Finite Group Databasebeta

← C2×#11849 #482172

Subgroups — C2×#11849 #482172

Order 512.

992 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2599C211yesmaximal, characteristic
2C2×#1822C211yesmaximal, characteristic
3C2×#1458C211yesmaximal, characteristic
4#11849C218yesmaximal, direct product factor
5C2×#1354C212yesmaximal, complemented
6C2×#1346C212yesmaximal, complemented
7C22×C4×Q8C411yescharacteristic
8C22×#117C2211yescharacteristic
9C2×#431—21no—
10C2×#407—21no—
11#2599—22no—
12#2599—22no—
13#2599C414yes—
14#2599C2214yes—
15C2×#398C2211yescharacteristic
16C2×#397C411yescharacteristic
17C2×#395C2211yescharacteristic
18C2×#393C411yescharacteristic
19C2×#388—21no—
20C2×#387—21no—
21C2×#387—21no—
22C2×#385—21no—
23C2×#383C411yescharacteristic
24#1822—22no—
25#1822C2214yes—
26C2×#343—21no—
27C2×#343C2212yes—
28#1458—22no—
29#1458C2214yes—
30C2×#324C2212yes—
31C2×#323—21no—
32#1354C2218yescomplemented
33#1346C2218yescomplemented
34C23×Q8C2×C411yescharacteristic
35C24×C4D411yescharacteristic
36C22×M16—22no—
37C2×C4×Q8—22no—
38C2×C4×Q8—42no—
39C2×C4×Q8—41no—
40C2×C4×Q8—41no—
41C2×C4×Q8C2×C414yes—
42C2×#117—41no—
43C2×#117—41no—
44C2×#117—42no—
45C2×#117C2×C412yes—
46C2×#117D412yes—
47C2×#117D412yes—
48C2×#117C2312yes—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4D411yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C22⋊C4—21no—
56C22×C22⋊C4—21no—
57C22×C22⋊C4C2×C411yescharacteristic
58C22×C22⋊C4D411yescharacteristic
59C22×C42—21no—
60C22×C42C2×C411yescharacteristic
61C22×C42D411yescharacteristic
62#431—24no—
63#431—42no—
64C2×C4⋊C8—41no—
65#407—24no—
66#407—42no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8—41no—
69C2×C22⋊C8—41no—
70C2×C4×C8—41no—
71#398—21no—
72#398—21no—
73#398C2×C412yes—
74#398C2312yes—
75#397—22no—
76#397C2×C414yes—
77#395—21no—
78#395—21no—
79#395C2×C412yes—
80#395C2312yes—
81#393—22no—
82#393C2×C414yes—
83#388—24no—
84#388—24no—
85#387—24no—
86#387—24no—
87#387—24no—
88#387—24no—
89#385—24no—
90#385—24no—
91#383—22no—
92#383C2×C414yes—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95C2×C411yescharacteristic
100C2×#95C2×C411yescharacteristic
101#343—22no—
102#343—24no—
103#343C2314yes—
104#324—22no—
105#324C2314yes—
106#323—24no—
107C25C22⋊C411yescharacteristic
108C22×Q8—21no—
109C22×Q8—44no—
110C22×Q8—41no—
111C22×Q8—41no—
112C22×Q8C22×C412yes—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—22no—
117C23×C4—24no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—42no—
122C23×C4C22⋊C412yes—
123C23×C4C22⋊C411yescharacteristic
124C23×C4C22⋊C411yescharacteristic
125C23×C4C22⋊C411yescharacteristic
126C23×C4C22×C411yescharacteristic
127C23×C4C2×D412yes—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C2×M16—24no—
131C2×M16—42no—
132C2×M16—44no—
133C2×M16—44no—
134C22×C8—42no—
135C22×C8—42no—
136C4×Q8—44no—
137C4×Q8—44no—
138C4×Q8—44no—
139C4×Q8—44no—
140#117—42no—
141#117—42no—
142#117—42no—
143#117—42no—
144#117—42no—
145#117—42no—
146#117—42no—
147#117—42no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4C22⋊C412yes—
166C2×C4⋊C4C22×C412yes—
167C2×C4⋊C4C22×C411yescharacteristic
168C2×C4⋊C4C2×D412yes—
169C2×C4⋊C4C2×D411yescharacteristic
170C2×C4⋊C4C2×D411yescharacteristic
171C2×C4⋊C4C2411yescharacteristic
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—44no—
179C2×C22⋊C4—44no—
180C2×C22⋊C4C22⋊C412yes—
181C2×C22⋊C4C22⋊C412yes—
182C2×C22⋊C4C22×C412yes—
183C2×C22⋊C4C2×D412yes—
184C2×C42—22no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—22no—
190C2×C42—24no—
191C2×C42—42no—
192C2×C42—42no—
193C2×C42C22⋊C412yes—
194C2×C42C22⋊C412yes—
195C2×C42C22×C412yes—
196C2×C42C2×D412yes—
197C4⋊C8—44no—
198C22⋊C8—24no—
199C22⋊C8—42no—
200C22⋊C8—44no—
201C22⋊C8—44no—
202C4×C8—44no—
203#95—24no—
204#95—24no—
205#95—22no—
206#95—24no—
207#95—24no—
208#95—22no—
209#95—21no—
210#95—21no—
211#95—22no—
212#95—22no—
213#95C22×C412yes—
214#95C22×C412yes—
215C24—21no—
216C24—22no—
217C24—21no—
218C24—22no—
219C24—42no—
220C24—42no—
221C24C2×C22⋊C411yescharacteristic
222C24C2×C22⋊C412yes—
223C2×Q8—44no—
224C2×Q8—42no—
225C2×Q8—44no—
226C2×Q8—42no—
227C2×Q8—48no—
228C2×Q8—44no—
229C2×Q8—44no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—48no—
259C22×C4—42no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4#9912yes—
277C22×C4#9911yescharacteristic
278C22×C4#9911yescharacteristic
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4#13514yes—
286C22×C4C23×C411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C22×C4C22×D411yescharacteristic
289M16—44no—
290M16—44no—
291M16—84no—
292C2×C8—44no—
293C2×C8—44no—
294C2×C8—84no—
295C2×C8—84no—
296C4⋊C4—42no—
297C4⋊C4—41no—
298C4⋊C4—42no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—42no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4—42no—
306C4⋊C4—44no—
307C4⋊C4—44no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—48no—
315C22⋊C4—48no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—48no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—22no—
330C23—42no—
331C23—42no—
332C23—48no—
333C23—42no—
334C23—42no—
335C23—42no—
336C23—42no—
337C23—81no—
338C23—81no—
339C23—84no—
340C23C2×#9911yescharacteristic
341C23C2×#9911yescharacteristic
342C23#41612yes—
343C23#42112yes—
344C23C22×C22⋊C411yescharacteristic
345Q8—44no—
346Q8—44no—
347Q8—48no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—24no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—84no—
398C2×C4—84no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4C2×#9912yes—
402C2×C4C2×#9912yes—
403C2×C4#41612yes—
404C2×C4C2×#13512yes—
405C8—84no—
406C8—84no—
407C22—21no—
408C22—21no—
409C22—24no—
410C22—24no—
411C22—21no—
412C22—21no—
413C22—44no—
414C22—42no—
415C22—42no—
416C22—42no—
417C22—82no—
418C22—84no—
419C22—84no—
420C22#151612yes—
421C22#151612yes—
422C22#212012yes—
423C22#213512yes—
424C22C22×#9911yescharacteristic
425C22C2×#41611yescharacteristic
426C22C2×#42111yescharacteristic
427C4—22no—
428C4—22no—
429C4—22no—
430C4—22no—
431C4—41no—
432C4—44no—
433C4—41no—
434C4—41no—
435C4—41no—
436C4—42no—
437C4—42no—
438C4—82no—
439C4—82no—
440C4—82no—
441C2—21no—
442C2—21no—
443C2—22no—
444C2—82no—
445C2#1184914yesdirect product factor
446C2C2×#151611yescharacteristic
447C2C2×#212011yescharacteristic
448C2C2×#213511yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11849C2direct (N × H)
5. C2×#1354C2semidirect
6. C2×#1346C2semidirect
32. #1354C22semidirect
33. #1346C22semidirect
445. C2#11849direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25992
C2×#18222
C2×#14582
#118492
C2×#13542
C2×#13462