C4—41no—
485C4—82no—
486C4—82no—
487C4—88no—
488C4—82no—
489C4—82no—
490C2—21no—
491C2—21no—
492C2—22no—
493C2—42no—
494C2—42no—
495C2—82no—
496C2#1173114yesdirect product factor
497C2C2×#150111yescharacteristic
498C2C2×#212111yescharacteristic
499C2C2×#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. #11731C2direct (N × H)
5. C2×#1351C2semidirect
6. C2×#1344C2semidirect
7. C22×C22⋊Q8C4semidirect
13. #2638C4semidirect
15. C2×#398C4semidirect
18. C2×#395C4semidirect
23. C2×#381C4semidirect
32. #1351C22semidirect
33. #1344C22semidirect
42. C2×C22⋊Q8C2×C4semidirect
72. #398C2×C4semidirect
80. #395C2×C4semidirect
92. #381C2×C4semidirect
496. C2#11731direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#26382
C2×C4×#992
C2×#14602
#117312
C2×#13512
C2×#13442