Finite Group Databasebeta

← C2×#11575 #480551

Subgroups — C2×#11575 #480551

Order 512.

1702 conjugacy classes of subgroups.

The subgroup lattice has 465 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2409C211yesmaximal, characteristic
2C2×#1466C211yesmaximal, characteristic
3C2×#1459C211yesmaximal, characteristic
4#11575C218yesmaximal, direct product factor
5C2×#1326C214yesmaximal, complemented
6C24×Q8C411yescharacteristic
7C22×C22⋊Q8—21no—
8C22×C4×D4C2211yescharacteristic
9C2×#432—21no—
10C22×C22⋊C8—21no—
11#2409—22no—
12#2409—22no—
13#2409C414yes—
14#2409C2214yes—
15C2×#395—21no—
16C2×#393—21no—
17C2×#391—21no—
18C2×#386C412yes—
19C2×#385C2212yes—
20C2×#380C411yescharacteristic
21#1466—22no—
22#1466C2214yes—
23#1459—22no—
24#1459C2214yes—
25C2×#324—21no—
26C2×#323—21no—
27C2×#323C2212yes—
28C2×#323C2212yes—
29#1326C22116yescomplemented
30C23×Q8—21no—
31C23×Q8—22no—
32C23×Q8—21no—
33C23×Q8—41no—
34C23×Q8—42no—
35C23×Q8—41no—
36C23×Q8—48no—
37C23×Q8C2×C414yes—
38C23×D4C2×C411yescharacteristic
39C24×C4C2×C411yescharacteristic
40C24×C4C2×C412yes—
41C24×C4D411yescharacteristic
42C23×C8—41no—
43C2×C22⋊Q8—24no—
44C2×C22⋊Q8—24no—
45C2×C22⋊Q8—48no—
46C2×C4×D4—41no—
47C2×C4×D4—41no—
48C2×C4×D4—42no—
49C2×C4×D4C2×C412yes—
50C2×C4×D4D412yes—
51C2×C4×D4D412yes—
52C2×C4×D4C2312yes—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4D411yescharacteristic
56C22×C22⋊C4—22no—
57C22×C22⋊C4C2×C412yes—
58C22×C22⋊C4D411yescharacteristic
59C22×C22⋊C4C2311yescharacteristic
60C22×C42D411yescharacteristic
61#432—24no—
62#432—42no—
63C2×C4⋊C8—41no—
64C2×C22⋊C8—22no—
65C2×C22⋊C8—24no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—42no—
68C2×C22⋊C8—41no—
69C2×#97—41no—
70#395—24no—
71#395—24no—
72#393—24no—
73#393—24no—
74#391—24no—
75#391—24no—
76#386—24no—
77#386C2×C418yes—
78#385—22no—
79#385—22no—
80#385C2×C414yes—
81#385C2314yes—
82#380—22no—
83#380C2×C414yes—
84C2×#95—21no—
85C2×#95—22no—
86C2×#95—22no—
87#324—24no—
88#323—24no—
89#323—22no—
90#323—22no—
91#323C2314yes—
92#323C2314yes—
93C25C22⋊C411yescharacteristic
94C25C22×C411yescharacteristic
95C22×Q8—28no—
96C22×Q8—24no—
97C22×Q8—24no—
98C22×Q8—44no—
99C22×Q8—44no—
100C22×Q8—44no—
101C22×Q8—48no—
102C22×Q8—42no—
103C22×Q8—42no—
104C22×Q8—44no—
105C22×Q8—44no—
106C22×Q8—416no—
107C22×Q8—416no—
108C22×Q8—48no—
109C22×Q8—432no—
110C22×Q8—416no—
111C22×Q8—48no—
112C22×Q8—42no—
113C22×Q8—41no—
114C22×Q8—41no—
115C22×D4—24no—
116C22×D4—24no—
117C22×D4—41no—
118C22×D4—41no—
119C22×D4C22⋊C412yes—
120C22×D4C22×C412yes—
121C23×C4—22no—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—44no—
131C23×C4—48no—
132C23×C4—44no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4C22⋊C411yescharacteristic
136C23×C4C22⋊C411yescharacteristic
137C23×C4C22⋊C412yes—
138C23×C4C22⋊C412yes—
139C23×C4C22⋊C411yescharacteristic
140C23×C4C22×C412yes—
141C23×C4C22×C414yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D412yes—
145C22×C8—42no—
146C22×C8—44no—
147C22×C8—42no—
148C22×C8—84no—
149C22⋊Q8—416no—
150C22⋊Q8—416no—
151C4×D4—42no—
152C4×D4—42no—
153C4×D4—42no—
154C4×D4—42no—
155C4×D4—42no—
156C4×D4—42no—
157C4×D4—42no—
158C4×D4—42no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—48no—
166C2×C4⋊C4C22⋊C412yes—
167C2×C4⋊C4C2×D412yes—
168C2×C22⋊C4—28no—
169C2×C22⋊C4—28no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—48no—
176C2×C22⋊C4C22⋊C412yes—
177C2×C22⋊C4C22×C414yes—
178C2×C22⋊C4C22×C411yescharacteristic
179C2×C22⋊C4C2×D412yes—
180C2×C22⋊C4C2×D411yescharacteristic
181C2×C22⋊C4C2×D411yescharacteristic
182C2×C22⋊C4C2411yescharacteristic
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—22no—
186C2×C42C22⋊C412yes—
187C2×C42C2×D412yes—
188C4⋊C8—44no—
189C22⋊C8—24no—
190C22⋊C8—24no—
191C22⋊C8—42no—
192C22⋊C8—44no—
193C22⋊C8—42no—
194#97—44no—
195#95—24no—
196#95—24no—
197#95—28no—
198#95—24no—
199C24—21no—
200C24—21no—
201C24—21no—
202C24—22no—
203C24—22no—
204C24—21no—
205C24—42no—
206C24—42no—
207C24—42no—
208C24—42no—
209C24#10114yes—
210C24C2×C22⋊C411yescharacteristic
211C24C2×C22⋊C411yescharacteristic
212C24C2×C22⋊C412yes—
213C24#13114yes—
214C24C23×C411yescharacteristic
215C2×Q8—216no—
216C2×Q8—48no—
217C2×Q8—48no—
218C2×Q8—416no—
219C2×Q8—416no—
220C2×Q8—48no—
221C2×Q8—416no—
222C2×Q8—432no—
223C2×Q8—432no—
224C2×Q8—416no—
225C2×Q8—416no—
226C2×Q8—432no—
227C2×Q8—416no—
228C2×Q8—48no—
229C2×Q8—44no—
230C2×Q8—44no—
231C2×D4—24no—
232C2×D4—24no—
233C2×D4—24no—
234C2×D4—24no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—48no—
238C2×D4—48no—
239C22×C4—24no—
240C22×C4—28no—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—44no—
262C22×C4—48no—
263C22×C4—48no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—48no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—416no—
272C22×C4—48no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—44no—
276C22×C4—44no—
277C22×C4—42no—
278C22×C4—44no—
279C22×C4—48no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—84no—
286C22×C4—82no—
287C22×C4#9912yes—
288C22×C4#9911yescharacteristic
289C22×C4#9911yescharacteristic
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C412yes—
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C2×C8—44no—
298C2×C8—44no—
299C2×C8—84no—
300C2×C8—84no—
301C2×C8—88no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—48no—
305C4⋊C4—48no—
306C4⋊C4—48no—
307C22⋊C4—28no—
308C22⋊C4—42no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—42no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—416no—
315C22⋊C4—44no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C23—22no—
321C23—22no—
322C23—21no—
323C23—21no—
324C23—22no—
325C23—21no—
326C23—21no—
327C23—24no—
328C23—24no—
329C23—24no—
330C23—22no—
331C23—24no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—48no—
336C23—41no—
337C23—42no—
338C23—48no—
339C23—42no—
340C23—42no—
341C23—48no—
342C23—42no—
343C23—42no—
344C23—42no—
345C23—42no—
346C23—42no—
347C23—81no—
348C23—84no—
349C23—81no—
350C23#40812yes—
351C23#40812yes—
352C23C2×#9911yescharacteristic
353C23C2×#9911yescharacteristic
354C23C2×#10112yes—
355C23C2×#10114yes—
356C23C22×C22⋊C411yescharacteristic
357C23C2×#13112yes—
358Q8—416no—
359Q8—432no—
360Q8—416no—
361D4—48no—
362D4—48no—
363C2×C4—22no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—28no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—48no—
379C2×C4—42no—
380C2×C4—416no—
381C2×C4—48no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—48no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—48no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—88no—
407C2×C4AΣL(1,𝔽16)14yes—
408C2×C4C2×#9912yes—
409C2×C4C2×#9912yes—
410C8—84no—
411C8—84no—
412C22—24no—
413C22—24no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—22no—
419C22—22no—
420C22—24no—
421C22—42no—
422C22—42no—
423C22—44no—
424C22—42no—
425C22—42no—
426C22—48no—
427C22—48no—
428C22—48no—
429C22—41no—
430C22—41no—
431C22—42no—
432C22—41no—
433C22—41no—
434C22—42no—
435C22—82no—
436C22—84no—
437C22#147212yes—
438C22#147212yes—
439C22C2×AΣL(1,𝔽16)12yes—
440C22#212312yes—
441C22C2×#40811yescharacteristic
442C22C22×#9911yescharacteristic
443C22C22×#10111yescharacteristic
444C4—22no—
445C4—41no—
446C4—48no—
447C4—42no—
448C4—41no—
449C4—41no—
450C4—41no—
451C4—42no—
452C4—84no—
453C4C2×AΣL(1,𝔽16)14yes—
454C2—21no—
455C2—21no—
456C2—22no—
457C2—42no—
458C2—42no—
459C2—44no—
460C2#1157514yesdirect product factor
461C2C2×#147211yescharacteristic
462C2C22×AΣL(1,𝔽16)11yescharacteristic
463C2C2×#212311yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC24C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11575C2direct (N × H)
5. C2×#1326C2semidirect
29. #1326C22semidirect
460. C2#11575direct (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×#14662
C2×#14592
#115752
C2×#13262