Finite Group Databasebeta

← C2×#11715 #481487

Subgroups — C2×#11715 #481487

Order 512.

1272 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2396C211yesmaximal, characteristic
2C2×#1740C211yesmaximal, characteristic
3C2×#1458C211yesmaximal, characteristic
4#11715C218yesmaximal, direct product factor
5C2×#1352C212yesmaximal
6C2×#1351C212yesmaximal
7C22×C4×Q8C411yescharacteristic, complemented
8C22×#117—21no—
9C22×#117C2211yescharacteristic
10C2×#431—21no—
11#2396—22no—
12#2396—22no—
13#2396C414yescomplemented
14#2396C2214yes—
15C2×#400C411yescharacteristic, complemented
16C2×#399C2211yescharacteristic
17C2×#398—21no—
18C2×#396—21no—
19C2×#395C411yescharacteristic, complemented
20C2×#393C2211yescharacteristic
21C2×C4×C4⋊C4—21no—
22C2×C4×C4⋊C4C411yescharacteristic, complemented
23C2×C4×C22⋊C4—21no—
24#1740—22no—
25#1740C2214yes—
26C2×#342—21no—
27C2×#342C2212yes—
28#1458—22no—
29#1458C2214yes—
30C2×#324—21no—
31C2×#323C2212yes—
32#1352C2218yes—
33#1351C2218yes—
34C23×Q8C2×C411yescharacteristic
35C24×C4D411yescharacteristic
36C2×C4×Q8—22no—
37C2×C4×Q8—42no—
38C2×C4×Q8—41no—
39C2×C4×Q8—41no—
40C2×C4×Q8C2×C414yescomplemented
41C2×#117—28no—
42C2×#117—24no—
43C2×#117—41no—
44C2×#117—42no—
45C2×#117—41no—
46C2×#117—42no—
47C2×#117—44no—
48C2×#117C2×C412yes—
49C2×#117D412yes—
50C2×#117D412yes—
51C2×#117C2312yes—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—41no—
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4D411yescharacteristic
60C22×C22⋊C4—22no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4D411yescharacteristic
64C22×C22⋊C4C2311yescharacteristic
65C22×C42—21no—
66C22×C42—21no—
67C22×C42—41no—
68C22×C42C2×C411yescharacteristic
69C22×C42D411yescharacteristic
70#431—24no—
71#431—42no—
72C2×C4⋊C8—41no—
73C2×C22⋊C8—22no—
74C2×C4×C8—41no—
75#400—22no—
76#400C2×C414yescomplemented
77#399—21no—
78#399—21no—
79#399C2×C412yes—
80#399C2312yes—
81#398—24no—
82#398—24no—
83#396—24no—
84#396—24no—
85#395—22no—
86#395C2×C414yescomplemented
87#393—21no—
88#393—21no—
89#393C2×C412yes—
90#393C2312yes—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—24no—
93C4×C4⋊C4—24no—
94C4×C4⋊C4C2×C414yescomplemented
95C4×C22⋊C4—24no—
96C4×C22⋊C4—24no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C411yescharacteristic
103#342—24no—
104#342—22no—
105#342C2314yes—
106#324—24no—
107#323—22no—
108#323C2314yes—
109C25C2×D411yescharacteristic
110C22×Q8—21no—
111C22×Q8—44no—
112C22×Q8—41no—
113C22×Q8—41no—
114C22×Q8C22×C412yes—
115C23×C4—21no—
116C23×C4—22no—
117C23×C4—21no—
118C23×C4—24no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—42no—
124C23×C4C22⋊C411yescharacteristic
125C23×C4C22⋊C412yes—
126C23×C4C22⋊C411yescharacteristic
127C23×C4C22⋊C411yescharacteristic
128C23×C4C22⋊C411yescharacteristic
129C23×C4SD1614yes—
130C23×C4Q1614yes—
131C23×C4C22×C411yescharacteristic
132C23×C4C2×D412yes—
133C23×C4C2×D411yescharacteristic
134C22×C8—42no—
135C4×Q8—44no—
136C4×Q8—44no—
137C4×Q8—44no—
138C4×Q8—44no—
139#117—216no—
140#117—42no—
141#117—42no—
142#117—42no—
143#117—42no—
144#117—42no—
145#117—42no—
146#117—42no—
147#117—42no—
148#117—416no—
149#117—48no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—48no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4C22⋊C412yes—
170C2×C4⋊C4C22⋊C412yes—
171C2×C4⋊C4C22×C412yes—
172C2×C4⋊C4C22×C412yes—
173C2×C4⋊C4C2×D412yes—
174C2×C22⋊C4—28no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
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—44no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—42no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4C22⋊C412yes—
189C2×C22⋊C4C22×C411yescharacteristic
190C2×C22⋊C4C2×D412yes—
191C2×C22⋊C4C2×D411yescharacteristic
192C2×C22⋊C4C2×D411yescharacteristic
193C2×C22⋊C4C2411yescharacteristic
194C2×C42—21no—
195C2×C42—22no—
196C2×C42—21no—
197C2×C42—22no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—24no—
201C2×C42—24no—
202C2×C42—48no—
203C2×C42—44no—
204C2×C42—41no—
205C2×C42—42no—
206C2×C42—41no—
207C2×C42—44no—
208C2×C42C22⋊C412yes—
209C2×C42C22⋊C412yes—
210C2×C42C22×C412yes—
211C2×C42C2×D412yes—
212C4⋊C8—44no—
213C22⋊C8—24no—
214C22⋊C8—42no—
215C4×C8—44no—
216#95—24no—
217#95—24no—
218#95—22no—
219#95—24no—
220#95—21no—
221#95—21no—
222#95—22no—
223#95C22×C412yes—
224#95C22×C412yes—
225C24—21no—
226C24—21no—
227C24—22no—
228C24—42no—
229C24—42no—
230C24C2×C22⋊C411yescharacteristic
231C24C2×C22⋊C411yescharacteristic
232C24C2×SD1612yes—
233C24C2×Q1612yes—
234C24C22×D411yescharacteristic
235C2×Q8—44no—
236C2×Q8—42no—
237C2×Q8—44no—
238C2×Q8—42no—
239C2×Q8—48no—
240C2×Q8—44no—
241C2×Q8—44no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—28no—
248C22×C4—24no—
249C22×C4—28no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—44no—
281C22×C4—48no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—84no—
290C22×C4—84no—
291C22×C4#9911yescharacteristic
292C22×C4#9912yes—
293C22×C4#9911yescharacteristic
294C22×C4#10314yes—
295C22×C4#10318yes—
296C22×C4#10314yes—
297C22×C4#10414yes—
298C22×C4C2×C22⋊C412yes—
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C22⋊C412yes—
302C22×C4C2×C22⋊C412yes—
303C22×C4C2×C22⋊C412yes—
304C22×C4C2×SD1614yes—
305C22×C4C2×Q1614yes—
306C22×C4C23×C411yescharacteristic
307C22×C4C22×D411yescharacteristic
308C2×C8—44no—
309C2×C8—84no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—44no—
316C4⋊C4—48no—
317C4⋊C4—44no—
318C4⋊C4—48no—
319C4⋊C4—48no—
320C4⋊C4—42no—
321C4⋊C4—416no—
322C22⋊C4—28no—
323C22⋊C4—42no—
324C22⋊C4—41no—
325C22⋊C4—42no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—41no—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C22⋊C4—44no—
333C22⋊C4—48no—
334C22⋊C4—44no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—416no—
344C42—44no—
345C42—44no—
346C42—48no—
347C23—22no—
348C23—22no—
349C23—21no—
350C23—21no—
351C23—22no—
352C23—22no—
353C23—21no—
354C23—21no—
355C23—22no—
356C23—22no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—44no—
362C23—44no—
363C23—42no—
364C23—42no—
365C23—81no—
366C23—81no—
367C23—84no—
368C23C2×#9911yescharacteristic
369C23C2×#9911yescharacteristic
370C23C2×#10312yes—
371C23C2×#10312yes—
372C23C2×#10412yes—
373C23C22×C22⋊C411yescharacteristic
374C23C22×SD1611yescharacteristic
375C23C22×Q1611yescharacteristic
376Q8—44no—
377Q8—44no—
378Q8—48no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—28no—
386C2×C4—28no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—21no—
397C2×C4—42no—
398C2×C4—48no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—48no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—84no—
426C2×C4—82no—
427C2×C4—84no—
428C2×C4—84no—
429C2×C4—816no—
430C2×C4—82no—
431C2×C4#32818yes—
432C2×C4#32814yes—
433C2×C4#32814yes—
434C2×C4C2×#9912yes—
435C2×C4C2×#9912yes—
436C2×C4C2×#10314yes—
437C2×C4C2×#10314yes—
438C2×C4C2×#10414yes—
439C8—84no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—24no—
445C22—24no—
446C22—21no—
447C22—21no—
448C22—22no—
449C22—41no—
450C22—41no—
451C22—42no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—44no—
457C22—44no—
458C22—81no—
459C22—81no—
460C22—82no—
461C22—82no—
462C22C2×#32812yes—
463C22C2×#32812yes—
464C22#212012yes—
465C22#212612yes—
466C22C22×#9911yescharacteristic
467C22C22×#10311yescharacteristic
468C22C22×#10411yescharacteristic
469C4—24no—
470C4—44no—
471C4—44no—
472C4—41no—
473C4—41no—
474C4—42no—
475C4—41no—
476C4—42no—
477C4—41no—
478C4—82no—
479C4—88no—
480C4—82no—
481C4C2×#32814yes—
482C4C2×#32814yes—
483C2—21no—
484C2—21no—
485C2—22no—
486C2—41no—
487C2—41no—
488C2—42no—
489C2#1171514yesdirect product factor
490C2C22×#32811yescharacteristic
491C2C2×#212011yescharacteristic
492C2C2×#212611yescharacteristic
subgroupG/H
centerC22×C4#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
socleC23C2×#99
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11715C2direct (N × H)
7. C22×C4×Q8C4semidirect
13. #2396C4semidirect
15. C2×#400C4semidirect
19. C2×#395C4semidirect
22. C2×C4×C4⋊C4C4semidirect
40. C2×C4×Q8C2×C4semidirect
76. #400C2×C4semidirect
86. #395C2×C4semidirect
94. C4×C4⋊C4C2×C4semidirect
489. C2#11715direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#23962
C2×#17402
C2×#14582
#117152
C2×#13522
C2×#13512