Finite Group Databasebeta

← C2×#11815 #481863

Subgroups — C2×#11815 #481863

Order 512.

1818 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#395C211yesmaximal, characteristic
2C2×#1821C211yesmaximal, characteristic
3C2×#1463C211yesmaximal, characteristic
4#11815C218yesmaximal, direct product factor
5C2×#1354C214yesmaximal, complemented
6C24×Q8C411yescharacteristic
7C23×C4⋊C4—21no—
8C23×C4⋊C4C2211yescharacteristic
9C2×#407—21no—
10C2×#407—21no—
11C2×#395—22no—
12C2×#395—21no—
13C2×#395—21no—
14C2×#395—22no—
15C2×#395C412yes—
16C2×#395C414yes—
17C2×#395C2214yes—
18C2×#395C2212yes—
19C22×#95—21no—
20C22×#95C411yescharacteristic
21#1821—22no—
22#1821C2214yes—
23C2×#343—21no—
24C2×#343C2212yes—
25#1463—22no—
26#1463C2214yes—
27C2×#324—21no—
28C2×#324C2212yes—
29#1354C22116yescomplemented
30C23×Q8—22no—
31C23×Q8—21no—
32C23×Q8—48no—
33C23×Q8—42no—
34C23×Q8—41no—
35C23×Q8—41no—
36C23×Q8C2×C414yes—
37C23×Q8C2×C412yes—
38C24×C4—21no—
39C24×C4—21no—
40C24×C4C2×C411yescharacteristic
41C24×C4D411yescharacteristic
42C24×C4D411yescharacteristic
43C22×M16—22no—
44C22×M16—41no—
45C22×C4⋊C4—24no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—24no—
49C22×C4⋊C4—41no—
50C22×C4⋊C4—41no—
51C22×C4⋊C4—42no—
52C22×C4⋊C4—48no—
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4C2×C412yes—
55C22×C4⋊C4D412yes—
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4D412yes—
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2311yescharacteristic
61#407—24no—
62#407—24no—
63#407—42no—
64#407—42no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—41no—
67C2×C22⋊C8—41no—
68C2×C22⋊C8—41no—
69#395—24no—
70#395—24no—
71#395—24no—
72#395—24no—
73#395—22no—
74#395—24no—
75#395—22no—
76#395C2×C418yes—
77#395C2×C414yes—
78#395C2314yes—
79C2×#95—22no—
80C2×#95—21no—
81C2×#95—22no—
82C2×#95—24no—
83C2×#95—24no—
84C2×#95—22no—
85C2×#95C2×C414yes—
86C2×#95C2×C412yes—
87#343—22no—
88#343—24no—
89#343C2314yes—
90#324—24no—
91#324—22no—
92#324C2314yes—
93C25C22⋊C411yescharacteristic
94C22×Q8—24no—
95C22×Q8—22no—
96C22×Q8—416no—
97C22×Q8—48no—
98C22×Q8—416no—
99C22×Q8—48no—
100C22×Q8—44no—
101C22×Q8—42no—
102C22×Q8—44no—
103C22×Q8—42no—
104C22×Q8—44no—
105C22×Q8—42no—
106C22×Q8—44no—
107C22×Q8—42no—
108C22×Q8—432no—
109C22×Q8—416no—
110C22×Q8—48no—
111C22×Q8—48no—
112C22×Q8C22×C414yes—
113C23×C4—28no—
114C23×C4—28no—
115C23×C4—21no—
116C23×C4—22no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—24no—
123C23×C4—22no—
124C23×C4—24no—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—24no—
129C23×C4—42no—
130C23×C4—44no—
131C23×C4—44no—
132C23×C4—44no—
133C23×C4—42no—
134C23×C4—41no—
135C23×C4—41no—
136C23×C4—44no—
137C23×C4—44no—
138C23×C4C22⋊C411yescharacteristic
139C23×C4C22⋊C412yes—
140C23×C4C22⋊C412yes—
141C23×C4C22⋊C411yescharacteristic
142C23×C4C22⋊C412yes—
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22×C412yes—
145C23×C4C22×C411yescharacteristic
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D412yes—
150C2×M16—24no—
151C2×M16—42no—
152C2×M16—44no—
153C2×M16—44no—
154C2×M16—44no—
155C2×M16—84no—
156C22×C8—42no—
157C22×C8—42no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—28no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—416no—
178C2×C4⋊C4—48no—
179C2×C4⋊C4—48no—
180C2×C4⋊C4—416no—
181C2×C4⋊C4C22⋊C412yes—
182C2×C4⋊C4C22⋊C412yes—
183C2×C4⋊C4C22⋊C412yes—
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C4⋊C4C2×D412yes—
187C2×C4⋊C4C2×D411yescharacteristic
188C2×C4⋊C4C2×D411yescharacteristic
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2411yescharacteristic
191C22⋊C8—24no—
192C22⋊C8—42no—
193C22⋊C8—44no—
194C22⋊C8—44no—
195C22⋊C8—44no—
196#95—28no—
197#95—24no—
198#95—24no—
199#95—22no—
200#95—24no—
201#95C22×C414yes—
202C24—24no—
203C24—24no—
204C24—22no—
205C24—22no—
206C24#9912yes—
207C24#9911yescharacteristic
208C24#9911yescharacteristic
209C24C2×C22⋊C412yes—
210C24C2×C22⋊C411yescharacteristic
211C2×Q8—48no—
212C2×Q8—416no—
213C2×Q8—48no—
214C2×Q8—44no—
215C2×Q8—48no—
216C2×Q8—44no—
217C2×Q8—432no—
218C2×Q8—416no—
219C2×Q8—432no—
220C2×Q8—416no—
221C2×Q8—416no—
222C2×Q8—48no—
223C2×Q8—432no—
224C2×Q8—416no—
225C2×Q8—416no—
226C2×Q8—48no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—216no—
231C22×C4—28no—
232C22×C4—216no—
233C22×C4—28no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—48no—
267C22×C4—48no—
268C22×C4—44no—
269C22×C4—432no—
270C22×C4—48no—
271C22×C4—48no—
272C22×C4—44no—
273C22×C4—416no—
274C22×C4—48no—
275C22×C4—48no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—48no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—42no—
282C22×C4—44no—
283C22×C4—42no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—42no—
287C22×C4—44no—
288C22×C4—42no—
289C22×C4—44no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—44no—
293C22×C4—82no—
294C22×C4—84no—
295C22×C4C2×C22⋊C412yes—
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C412yes—
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C2×C22⋊C412yes—
301C22×C4#13714yes—
302C22×C4C23×C411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4C22×D411yescharacteristic
305M16—44no—
306M16—44no—
307M16—84no—
308M16—88no—
309C2×C8—44no—
310C2×C8—44no—
311C2×C8—84no—
312C2×C8—84no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—41no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—42no—
322C4⋊C4—41no—
323C4⋊C4—48no—
324C4⋊C4—416no—
325C4⋊C4—48no—
326C23—216no—
327C23—216no—
328C23—24no—
329C23—22no—
330C23—24no—
331C23—22no—
332C23—22no—
333C23—24no—
334C23—22no—
335C23—24no—
336C23—22no—
337C23—22no—
338C23—24no—
339C23—22no—
340C23—22no—
341C23AΣL(1,𝔽16)14yes—
342C23#35614yes—
343C23C2×#9912yes—
344C23C2×#9912yes—
345C23C2×#9911yescharacteristic
346C23C2×#9911yescharacteristic
347C23#41912yes—
348C23#42112yes—
349C23C22×C22⋊C411yescharacteristic
350Q8—48no—
351Q8—48no—
352Q8—416no—
353Q8—432no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—28no—
361C2×C4—28no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—432no—
377C2×C4—416no—
378C2×C4—44no—
379C2×C4—416no—
380C2×C4—48no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—44no—
388C2×C4—416no—
389C2×C4—48no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—81no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—81no—
409C2×C4#41912yes—
410C2×C4C2×#13712yes—
411C8—84no—
412C8—84no—
413C22—28no—
414C22—28no—
415C22—216no—
416C22—24no—
417C22—28no—
418C22—28no—
419C22—24no—
420C22—21no—
421C22—22no—
422C22—21no—
423C22—21no—
424C22—22no—
425C22—21no—
426C22—22no—
427C22—22no—
428C22#151112yes—
429C22#151112yes—
430C22C2×AΣL(1,𝔽16)14yes—
431C22C2×AΣL(1,𝔽16)12yes—
432C22C2×#35614yes—
433C22C2×#35612yes—
434C22C22×#9911yescharacteristic
435C22C2×#41911yescharacteristic
436C22C2×#42111yescharacteristic
437C4—22no—
438C4—22no—
439C4—48no—
440C4—41no—
441C4—48no—
442C4—41no—
443C4—42no—
444C4—41no—
445C4—41no—
446C4—42no—
447C4—42no—
448C2—21no—
449C2—24no—
450C2—21no—
451C2—24no—
452C2—22no—
453C2#1181514yesdirect product factor
454C2C2×#151111yescharacteristic
455C2C22×AΣL(1,𝔽16)11yescharacteristic
456C2C22×#35611yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11815C2direct (N × H)
5. C2×#1354C2semidirect
29. #1354C22semidirect
453. C2#11815direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#3952
C2×#18212
C2×#14632
#118152
C2×#13542