Finite Group Databasebeta

← C2×#11778 #481608

Subgroups — C2×#11778 #481608

Order 512.

1220 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2536C211yesmaximal, characteristic
2C2×C4×#101C211yesmaximal, characteristic, complemented
3C2×#1467C211yesmaximal, characteristic, complemented
4#11778C218yesmaximal, direct product factor
5C2×#1346C212yesmaximal, complemented
6C2×#1345C212yesmaximal
7C22×C22⋊Q8—21no—
8C22×C4×Q8C2211yescharacteristic
9C22×C4×D4C411yescharacteristic
10C2×#432—21no—
11C22×#101—21no—
12C2×C4×M16—21no—
13#2536—22no—
14#2536—22no—
15#2536C414yes—
16#2536C2214yes—
17C2×#398C411yescharacteristic
18C2×#398C2211yescharacteristic
19C2×#394C411yescharacteristic
20C2×#394C2211yescharacteristic
21C2×#388—21no—
22C2×#388—21no—
23C2×#382C411yescharacteristic
24C2×#380—21no—
25C4×#101—22no—
26C4×#101C2214yescomplemented
27C2×#343C2212yes—
28#1467—22no—
29#1467C2214yescomplemented
30C2×#324—21no—
31C2×#324C2212yes—
32#1346C2218yescomplemented
33#1345C2218yes—
34C23×Q8D411yescharacteristic
35C23×D4C2×C411yescharacteristic
36C24×C4—21no—
37C22×M16—22no—
38C2×C22⋊Q8—24no—
39C2×C22⋊Q8—24no—
40C2×C22⋊Q8—48no—
41C2×C4×Q8—41no—
42C2×C4×Q8—41no—
43C2×C4×Q8—42no—
44C2×C4×Q8C2×C412yes—
45C2×C4×Q8D412yes—
46C2×C4×Q8D412yes—
47C2×C4×Q8C2312yes—
48C2×C4×D4—22no—
49C2×C4×D4—42no—
50C2×C4×D4—41no—
51C2×C4×D4—41no—
52C2×C4×D4C2×C414yes—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C22⋊C4—21no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C42C2×C411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42D411yescharacteristic
68#432—24no—
69#432—42no—
70C2×C4⋊C8—41no—
71C2×#101—28no—
72C2×#101—24no—
73C2×C22⋊C8—22no—
74C4×M16—24no—
75C4×M16—42no—
76C2×#97—41no—
77C2×#97—41no—
78C2×C4×C8—41no—
79#398—22no—
80#398—21no—
81#398—21no—
82#398C2×C414yes—
83#398C2×C412yes—
84#398C2312yes—
85#394—22no—
86#394—21no—
87#394—21no—
88#394C2×C414yes—
89#394C2×C412yes—
90#394C2312yes—
91#388—24no—
92#388—24no—
93#388—24no—
94#388—24no—
95#382—22no—
96#382C2×C414yes—
97#380—24no—
98#380—24no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95C2×C411yescharacteristic
103C2×#95C2×C411yescharacteristic
104#343—22no—
105#343C2314yes—
106#324—24no—
107#324—22no—
108#324C2314yes—
109C25—21no—
110C22×Q8—24no—
111C22×Q8—24no—
112C22×Q8—41no—
113C22×Q8—41no—
114C22×Q8C22⋊C412yes—
115C22×Q8C2×D412yes—
116C22×D4—21no—
117C22×D4—41no—
118C22×D4—41no—
119C22×D4—44no—
120C22×D4C22×C412yes—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—24no—
125C23×C4—42no—
126C23×C4—42no—
127C23×C4—44no—
128C23×C4—44no—
129C23×C4C22⋊C411yescharacteristic
130C23×C4C22⋊C411yescharacteristic
131C23×C4C22⋊C411yescharacteristic
132C23×C4C22⋊C411yescharacteristic
133C23×C4C22×C411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C2×M16—28no—
137C2×M16—24no—
138C2×M16—42no—
139C2×M16—44no—
140C22×C8—42no—
141C22×C8—42no—
142C22⋊Q8—416no—
143C22⋊Q8—416no—
144C4×Q8—42no—
145C4×Q8—42no—
146C4×Q8—42no—
147C4×Q8—42no—
148C4×Q8—42no—
149C4×Q8—42no—
150C4×Q8—42no—
151C4×Q8—42no—
152C4×D4—44no—
153C4×D4—44no—
154C4×D4—44no—
155C4×D4—44no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4C22⋊C412yes—
173C2×C4⋊C4C22⋊C412yes—
174C2×C4⋊C4C22×C412yes—
175C2×C4⋊C4C22×C412yes—
176C2×C4⋊C4C22×C411yescharacteristic
177C2×C4⋊C4C2×D412yes—
178C2×C4⋊C4C2×D411yescharacteristic
179C2×C4⋊C4C2×D411yescharacteristic
180C2×C4⋊C4C2411yescharacteristic
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—44no—
195C2×C42—22no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—22no—
202C2×C42—21no—
203C2×C42—22no—
204C2×C42C22⋊C412yes—
205C2×C42C22⋊C412yes—
206C2×C42C22⋊C412yes—
207C2×C42C22×C412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D412yes—
210C4⋊C8—44no—
211#101—216no—
212C22⋊C8—24no—
213C22⋊C8—42no—
214#97—44no—
215#97—44no—
216C4×C8—44no—
217#95—22no—
218#95—24no—
219#95—24no—
220#95—21no—
221#95—21no—
222#95—22no—
223#95C22×C412yes—
224#95C22×C412yes—
225C24—24no—
226C24—22no—
227C24—42no—
228C24—42no—
229C24—42no—
230C24—41no—
231C24—41no—
232C24—42no—
233C24—42no—
234C24C2×C22⋊C411yescharacteristic
235C2×Q8—24no—
236C2×Q8—24no—
237C2×Q8—24no—
238C2×Q8—24no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—48no—
242C2×Q8—48no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—44no—
246C2×D4—44no—
247C2×D4—48no—
248C2×D4—44no—
249C2×D4—44no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—42no—
270C22×C4—48no—
271C22×C4—48no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—48no—
286C22×C4—88no—
287C22×C4—88no—
288C22×C4#9911yescharacteristic
289C22×C4#9911yescharacteristic
290C22×C4#9912yes—
291C22×C4#10414yes—
292C22×C4C2×C22⋊C412yes—
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C412yes—
298C22×C4C2×C22⋊C412yes—
299C22×C4AGL(1,ℤ/8)14yes—
300C22×C4C23×C411yescharacteristic
301C22×C4C22×D411yescharacteristic
302C22×C4C22×D411yescharacteristic
303M16—28no—
304M16—48no—
305M16—44no—
306C2×C8—44no—
307C2×C8—48no—
308C2×C8—44no—
309C2×C8—84no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—41no—
313C4⋊C4—42no—
314C4⋊C4—41no—
315C4⋊C4—42no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—48no—
321C4⋊C4—48no—
322C4⋊C4—44no—
323C4⋊C4—44no—
324C22⋊C4—48no—
325C22⋊C4—48no—
326C22⋊C4—44no—
327C22⋊C4—44no—
328C22⋊C4—44no—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C42—22no—
333C42—22no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—22no—
344C23—22no—
345C23—22no—
346C23—24no—
347C23—42no—
348C23—44no—
349C23—48no—
350C23—48no—
351C23—42no—
352C23—42no—
353C23—42no—
354C23—44no—
355C23—42no—
356C23—42no—
357C23—84no—
358C23—84no—
359C23—82no—
360C23—82no—
361C23—82no—
362C23—82no—
363C23C2×#9911yescharacteristic
364C23C2×#9911yescharacteristic
365C23#41812yes—
366C23C2×#10412yes—
367C23C22×C22⋊C411yescharacteristic
368Q8—48no—
369Q8—48no—
370D4—44no—
371D4—44no—
372D4—48no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—816no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—84no—
419C2×C4—84no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4C2×#9912yes—
423C2×C4C2×#9912yes—
424C2×C4#41812yes—
425C2×C4C2×#10414yes—
426C2×C4C2×AGL(1,ℤ/8)12yes—
427C8—48no—
428C8—84no—
429C22—24no—
430C22—24no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—44no—
436C22—44no—
437C22—44no—
438C22—82no—
439C22—82no—
440C22—88no—
441C22—88no—
442C22—82no—
443C22—82no—
444C22#150712yes—
445C22#150712yes—
446C22#212012yes—
447C22#213312yes—
448C22C22×#9911yescharacteristic
449C22C2×#41811yescharacteristic
450C22C22×#10411yescharacteristic
451C4—22no—
452C4—22no—
453C4—22no—
454C4—24no—
455C4—22no—
456C4—41no—
457C4—41no—
458C4—41no—
459C4—41no—
460C4—42no—
461C4—42no—
462C4—82no—
463C4—82no—
464C4—82no—
465C2—21no—
466C2—21no—
467C2—22no—
468C2—84no—
469C2#1177814yesdirect product factor
470C2C2×#150711yescharacteristic
471C2C2×#212011yescharacteristic
472C2C2×#213311yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×#101C2semidirect
3. C2×#1467C2semidirect
4. #11778C2direct (N × H)
5. C2×#1346C2semidirect
26. C4×#101C22semidirect
29. #1467C22semidirect
32. #1346C22semidirect
469. C2#11778direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25362
C2×C4×#1012
C2×#14672
#117782
C2×#13462
C2×#13452