Finite Group Databasebeta

← C2×#11570 #480539

Subgroups — C2×#11570 #480539

Order 512.

1334 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic
2C2×#1465C211yesmaximal, characteristic
3C2×#1459C211yesmaximal, characteristic
4#11570C218yesmaximal, direct product factor
5C2×#1325C214yesmaximal, complemented
6C22×C4⋊Q8C411yescharacteristic
7C22×C22⋊Q8—21no—
8C22×C22⋊Q8—21no—
9C22×C4×D4C2211yescharacteristic
10C2×#407—21no—
11C22×C22⋊C8—21no—
12#2542—22no—
13#2542—22no—
14#2542C414yes—
15#2542C2214yes—
16C2×#399—21no—
17C2×#399—21no—
18C2×#390C412yes—
19C2×#384C2212yes—
20C2×C4×C4⋊C4C411yescharacteristic
21#1465—22no—
22#1465C2214yes—
23#1459—22no—
24#1459C2214yes—
25C2×#324C2212yes—
26C2×#324C2212yes—
27C2×#323—21no—
28C2×#323—21no—
29#1325C22116yescomplemented
30C23×Q8—21no—
31C23×D4C2×C411yescharacteristic
32C24×C4D411yescharacteristic
33C24×C4D411yescharacteristic
34C22×M16—41no—
35C23×C8—41no—
36C2×C4⋊Q8—22no—
37C2×C4⋊Q8—42no—
38C2×C4⋊Q8—41no—
39C2×C4⋊Q8—41no—
40C2×C4⋊Q8C2×C414yes—
41C2×C22⋊Q8—24no—
42C2×C22⋊Q8—24no—
43C2×C22⋊Q8—24no—
44C2×C22⋊Q8—24no—
45C2×C22⋊Q8—48no—
46C2×C22⋊Q8—48no—
47C2×C4×D4—41no—
48C2×C4×D4—41no—
49C2×C4×D4—42no—
50C2×C4×D4C2×C412yes—
51C2×C4×D4D412yes—
52C2×C4×D4D412yes—
53C2×C4×D4C2312yes—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4C2×C412yes—
59C22×C4⋊C4C2311yescharacteristic
60C22×C22⋊C4—22no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4D411yescharacteristic
63C22×C22⋊C4D411yescharacteristic
64C22×C42C2×C412yes—
65C22×C42C2×C411yescharacteristic
66#407—24no—
67#407—42no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—24no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—42no—
72C2×C22⋊C8—41no—
73C2×C22⋊C8—41no—
74#399—24no—
75#399—24no—
76#399—24no—
77#399—24no—
78#390—24no—
79#390C2×C418yes—
80#384—22no—
81#384—22no—
82#384C2×C414yes—
83#384C2314yes—
84C4×C4⋊C4—22no—
85C4×C4⋊C4C2×C414yes—
86C2×#95—21no—
87#324—22no—
88#324—22no—
89#324C2314yes—
90#324C2314yes—
91#323—24no—
92#323—24no—
93C25C22⋊C411yescharacteristic
94C25C22⋊C411yescharacteristic
95C22×Q8—24no—
96C22×Q8—44no—
97C22×Q8—44no—
98C22×Q8—42no—
99C22×Q8—41no—
100C22×Q8—41no—
101C22×D4—24no—
102C22×D4—24no—
103C22×D4—41no—
104C22×D4—41no—
105C22×D4C22⋊C412yes—
106C22×D4C22×C412yes—
107C23×C4—22no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—44no—
116C23×C4—44no—
117C23×C4C22⋊C411yescharacteristic
118C23×C4C22⋊C412yes—
119C23×C4C22⋊C412yes—
120C23×C4C22⋊C411yescharacteristic
121C23×C4C22×C411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D412yes—
125C23×C4C2×D412yes—
126C2×M16—44no—
127C2×M16—84no—
128C22×C8—42no—
129C22×C8—44no—
130C22×C8—42no—
131C22×C8—84no—
132C4⋊Q8—44no—
133C4⋊Q8—44no—
134C4⋊Q8—44no—
135C4⋊Q8—44no—
136C22⋊Q8—416no—
137C22⋊Q8—416no—
138C22⋊Q8—416no—
139C22⋊Q8—416no—
140C4×D4—42no—
141C4×D4—42no—
142C4×D4—42no—
143C4×D4—42no—
144C4×D4—42no—
145C4×D4—42no—
146C4×D4—42no—
147C4×D4—42no—
148C2×C4⋊C4—28no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—48no—
161C2×C4⋊C4—48no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4C22×C414yes—
165C2×C4⋊C4C22×C411yescharacteristic
166C2×C4⋊C4C2×D411yescharacteristic
167C2×C4⋊C4C2×D411yescharacteristic
168C2×C4⋊C4C2411yescharacteristic
169C2×C22⋊C4—28no—
170C2×C22⋊C4—28no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—48no—
176C2×C22⋊C4—48no—
177C2×C22⋊C4C22⋊C412yes—
178C2×C22⋊C4C22⋊C412yes—
179C2×C22⋊C4C2×D412yes—
180C2×C22⋊C4C2×D412yes—
181C2×C42—22no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—28no—
185C2×C42—22no—
186C2×C42C22⋊C412yes—
187C2×C42C22×C414yes—
188C2×C42C22×C412yes—
189C22⋊C8—24no—
190C22⋊C8—24no—
191C22⋊C8—42no—
192C22⋊C8—44no—
193C22⋊C8—42no—
194C22⋊C8—44no—
195#95—24no—
196C24—21no—
197C24—22no—
198C24—21no—
199C24—21no—
200C24—22no—
201C24—22no—
202C24—21no—
203C24—42no—
204C24—42no—
205C24—42no—
206C24—42no—
207C24#10114yes—
208C24C2×C22⋊C411yescharacteristic
209C24C2×C22⋊C412yes—
210C24C2×C22⋊C412yes—
211C2×Q8—48no—
212C2×Q8—48no—
213C2×Q8—44no—
214C2×Q8—44no—
215C2×Q8—84no—
216C2×Q8—88no—
217C2×Q8—84no—
218C2×D4—24no—
219C2×D4—24no—
220C2×D4—24no—
221C2×D4—24no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4—48no—
225C2×D4—48no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—28no—
231C22×C4—24no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—84no—
263C22×C4—84no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4#9912yes—
267C22×C4#9911yescharacteristic
268C22×C4#9911yescharacteristic
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4#13114yes—
275C22×C4C23×C411yescharacteristic
276C22×C4C22×D411yescharacteristic
277C22×C4C22×D411yescharacteristic
278M16—88no—
279C2×C8—44no—
280C2×C8—44no—
281C2×C8—84no—
282C2×C8—88no—
283C2×C8—84no—
284C4⋊C4—216no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—44no—
290C4⋊C4—48no—
291C4⋊C4—48no—
292C4⋊C4—48no—
293C4⋊C4—48no—
294C4⋊C4—44no—
295C4⋊C4—48no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C22⋊C4—416no—
301C22⋊C4—416no—
302C42—22no—
303C42—22no—
304C42—22no—
305C42—22no—
306C42—28no—
307C42—44no—
308C23—22no—
309C23—22no—
310C23—22no—
311C23—24no—
312C23—22no—
313C23—24no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—42no—
318C23—42no—
319C23—48no—
320C23—42no—
321C23—42no—
322C23—48no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23—42no—
329C23—81no—
330C23—81no—
331C23—84no—
332C23—81no—
333C23—81no—
334C23—84no—
335C23#40812yes—
336C23C2×#9911yescharacteristic
337C23C2×#9911yescharacteristic
338C23C2×#10112yes—
339C23C2×#10114yes—
340C23C22×C22⋊C411yescharacteristic
341Q8—88no—
342Q8—88no—
343D4—48no—
344D4—48no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—28no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—416no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—84no—
386C2×C4—84no—
387C2×C4#35314yes—
388C2×C4#40812yes—
389C2×C4C2×#9912yes—
390C2×C4C2×#9912yes—
391C2×C4C2×#13112yes—
392C8—84no—
393C8—84no—
394C22—24no—
395C22—24no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—24no—
401C22—44no—
402C22—44no—
403C22—48no—
404C22—42no—
405C22—42no—
406C22—42no—
407C22—42no—
408C22—42no—
409C22—82no—
410C22—84no—
411C22—82no—
412C22—84no—
413C22—84no—
414C22#147212yes—
415C22#147212yes—
416C22C2×#35312yes—
417C22#213512yes—
418C22C2×#40811yescharacteristic
419C22C22×#9911yescharacteristic
420C22C22×#10111yescharacteristic
421C4—22no—
422C4—22no—
423C4—22no—
424C4—48no—
425C4—41no—
426C4—41no—
427C4—41no—
428C4—41no—
429C4—42no—
430C4—84no—
431C4C2×#35314yes—
432C2—21no—
433C2—21no—
434C2—22no—
435C2—44no—
436C2—82no—
437C2#1157014yesdirect product factor
438C2C2×#147211yescharacteristic
439C2C22×#35311yescharacteristic
440C2C2×#213511yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11570C2direct (N × H)
5. C2×#1325C2semidirect
29. #1325C22semidirect
437. C2#11570direct (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×#14652
C2×#14592
#115702
C2×#13252