Finite Group Databasebeta

← C2×#11878 #482047

Subgroups — C2×#11878 #482047

Order 512.

1288 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2547C211yesmaximal, characteristic
2C2×#1799C211yesmaximal, characteristic
3C2×#1459C211yesmaximal, characteristic
4#11878C218yesmaximal, direct product factor
5C2×#1344C214yesmaximal, complemented
6C22×C4⋊Q8C411yescharacteristic, complemented
7C22×#124—21no—
8C22×C4×D4C2211yescharacteristic
9C22×#99—21no—
10C22×C22⋊C8—21no—
11#2547—22no—
12#2547—22no—
13#2547C414yescomplemented
14#2547C2214yes—
15C2×#397C412yescomplemented
16C2×#397C2212yes—
17C2×#391—21no—
18C2×#380—21no—
19C2×#380—21no—
20C2×#380—21no—
21C2×C4×C4⋊C4C411yescharacteristic, complemented
22#1799—22no—
23#1799C2214yes—
24C2×#342C2212yes—
25#1459—22no—
26#1459C2214yes—
27C2×#324—21no—
28C2×#323C2212yes—
29#1344C22116yescomplemented
30C23×Q8—21no—
31C23×D4D411yescharacteristic
32C24×C4D411yescharacteristic
33C24×C4D411yescharacteristic
34C23×C8—41no—
35C2×C4⋊Q8—22no—
36C2×C4⋊Q8—42no—
37C2×C4⋊Q8—41no—
38C2×C4⋊Q8—41no—
39C2×C4⋊Q8C2×C414yescomplemented
40C2×#124—24no—
41C2×#124—24no—
42C2×#124—44no—
43C2×#124—44no—
44C2×C4×D4—41no—
45C2×C4×D4—41no—
46C2×C4×D4—42no—
47C2×C4×D4C2×C412yes—
48C2×C4×D4D412yes—
49C2×C4×D4D412yes—
50C2×C4×D4C2312yes—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—41no—
54C22×C4⋊C4C2×C412yes—
55C22×C4⋊C4D411yescharacteristic
56C22×C22⋊C4—22no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4—22no—
59C22×C22⋊C4C2×C411yescharacteristic
60C22×C22⋊C4C2311yescharacteristic
61C22×C42—21no—
62C22×C42C2×C411yescharacteristic
63C2×#99—24no—
64C2×#99—44no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—24no—
67C2×C22⋊C8—42no—
68C2×C22⋊C8—41no—
69#397—24no—
70#397—22no—
71#397—22no—
72#397C2×C414yes—
73#397C2×C418yescomplemented
74#397C2314yes—
75#391—24no—
76#391—24no—
77#380—24no—
78#380—24no—
79#380—24no—
80#380—24no—
81#380—24no—
82#380—42no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4C2×C414yescomplemented
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—41no—
88C2×#95C2×C412yes—
89#342—22no—
90#342C2314yes—
91#324—24no—
92#323—22no—
93#323C2314yes—
94C25C22⋊C411yescharacteristic
95C25C2×D411yescharacteristic
96C22×Q8—24no—
97C22×Q8—42no—
98C22×Q8—42no—
99C22×Q8—44no—
100C22×Q8—44no—
101C22×D4—41no—
102C22×D4—41no—
103C22×D4—42no—
104C22×D4C22⋊C412yes—
105C22×D4D814yes—
106C22×D4SD1614yes—
107C22×D4C2×D412yes—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—22no—
112C23×C4—21no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—44no—
117C23×C4—44no—
118C23×C4—42no—
119C23×C4C22⋊C411yescharacteristic
120C23×C4C22⋊C411yescharacteristic
121C23×C4C22⋊C412yes—
122C23×C4C22⋊C412yes—
123C23×C4C22⋊C411yescharacteristic
124C23×C4C22×C411yescharacteristic
125C23×C4C2×D412yes—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D412yes—
128C22×C8—42no—
129C22×C8—44no—
130C22×C8—84no—
131C4⋊Q8—44no—
132C4⋊Q8—44no—
133C4⋊Q8—44no—
134C4⋊Q8—44no—
135#124—416no—
136#124—48no—
137#124—48no—
138C4×D4—42no—
139C4×D4—42no—
140C4×D4—42no—
141C4×D4—42no—
142C4×D4—42no—
143C4×D4—42no—
144C4×D4—42no—
145C4×D4—42no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—84no—
159C2×C4⋊C4C22⋊C412yes—
160C2×C4⋊C4C22×C414yes—
161C2×C4⋊C4C2×D412yes—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—28no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—41no—
170C2×C22⋊C4—48no—
171C2×C22⋊C4—44no—
172C2×C22⋊C4—44no—
173C2×C22⋊C4—44no—
174C2×C22⋊C4—44no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4C22⋊C412yes—
177C2×C22⋊C4C22×C412yes—
178C2×C22⋊C4C22×C411yescharacteristic
179C2×C22⋊C4C2×D411yescharacteristic
180C2×C22⋊C4C2×D411yescharacteristic
181C2×C22⋊C4C2411yescharacteristic
182C2×C42—22no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—24no—
186C2×C42—44no—
187C2×C42C22⋊C412yes—
188C2×C42C22×C412yes—
189#99—48no—
190C22⋊C8—24no—
191C22⋊C8—42no—
192C22⋊C8—44no—
193#95—24no—
194#95—28no—
195#95—22no—
196#95—24no—
197#95—44no—
198#95C22×C414yes—
199C24—21no—
200C24—21no—
201C24—21no—
202C24—21no—
203C24—22no—
204C24—22no—
205C24—42no—
206C24—42no—
207C24—42no—
208C24—42no—
209C24#10214yes—
210C24C2×C22⋊C411yescharacteristic
211C24C2×C22⋊C412yes—
212C24C2×C22⋊C411yescharacteristic
213C24C2×D812yes—
214C24C2×SD1612yes—
215C24C22×D411yescharacteristic
216C2×Q8—48no—
217C2×Q8—48no—
218C2×Q8—48no—
219C2×Q8—88no—
220C2×Q8—88no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4—44no—
225C2×D4—44no—
226C2×D4—44no—
227C2×D4—44no—
228C2×D4—42no—
229C2×D4#10218yes—
230C2×D4C2×D814yes—
231C2×D4C2×SD1614yes—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—84no—
270C22×C4—84no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4—84no—
274C22×C4—84no—
275C22×C4—84no—
276C22×C4#9912yes—
277C22×C4#9911yescharacteristic
278C22×C4#9911yescharacteristic
279C22×C4#10214yes—
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C23×C411yescharacteristic
286C22×C4C22×D411yescharacteristic
287C2×C8—44no—
288C2×C8—84no—
289C2×C8—88no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—44no—
293C4⋊C4—48no—
294C4⋊C4—44no—
295C4⋊C4—44no—
296C4⋊C4—44no—
297C4⋊C4—88no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—44no—
305C22⋊C4—44no—
306C22⋊C4—48no—
307C22⋊C4—416no—
308C22⋊C4—44no—
309C22⋊C4—44no—
310C22⋊C4—84no—
311C42—22no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—48no—
316C23—22no—
317C23—22no—
318C23—21no—
319C23—22no—
320C23—21no—
321C23—22no—
322C23—24no—
323C23—22no—
324C23—21no—
325C23—21no—
326C23—22no—
327C23—22no—
328C23—24no—
329C23—22no—
330C23—41no—
331C23—42no—
332C23—41no—
333C23—48no—
334C23—42no—
335C23—41no—
336C23—44no—
337C23—44no—
338C23—41no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—81no—
344C23—81no—
345C23—81no—
346C23—81no—
347C23—84no—
348C23—84no—
349C23C2×#9911yescharacteristic
350C23C2×#9911yescharacteristic
351C23C2×#10212yes—
352C23C2×#10212yes—
353C23C2×#10214yes—
354C23#42112yes—
355C23C22×C22⋊C411yescharacteristic
356C23C22×D811yescharacteristic
357C23C22×SD1611yescharacteristic
358Q8—816no—
359D4—48no—
360D4—44no—
361D4—44no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—88no—
400C2×C4—84no—
401C2×C4—84no—
402C2×C4—84no—
403C2×C4—84no—
404C2×C4—84no—
405C2×C4—168no—
406C2×C4#35414yes—
407C2×C4C2×#9912yes—
408C2×C4C2×#9912yes—
409C2×C4C2×#10214yes—
410C8—84no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—24no—
416C22—24no—
417C22—21no—
418C22—21no—
419C22—22no—
420C22—24no—
421C22—41no—
422C22—41no—
423C22—44no—
424C22—42no—
425C22—41no—
426C22—41no—
427C22—42no—
428C22—41no—
429C22—42no—
430C22—41no—
431C22—44no—
432C22—48no—
433C22—44no—
434C22—81no—
435C22—81no—
436C22—82no—
437C22—82no—
438C22—84no—
439C22—82no—
440C22#151912yes—
441C22#151912yes—
442C22#212312yes—
443C22C2×#35412yes—
444C22C22×#9911yescharacteristic
445C22C22×#10211yescharacteristic
446C22C2×#42111yescharacteristic
447C4—22no—
448C4—24no—
449C4—41no—
450C4—41no—
451C4—41no—
452C4—41no—
453C4—42no—
454C4—84no—
455C4—84no—
456C4—164no—
457C4C2×#35414yes—
458C2—21no—
459C2—21no—
460C2—22no—
461C2—41no—
462C2—41no—
463C2—42no—
464C2—44no—
465C2#1187814yesdirect product factor
466C2C2×#151911yescharacteristic
467C2C2×#212311yescharacteristic
468C2C22×#35411yescharacteristic
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. #11878C2direct (N × H)
5. C2×#1344C2semidirect
6. C22×C4⋊Q8C4semidirect
13. #2547C4semidirect
15. C2×#397C4semidirect
21. C2×C4×C4⋊C4C4semidirect
29. #1344C22semidirect
39. C2×C4⋊Q8C2×C4semidirect
73. #397C2×C4semidirect
84. C4×C4⋊C4C2×C4semidirect
465. C2#11878direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25472
C2×#17992
C2×#14592
#118782
C2×#13442