Finite Group Databasebeta

← C2×#11717 #481489

Subgroups — C2×#11717 #481489

Order 512.

1216 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2628C211yesmaximal, characteristic
2C2×#1740C211yesmaximal, characteristic
3C2×#1458C211yesmaximal, characteristic
4#11717C218yesmaximal, direct product factor
5C2×#1351C214yesmaximal
6C22×C4⋊Q8C411yescharacteristic, complemented
7C22×#117—21no—
8C22×#117C2211yescharacteristic
9C2×#431—21no—
10#2628—22no—
11#2628—22no—
12#2628C414yescomplemented
13#2628C2214yes—
14C2×#400—21no—
15C2×#399—21no—
16C2×#395C412yescomplemented
17C2×#393C2212yes—
18C2×#389—21no—
19C2×#382C411yescharacteristic, complemented
20C2×#380—21no—
21#1740—22no—
22#1740C2214yes—
23C2×#342—21no—
24C2×#342C2212yes—
25#1458—22no—
26#1458C2214yes—
27C2×#324—21no—
28C2×#323C2212yes—
29#1351C22116yes—
30C23×Q8C2×C412yes—
31C24×C4D411yescharacteristic
32C2×C4⋊Q8—22no—
33C2×C4⋊Q8—42no—
34C2×C4⋊Q8—41no—
35C2×C4⋊Q8—41no—
36C2×C4⋊Q8C2×C414yescomplemented
37C2×#117—28no—
38C2×#117—24no—
39C2×#117—41no—
40C2×#117—41no—
41C2×#117—42no—
42C2×#117—42no—
43C2×#117—44no—
44C2×#117C2×C412yes—
45C2×#117D412yes—
46C2×#117D412yes—
47C2×#117C2312yes—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—41no—
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4D411yescharacteristic
55C22×C22⋊C4—22no—
56C22×C22⋊C4—22no—
57C22×C22⋊C4D411yescharacteristic
58C22×C22⋊C4C2311yescharacteristic
59C22×C42—41no—
60C22×C42C2×C411yescharacteristic
61C22×C42D411yescharacteristic
62#431—24no—
63#431—42no—
64C2×C4⋊C8—41no—
65C2×C22⋊C8—22no—
66C2×C4×C8—41no—
67#400—24no—
68#400—24no—
69#399—24no—
70#399—24no—
71#395—24no—
72#395C2×C418yescomplemented
73#393—22no—
74#393—22no—
75#393C2×C414yes—
76#393C2314yes—
77#389—24no—
78#389—24no—
79#382—22no—
80#382C2×C414yescomplemented
81#380—24no—
82#380—24no—
83C2×#95—22no—
84C2×#95—21no—
85C2×#95—22no—
86C2×#95C2×C412yes—
87#342—24no—
88#342—22no—
89#342C2314yes—
90#324—24no—
91#323—22no—
92#323C2314yes—
93C25C2×D411yescharacteristic
94C22×Q8—22no—
95C22×Q8—42no—
96C22×Q8—42no—
97C22×Q8—84no—
98C22×Q8C22×C414yes—
99C23×C4—21no—
100C23×C4—22no—
101C23×C4—21no—
102C23×C4—24no—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4—42no—
106C23×C4C22⋊C411yescharacteristic
107C23×C4C22⋊C412yes—
108C23×C4C22⋊C411yescharacteristic
109C23×C4C22⋊C411yescharacteristic
110C23×C4C22⋊C411yescharacteristic
111C23×C4SD1614yes—
112C23×C4Q1614yes—
113C23×C4C22×C411yescharacteristic
114C23×C4C2×D412yes—
115C23×C4C2×D411yescharacteristic
116C22×C8—42no—
117C4⋊Q8—44no—
118C4⋊Q8—44no—
119C4⋊Q8—44no—
120C4⋊Q8—44no—
121#117—216no—
122#117—42no—
123#117—42no—
124#117—42no—
125#117—42no—
126#117—42no—
127#117—42no—
128#117—42no—
129#117—42no—
130#117—416no—
131#117—48no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—28no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4—48no—
143C2×C4⋊C4—44no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—44no—
148C2×C4⋊C4—84no—
149C2×C4⋊C4C22⋊C412yes—
150C2×C4⋊C4C22⋊C412yes—
151C2×C4⋊C4C22×C412yes—
152C2×C4⋊C4C2×D412yes—
153C2×C22⋊C4—28no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—41no—
158C2×C22⋊C4—41no—
159C2×C22⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C22⋊C4—42no—
162C2×C22⋊C4—44no—
163C2×C22⋊C4—84no—
164C2×C22⋊C4C22⋊C412yes—
165C2×C22⋊C4C22×C411yescharacteristic
166C2×C22⋊C4C2×D412yes—
167C2×C22⋊C4C2×D411yescharacteristic
168C2×C22⋊C4C2×D411yescharacteristic
169C2×C22⋊C4C2411yescharacteristic
170C2×C42—22no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—22no—
176C2×C42—48no—
177C2×C42—44no—
178C2×C42C22⋊C412yes—
179C2×C42C22⋊C412yes—
180C2×C42C22×C412yes—
181C2×C42C2×D412yes—
182C4⋊C8—44no—
183C22⋊C8—24no—
184C22⋊C8—42no—
185C4×C8—44no—
186#95—28no—
187#95—24no—
188#95—24no—
189#95—22no—
190#95—24no—
191#95C22×C414yes—
192C24—21no—
193C24—21no—
194C24—22no—
195C24—42no—
196C24—42no—
197C24C2×C22⋊C411yescharacteristic
198C24C2×C22⋊C411yescharacteristic
199C24C2×SD1612yes—
200C24C2×Q1612yes—
201C24C22×D411yescharacteristic
202C2×Q8—44no—
203C2×Q8—44no—
204C2×Q8—84no—
205C2×Q8—84no—
206C2×Q8—88no—
207C2×Q8—88no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—28no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—28no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—44no—
238C22×C4—48no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—44no—
246C22×C4—84no—
247C22×C4—84no—
248C22×C4#9912yes—
249C22×C4#9911yescharacteristic
250C22×C4#9911yescharacteristic
251C22×C4#10314yes—
252C22×C4#10318yes—
253C22×C4#10314yes—
254C22×C4#10414yes—
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×SD1614yes—
262C22×C4C2×Q1614yes—
263C22×C4C23×C411yescharacteristic
264C22×C4C22×D411yescharacteristic
265C2×C8—44no—
266C2×C8—84no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—48no—
272C4⋊C4—416no—
273C4⋊C4—44no—
274C4⋊C4—44no—
275C4⋊C4—48no—
276C4⋊C4—88no—
277C22⋊C4—28no—
278C22⋊C4—42no—
279C22⋊C4—41no—
280C22⋊C4—42no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—48no—
285C22⋊C4—44no—
286C22⋊C4—84no—
287C22⋊C4—84no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—22no—
293C42—22no—
294C42—22no—
295C42—22no—
296C42—416no—
297C23—22no—
298C23—22no—
299C23—21no—
300C23—21no—
301C23—22no—
302C23—22no—
303C23—21no—
304C23—21no—
305C23—22no—
306C23—22no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—44no—
311C23—44no—
312C23—41no—
313C23—42no—
314C23—42no—
315C23—81no—
316C23—81no—
317C23—84no—
318C23C2×#9911yescharacteristic
319C23C2×#9911yescharacteristic
320C23C2×#10312yes—
321C23C2×#10312yes—
322C23C2×#10412yes—
323C23C22×C22⋊C411yescharacteristic
324C23C22×SD1611yescharacteristic
325C23C22×Q1611yescharacteristic
326Q8—88no—
327Q8—88no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—28no—
335C2×C4—28no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—48no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—88no—
366C2×C4—84no—
367C2×C4—84no—
368C2×C4—84no—
369C2×C4—816no—
370C2×C4—84no—
371C2×C4#32818yes—
372C2×C4#32814yes—
373C2×C4#32814yes—
374C2×C4C2×#9912yes—
375C2×C4C2×#9912yes—
376C2×C4C2×#10314yes—
377C2×C4C2×#10314yes—
378C2×C4C2×#10414yes—
379C8—84no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—24no—
385C22—24no—
386C22—21no—
387C22—21no—
388C22—22no—
389C22—41no—
390C22—41no—
391C22—42no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—44no—
397C22—44no—
398C22—81no—
399C22—81no—
400C22—82no—
401C22—82no—
402C22C2×#32812yes—
403C22C2×#32812yes—
404C22#212212yes—
405C22#212912yes—
406C22C22×#9911yescharacteristic
407C22C22×#10311yescharacteristic
408C22C22×#10411yescharacteristic
409C4—24no—
410C4—41no—
411C4—41no—
412C4—42no—
413C4—41no—
414C4—42no—
415C4—41no—
416C4—84no—
417C4—88no—
418C4—84no—
419C4C2×#32814yes—
420C4C2×#32814yes—
421C2—21no—
422C2—21no—
423C2—22no—
424C2—41no—
425C2—41no—
426C2—42no—
427C2#1171714yesdirect product factor
428C2C22×#32811yescharacteristic
429C2C2×#212211yescharacteristic
430C2C2×#212911yescharacteristic
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. #11717C2direct (N × H)
6. C22×C4⋊Q8C4semidirect
12. #2628C4semidirect
16. C2×#395C4semidirect
19. C2×#382C4semidirect
36. C2×C4⋊Q8C2×C4semidirect
72. #395C2×C4semidirect
80. #382C2×C4semidirect
427. C2#11717direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#26282
C2×#17402
C2×#14582
#117172
C2×#13512