Finite Group Databasebeta

← C2×#11846 #482155

Subgroups — C2×#11846 #482155

Order 512.

994 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2611C211yesmaximal, characteristic
2C2×#1820C211yesmaximal, characteristic
3C2×#1458C211yesmaximal, characteristic
4#11846C218yesmaximal, direct product factor
5C2×#1352C212yesmaximal
6C2×#1344C212yesmaximal
7C22×#125C411yescharacteristic, complemented
8C22×#117C2211yescharacteristic
9C2×#431—21no—
10#2611—22no—
11#2611—22no—
12#2611C414yescomplemented
13#2611C2214yes—
14C2×#400C411yescharacteristic, complemented
15C2×#399C2211yescharacteristic
16C2×#398C411yescharacteristic, complemented
17C2×#397C2211yescharacteristic
18C2×#389—21no—
19C2×#388—21no—
20C2×#387—21no—
21C2×#385—21no—
22C2×#381—21no—
23C2×#376C411yescharacteristic, complemented
24#1820—22no—
25#1820C2214yes—
26C2×#342—21no—
27C2×#342C2212yes—
28#1458—22no—
29#1458C2214yes—
30C2×#324—21no—
31C2×#323C2212yes—
32#1352C2218yes—
33#1344C2218yes—
34C24×C4D411yescharacteristic
35C2×#125—22no—
36C2×#125—42no—
37C2×#125—41no—
38C2×#125—41no—
39C2×#125C2×C414yescomplemented
40C2×#117—41no—
41C2×#117—41no—
42C2×#117—42no—
43C2×#117C2×C412yes—
44C2×#117D412yes—
45C2×#117D412yes—
46C2×#117C2312yes—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4D411yescharacteristic
55C22×C22⋊C4—21no—
56C22×C22⋊C4—21no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4D411yescharacteristic
59C22×C22⋊C4C2311yescharacteristic
60C22×C42—21no—
61C22×C42C2×C411yescharacteristic
62C22×C42D411yescharacteristic
63#431—24no—
64#431—42no—
65C2×C4⋊C8—41no—
66C2×C22⋊C8—22no—
67C2×C4×C8—41no—
68#400—22no—
69#400C2×C414yescomplemented
70#399—21no—
71#399—21no—
72#399C2×C412yes—
73#399C2312yes—
74#398—22no—
75#398C2×C414yescomplemented
76#397—21no—
77#397—21no—
78#397C2×C412yes—
79#397C2312yes—
80#389—24no—
81#389—24no—
82#388—24no—
83#388—24no—
84#387—24no—
85#387—24no—
86#385—24no—
87#385—24no—
88#381—24no—
89#381—42no—
90#376—22no—
91#376C2×C414yescomplemented
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—41no—
98C2×#95—41no—
99C2×#95C2×C411yescharacteristic
100C2×#95C2×C411yescharacteristic
101#342—22no—
102#342—24no—
103#342C2314yes—
104#324—24no—
105#323—22no—
106#323C2314yes—
107C25C2×D411yescharacteristic
108C23×C4—21no—
109C23×C4—24no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—22no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—42no—
117C23×C4—42no—
118C23×C4C22⋊C411yescharacteristic
119C23×C4C22⋊C412yes—
120C23×C4C22⋊C411yescharacteristic
121C23×C4C22⋊C411yescharacteristic
122C23×C4C22⋊C411yescharacteristic
123C23×C4C22×C411yescharacteristic
124C23×C4C2×D412yes—
125C23×C4C2×D411yescharacteristic
126C22×C8—42no—
127#125—44no—
128#125—44no—
129#125—44no—
130#125—44no—
131#117—42no—
132#117—42no—
133#117—42no—
134#117—42no—
135#117—42no—
136#117—42no—
137#117—42no—
138#117—42no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4C22⋊C412yes—
162C2×C4⋊C4C22⋊C412yes—
163C2×C4⋊C4C22×C412yes—
164C2×C4⋊C4C22×C412yes—
165C2×C4⋊C4C22×C412yes—
166C2×C4⋊C4C2×D412yes—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—44no—
178C2×C22⋊C4—44no—
179C2×C22⋊C4—44no—
180C2×C22⋊C4—42no—
181C2×C22⋊C4C22⋊C412yes—
182C2×C22⋊C4C22×C411yescharacteristic
183C2×C22⋊C4C2×D412yes—
184C2×C22⋊C4C2×D411yescharacteristic
185C2×C22⋊C4C2×D411yescharacteristic
186C2×C22⋊C4C2411yescharacteristic
187C2×C42—22no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—22no—
193C2×C42—24no—
194C2×C42—44no—
195C2×C42C22⋊C412yes—
196C2×C42C22⋊C412yes—
197C2×C42C22×C412yes—
198C2×C42C2×D412yes—
199C4⋊C8—44no—
200C22⋊C8—24no—
201C22⋊C8—42no—
202C4×C8—44no—
203#95—24no—
204#95—24no—
205#95—24no—
206#95—24no—
207#95—22no—
208#95—21no—
209#95—21no—
210#95—22no—
211#95—44no—
212#95—44no—
213#95C22×C412yes—
214#95C22×C412yes—
215C24—21no—
216C24—21no—
217C24—22no—
218C24—42no—
219C24—42no—
220C24C2×C22⋊C411yescharacteristic
221C24C2×C22⋊C411yescharacteristic
222C24#13514yes—
223C24C22×D411yescharacteristic
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—24no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—22no—
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—42no—
250C22×C4—42no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4—48no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—44no—
272C22×C4—84no—
273C22×C4—84no—
274C22×C4—84no—
275C22×C4#9912yes—
276C22×C4#9911yescharacteristic
277C22×C4#9911yescharacteristic
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C412yes—
284C22×C4C23×C411yescharacteristic
285C22×C4C22×D411yescharacteristic
286C2×C8—44no—
287C2×C8—84no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—44no—
295C4⋊C4—44no—
296C4⋊C4—48no—
297C4⋊C4—48no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C22⋊C4—42no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—42no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—44no—
309C22⋊C4—44no—
310C22⋊C4—44no—
311C22⋊C4—44no—
312C22⋊C4—44no—
313C22⋊C4—44no—
314C22⋊C4—84no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—48no—
324C23—22no—
325C23—22no—
326C23—21no—
327C23—21no—
328C23—24no—
329C23—21no—
330C23—21no—
331C23—24no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—48no—
336C23—41no—
337C23—42no—
338C23—42no—
339C23—81no—
340C23—84no—
341C23—81no—
342C23C2×#9911yescharacteristic
343C23C2×#9911yescharacteristic
344C23#41612yes—
345C23#41612yes—
346C23#42112yes—
347C23C22×C22⋊C411yescharacteristic
348C23C2×#13512yes—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—24no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—24no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—48no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—42no—
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—81no—
395C2×C4—81no—
396C2×C4—84no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—84no—
402C2×C4—84no—
403C2×C4—84no—
404C2×C4—168no—
405C2×C4C2×#9912yes—
406C2×C4C2×#9912yes—
407C8—84no—
408C22—21no—
409C22—21no—
410C22—24no—
411C22—24no—
412C22—21no—
413C22—21no—
414C22—22no—
415C22—22no—
416C22—42no—
417C22—42no—
418C22—41no—
419C22—41no—
420C22—48no—
421C22—41no—
422C22—41no—
423C22—81no—
424C22—81no—
425C22—84no—
426C22#151612yes—
427C22#151612yes—
428C22#212312yes—
429C22#212612yes—
430C22C22×#9911yescharacteristic
431C22C2×#41611yescharacteristic
432C22C2×#42111yescharacteristic
433C4—22no—
434C4—22no—
435C4—44no—
436C4—41no—
437C4—41no—
438C4—42no—
439C4—41no—
440C4—42no—
441C4—41no—
442C4—42no—
443C4—82no—
444C4—82no—
445C4—82no—
446C4—164no—
447C2—21no—
448C2—21no—
449C2—22no—
450C2—42no—
451C2—42no—
452C2#1184614yesdirect product factor
453C2C2×#151611yescharacteristic
454C2C2×#212311yescharacteristic
455C2C2×#212611yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11846C2direct (N × H)
7. C22×#125C4semidirect
12. #2611C4semidirect
14. C2×#400C4semidirect
16. C2×#398C4semidirect
23. C2×#376C4semidirect
39. C2×#125C2×C4semidirect
69. #400C2×C4semidirect
75. #398C2×C4semidirect
91. #376C2×C4semidirect
452. C2#11846direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#26112
C2×#18202
C2×#14582
#118462
C2×#13522
C2×#13442