Finite Group Databasebeta

← C2×#11788 #481633

Subgroups — C2×#11788 #481633

Order 512.

1430 conjugacy classes of subgroups.

The subgroup lattice has 461 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic
2C2×C4×#99C211yesmaximal, characteristic
3C2×#1465C211yesmaximal, characteristic
4#11788C218yesmaximal, direct product factor
5C2×#1352C214yesmaximal, complemented
6C22×C4⋊Q8C411yescharacteristic, complemented
7C22×C22⋊Q8—21no—
8C22×C22⋊Q8—21no—
9C22×C4×D4C2211yescharacteristic
10C22×#99—21no—
11C2×#407—21no—
12#2542—22no—
13#2542—22no—
14#2542C414yescomplemented
15#2542C2214yes—
16C2×#399C412yescomplemented
17C2×#399C2212yes—
18C2×#390—21no—
19C2×#384—21no—
20C2×C4×C4⋊C4C411yescharacteristic, complemented
21C2×C4×C22⋊C4—21no—
22C4×#99—22no—
23C4×#99C2214yes—
24C2×#342C2212yes—
25#1465—22no—
26#1465C2214yes—
27C2×#324—21no—
28C2×#323C2212yes—
29#1352C22116yescomplemented
30C23×Q8—21no—
31C23×D4D411yescharacteristic
32C24×C4D411yescharacteristic
33C24×C4D411yescharacteristic
34C22×M16—41no—
35C2×C4⋊Q8—22no—
36C2×C4⋊Q8—42no—
37C2×C4⋊Q8—41no—
38C2×C4⋊Q8—41no—
39C2×C4⋊Q8C2×C414yescomplemented
40C2×C22⋊Q8—24no—
41C2×C22⋊Q8—24no—
42C2×C22⋊Q8—24no—
43C2×C22⋊Q8—24no—
44C2×C22⋊Q8—48no—
45C2×C22⋊Q8—48no—
46C2×C4×D4—41no—
47C2×C4×D4—41no—
48C2×C4×D4—42no—
49C2×C4×D4C2×C412yes—
50C2×C4×D4D412yes—
51C2×C4×D4D412yes—
52C2×C4×D4C2312yes—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4C2×C412yes—
58C22×C4⋊C4D411yescharacteristic
59C22×C22⋊C4—22no—
60C22×C22⋊C4—22no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4C2×C411yescharacteristic
63C22×C22⋊C4C2311yescharacteristic
64C22×C42—21no—
65C22×C42—41no—
66C22×C42C2×C411yescharacteristic
67C2×#99—28no—
68C2×#99—24no—
69#407—24no—
70#407—42no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—41no—
73#399—24no—
74#399—22no—
75#399—22no—
76#399C2×C414yes—
77#399C2×C418yescomplemented
78#399C2314yes—
79#390—24no—
80#390—24no—
81#384—24no—
82#384—24no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4C2×C414yescomplemented
85C4×C22⋊C4—24no—
86C4×C22⋊C4—42no—
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—44no—
98C22×Q8—44no—
99C22×Q8—42no—
100C22×Q8—41no—
101C22×Q8—41no—
102C22×D4—24no—
103C22×D4—24no—
104C22×D4—41no—
105C22×D4—41no—
106C22×D4C22⋊C412yes—
107C22×D4C2×D412yes—
108C23×C4—22no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—22no—
113C23×C4—21no—
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—
128C2×M16—44no—
129C2×M16—84no—
130C22×C8—42no—
131C4⋊Q8—44no—
132C4⋊Q8—44no—
133C4⋊Q8—44no—
134C4⋊Q8—44no—
135C22⋊Q8—416no—
136C22⋊Q8—416no—
137C22⋊Q8—416no—
138C22⋊Q8—416no—
139C4×D4—42no—
140C4×D4—42no—
141C4×D4—42no—
142C4×D4—42no—
143C4×D4—42no—
144C4×D4—42no—
145C4×D4—42no—
146C4×D4—42no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—48no—
159C2×C4⋊C4—48no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4C22⋊C412yes—
163C2×C4⋊C4C22×C414yes—
164C2×C4⋊C4C2×D412yes—
165C2×C22⋊C4—28no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—28no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—48no—
175C2×C22⋊C4—48no—
176C2×C22⋊C4—42no—
177C2×C22⋊C4—44no—
178C2×C22⋊C4C22⋊C412yes—
179C2×C22⋊C4C22×C412yes—
180C2×C22⋊C4C22×C411yescharacteristic
181C2×C22⋊C4C2×D411yescharacteristic
182C2×C22⋊C4C2×D411yescharacteristic
183C2×C22⋊C4C2411yescharacteristic
184C2×C42—22no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—24no—
188C2×C42—44no—
189C2×C42—44no—
190C2×C42—84no—
191C2×C42C22⋊C412yes—
192C2×C42C22×C412yes—
193#99—216no—
194C22⋊C8—24no—
195C22⋊C8—42no—
196C22⋊C8—44no—
197#95—22no—
198#95—44no—
199#95C22×C414yes—
200C24—21no—
201C24—21no—
202C24—22no—
203C24—21no—
204C24—22no—
205C24—21no—
206C24—22no—
207C24—42no—
208C24—42no—
209C24—42no—
210C24—42no—
211C24C2×C22⋊C411yescharacteristic
212C24C2×C22⋊C412yes—
213C24C2×C22⋊C411yescharacteristic
214C24#13714yes—
215C24C22×D411yescharacteristic
216C2×Q8—48no—
217C2×Q8—48no—
218C2×Q8—44no—
219C2×Q8—44no—
220C2×Q8—84no—
221C2×Q8—88no—
222C2×Q8—84no—
223C2×D4—24no—
224C2×D4—24no—
225C2×D4—24no—
226C2×D4—24no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—48no—
230C2×D4—48no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—28no—
236C22×C4—24no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—44no—
264C22×C4—48no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—84no—
270C22×C4—84no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4—84no—
274C22×C4—84no—
275C22×C4#9912yes—
276C22×C4#9911yescharacteristic
277C22×C4#9911yescharacteristic
278C22×C4#10414yes—
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C23×C411yescharacteristic
285C22×C4C22×D411yescharacteristic
286M16—88no—
287C2×C8—44no—
288C2×C8—84no—
289C4⋊C4—216no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—44no—
293C4⋊C4—48no—
294C4⋊C4—416no—
295C4⋊C4—48no—
296C4⋊C4—48no—
297C4⋊C4—44no—
298C22⋊C4—28no—
299C22⋊C4—42no—
300C22⋊C4—41no—
301C22⋊C4—42no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—48no—
306C22⋊C4—416no—
307C22⋊C4—48no—
308C22⋊C4—48no—
309C22⋊C4—44no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—22no—
314C42—48no—
315C42—88no—
316C23—22no—
317C23—22no—
318C23—21no—
319C23—21no—
320C23—24no—
321C23—21no—
322C23—21no—
323C23—24no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—41no—
328C23—42no—
329C23—41no—
330C23—41no—
331C23—42no—
332C23—48no—
333C23—41no—
334C23—48no—
335C23—42no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—81no—
342C23—81no—
343C23—84no—
344C23—81no—
345C23—81no—
346C23—84no—
347C23C2×#9911yescharacteristic
348C23C2×#9911yescharacteristic
349C23#41912yes—
350C23#41912yes—
351C23C2×#10412yes—
352C23C22×C22⋊C411yescharacteristic
353C23C2×#13712yes—
354Q8—88no—
355Q8—88no—
356D4—48no—
357D4—48no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—28no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—416no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—44no—
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—48no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—84no—
398C2×C4—84no—
399C2×C4—816no—
400C2×C4—84no—
401C2×C4—84no—
402C2×C4#35314yes—
403C2×C4C2×#9912yes—
404C2×C4C2×#9912yes—
405C2×C4C2×#10414yes—
406C8—84no—
407C22—24no—
408C22—24no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—22no—
414C22—22no—
415C22—42no—
416C22—42no—
417C22—44no—
418C22—48no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—42no—
424C22—42no—
425C22—42no—
426C22—81no—
427C22—81no—
428C22—84no—
429C22—82no—
430C22—84no—
431C22—84no—
432C22#150812yes—
433C22#150812yes—
434C22#212312yes—
435C22C2×#35312yes—
436C22C22×#9911yescharacteristic
437C22C2×#41911yescharacteristic
438C22C22×#10411yescharacteristic
439C4—22no—
440C4—24no—
441C4—48no—
442C4—41no—
443C4—41no—
444C4—41no—
445C4—41no—
446C4—42no—
447C4—88no—
448C4—84no—
449C4C2×#35314yes—
450C2—21no—
451C2—21no—
452C2—22no—
453C2—42no—
454C2—42no—
455C2—82no—
456C2#1178814yesdirect product factor
457C2C2×#150811yescharacteristic
458C2C2×#212311yescharacteristic
459C2C22×#35311yescharacteristic
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. #11788C2direct (N × H)
5. C2×#1352C2semidirect
6. C22×C4⋊Q8C4semidirect
14. #2542C4semidirect
16. C2×#399C4semidirect
20. C2×C4×C4⋊C4C4semidirect
29. #1352C22semidirect
39. C2×C4⋊Q8C2×C4semidirect
77. #399C2×C4semidirect
84. C4×C4⋊C4C2×C4semidirect
456. C2#11788direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25422
C2×C4×#992
C2×#14652
#117882
C2×#13522