Finite Group Databasebeta

← C2×#11786 #481636

Subgroups — C2×#11786 #481636

Order 512.

1250 conjugacy classes of subgroups.

The subgroup lattice has 445 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2487C211yesmaximal, characteristic
2C2×C4×#99C211yesmaximal, characteristic
3C2×#1466C211yesmaximal, characteristic
4#11786C218yesmaximal, direct product factor
5C2×#1352C214yesmaximal, complemented
6C22×C4×D4C2211yescharacteristic
7C23×C4⋊C4C411yescharacteristic, complemented
8C2×#432—21no—
9C22×#99—21no—
10#2487—22no—
11#2487—22no—
12#2487C414yescomplemented
13#2487C2214yes—
14C2×#400C412yescomplemented
15C2×#399C2212yes—
16C2×#389—21no—
17C2×#386—21no—
18C2×#385—21no—
19C2×#380—21no—
20C2×#380C411yescharacteristic, complemented
21C2×C4×C22⋊C4—21no—
22C4×#99—22no—
23C4×#99C2214yes—
24C2×#342C2212yes—
25#1466—22no—
26#1466C2214yes—
27C2×#323—21no—
28C2×#323C2212yes—
29#1352C22116yescomplemented
30C23×D4D411yescharacteristic
31C24×C4—21no—
32C24×C4C2×C411yescharacteristic
33C24×C4D411yescharacteristic
34C2×C4×D4—41no—
35C2×C4×D4—41no—
36C2×C4×D4—42no—
37C2×C4×D4C2×C412yes—
38C2×C4×D4D412yes—
39C2×C4×D4D412yes—
40C2×C4×D4C2312yes—
41C22×C4⋊C4—22no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—42no—
46C22×C4⋊C4—41no—
47C22×C4⋊C4—41no—
48C22×C4⋊C4C2×C412yes—
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C414yescomplemented
51C22×C22⋊C4—22no—
52C22×C22⋊C4—21no—
53C22×C22⋊C4—22no—
54C22×C22⋊C4D411yescharacteristic
55C22×C22⋊C4C2311yescharacteristic
56C22×C42—41no—
57C22×C42D411yescharacteristic
58#432—24no—
59#432—42no—
60C2×C4⋊C8—41no—
61C2×#99—28no—
62C2×#99—24no—
63C2×C22⋊C8—22no—
64C2×#97—41no—
65#400—24no—
66#400C2×C418yescomplemented
67#399—22no—
68#399—22no—
69#399C2×C414yes—
70#399C2314yes—
71#389—24no—
72#389—24no—
73#386—24no—
74#386—24no—
75#385—24no—
76#385—24no—
77#380—22no—
78#380—24no—
79#380—24no—
80#380C2×C414yescomplemented
81C4×C22⋊C4—24no—
82C4×C22⋊C4—42no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—41no—
86C2×#95C2×C412yes—
87#342—22no—
88#342C2314yes—
89#323—24no—
90#323—22no—
91#323C2314yes—
92C25C22⋊C411yescharacteristic
93C25C2×D411yescharacteristic
94C22×D4—24no—
95C22×D4—24no—
96C22×D4—41no—
97C22×D4—41no—
98C22×D4C22⋊C412yes—
99C22×D4C2×D412yes—
100C23×C4—22no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—24no—
109C23×C4—44no—
110C23×C4—44no—
111C23×C4—42no—
112C23×C4—42no—
113C23×C4—42no—
114C23×C4—84no—
115C23×C4C22⋊C411yescharacteristic
116C23×C4C22⋊C411yescharacteristic
117C23×C4C22⋊C412yes—
118C23×C4C22⋊C412yes—
119C23×C4C22⋊C411yescharacteristic
120C23×C4C22×C411yescharacteristic
121C23×C4C22×C412yes—
122C23×C4C2×D412yes—
123C23×C4C2×D411yescharacteristic
124C22×C8—42no—
125C4×D4—42no—
126C4×D4—42no—
127C4×D4—42no—
128C4×D4—42no—
129C4×D4—42no—
130C4×D4—42no—
131C4×D4—42no—
132C4×D4—42no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4—44no—
143C2×C4⋊C4—44no—
144C2×C4⋊C4—44no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—44no—
149C2×C4⋊C4—44no—
150C2×C4⋊C4—84no—
151C2×C4⋊C4C22⋊C412yes—
152C2×C4⋊C4C22×C414yes—
153C2×C4⋊C4C22×C412yes—
154C2×C22⋊C4—28no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C22⋊C4—41no—
162C2×C22⋊C4—41no—
163C2×C22⋊C4—42no—
164C2×C22⋊C4—44no—
165C2×C22⋊C4—44no—
166C2×C22⋊C4—84no—
167C2×C22⋊C4C22⋊C412yes—
168C2×C22⋊C4C22×C411yescharacteristic
169C2×C22⋊C4C2×D411yescharacteristic
170C2×C22⋊C4C2×D412yes—
171C2×C22⋊C4C2×D411yescharacteristic
172C2×C22⋊C4C2411yescharacteristic
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—22no—
176C2×C42—44no—
177C2×C42—84no—
178C2×C42C22⋊C412yes—
179C2×C42C2×D412yes—
180C4⋊C8—44no—
181#99—216no—
182C22⋊C8—24no—
183C22⋊C8—42no—
184#97—44no—
185#95—28no—
186#95—28no—
187#95—22no—
188#95—44no—
189#95C22×C414yes—
190C24—21no—
191C24—21no—
192C24—21no—
193C24—22no—
194C24—21no—
195C24—22no—
196C24—42no—
197C24—42no—
198C24—42no—
199C24—42no—
200C24#10414yes—
201C24C2×C22⋊C411yescharacteristic
202C24C2×C22⋊C411yescharacteristic
203C24C2×C22⋊C412yes—
204C24#13714yes—
205C24C22×D411yescharacteristic
206C2×D4—24no—
207C2×D4—24no—
208C2×D4—24no—
209C2×D4—24no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4—48no—
213C2×D4—48no—
214C22×C4—24no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—44no—
236C22×C4—48no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—44no—
247C22×C4—48no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—48no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—44no—
256C22×C4—84no—
257C22×C4—82no—
258C22×C4—88no—
259C22×C4—84no—
260C22×C4—84no—
261C22×C4—88no—
262C22×C4—88no—
263C22×C4#9912yes—
264C22×C4#9911yescharacteristic
265C22×C4#9911yescharacteristic
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C23×C411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C2×C8—44no—
274C2×C8—84no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—44no—
278C4⋊C4—48no—
279C4⋊C4—44no—
280C4⋊C4—44no—
281C4⋊C4—44no—
282C4⋊C4—84no—
283C4⋊C4—84no—
284C22⋊C4—28no—
285C22⋊C4—42no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—42no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—48no—
292C22⋊C4—44no—
293C22⋊C4—48no—
294C22⋊C4—84no—
295C22⋊C4—84no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42—22no—
300C42—88no—
301C23—22no—
302C23—22no—
303C23—22no—
304C23—21no—
305C23—21no—
306C23—22no—
307C23—24no—
308C23—24no—
309C23—24no—
310C23—21no—
311C23—21no—
312C23—24no—
313C23—42no—
314C23—42no—
315C23—48no—
316C23—42no—
317C23—42no—
318C23—48no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—48no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23—81no—
329C23—84no—
330C23—81no—
331C23C2×#9911yescharacteristic
332C23C2×#9911yescharacteristic
333C23#41912yes—
334C23#41912yes—
335C23C2×#10412yes—
336C23C2×#10414yes—
337C23C22×C22⋊C411yescharacteristic
338C23C2×#13712yes—
339D4—48no—
340D4—48no—
341C2×C4—22no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—48no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—88no—
381C2×C4—84no—
382C2×C4—84no—
383C2×C4—816no—
384C2×C4—84no—
385C2×C4—816no—
386C2×C4—84no—
387C2×C4#35214yes—
388C2×C4C2×#9912yes—
389C2×C4C2×#9912yes—
390C8—84no—
391C22—24no—
392C22—24no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—22no—
398C22—22no—
399C22—24no—
400C22—42no—
401C22—42no—
402C22—44no—
403C22—44no—
404C22—48no—
405C22—48no—
406C22—48no—
407C22—42no—
408C22—41no—
409C22—41no—
410C22—42no—
411C22—41no—
412C22—41no—
413C22—81no—
414C22—81no—
415C22—84no—
416C22#150812yes—
417C22#150812yes—
418C22C2×#35212yes—
419C22#212812yes—
420C22C22×#9911yescharacteristic
421C22C2×#41911yescharacteristic
422C22C22×#10411yescharacteristic
423C4—22no—
424C4—41no—
425C4—41no—
426C4—42no—
427C4—41no—
428C4—41no—
429C4—42no—
430C4—84no—
431C4—88no—
432C4—84no—
433C4C2×#35214yes—
434C2—21no—
435C2—21no—
436C2—22no—
437C2—42no—
438C2—44no—
439C2—42no—
440C2#1178614yesdirect product factor
441C2C2×#150811yescharacteristic
442C2C22×#35211yescharacteristic
443C2C2×#212811yescharacteristic
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. #11786C2direct (N × H)
5. C2×#1352C2semidirect
7. C23×C4⋊C4C4semidirect
12. #2487C4semidirect
14. C2×#400C4semidirect
20. C2×#380C4semidirect
29. #1352C22semidirect
50. C22×C4⋊C4C2×C4semidirect
66. #400C2×C4semidirect
80. #380C2×C4semidirect
440. C2#11786direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24872
C2×C4×#992
C2×#14662
#117862
C2×#13522