Finite Group Databasebeta

← C2×#11556 #480474

Subgroups — C2×#11556 #480474

Order 512.

958 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2499C211yesmaximal, characteristic
2C2×#1464C211yesmaximal, characteristic
3C2×#1458C211yesmaximal, characteristic
4#11556C218yesmaximal, direct product factor
5C2×#1324C212yesmaximal, complemented
6C2×#1319C212yesmaximal, complemented
7C22×C22⋊Q8C411yescharacteristic
8C22×#117C2211yescharacteristic
9C2×#432—21no—
10C2×#431—21no—
11#2499—22no—
12#2499—22no—
13#2499C414yes—
14#2499C2214yes—
15C2×#397—21no—
16C2×#397—21no—
17C2×#391C411yescharacteristic
18C2×#391C2211yescharacteristic
19C2×#387—21no—
20C2×#387—21no—
21C2×#384C411yescharacteristic
22C2×#382C2211yescharacteristic
23C2×C4×C22⋊C4C411yescharacteristic
24#1464—22no—
25#1464C2214yes—
26#1458—22no—
27#1458C2214yes—
28C2×#324C2212yes—
29C2×#324C2212yes—
30C2×#323—21no—
31C2×#323—21no—
32#1324C2218yescomplemented
33#1319C2218yescomplemented
34C23×Q8C2×C411yescharacteristic
35C24×C4C2×C411yescharacteristic
36C2×C22⋊Q8—22no—
37C2×C22⋊Q8—42no—
38C2×C22⋊Q8—41no—
39C2×C22⋊Q8—41no—
40C2×C22⋊Q8C2×C414yes—
41C2×#117—41no—
42C2×#117—41no—
43C2×#117—42no—
44C2×#117C2×C412yes—
45C2×#117D412yes—
46C2×#117D412yes—
47C2×#117C2312yes—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C22⋊C4—21no—
55C22×C22⋊C4—21no—
56C22×C22⋊C4D411yescharacteristic
57C22×C22⋊C4D411yescharacteristic
58C22×C42C2×C411yescharacteristic
59C22×C42C2×C411yescharacteristic
60C22×C42D411yescharacteristic
61C22×C42D411yescharacteristic
62#432—24no—
63#432—42no—
64#431—24no—
65#431—42no—
66C2×C4⋊C8—41no—
67C2×C4⋊C8—41no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×#97—41no—
71C2×C4×C8—41no—
72#397—24no—
73#397—24no—
74#397—24no—
75#397—24no—
76#391—22no—
77#391—21no—
78#391—21no—
79#391C2×C412yes—
80#391C2×C414yes—
81#391C2312yes—
82#387—24no—
83#387—24no—
84#387—24no—
85#387—24no—
86#384—22no—
87#384C2×C414yes—
88#382—21no—
89#382—21no—
90#382C2×C412yes—
91#382C2312yes—
92C4×C22⋊C4—22no—
93C4×C22⋊C4C2×C414yes—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101#324—22no—
102#324—22no—
103#324C2314yes—
104#324C2314yes—
105#323—24no—
106#323—24no—
107C25C22⋊C411yescharacteristic
108C22×Q8—21no—
109C22×Q8—24no—
110C22×Q8—41no—
111C22×Q8—41no—
112C22×Q8—42no—
113C22×Q8C22×C412yes—
114C23×C4—21no—
115C23×C4—22no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—24no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—42no—
123C23×C4C22⋊C412yes—
124C23×C4C22⋊C411yescharacteristic
125C23×C4C22⋊C411yescharacteristic
126C23×C4C22⋊C411yescharacteristic
127C23×C4C22×C412yes—
128C23×C4C22×C411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C22×C8—42no—
132C22×C8—42no—
133C22⋊Q8—44no—
134C22⋊Q8—44no—
135C22⋊Q8—44no—
136C22⋊Q8—44no—
137#117—42no—
138#117—42no—
139#117—42no—
140#117—42no—
141#117—42no—
142#117—42no—
143#117—42no—
144#117—42no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4C22⋊C412yes—
161C2×C4⋊C4C22×C412yes—
162C2×C4⋊C4C22×C412yes—
163C2×C4⋊C4C22×C411yescharacteristic
164C2×C4⋊C4C2×D411yescharacteristic
165C2×C4⋊C4C2×D411yescharacteristic
166C2×C4⋊C4C2411yescharacteristic
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—41no—
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—44no—
174C2×C22⋊C4—42no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4C22⋊C412yes—
177C2×C22⋊C4C22⋊C412yes—
178C2×C22⋊C4C2×D412yes—
179C2×C22⋊C4C2×D412yes—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—24no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—42no—
190C2×C42C22⋊C412yes—
191C2×C42C22⋊C412yes—
192C2×C42C22×C412yes—
193C2×C42C22×C412yes—
194C2×C42C2×D412yes—
195C2×C42C2×D412yes—
196C4⋊C8—44no—
197C4⋊C8—44no—
198C22⋊C8—24no—
199C22⋊C8—24no—
200C22⋊C8—42no—
201C22⋊C8—42no—
202#97—44no—
203C4×C8—44no—
204#95—24no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—24no—
210#95—24no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215C24—21no—
216C24—21no—
217C24—22no—
218C24—22no—
219C24—42no—
220C24—42no—
221C24C2×C22⋊C411yescharacteristic
222C24C2×C22⋊C412yes—
223C2×Q8—24no—
224C2×Q8—42no—
225C2×Q8—42no—
226C2×Q8—42no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8—84no—
230C2×Q8—84no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—48no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—42no—
261C22×C4—42no—
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—42no—
271C22×C4—42no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4#9912yes—
278C22×C4#9911yescharacteristic
279C22×C4#9911yescharacteristic
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C412yes—
286C22×C4#13114yes—
287C22×C4C23×C411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C2×C8—44no—
291C2×C8—44no—
292C2×C8—84no—
293C2×C8—84no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—42no—
301C4⋊C4—48no—
302C4⋊C4—48no—
303C4⋊C4—42no—
304C4⋊C4—44no—
305C4⋊C4—44no—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C22⋊C4—48no—
311C22⋊C4—48no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—24no—
321C42—42no—
322C42—42no—
323C42—42no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—42no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—42no—
336C23—42no—
337C23—84no—
338C23—84no—
339C23#40812yes—
340C23C2×#9911yescharacteristic
341C23C2×#9911yescharacteristic
342C23#41312yes—
343C23C22×C22⋊C411yescharacteristic
344Q8—84no—
345Q8—84no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—24no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—48no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—84no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4#40812yes—
400C2×C4C2×#9912yes—
401C2×C4C2×#9912yes—
402C2×C4C2×#13112yes—
403C8—84no—
404C8—84no—
405C22—24no—
406C22—24no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—42no—
412C22—42no—
413C22—42no—
414C22—82no—
415C22—82no—
416C22—84no—
417C22—84no—
418C22#146812yes—
419C22#146812yes—
420C22#212912yes—
421C22#213212yes—
422C22C2×#40811yescharacteristic
423C22C22×#9911yescharacteristic
424C22C2×#41311yescharacteristic
425C4—22no—
426C4—22no—
427C4—22no—
428C4—22no—
429C4—44no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—41no—
434C4—42no—
435C4—42no—
436C4—82no—
437C4—82no—
438C4—82no—
439C2—21no—
440C2—21no—
441C2—22no—
442C2—82no—
443C2#1155614yesdirect product factor
444C2C2×#146811yescharacteristic
445C2C2×#212911yescharacteristic
446C2C2×#213211yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ 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. #11556C2direct (N × H)
5. C2×#1324C2semidirect
6. C2×#1319C2semidirect
32. #1324C22semidirect
33. #1319C22semidirect
443. C2#11556direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24992
C2×#14642
C2×#14582
#115562
C2×#13242
C2×#13192