Finite Group Databasebeta

← C2×#11904 #482127

Subgroups — C2×#11904 #482127

Order 512.

1742 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×#2409C211yesmaximal, characteristic
2C2×#1800C211yesmaximal, characteristic
3C2×#1459C211yesmaximal, characteristic
4#11904C218yesmaximal, direct product factor
5C2×#1354C214yesmaximal, complemented
6C24×Q8C411yescharacteristic
7C22×C22⋊Q8—21no—
8C22×C4×D4C2211yescharacteristic
9C2×#423—21no—
10C22×#100—21no—
11C22×C22⋊C8—21no—
12#2409—22no—
13#2409—22no—
14#2409C414yes—
15#2409C2214yes—
16C2×#395C2212yes—
17C2×#393C412yes—
18C2×#391—21no—
19C2×#386—21no—
20C2×#385—21no—
21C2×#380C411yescharacteristic
22#1800—22no—
23#1800C2214yes—
24C2×#343C2212yes—
25#1459—22no—
26#1459C2214yes—
27C2×#324C2212yes—
28C2×#323—21no—
29#1354C22116yescomplemented
30C23×Q8—22no—
31C23×Q8—21no—
32C23×Q8—41no—
33C23×Q8—42no—
34C23×Q8—41no—
35C23×Q8—48no—
36C23×Q8C2×C412yes—
37C23×Q8C2×C414yes—
38C23×D4D411yescharacteristic
39C24×C4—21no—
40C24×C4C2×C411yescharacteristic
41C24×C4D411yescharacteristic
42C22×M16—22no—
43C23×C8—41no—
44C2×C22⋊Q8—24no—
45C2×C22⋊Q8—24no—
46C2×C22⋊Q8—48no—
47C2×C4×D4—41no—
48C2×C4×D4—41no—
49C2×C4×D4—42no—
50C2×C4×D4C2×C412yes—
51C2×C4×D4D412yes—
52C2×C4×D4D412yes—
53C2×C4×D4C2312yes—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4C2311yescharacteristic
57C22×C22⋊C4—22no—
58C22×C22⋊C4—21no—
59C22×C22⋊C4C2×C411yescharacteristic
60C22×C22⋊C4D411yescharacteristic
61C22×C42D411yescharacteristic
62#423—24no—
63#423—42no—
64C2×C4⋊C8—41no—
65C2×C4⋊C8—41no—
66C2×#100—24no—
67C2×#100—44no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—24no—
70C2×C22⋊C8—42no—
71C2×C22⋊C8—41no—
72#395—22no—
73#395—22no—
74#395C2×C414yes—
75#395C2314yes—
76#393—24no—
77#393C2×C418yes—
78#391—24no—
79#391—24no—
80#386—24no—
81#386—24no—
82#385—24no—
83#385—24no—
84#380—22no—
85#380C2×C414yes—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95C2×C412yes—
89#343—22no—
90#343C2314yes—
91#324—22no—
92#324C2314yes—
93#323—24no—
94C25C22⋊C411yescharacteristic
95C25C22⋊C411yescharacteristic
96C22×Q8—28no—
97C22×Q8—24no—
98C22×Q8—22no—
99C22×Q8—48no—
100C22×Q8—44no—
101C22×Q8—44no—
102C22×Q8—44no—
103C22×Q8—44no—
104C22×Q8—44no—
105C22×Q8—42no—
106C22×Q8—42no—
107C22×Q8—416no—
108C22×Q8—416no—
109C22×Q8—48no—
110C22×Q8—432no—
111C22×Q8—416no—
112C22×Q8—48no—
113C22×Q8—42no—
114C22×Q8—41no—
115C22×Q8—41no—
116C22×Q8C22×C414yes—
117C22×D4—41no—
118C22×D4—41no—
119C22×D4—42no—
120C22×D4C22⋊C412yes—
121C22×D4SD1614yes—
122C22×D4Q1614yes—
123C22×D4C2×D412yes—
124C23×C4—22no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—24no—
133C23×C4—48no—
134C23×C4—44no—
135C23×C4—44no—
136C23×C4—42no—
137C23×C4—42no—
138C23×C4C22⋊C411yescharacteristic
139C23×C4C22⋊C412yes—
140C23×C4C22⋊C412yes—
141C23×C4C22⋊C411yescharacteristic
142C23×C4C22×C411yescharacteristic
143C23×C4C22×C412yes—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D412yes—
147C2×M16—24no—
148C2×M16—42no—
149C2×M16—44no—
150C2×M16—44no—
151C22×C8—42no—
152C22×C8—44no—
153C22×C8—42no—
154C22×C8—84no—
155C22⋊Q8—416no—
156C22⋊Q8—416no—
157C4×D4—42no—
158C4×D4—42no—
159C4×D4—42no—
160C4×D4—42no—
161C4×D4—42no—
162C4×D4—42no—
163C4×D4—42no—
164C4×D4—42no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—48no—
172C2×C4⋊C4C22×C411yescharacteristic
173C2×C4⋊C4C2×D411yescharacteristic
174C2×C4⋊C4C2×D411yescharacteristic
175C2×C4⋊C4C2411yescharacteristic
176C2×C22⋊C4—28no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C22⋊C4—48no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4C22⋊C412yes—
185C2×C22⋊C4C22⋊C412yes—
186C2×C22⋊C4C22×C412yes—
187C2×C22⋊C4C2×D412yes—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—22no—
191C2×C42C22⋊C412yes—
192C2×C42C2×D412yes—
193C4⋊C8—44no—
194C4⋊C8—44no—
195#100—48no—
196C22⋊C8—24no—
197C22⋊C8—42no—
198C22⋊C8—44no—
199#95—28no—
200#95—24no—
201#95—22no—
202#95—24no—
203#95C22×C414yes—
204C24—21no—
205C24—21no—
206C24—21no—
207C24—22no—
208C24—21no—
209C24—22no—
210C24—42no—
211C24—42no—
212C24—42no—
213C24—42no—
214C24#10314yes—
215C24#10314yes—
216C24C2×C22⋊C411yescharacteristic
217C24C2×C22⋊C412yes—
218C24C2×C22⋊C412yes—
219C2×Q8—216no—
220C2×Q8—48no—
221C2×Q8—48no—
222C2×Q8—48no—
223C2×Q8—416no—
224C2×Q8—48no—
225C2×Q8—44no—
226C2×Q8—44no—
227C2×Q8—416no—
228C2×Q8—432no—
229C2×Q8—432no—
230C2×Q8—416no—
231C2×Q8—416no—
232C2×Q8—432no—
233C2×Q8—48no—
234C2×Q8—48no—
235C2×Q8—44no—
236C2×Q8—44no—
237C2×Q8—48no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—44no—
242C2×D4—44no—
243C2×D4—44no—
244C2×D4—44no—
245C2×D4—42no—
246C2×D4#10318yes—
247C2×D4C2×SD1614yes—
248C2×D4C2×Q1614yes—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—28no—
254C22×C4—24no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—44no—
271C22×C4—48no—
272C22×C4—416no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—48no—
279C22×C4—416no—
280C22×C4—416no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—84no—
292C22×C4—82no—
293C22×C4#9912yes—
294C22×C4#9911yescharacteristic
295C22×C4#9911yescharacteristic
296C22×C4C2×C22⋊C412yes—
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C412yes—
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×SD1612yes—
302C22×C4C2×Q1612yes—
303C22×C4C23×C411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306M16—44no—
307M16—44no—
308M16—84no—
309C2×C8—44no—
310C2×C8—44no—
311C2×C8—84no—
312C2×C8—84no—
313C2×C8—88no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—48no—
319C4⋊C4—48no—
320C4⋊C4—48no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C22⋊C4—42no—
325C22⋊C4—416no—
326C22⋊C4—48no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—24no—
337C23—24no—
338C23—24no—
339C23—22no—
340C23—24no—
341C23—42no—
342C23—48no—
343C23—42no—
344C23—42no—
345C23—42no—
346C23—48no—
347C23—42no—
348C23—48no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23—42no—
354C23—81no—
355C23—81no—
356C23—84no—
357C23C2×#9911yescharacteristic
358C23C2×#9911yescharacteristic
359C23C2×#10312yes—
360C23C2×#10314yes—
361C23C2×#10314yes—
362C23#42112yes—
363C23C22×C22⋊C411yescharacteristic
364Q8—416no—
365Q8—48no—
366Q8—48no—
367Q8—416no—
368Q8—416no—
369D4—48no—
370D4—44no—
371D4—44no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—28no—
384C2×C4—22no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—48no—
393C2×C4—42no—
394C2×C4—432no—
395C2×C4—44no—
396C2×C4—42no—
397C2×C4—48no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—88no—
417C2×C4AΣL(1,𝔽16)14yes—
418C2×C4C2×#9912yes—
419C2×C4C2×#9912yes—
420C2×C4C2×#10312yes—
421C2×C4C22×SD1611yescharacteristic
422C2×C4C22×Q1611yescharacteristic
423C8—84no—
424C8—84no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—24no—
430C22—24no—
431C22—24no—
432C22—24no—
433C22—44no—
434C22—44no—
435C22—44no—
436C22—48no—
437C22—42no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—48no—
442C22—48no—
443C22—82no—
444C22—84no—
445C22#152012yes—
446C22#152012yes—
447C22C2×AΣL(1,𝔽16)12yes—
448C22#213512yes—
449C22C22×#9911yescharacteristic
450C22C22×#10311yescharacteristic
451C22C2×#42111yescharacteristic
452C4—22no—
453C4—21no—
454C4—21no—
455C4—22no—
456C4—41no—
457C4—48no—
458C4—41no—
459C4—41no—
460C4—41no—
461C4—42no—
462C4—84no—
463C4C2×AΣL(1,𝔽16)14yes—
464C2—21no—
465C2—21no—
466C2—22no—
467C2—44no—
468C2—44no—
469C2#1190414yesdirect product factor
470C2C2×#152011yescharacteristic
471C2C22×AΣL(1,𝔽16)11yescharacteristic
472C2C2×#213511yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11904C2direct (N × H)
5. C2×#1354C2semidirect
29. #1354C22semidirect
469. C2#11904direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24092
C2×#18002
C2×#14592
#119042
C2×#13542